Reaeration in 

Open-Channel Flow

GEOLOGICAL SURVEY PROFESSIONAL PAPER 737
Reaeration in 

Open-Channel Flow

By J. P. BENNETT and R. E. RATHBUN

GEOLOGICAL SURVEY PROFESSIONAL PAPER 737

An evaluation of procedures for 

measuring and predicting the 

reaeration coefficient of 

open-channel flows

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1972
UNITED STATES DEPARTMENT OF THE INTERIOR 

ROGERS C. B. MORTON, Secretary

GEOLOGICAL SURVEY 

V. E. McKelvey, Director

Library of Congress catalog-card No. 72-600020

For sale by the Superintendent of Documents, U.S. Government Printing Office 

Washington, D.C. 20402 - Price $1 (Paper Cover)

Stock Number 2401-2146
CONTENTS

Page

Abstract ................................................................................. 1

Introduction....... .... .........- .. . 1

Theories for predicting the reaeration coefficient. . 7

Diffusion equation............................................................ 7

Conceptual models for the gas-liquid

mass-transfer process.......................................... 9

Film model. . . .. 9

Renewal models-..-..-...-.- 10

Penetration model.... 10

Film-penetration model................................... 11

Surface renewal-damped-eddy

diffusivity model 13

Kinetic theory..-.--.....--....... 14

Large-eddy model...--...-.-..-.. . . 15

Turbulent-diffusion model... . 16

Semiempirical models .............................................. 17

Krenkel model..--..-....-............ ... 18

Thackston model.-.....-........-...-..- . . 18

Empirical prediction equations.. .. 19

Streeter-Phelps equations.-. . 19

Churchill-Elmore-Buckingham equations. 20

Owens-Edwards-Gibbs equation. . 20

Langbein-Durum equation...................................... 21

Isaacs-Gaudy equations.. .. . 21

Negulescu-Rojanski equations............................... 21

Experimental determination of the reaeration

coefficient....................................................................... 21

Measurement of dissolved oxygen in water. . 22 

Techniques for determining the reaeration

coefficient-.....-.............- . 22

Dissolved-oxygen balance technique. . 22

BOD test and KI............................................... 23

Bottom deposits, KS, and m............................ 24

Photosynthesis and respiration...................... 26

Other parameters............................................. 28

Calculation of k% in a recirculating flume. 29

Page

Experimental determination of the reaeration

coefficient Continued 

Techniques for determining the reaeration

coefficient Continued 

Disturbed-equilibrium technique 30

Tracer technique. .............................................. 31

Analysis of errors in reaeration-coefficient

calculation techniques-.-- 33

Accuracy of measured dissolved-oxygen

balance parameters.-.-.-...-... 33

Dissolved-oxygen balance technique. .. 34 

Disturbed-equilibrium technique. .. 39 

Tracer technique .. 42

Laboratory measurements...................................... 43

Prediction of the reaeration coefficient - . 44 

Reaeration-coefficient data sets-.... 44

Field data.-......-..--....--.-...-.. 44

Flume data.-......-.....-..-.......- 46

Evaluation of reaeration-coefficient prediction

equations..................................... ........................... 47

Development of a reaeration-coefficient

prediction equation ...................................... 48

Regression analysis.-. , . 51 

Results of regression analysis. 52

Effect of detergents and wind on the reaeration

coefficient. .....................................................-.. .... . 56

Conclusions and recommendations 58

Conclusions.--..-..-....-.. .. 58

Recommendations.--...-.--...-.....-.. 58

References....--.. ........................................ ..... .. . 59

Appendix A: Effect of a corrosion reaction on 

reaeration-coefficient measurements in a 

laboratory flume--...-....-..-...---.. 67

Symbols used in appendix A. .................................... 70

Appendix B: Collected data. ..................................... 73

ILLUSTRATIONS

[All illustrations are graphs]

Page

FIGURE 1. Example of a DO deficit sag curve................................................................................. ... .... .. . 3

2. Computation of reaeration coefficient from recirculating-flume data.............. -. . 30

3-10. Dissolved-oxygen balance technique:

3. Sensitivity of reaeration coefficient to rate constant for BOD decay.-. 35

4. Sensitivity of reaeration coefficient to photosynthetic-oxygen production rate . 35

5. Sensitivity of reaeration coefficient to upstream DO deficit.. . 35

6. Sensitivity of reaeration coefficient to upstream BOD.-.-..-...-.........--.... - 36

7. Sensitivity of reaeration coefficient to traveltime ............................................................ 36

8. Sensitivity of reaeration coefficient to downstream DO deficit ............................................... ........ 37

9. Effect of a systematic error in DO measurement procedure on

calculated reaeration coefficient--........-.....---..-.-...-......---...-- . - 38

10. Effect of ignoring photosynthetic-oxygen production rate-.......--.-. - -- 38

in
IV CONTENTS

FIGURES 11-16. Disturbed-equilibrium technique: Page

11. Sensitivity of reaeration coefficient to photosynthetic-oxygen production rate................................... 39

12. Sensitivity of reaeration coefficient to upstream DO deficit................................................................... 40

13. Sensitivity of reaeration coefficient to downstream DO deficit.............................................................. 40

14. Sensitivity of reaeration coefficient to traveltime..................................................................................... 40

15. Effect on calculated reaeration coefficient of ignoring a respiration rate difference....................... 40

16. Variation of calculated reaeration coefficient with respiration-rate ratio......................................... 41

17. Reaeration coefficient as a function of depth, from prediction equations of table 10 ................................ 50

18. Synthetic flume data from the upstream sampling station.................................................................................. 70

19. Determination of 61 and K2 from M values calculated at two different DO deficits....................................... 71

TABLES

Page

TABLE 1. Basic parameter sets used in the sensitivity analyses of the DO balance equation......................................... 34

2. Assumed errors in measurements of DO balance parameter................................................................................ 37

3. Slopes of the sensitivity-analysis plots........................................................................................................................ 37

4. Errors in the calculated reaeration coefficient resulting from errors in parameter

measurement, DO balance technique.................................................................................................................... 37

5. Basic parameter sets for the sensitivity analysis of the disturbed-equilibrium technique............................... 39

6. Assumed parameter errors and slopes of sensitivity-analysis plots for

disturbed-equilibrium technique............................................................................................................................ 41

7. Errors in the calculated reaeration coefficient resulting from errors in parameter

measurement, disturbed-equilibrium technique................................................................................................. 41

8. Statistical data concerning measured gas-transfer coefficient........................................................................... . 42

9. Ranges of experimental data from the literature-.--.-..-.--.--.............-.--..--.....-.....................................--.-..-....-....... 45

10. Performance of k2 prediction equations in terms of standard error of estimate............................................... 49

11. Results from regression analysis.................................................................................................................................. 52

12. Values of Student t for comparison of the exponents in equation 162 with those for the two

corresponding field and flume equations of table 11......................................................................................... 53

13. Dependence of turbulent intensity on the mean hydraulic parameters................................................................ 54

14. Results of the Student t test of the exponents in the flume equations and in the

field equations of table 13............... ................................................................................................................... 54

15. Prediction equations for mean velocity, obtained from the data used in table 11............................................. 54

16. Results of the Student t test of exponents in the regression equations of this study and

corresponding equations in the literature......... ............................................................................................... 55

17. Parameters used in calculating the synthetic flume data and the resulting calculated

reaeration coefficient............................................................................................................................................ 69

18-26. Reaeration coefficients and hydraulic data from:

18. Churchill, Elmore, and Buckingham (1962)............................................................................................. 73

19. Owens, Edwards, and Gibbs (1964).... .............. . ................................................................... 73

20. Gameson, Truesdale, and Downing (1955)..........-............... ................................................................ 74

21. O'Connor and Dobbins (1958) (Streeter and Phelps, 1925)................................................................. 74

22. O'Connor and Dobbins (1958) (various sources)................_.... .......................................................... 74

23. Tsivoglou and others (1967, 1968)............................................................................................................... 74

24. Thackston (1966)............................................................................................................................................ 75

25. Krenkel (I960)................................................................................................................................................ 75

26. Negulescu and Rojanski (1969)............. ............................................................................................... 74

SYMBOLS

Symbol Definition

a Parameter in the surface-renewal-damped-eddy diffusivity model (King, 1966).

fli Constant in the equation for consumption of dissolved oxygen by bottom deposits.

a-i Constant in the Edwards and Rolley (1965) equation for rate of consumption of dissolved oxygen by bottom deposits.

flo Rate of addition of dissolved oxygen to the reach by drainage accrual, in milligrams per liter per day.

A Temperature relation in the equation of Dobbins (1965); ^ = 9.68+0.054 (7-20°), where T is temperature, in degrees

	Celsius. 

A^A\.. .AN Coefficients in the linear-regression equation.
CONTENTS V

Symbol Definition

4e Parameter in the large-eddy model (Fortescue and Pearson, 1967), in centimeters per second.

^o Rate of addition of dissolved oxygen to the reach by drainage accrual, in grams per square meter per hour.

4 S Water surface area, in square meters.

Ax Cross-sectional area of the reach at longitudinal position jc, in square feet.

b Proportion of available molecules in the surface layer that actually escape.

b\ Exponent on dissolved-oxygen concentration in the Edwards and Rolley (1965) equation for rate of consumption of

dissolved oxygen by bottom deposits. 

B Temperature relation in the equation of Dobbins (1965); 5 = 0.976 + 0.0137 (30°-7):!/2 , where T is temperature, in

degrees Celsius.

BI Population regression coefficient.

^F Factor in the Dobbins (1965) equation, equal to 1.0 + F2 , where F is the Froude number. 

BOD Biochemical oxygen demand, in milligrams per liter. 

BPS Basic parameter set; a set of values assumed for the parameters of the dissolved-oxygen balance equation or the

disturbed-equilibrium technique equation, 

c Concentration fluctuation, or the difference between the instantaneous concentration and the time-averaged

concentration, in milligrams per liter or in grams per cubic meter, 

ci, c2 , c;i, c4 , c-r, Constants in the equations of Dobbins (1964b, 1965). 

fi, c3', cs' Constants in the Metzger-Dobbins (1967) model.

C Dissolved-oxygen concentration, in milligrams per liter or in grams per cubic meter. 

C _ Mean concentration of dissolved oxygen in the reach, in milligrams per liter. 

C, d Mean concentrations of dissolved oxygen in the reach for the two levels of dissolved-oxygen concentration,

disturbed-equilibrium technique, in milligrams per liter. 

Ca, Ch Dissolved-oxygen concentrations at the upstream and downstream ends of the reach, respectively, at the first level

of dissolved-oxygen concentration of the disturbed-equilibrium technique, in milligrams per liter. 

Ca, C'b Dissolved-oxygen concentration at the upstream and downstream ends of reach, respectively, at the second level of

dissolved-oxygen concentration of the disturbed-equilibrium technique, in milligrams per liter. 

Ci Dissolved-oxygen concentration at the gas-liquid interface, in milligrams per liter. 

C/i, Co/ Initial and final concentrations, respectively, of dissolved oxygen in the dark bottle of the light- and dark-bottle

test for photosynthesis, in milligrams per liter. 

CH, Cif Initial and final concentrations, respectively, of dissolved oxygen in the light bottle of the light- and dark-bottle

test for photosynthesis, in milligrams per liter.

Dissolved-oxygen concentration at saturation, in milligrams per liter or in grams per cubic meter. 

Saturation concentrations of dissolved oxygen for the two levels of dissolved-oxygen concentration of the disturbed- 

_ equilibrium technique, in milligrams per liter.

Cs Mean saturation concentration of dissolved oxygen in the reach, in milligrams per liter. 

Ct Dissolved-oxygen concentration, in milligrams per liter; subscript t indicates variability with respect to time at a

particular point in space.

CS Concentration of the tracer gas in the water, in milligrams per liter. 

CL Dissolved-oxygen concentration in the bulk liquid phase, in milligrams per liter. 

Ct Concentration of the conservative liquid tracer in the water, in milligrams per liter. 

AC Difference in dissolved-oxygen concentration between upstream and downstream ends of the reach, in milligrams

per liter. 

AC Difference in dissolved-oxygen concentration between upstream and downstream ends of the reach at sunrise, in

milligrams per liter.

d\, dt Molecular diameters of gases 1 and 2, in angstroms.

do Rate of transfer of oxygen through the water surface, in milligrams per liter per day. 

dia Depth of bottom deposit, in centimeters. 

D Dissolved-oxygen deficit, or the difference between the saturation concentration and the concentration of dissolved

oxygen, in milligrams per liter.

^a Dissolved-oxygen deficit at upstream end of reach, in milligrams per liter. 

D'a Dissolved-oxygen deficit at upstream end of reach at the second level of dissolved-oxygen concentration of the

disturbed-equilibrium technique, in milligrams per liter.

A> Dissolved-oxygen deficit at downstream end of reach, in milligrams per liter. 

£>b Dissolved-oxygen deficit at downstream end of reach at the second level of dissolved-oxygen concentration of the

disturbed-equilibrium technique, in milligrams per liter. 

Di Combination of the molecular- and turbulent-diffusion coefficients at the surface, Krenkel and Orlob (1963) model,

in square feet per day.

Dm Molecular-diffusion coefficient, in square feet per day.

Do Rate of transfer of oxygen through the water surface, in grams per square meter per hour. 

Ar Longitudinal dispersion coefficient, in square feet per day. 

DH Parameter that includes the effect of photosynthesis, plant respiration, and removal of dissolved oxygen by benthal

layer; in milligrams per liter per day. 

DM, DK Average dissolved-oxygen deficit in the reach at that time in the morning and evening, respectively, when the

production of oxygen by photosynthesis is zero, in milligrams per liter.
VI CONTENTS

Symbol Definition

DO Dissolved oxygen.

E Rate of energy dissipation per unit mass of liquid, equal to USg for open-channel flow, in ergs per second per gram.

E Combination of the activation energies for molecular diffusion and viscous flow, in degrees Kelvin.

~E Average kinetic energy of a roll cell, large-eddy model (Fortescue and Pearson, 1967), in square centimeters per

second per second.

Ea Activation energy for the absorption of oxygen by water, in calories per gram mole. 

EP Percent standard error of estimate for the reaeration coefficient, k2 , in percent. 

Es Standard error of estimate for the reaeration coefficient, k2 , in reciprocal days. 

ESL Standard error of estimate for the reaeration coefficient, £2, based on common logarithms. 

Ev Rate of energy dissipation per unit volume of liquid, in ergs per second per cubic centimeter. 

E( Rate of energy dissipation per unit mass of liquid at the surface, in ergs per second per gram. 

/ Dissolved-oxygen exchange coefficient, in meters per hour. 

f(k2) Functional relation of the reaeration coefficient, common logarithm base; used in the sensitivity analyses of the

dissolved-oxygen balance and disturbed-equilibrium techniques. 

f'(k2 ) First derivative of the functional relation of the reaeration coefficient, common logarithm base; used in the NewtonRaphson iterative method for solving the dissolved-oxygen balance and disturbed-equilibrium equations for

the sensitivity analyses.

fi(s) Rate-of-surface-renewal function, Thackston and Krenkel (1969a) model. 

f2 (H) Depth-of-flow function, Thackston and Krenkel (1969a) model. 

F Volumetric water flow rate, in cubic meters per hour or cubic feet per second. 

Fc Source-sink term, in milligrams per liter per day.

Tc Source-sink term averaged over the depth of flow, in milligrams per liter per day. 

F Froude number, equal to U/VgH. 

g Acceleration of gravity, in feet per second per second.

G General functional relation between dependent and independent variables in linear regression analysis. 

h Henry's Law constant, in liters per milligram. 

H Mean depth of flow, in feet or meters.

H' Depth above minimum low-water stage, Streeter and Phelps (1925) prediction equation for the reaeration coefficient, in feet.

j Exponent on depth of flow in the general form of the empirical prediction equation for the reaeration coefficient. 

J Constant in the Thackston and Krenkel (1969a) model. 

kt Deoxygenation coefficient or rate constant for biochemical oxidation of carbonaceous material, common logarithm

base, in reciprocal days. 

k2 Reaeration coefficient or rate constant for oxygen absorption from the atmosphere, common logarithm base, in

reciprocal days, or in reciprocal minutes. 

(£2)0 Reaeration coefficient calculated from the results of an experimental test, common logarithm base, in reciprocal

days. 

(k2)e Reaeration coefficient estimated from a linear regression analysis equation, common logarithm base, in reciprocal

days.

k3 Rate constant for the removal of BOD by sedimentation and adsorption, common logarithm base, in reciprocal days. 

kL Overall mass-transfer coefficient based on liquid phase, equivalent to k2H, in centimeters per second or in feet per

day. 

Ki Deoxygenation coefficient or rate constant for biochemical oxidation of carbonaceous material, natural logarithm

base, in reciprocal days. 

K2 Reaeration coefficient or rate constant for oxygen absorption from the atmosphere, natural logarithm base, in

reciprocal days.

K2(, Mass-transfer coefficient for the tracer gas, in reciprocal days. 

K2j, Reaeration coefficient at 7° Celsius, in reciprocal days. 

K2lt Reaeration coefficient at 20° Celsius, in reciprocal days.

K3 Rate constant for the removal of BOD by sedimentation and adsorption, in reciprocal days. 

KG Overall mass-transfer coefficient based on the gas phase, in milligrams per square centimeter per second per

atmosphere.

KL Overall mass-transfer coefficient based on the liquid phase, in centimeters per second or in feet per day. 

KN Rate constant for the oxidation of nitrogenous material, in reciprocal days. 

L Biochemical oxygen demand of carbonaceous material, in milligrams per liter.

La Biochemical oxygen demand of carbonaceous material at upstream end of reach, in milligrams per liter. 

Lb Biochemical oxygen demand of carbonaceous material at downstream end of the reach, in milligrams per liter. 

Lf Film thickness, in feet or meters. 

LI Average film thickness, in feet or meters. 

Lt Biochemical oxygen demand of carbonaceous material, in milligrams per liter; subscript / indicates variability with

respect to time at a particular point in space. 

LK Rate of addition of BOD along the reach, in milligrams per liter per day. (Includes addition of BOD with runoff,

the diffusion of partially degraded waste from the benthal layer into the water, and the scour and resuspensibn

of bottom deposits.)
CONTENTS VII

Symbol Definition

m Rate of addition of BOD to water from bottom deposits, in milligrams per liter per day.

M Molecular weight.

Ms Modulus of compression at the surface, in grams per second per second.

n An integer.

ns Number of new surface layers exposed per unit of time, in reciprocal seconds.

A7 Mass-transport rate of oxygen, in milligrams per square centimeter per second.

N Average rate of mass transport of oxygen, in milligrams per square centimeter per second.

A7 Number of data sets in linear regression analysis.

A'jr Concentration of nitrogenous material at longitudinal position x, in milligrams per liter.

P Rate of production of dissolved oxygen by photosynthesis, in milligrams per liter per day.

P Rate of production of dissolved oxygen by photosynthesis at the second level of dissolved-oxygen concentration of

the disturbed-equilibrium -technique, in milligrams per liter per day. 

p(f) Distribution function for surface-renewal time or for distance below the interface to which renewal occurs (/ is

dummy variable for either t or yO, Harriott (1962) model.

PI Partial pressure of the diffusing gas, oxygen, at the gas-liquid interface, in atmospheres. 

Pt Rate of production of dissolved oxygen by photosynthesis, in milligrams per liter per day; subscript / indicates

variability with respect to time at a particular point in space. 

PJ-,I Rate of production of dissolved oxygen by photosynthesis at longitudinal position x at time /, in milligrams per liter

per day.

PH Barometric pressure, in atmospheres.

PC Partial pressure of the diffusing gas, oxygen, in the gas phase, in atmospheres.

P Rate of gross primary production of dissolved oxygen by photosynthesis, in grams per square meter per hour. 

PI Rate of production of dissolved oxygen by photosynthesis in the light bottle of the light- and dark-bottle test, in

milligrams per liter per day.

PH Gross rate of photosynthetic production of dissolved oxygen by attached plants, in grams per square meter per hour. 

PP Gross rate of photosynthetic production of dissolved oxygen by phytoplankton, in grams per square meter per hour. 

P Peclet number, equal to AAf/Dm .

q Rate of change of dissolved-oxygen content in the reach, in milligrams per liter per day. 

q\ Rate of change of dissolved-oxygen content in the reach at sunrise, in milligrams per liter per day. 

q\t, qn Rate of change of dissolved-oxygen content in the reach at that time in the morning and evening, respectively,

when the production of oxygen by photosynthesis is zero, in milligrams per liter per day. 

Q0 Rate of change of dissolved-oxygen content in the reach, in grams per square meter per hour. 

Qi, Qa Average rate of change of dissolved-oxygen concentration per unit area in the reach for the two levels of dissolvedoxygen concentration, disturbed-equilibrium technique, in grams per square meter per hour. 

r Rate of consumption of dissolved oxygen by plant respiration, in milligrams per liter per day. 

re Rate of entry of gas molecules into the surface layer, in grams per second. 

r0 Rate of loss of gas molecules from a surface layer, in grams per second. 

r, Rate of consumption of dissolved oxygen by plant respiration, in milligrams per liter per day; subscript t indicates

variability with respect to time at a particular point in space. 

rf, t Rate of consumption of dissolved oxygen by plant respiration at longitudinal position x at time /, in milligrams per

liter per day.

R Rate of dissolved-oxygen utilization by all processes, in grams per square meter per hour,

R Ratio of the oxygen consumed by respiration at the two levels of dissolved-oxygen concentration, disturbedequilibrium technique. 

Rd, RI Rate of consumption of dissolved oxygen by respiration in the dark and light bottles, respectively, of the light- and

dark-bottle test for photosynthesis, in milligrams per liter per day.

Ri Equals the quantity (Ai B>)jSbi, used to test the significance of the regression coefficient A-,.

RB Rate of consumption of dissolved oxygen by respiration by attached plants, in grams per square meter per hour. 

RBOD Rate of consumption of dissolved oxygen by the oxidation of organic wastes in the stream water, in grams per square

meter per hour.

Rc Ideal-gas constant, in calories per gram mole per degree Kelvin.

RH Hydraulic radius, equal to the cross-sectional area of flow divided by the wetted perimeter, in feet. 

RM Rate of consumption of dissolved oxygen by bottom deposits, in grams per square meter per hour. 

RP Rate of consumption of dissolved oxygen by respiration of phytoplankton, in grams per square meter per hour. 

RMS Root-mean-square, or square root of the mean value of the square of the deviations of the measured parameter

from the mean parameter.

* Average rate of surface renewal, in reciprocal hours. 

5 Water-surface or channel slope, in feet per foot. 

Sbt Standard error of the regression coefficient A-,.

SEf, t Rate of removal of BOD by sedimentation at longitudinal position x at time /, in milligrams per liter per day. 

t Time, in days or hours. 

t' Flow time, or the time that must elapse between dissolved-oxygen measurements at the two sampling points in a

recirculating flume, in seconds.
VIII CONTENTS

Symbol Definition

ti, tz The time at which sunrise occurred, and the time in the evening when the rate of change of dissolved-oxygen content in the reach returned to its sunrise value. 

ta, / Times at which the concentrations of dissolved oxygen are measured at the upstream and downstream ends,

respectively, of the reach; t^-ta is the flow time, or the time required for a slug of water to flow through the

reach, in hours.

te Age of surface at which surface elements are replaced, in days or hours. 

T Temperature, in degrees Celsius. 

T Absolute temperature, in degrees Kelvin. 

M* Shear velocity, equal to Vg//5, in feet per second. 

7T' Longitudinal turbulent intensity, in centimeters per second. 

ue Eddy velocity in the longitudinal direction, large-eddy model (Fortescue and Pearson, 1967), in centimeters per

second.

M, Velocity fluctuation in the xi direction (tensor notation), or the difference between the instantaneous velocity and 

__ the time-averaged velocity, in feet per day. 

~i«c~ Product of the velocity fluctuation and the concentration fluctuation, in milligrams per liter times centimeters per

second. (Overbar indicates average with respect to time.)

U Mean flow velocity in the longitudinal direction, in feet per day. 

Ui Velocity in the jc, direction (tensor notation), in feet per day.

ve Eddy velocity in the vertical direction, large-eddy model (Fortescue and Pearson, 1967), in centimeters per second. 

V Volume of the system, in cubic centimeters.

w Exponent on velocity in the general form of the empirical prediction equation for the reaeration coefficient. 

W Width, in feet or meters. 

WG Weight of gaseous tracer (krypton) injected for a tracer-procedure determination of the reaeration coefficient, in

pounds. 

WL Weight of conservative liquid tracer (tritiated water) injected for a tracer-procedure determination of the reaeration

coefficient, in pounds. 

W0(i Weight of gaseous tracer (krypton) injected per unit of cross-sectional area of flow, equal to Wc,IAx, in pounds per

square foot. 

WVL Weight of conservative liquid tracer (tritiated water) injected per unit of cross-sectional area of flow, equal to

WLJA X, in pounds per square foot.

x Longitudinal position or direction, in feet.

x Longitudinal position in a coordinate system moving with the mean flow velocity, equal to x Ut. 

*a> *b Longitudinal position of the upstream and downstream ends of the reach, in feet. 

Xi Coordinate direction in tensor notation: Xi, longitudinal direction; x2 , vertical direction; .v:i , lateral direction. 

A* Length of reach, or Xb-xa , in feet.

X Distance between measurement points in a recirculating flume, in feet. 

Xt Amount of carbonaceous waste oxidized up to time t, in milligrams per liter. 

Xn, X-2\.. .Ajv, Independent variables in the linear regression equation. 

y Vertical position or direction, in feet.

y\ Vertical distance below the interface to which surface renewal occurs, Harriott (1962) model, in feet. 

y Dependent variable in linear regression equation.

Ye> Estimate of the dependent variable, Yi, from the linear regression equation. 

^f Dependent variable / in linear regression equation.

Z Constant in the Streeter and Phelps (1925) prediction equation for the reaeration coefficient.

<* >$ Parameters in the distribution function for surface-renewal time or for distance below the interface to which surface renewal occurs, Harriott (1962) model. 

ft Turbulence parameter for a flow layer in the Marchello and Toor (1963) model, equal to s(LfH-2)2, in square feet per

hour.

F Gamma function, T(a) = f^ e -«ya - l dy. 

8 Mean relative increase in flow velocity per 5-foot increase in gage height, Streeter and Phelps (1925) prediction

equation for the reaeration coefficient. 

d Indicates partial differentiation, 

e Eddy-diffusion coefficient for mass, equal to eT+Dm , in square feet per day (y and x subscripts indicate vertical and

longitudinal directions, respectively; i and j subscripts are tensor notation).

 T Turbulent diffusion coefficient, in square feet per day (/' and j subscripts are tensor notation). 

 Eddy-diffusion coefficient for momentum, in square feet per day. 

17 Exponent on velocity in the Streeter and Phelps (1925) prediction equation for the reaeration coefficient.

6 Temperature coefficient, equal to (K2T/K220) T -'2(f

K Slope parameter in logarithmic velocity-distribution law of von Karman.

K(; Gas-film coefficient for mass transfer, in milligrams per square centimeter per second per atmosphere.

KL Liquid-film coefficient for mass transfer, in liters per square centimeter per second.

A Macroscale of turbulence, in centimeters.

/i Absolute viscosity of the liquid, in grams per centimeter-second.
CONTENTS IX

Symbol Definition

/AO Velocity normal to the interface that a gas molecule must have to be able to leave the liquid, in centimeters per

	second.

v Kinematic viscosity of the liquid, in square centimeters per second.

IT The constant 3.1416.

p Mass density of the liquid, in grams per cubic centimeter,

cr Surface tension of the liquid, in dynes per centimeter.

</>(/) Distribution function for surface ages.

X Constant in the Isaacs and Gaudy (1968) equation.

i/; Constant in the empirical prediction equation for the reaeration coefficient.

o> Dissolved-oxygen consumption rate by bottom deposits, in grams per square meter per hour.

466-751 O - 72 - 2

REAERATION IN OPEN-CHANNEL FLOW

By J. P. BENNETT and R. E. RATHBUN

ABSTRACT

Reaeration, the physical absorption of oxygen from the 

atmosphere, is the primary process by which a stream replaces oxygen consumed in the biodegradation of organic 

wastes. The reaeration process in a stream is characterized 

by its surface reaeration coefficient. Hence, knowledge of the 

reaeration coefficient permits determination of the quantity 

of waste that can be discharged into a stream without causing 

serious depletion of the dissolved-oxygen content of the stream.

The three basic methods for measuring the reaeration coefficient are the dissolved-oxygen balance, disturbed-equilibrium, and tracer procedures. The dissolved-oxygen balance 

method consists of measuring the various sources and sinks 

of dissolved oxygen and determining by difference the amount 

of reaeration needed to balance the equation. The disturbedequilibrium method consists of artificially producing dissolvedoxygen deficits by adding sodium sulfite to the stream and 

subsequently measuring upstream and downstream concentrations of dissolved oxygen at two different concentration levels. 

The tracer method consists of using an inert radioactive gas 

as a tracer for oxygen and correlating the rate of desorption 

of the tracer gas with the rate of absorption of oxygen.

Various theoretical models of the oxygen-absorption process 

exist; however, these models are generally not suited for prediction of the reaeration coefficient in streams because the 

model parameters have not been adequately related to bulkflow hydraulic variables. Semiempirical and empirical equations developed from experimental data adequately predict 

reaeration coefficients for streams of the type on which the 

equations were based, but large errors may occur when the 

equations are applied to other types of streams or to conditions outside the range of variables considered in the original 

correlation.

INTRODUCTION

Rivers and streams are used for the disposal of 

municipal and industrial wastes, and Hull (1963) 

suggested that this use is one of the most important factors contributing to the general health and 

welfare of the people of the United States. Discharge of wastes into a stream, however, does not 

simply dilute the wastes. Each stream and river 

has a natural capacity for oxidizing biodegradable 

wastes, thus purifying the waters. This purification capacity is dependent upon many factors, including the water discharge, the depth of flow, the 

velocity of flow, and the various sources and sinks 

of dissolved oxygen along the stream. The natural

purification capacity of a stream therefore will 

vary as the hydraulic conditions in the stream 

change with time.

Although the flow in a specific stream may have 

varied greatly from day to day and from year to 

year, the total annual runoff for the United States 

as a whole showed only a small, apparently cyclic, 

variation with time for the years 1895 to 1955 (Leopold and Langbein, 1960). On the other hand, the 

population and the number of industrial facilities 

have increased tremendously, and this growth has 

placed an ever increasing burden on the rivers and 

streams of the United States. In many instances, 

the purification capacities of streams and rivers 

have been exceeded.

The fact that municipal and industrial wastes 

are discharged into a stream does not necessarily 

mean that the stream is polluted, because the 

stream does have the capacity for self-purification. The definition of pollution has received 

considerable attention. The Colorado Supreme 

Court in 1934 (Gindler, 1967, p. 5) defined pollution 

as follows:

For the purpose of this case, the word "pollution" means an 

impairment, with attendant injury, to the use of water that the 

plaintiffs are entitled to make. Unless the introduction of 

extraneous matter so unfavorably affects such use, the condition is short of pollution. In reality, the thing forbidden is injury. The quantity introduced is immaterial.

Other definitions were discussed by Haney (1966), 

and he proposed (p. 110) that "pollution is the impairment of water quality, with resultant significant interference with beneficial water use." 

Apparently, from these definitions, the determination of when pollution exists depends both on the 

intended use of the stream and on some water 

quality parameter that will insure that the water 

is of a quality adequate for the intended purpose. 

Of the various indicators of water quality (such 

as the coliform bacteria count, the hydrogen ion 

concentration, and the concentrations of phosphates, nitrates, synthetic organics, and industrial 

chemicals), the one most often used as a measure 

of pollution by biodegradable organic wastes is 

DO (dissolved oxygen) concentration (Wolman,
REAERATION IN OPEN-CHANNEL FLOW

1960). This report uses DO concentration as the 

principal measure of water quality. Thus the condition of a stream is determined from a DO balance 

which includes all the sources and sinks of DO 

along the reach of interest. The various types of 

DO balances for streams and rivers that have 

appeared in the literature are discussed in the 

following paragraphs.

The purification capacity of a stream depends 

upon its DO resources and its ability to replace the 

oxygen consumed in the oxidation of organic 

wastes. One of the first studies of the DO balance 

was the classical study of Streeter and Phelps 

(1925). These authors assumed one sink and one 

source of DO.

The oxygen sink was the biochemical oxidation 

of carbonaceous wastes by bacteria in the stream. 

It is generally assumed that biochemical oxidation 

is a first-order process; that is, the rate of biochemical oxidation is proportional to the amount 

of carbonaceous material remaining to be oxidized. 

This process may be described by

f K, L, (1)

where L is the BOD (biochemical oxygen demand) 

of the carbonaceous material, / is time, and Ki is 

the deoxygenation coefficient or the rate constant 

for the oxidation of carbonaceous wastes. Note 

that the concentration of biodegradable waste is 

expressed in terms of the oxygen demand. The concentration of biodegradable waste in the stream is 

not directly measurable and is of importance only 

because it controls the demand for DO. The DO 

demand is determined by a BOD test, and standard 

procedures for the test were described by the 

American Public Health Association (1965). If the 

DO concentration is expressed in terms of the saturation deficit, D, or the difference between the 

DO saturation concentration and the actual concentration at time /, then

~dt =~~di' (2) 

The oxygen source considered by Streeter and 

Phelps (1925) was reaeration, or the physical 

absorption of oxygen from the atmosphere by the 

flowing stream. Adeney and Becker (1919, 1920) 

showed that the absorption of oxygen by water is 

a first-order process; that is, the rate of absorption 

is directly proportional to the saturation deficit. 

This process may be described by

Streeter and Phelps (1925) assumed that the DO 

deficit was determined by these two independent 

processes. Therefore, the DO balance was

dt - KK * Dn > (3)

. M\. \ l-i l^ dt 

The solution of equation 1 is

(4)

L=La exp(-/M f). (5)

Substituting equation 5 in equation 4, the solution 

of equation 4 is

jy T 

T^ *^l L",

A.2 A.I

[exp(- ATi 0~exp(- AT2 0] +Da exp(-

(6)

where K2 is the reaeration coefficient or the rate 

constant for the absorption of oxygen from the 

atmosphere and D is the DO deficit.

where La is the BOD of the carbonaceous material 

and Da is the DO deficit just upstream from the 

point at which the waste is added to the stream. 

The time, /, is expressed in terms of flow time along 

the stream. If / is flow time, then dD/dt is equivalent to U dD/dx, where U is the mean flow velocity, 

in feet per day, and x is longitudinal distance, in 

feet. The factors D and L are usually expressed in 

parts per million or milligrams per liter, and K\ 

and K2 are usually expressed in reciprocal days. 

A plot of D as a function of t gives the classical sag 

curve for DO deficit. An assumption inherent in 

equation 6 is that the initial pollution load is added 

at the upstream end of the reach and thereafter 

is affected only by biochemical oxidation and not 

by the addition of either pollution or dilution flow 

along the reach.

Streeter and Phelps (1925) were aware that the 

assumption of a slug injection of BOD was an oversimplification of the actual situation; hence, they 

also solved equation 4 for a BOD source uniformly 

distributed along the reach of interest. This BOD 

distribution resulted in a much more complicated 

solution. Other assumptions regarding the distribution of the BOD load are possible, and the complexity of the solution, equation 6, increases as 

the complexity of the assumed distribution for 

L increases.

An example of an oxygen sag curve for an La of 

9.0 mg/1 (milligrams per liter), a Da of 1.0 mg/1, a 

Ki of 0.1 days- 1 (reciprocal days), and a K2 of 

0.2 days" 1 is presented in figure 1. Just below 

the point of addition of the pollution load, the oxidation of the waste predominates, and the DO deficit increases. However, reaeration begins as 

soon as a DO deficit develops, and the rate of reaeration increases as the deficit increases. Finally, 

at the minimum deficit of the oxygen sag curve, 

the rate of supply of oxygen by reaeration equals 

the rate of consumption of oxygen by biochemical 

oxidation. Thereafter, the reaeration process predominates, and the DO deficit is gradually reduced
INTRODUCTION

10 15 20 

TIME, IN DAYS

FIGURE 1. Example of a DO deficit sag curve.

to zero. Also shown in figure 1 is the increase in 

the DO deficit in the absence of reaeration (#2 = 0).

The oxygen sag equation of Streeter and Phelps 

(1925) has been widely used, and different forms 

of the equation have appeared in the literature. 

Some authors have used common logarithms (base 

10), and the deoxygenation and reaeration coefficients under these circumstances are generally 

represented by ki and A 2 , respectively, where

Ki = 2.30*i (7) 

and

#2 = 2.30 #2 . (8) 

A mathematical analysis of the Streeter-Phelps 

equations was presented by Fair (1939).

Processes other than biochemical oxidation and 

reaeration also may be important in the DO 

balance of a stream. Although Streeter and Phelps 

(1925) chose to include only these two fundamental 

processes in their oxygen balance, they were aware 

of the possible effects of inflow along the reach, 

sedimentation and scouring in the reach, and 

photosynthesis (Streeter and Phelps, 1925, p. 10, 

22). Other authors, notably Dobbins (1964b) and 

Camp (1963), have expanded the fundamental 

Streeter-Phelps equation to include in the oxygen 

balance these and other sources and sinks of DO.

Dobbins (1964b) included the removal of BOD by 

sedimentation or adsorption on bottom sediments, 

the removal of DO from the water by diffusion into 

the benthal layer and by plankton and fixed-plant 

respiration, the addition of BOD along the reach 

by runoff and by scour of bottom deposits, the 

addition of DO by the photosynthetic process of 

plankton and fixed plants, and the continuous 

redistribution of the BOD and DO by the effects 

of longitudinal dispersion.

With these sources and sinks considered, the DO 

balance has the form

"^ i rr"*-' _ rj i If (f f^\ IS I n (Q\

 T U Lfy 2 ~rA.2l,Cs C,J /Vi/-y L>BI (.&)

where C is the DO concentration, Cs is the saturation DO concentration, x is longitudinal distance 

along the reach, U is the mean stream velocity, 

Do- is the longitudinal-dispersion coefficient, and 

DB includes the removal of DO by diffusion into 

the benthal layer and the effect of photosynthesis 

and respiration by plants. Therefore, the algebraic 

value of DB may be positive or negative, depending 

on the relative magnitudes of the three processes. 

Dobbins (1964b) used the convention that DB is 

positive when the rate of removal of DO exceeds 

the rate of addition.

A BOD balance analogous to the DO balance may 

be written as follows:

where K3 is the rate constant for the removal of 

BOD by sedimentation and adsorption, and LR is 

the rate of addition of BOD along the reach, including the addition of BOD with runoff, the diffusion of partially degraded waste from the benthal 

layer into the water, and the scour and resuspension of bottom deposits.

In general, steady-state and uniform-flow conditions are assumed in a DO balance study. With 

these assumptions, the dC/dt and dL/dt terms are 

zero, and the water discharge, the input BOD load, 

the temperature, and the rate constants at each 

point in the reach are assumed independent of 

time. Other assumptions inherent in equations 9 

and 10 are as follows:

1. The DO and BOD are uniformly distributed 

over each cross section so that equations 9 

and 10 can be written in the one-dimensional 

form.

2. The processes described by the rate constants 

Ki, K2, and K3 are first-order processes; that 

is, the rate of removal of BOD is proportional 

to the amount of BOD remaining, and the 

rate of reaeration is proportional to the DO 

deficit.

3. The processes represented by LR and DK are uniformly distributed along the reach.

4. Only the carbonaceous demand of the waste is 

significant. If the nitrogenous demand is 

important, an additional term must be added 

to equations 9 and 10.

Each of the terms in equations 9 and 10 represents a time-rate of change of the BOD or DO 

concentration, and the units used are generally
REAERATION IN OPEN-CHANNEL FLOW

parts per million per day or milligrams per liter 

per day.

Dobbins (1964b) solved equations 9 and 10 using 

these assumptions and the added assumption that 

the effect of longitudinal dispersion is negligible 

compared with the other terms for natural 

streams. For the BOD equation, the solution 

showed that the BOD may either decrease or increase along the reach, depending on whether La 

is greater or less than LnKKi + Ks), where La is the 

initial BOD load at the upstream end of the reach. 

Dobbins (1964b) suggested a procedure for the 

evaluation of the various constants, and he also 

discussed the DB term, which includes several 

different processes.

Camp (1963) presented DO and BOD balance 

equations similar in form to equations 9 and 10. 

Primary differences are in the use of dC/dt and 

dL/dt as opposed to UdC/dx and UdL/dx and in the 

treatment of the DB term. With the assumptions 

of steady-state and uniform-flow conditions,

jru i- (11)

Therefore, Camp's equations (1963, p. 295,297) may 

be written as

 =/C2(Cs -C)-/CiL+/7 (12) dt

for the DO balance and

for the BOD balance. In equation 12, p is the rate 

of production of oxygen by photosynthesis; and 

in equation 13, m is the rate of addition of BOD to 

the water from bottom deposits. For streams 

where longitudinal mixing is important, Camp 

(1963, p. 311) included a Dx(dz/dx2) term on the righthand side of equations 12 and 13.

Other forms of DO balances have been used. The 

Water Pollution Research Laboratory group at 

Stevenage, England (Edwards and Owens, 1962; 

Edwards, 1962; Edwards and Owens, 1965; Owens, 

1965), has used a DO balance which consists simply 

of an algebraic summation of the contributions or 

debits of the various sources and sinks of DO. The 

equation is

Qo=P±Do-R, (14) 

where P is the rate of production of oxygen by 

photosynthesis, R is the rate of oxygen utilization, 

Do is the rate of transfer of oxygen through the 

surface of the stream, and Qo is the rate of change 

of DO content in the reach of interest. Equation 

14 is an integral form of the DO balance in which 

the various coefficients are assumed to be independent of time and longitudinal position within

the reach. Stream flow is also assumed to be constant and uniform over the reach of interest. 

The factor QQ is defined by

O = (C C ) (15)

where F is the volumetric rate of flow of water, 

As is the water surface area of the reach, and Cb 

and Ca are the concentrations of DO at the downstream and upstream ends, respectively, of the 

reach at times tb and ra, where tb ta is the time 

required for a slug of water to flow through the 

reach. 

The factor Do is defined by

D9=fi£.-C), (16) 

where / is the exchange coefficient for the reach, 

and Cs and C are the mean saturation and the mean 

concentration of dissolved oxygen in the reach, 

respectively.

Each term in equations 14 to 16 has the dimensions of grams per square meter per hour, or 

equivalent, and is a rate per unit of surface area. 

To put these equations in dimensions comparable 

to the dimensions of the Streeter and Phelps (1925), 

Camp (1963), and Dobbins (1964b) equations, each 

term must be divided by the mean depth of flow. 

Thus, the exchange coefficient, /, divided by the 

mean depth of flow is equivalent to the reaeration 

coefficient, K2 .

The basic DO balance, equation 14, was modified 

as needed to meet the conditions in the stream 

under study. For example, Owens and Edwards 

(1966) used

QO=PB +PP-RB-RP±DO, a?)

where the B subscript refers to bottom deposits 

and attached plants, and the P subscript refers to 

phytoplankton, which live unattached in the water. 

In the River Lark, in England, just below a sewage 

works, Owens and Edwards (1963) found that the 

consumption of oxygen by the bottom deposits or 

mud was a very important factor. Thus the respiration term, RB, for bottom deposits was divided into 

two terms, RB and RM, where the subscript B refers 

to attached bottom plants (benthos) and the subscript M refers to mud or bottom deposits. The 

resultant equation,

was discussed by Owens and Knowles (1968). If 

significant organic pollution exists in the water 

of the stream, then an additional oxygen sink, /?BOD> 

must be added to the right-hand side of equation 18. 

Bain (1968) used a modified form of the StreeterPhelps equation for predicting the variation of the 

DO concentration with time at a point. His DO 

balance was
INTRODUCTION

dCt 

dt

=pt-rt-Kl (19)

where the t subscript indicates variability with 

time at a particular point in space. Bain (1968) 

wrote equation 19 for each of six 4-hour periods, 

giving a set of six simultaneous equations for each 

day of observation. These equations were solved 

by a numerical technique using sequential substitution, and the result was a dynamic DO balance 

for a 24-hour period at a specific point in the stream. 

O'Connell and Thomas (1965) considered the 

assumption of steady-state conditions as the 

major fault of the Streeter-Phelps equation and 

therefore used

(20)

They solved equation 20 using the assumption 

that the p and r factors were constant with respect 

to time and position. Hence, the solution is applicable only at a point for a short interval of flow 

time.

Li (1962) presented a generalized solution of the 

Streeter-Phelps equation that was expanded to 

include the effects of BOD and runoff additions 

along the stream and of fluctuations of the BOD 

load and DO concentration at the point of addition 

of the pollution to the stream. He assumed that the 

variations in BOD load and DO concentration were 

cyclic and were repeated cycle after cycle.

O'Connor (1967) wrote a general DO balance for 

a one-dimensional stream in which the oxygen demand of the waste is divided into the requirements 

for the carbonaceous material and those for the 

nitrogenous material. The result is

-KN Nx -rx,t-SEx,t, (21)

where F is the volumetric rate of flow, Ax is the 

cross-sectional area of flow at longitudinal position 

x, KM is the rate constant for the oxidation of nitrogenous material, Nx is the concentration of nitrogenous material at x, SE is the rate of removal of 

BOD by sedimentation, p is the rate of production 

of DO by photosynthesis, r is the rate of consumption of dissolved oxygen by plant respiration, and 

the x,t subscript indicates the longitudinal position x at time t. The oxygen demand of the nitrogenous material has not received as much attention 

as the oxygen demand of the carbonaceous material. However, work by Courchaine (1968), 

O'Connell and Thomas (1965), Wezernak and 

Gannon (1968), and O'Connor, St. John, and DiToro 

(1968) has shown the importance of nitrification

on the DO balance of streams receiving discharges 

of municipal waste.

O'Connor (1967) solved the general DO balance 

(eq 21) analytically using the assumptions that the 

photosynthetic rate varied sinusoidally with time 

and that the maximum rate of photosynthesis and 

the factors rx,, and SEX,, were independent of position and time. The solution was applicable to a sunlight period of 12 hours. O'Connor discussed in 

general terms two limiting forms of the solution. In 

the first situation the stream is heavily polluted, is 

shaded, and contains large amounts of suspended 

materials and (or) small amounts of nutrient 

materials. Under these conditions, p*,, and rflt are 

small and are negligible with respect to the BOD 

factors, and the assumption of steady-state conditions is reasonably valid. The solution reduces to 

the oxygen sag equation of Streeter and Phelps 

(1925) with a term added for the oxygen demands 

of the bottom deposits. In the second situation, the 

stream is in the latter stage of self-purification. 

Oxidation of organic material is essentially complete, and bacterial respiration is negligible in 

comparison with algae respiration. The assumption of steady-state conditions is not valid, and the 

solution reduces to an equation that describes the 

temporal distribution of DO as a result of photosynthetic activity.

The variety of DO balances that have appeared 

in the literature demonstrates an important fact: 

each stream and each reach of a stream must be 

considered as a separate problem. Generalizations 

are possible, however. For example, McConnell 

and Sigler (1959) found in a shallow, swift stream 

that the effect of the stream metabolism on the 

DO content was negligible compared with the 

effect of reaeration. On the other hand, in the 

River Ivel, in England, where the water velocity 

seldom was greater than 6 centimeters per second, 

reaeration was relatively unimportant (Owens, 

1965). Therefore, in an initial DO balance on a 

stream, all possible sources and sinks of DO should 

be considered. The unimportant factors may then 

be eliminated as they are shown to be negligible.

With the development of computers, the computation of DO distributions from complex analytical 

solutions of DO balances was simplified greatly. 

Frankel (1965) developed a dynamic model of the 

sag equation for computer solution. His model was 

an extension of the model of Li (1962) to include a 

variable DO source or sink caused by the presence 

of algae.

In addition to simplifying computations, computers made possible numerical solutions that
REAERATION IN OPEN-CHANNEL FLOW

previously would not have been feasible. Dresnack 

and Dobbins (1968) presented numerical solutions 

of the BOD and DO equations where BOD and DO 

inputs varying with time and space prevented analytical solutions. These solutions were limited to 

steady-state and uniform-flow conditions. Three 

examples were considered. These were: (1) The 

effect of the longitudinal-dispersion coefficient on 

the concentration profile of a conservative pollutant introduced at the upstream end of the reach 

in a periodically varying manner; (2) the effect of 

sinusoidal inputs of BOD and DO at the upstream 

end of the reach on the BOD and DO profiles; and 

(3) the effects of sinusoidally varying inputs of 

BOD and DO and a sinusoidal variation in the 

algae effects. Bella and Dobbins (1968) presented 

a modeling procedure using finite-difference 

techniques for BOD and DO profiles in streams 

and estuaries; this procedure considered temporal 

and spatial variations in hydraulic conditions.

The development of computers also has resulted 

in the advancement of the concepts of systems 

analysis, and these concepts in turn have been 

applied to the modeling of the DO resources of a 

stream. A system may in general be defined as an 

entity which operates on or transforms an input 

to produce an output. Thus the system consists of 

the input, the output, and the transformation between the two. Thomann (1963) presented a general 

model for describing the variations of DO with 

time in a finite number of sections of a body of 

water and applied this model (1965) to the Delaware River Estuary. He determined the transient 

response of the estuary to an instantaneous injection of waste, demonstrated the ease of extending the systems-analysis model to two dimensions, 

and determined the steady-state conditions for the 

one-dimensional model. He also applied the linear 

programing concept to the steady-state conditions 

to determine the most economical combination of 

waste treatments for the various waste sources 

that would assure a minimum steady-state concentration of DO of 2 mg/1. The latter problem was 

solved for two values of the reaeration coefficient.

Liebman and Lynn (1966) applied dynamic programing in a simplified example of river-basin 

planning based on data from the Willamette River, 

Oreg. The objective was to determine the amount 

of BOD removal required of each waste discharger 

so that the water-quality standards for DO would 

be met with minimum total cost to the basin. The 

solution of the Streeter-Phelps equation was used 

with Ki and K2 assumed known and constant in 

each reach. The equation was modified, however, 

to account for changes in the saturation concentration, Cs, as a result of temperature changes. 

Cs was assumed to vary linearly with distance 

downstream. Three situations were considered:

(1) the configuration of treatment plants that 

would meet the DO standards with minimum cost;

(2) same as No. 1 except that the minimum DO 

standard was reduced 0.1 mg/1; and (3) same as No. 1 

except the requirement was added that all plants 

provide the same percentage of BOD removal.

The general application of the concepts of linear 

programing to the solution of the DO balance 

equation and the determination of the degrees of 

treatment required for minimum basin cost were 

discussed by ReVelle, Loucks, and Lynn (1967). The 

application of linear programing to the data from 

the Willamette River, Oreg., was discussed by 

Revelle, Loucks, and Lynn (1968). The results were 

essentially the same as those obtained with 

dynamic programing.

Most DO balances give one value for the DO concentration at a downstream point. Similarly, most 

DO quality standards do not distinguish as to the 

degree or duration of a pollution violation. This 

consideration is important because a DO concentration that is sufficient on the average to assure 

healthy fish does not prevent fish kills. Thayer and 

Krutchkoff (1967) presented a stochastic model 

applicable to steady-state flow and input BOD and 

DO conditions that gives the probability distributions for the BOD and DO concentrations at 

downstream points. With the confidence limits 

determined from the distributions, the proportion 

of the time that the BOD will be greater than any 

specified concentration can be determined. The 

model can be used with any flow condition, such as 

the expected minimum flow condition, provided 

the parameters ATi, K2 , K3 , and DB associated with 

the flow condition are known.

Loucks and Lynn (1966) developed probabilistic 

models for predicting the probability distribution 

of minimum DO concentration at points downstream from a waste-treatment plant. The models 

were based on different mean daily stream and 

waste flow conditions. Loucks and Lynn suggested 

that the use of a maximum allowable probability 

of the DO concentration falling below a specified 

concentration for a given number of consecutive 

days was a more realistic approach than the deterministic DO standards generally used.

Kothandaraman and Ewing (1969) considered 

the effect of variations in Ki and K2 on the downstream DO distributions and developed procedures 

for taking the expected variations into account 

in predicting the DO. A Monte Carlo simulation 

technique was used to analogize the variations in
THEORIES FOR PREDICTING THE REAERATION COEFFICIENT

Ki and K2 . The variations of Ki and K2 within their 

practical ranges were found to have significant 

effects in the prediction of the DO concentrations, 

and the errors increased with temperature 

increase.

DiToro and O'Connor (1968) studied the effect of 

time-varying discharge and mean velocity on the 

BOD and DO distributions. They found that the 

standard deviations of tlie resultant BOD and DO 

distributions were approximately equal to the coefficient of variation of the velocity multiplied by 

the BOD or DO distribution calculated using the 

mean velocity.

Despite the sophistication of the mathematical 

techniques permitted by the development of computers, the various parameters of the DO balance 

still must be measured or estimated. Thomann 

(1965) stated that his systems-analysis study of the 

Delaware River Estuary pointed out "the importance of good engineering evaluation of the system 

parameters of flow, diffusion, reaeration, and 

decay coefficients."

A critical analysis of previous theoretical and 

experimental work concerning the reaeration coefficient in open-channel flows is given on the 

following pages. The discussion covers theoretical 

and empirical models for prediction of K2 , techniques for experimental determination of K2, an 

analysis of previous experimental data, a suggested prediction equation, and suggestions for 

future research. From the introductory discussion of the DO balance for open-channel flows, it 

is apparent that if K2 is to be calculated using a 

DO balance equation, all other parameters appearing in the equation must be known. The procedures 

for determining these parameters are therefore 

discussed also.

THEORIES FOR PREDICTING THE 

REAERATION COEFFICIENT

The gas-absorption process at the interface between a flowing liquid and a gas has received considerable attention in recent years. The process 

is very complicated and only superficially understood; so theoretical treatments have been only 

partially successful. Still, considerable useful information concerning this phenomenon is given in 

the technical literature of chemical and sanitary 

engineering. The purpose of this section is to describe the various theories for predicting the 

reaeration coefficient or the gas-liquid masstransfer coefficient for open-channel flows.

DIFFUSION EQUATION

A gas moves into and through a flowing liquid 

by the process of diffusion. This process may be 

described by the general diffusion equation 

(Hinze, 1959)

ac , rr ac a / ac \, ~ /r»r»\ -Z7+Ui = [Dm - uic\ + Fc, (22) dt dxt dxt\ dXi I

given here in tensor notation. The capital letters 

represent time-averaged quantities, the lowercase 

letters represent fluctuating components or the 

instantaneous deviations from the time-averaged 

quantities, C is the concentration of the diffusing 

substance, LJi is the velocity in the xi direction, Dm 

is the molecular diffusivity of the dissolved gas 

in the liquid, Fc is a source-sink term, and the overbar on Hie indicates an averaging procedure with 

respect to time. Equation 22 applies to a dispersant 

which is in solution in a steady-state flow of an 

incompressible liquid.

With the common assumptions that (1) turbulent 

diffusion may be described by a turbulent-diffusion 

coefficient, er0 , such that

d^fT iTc' (9Vi it MI*- . \&tj)

(2) the total diffusion may be described by a diffusion tensor, ey, where the turbulent and molecular 

diffusion are independent and therefore additive, 

or

ey= ry +Dm ; (24) 

and (3) the coordinate axes coincide with the principal axes of the diffusion tensor, Sayre and Chang 

(1968) reduced equation 22 to

?JL+ Ut |f=~(e, |f ) + /V (25)

For uniform two-dimensional flow in a straight 

channel of constant cross section, equation 25 may 

be expressed as

ac ac = a2C ( dt dx~ x dx2 .. 

Equation 26 is expressed in the coordinate system 

usually used for open-channel flow situations. 

That is, the mean flow is in the x, or longitudinal, 

direction, and U is the flow velocity in that direction; y is in the vertical direction, or the direction 

of the channel depth; and the mean flow in the y 

direction is zero.

 1 CH 

Defining averages of the type C=T7 Cdy, and "Jo 

noting that the mass transport across the lower

flow boundary is zero UJ/-T- =0],

age equation 26 over the flow depth H and obtain

(26)

one can aver-

^-+T-| , J7r c/C J=r-^l dt dx\ / dx2

l (* dC\\ 

 ley^ I H\ y dy/\y = H

466-751 O - 72 - 3
8 REAERATION IN OPEN-CHANNEL FLOW

correOver much of the flow depth of a turbulent openchannel flow, mass transport in the y direction as 

a result of vertical diffusion is very large compared 

with the mass transport through the surface. 

Hence, the vertical gradient of C generally is very 

small except near the surface and within the bulk 

of the flow the value of C should approach C. This 

has been verified by W. W. Sayre (written commun. 

1968), who numerically solved the equation

 = ( 2, ). He assumed ey to be equal to the mo- di oy \ oyI

mentum transport coefficient of a flow having a 

logarithmic velocity distribution and solved the 

equation for C as a function of time at 50 points in

the vertical using values of \ y )\

\ dy/\y = H

spending to the range of such values observed for 

streams in nature. Initially, the flow was assumed 

to be unsaturated except for the top increment, 

and he found that even for very short times, there 

was no appreciable concentration gradient in the 

vertical. In support of Sayre's work, Churchill, 

Elmore, and Buckingham (1962) found no vertical 

concentration gradients in a series of DO measurements in a deep, low-velocity stream, and Edwards 

and Owens (1962) found that the DO concentration 

of a slow-flowing English stream was greater than 

98 percent of the surface concentration except in 

a relatively small area near the bed. Also, Whipple, 

Hunter, Davidson, Dittman, and Yu (1969, p. 116) 

concluded that the centerline DO concentration at 

a depth of 1 foot was a good estimate of the mean 

cross-sectional concentration of DO for the slowflowing Passaic River in New Jersey. When the 

concentration of the dissolved substance is constant in the vertical, equation 27 may be written

dt ^ B ^ + /V (28)

It is generally assumed that the diffusion equation in this form is applicable to computing the 

longitudinal distribution of DO in streams.

Because of the experimental and theoretical 

evidence indicating no concentration gradient of 

DO in the vertical direction of an open-channel 

flow, the use of the overbar on concentration in 

equation 28 to indicate an average with respect tq 

the depth of flow in unnecessary. Hence, this overbar on concentration will not be used in the 

remainder of the report. For consistency, the overbar on the velocity will also be dropped, with the 

understanding, however, that U is the flow velocity 

averaged over the depth of flow or the mean velocity of flow in the longitudinal direction.

For reasons of convenience and tradition, completely aside from the fact that the turbulenttransport process in open-channel flows is not 

sufficiently understood to permit evaluation of 

y near a free surface, the second term on the right 

in equation 28 is generally expressed using one 

of two mass-transfer coefficients, KL or K2 . The 

more general coefficient KL, the liquid-film coefficient, is defined as follows:

' (29)

where CL is the DO concentration in the bulk liquid 

phase. Comparison of equations 9 and 28 shows 

that the reaeration coefficient, K2 , can be expressed in terms of the film coefficient as

K2 = KJH. (30)

(Eq 9 is the same as eq 28, except the source-sink 

terms have been lumped into the term Fc in eq 28.) 

Theories for the prediction of KL or K2 can be 

checked in certain idealized situations by measurements of CL, evaluation of the source-sink 

terms, and substitution into an integrated version 

of equation 28.

The prediction of mass-transfer coefficients 

from solute and solvent properties and from the 

parameters of the flow is a problem which has 

received much discussion in the fluid-mechanics 

literature. At the present time (1971) fairly reliable 

theories are available for the prediction of heatand mass-transfer coefficients for flows near the 

solid boundaries in straight circular pipes (Hinze, 

1959). These fall into two general categories: the 

eddy-diffusion or gradient-diffusion theories and 

the renewal theories.

Because of the gas-liquid interface, side-wall 

effects, and secondary currents, the mechanics 

of flow in straight open channels with uniform 

cross sections are considerably more complex 

than the mechanics of flow in straight circular 

pipes. Furthermore, mass-transfer coefficients 

usually are desired not for straight channels with 

uniform cross sections but for natural channels, 

which are neither straight nor of uniform cross 

section. Because of these complexities, reliable 

theories comparable to the eddy-diffusivity 

theories for pipe flows have not been developed for 

the prediction of mass-transfer coefficients from 

mean flow parameters in open-channel flows. The 

theories which have arisen are in general less 

rigorous, in the fluid-mechanics sense, than are 

the eddy-diffusivity theories. They fall into three 

general groups: (1) the conceptual theories, based 

on assumed mathematical models of the masstransfer processes within the flow; (2) the semi-
THEORIES FOR PREDICTING THE REAERATION COEFFICIENT

empirical theories, qualitatively relating the 

mass-transfer coefficients to some measurable 

flow property, the constants in the relation being 

determined by multiple regression; and (3) 

empirical theories resulting from dimensional 

analysis, again with the constants being determined by multiple-regression techniques. The 

remainder of this section is devoted to a discussion of the models of these types which have 

appeared in the literature.

CONCEPTUAL MODELS FOR THE GAS-LIQUID 

MASS-TRANSFER PROCESS

Conceptual models are derived from equations 

describing an interpretation of the physical conditions governing mass transfer at a gas-liquid 

interface. Of these equations, the one generally 

used is the differential equation describing nonconvective diffusion. The boundary conditions 

chosen vary from model to model. In some models, 

the parameters arising from the solution of the 

differential equation with the given boundary conditions are not clearly related to the mean flow 

parameters such as mean velocity, depth of flow, 

or turbulent intensity. In such models, discovering 

the relation of the former parameters to the latter 

is a problem requiring considerable thought and 

experimental verification. The models may be 

divided into the following general groups: (1) Film 

models; (2) renewal, including penetration, filmpenetration, and surface-renewal-damped-eddy 

diffusivity models; (3) kinetic-theory models; (4) 

large-eddy models; and (5) turbulent-diffusion 

models. The usefulness of a particular model ultimately must be determined by how well it predicts 

reaeration coefficients for natural streams.

FILM MODEL

The earliest model for mass transfer at a gasliquid interface was the two-film model of Lewis 

and Whitman (1924). They postulated that laminar 

films of gas and liquid exist at the interface 

between the two phases. Because mass transport 

in a direction perpendicular to the mean-flow direction in a laminar film is by molecular diffusion 

alone, a process much slower than the turbulent 

diffusion within the phases, the resistance to mass 

transfer is concentrated in the two films. The 

interface between the gas and the liquid films 

seems to offer no resistance to mass transfer 

(Sherwood and Pigford, 1952). If the gas and the 

liquid phases outside of the film region are 

assumed to be completely mixed, then the mass 

transport, N, of oxygen through the films for the

oxygen-water system is

\ T ( 4~* 4~* \ / . \ /O 1 \

N = KL(CI CL) = KG (pG pi), (31) 

where CL is the concentration of oxygen in the 

water phase, d is the concentration of oxygen at 

the interface, pc and p\ are, respectively, the 

partial pressures of oxygen in the gas phase and 

at the interface, and KL and KG are, respectively, 

the liquid-film and gas-film coefficients. Equilibrium conditions usually are assumed at the interface, so that C, and /?, are related by Henry's Law, 

or pi=hd, where h is the Henry's Law constant.

The film coefficients are expected to be dependent on the properties of the liquid and the 

gas and on the flow parameters, but independent 

of the concentration of the dissolved gas in the 

liquid (Sherwood and Pigford, 1952).

Concentrations and partial pressures at the 

interface are difficult to measure, however; so 

information on the individual film coefficients is 

difficult to obtain. Thus, equation 31 usually is 

written in terms of the overall liquid-film and gasfilm coefficients, or

N = KL(CS CL) = KG (PG PS), (32) 

where ps is the partial pressure corresponding to 

the saturation concentration, Cs, and KL and KG 

are, respectively, the overall liquid-film and gasfilm coefficients for mass transfer.

To be of use in engineering calculations, the 

overall coefficients should be dependent on the 

properties of the liquid and of the gas and on the 

flow parameters, but be independent of the concentration of dissolved gas. From equations 31 

and 32 and Henry's Law,

-L=1+J-. (33) 

KL KL n KG

Thus, for any gas-liquid pair that obeys Henry's 

Law, the overall liquid-film coefficient is independent of the gas concentration in the liquid. 

For a slightly soluble gas that does not react chemically with the liquid, h is large, so that equation 

33 reduces to

KL = KL. (34)

Equation 34 is equivalent to saying that the gasfilm resistance is negligible with respect to the 

liquid-film resistance and that KL is independent 

of the concentration of dissolved gas for a slightly 

soluble gas that obeys Henry's Law. The oxygenwater system is such a gas-liquid pair.

Because mass transfer in the laminar film is by 

molecular diffusion, and if the concentration 

gradient across the film is assumed to be linear, 

then

N = Dm (Cs ~ CL\ (35)
10 REAERATION IN OPEN-CHANNEL FLOW

where L/ is the thickness of the film. By comparing 

equations 32 and 35, it follows that

KL =2*. (36)

Ar

In equation 36, the solute and solvent properties 

are accounted for by the molecular diffusivity, Dm, 

and the hydrodynamics of the flow are accounted 

for by the layer thickness, L/. For a truly laminar 

layer to exist on the surface of an open-channel 

flow, the flow must be very tranquil. Such flows 

are seldom found in field situations, however, 

so that equation 36 does not suffice to describe 

the variation of the reaeration coefficient with 

flow parameters. The concept of diffusion into 

laminar patches of infinitesimal area existing for 

short times on the water surface before being replaced by liquid from the bulk liquid has been used, 

however, in subsequent models describing the 

reaeration process.

RENEWAL MODELS

The renewal models are analogous to those for 

predicting mass-transfer coefficients for fluid 

flows near solid boundaries. In the renewal models, 

diffusion into a surface layer is generally assumed 

to be purely molecular. The surface layer is sometimes assumed to be of infinite thickness, and 

sometimes to have a thickness, L/, which may or 

may not be randomly distributed about an average 

thickness, L/. Below the layer, the diffusivity in 

the liquid is assumed to be infinite, the concentration independent of depth. After diffusion into an 

element of the surface layer has been taking place 

for some time, the element is assumed to be 

replaced by liquid from the bulk flow. The frequency of renewal is governed by an assumed 

probability density function. The mass-transfer 

coefficients predicted by these models are functions of the molecular diffusivity, an average 

renewal frequency, and, generally, an average 

thickness of the surface layer. The renewal models 

include (1) the penetration model, (2) the filmpenetration model, and (3) the surface-renewaldamped-eddy diffusivity model.

PENETRATION MODEL

The penetration model was first postulated by 

Higbie (1935) and later refined by Danckwerts 

(1951). Both authors visualized the surface of the 

liquid as being a laminar film, the thickness of 

which is large with respect to the depth which 

can be penetrated by molecular diffusion during 

the life of the film. The film lifetime is the time

elapsed between the formation of the film on the 

surface and its replacement by fresh liquid from 

the bulk flow. The difference between the Higbie 

and the Danckwerts theories lies in the replacement process visualized by the authors.

Molecular diffusion in a laminar layer is 

governed by the differential equation

*x j. ^^ * ft -\ 2 ' ' (37) dt dy

For an infinitely deep layer the boundary conditions are:

C = d, t=Q,

C=CS, (38)

where CL is the concentration in the bulk-liquid 

phase, y is the distance from the interface, and t is 

the time elapsed since the formation of the interface. The solution to equations 37 and 38 is 

(Carslaw and Jaeger, 1959)

= C erfc (39)

and the rate of transport of solute through the 

surface is

dy (40)

Higbie (1935) visualized the surface elements as 

all being of the same age and as all being replaced 

simultaneously on reaching an age te . Alternatively, the ages of the differential elements of 

the laminar surface film can be visualized as being 

distributed according to the distribution function 

such that

(41)

0 t>te.

The average rate of mass transport, N, through a 

unit of surface area for either case is

<=ff'"«;.-,

tejo

N=-\ (C»-CL\ 

Because N = KL (Cs -Ci), one has

Dm Jt ()(/-> r> \ 1 * dt = 2(Cs-CL) I -

TTt \\TTte

(42)

(43)

Danckwerts (1951) assumed that the surface 

consists of an infinite number of differential elements of area which have their ages distributed 

according to

0(0=«?-*, (44)

where s is the average rate of surface renewal. 

The average rate of mass transport through a unit 

of surface area is then
THEORIES FOR PREDICTING THE REAERATION COEFFICIENT 11

N

=/"

Jo

(C.-Ci)

so that

(46)

Neither author attempted to relate either l/te 

or s to the mechanics of flow in open channels, but 

it is clear that each of these variables must represent a parameter related to the average residence 

time of the large flow eddies at the surface.

FILM-PENETRATION MODEL

The film-penetration model was developed by 

Dobbins (1956) and later by Toor and Marchello 

(1958). Certain modifications have been discussed 

in the literature (see, for example, Marchello and 

Toor, 1963, and Harriott, 1962), but essentially, the 

film-penetration theory consists of assuming 

molecular diffusion into laminar surface elements 

of differential area and thickness, L/, which exist 

on top of a turbulent liquid which has uniform 

solute concentration, CL- The ages of the surface 

elements are assumed to be distributed according 

to Danckwerts' (1951) distribution function for 

surface age as given by equation 44.

The differential equation governing the diffusion process is equation 37. The boundary conditions for the film-penetration model are

C=CL, /=0, 0<y^Lf, 

C=CS , fi=0, y=Q, 

C = CL, t^O, y = Lf. (47) 

Dobbins (1956) obtained as the solution to equations 37 and 47

erf C-G f-erf

and using the age-distribution function of Danckwerts (1951), he obtained

(49)

When the film-renewal rate, s, becomes very 

small, equation 49 approaches the film limit, or 

KL =Dm/Lf, while for VsL2//D,n >3.0, equation 49 

gives essentially the penetration limit, or

Marchello and Toor (1963) presented a modification of the film-penetration model in which the 

surface layer is assumed not to be replaced but to 

be instantaneously mixed at times which are 

distributed according to the age-distribution function of Danckwerts (1951). In the periods between 

mixings, diffusion is governed by equation 37. They 

obtained for the mass-transfer coefficient

>l/2 (50)

For small renewal rates, equation 50 approaches 

the film-model limit, and for large renewal rates it 

approaches one-half the penetration-model prediction for KL.

Another modification of the film-penetration 

model was presented by Harriott (1962). His theory 

was designed for application to mass transport at 

fluid-solid interfaces, but it could probably be 

applied also to open-channel flows, especially in 

the presence of surfactants. In this theory it is 

postulated that diffusion is molecular, according 

to equation 37, and that renewal occurs not necessarily all the way to the surface but to some distance yi below the surface. In applying this theory, 

one must solve equation 37 with the boundary 

conditions

C=CS , y=0, t^O, 

C=CL, j^°o, /i=0, 

C=f(y) O^y^ , f=0. (51)

The probability density functions used by Harriott 

(1962) to describe the distributions of the renewal 

period, /, and of the distance below the surface to 

which renewal occurs, yi, were of the form

(52)

where / is a dummy variable (signifying t or yi), a 

and (3 are distribution parameters, and F denotes 

the gamma function. In this model, KL cannot be 

obtained analytically, so it must be obtained using 

numerical techniques. Harriott found that at the 

penetration-model limit, the numerical procedures 

gave KL cc \/Dm s as expected; however, at what 

would ordinarily be the film-model limit, he found 

KL ccyin ~ l Dml ~ nlz 5n/2, where 0^«^1 and n is highly 

sensitive to the distribution parameters a and (3.

Several methods have been presented for determining the parameters in the Dobbins model in 

terms of the parameters of the turbulent-flow system. Not much has been done along these lines 

with the Marchello and Toor model or with the 

Harriott model; however, the concepts used in the 

case of the Dobbins model should be adaptable to 

these models as well.

The earliest method for relating the parameters 

of the Dobbins model to the parameters of openchannel flow was presented by O'Connor and 

Dobbins (1958). They pointed out that for rivers,

A 1/2 

I =1, so that

/

coth

\

L = VD,n s, and all that

needs to be determined in terms of the flow param-
12 REAERATION IN OPEN-CHANNEL FLOW

eters is s, the mean rate of surface renewal. For 

determining s, river turbulence is divided into two 

classes: nonisotropic, and essentially isotropic at 

the water surface. For both types of turbulence, 

s is defined as the ratio of the vertical turbulent 

intensity to the mixing length, at the water surface. For a Chezy C less than 17, O'Connor and 

Dobbins assumed nonisotropic turbulence to exist, 

and using the definition of the Prandtl mixing 

length and the von Karman logarithmic velocity 

law, they obtained

s- (53)

where H is the flow depth, S is the channel slope, 

g is the gravitational constant, and K is the slope 

parameter in von Karman's logarithmic velocity 

law. Substitution of equation 53 into equation 49 

leads to

. £L = nror &2 = (54)

where kL is in feet per day, // is in feet, and Dm is 

in square feet per day. For a Chezy C greater than 

17, isotropic turbulence is assumed to exist. For 

this case, O'Connor and Dobbins (1958) used measurements made by Kalinske (1943) on the Mississippi River between Dubuque, la., and Cairo, 111., 

which showed that the vertical turbulent intensity 

was approximately 0.1 U and that the mixing 

length was approximately 0.1 //; hence,

°-1C/ U (55)

O.I// //'

Thus

m

u I , or kz - , (56)fi

where the mean flow velocity U is in feet per second 

and all other units are the same as in equation 54. 

In equations 55 and 56, kL and kz are for use with 

the base 10.

Another theory relating the parameters of the 

Dobbins model to the mean flow parameters was 

presented by Dobbins (1946b). He postulated that 

the effective film thickness must be related to the 

smallest surface eddies into which the gas can 

penetrate by molecular diffusion before being 

carried into the flow below. Using Kolmogoroff s 

limit for the size of the energy-dissipating eddies, 

he obtained

w _ I *^ I _ I ** \ / ff n\

where cs and c4 are constants, v is the kinematic 

viscosity of the liquid, and Eg and E are, respectively, the energy dissipated per unit mass of 

liquid at the surface, and the energy dissipated 

per unit mass of liquid by the entire flow. Dobbins 

also postulated that the rate of expenditure of

energy required to overcome the surface tension 

and replace the surface liquid is proportional to 

the turbulent energy per unit mass of liquid near 

the surface. Thus

5=iI£M£l=liEI_£L_ (58)

(T C43 (T

where ct and c5 are constants, p the mass density 

of the liquid, and cr its surface tension. It can 

further be shown that

(59)

IT

In equations 57 and 58, c5 is dependent only on 

solute and solvent properties, whereas c4 is dependent on the dynamics of the flow system. Dobbins 

reported c5 to be 14.3 for the oxygen-water system 

at 20°C. He obtained

. 1/4" 2

c4 =0.65+15,000 H

(60)

from a plot of experimental results that contains 

considerable scatter. In the closure to his 1964b 

paper, Dobbins (1965) presented a prediction equation derived from the same data, but which is more 

convenient to use than equations 60, 59, and 57 in 

combination with equation 49. The prediction 

equation for kL is

kL =-

3/2 (61)

where

c4 =0.9+F

,4=9.68+0.054(7-20°) 

5 = 0.976+0.0137 (30°-7)3/2 

E=30.05C7

and where the slope, S, is in feet per thousand feet; 

T is the temperature, in degrees Celsius; and kL is 

in feet per day.

In a modification of Dobbins' (1964b) theory, 

Metzger and Dobbins (1967) used a surface compressional modulus, Ms, in place of the surface 

tension, <r. Following arguments similar to those 

used by Dobbins (1964b), they obtained

(63) Ms

and

(64)

(65)

where c2 and c4 are the same constants as in the 

Dobbins theory and ci, c3, and cs are new constants, 

and where c5 is independent of the flow dynamics. 

For distilled water, Metzger and Dobbins (1967)
THEORIES FOR PREDICTING THE REAERATION COEFFICIENT 13

found -=1. Ms gram

, and Metzger (1968) found this

same value in the presence of 0.5 mg/1 of alkyl benzene sulfonate (a surfactant) in water. The 

significance of having Ms rather than a- in the denominator of equation 65 is that Ms increases in the 

presence of surface-active agents while cr decreases. Thus equation 65 is in qualitative agreement with the observation that surface-active 

agents retard surface renewals.

The localized-mixing model of Marchello and 

Toor (1963) can be applied not only to the surface 

layer of an open-channel flow but also to lower 

layers within this type of flow and to all layers in 

other types of flow, such as pipe flows. When this 

is done, for any layer:

(66)

where ey is the eddy diffusivity for mass, and /8' is 

the turbulence parameter for the layer, such that

\2

(67)

Similarly, after applying the localized-mixing 

model to momentum transfer, Marchello and Toor 

obtained

(68)

where ev is the eddy diffusivity for momentum. 

Thus, if one can determine the momentum eddy 

diffusivity for a particular layer, one can solve 

equation 68 for /8' and determine the eddy diffusivity for mass for the layer from equation 66.

The combination of the film concept of Lewis and 

Whitman (1924) and the penetration concept of 

Higbie (1935) and Danckwerts (1951) into one model 

has considerable merit in that the combined model 

can accurately predict reaeration coefficients for 

natural streams over a much wider range of conditions than can either of the models alone. The 

film-penetration models, however, are idealized 

in that the diffusion process is visualized as taking 

place in two separate steps, rather than as being 

governed by the classical diffusion equation embodying both molecular and turbulent diffusion. 

The first step is molecular diffusion; the second 

step is turbulent diffusion, or nongradient diffusion, embodied by a renewal or mixing process. 

This idealization will, of course, lead to difficulties 

in relating model parameters to flow parameters,

and lead to deviations of the predicted from the 

true mass-transfer coefficients in situations where 

the flow conditions differ appreciably from those 

under which the experimental model coefficients 

were determined. For the Dobbins (1956) model, 

the O'Connor and Dobbins (1958) method for determining s is perhaps overly simplified, and it 

requires no experimental coefficients to be evaluated. That this method predicts kL values for 

natural streams as well as it does is therefore a 

little surprising. The Dobbins (1964b) and Metzger 

and Dobbins (1967) methods are more rational and 

more sophisticated, and they require the evaluation of two experimental coefficients. Equation 60 

used by Dobbins (1964b) to predict c4 was obtained 

from a plot of field data that contains considerable 

scatter. This scatter is, however, apparently of 

minor importance, because if one checks the prediction equation using the data from which equation 60 was evaluated, there is only a small amount 

of scatter about the line of perfect agreement 

between predicted and observed k2 values.

SURFACE-RENEWAL DAMPED-EDDY DIFFUSIVITY MODEL

The surface-renewal-damped-eddy diffusivity 

model was presented by King (1966), who postulated that the mass transport near the surface is 

due not only to Dm, but also to turbulent diffusion 

caused by small-scale eddies. The coefficient 

characterizing the turbulent diffusion caused by 

the small-scale eddies is the eddy diffusivity for 

mass, y, which is assumed to be of the form 

y=ayn , where a and n are parameters independent 

of time, and y is the distance from the interface. 

The surface renewals are assumed to be due to the 

large-scale eddies, and the surface ages are 

assumed to be distributed according to the distribution function for uniform surface ages (eq 41).

The diffusion equation applicable to the above 

model is

«\ acl

TO'

and the boundary conditions used by King (1966)

are

C = CL, t=Q, y>0, 

C \_^ ^-'Lj Ci * f>0 ^9 J v »°° 9

C=CS , r>0, y=0. (70) 

Equation 69 along with the boundary conditions 

70 can be solved analytically only in special cases, 

such as for a = Q and all «; for all a and « = 0, 1, °°; 

and for all a and all n with large t. For large t, the 

solution of equations 69 and 70 (King, 1966) gives

KL n . TT 7-77-= sin-,

i-i/n -. n 7

(71)
14 REAERATION IN OPEN-CHANNEL FLOW

where t may be interpreted as the surface age at 

renewal.

If the surface-renewal-damped-eddy diffusivity 

model is to be useful in predicting film coefficients, 

the parameters a and n in equation 69 as well as an 

average surface age, 7, must be estimated. From 

equation 71 it is seen for large surface ages that 

the exponent n can be estimated from the variation of KL with molecular diffusivity, if the other 

factors remain constant. Kozinski and King (1966) 

concluded that KL^Dm0- 75 for gas-liquid transfer 

at large surface ages; thus « = 4. From the work of 

Calderbank and Moo-Young (1961), King (1966) concluded that

a = 0.006 Eyp

2 >

(72)

where Ev is the energy-dissipation rate per unit of 

liquid volume, p is the mass density of the liquid, 

and fji is the absolute viscosity of the liquid. From 

figure 3 of King (1966) and from equation 71 for 

n = 4, it is seen that if surface-renewal effects control, the surface-renewal-damped-eddy diffusivity model yields film-coefficient predictions 

of the same order as those measured in large open 

channels and rivers.

The surface-renewal-damped-eddy diffusivity 

model is physically more realistic than previous 

models because the artifice of a surface layer is 

avoided. Also, the eddy diffusivity is specifically 

accounted for in the diffusion equation, rather 

than by being lumped into surface renewal, so that 

gradient and nongradient diffusion are more 

clearly separated. Practically, however, the model 

is cumbersome in that three parameters must be 

evaluated, and equation 69 must be solved numerically in most cases. King (1966) showed that if 

n = 4 and a is given by equation 72, s should be

'£F//12/3

. P J

proportional to for open-channel flows.

This has not been checked experimentally.

KINETIC THEORY

The kinetic theory as developed by Miyamoto 

(1932a, b) was presented by Krenkel and Orlob 

(1963). The model is based on the frequencies and 

velocities of bombardment of the interface by 

solute molecules from both the liquid and the gas 

phases. The resulting equation is

i to 7"'\i/2

, (73)

where RG is the ideal-gas constant, T' is the absolute temperature, M is the molecular weight of the 

gas, and /x0 is the velocity which a solute molecule

must attain normal to the interface to be able to 

leave the liquid.

If equation 73 is to account for the turbulence 

properties of the flow as well as for solute and 

solvent properties, the critical parameter would 

have to be /x0'- That ^ could account for the turbulence properties of tHe flow is reasonable, because 

M(/A0')2 /2 may be viewed as an activation energy 

term, and as pointed out by Bird, Stewart, and 

Lightfoot (1960, p. 27), the activation energy 

of liquids is influenced by local values of the shear 

stress, which can in turn be related to local turbulence properties. Apparently, no studies, either 

theoretical or experimental, have been made in 

which such relations have been presented.

Another kinetic-theory model was presented by 

Tsivoglou (1967). It too arises from consideration 

of the difference between the rate of entry and the 

rate of loss of gas (oxygen) molecules from a surface layer of liquid (water). The loss rate, r0, is

r0 =b(LfCnsAs) (74) 

where

C=concentration of gas in the liquid, 

Lf= thickness of surface layer from which gas 

molecules can escape to the atmosphere, 

ns = number of new surface layers exposed per

unit of time, 

As = surf ace area, and 

b=the proportion of available molecules in the

surface layer that actually escape. 

The entry rate, re, which is dependent on the gas 

concentration in the air above the water, is a constant. It is equal to the loss rate at C = CS , so that

re = b(LfCs nsAs). (75) 

The time rate of change of the liquid concentration 

is then

dC 

dt

so that

re r0

K2 =

l(Cs -C\ (76)

(77)

A 1 Because -rr=77» in which Kis the volume of the sysV n

tern and H is its depth,

KL=bLfns. (78)

Equation 78 is mostly of theoretical interest, 

because at the present time none of the parameters on the right-hand side can be measured. 

However, some of the results obtained by Tsivoglou 

(1967) by manipulation of the equations of this 

model are of interest. First, because the entry rate 

is a function only of the gas concentration in the 

layer above the fluid, Tsivoglou concluded that the
THEORIES FOR PREDICTING THE REAERATION COEFFICIENT 15

factor b varies with temperature according to the 

relation

fL.\ /l_\ (Cs)r, fr7Q\ (b)Tt = (b)T, (r , ', (79)

(<-s)T2

where the subscripts indicate two different temperatures. Combining equation 79 with equation 

77 yields

-^=-^Ii=e (T2- Ti\ (80)

in which 6 is a temperature-correction coefficient 

for the mass-transfer coefficient. The temperature-correction coefficient will be discussed 

in greater detail later. At present, the value 

usually accepted for 0 is 1.0241, given by Elmore 

and West (1961). The values calculated by Tsivoglou 

for B from equation 80 for oxygen and water for 

temperatures from 0° to 30°C range from 1.018 to 

1.026, which lends credence to Tsivoglou's model. 

Tsivoglou (1967) presented further support for 

his model. Assuming the average molecular transport within the body of a turbulent liquid to be due 

to an average concentration gradient between 

volume elements, and equating the mass transfer at the surface to the average transfer between 

the volume elements, Tsivoglou showed that for 

the simultaneous transfer of two gases,

When considering the ratio of the mass-transfer 

coefficients of two gases, equation 81 leads to the 

same conclusion as the film-model equation (36) 

namely that the ratio of the mass-transfer 

coefficients of the two gases is directly proportional to the first power of the ratio of their molecular diffusivities.

In further developing his model, Tsivoglou (1967) 

pointed out that if mass transport through the 

surface is governed by Einstein's law of diffusion, 

then

4 (82) (D«)2 </i'

in which di and d-i are the molecular diameters of 

the two gases. From equations 82 and 81,

di (83)

Tsivoglou, O'Connell, Walter, Godsil, and Logsdon 

(1965) and Tsivoglou (1967) completed an extensive 

series of measurements of the simultaneous transfer of several pairs of gases across a water interface. Two types of apparatus were used: one in 

which both gases were transferred simultaneously 

into the water, and one in which a second gas was 

transferred out of the water as oxygen was transferred into it. A range of temperature and turbulence conditions was studied for each pair of gases 

and each type of apparatus. On the basis of these 

measurements, Tsivoglou (1967) concluded that 

the ratio of the gas-transfer coefficients was a constant within the limits of experimental error for 

the range of temperature and turbulence conditions studied. This supports his model as given in 

equation 77.

Tsivoglou (1967) suggested that his experimental 

measurements of the gas-transfer ratios for the 

different pairs of gases together with moleculardiameter data from the literature support equation 83. However, a linear regression analysis of 

the logarithms of the transfer-coefficient ratios 

and of the molecular-diameter ratios yielded

:6

(84) OQ2

where the 95-percent confidence limits on the 

exponent are 1.09 and 1.44. Other experimental 

evidence suggests that equation 83 and the argument that Dm can be obtained from Einstein's law 

of diffusion are not strictly valid. Kozinski and 

King (1966) found that the exponent on Dm in the 

relation kL « Dmn varied from about 0.50 to 0.75 for 

the stirred-tank data of Dobbins (1964a); these 

values are considerably different from the 1.26 

indicated by equations 84 and 82.

These apparent discrepancies suggest that 

Einstein's law of diffusion is not strictly applicable to this situation, but note that these discrepancies do not in any way reflect on the validity of 

the, model as represented by equation 81. The only 

requirement of the model here is that the gastransfer-coefficient ratio be a constant, as 

was found in a laboratory study by Tsivoglou, 

O'Connell, Walter, Godsil, and Logsdon (1965) and 

Tsivoglou (1967).

LARGE-EDDY MODEL

The large-eddy model as presented by Fortescue 

and Pearson (1967) postulates that the masstransfer properties of a turbulent surface can be 

modeled by a series of steady square roll cells. The 

cells touch the surface of the flow and are convected downstream with a velocity equal to the 

mean flow velocity at the surface. The roll cells 

are taken to be A on a side, where A is the turbulent 

macroscale of the flow. The velocity pattern within 

a cell is given by

, . TTX Try

(85) . TTX . try ve =-Ae cos sir\-fL. A A

466-751 O - 72 - 4
16 REAERATION IN OPEN-CHANNEL FLOW

The parameter Ae is a measure of E, the average 

kinetic energy of the roll cell, because

1 f A f A :=^b

41* JO JO

(86)

The differential equation describing the gas concentration distribution in the large-eddy model is

C/O C/O j-k

W#>~T r Vt> ^/-/M

dx dy

with boundary conditions 

C=CS, y=Q,

 =0, x=n\, dx

C=CL, y=\, O^Jc^A, (88) 

where n is an integer. Equation 87 with boundary 

conditions 88 must be solved numerically.

Dimensional analysis of equation 87 along with 

equation 85 shows that the characteristic parameter describing the large-eddy model is P, 

the Peclet number, where P=A.AeIDm. Fortescue 

and Pearson (1967) reported that for P values "over 

the range that was revelant," the numerical solution of equations 85, 87, and 88 yields

kL =

Dm u'\ 1/2 

A ) '

(89)

where u' is the longitudinal turbulent intensity. 

Using the O'Connor and Dobbins (1958) relation for 

u'l\ = UIH, Fortescue and Pearson reported that 

their model gives "rather better" predictions than 

the Higbie (1935) penetration model (eq 43) with 

llte=u'l\=U/H, for the river data used by O'Connor 

and Dobbins (1958).

The large-eddy model is different in concept 

from the other models presented in this section, 

but the concept is as realistic as, for example, the 

film-penetration concept. An individual cell is a 

considerably idealized turbulent eddy, and the surface film is a film of eddies, rather than a stagnant 

film. The model avoids the question of surface age 

by allowing the age to vary in a set pattern across 

the top of the cell. It would be interesting to find 

the effect on the computed film coefficient of 

assumed distribution functions for varying cell 

sizes and kinetic energy. A more realistic relation 

for surface values of w'/A might improve the predictions made by the model.

TURBULENT-DIFFUSION MODEL

In discussing the turbulent-diffusion model, 

Kishinevsky (1955) postulated for high turbulence 

levels that the turbulent diffusivity, ey, is not 

damped out at the water surface. He further stated 

that the turbulent diffusivity at the surface is

much larger than the molecular diffusivity, so that

2

(90)

An example in which equation 90 can be said to 

govern was presented by Kishinevsky and Serebryansky (1956). They reported studies in a rapidly 

stirred tank in which the measured film coefficients for hydrogen, oxygen, and nitrogen were 

all roughly the same, despite the fact that the 

molecular diffusivity of hydrogen is about twice 

that of oxygen and nitrogen. They also reported, 

however, that at lower stirring speeds the molecular diffusivity influenced the measured film coefficient. The fact that in open-channel flows the 

measured values of kL are a function of temperature indicates that in this type of flow turbulence 

levels do not reach a magnitude sufficient to make 

equation 90 applicable.

Note that the models discussed predict a range 

of dependence of kL on Dm varying from an exponent of 1 for the film model to 0 for the turbulentdiffusion model. The film-penetration model 

predicts an exponent on Dm which varies from 0.5 

to 1.0, whereas the surface-renewal-damped-eddy 

diffusivity model can predict an exponent anywhere from 0 to 1.0. Experimental data taken in 

stirred tanks (Kozinski and King, 1966) indicated 

that the exponent is in the 0.5 to 0.75 range. Therefore, each model (with the possible exception of the 

film model) seemingly is applicable to a limited 

range of turbulence conditions. However, the 

surface-renewal-damped-eddy diffusivity model 

is the only one which is applicable over the entire 

range of turbulence conditions.

All the conceptual models for the gas-liquid 

mass-transfer process involve two physical properties of the oxygen-water system. These are the 

molecular diffusivity and the saturation concentration for oxygen in water. Each of these properties has been studied in considerable detail, and 

the following paragraphs summarize briefly the 

results of these studies.

The determination of the molecular diffusivities 

of dissolved gases in liquids was reviewed by 

Himmelblau (1964), and the experimental determinations of the molecular diffusivity of oxygen 

in water were discussed by Duda and Vrentas 

(1968).

The experimental data for molecular diffusivity 

at 25°C fall into two groups, one group indicating 

a value of about 2.5xlO~5 cm2 per sec (square centimeters per second) and the second group indicating a value of about 2.0xlO~5 cm2 per sec. The 

larger values in general have been used (Isaacs
THEORIES FOR PREDICTING THE REAERATION COEFFICIENT 17

and Gaudy, 1968; Metzger and Dobbins, 1967; 

Kozinski and King, 1966; Dobbins, 1964b; O'Connor 

and Dobbins, 1958). Himmelblau (1964) also believed 

that the smaller values at 25°C were probably incorrect, and Baird and Davidson (1962) believed 

that the larger values obtained with the wetted 

sphere procedure were more accurate than the 

values obtained with the annular jet. On the other 

hand, Kolthoff and Izutsu (1964) concluded that a 

modified equation incorporating a second perturbation term should be used in the calculation of 

oxygen diffusivities from polarographic measuremerits. This modification resulted in diffusivities 

of 1.94xlO-5 and 1.97xlO'5 cm2 per sec at 25°C as 

compared with values of 2.56xlO~5 and 2.50 xlO~5 

cm2 per sec obtained without the second perturbation term. Duda and Vrentas (1968) concluded from 

their consideration of the available experimental 

data that the evidence tended to support the 

smaller diffusivity values, in agreement with their 

laminar-jet result. Controversy still exists concerning the correct value for the molecular diffusivity 

of oxygen in water; so diffusivity data for the 

oxygen-water system should be used with this consideration in mind.

The saturation concentrations presented by the 

American Public Health Association (1960, 1965) 

and the Committee on Sanitary Engineering Research (1960) in general have been accepted and 

used as standards. Truesdale, Downing, and 

Lowden (1955) obtained values that were significantly different from the American Public Health 

Association values, but later work by Montgomery, 

Thorn, and Cockburn (1964) showed that the values 

were low because of losses of iodine vapor during 

analysis for DO. More recent studies of the saturation concentration were discussed by Montgomery 

(1969). He concluded that the values from these 

studies were in general slightly smaller than the 

values reported by the American Public Health 

Association (1960, 1965) and, for temperatures 

above 15°C, were slightly larger than values obtained by the Committee on Sanitary Engineering 

Research (1960).

The saturation concentration decreases with 

increasing temperature and decreases with decreasing barometric pressure. The temperature 

dependence of the saturation concentrations obtained by the Committee on Sanitary Engineering 

Research (1960) was expressed in terms of a cubic 

equation. The dependence on barometric pressure, 

pB, may be expressed in terms of a simple ratio, or

i .

PB,

(91)

For elevations above 3,000 feet and temperatures 

above 25°C, corrections for the water vapor pressure should be made (Am. Public Health Assoc., 

1965, p. 409).

Equation 91 demonstrates a fact perhaps not 

generally recognized. For a temperature of 20°C 

and a pressure of 760 mm Hg (millimeters of 

mercury), the saturation concentration is 9.02 

mg/1 (Committee on Sanitary Engineering Research, 1960). If a concentration of 5.0 mg/1 is required for prevention of fish kills, then a range of 

about 4 mg/1 of DO is available for a stream at an 

elevation of about sea level for example, along 

the east coast of the United States. However, for a 

stream at an elevation of about 5,000 feet in the 

Rocky Mountain region, the saturation concentration is about 7.5 mg/1; hence, a range of only 

about 2.5 mg/1 of DO is available for use in oxidation of wastes.

The saturation concentration decreases with increasing chloride concentration in the water. The 

effect ranges from a decrease of 0.017 mg/1 of DO 

per 100 mg/1 of chloride at 0°C to a decrease of 

0.008 mg/1 of DO per 100 mg/1 of chloride at 30°C 

(Am. Public Health Assoc., 1965, p. 409). Sewage in 

the water may affect the DO saturation value. 

Moore (1938) found that the solubility of oxygen in 

domestic sewage was about 95 percent of the 

solubility in distilled water, and Rand (1959) concluded that the solubility of oxygen in sewage was 

not significantly different from the solubility in 

pure water.

SEMIEMPIRICAL MODELS

The semiempirical models result from a consideration and combination of the factors that should 

affect mass transfer at a free surface. These 

models rely on regression-correlation analysis of 

experimental data for establishment of a prediction equation and for verification of the assumed 

relations between the mass-transfer coefficient 

and the flow parameters. In general, the steps in 

the derivation of these models are: (1) From intuition, a knowledge of previous conceptual models, a 

knowledge of the results of previous empirical 

studies, or a combination of these, to postulate that 

the film coefficient kL (or the reaeration coefficient, 

A:2) is a function of certain of the parameters of the 

flow; (2) if necessary, to manipulate the assumed 

relation to obtain a relation for kL (or /c2) in terms 

of easily measured mean-flow parameters; and 

(3) to use regression-correlation analyses to fit 

experimental data to the assumed relation and to
18 REAERATION IN OPEN-CHANNEL FLOW

check its validity. The determination of the usefulness of these models must be based on the reliability of the data from which they were derived, 

on how well they predict mass-transfer coefficients 

for natural streams, and on ease of use. The first 

two points are discussed in a later section of this 

paper; the last, as the models are presented. The 

models discussed in this section are those presented by Krenkel (1960) and by Thackston (1966). 

The limits on the ranges of data from which the 

semiempirical and empirical models were derived 

are discussed in a later section.

KRENKEL MODEL

The Krenkel model was presented by Krenkel 

(1960) and Krenkel and Orlob (1963), who reasoned 

that

^[^Ib^]' (92)

where the first term in brackets is characteristic 

of the rate of absorption at the surface, and the 

second term is characteristic of the rate of surface 

renewal. In equation 92, £>, is a combination of 

molecular and turbulent diffusivity at the surface, 

and e is an eddy-diffusion coefficient. However, 

both Krenkel (1960) and Krenkel and Orlob (1963) 

believed that the reaeration coefficient is better 

represented by

kz <* s exp [- EJRc T' ], (93) 

where s=e///2 , Ea is the activation energy for 

oxygen absorption, RG is the ideal-gas constant, 

and T' is the absolute temperature. In using equations 92 and 93, the parameter most characteristic 

of the eddy-diffusion coefficient, e, is Dx, the longitudinal dispersion coefficient of the flow.

Regression analysis by Krenkel and Orlob (1963) 

of 58 laboratory measurements of k-2 made at 

depths from 0.08 to 0.20 foot in a 1-foot-wide flume 

yielded

/:2 = 1.418exp [ EalRcT'^-j^ 0.0069, (94) 

where E« = 5,875 ^, *c =1.99 m<^oK, T' is the

absolute temperature in degrees Kelvin, and the 

units of k2 are reciprocal minutes. Because the procedure for measuring Dx is difficult and time consuming, this parameter is not one of the commonly 

available stream-flow parameters. However, for 

uniform open-channel flows in the depth range 

given above, Krenkel (1960) showed by a regression 

analysis that Dx can be related to the commonly 

measured mean flow parameters by

Dx = 6.425 E0- 296 //1 - 243, (95) 

where Dx is in square feet per minute, E-USg,

where U is in feet per minute, g is in feet per 

minute per minute, and E is the energy dissipated 

per unit mass of flowing fluid. Alternatively, from 

a separate regression analysis of the same data, 

Krenkel (1960) and Krenkel and Orlob (1963) reported that kz at 20°C can be determined from

(£2)20°= 1-141 xlO- 4 E°-408 //-°-66 (96) 

where, again, kz is in reciprocal minutes. When 

equation 96 is to be used for flow at some temperature other than 20°C, adjustment of kz must be 

made using a suitable temperature correction 

formula. (See eq 104.)

Experimental differentiation between equations 

92 and 93 may be very difficult if Dt = Dm because 

for a small temperature range (for example, ±10°C)

 -ocexpC E'/7'), (97)

where E' is a combination of the activation 

energies for molecular diffusion and viscous flow 

and is independent of temperature. Thus, a plot 

of log k2 versus 1/7' for constant e///2 will show a 

linear variation, whether molecular diffusivity or 

the activation energy for oxygen absorption is the 

controlling factor. Most likely, Dm (or eq 92) controls, because equation 93 in effect states that 

interfacial resistance controls; but Davies and 

Rideal (1963) and Sherwood and Pigford (1952) 

stated that interfacial resistance is negligible for 

the absorption of slightly soluble gases.

An important factor to be considered when using 

equations such as 94 and 95 is that they were 

developed for flows in straight channels with uniform cross section, whereas natural streams 

generally have nonuniform cross sections and considerable curvature in plan. In such channels, Dx 

can be considerably greater than the £>x for an 

equivalent straight channel with uniform cross 

section, so equations 94 and 95 might not apply. 

Equations of the type of 96 might, however, be 

more valid in natural streams because the rate of 

energy dissipation includes the effects of nonuniform cross sections and curvature. Finally, as 

mentioned above, Dx is difficult to measure or predict, whereas equations of the type of 96 use readily 

measurable mean-flow parameters. Thus, from 

this point of view, use of equation 96 would be recommended over the use of equation 94.

THACKSTON MODEL

The Thackston model was presented by Thackston (1966) and Thackston and Krenkel (1969a), 

who postulated that

'- --^l(5) (98)
THEORIES FOR PREDICTING THE REAERATION COEFFICIENT 19

where s is the average surface renewal rate and H 

is the flow depth. They further postulated that 

fi(s)cc y, where e^ is the vertical eddy diffusivity 

at the water surface, and that / >(//) a://2 . They 

assumed that

u

, CC y =-//«,, (99)

where ey is the average value of the vertical eddy 

diffusivity, «* is the shear velocity, \/gHS, and K is 

von Karman's constant. The resulting prediction 

equation is of the form

-/"* ./ , (100)

where J is a constant to be determined from regression analysis.

Based on 40 values of A2 measured by Thackston 

(1966) in a 2-foot-wide flume at flow depths ranging 

from 0.05 to 0.23 foot, a regression analysis yielded

= 0.000215 H(101)

where A2 is in reciprocal seconds, «* is in feet per 

second, and H is in feet. More importantly, equation 101 fits rather well to the field data used by 

O'Connor and Dobbins (1958) and the data collected 

by Churchill, Elmore, and Buckingham (1962). 

Furthermore, the fit is improved if a Froudenumber relation is included in the regression analysis. A regression analysis which included these 

field data plus the flume data of Thackston (1966) 

yielded

T, H

(102)

where F is the Froude number.

Some question exists as to the universal applicability of an equation of the type of 102, because 

when the data of Krenkel (1960) were analyzed in 

the same fashion as those yielding equation 101, 

the prediction equation

(103)

was obtained. Thackston and Krenkel (1969a) explained the difference between equations 101 and 

103 by stating that because Krenkel's 1-foot-flume 

data do not fit field data, whereas Thackston's 2- 

foot-flume data do, the conditions in the 1-footflume were not really characteristic of field 

conditions. More likely, however, this discrepancy 

indicates that prediction equations of the type of 

101 and 102 are not general enough and that some 

additional parameter such as the width-depth 

ratio, WIH, should be included in the prediction 

equation.

EMPIRICAL PREDICTION EQUATIONS

The empirical prediction equations result from 

least-squares multiple-regression analyses of the 

variables considered pertinent by the investigator 

or from dimensional analysis of the mean-flow and 

gas-liquid parameters. For water and air in the 

temperature range encountered in the field, it is 

generally assumed that the temperature dependency of k-2 resulting from the temperature dependence of the molecular diffusivity, kinematic 

viscosity, surface tension, and other physical 

properties can be embodied in a single function of 

the temperature. This dependency is generally 

removed before the least-squares analysis is 

begun. The empirical temperature-correction 

equation (the form of which can be derived from 

rate process considerations) is of the form

K2T=K2w e (T~ 2(f\ (104) 

where T is in degrees Celsius and the 0 generally 

used is the 1.0241 suggested by Elmore and West 

(1961).

The empirical prediction equations are generally 

of the form

TJW

20°^7r, (105)

where i// is a constant obtained from a regression 

analysis of experimental data, and w and j may be 

chosen in this fashion; or w and./ may be chosen for 

dimensional homogeneity of the prediction equation. Judgment of a particular prediction equation must be based on the quality of the data from 

which it is derived, and on how well it predicts 

field reaeration coefficients. The equations discussed in this section are based generally on fieldscale data and utilize readily measurable 

mean-flow parameters for prediction of reaeration 

coefficients. Presented in the following sections 

are equations by Streeter and Phelps (1925), 

Churchill, Elmore, and Buckingham (1962), Owens, 

Edwards, and Gibbs (1964), Langbein and Durum 

(1967), Isaacs and Gaudy (1968), and Negulescu 

and Rojanski (1969).

STREETER-PHELPS EQUATIONS

The pioneering study of reaeration coefficients 

of large-scale natural streams was presented by 

Streeter and Phelps (1925). In this study, the classical oxygen sag equation

(m is t is T\ ft (\a\ j-=KiL-K2 D UOo)

was developed and was used in its integrated form 

to calculate reaeration coefficients from measured
20 EEAERATION IN OPEN-CHANNEL FLOW

values of the time of travel, upstream and downstream values of the DO deficit and BOD, and the 

BOD reaction rate constant, Ki.

The prediction equations were developed from 

data taken in 1914 and 1915 on the Ohio River 

between Pittsburgh, Pa., and Louisyille, Ky. The 

study reach was 611 river miles longj; it was separated into 11 sub-reaches for experimental measurements. During the study period at the various 

points in the study reach, the mean monthly values 

of discharge, depth, and velocity varied, respectively, from 1,670 to 309,000 cfs (cubic feet per 

second), from 16 to 48 feet, and from 0.1 to 4.6 fps 

(feet per second). Eighty-six average values of kz 

were reported as corresponding to the monthly 

averages of the flow parameters existing over the 

period of measurement on the various sub-reaches.

The prediction equation suggested by Streeter 

and Phelps (1925) was

(*2V=pp (107)

where //' is the depth above minimum low water 

stage and kz is in reciprocal days. The coefficient Z 

is a function of what Streeter and Phelps called 

an irregularity factor, which is defined as the 

number of changes in channel bottom slope producing a change in elevation greater than 1 foot, 

expressed on a per-mile basis. For irregularity 

factors of 2.0 to 3.0, Z=[0.39(101 - 16)5 + 17], and for 

irregularity factors of 3.0 to 5.0, Z = 1.1S2 - 3, where 5 

is the channel water-surface slope, in feet per mile. 

The coefficient 17 is a function of 8, or T7=(8-0.17)/ 

(8 1.17), where 8 is the mean relative increase in 

flow velocity per 5-foot increase in gage height.

Because of the need for computing the irregularity factor and the coefficient 8, equation 107 is 

difficult to use. Also, because //' is not a true 

streamflow parameter, a prediction equation in 

terms of this parameter probably is not completely 

general. Other authors (for example, O'Connor and 

Dobbins, 1958) have, however, reanalyzed some of 

the Streeter-Phelps data to yield prediction equations with a little more generality.

CHURCHILL-ELMORE-BUCKINGHAM EQUATIONS

Probably the best field study of the reaeration 

coefficient has been that by Churchill, Elmore, 

and Buckingham (1962). Their measurements were 

made in stretches below dams, where the water 

released was deficient in DO because of prolonged 

storage under conditions of thermal stratification, 

and where generally the BOD was negligible.

The study covered 16 different reaches on five 

rivers. Five hundred and nine individual kz determinations were made for 30 different mean-flow 

situations. The discharge, mean depth, and mean 

velocity ranged, respectively, from 952 to 17,270 

cfs, from 2.12 to 11.41 feet, and from 1.85 to 5.00 fps.

Churchill, Elmore, and Buckingham (1962) performed many multiple-regression analyses on the 

pertinent stream flow, gas, and liquid parameters 

corresponding to the kz measurements. None of the 

prediction equations was found to be statistically 

any more significant than any of the others, so 

that the simplest one was suggested by these 

authors for general usage. This equation, which 

has a coefficient of multiple correlation of 0.82, is 

#2 = 5.026 £/°-969 //-1 -673 (1.0241)r- 20°, (108) 

where kz is in reciprocal days, V is in feet per 

second, H is in feet, and T is in degrees Celsius.

Equation 108 is much more convenient to use 

than equation 107 because it utilizes only the 

readily available parameters U, H, and 7.

OWENS-EDWARDS GIBBS EQUATION

Owens, Edwards, and Gibbs (1964) reported 

reaeration-coefficient measurements for several 

English streams. These coefficients were determined using the disturbed-equilibrium method of 

deaeration with sodium sulfite and cobalt catalyst. 

Use of this procedure eliminates the necessity of 

considering parameters of the DO balance that 

are independent of the DO concentration.

The study by Owens, Edwards, and Gibbs (1964) 

included measurements on 21 different reaches of 

six streams. They listed 32 separate observed 

values of kz for stream discharges, depths, and 

mean velocities ranging, respectively, from 1.50 

to 36.2 cfs, from 0.34 to 2.44 feet, and from 0.13 to 

1.83 fps. They also listed six values of A2 from 

Gameson, Truesdale, and Downing (1955) for two 

reaches of a different stream for discharges, 

depths, and mean velocities within the above 

ranges.

Multiple-regression analysis of the data by 

Owens, Edwards, and Gibbs (1964) yielded

A:2 = 10.90 £/°-73 //- 1 - 75 (1.0241)r- 20°. (109) 

When Owens, Edwards, and Gibbs (1964) included 

the data of Gameson, Truesdale, and Downing 

(1955) and that of Churchill, Elmore, and Buckingham (1962) in the analysis, the resultant equation 

was

£2 = 9.41 £/« «* #-»-M (1.0241)r- 20P. (110) 

In these equations, kz is in reciprocal days, U is in 

feet per second, H is in feet, and T is in degrees 

Celsius. Equation 110 was obtained from 68 kz 

values, which had a 300-fold range from about 0.25 

to 75 days" 1 . Again, equations 109 and 110 are very
EXPERIMENTAL DETERMINATION OF THE REAERATION COEFFICIENT 21

convenient to use because they utilize only the 

readily available parameters U, H, and T.

LANGBEIN-DURUM EQUATION

Langbein and Durum (1967) combined the river 

data of O'Connor and Dobbins (1958) and of Churchill, Elmore, and Buckingham (1962) with the 

laboratory data of Krenkel and Orlob (1963) and of 

Streeter, Wright, and Kerr (1936) and obtained the 

equation

U = 3.3- (111)

where fe is in reciprocal days, U is in feet per 

second, and H is in feet. They also considered in a 

general way the effect of downstream and regional 

variations of stream discharge on the reaeration 

coefficient computed from equation 111. They 

found that the computed reaeration coefficients 

decreased in the downstream direction with about 

the square root of the discharge.

ISAACS GAUDY EQUATIONS

Isaacs and Gaudy (1968) made a study of reaeration in a circular tank with moving walls intended 

to simulate streamflow conditions. They reported 

52 individual fe observations for 20 different 

simulated streamflow conditions. The simulated 

depth ranged from 0.50 to 1.50 feet, and the simulated mean velocity, from 0.55 to 1.63 fps.

A multiple-regression analysis of the data by 

Isaacs and Gaudy (1968) yielded exponents of 

1.0027 and -1.4859 on mean velocity and depth, 

respectively; so a second analysis was performed 

in which these exponents were held to 1.0 and 

-1.5. The results of this analysis were

2«°. (112)

Using 29 of the 30 data points of Churchill, Elmore, 

and Buckingham (1962), Isaacs and Gaudy obtained

20P , (113)

but using Krenkel's (1960) flume data they 

obtained

(114)

In equations 112, 113, and 114, fe is in reciprocal 

days, U is in feet per second, and H is in feet.

Isaacs and Gaudy (1968) pointed out that the 

multiplicative constants in equations 112, 113, and

114 can be interpreted as / .>' \

(115)

Equations of the type of 112 through 114 are convenient to use, and if the constant is expressed as 

in equation 115, they are dimensionally homogeneous. There is some question as to whether the 

data from which equation 112 is derived truly 

represent streamflow conditions (Thackston and 

Krenkel, 1969b; Bella, 1969), so that use of this 

equation for prediction of stream reaeration coefficients is questionable. However, equation 113 

should be applicable to streamflows, at least for the 

range of flow parameters over which the measurements were taken, because it is based on field data. 

The disparity between the constants in equations 

113 and 114 probably indicates the effects of various 

nonhomogeneities in the data collection and 

modeling procedures and also suggests that all the 

pertinent variables have not been included in the 

prediction equations.

NEGULESCU-ROJANSKI EQUATIONS

The equations given by Negulescu and Rojanski 

(1969) were derived by regression analysis from a 

series of 18 fe and Dx measurements in a 66-footlong by 0.66-foot-wide recirculating flume. Not all 

the data are given, but for those available, the 

depths range from 0.164 to 0.492 foot, and the 

velocities range from 0.656 to 1.903 fps. The equations obtained by Negulescu and Rojanski are

and

fe = 1.423 D.

/fAl.63

\H) '

(116)

(117)

8 

where the constant X is dimensionless.

where fe is in reciprocal days, U/H is in reciprocal 

seconds, and Dx is in square feet per day. Comparing values of fe predicted by equations 116 and 117 

and values from a set of river measurements 

(which are not reported), Negulescu and Rojanski 

(1969) concluded that equation 117 is the more 

accurate.

Again, from the standpoint of convenience in 

obtaining measured parameters for prediction of 

fe in field streams, equation 116 is the desirable one 

to use.

EXPERIMENTAL DETERMINATION OF THE 

REAERATION COEFFICIENT

All the experimental techniques for determining 

the reaeration coefficient in open-channel flow, 

except the tracer technique of Tsivoglou (1967) and 

Tsivoglou and others (1965, 1968), use some integrated form of a DO balance equation for 

example, an integrated form of equation 12. The 

determination of K> requires, therefore, that all
22 REAERATION IN OPEN-CHANNEL FLOW

other parameters in the equation be measured. 

These parameters are determined by using a 

variety of techniques, all of which depend on the 

measurement of the amount of oxygen dissolved 

in a volume of water.

The purpose of this section is to discuss (1) the 

basic DO measurement techniques, (2) the various 

techniques for determining K2 , and (3) an analysis 

of the errors expected from the different techniques for determining the reaeration coefficient.

MEASUREMENT OF DISSOLVED OXYGEN IN WATER

The basic method for the determination of DO in 

water is the Winkler procedure, which was first 

described in the literature in 1888. (See Am. Public 

Health Assoc., 1965, p. 414, for reference.) Since 

that time, the procedure has been improved by 

advances in equipment, technique, and instrumentation, and it is still the basis for most analytical 

methods for the determination of DO in water.

Oxidizing or reducing substances in the sample 

interfere with the procedure, however, and various 

modifications of the basic Winkler method have 

been developed. A common interference in waters 

receiving sewage and effluents from sewage treatment plants is nitrate nitrogen, and the Alsterberg 

modification is used under these conditions. 

Details of the procedure were given by the American Public Health Association (1965) and by 

Rainwater and Thatcher (1960). The Alsterberg 

modification is the procedure generally used for 

routine analysis of natural waters. Details of other 

modifications of the basic Winkler procedure were 

also given by the American Public Health Association (1965), and several of these modifications were 

compared by Montgomery, Thorn, and Cockburn 

(1964).

Various instrumental methods for the determination of DO in water have also been developed. 

These include, for example, gas-exchange, gaschromatographic, radiometric, voltammetric, and 

membrane-electrode methods. The procedures 

used in these methods and the advantages and disadvantages of these methods and of titrimetric 

methods were discussed by Mancy and Jaffe (1966).

TECHNIQUES FOR DETERMINING THE REAERATION 

COEFFICIENT

Three general techniques are used for determining the reaeration coefficient of open-channel 

flow. These are the dissolved-oxygen balance 

technique, the disturbed-equilibrium technique, 

and the tracer technique. The first two techniques 

utilize an integrated form of the equation of nondispersive conservation of mass for a gas dissolved 

in a liquid (eq 28 with ex negligibly small). The third 

technique utilizes an integrated form of this equation for an instantaneous input of tracer material 

which is convected and dispersed longitudinally 

by the flow. The basis and procedures of these 

techniques are discussed in the following 

paragraphs.

DISSOLVED-OXYGEN BALANCE TECHNIQUE

Studies utilizing the DO balance technique are 

of two types: those in recirculating flumes, and 

those in natural streams or in nonrecirculating 

flumes. Studies of the first type utilize the nonsteady-state nondispersive form of the conservation of mass equation, equation 28 with e.r=0. Those 

of the second type utilize the steady-state nondispersive form of this equation that is, equation 28 

with the x and the dC/dt terms equal to zero. We 

know of no techniques which have been applied to 

the calculation of K2 in nonsteady dispersive conditions; however, the numerical techniques of 

Thomann (1963, 1965) and Dresnack and Dobbins 

(1968) could be modified to compute K2 from measured variations of DO with time.

The general DO balance technique for natural 

streams is based on a more sophisticated version 

of the equations originally presented by Streeter 

and Phelps (1925) for example, equations 12 and 

13, as presented by Camp (1963). These onedimensional steady-state equations are applicable 

to conditions which meet the assumptions previously given for equations 9 and 10. In addition, 

longitudinal dispersion is generally assumed to 

have a negligible effect on the longitudinal DO 

profiles in natural streams.

With these assumptions, the solutions to equations 12 and 13 yield the BOD and the DO deficit at 

a downstream station, Jt=jcb, in terms of the BOD 

and DO deficit at an upstream station, x=xa. The 

solutions are

(118)

and

io-**+o.io-'.«. (119)

where the flow time is f=*b «? the rate constants

are to the base 10, and equation 12 has been solved 

in deficit form.
EXPERIMENTAL DETERMINATION OF THE REAERATION COEFFICIENT 23

Equation 119, in its most general form, must be 

solved for k2 by trial and error. Before this can be 

done, however, the other parameters appearing in 

equations 118 and 119 must be evaluated. The 

techniques used in evaluating these parameters 

are discussed briefly in the following paragraphs.

BOD TEST AND K^

Of the various parameters in a DO balance, the 

rate constant, Ki, for the oxidation of organic 

wastes probably has been studied to the greatest 

extent. The oxidation process is generally considered to be a first-order process as represented 

by equation 1. The two parameters in equation 1, 

KI and L, may be determined from a BOD test. The 

earliest recorded use of a BOD test was in 1870 

(Theriault, 1926). Since that time, numerous 

studies have been made of the BOD test. The literature prior to 1927 on the BOD test was summarized by Theriault (1927), and a bibliography of 

the literature on the BOD test since 1927 was compiled by Hull (1959).

The BOD test consists basically of obtaining two 

samples of the water containing organic wastes, 

determining the DO concentration in one of the 

samples at the beginning of the test, incubating 

the second sample in darkness at 20°C for a period 

of time, usually 5 days, and determining the DO 

concentration in the second sample at the end of 

the incubation period. The difference in DO concentrations is the 5-day BOD of the sample.

In practice, however, the procedure is not so 

simple. To prevent complete depletion of the DO 

in the test sample, the sample must be properly 

diluted before the start of the test. The dilution 

water must be properly prepared and seeded. The 

type and amount of seed to be used in the dilution 

water for the particular waste of interest must be 

determined by experience. If the sample is alkaline 

or acidic or contains residual chlorine or other 

toxic substances, then it also must be treated 

before the start of the test. The complete details 

of the standard 5-day, 20°C BOD test were given 

by the American Public Health Association (1965).

The reaeration technique of Elmore (1955) for 

BOD determinations has the advantage that dilution water and seeding are not required. This procedure, however, has the disadvantages that the 

5-day BOD must be less than 25 mg/1, large volumes 

of sample are required, and considerable attention 

on the part of the technician is necessary during 

the initial stages of the test.

A respirometer assembly for BOD determinations was described by Gannon, Pelton, and

Westfield (1965). The sample is placed in a sealed 

cell equipped with a DO measurement probe, and 

the DO concentration is measured continuously. 

When the DO concentration decreases to 2 mg/1, 

the cell is aerated with compressed air. This process 

is continued for as long as desired. The respirometer has the same advantages as the reaeration 

technique in that no dilution and seeding are 

necessary, and it has the additional advantage that 

the original sample is kept intact. The literature 

on other types of improved techniques for BOD 

determinations was reviewed by Gannon, Pelton, 

and Westfield (1965), and a complete review of 

respirometric methods was presented by Montgomery (1967).

The results of a BOD test are usually in the form 

of either BOD or amount of waste oxidized at time 

t as a function of t. The applicable equation is

^ (120)

where Xt is either the BOD or the amount of waste 

oxidized up to time t, and L is the total first stage, 

or carbonaceous, BOD of the sample. Because both 

L and k\ are unknown, several different techniques 

have been developed for determining L and k\ from 

the results of BOD tests. Among these techniques 

are the least-squares procedure of Reed and 

Theriault (1931), the log-difference method of 

Fair (1936), the slope method of Thomas (1937), the 

moments method of Moore, Thomas, and Snow 

(1950), the graphical procedure of Lee (1951), and 

the ratio methods of Williams (1951) and Sheehy 

(1960). The Reed-Theriault procedure is the most 

accurate, but it also requires the most time for 

the calculations because it is a trial-and-error procedure. Of the other procedures, the moments 

method gives the most accurate results for the time 

consumed (Ludzack and others, 1953; Schroepfer 

and others, 1960; Gannon, 1966; Zanoni, 1967).

The extensive series of BOD measurements by 

Streeter and Phelps (1925) suggested that ki has a 

value of 0.1 day- 1 at 20°C for all types of sewage 

and organic wastes. Later work, however, has 

shown that ki at 20°C may range from about 0.01 

day- 1 for some industrial wastes and well-oxidized 

treatment-plant effluents to about 0.30 day- 1 for 

readily oxidizable wastes (Camp, 1963, p. 247). 

Schroepfer, Robins, and Susag (1960) found that 

the ki values for raw wastewaters and primary 

settled effluents from treatment plants in the 

Minneapolis-St. Paul area varied from 0.14 day- 1 

on Sundays and holidays to 0.24 day- 1 on weekdays. 

They concluded that k\ depended greatly on the 

degree of stabilization of the effluent. Kothandaraman and Ewing (1969) found that k\ values for

466-751 O - 72 - 5
24 REAERATION IN OPEN-CHANNEL FLOW

the Ohio River downstream of Cincinnati, Ohio, 

were normally distributed and had a mean value 

of 0.075 day- 1 and a standard deviation of 0.028 

day- 1 . Camp (1965) also noted considerable daily 

variation of ki values for several stations on the 

Merrimack River in Massachusetts. Thus, ki 

apparently cannot be considered a constant, but 

ki values must be determined for the waste, the 

reach, and the time of interest.

The ki values determined at 20°C in the laboratory are converted to the appropriate river temperature by means of the equation

(if \ tif\ Q(T-20P) n 9~n \H-lJrp \*Vl/0|\o" , V-*-^-1 -*-/

where 6 is the temperature coefficient, which has 

a value of 1.047 for temperatures from 10° to 37°C 

(Camp, 1963, p. 247). Zanoni (1967) found the same 

value for 0 for a somewhat smaller temperature 

range (15° to 32°C) and 6 values of 0.985 for temperatures from 32° to 40°C and 1.126 for temperatures from 2° to 15°C. He also found that L increased 

with temperature.

Widespread use of the laboratory 5-day 20°C 

BOD test has made apparent several aspects of 

the standard BOD bottle test, as well as the aforementioned problems with dilution and seeding, 

that require additional study. These aspects are 

concerned with how well the diluted and seeded 

sample incubated in darkness under quiescent 

conditions in a constant temperature bath approximates the behavior of a slug of water of similar 

size under dynamic streamflow conditions. For 

example, the turbulence conditions are obviously 

different, and increased values of ki have been 

found when the BOD bottles were stirred or rotated 

(Owens and Edwards, 1966; Gannon, 1966). The increased turbulence prevents deposition of suspended material and facilitates the transport of 

the wastes to the organisms. The large surface-tovolume ratio of the bottle may result in increased 

growth of organisms on the inside of the bottle 

and, hence, faster DO consumption than would occur in an unconfined slug of water of the same size 

(Symons, 1966; Hull, 1969). The bottle test does not 

allow for the possible decrease in the respiration 

rate of organisms with decreasing DO concentration in the bottle (Owens and Edwards, 1966). When 

algae are present, the bottle test conducted in 

darkness could give a high value for the BOD of 

the sample because of respiration demands of the 

algae (Owens and Edwards, 1966). In the stream, 

however, the algae replenish the oxygen consumed 

by their respiration by the process of photosynthesis during daylight hours. The 5-day period 

required for the standard bottle test is also a disadvantage because the conditions in the streams 

may change considerably while the k\ value is 

being determined. Camp (1965) found that the 

water temperatures varied widely during the daylight hours on the Merrimack River in Massachusetts and concluded that it was pointless to 

determine the BOD in the laboratory at a constant 

temperature. Hull (1966, 1969) and Camp (1965) 

recommended using a dark bottle suspended in 

the stream at the point at which the sample was 

obtained. This procedure eliminates delay and the 

need for a temperature correction, and it gives a 

BOD value for the period of sample incubation. To 

determine ki, however, requires BOD values for 

different times or continuous recording of the DO 

concentration in the submerged bottle, and one is 

again faced with the problem of changing conditions in the stream.

BOTTOM DEPOSITS, Ks, AND m

In addition to the biochemical oxidation of 

organic wastes in the flowing stream, wastes may 

also be removed from the stream by settling to the 

bottom. Bottom deposits form in three general 

ways (Velz, 1958): (1) deposition of settleable solids 

from the wastes; (2) deposition resulting from the 

flocculation or coagulation of solids within the 

stream; and (3) biological extraction and accumulation of wastes by growths attached to the 

bottom of the stream.

The first type of deposit is the result strictly of 

a sedimentation process. The second type may be 

considered as resulting from a process analogous 

to an activated sludge system in which the growth 

is dispersed throughout the flowing stream. Under 

certain conditions, the growths flocculate, and the 

floes increase in size until sedimentation occurs. 

The third type of deposit may be considered as resulting from a process analogous to that occurring 

in a trickling filter. Biological extraction and 

accumulation of wastes were studied by Velz and 

Gannon (1963). The rate of sedimentation is generally assumed to be proportional to the amount 

of BOD, or

rate of sedimentation=K3 L. (122)

A velocity of 0.6 fps was considered by Velz (1958) 

to be the critical velocity at which organic wastes 

begin to deposit.

Even though the formation of a bottom deposit 

may result in a reduction of the BOD of the water 

from which the waste was removed, the deposit 

may be a sink for DO to other masses of water
EXPERIMENTAL DETERMINATION OF THE REAERATION COEFFICIENT 25

passing over the deposit. Several processes may be 

involved:

1. BOD additions to the overlying water as a result 

of resuspension of the bottom deposits. A 

velocity of 1.0 to 1.5 fps is considered necessary 

for resuspension, the actual velocity necessary 

being dependent upon the degree of decomposition and compaction of the deposit (Velz, 

(1958).

2. Diffusion of the anaerobic decomposition products of the deposit into the overlying water.

3. Diffusion of DO from the overlying water into 

the aerobic zone of the deposit.

Fair, Moore, and Thomas (1941) concluded that the 

oxygen demand of sludge deposits was controlled 

by process 2 and that process 3 was relatively 

unimportant. The relative importance of processes 

1 and 2 may be seen qualitatively from an experiment of Edwards and Rolley (1965) in which the 

DO consumption rate of a mud sample increased 

approximately sixfold when the sample began to 

scour. Similarly Brown and Clark (1968) reported 

large decreases in the DO concentrations during 

dredging operations in a tidal estuary. They also 

pointed out that resuspension of bottom deposits 

decreases light penetration and, consequently, the 

additions of DO by photosynthesis.

The importance of bottom deposits on DO balances has been demonstrated by several field 

studies, including those by Velz (1958), Owens and 

Edwards (1963), and Camp (1965). Velz (1958) discussed the effect of sludge deposits on DO profiles 

for several situations. Owens and Edwards (1963) 

found on the River Lark in England that they 

obtained very poor agreement between measured 

and observed DO profiles until the DO demands of 

the mud were included in the DO balance. Edwards 

and Owens (1965, fig. 1) presented the percentage 

of the total DO consumed by mud as a function of 

the water depth for BOD and mud consumption 

values typical of English streams. Their graph 

shows that for depths of flow less than 1.25 meters, 

the mud consumed more oxygen than was consumed by the biochemical oxidation of the wastes 

in the water. Camp (1965) found on the Merrimack 

River in Massachusetts that 50 to 80 percent of the 

BOD removal was by sedimentation. He also noted 

that most of the BOD removed by sedimentation 

consisted of solids formed by bioflocculation.

The effects of removal of BOD by sedimentation 

and addition of BOD to the overlying water by 

bottom deposits have been included in the DO 

balances of Dobbins (1964b) and Camp (1963). 

Dobbins (1964b) described three situations for

which K3 can be determined if Ki has been determined from laboratory measurements. The term 

DB, which includes the addition of BOD to the 

water from bottom deposits, in general must be 

determined by difference from the DO profile 

equation. This procedure, however, requires an 

independent estimate of the reaeration coefficient, 

K-Z. Camp (1965) recommended that m, the rate of 

addition of BOD to the overlying water, be determined from in situ measurements by the technique 

of Fair, Moore, and Thomas (1941) and Camp (1963, 

p. 310), and that Ks be determined from the BOD 

profile equation after K\ has been determined from 

laboratory BOD tests.

In addition to the studies of the overall effect 

of bottom deposits on the DO and BOD balances, 

the consumption of DO by muds has been studied 

in some detail with somewhat contradictory results. For example, Baity (1938) and Fair, Moore, 

and Thomas (1941) found that the DO consumption 

rate, w, was dependent on the depth, C/M, of the 

deposit according to

a) = ai UM0-485, (123)

in which ai is a constant. On the other hand, 

Edwards and Rolley (1965) found that the DO consumption rate was independent of the deposit 

depth for depths greater than 2 cm, and McDonnell 

and Hall (1969) concluded that the consumption 

rate was essentially independent of the sample 

depth.

With respect to the dependence of the DO consumption rate on the DO concentration of the 

overlying water, Baity (1938) found that the consumption rate was independent of the DO concentration for the range of 2 to 5 mg/1 for deposits 0.5 

cm thick. Edwards and Rolley (1965), however, 

found that the DO consumption rate was dependent on the DO concentration in the overlying 

water according to

(124)

where bi is 0.45 for the muds studied. This relation 

was valid for DO concentrations in the range from 

2 to 8 mg/1 but was not satisfactory for muds containing large populations of midge larvae. Similar 

results were obtained by McDonnell and Hall (1969). 

An attempt by Rolley and Owens (1967) to correlate the DO consumption rates with various chemical properties of the muds was unsuccessful. They 

attempted to correlate five chemical properties 

with DO consumption rates for both winter and 

summer samples from 12 English rivers but found 

no correlation, although a strong correlation was 

found among all the chemical properties.
26 REAERATION IN OPEN-CHANNEL FLOW

PHOTOSYNTHESIS AND RESPIRATION

The presence of phytoplankton, benthic algae, 

and rooted aquatic plants in a stream or river results in the production of oxygen through the process of photosynthesis. The plants combine carbon 

dioxide and water in the presence of sunlight to 

produce carbohydrates and oxygen. At the same 

time, however, the plants require oxygen in their 

respiration process. In general, the rate of production of oxygen exceeds the rate of consumption 

during daylight hours whereas consumption is 

predominant during nighttime hours. This may 

result in a pronounced diurnal variation of the 

DO concentration in the stream, the maximum DO 

concentration occurring in the afternoon and the 

minimum DO concentration occurring just before 

dawn.

In the past, the photosynthesis and respiration 

processes have been of interest only to biologists, 

who in their studies attempted in a very general 

way to allow for additions of DO by reaeration. 

Engineers, on the other hand, with the exception 

of a few such as Hull (1960), ignored photosynthesis and concentrated their efforts on the 

reaeration process. When photosynthesis is ignored, grossly incorrect values of the reaeration 

coefficient, K2, may be obtained if the usual computation procedure is used. For example, Gunnerson and Bailey (1963) found negative values of K2 

for two sampling periods on the Sacramento River, 

Calif. Lynch (1965) also found negative K2 values 

for nocturnal sampling periods on the Tioughnioga 

River in New York, and he believed that the computed K2 values for daytime sampling periods 

would be larger than the true values if a separate 

factor for photosynthesis were not included in the 

DO balance. Similarly, Camp (1965) concluded that 

the Kz values for streams of the type of the Merrimack River were considerably smaller than had 

been previously thought because of including 

photosynthetic effects with reaeration.

Opinions as to whether the DO produced by 

photosynthesis is an asset or a liability to a stream 

vary widely. Camp (1963) stated that the DO produced by photosynthesis should not be relied upon 

because there is no assurance that this DO will be 

available during the critical low-flow periods of 

summer when it is needed most. Similarly, Imhoff 

and Fair (1956) suggested that the DO produced 

by photosynthesis should not be relied upon in 

engineering calculations because it is available 

only during daylight hours and the warmer seasons of the year. On the other hand, Camp (1965)

found in a summer study on the Merrimack River 

in Massachusetts that about 67 percent of the DO 

was furnished by photosynthesis and that reaeration was relatively unimportant, and he concluded 

"that reliance must be placed on the photosynthetic 

production of DO." Merritt, McDonald, and Paulson (1968) found on the Iowa River that photosynthesis was a contributing factor in the DO 

balance, and they concluded that "whether or not 

the photosynthetic effect can be relied upon 

quantitatively, it remains an important factor of 

the oxygen balance in streams." Similar photosynthetic effects have been observed in European 

rivers, and this literature has been summarized 

by Edwards and Owens (1965). Leffel (1968) found 

on the Chao Phraya River in Thailand that photosynthetic production of DO was small in comparison with reaeration. Bain (1968) stated that in a 

deep turbid stream the photosynthetic production 

of DO_ often is not sufficient to offset the respiration demands of the plants, even if the stream is 

well mixed. O'Connell and Thomas (1965) pointed 

out that if the photosynthetic production of DO 

is by benthic algae or rooted aquatic plants rather 

than phytoplankton, then photosynthesis is 

a liability because the oxygen produced during 

daylight hours by the attached plants is not in the 

same mass of water from which oxygen is consumed by nighttime respiration. An example of 

this cited by O'Connell and Thomas is the shallow 

rapidly-flowing Truckee River below the outfall of 

the sewage treatment plant of Reno, Nev. In a deep 

slowly-flowing stream, the response of the stream 

to a nutrient source is the development of phytoplankton which move with the stream. Hence, the 

production and consumption of oxygen occur at 

different points in space but in essentially the 

same mass of water. In streams with attached 

plants, the production and consumption of oxygen 

occur at the same point in space but in different 

masses of water. Edwards (1962) suggested that 

sometimes plant growths may be beneficial even 

when there is no net gain of DO because the plants 

produce oxygen during the summer conditions of 

low flows and high temperatures and remove oxygen during the decay process in winter when high 

flows and low temperatures exist. Gunnerson 

(1966) suggested that photosynthesis may be neglected if the time period over which the data are 

averaged is sufficiently long-for example, longer 

than 1 year.

Obviously, then, considerable controversy exists 

concerning photosynthesis and respiration in a 

natural stream. Hull (1966) pointed out, however,
EXPERIMENTAL DETERMINATION OF THE REAERATION COEFFICIENT 27

that the controversy generally is whether the oxygen production by photosynthesis is significant 

and not whether a term for photosynthesis should 

be included in the DO balance. Thus Camp (1963), 

Dobbins (1964b), the Water Pollution Research 

Laboratory group (Edwards and Owens, 1962,1965; 

Edwards, 1962; Owens, 1965), O'Connor (1967), and 

others have included terms for the photosynthetic 

production of oxygen in their DO balances.

The production of oxygen by plant photosynthesis depends on several factors, including light 

intensity at the water surface, optical density of 

the water, and weight and distribution of plants 

in the stream (Owens and Edwards, 1964). The 

depth of the stream affects the photosynthetic 

production rate in that the extent of light penetration varies inversely with depth. Photosynthesis is 

limited to the euphotic zone, which is generally 

assumed to extend to the depth at which the light 

intensity is 1 percent of the light intensity at the 

surface (Owens and Edwards, 1966). The optical 

density of the water depends on the concentration, type, and size of suspended particles and on 

the color of the water produced by dissolved 

materials. An example of a situation where the 

environmental conditions were suitable for a high 

level of photosynthetic oxygen production but 

where actual production was limited to a very low 

level because of the turbidity of the water is the 

Chao Phya River in Thailand (Pescod, 1969). The 

weight and the distribution of the plants affect the 

net DO production mainly by affecting the total 

respiration rate. Thus, as the plant density 

increases, the photosynthetic production rate remains approximately constant because of selfshading by the plants, although the respiration 

rate continues to increase (Edwards, 1968).

Several procedures have been developed for the 

measurement of photosynthetic and respiration 

effects in natural streams. These include the 

upstream-downstream method (Odum, 1956; 

Owens, 1965), the single-curve procedure (Odum, 

1956), which is a simplification of the upstreamdownstream procedure; and the light- and darkbottle technique.

The light- and dark-bottle technique requires 

collection of water samples at a number of verticals 

in the cross section and at a number of points at 

each vertical so that the number of samples is 

sufficient to yield an accurate value of photosynthesis and respiration for the cross section. Each 

sample is divided into three parts; one part is 

placed in a transparent glass bottle, another part 

is placed in an opaque bottle so that all light is

excluded, and the third part is analyzed for the DO 

concentration. The light and dark bottles are suspended in the stream at the sampling point for a 

period of time, usually 24 hours or a multiple 

thereof. At the end of the desired period, the light 

and dark bottles are analyzed for their DO concentrations. Equations for the light-bottle and darkbottle DO concentration are, respectively,

Clf-Cn=Pi-Ri (125) 

Cd,-CV=/?d, (126) 

where the / and d subscripts refer to light and dark 

bottles, respectively; / and / are subscripts referring to initial and final DO concentrations; R is the 

gross respiration, or BOD; C is DO concentration; 

and P is the photosynthetic oxygen production. For 

reasonably short periods of time, Ri = Rd, and because Cu = CM, it follows that

G/-Cd/ =P*. (127)

Hull (1963) believed that the light- and dark-bottle 

technique probably is more accurate than the 

standard BOD test and that the technique in general gives conservatively low estimates of the 

oxygen production by photosynthesis.

Symons (1966) discussed several problems with 

the classical light- and dark-bottle technique, including unnaturally large surface-to-volume ratios 

in the bottles, the relatively long time required for 

the test, the lack of mixing in the bottles, and the 

necessity for removal of samples from the stream, 

which may result in harm to the algae. He discussed a three-probe chamber device designed to 

eliminate these problems. O'Connell and Thomas 

(1965) also used a chamber device but did not 

measure DO continuously.

The upstream-downstream method uses measurements of the DO concentrations at the ends of 

the reach of interest for a 24-hour period. The basis 

of the procedure is a DO balance of the form 

(Odum, 1956)

QO =P-R+DO+AO, (128)

where Q0 is the rate of change of DO content, P 

is the gross primary production, R is the respiration rate, D0 is the rate of reaeration, and A0 is the 

rate of addition of DO to the reach by drainage 

accrual. Each term of equation 128 is on an area 

basis for example, in units of grams per square 

meter per hour.

If each term in equation 128 is divided by H, the 

mean depth of flow, the result is

q=p-r+d0 + Si(h (129) 

where the terms are now on a concentration basis 

 for example, in units of milligrams per liter per 

day.

The gross primary production, P, is defined as
28 REAERATION IN OPEN-CHANNEL FLOW

the sum of the net DO plant production and community respiration during the daylight hours. 

Therefore, it follows that if r and do are independent of time and a0 is negligible or independent of 

time (Odum, 1956),

P=H (130)

where F is the volumetric water flow rate, As is 

the surface area of the reach, AC is the DO concentration difference between the upstream and 

downstream stations, and the subscripts 1 and 2 

refer, respectively, to the time at which sunrise 

occurred and the time in the evening at which q 

or AC returned to its sunrise value. When equation 

130 is used to determine P, equation 128 must have 

units of weight per unit of area per day.

In general, the oxygen supplied by reaeration, 

do, is not independent of time because the rate of 

reaeration depends on the DO deficit. If a reaeration coefficient is available for the reach, then a 

correction can be applied for each hour in the calculation of the q curve from the DO measurements.

The reaeration coefficient can be calculated from 

the diurnal DO measurements if r and ao are 

assumed independent of time. From measurements before dawn and in the evening, when the 

p values are zero, it can be shown (Odum, 1956) that

M E

where D is now the average DO deficit in the reach, 

and the M and E subscripts refer to the morning 

and evening times, respectively. The reaeration 

coefficient calculated from equation 131 is the Kz 

for the average deficit of the reach as compared 

with an integral K2 obtained by integrating the 

deficit over the DO change.

Both equations 130 and 131 assume that r is 

independent of time and, hence, independent of 

the DO concentration level. Opinions differ as to 

the validity of this assumption. Odum (1956) concluded on the basis of the available information 

that it was simplest to assume r constant. On the 

other hand, Edwards, Owens, and Gibbs (1961) 

concluded that the respiration rate of plants 

depended on the DO concentration level. Therefore, corrections for the dependence of r on DO 

concentration may be necessary.

The respiration rate can be determined from the 

results of an upstream-downstream experiment in 

the following ways (Odum, 1956). First, if the rate 

of reaeration is known and A0 is negligible, then 

for the hours of darkness the respiration rate can 

be calculated from the observed rate of change of 

DO in the reach. Second, if during the hours of 

darkness the water is saturated with DO at some

time, as usually occurs after sunset, then at that 

instant, because the reaeration is zero, the respiration rate is equal to the rate of change of DO in 

the reach, assuming A Q is negligible.

The single-curve procedure (Odum, 1956) is a 

simplification of the upstream-downstream procedure in which the diurnal variation of DO concentration at a single station is used in the 

calculation of P and R. This procedure assumes 

that the entire reach experiences uniform changes 

in the DO concentration, and hence the curves at 

the upstream and downstream stations would be 

identical. Owens (1965) compared the two calculation procedures for two stations on the River Ivel 

in England which had nearly identical diurnal DO 

curves and found that the calculated P and R 

values were different for the two procedures and 

also different for the two stations. Edwards and 

Owens (1962) also found differences between the 

two procedures.

In addition to their direct effects on the DO 

balance, plants also exert indirect effects on the 

DO balance. For example, heavy plant growths 

can increase the depth of flow and reduce the 

water velocity, which in turn reduces the addition 

of oxygen by reaeration. At the same time, the 

reduced velocity increases the deposition of solids. 

If the plants extend through the surface, then turbulence at the air-water interface may be increased, thereby facilitating the absorption of 

oxygen by the water. Finally, the presence of the 

plants increases the surface area on which the 

bacteria can live and oxidize wastes present in 

the stream.

OTHER PARAMETERS

Other parameters that are necessary in a DO 

balance study include the DO concentrations at 

the upstream and downstream ends of the study 

reach and the flow time through the reach.

The flow time can be determined from mean 

velocity measurements and the distance between 

the ends of the study reach, or

A,=-. A (132)

Distances may be obtained from topographic maps, 

and mean velocities from current meters or from 

stage-discharge ratings and cross-section measurements. Long study reaches are desirable; however, 

practical considerations such as variations in the 

channel cross section, location of sewage outfalls, 

and personnel requirements generally limit the 

length of reach that can be studied.
EXPERIMENTAL DETERMINATION OF THE REAERATION COEFFICIENT 29

The flow time can also be determined from tracer 

measurements, using either radioactive or fluorescent tracers. The procedure consists of instantaneously injecting a quantity of the tracer far 

enough upstream from the reach of interest so that 

the tracer can be uniformly mixed in both the 

lateral and vertical directions before it enters the 

reach. Procedures for estimating the distances 

required for complete mixing have been presented 

by Sayre (1965) and Fischer (1967). The flow time 

is the time required for the tracer cloud to move 

through the study reach. In some instances, the 

peaks of the tracer concentration versus time 

curves have been used in flow-time determinations; however, it is more correct to use the centroids of the curves. The flow time is the difference 

between the times at which the centroid of the dye 

cloud passes the downstream and upstream ends 

of the reach.

The upstream and downstream DO concentrations are determined from samples obtained at the 

ends of the study reach. One of the procedures 

previously described is used to determine the DO 

concentrations of the samples. The samples at the 

downstream point usually are obtained at a time 

lag of A*/Lf after the samples at the upstream 

point. This is done because DO balance equations 

generally are written in a coordinate system moving with the mean flow velocity.

The number of samples obtained at each cross 

section must be sufficient to define a mean DO 

concentration for the cross section. As mentioned 

previously, gradients of DO in the vertical direction in a flowing stream are generally negligible. 

The lateral DO distribution depends on many factors, including distance from sewage outfalls and 

stream tributaries, cross-section shape and tortuosity of the channel, and size, shape, and spacing of bed roughness elements. Churchill, Elmore, 

and Buckingham (1962) observed consistent 

U-shaped DO distributions across the stream 

channel, and they attributed these to (1) incomplete mixing of tributary inflows, (2) increased 

reaeration near the banks because of the longer 

traveltime, and (3) unequal effects across the 

stream width of photosynthesis and respiration by 

attached aquatics. Thus the possibility of lateral 

variations in the DO concentration should be 

considered.

To determine k2 in the most general field situations, all the parameters in equation 119 must be 

evaluated using the methods previously described. 

In specific reaches, however, some of the factors 

in equation 119 may be of negligible importance. 

For example, if p, ni, and k3 are negligible, equation

119 reduces to the classical Streeter-Phelps sag 

equation, equation 6. Examples of reaches in which 

p, m, and k3 were not negligible have been given 

previously. In the reaches chosen specifically for 

the study of reaeration by Churchill, Elmore, and 

Buckingham (1962), the BOD, k3, m, and p parameters were all negligible, so that equation 119 

reduced to

Db =Da 10-fc2<. (133)

Thus, kz could be calculated simply from DO measurements on samples obtained at two points in 

the reach separated by the flow time, t. In general, 

however, equation 119 or a similar equation should 

be the starting point in a DO balance calculation 

of k2 for a natural stream, and parameters should 

be eliminated only after they have been shown to 

be negligible for the reach and hydraulic conditions of interest.

CALCULATION OF k2 IN A RECIRCULATING FLUME

In a laboratory flume the sources and sinks of 

DO can be controlled so that the BOD, k3, m, and p 

parameters will in general be zero. With these 

conditions, the reaeration coefficient can be calculated from equation 133. Flume lengths, 

however, are limited by space and economical considerations so that the available flow times and 

the consequent change in DO deficit generally will 

be very small. Thus the calculated reaeration 

coefficients will be subject to large errors. This 

problem can be alleviated by recirculating the 

water through the flume.

In recirculating flumes, the water is partly deaerated before the start of an experiment, usually 

using sodium sulfite in the presence of a cobalt 

catalyst. As the water circulates through the 

flume, it reaerates at a rate which is dependent on 

the hydraulic conditions in the flume and on the 

characteristics of the flume return-flow system. 

For steady-state hydraulic conditions, the form of 

equation 9 generally assumed to be applicable is

<*Q +U<*P-=-2.3k2 D, (134)

where the DO concentrations have been expressed 

as deficits. Making the substitutions x'=x Ut and 

t' = t into equation 134, one obtains

 9^Lr n n^i / £-dK2 LJ) \J-OO}

and integrating equation 135 with D = Da at f'=0 

gives

or

D=Da .10-fc2<,

1

(136)

(137)
30 REAERATION IN OPEN-CHANNEL FLOW

Equation 137 is identical in form with equation 

133; however, in equation 137, because conditions 

are not steady state, the flow time t' is a time lag 

which must elapse between the DO measurements 

at the two sampling stations, whereas in equation 

133, because conditions are steady state, the DO 

measurements at the two stations may be made at 

any time. In a recirculating flume, it is common 

experimental practice to obtain an average log 

deficit difference as compared with the single 

measurement of the difference obtained in a 

natural stream or a nonrecirculating flume. The 

average log deficit is determined by the procedure 

illustrated in figure 2. The parallel lines show the 

variation with time of the logarithm of the deficit 

at two points in the flume a distance X apart. Because the water is recirculated, several points can 

be obtained on each line, and thus the positions and slopes of the lines can be accurately 

determined.

TIME

FIGURE 2. Computation of reaeration coefficient, K%, from 

recirculating-flume data.

DISTURBED-EQUILIBRIUM TECHNIQUE

The disturbed-equilibrium technique developed 

by the Water Pollution Research Laboratory 

(Gameson and Truesdale, 1959) is based on a DO 

balance described by equation 14. The procedure 

consists of measuring the DO concentrations at 

the upstream and downstream ends of the reach 

of interest at two different levels of DO concentration. The second DO concentration level is 

obtained usually by adding sodium sulfite and a 

cobalt catalyst to the stream, although the diurnal

variations in DO concentration caused by photosynthesis may be used. If P, /?, K2, U, and Cs are 

constant for the reach during the period of the 

experiment and therefore independent of the 

DO concentration, then by manipulation of equation 14:

K2 =-ln^a "^a , (138)

where Ca and Ca are the upstream DO concentrations at the two levels of concentration, Cb and Cb' 

are the downstream DO concentrations measured 

at a time t after the upstream concentrations, and 

t is the flow time for the reach of interest.

Although the respiration rates of bacteria are 

independent of the DO concentration for concentrations larger than 1 mg/1 (Edwards and others, 

1961), the respiration rates of plants (Edwards and 

others, 1961), and of bottom deposits (McDonnell 

and Hall, 1969; Edwards and Rolley, 1965) are not. 

Furthermore, photosynthetic DO production can 

cause large variations in DO concentration with 

time on a specific day and from day to day. Therefore, when plants are present, Edwards, Owens, 

and Gibbs (1961) suggested that the experiments 

be conducted at night, when the photosynthetic 

contributions are negligible. To account for the 

dependence of the plant respiration on DO concentration, they showed that

K2= _J^-^Qi ^_ t (139)

where Ci and C2 are the average DO concentrations for the study reach for the two levels of DO 

concentration, Q\ and Q2 are the average rates of 

change per unit area of the DO concentration in 

the reach at each of the two levels of DO concentration, CS2 and Ctl are the DO saturation concentrations at the times of the measurements for the 

DO concentrations at the two levels, and R is the 

ratio of the oxygen consumed by respiration at 

DO concentration C2 to that consumed at DO concentration Ci. The reaeration coefficient obtained 

from equation 139 is the K2 for the average DO 

deficit in the study reach.

The present authors derived from equation 119 

an equation that will give an integral reaeration 

coefficient rather than one based on the average 

DO deficit for the reach. If equation 119 is written 

for upstream deficits of D a and D a and the 

difference taken, the result is

(140)

where p and p include the respiration rates of both
EXPERIMENTAL DETERMINATION OF THE REAERATION COEFFICIENT 31

plants and bottom deposits. In equation 119, the 

parameter p is the net production of oxygen by 

photosynthesis. However, the production of DO by 

photosynthesis is assumed to be constant or zero 

during the period of the measurement, so that the 

p parameter becomes strictly a respiration term. 

The derivation of equation 140 also assumes that 

Ki, Ks, La and m are constant during the period of 

the measurements. Because K2 appears on both 

sides of equation 140, a trial-and-error solution is 

necessary.

The disturbed-equilibrium technique requires 

measuring the upstream and downstream DO concentrations and the respiration rate in the reach 

at two different values of the upstream DO deficit. 

The different values of the upstream deficit may 

be produced by diurnal variations in photosynthesis and respiration or in BOD inflow, or by the 

addition of sodium sulfite along with a cobalt 

catalyst. In general, the technique is very convenient to use, especially if the DO variations are 

produced by natural conditions in the stream. In 

this case, however, accurate measurements of the 

respiration rates p and/?' are necessary, quantities 

which are often not easily obtained. If the deficit 

is produced by sulfite dosing, where p and p' are 

small, there is less uncertainty in the measurements of p and p'; however, their variation may 

still be significant. The amount of sulfite required 

places limitations on the size of stream that can 

be dosed and on the length of time that the dose 

can be applied. Furthermore, it must be assured 

that the sulfite is completely mixed with the flow 

and that the reaction has proceeded to completion 

before the flow enters the study reach. Finally, in 

either method, the experimenter must assure himself that the effect of the time variations in DO 

and BOD do not cause serious errors in the computation of the reaeration coefficient.

TRACER TECHNIQUE

Equations 81 and 83 resulting from the kinetictheory model of Tsivoglou (1967) are the basis of the 

tracer technique for determination of the reaeration coefficient. As discussed previously, Tsivoglou 

(1967) concluded on the basis of laboratory measurements that the gas-transfer-coefficient ratio 

in equations 81 and 83 was a constant within the 

limits of experimental error for the range of temperature and turbulence conditions studied. 

Because the ratio is constant, an inert gas that 

does not participate in the various biochemical 

and biological processes in a stream can be used 

as a tracer for oxygen. This permits the determination of a true reaeration coefficient in the 

presence of pollution and photosynthetic effects, 

provided the value of the ratio of gas-transfer 

coefficients is known for the inert gas and oxygen. 

Thus, the tracer technique is unique in that it does 

not require the measurement of any of the parameters in a DO balance. The technique was first 

described by Tsivoglou, O'Connell, Walter, Godsil, 

and Logsdon (1965), a field application was 

described by Tsivoglou, Cohen, Shearer, and Godsil 

(1968), and a complete discussion of the technique 

was presented by Tsivoglou (1967).

The technique uses an instantaneous injection 

of three tracers at a point upstream from the reach 

over which the reaeration coefficient is to be measured. The tracers are (1) a fluorescent dye, the 

purpose of which is to enable field personnel to 

follow the movement of the tracers; (2) tritiated 

water, which is used as a conservative dispersion 

tracer; and (3) the radioactive tracer gas krypton85, which is used to measure the gas transfer capacity of the flow. Samples of stream water are 

removed from the flow as the dye peak passes the 

upstream end and again as it passes the downstream end of the reach. The transfer capacity of 

the flow for the tracer gas is obtained from the 

relative concentrations of the gas and tritiated 

water tracers at the two ends of the reach. 

Tsivoglou (1967) showed on the basis of an argument involving numbers of krypton and tritium 

atoms and an exponential expression for gas 

exchange that

(141) (Co]

\C£/a

where C* is concentration; t is flow time, or 

where A* is the distance between the downstream 

and upstream stations indicated respectively by 

the subscripts "b" and "a"; the term K2c is the gastransfer coefficient of the tracer gas; and the G 

and L subscripts refer respectively to the krypton 

and tritium tracers.

The tracer-gas transfer coefficient, KzG , can be 

converted to a reaeration coefficient, K2 , by means 

of the gas-transfer ratio determined by Tsivoglou 

(1967). He found that the average ratio of the rate 

of krypton desorption to the rate of oxygen absorption was 0.83. Hence,

K'=m- (142)

The crucial assumption in the tracer technique 

is that this ratio is a constant that is, independent of the flow conditions, temperature, and 

pollutants present.

466-751 O - 72 - 6
32 REAERATION IN OPEN-CHANNEL FLOW

We developed a theoretical basis for the technique from solutions of the conservation-of-mass 

equation for the two tracers. The one-dimensional 

diffusion equation in a steady-uniform flow for a 

conservative tracer is

^r+u^r=D*^' (143)

where C\ is the tracer concentration. The corresponding equation for the gaseous tracer is

,c * aa a ^ ^

dt ~ dx dx2 "'<> CG' (144) 

where C*G is the tracer gas concentration. Implicit 

in the use of the one-dimensional diffusion equation is the assumption that the tracers are uniformly distributed in the cross section of the 

stream. Therefore, the upstream sampling point 

must be far enough downstreanrfrom the injection point so that complete mixing in both the 

lateral and vertical directions has been achieved. 

The solutions to equations 143 and 144 for instantaneous inputs of weight WL of the liquid and WG 

of the gaseous tracer at ;c=0 and t=0 were presented by Camp (1963). For equation 143, the 

solution is

and for equation 144,

(145)

(146)

in which WoL=WLIAx and W*G =WG\AX, where Ax is 

the flow area. For any longitudinal position x at 

time t, the ratio of the gas concentration to the 

liquid concentration is

e-'V . (147)

If one now takes the ratio of equation 147 for the 

downstream station at a time t=bx!U to equation 

147 for the upstream station at t=Q, the result is 

identical with equation 141 given previously by 

Tsivoglou (1967). Using equations 141 and 142 and 

the four measured tracer concentrations, the 

reaeration coefficient K2 can be computed.

The arguments used by Tsivoglou (1967) in 

developing equation 141 require no assumptions 

regarding vertical and lateral mixing of the 

tracers. However, the reaeration coefficient obtained from the tracer technique using a pointsource injection and samples from the centerline 

of the channel is representative only of the filament of the stream traversed by the tracers in 

moving from the injection point to the sampling

point. To obtain a K2 representative of the entire 

cross section of a reach, the tracers must be injected sufficiently far upstream so that they will be 

uniformly distributed in the cross section before 

entering the reach. Under these conditions, the 

one-dimensional diffusion equation is applicable, 

and as shown previously, a theoretical basis exists 

for the tracer-technique equation obtained from 

intuitive arguments by Tsivoglou (1967).

The assumptions on which the tracer technique 

is based were listed by Tsivoglou, Cohen, Shearer, 

and Godsil (1968, p. 302) as follows:

(1) The tritiated water undergoes dispersion only in the 

stream, and is not lost from the stream water in any significant 

amount; (2) the dissolved krypton-85 undergoes the same dispersion as the tritiated water, in addition is lost to the atmosphere, and is not otherwise lost in any significant amount; 

(3) the ratio (0.83) of gas transfer coefficients for dissolved 

krypton-85 and oxygen is not affected significantly by temperature, turbulence, or the presence of usual water pollutants, in 

the range of interest***.

The tritium and krypton tracers were chosen with 

the first two assumptions in mind, and therefore 

these assumptions should be valid. The third 

assumption was verified experimentally, as discussed previously.

The tracer technique has several advantages 

over the DO balance and disturbed-equilibrium 

techniques for determination of reaeration coefficients. The most obvious advantage is that 

only the concentrations of two radioactive substances need be determined as compared with all 

the various parameters that must be measured in 

the other two techniques. In addition, the nuclear 

counting techniques are very accurate, and because only ratios of the counts are used, determination of the counting efficiency is not necessary, 

provided it is the same for the counting of all 

samples. The possibility exists also of determining 

with the tracers the discharge and longitudinaldispersion characteristics of the stream at the 

same time that the reaeration coefficient is measured. This would require additional effort, 

however, because the complete concentrationversus-time curve at each cross section is necessary whereas only the peak concentrations 

are needed for the reaeration-coefficient determination.

There are also disadvantages to the tracer technique. Radiation exposure received by the personnel conducting the study and the public along 

the stream may be significant. Tsivoglou (1967) 

reported that radiation exposure could be eliminated with a minimum of trouble for discharges 

less than 200 cfs. However, for larger flows, more 

stringent precautions are required because of the
EXPERIMENTAL DETERMINATION OF THE REAERATION COEFFICIENT 33

larger amounts of tracer necessary. Another disadvantage of using radioactive tracers is that the 

process for obtaining a license to use radioactive 

material in a public waterway is quite involved. 

Finally, the nuclear counting equipment necessary for tritium and krypton is expensive and not 

field-type equipment; hence, samples must be 

returned to a laboratory for analysis.

ANALYSIS OF ERRORS IN REAERATION-COEFFICIENT 

CALCULATION TECHNIQUES

The determination of the reaeration coefficient 

by any of the techniques described previously is 

subject to many sources of errors. The major 

sources of error are (1) differences between the 

dynamic stream condition and the quiescent condition in the various bottle techniques developed 

for measuring the DO balance parameters, (2) the 

difficulty and expense of obtaining a sufficient 

number of representative samples to describe 

accurately the stream conditions at the cross 

section of interest, (3) deviations from the assumptions inherent in the derivation of the DO balance 

equation, and (4) the accuracy of the DO measurement procedure used in the various techniques 

for determining the DO balance parameters.

A general statistical analysis of the effect of 

these various sources of error on the calculation 

of the reaeration coefficient is beyond the scope of 

this report and probably is not feasible with the 

present state of our knowledge. However, it is 

instructive and important to consider the range 

of possible errors in the measured DO balance 

parameters and the effect of these errors on the 

DO balance equation used to calculate the reaeration coefficient.

The purpose of this section is to present a discussion of (1) the sensitivity of the DO balance 

technique to errors in the measured parameters;

(2) the sensitivity of the disturbed-equilibrium 

technique to errors in the measured parameters;

(3) possible errors in the tracer technique; and

(4) reproducibility of reaeration coefficients measured in flumes.

ACCURACY OF MEASURED DISSOLVED-OXYGEN BALANCE 

PARAMETERS

The accuracy of the measured value of a DO 

balance parameter is dependent not only on the 

accuracy of the DO measurements but also on the 

extent to which the measurement conditions under 

which the measurements are made simulate the 

conditions in the flow, and on how representative 

the samples are of the total flow.

For example, as has already been mentioned, 

not only is accurate measurement of BOD in 

bottles difficult due to dilution and seeding problems, but also there is considerable question as to 

how accurately the behavior of a sample of water 

incubated under quiescent conditions in a constant 

temperature bath simulates the behavior of a 

similar slug of water under dynamic streamflow 

conditions and varying temperature. There are 

similar problems with the techniques for determining the rate of oxygen consumption by the 

benthos and for determining the rates of photosynthesis and respiration of suspended and 

attached plants.

Completely aside from the problem of simulating 

field conditions with measurement techniques is 

the problem of obtaining representative samples 

in the field. The uniformity of DO and BOD distribution in a cross section will depend on a multiplicity of factors, among which the most important 

are distance from sewage outfalls and stream tributaries, shape of cross section, tortuosity of 

channel, and bed roughness size, shape, and spacing. For example, Churchill, Elmore, and Buckingham (1962, figs. 3-9) found a consistent U-shaped 

DO distribution across the stream channel and 

attributed this distribution to (1) incomplete mixing into the main stream of flow from upstream 

tributaries, (2) increased reaeration near the banks 

due to a longer traveltime, and (3) unequal effects 

across the stream of photosynthesis and respiration by attached aquatics. Considering the large 

cross-sectional DO variation and the small longitudinal variation in average DO illustrated in 

these figures, one would expect considerable error 

in the calculated k2 if DO were sampled at only one 

point in the cross section. Another example of the 

difficulty of obtaining representative samples for 

the computation of the DO balance parameters 

was given by Owens and Edwards (1961), who took 

extensive samples of attached aquatic plants from 

several English streams for analysis of the photosynthesis-respiration parameter. In their studies, 

the standard deviation of the amount of the 

attached aquatics per unit area averaged 32 percent of the mean. The size of this standard deviation in terms of the mean indicates the difficulty 

of obtaining a reliable estimate of photosynthesis 

or respiration without sampling the plants on a 

large area of the stream bed.

Because of the difficulty of reproducing the 

stream environment in the measurement techniques for DO balance parameters, and because of 

the difficulty and expense of obtaining representa-
34 REAERATION IN OPEN-CHANNEL FLOW

tive samples, it is apparent that if all parameters 

in the DO balance equation are significant, then 

the two conditions discussed above, and not DO 

measurements themselves, limit the accuracy 

which can be obtained in the calculation of k2 . 

Because of the present state of uncertainty about 

these two situations, the accuracy obtainable from 

DO balance calculations is largely a matter of conjecture. The error analysis techniques used in this 

section have been designed with this in mind. Reasonable average values for the various DO balance 

parameters and what the authors consider to be 

reasonable values of errors are.given in this 

section.

DISSOLVED OXYGEN BALANCE TECHNIQUE

Because of the complexity of the general DO 

balance equation, the number of parameters involved, and the difficulty of reproducing field 

conditions for replications of k2 determinations, a 

rigorous statistical analysis of the effect of errors 

on the calculated k2 cannot be performed at the 

present time. Alternatively, however, one can 

perform for k2 a sensitivity analysis, similar to that 

of Kothandaraman and Ewing (1969), on the downstream DO deficit calculated from the DO balance 

equation. The sensitivity analysis is designed to 

yield the error in the calculated value of k2, when 

the error in another parameter of the DO balance 

equation is known. When the distribution functions of the errors in the parameters of the DO 

balance equation are known, a Monte Carlo 

approach similar to that used by Kothandaraman 

and Ewing (1969) can be used to determine the 

distribution function of the calculated k2 values.

A sensitivity analysis consists of observing the 

behavior of the output of a system with all except 

one of its parameters constant. This parameter 

is varied throughout the range of possible values, 

and at any given value of the parameter, the sensitivity of the system to the parameter is the 

change in output per unit change in the parameter. 

The sensitivity analysis was performed on the general DO balance equation, equation 119, written in 

the form

m

2.3*1 +ZX

(148)

Equation 148 was solved for k2 using the NewtonRaphson iterative method (Kunz, 1957) in the form (149)

and calculations were stopped when the absolute 

value of the ratio on the right-hand side of equation 149 was less than 0.001 times the true value 

of k2 .

The sensitivity analysis consisted of the following steps:

1. Assume a typical set of parameters for use in 

the DO balance equation, equation 119. This 

set of parameters is called a BPS (basic 

parameter set).

2. Substitute the BPS into equation 119, and calculate the corresponding BPS value of £>b .

3. Use equations 148 and 149 to compute the sensitivity of equation 119 to each of the parameters in turn.

In some calculations, the classical sensitivity 

analysis procedure was changed in that several 

of the parameters were varied simultaneously.

The BPS values are given in table 1. Typical sensitivity analysis plots are given in figures 3

TABLE 1. Basic parameter sets used in the sensitivity analyses of the DO balance equation

BPS ki *2 fa 

No. (days-') (days-') (days- 1 ) pe^daV) (mgE/1) <mg3/1) (days) (mgl>/1)

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

0.1

.1

.1

.1

.1

.1

.1

.1

.1

.1

.1

.4

,4

.1

.1

.1

.1

.1

.1

1

1

1

1

1

.2

10

10

.2

1

1

1

1

1

1

1

1

1

1

0

0

.1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

0

0

0

20

20

10

5

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

3

3

3

3

3

3

3

3

3

3

5

5

5

3

3

3

3

3

3

5

0

5

5

200

5

5

200

200

200

5

5

50

5

50

5

200

5

5

0.05

.05

.05

.05

.05

.05

.05

.05

.05

.05

.05

.05

.05

.05

.05

.05

.05

.05

.05

2.73

2.67

2.73

2.68

4.84

2.99

.98

2.31

5.20

4.84

4.51

4.67

6.58

2.73

3.21

1.78

3.89

2.26

2.49

through 8. The ordinates in these figures are expressed as a percentage of the BPS value of k2 , and 

the abscissas are in the dimensions of the parameter that is being varied in the sensitivity 

analyses. Of the 19 BPS considered in the sensitivity analyses, six, in general, were selected for 

presentation in figures 3 through 8. Numbers 1, 5, 

6, 7, 15, and 16 were selected as being typical of 

the following field conditions: 1, A medium-sized 

unpolluted river; 5, a medium-sized heavily 

polluted river; 6, a large unpolluted river; 7, a 

shallow, swift-flowing stream; 15, a medium-size 

moderately polluted river; and 16, a medium-sized 

unpolluted river in which the effect of ignoring 

a photosynthetic-oxygen production rate of 20 

mg/1 per day is investigated.
EXPERIMENTAL DETERMINATION OF THE REAERATION COEFFICIENT 35

900 500

0.2 0.4 0.6 0.8 

BIOCHEMICAL OXYGEN DEMAND DECAY 

RATE CONSTANT, IN RECIPROCAL DAYS

FIGURE 3. Sensitivity of reaeration coefficient to rate constant for BOD decay, dissolved-oxygen balance technique.

The sedimentation rate constant, k3, and m, the 

rate of addition of BOD from bottom deposits, were 

not considered in figures 3 through 8 because the 

sensitivity analyses showed that the dependence 

of k2 on these parameters was very small. For k3, 

the most sensitive example was for BPS 5, where 

kz changed only 23 percent of its BPS value for a 

change of k3 from 0 to 0.8. For m, the most sensitive 

example was for BPS 10, where k2 changed only 13 

percent as m varied from 0 to 200 mg/1 per day.

The results shown in figures 3 through 8 can be 

clarified using an approximate form of equation 

119. Because the exponents in this equation are 

much less than 1, the exponential terms may be

PHOTOSYNTHETIC PRODUCTION RATE, IN 

MILLIGRAMS PER LITER PER DAY

FIGURE 4. Sensitivity of reaeration coefficient to photosynthetic-oxygen production rate, dissolved-oxygen balance 

technique.

1-0 7f>

UPSTREAM DISSOLVED-OXYGEN DEFICIT, 

IN MILLIGRAMS PER LITER

FIGURE 5. Sensitivity of reaeration coefficient to upstream 

DO deficit, dissolved-oxygen balance technique.
36 REAERATION IN OPEN-CHANNEL FLOW

-J1200

UJ tniooo

800

600

400

BPS 16

BPS 6

BPS 5.

100 150 200

UPSTREAM BIOCHEMICAL OXYGEN DEMAND, 

IN MILLIGRAMS PER LITER

FIGURE 6. Sensitivity of reaeration coefficient to upstream 

BOD, dissolved-oxygen balance technique.

0.05 0.10 0.15 

TRAVELTIME, IN DAYS

0.20 0.25

FIGURE 7. Sensitivity of reaeration coefficient to traveltime, 

dissolved-oxygen balance technique.

approximated by the first two terms in their 

Taylor-series expansions. In this case, equation 

119 reduces to

The first term in the brackets of equation 150 

explains why the sensitivity of kz to variations in 

flow time, t, decreases as Db approaches Da (fig. 7). 

The second and third terms in the brackets 

explain the linear variation of kz in terms of ki, p, 

and La (figs. 3, 4, and 6). The second term in the 

brackets explains the changes in sensitivity of kz 

to ki with changes in La. Finally, the absence of k3 

and m from equation 150 explains the lack of sensitivity of the calculated kz to these parameters. 

The solution of equation 150 for kz using BPS 1 is 

0.94, and that using BPS 5 is 1.37; thus, the 

approximations in the equation are too great for 

it to be used for calculation of kz directly, but it is 

valuable for showing quickly the relative seriousness of the errors in the calculated kz which can 

result from errors in the other parameters.

The extension of the sensitivity analysis to 

the determination of the error in a reaerationcoefficient calculation resulting from an error in 

a measured parameter requires information on 

the magnitudes of the errors to be expected in the 

measured parameters. However, as discussed 

previously, the different techniques developed for 

measuring the DO balance parameters are subject 

to various types of errors, and the determination 

of such errors is difficult. Therefore, for the purpose of an error analysis we have made what we 

consider to be reasonable assumptions of the RMS 

(root-mean-square) and maximum errors to be 

expected in the various measured parameters. 

These estimates are presented in table 2. The RMS 

error is the square root of the mean value of the 

square of the deviations of the measured parameter from the true value of the parameter.

The errors in a calculated kz corresponding to 

the estimated parameter errors given in table 2 

are given by the product of the estimated error 

and the slope of the sensitivity-analysis plot. It is 

apparent from figures 3 through 8 that the slopes 

of the sensitivity-analysis plots depend on the 

particular BPS considered. Therefore, the minimum, average, and maximum slopes were deter-
EXPERIMENTAL DETERMINATION OF THE REAERATION COEFFICIENT 37

1200

hiooo -

800 -

200

600 -

400 -

DOWNSTREAM DISSOLVED-OXYGEN DEFICIT, 

IN MILLIGRAMS PER LITER

FIGURE 8. Sensitivity of reaeration coefficient to downstream 

DO deficit, dissolved-oxygen balance technique.

TABLE 2. Assumed errors in measurements of DO balance 

parameters

Parameter

fcl

ks

Da

Db

La 

P 

m 

t

Assumed

Root-mean-square

0.05 days-1 

.10 days"1 

.10 mg/1 

.10 mg/1 

2.0 mg/1 

2.0 mg/1 per day 

5.0 mg/1 per day 

.002 day

error

Maximum

0.30 days-1 

.50 days"1 

1.0 mg/1 

1.0 mg/1 

10 mg/1 

10 mg/1 per day 

20 mg/1 per day 

.005 day

mined for each parameter so that the entire range 

of conditions covered by the 19 different BPS 

would be considered. The slopes are presented in 

table 3. The minimum, average, and maximum 

slopes were combined, each in turn, with the RMS 

and maximum error estimates from table 2 to 

give the expected errors in the calculated reaeration coefficient. These errors are presented in table 

4. It is apparent that, with the exception of m, 

errors in all the parameters may cause considerable error in the calculated kz value. As the value 

of k2 decreases, the percentage of error increases.

TABLE 3. Slopes of the sensitivity-analysis plots

*,

*3

Da 

0b 

J-a

P

1

Minimum

1.6

0.01

1.4 (mg/1 per day)" 1 

1.6 (mg/1 per day)" 1 

0.02 (mg/1 per day)" 1

0.11 (mg/1)-' 1x10-* (mg/1)- 1

5.0 days" 2

Slope

Average

10

0.05

2.0 (mg/1 per day)" 1 

2.5 (mg/1 per day)" 1 

0.035 (mg/1 per day)" 1

0.20 (mg/1)" 1

5x10"' (mg/1)" 1

40 days" 2

Maximum

80

0.50

3.0 (mg/1 per day)" 1 

6.9 (mg/1 per day)- 1 

0.08 (mg/1 per day)- 1

0.30 (mg/1)- 1

2x10"" (mg/1)- 1

150 days- 2

TABLE 4. Errors in the calculated reaeration coefficient 

resulting from errors in parameter measurement, DO balance technique

Error in calculated klt in days- 1

Minimum slope Average slope Maximum slope

*,

*3

oa

0b

i-a

P

m

t

RMS

0.08

.001

.14

.16

.04

.22 5xlO" 4

.01

Maximum

0.48

.005

1.4

1.6

.20

1.1 2xlO"3

.025

RMS

0.50

.005

.20

.25

.07

.40

2.5 xlO" 3

.08

Maximum

3.0

.025

2.0

2.5

.35

2.0

.01

.20

RMS

4.0

.05

.30

.69

.16

.60

.01

.30

Maximum

24

.25

3.0

6.9

.80

3.0

.04

.75

Figures 5 and 8 show the sensitivity of the calculated kz to the upstream and downstream DO 

deficits, respectively. Because the lines slope in 

opposite directions, the effect of systematic errors 

in the procedure used to determine the DO should 

tend to cancel. This was verified with calculations 

based on BPS 14. The reaeration coefficient was 

calculated from equations 148 and 149 using a 

systematic error in the upstream and downstream 

DO deficits, and the results are presented in figure 

9. The abscissa is in terms of the systematic error 

in DO and is expressed as a percentage of the upstream deficit or a percentage of the deficit range. 

It is apparent from comparison of figures 9 and 5 

or figures 9 and 8 that the effect of the systematic 

error is much less severe than the effect of a 

random error of the same magnitude in one of the 

DO deficits. The use of an incorrect value of the 

saturation concentration, Cs, in the computation 

of the DO deficits will have an effect identical with 

a systematic error in the DO measurement 

procedure.

The effect on the calculated k2 of ignoring photosynthetic production is shown in figure 10. The 

downstream DO deficits for BPS 16,18, and 19 were 

calculated with the photosynthetic production
38 REAERATION IN OPEN-CHANNEL FLOW

LJ

D

2 160 

t

C/5

£150 

t

a 140

£L

O

00

LL. 

O

LJ 120 

0

PERCEN' o

- 100

H

Z 

LJ

g 90

LL. 

LJ 

O

O

z 80 

o

o: 7n

DISSOLVED-OXYGEN ERROR, IN PERCENTAGE 

OF DISSOLVED-OXYGEN DEFICIT RANGE 

-400 -300 -200 -100 0 +100 +200 +300 +400

1

-\

-

-

-

-

-

1

1 

BPS 14

\

\

\

\

1

1

\

1

1

\

>

1

1

x^^

1

1

X

1

1

1

1

_

-

-

-

-

I

-40 -30 -20 -10 0 +10 +20 +30 

DISSOLVED-OXYGEN ERROR, IN PERCENTAGE 

OF INITIAL DISSOLVED-OXYGEN DEFICIT

+40

FIGURE 9. Effect of a systematic error in DO measurement 

procedure on calculated reaeration coefficient, dissolvedoxygen balance technique.

500

5 10 15

PHOTOSYNTHETIC PRODUCTION RATE,

IN MILLIGRAMS PER LITER PER DAY

20

FIGURE 10. Effect of ignoring photosynthetic-oxygen production rate, dissolved-oxygen balance technique.

term in the equations. Then the k2 was calculated 

for the same conditions except that photosynthesis 

was ignored. The effect is very definite; for 

example, for a photosynthetic production rate of 

2 mg/1 per day, the calculated kz is 130 percent of 

the BPS value of 1.0 days- 1 . Thus, if a term for 

photosynthetic production is not included in the 

DO balance equation and this production is significant, the calculated k-2 will be significantly larger 

than the true value. The effect of photosynthesis 

was discussed previously by Camp (1965) and 

Lynch (1965).

The effect on the calculated reaeration coefficient of simultaneous errors in two or more of the 

parameters was investigated in this study to a 

limited extent, and it was found that the overall 

errors were in general not equal to the sums of 

the individual errors. Thus each individual error 

set must be investigated as an entity; however, 

this is beyond the scope of the present report.

From the results of the sensitivity analysis presented in figures 3 through 10 and in table 4, 

several general conclusions are possible. Because 

of the limited scope of the sensitivity analysis, 

estimates of the errors will be qualitative only. It 

is expected in the calculation of the reaeration 

coefficient from equation 119 that:

1. If the BOD levels are small, and if attached or 

floating plants are present in significant quantities, then neglect of the photosynthetic production is probably the largest source of error in the 

calculation of the reaeration coefficient.

2. If the BOD levels are small, and if errors in 

the upstream and downstream DO deficit are independent, then these errors are probably the second 

largest source of error in the calculation of the 

reaeration coefficient.

3. For small to moderate upstream BOD levels, 

the errors in the calculated £2 resulting from 

errors in kt and La are generally small but significant for £2 ^1.0 days- 1 . For an upstream BOD of 

greater than about 50 mg/1, the errors in the calculation of k2 resulting from errors in /c t are 

especially significant.

4. The other parameters should cause no significant errors in the calculated k2 unless ta^O.l 

days- 1 .

5. By summing the squares of the RMS errors 

for the average slope column of table 4 and taking 

the square root, an expected relative RMS error 

of 65 percent of a true k2 value of 1.0 days- 1 is 

obtained.

6. If errors in the upstream and downstream DO 

deficits are the result of a systematic error in the 

DO measurement procedure or of the use of an
EXPERIMENTAL DETERMINATION OF THE REAERATION COEFFICIENT 39

incorrect DO saturation value, then these errors 

compensate each other to some extent so that the 

error in the calculated reaeration coefficient is not 

as great as it would be if the error had occurred 

independently in Da or £>b.

7. The effects of simultaneous errors in two or 

more parameters are not linearly additive, so that 

the results of this study cannot be used to determine the effects of more than one parameter error 

on the calculated k2 values. Future studies should 

be directed to determining the sensitivity of the DO 

balance equation to errors in two or more parameters. The most ideal study would be to use Monte 

Carlo techniques, which would give the distribution function of the reaeration coefficient calculated from the DO balance equation for known 

distributions of the other parameters.

DISTURBED-EQUILIBRIUM TECHNIQUE

A rigorous statistical analysis of the disturbedequilibrium technique is presently impractical for 

the same reasons that such an analysis is impractical for the DO balance technique. However, a 

sensitivity analysis was performed, using the same 

approach as for the DO balance technique. The 

results of the sensitivity analysis and a comparison of the results of calculating reaeration coefficients from equations 139 and 140 are discussed 

in the following paragraphs.

The equations used in the sensitivity analysis 

were equation 140 written in the form

l ~ Da+ 2JUj'

(151)

and a Newton-Raphson iterative solution as given 

by equation 149. The EPS values studied are listed 

in table 5. These were chosen to correspond to the 

EPS used in the sensitivity analysis of the DO 

balance technique.

TABLE 5. Basic parameter sets for the sensitivity analysis 

of the disturbed-equilibrium technique

EPS ft, / Da Da P P. Db Db 

No. (days-') (days) (mg/1) (mg/1) p^m|;'y) p^ay) (m*ll) lm«M

1

2

3

4

5

6

7

8

9

10

11

12

13

1

1

1

1

.2

.2

10

10

1

1

1

1

1

0.05

.05

.05

.05

.05

.05

.05

.05

.05

.05

.05

.05

.05

3

3

3

3

3

3

3

3

3

3

3

3

3

5

5

8

8

5

5

5

5

5

5

5

8

8

0

-1

0

-1

0

-1

0

-1-1

-5

-10

-5

-10

0

0

0

0

0

0

0

0

0

0

0

0

0

2.73

2.73

2.73

2.73

2.99

2.99

.98

.98

2.73

2.73

2.73

2.73

2.73

4.51

4.46

7.18

7.14

4.94

4.89

1.62

1.59

4.46

4.27

4.04

6.95

6.71

The results of the sensitivity analyses of equation 151 are presented in figures 11 through 14, in 

which the ordinates are in terms of percentage of 

the EPS value of the reaeration coefficient, and 

the abscissas are in the dimensions of the variable 

being investigated. In general, the sensitivity increases as the deficit range decreases.

250

-8 -6 -4 -2 0 

PHOTOSYNTHETIC PRODUCTION RATE, 

IN MILLIGRAMS PER LITER PER DAY

FIGURE 11. Sensitivity of reaeration coefficient to photosynthetic-oxygen production rate, disturbed-equilibrium technique.

With the exception of the traveltime, two measurements of each parameter are necessary in the 

disturbed-equilibrium technique. Because of the 

additive nature of the variances, we have multiplied by 1.5 the estimated parameter errors presented previously, in table 2, and used in the 

sensitivity analysis of the DO balance technique. 

These estimates of the parameter errors and the 

slopes of sensitivity-analyses plots for the 

disturbed-equilibrium technique (figs. 11 through 

14) are presented in table 6. The possible errors in 

the calculated reaeration coefficient obtained from 

the different combinations of RMS and maximum errors and minimum, average, and maximum 

slopes are presented in table 7.
40 REAERATION IN OPEN-CHANNEL FLOW

1400

0

FIGURE

1200

1234

UPSTREAM DISSOLVED-OXYGEN DEFICIT,

IN MILLIGRAMS PER LITER

12. Sensitivity of reaeration coefficient to upstream 

DO deficit, disturbed-equilibrium technique.

0.10 0.15 

TRAVELTIME, IN DAYS

0.20 0.25

FIGURE 14. Sensitivity of reaeration coefficient to traveltime, disturbed-equilibrium technique.

FIGURE 13.- 

stream

123456 

DOWNSTREAM DISSOLVED-OXYGEN 

DEFICIT, IN MILLIGRAMS PER LITER

- Sensitivity of reaeration coefficient to downDO deficit, disturbed equilibrium technique.

RESPIRATION RATE DIFFERENCE, IN 

MILLIGRAMS PER LITER PER DAY

FIGURE 15. Effect on calculated reaeration coefficient of 

ignoring a respiration rate difference, disturbed-equilibrium 

technique.
EXPERIMENTAL DETERMINATION OF THE REAERATION COEFFICIENT 41 

TABLE 6. Assumed parameter errors and slopes of sensitivity-analysis plots for disturbed-equilibrium technique

Parameter -

Assumed error Slope

Root-mean-square Maximum Minimum Average Maximum

p 3.0 mg/1 per day 15 mg/1 per day

Da .15 mg/1 1.5 mg/1

Db .15 mg/1 1.5 mg/1

t .002 day .005 day

0.09

.1.5 (mg/1 per day)"1 

2.0 (mg/1 per day)"1 

2 days-2

0.20

2.5 (mg/1 per day)"1 

6.0 (mg/1 per day)"1 

20 days"2

0.40 (mg/1)-1 

4.0 (mg/1 per day) -1 

14 (mg/1 per day)"1 

200 days"2

TABLE 7. Errors in the calculated reaeration coefficient resulting from errors in parameter measurement, disturbedequilibrium technique

700

Error in calculated kt, in days" 1

Minimum slope

RMS

p 0.27

£> .22

Db .30

l .004

Maximum

1.35

2.25

3.0

.01

Average slope

RMS

0.60

.38

.90

.04

Maximum

3.0

3.8

9.0

.10

Maximum slope

RMS

1.2

.60

2.1

.40

Maximum

6.0

6.0

21

1.0

Figure 15 shows the effect of ignoring a difference in the respiration rates for the two levels of 

DO concentration of the disturbed-equilibrium 

technique. Reaeration coefficients were calculated from equation 151 for various respiration rate 

differences, and then the calculations were repeated with no respiration rate difference. It is 

apparent from figure 15 that the effect of ignoring 

the difference between respiration rates is definitely significant. For example, neglecting a respiration rate difference of 1.0 mg/1 per day causes 

about 12 percent error in the calculated reaeration 

coefficient, and the error increases rapidly as the 

difference in respiration rate increases.

Either equation 139 or 140 can be used to calculate k2 from measurements by the disturbedequilibrium technique when the respiration rates 

are different for the two levels of DO concentration. Equation 140 gives an integral reaeration 

coefficient, whereas equation 139 gives a reaeration coefficient based on the average DO deficit 

for the reach. Figure 16 shows the variation with 

respiration-rate ratio of /c2 as calculated from equation 139 for EPS 2, 4, and 13. A comparison of 

figures 15 and 16 shows that the sensitivity of the 

reaeration coefficient calculated from equation 

140 to the difference in respiration rates is considerably less than the sensitivity of the reaeration coefficient calculated from equation 139 to the 

respiration-rate ratio.

In general, the respiration-rate ratio for a 

disturbed-equilibrium-technique experiment 

should not deviate too much from a value of 1.0. 

From figure 7 of Edwards, Owens, and Gibbs (1961), 

it is estimated that R' = 0.85 for a DO concentration change from 2 to 4 mg/1, and that R' = 0.65 for

600

500

400

300

BPS 2

0.2 0.4 0.6 

RATIO OF RESPIRATION RATE

0.8

FIGURE 16. Variation of calculated reaeration coefficient 

with respiration-rate ratio, disturbed-equilibrium technique.

a change from 2 to 10 mg/1. From figure 16, however, it is apparent that equation 139 cannot give 

the correct reaeration coefficient of BPS 2, 4, and 

13 if fl>0.64. Therefore, because the BPS values 

were chosen as representative of conditions in 

natural streams, the use of equation 140 is 

preferred.
42 REAERATION IN OPEN-CHANNEL FLOW

.From the results of the sensitivity analysis presented in figures 11 through 16 and in table 7, 

several general conclusions are possible. These 

conclusions, as follows, are qualitative only, 

because of the limited scope of the sensitivity 

analysis.

1. Errors in all parameters of the disturbedequilibrium technique are significant in the calculation of the reaeration coefficient. The most 

significant is the error in the downstream DO 

deficit, followed by errors in the net photosynthetic production parameter, upstream DO deficit, 

and traveltime.

2. Neglecting a respiration-rate difference of 1.0 

mg/1 per day causes about 12 percent error in the 

calculated reaeration coefficient; therefore, 

neglecting the effect of DO concentration on the 

respiration rate may be significant.

3. The use of equation 140 for disturbed- equilibrium-technique calculations is preferred.

4. The square root of the sum of the squares of 

the RMS errors for the average-slope column of 

table 7 yields an expected relative RMS error of 

115 percent for a reaeration coefficient of 

1.0 days- 1 .

TRACER TECHNIQUE

Two major sources of error must be considered 

in an error analysis of the tracer technique. The 

first is the accuracy of the value of the ratio of the 

mass-transfer coefficient of the tracer gas to that 

of oxygen. The second is the accuracy of the procedures involved in measuring the mass-transfer 

coefficient of the tracer gas in an actual reaeration-coefficient determination.

The value of the mass-transfer coefficient ratio 

can cause errors in the calculated reaeration coefficient in two ways. First, there are errors inherent 

in the ratio because of errors in the laboratory 

procedures used to determine the ratio. Second, 

the conditions of turbulence and temperature and 

the presence of surfactants and pollutants in the 

measurement situation may be different from the 

conditions under which the ratio was determined, 

and this may result in a different ratio value.

The mass-transfer coefficient ratio for the 

krypton-oxygen gas pair was determined under a 

variety of laboratory conditions by Tsivoglou, 

O'Connell, Walter, Godsil, and Logsdon (1965) and 

by Tsivoglou (1967). The results of these tests are 

summarized in table 8. Also shown in table 8 are 

the maximum and minimum reaeration coefficients observed in the two series of tests. A statistical analysis showed that the mean values were

not significantly different at the 95-percent level 

of significance, and weighting the means in inverse 

proportion to their variances yielded a mean ratio 

of 0.83. The 95-percent confidence limits for the 

weighted mean were ±0.034, and the most probable 

error was 0.011, or about 1.3 percent of the mean 

ratio. Thus, the value of 0.83 may be regarded as 

an accurate estimate of the mass-transfer coefficient ratio for the krypton-oxygen gas pair.

TABLE 8i. Statistical data concerning measured gas-transfer 

coefficient

Transfer

coefficient Mean Minimum Maximum Median 

ratio

Range of K2 (days-')

Minimum Maximum

~i<7 0.82 

.846

0.76 

.789

0.91 

.909

0.81 

.846

0.04 

.045

1.5 

4.4

12.5 

68.6

fTsivoglou, O'Connell, Walter, Godsil, and Logsdon (1965). 

JTsivoglou (1967).

The effect of different conditions on the ratio 

was investigated extensively by Tsivoglou, 

O'Connell, Walter, Godsil, and Logsdon (1965) and 

Tsivoglou (1967). They measured mass-transfer 

coefficient ratios in two different experimental 

systems for seven different gases for transfer 

both into and out of the water. They also used a 

range of temperature and turbulence conditions 

that produced reaeration coefficients comparable 

to the upper range of reaeration coefficients 

observed in natural streams. They concluded that 

the mass-transfer coefficient ratio was not influenced significantly by (1) temperature in the 

range from 10° to 30°C, (2) the degree of turbulent 

mixing over a wide range of mixing conditions, 

(3) the presence or absence of a broken water surface, and (4) the direction of gas transfer.

In addition to this experimental evidence, some 

of the conceptual models discussed previously and 

a theoretical argument presented by Tsivoglou 

(1967) suggest that the transfer coefficient ratio 

is equal to the ratio of molecular diffusivities. 

Thus, this observation also suggests that the 

transfer coefficient ratio should be a constant, if 

differences in the temperature dependences of the 

diffusivities are neglected.

The accuracy of the procedure used to measure 

the mass-transfer coefficient of the tracer gas in 

an actual reaeration-coefficient determination 

depends on the accuracy of the nuclear counting 

techniques, the measurement of the flow time, and 

sampling errors. The counting techniques used by 

Tsivoglou (1967) involved splitting each sample 

into three parts and determining the number of
EXPERIMENTAL DETERMINATION OF THE REAERATION COEFFICIENT 43

counts of each sample for at least two 20-minute 

periods. Tsivoglou (1967) reported, 

This procedure, involving considerable replication and long 

total counts, provided excellent counting statistics, and a high 

level of accuracy for river samples whose count rates might be 

in the neighborhood of only twice the background count rate.

Flow time in the tracer technique is determined 

from measurements of the concentrations of the 

water-soluble fluorescent dye and, hence, should 

be very accurate. Furthermore, the flow time determined by this procedure should be identical 

with the flow time for the conservative liquid and 

gas tracers. Errors resulting from sampling should 

be small because the only quantity determined 

from the samples is the ratio of the concentrations 

of the gas and liquid tracers. Care must be exercised, however, to insure that the gas tracer is not 

lost from the samples during handling and 

counting.

The overall accuracy of the tracer technique is 

difficult to estimate. A qualitative indication of 

the reproducibility of k2 values determined from 

the tracer technique can be obtained from the work 

of Tsivoglou and others (1967, 1968). This work 

included two or three repetitions of k2 measurements for 14 river reaches. The average deviation 

from the mean for nine reaches was less than ±10 

percent; the average deviation was less than ±6 

percent for six of the reaches; and the maximum 

deviation for any reach was±18.2 percent. Because 

two days generally elapsed between measurements 

on a specific reach, some of the variation could 

have been the result of actual changes in the 

reaeration coefficient. Thus, on the basis of these 

limited data, errors in reaeration coefficients 

determined by the tracer technique may be of the 

order of ±15 percent of the mean.

LABORATORY MEASUREMENTS

In the laboratory, most of the sources and sinks 

in the DO balance can be eliminated, so that the 

complete DO balance, equation 119, reduces to the 

simple equation 133 for a laboratory flume. Hence, 

laboratory flume measurements of the reaeration 

coefficient should provide a limit on the reproducibility that can be expected from the DO 

balance technique in general. Unfortunately, the 

literature does not contain much information concerning reproducibility of reaeration-coefficient 

measurements in flumes. It does contain, however, 

some information for mixing-tank studies, an 

experimental situation considerably simpler than 

the flume situation because only one plot of log D 

versus time is involved. Dobbins (1964a) reported

measurements on three different gases at constant 

turbulence conditions in a mixing tank; in two 

cases, four repetitions were made, and the resulting variances were 3.6 percent and 8.6 percent of 

the mean. In one case, involving seven repetitions, 

the variance was 8.5 percent of the mean. Isaacs 

and Gaudy (1968) reported measurements in a 

"simulated" stream which in essence was a mixing 

tank with moving walls, used in an attempt to duplicate velocity conditions in streams. They reported 

two situations where five repetitions were made in 

which the variance was 6.3 and 6.9 percent of the 

mean. In five situations where three repetitions 

were used, the variance was from 1.6 to 4.1 percent 

of the calculated k2 value. Dobbins' k2 determinations were based on measurements of the volume 

of gas absorbed, whereas Isaacs and Gaudy's were 

made using the azide modification of the Winkler 

test. If one compares the average variance of about 

5 percent of Isaacs and Gaudy's measurements 

with the 0.1 percent of Cs reproducibility expected 

in the DO measurements with the Winkler method, 

the comparison indicates that as the complexity of 

the measurement procedure increases, the error 

increases considerably, so that the variance of 

flume measurements would be expected to be even 

larger than 5 percent. Furthermore, the comparison suggests that the accuracy of the DO measurement procedure should not be considered as 

indicative of the accuracy of the complete determination procedure for the reaeration coefficient.

The difference between the reproducibility of 

the Winkler method for DO determination and the 

expected reproducibility of flume measurements 

shows that other factors are important in the 

flume measurement procedures. These factors 

may be errors in reproducing or measuring the 

hydraulic conditions describing the flow. In addition, another factor not previously considered by 

researchers is the possibility of a corrosion reaction in the flume system.

Many laboratory flumes are constructed from 

metals, and because metals corrode or react with 

oxygen to varying degrees, a DO sink may be 

present that the researcher has not considered. 

The reaeration coefficient calculated by the procedure illustrated in figure 2 may be incorrect 

when a corrosion reaction is occurring in the 

channel. Furthermore, the magnitude of the error 

has been shown to depend on the relative magnitudes of the reaeration coefficient and the rate 

constant for the corrosion reaction and also on the 

initial DO deficit. (See appendix A.)

Therefore, the first step of any laboratory study 

of the reaeration coefficient should probably be
44 REAERATION IN OPEN-CHANNEL FLOW

the determination of the significance of possible 

corrosion reactions together with a determination 

of the reproducibility of the experimental measurements. The reproducibility of the measurement 

procedure can be determined by replication of the 

experimental runs, and the significance of possible 

corrosion reactions can be determined by the procedures outlined in appendix A.

The presence of an oxygen-consuming reaction 

in the laboratory equipment shows one way in 

which systematic errors can be introduced into 

flume measurements of the reaeration coefficient. 

This might explain one of the puzzling facts 

present in the literature on reaeration in flumes. 

This puzzle is the inconsistency between prediction equations obtained from data taken in two 

different but supposedly hydraulically equivalent 

flumes. The work of Krenkel (1960) and Thackston 

(1966) illustrates this inconsistency. The flumes 

used by these investigators were not significantly 

different hydraulically, except that Krenkel's was 

1.0 foot wide and Thackston's was 2.0 feet wide. 

The results, however, obtained from their studies 

over similar ranges of hydraulic variables may be 

given as (Thackston, 1966):

(Krenkel) &2 =0.000469^ (152)

H

and

(Thackston) k2 = 0.000215^-. (153)

H

The constants in the equations differ by a factor 

of 2.2.

The reasons for the discrepancy between these 

equations is not known. However, it might be due 

to the systematic error introduced by an oxygenconsuming reaction proceeding in the channel of 

one or the other of the experimental flumes.

Another possibility for the discrepancy is the 

failure to include some pertinent hydraulic variable in the equation for example, the width-depth 

ratio to account for the different widths. However, 

Thackston and Krenkel (1969a) wrote: "It is possible that some sort of wall effect is involved, inasmuch as the values of the width-to-depth ratio 

were essentially the same in both studies***." 

Additional experimental work will be necessary to 

determine the effect of the width-depth ratio on 

the reaeration coefficient.

PREDICTION OF THE REAERATION 

COEFFICIENT

None of the presently available models of the 

oxygen-absorption process in open-channel flows

are well enough developed or have been sufficiently verified so that they can be used to 

accurately predict reaeration coefficients from the 

mean hydraulic parameters. Because of this, one 

must rely for the prediction of reaeration coefficients on empirical regression equations based on 

experimental data. The mean parameters commonly available for use as the independent variables in such equations are width, depth, slope, 

and velocity. Another parameter which has been 

used as an independent variable in &2 prediction 

equations is the longitudinal-dispersion coefficient, Dx, but because of a scarcity of experimental 

Dx data, this parameter has not been used herein. 

The purpose of this section is to (1) present a 

summary of the known experimental data concerning the reaeration coefficient in open-channel 

flows, (2) evaluate the presently available equations for prediction of reaeration coefficients, and 

(3) present what we consider to be the most reasonable prediction equations that can be obtained 

by regression analysis from the available data.

REAERATION-COEFFICIENT DATA SETS

The reaeration-coefficient data sets available 

in the literature can be judged against two basic 

criteria. The first of these is the accuracy with 

which the basic data were measured and the carefulness with which the coefficients were calculated. The measurement techniques used must be 

considered, and if the DO balance technique or 

the disturbed-equilibrium technique is used, the 

effects of all neglected parameters must be 

evaluated.

The second criterion is the completeness of the 

data in the set. Specifically, this concerns the 

number of independent variables reported and the 

ranges of the independent and dependent 

variables.

From the points of view of scale and of datacollection methods, the reaeration-coefficient data 

fall into two general classes. These classes are data 

from laboratory flumes and data from natural 

streams. The flume data were in general collected 

at flow depths one to two orders of magnitude 

lower than the field-data flow depths, and at slopes 

one to two orders of magnitude higher. In terms of 

scaling, the data of Owens, Edwards, and Gibbs 

(1964) to some extent bridge the gap between the 

rest of the field data and the flume data.

FIELD DATA

The best available data were presented by 

Churchill, Elmore, and Buckingham (1962). The
PREDICTION OF THE REAERATION COEFFICIENT 45

DO balance technique was used, for the most part 

under such idealized conditions, and with such 

attention to detail that a high degree of accuracy 

is expected in the reaeration coefficients. These 

measurements were made in river reaches downstream from deep storage impoundments. A DO 

deficit was present because the water released was 

low in DO concentration as a result of prolonged 

storage under thermally stratified conditions. For 

the reservoirs studied, the BOD of the water was 

satisfied during storage, so that it was zero at 

release time.

Over the study period on each reach, the stream 

discharge was held constant by keeping the discharge constant at the upstream hydroelectric 

plant. The discharge was determined at local 

stream gaging stations, and the mean velocity was 

determined by dividing the discharge by the average cross-sectional area. The flow time was determined by dividing the length of the test reach by 

the mean velocity. Because the test reaches were 

carefully selected for uniformity, and because 

cross sections were measured at intervals roughly 

equivalent to the stream width, the accuracy of 

this method of determining the flow time is 

probably consistent with the accuracy of the other 

steps in the DO balance procedure used for the 

calculation of k2 .

The Winkler method, with a special end-point 

determination procedure which produced a standard deviation of only 0.007 mg/1, was used to 

determine the DO concentrations. The measured 

DO concentrations showed no variation in the vertical, but for many cross sections, the lateral DO

variation exceeded the variation in DO between 

the ends of the test reach. This necessitated taking 

DO samples at several points in the cross section 

and using the average of the measurements as the 

cross section DO concentration. For most cross 

sections, seven samples were used.

For all but seven of the 30 reaches studied, 

Churchill, Elmore, and Buckingham (1962) believed 

that except for k2 , all of the parameters in the DO 

balance equation were negligible. For the seven 

reaches they felt that correction was necessary 

for the photosynthesis and respiration of attached 

plants. The corrections were made using laboratory 

measurements of the photosynthesis and respiration rates of plants attached to rocks removed 

from the river bottom. For reasons discussed previously, and as pointed out by Hull and DeFilippi 

(1963), this type of correction is dubious, so that 

these seven values are less reliable than the other 

values reported. However, the problem was recognized, and an attempt was made to compensate for 

it. In addition, the corrections were for less than 

25 percent of the calculated coefficients, which, as 

will be shown later, is of the order of magnitude 

of the scatter around the best-fit regression equations obtainable from the field data.

The data obtained by Churchill, Elmore, and 

Buckingham (1962) are listed in table 18 of appendix B. In terms of the commonly measured mean 

hydraulic parameters, this data set is considered 

complete, because all the variables U, H, S, and W 

are reported. The ranges of the measured variables are listed in table 9. These data are representative of moderate-size streams.

TABLE 9. Ranges of experimental data from the literature

Reference

Churchill and 

others (1962)...............

Owens and 

others (1964)...............

Gameson and

O'Connor and

Tsivoglou and

Negulescu and

*2 (days- 1 )

Maximum

5.558

49.17

57.7

.92

3.373

52.50

115.3

18.70

Minimum

0.225 

.31 

34.3 

.14

.646 

5.78 

10.63

8.64

H

(feet)

Maximum

11.41 

2.44 

.42 

24.2

3.11 

.232 

.200

.492

Minimum

2.12 

.39 

.34 

4.0

1.16 

.052 

.080

.164

U 

(fps)

Maximum

5.00 

1.83 

.73 

4.20

.636 

2.320 

2.140

1.902

Minimum

1.85 

.13 

1.12 

.19

.286 

.365 

.243

.656

W 5x10" 

(feet) (ft per ft)

Maximum

667 

39.8 

5.7

110 

2.0 

1.0

.66

Minimum Maximum

90 23.514 

11.0 106.0

1.97

58 

201.2 

239.9

Minimum

1.2571 

1.56

.68

6.5' 

7.5

tStreeter and Phelps (1925) data only.

The reaeration-coefficient study presented 

by Owens, Edwards, and Gibbs (1964) was performed under conditions much less ideal than 

those prevailing for the study by Churchill, 

Elmore, and Buckingham (1962). In 11 of the 32 

cases presented, Owens, Edwards, and Gibbs noted

that either considerable weed growth or mild 

organic pollution was present. However, they did 

not state whether phytoplankton were present. 

The presence of plants is an important factor, 

because the disturbed-equilibrium technique was 

used to calculate the reaeration coefficients, and
46 REAERATION IN OPEN-CHANNEL FLOW

different DO consumption rates of the plants at 

the two DO levels used in this test could have 

significantly influenced the calculated reaeration 

coefficients.

In the study by Owens, Edwards, and Gibbs 

(1964), the discharge was determined by a dilution 

technique. The flow time was determined from the 

time of passage of the centroid of the tracer pulse 

through the reach. The reliability of the reported 

depths cannot be determined because the frequency of cross-sectioning was not reported. DO 

measurements were made using the Alsterberg 

modification of the Winkler method.

The paper by Owens, Edwards, and Gibbs (1964) 

contains two sets of data. The first set consists of 

the 32 kz values determined by these workers. 

Eleven of the k2 values were corrected for oxygen 

consumption by the bottom mud or for oxygen consumption by attached plants. This data set does 

not include the stream width, but this parameter 

can be calculated. The second data set, consisting 

of six kz values, was taken from Gameson, Truesdale, and Downing (1955). In this data set, neither 

the width nor slope values are given, but width 

values can be calculated. The two data sets are 

given in tables 19 and 20 in appendix B, and the 

ranges of the variables are given in table 9. The 

hydraulic parameters would be considered typical 

for steep, shallow streams.

A summary of the pre-1958 field data was presented by O'Connor and Dobbins (1958). The data 

consist for the most part of those of Streeter and 

Phelps (1925), screened by O'Connor and Dobbins 

(1958) to minimize the effects of algae and sludge 

deposits and modified so that the reported depth 

values approximate the true stream depth rather 

than the distance above mean low water.

The modified Streeter and Phelps (1925) data 

must be viewed with some caution, because they 

were obtained from a river containing organic 

pollution and probably phytoplankton, and kz 

values were computed from the Streeter-Phelps 

form of the DO balance equation, which ignores all 

but the suspended and dissolved BOD oxidation 

and the surface reaeration. In many of the test 

reaches the other terms in the DO balance equation probably would have influenced the calculated &2 values, had they been available for use in 

a more general form of the DO balance equation.

The data of Streeter and Phelps (1925) as modified by O'Connor and Dobbins (1958) are reproduced 

in table 21, appendix B. The ranges of the variables are given in table 9. The data set is incomplete

in that width values are not given. The StreeterPhelps data typify moderate-size rivers on gentle 

slopes. Table 22 of appendix B lists data collected 

by O'Connor and Dobbins from sources other than 

Streeter and Phelps. Stream width is not included 

in this data set, nor in three cases is stream slope. 

Because the sources of the individual data points 

vary, the ranges of the variables in this data set 

have not been included in table 9.

The only reaeration-coefficient data for streams 

that have been determined using the tracer technique are those given by Tsivoglou and others 

(1967, 1968). The data are rather limited in scope 

because the prime purpose of the study was not k-2 

determination, but verification of the measurement technique itself.

In the study by Tsivoglou and others (1967,1968), 

the traveltime was determined from the time of 

passage of the dye peak between the sampling 

stations. The stream was cross-sectioned every 

1000 feet, but as Tsivoglou stated (1967, p. 69), 

"The stream was characterized by a series of riffles 

and pools and an associated considerable range of 

velocities and depths," so the average depth measurements are questionable. The nuclear counting 

techniques used were highly reproducible, and 

sufficient replications were used so that even at 

low krypton concentrations the tracer-gas to 

liquid-concentration ratio should have been very 

accurate. The tracers were introduced at the 

centerline of the injection cross-section and 

sampled from the flow at the centerline of each 

sampling cross section. In light of the results of 

the study by Churchill, Elmore, and Buckingham 

(1962), this sampling procedure would seem to be 

questionable if the k? values obtained are to be 

representative of the entire cross section. Despite 

the questionable sampling technique, the reproducibility of calculated k-2 values was good, with a 

maximum deviation from the mean of 18 percent 

and with nine of 14 values having an average 

deviation from the mean of less than 10 percent.

The experimental data obtained by Tsivoglou 

and others (1967, 1968) are listed in table 23, 

appendix B. The channel slope is not given, and the 

width value was obtained by the present authors 

by dividing the depth-velocity product into 82.5 

cf s, the average discharge for the time period. The 

ranges of the individual variables are listed in 

table 9.

FLUME DATA

In all the laboratory studies reported herein, 

the flumes used were the recirculating type, and
PREDICTION OF THE REAERATION COEFFICIENT 47

the water in the flume was deaerated with sodium 

sulfite before the start of the experimental 

measurements.

Thackston (1966) used a flume 60 feet long and 2 

feet wide, and three different bottom roughnesses. 

All DO measurements were made using the Azide 

Winkler method, and the traveltime measurements 

were made using dissolved sodium chloride as a 

tracer. The sampling stations in the flume were 42 

feet apart; at each station, 10 samples were taken 

at 2-minute intervals. With at least two of the 

bottom roughnesses used, an oxygen-consuming 

reaction may have influenced the calculated 

reaeration coefficients. See appendix A for a discussion of this problem.

The data of Thackston (1966), also reported by 

Thackston and Krenkel (1969a), are given in table 

24, appendix B. The data set is complete, in that all 

the mean hydraulic parameters commonly measured are available. The ranges in the values of 

the variables are listed in table 9.

The flume used by Krenkel (1960) was 60 feet 

long and 1 foot wide, and the bottom roughness 

consisted of expanded metal mesh. In terms of 

experimental methods and techniques, and in 

terms of quality of measurements, the study was 

comparable to that by Thackston (1966). Again, 

oxidation reactions within the flume channel may 

have influenced the calculated reaeration coefficients. See appendix A.

The data of Krenkel (1960), which were also used 

by Krenkel and Orlob (1963), are given in table 25, 

appendix B. Again, all the commonly measured 

mean hydraulic parameters are listed, so the data 

set is considered to be complete. The ranges in 

values of the variables are reported in table 9.

The flume used by Negulescu and Rojanski 

(1969) was 66 feet long and 0.66 foot wide and had 

a 40-foot test section. The bed roughness consisted 

of 1-mm sand grains cemented to the flume floor. 

The DO measurements were made using the 

Winkler method. A 20-minute measuring period 

was used, and about 10 samples were collected at 

each sampling station.

Negulescu and Rojanski (1969) tabulated only 

eight of the 18 values they determined. These are 

listed in table 26, appendix B. Values for slope and 

temperature were not included in the original 

reference, so a temperature of 20°C was assumed. 

The ranges in values of the variables are listed 

in table 9.

EVALUATION OF REAERATION-COEFFICIENT PREDICTION 

EQUATIONS

The conceptual models discussed earlier have, in 

general, not been carried to the point where they 

can be used by themselves to predict reaeration 

coefficients from the mean stream parameters of 

width, depth, slope, and velocity. The most useful 

reaeration coefficient prediction equations are, 

therefore, those obtained from the semiempirical 

models. These equations have coefficients derived 

in some fashion from field data, or derived entirely 

by least-squares regression analysis from field 

data. Of course, such prediction equations should 

be used with caution in situations for which the 

mean stream parameters are outside the range of 

data from which the prediction equations were 

derived.

In this section, we will, for the prediction equations available in the literature, evaluate their 

performance over the entire range of available 

data. We will also assess on a common basis the 

performance of the various equations when 

applied to the data sets from which the equations 

were derived.

The basis chosen for comparison of the performance of the various prediction equations is the 

standard error of estimate. For a prediction equation yielding an estimated (k^)e for which the calculated reaeration coefficient is (k2\, the standard 

error of estimate, Es, is

 ± n '-, (154)

where n is the number of kz values used. Another 

definition of the standard error of estimate which 

is convenient to use in conjunction with leastsquares prediction equations derived from the 

logarithms of k-2 is

~ n ' < 155> 

The term ESL can be expressed as a percent error, 

EP, defined as

where EP is the percent standard error of estimate 

in terms of (A:2 )e . The term EP will always be between 

0 and 100 percent. When compared with Es as a percent, it is a conservative estimate of the percent 

error of the prediction equation. It must be emphasized that EP is not equal to the percent standard 

error of estimate that would be calculated from 

the regression of k2 (rather than the transformed 

data log kz).
48 REAERATION IN OPEN-CHANNEL FLOW

In table 10 are reported the quantities Es 

and EP for 13 of the equations discussed previously. 

For each of the equations, Es and EP have been 

computed first, for the data sets used by the 

original authors, and second, for all of the data 

sets discussed in the previous section which are 

sufficiently complete for use in the particular 

equation being evaluated. Table 10 also contains, 

for both groups of data, the number of data points 

and the ranges of the individual variables.

On the basis of the percent standard error of 

estimate, the equations which seem to fit best 

their original data sets are equation 96, with a 

standard error of estimate of 15 percent, and equation 101, with a standard error of estimate of 25 

percent. These equations were presented, respectively, by Krenkel (1960) and by Thackston (1966). 

The relatively good fit of these equations is not 

surprising, because they were obtained from 

laboratory studies in which the interfering factors and the hydraulic variables could be much 

better controlled than under the field conditions 

represented by most of the other equations.

Again on the basis of the percent standard error 

of estimate, the equations which best fit the entire 

range of available data are equation 61, with a 

standard error of estimate of 43 percent; equations 

101 and 102, with standard errors of estimate of 45 

percent, and equation 54, with a standard error of 

estimate of 46 percent. Equation 61, given by 

Dobbins (1964b), was derived from a consideration 

of two sets of field data and one set of flume data; 

equation 101, presented by Thackston (1966), was 

derived from flume data; equation 102, again from 

Thackston (1966), was derived from flume data, 

along with two sets of field data; equation 54, presented by O'Connor and Dobbins (1958), was 

derived without resorting to any experimental 

data. As might be expected, two of the best-fit 

equations were derived from data which covered 

essentially the same range as the data in 

appendix B.

In the 10th column under "Total applicable data" 

in table 10, it is seen that five of the equations, 

when applied to the entire available data range, 

have standard errors of estimate greater than 100 

per day. With the exception of equations 56 and 89, 

these five equations were derived from river data 

and give much too high k2 predictions at the shallow 

depths used in the flume studies. This fact illustrates the caution which must be used when applying kz prediction equations to situations in which 

the mean flow parameters are well outside the 

range of the data from which the prediction equations were obtained. Further caution is indicated

when it is realized that none of the values of Es 

listed in column 10 are less than 16 per day, a considerable magnitude when compared with the kz 

values of less than 5 per day usually determined 

for natural streams.

The reason for the large Es values given in the 

10th column under "Total applicable data" in table 

10 is further emphasized by reference to figure 17. 

In this figure, all the equations of table 10 have 

been plotted for an assumed slope of 0.0001 feet 

per foot and a mean velocity of 1.0 fps. The square 

ruled on the graph delimits the range of experimental data used. For the smaller depths at the 

extreme left edge of the depth range, -there is 

sometimes more than two orders of magnitude 

difference between a given pair of prediction equations. At the smaller depth values, the experimental data fall into a range representative of the 

bottom six prediction equations shown in the 

figure, so that large standard errors of estimate 

would be expected for the prediction equations 

which yield large k-z values at shallow depths.

In general, two different sets of equations are 

apparent in figure 17: those derived from flume 

data, which have slopes on the order of -1/2, and 

those derived from field data, which have slopes 

on the order of -3/2. It becomes apparent that 

there is a basic difference between the data sets 

obtained in the two different situations. Because 

of this, then the use of equations derived from 

flume data for the prediction of reaeration coefficients in natural streams is dubious. This matter 

is discussed further in the following sections.

DEVELOPMENT OF A REAERATION-COEFFICIENT 

PREDICTION EQUATION

As discussed previously, none of the available 

prediction equations can be considered applicable 

to the entire range of available data. Assuming the 

data to be bias-free, there are two possible reasons 

for this. First, none of the available equations were 

derived from a wide enough data range, and 

second, and more likely, some basic difference 

exists between the two data sets which precludes 

writing an equation which is applicable to the 

entire range of data. In this section, these alternatives are investigated using regression-analysis 

procedures. The data used are those discussed 

previously. The criterion for judging the best equation to use is the standard error of estimate of the 

prediction equation as obtained from the 62 field 

and (or) 110 flume data sets which contain all the 

variables (/, //, W, and 5.
PREDICTION OF THE REAERATION COEFFICIENT 49

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DEPTH, IN FEET

FIGURE 17. Reaeration coefficient as a function of depth, from prediction equations in table 10.

100
PREDICTION OF THE REAERATION COEFFICIENT 51

REGRESSION ANALYSIS

In general, regression analysis consists of computing the coefficients for an assumed equation 

form so that the sum of the squares of the departures of the experimental data points from the 

curve given by the equation is a minimum. For 

example, if for the data set Yi = G (Xu, X2i , ..., Xn i), 

the linear model

Y A 4- A. Y -4- A Y -4- 4- A Y (~\ x.H\ I ei Sloislis\.iiTSi2-A-2ii''*iStn-A-ni \±O I)

is used to obtain estimates Ye of the dependent 

variable Y, then the ,4,, /=0,... n, are chosen so that 

the quantity

A"

N' F<2 T [Y Y-1 2 CITS'1 j\ L-iS ^w L * ft * ll V-L'-'Oy

i = 1

is minimized. In equation 158, N' is the number of 

data sets used.

The best measure of the goodness of fit of a 

regression equation to a particular data group is 

the standard error of estimate Es, as defined by 

equation 154. A parameter which can be used to 

determine the relative significance of a particular 

variable to a given regression equation is the 

partial correlation coefficient. The partial correlation coefficient is the correlation coefficient 

between the dependent variable and the chosen 

independent variable, all other independent variables being held constant. Its value ranges 

between 1 and +1, and the larger its magnitude, 

the more significant is the chosen variable in the 

regression equation. The standard error of estimate and the partial correlation coefficient are 

discussed at much greater length in standard 

statistical-methods texts such as that by Snedecor 

and Cochran (1967).

The significance of a particular regression coefficient, At, may be tested using the fact that the 

distribution of the quantity

p _(Aj DJ) Ki~ ^ ^ (159) obi

is Student-r distribution with N'-n degrees of 

freedom. In equation 159, fi, is the population 

regression coefficient, and 5&i is the standard error 

of the regression coefficient At. (See Snedecor and 

Cochran (1967) for more detail.)

In conducting a regression analysis, the limitations of the method must be remembered. First, 

the model to which the data are fit is preselected 

so that there is freedom only to adjust the coefficients in the model. If the model is incomplete, a 

particular variable may appear to be of negligible 

significance, when in reality it may be of great 

significance. On the other hand, even though the 

model is incomplete, prediction equations might be

obtained which are entirely satisfactory within the 

range of data used in deriving them. These equations could, however, yield highly erroneous predictions if applied outside the data range. Second, 

even though the model is correct, the prediction 

equation will be incorrect if the data are incorrect 

or incomplete. Finally, in a regression analysis two 

types of significance must be considered: physical 

and statistical. For example, a variable may be 

highly significant in the physical process yet have 

little significance in the regression equation because the range of this particular variable was not 

wide enough in the data group chosen. Furthermore, it is possible to have a high statistical significance in the regression equation and yet to 

have assumed the wrong physical relation for that 

particular variable in the model. For these reasons, 

it is best before beginning a regression analysis 

to have a reliable physical model available. Unfortunately, as will be emphasized later, there are 

no such models presently available for the reaeration coefficient in terms of the mean values of the 

flow parameters which will accurately predict 

reaeration coefficients for the entire range of 

available data.

In this report, most of the regression analyses 

were performed using a general linear model and 

the logarithms of individual variables. The general 

model assumed was

(160)

(161)

In several cases, one or more of the At were 

assumed to be zero, and the effect on the standard 

error of estimate of the prediction equation was 

investigated. The coefficients were computed 

using the regression program BMD03R of the University of California, Los Angeles, Health Sciences 

Computing Facility (Dixon, 1965).

The results of a linear least-squares regression 

analysis on an equation of the form of 160 are 

generally slightly different from the results of a 

least-squares regression analysis of an equation 

of the form of equation 161, because the distribution of the variance of the logarithms of kz from 

equation 160 will be different from the distribution 

of the variance of kz from equation 161. If the 

analysis is conducted on an equation of the form 

of 160 and the result is converted to the form of 

equation 161, the average width of the scatter band 

about the prediction equation will increase directly 

in proportion with k-2 , whereas if the analysis is 

conducted directly using equation 161, the scatter

which is more commonly expressed as
52 REAERATION IN OPEN-CHANNEL FLOW

band width will be a constant, independent of k2 . 

In this study, it was decided to use an equation of 

the type of 160 and to accept deviation from the 

pure least-squares regression analysis procedures 

because of the relative simplicity of the analytical 

procedures involved in this type of analysis as compared with a least-squares analysis based on 

equation 161.

A prediction equation of the type of equation 

161 was chosen for comparison primarily because 

this type of k2 prediction equation has been used 

frequently in the past. Furthermore, most of the 

conceptual models available can be reduced, at 

least for natural streams, to this type of equation. 

A number of equations of different forms from 

equation 161 were tried with negligible success in 

reducing the standard error of estimate.

RESULTS OF REGRESSION ANALYSIS

The available experimental data may be divided 

into two general groups: first, those from natural 

streams, and second, those from laboratory flumes. 

Basic data groups used in the analysis were the 

data sets for which all of the variables k2 , H, U, W, 

and S were available. These consisted of the 62 

field data sets from Churchill, Elmore, and Buckingham (1962) and Owens, Edwards, and Gibbs 

(1964) and the 110 flume data sets from Krenkel 

(1960) and Thackston (1966). Regression analyses 

were performed separately and jointly on the two 

groups of data. Partial results of these regression 

analyses are listed in table 11.

Table 11 lists several comparable equations obtained from regression analyses of the field and 

flume data. The flow depth, H, and hydraulic 

radius, RH, were used in the field and flume cases, 

respectively, because in each case this particular 

variable resulted in a slightly lower standard 

error of estimate. To show the importance of each 

of the variables in the various prediction equations, the partial correlation coefficients are given 

in table 11. The standard error of estimate has been 

listed both in terms of logarithms and in terms of 

percentage of the predicted k2 value.

From an analysis of the results presented in 

table 11, it can be shown that all the prediction 

equations obtained from the flume data sets are 

significantly different statistically from the corresponding prediction equations obtained for the 

field data sets. ("Correspondence" means that the 

two equations contain the same independent 

variables.) For example, on comparison of the 

exponents in the first equation in column nine with 

those in the second equation in column one, for 

the variables (/, W, RH, and S, the Student-?

os

£ S3 2

rt rt O

IH IH oo

O O 

<N N

I I 1 5>s s,

O *ft IN en

--I 00 t- till |B 5,8 §

to «d

<n * o -H

CO
PREDICTION OF THE REAERATION COEFFICIENT 53

values (see eq 159) are -5.70, -11.88,10.42, and 7.18. 

For 106 degrees of freedom, these values indicate 

that the exponents in the two equations are 

different at a significance level less than 0.005. 

For the fifth equation in column nine and its

(

C0.624 \ 

k-i = 317.2 p n 75o)> the

Student-f values on S and RH are -0.69 and 3.56. 

Thus, the exponents on RH are significantly 

different at some level less than 0.005, while those 

on S are different at the 0.25 level. Similar results 

are obtained for the other equation forms listed. 

It must be concluded that all the kz prediction 

equations obtained from flume data are significantly different statistically from those obtained 

from field data.

The difference between the regression equations from the two groups of data is further 

illustrated in the standard error of estimate. For 

example, omitting W from the first equation of 

column nine, table 11, causes an increase in the 

standard error of estimate of 8.7 percent, but there 

is no noticeable effect when this parameter is 

omitted from the first equation in column one. 

Furthermore, the largest standard error of estimate occurs for the field equations when the mean 

velocity, U, is omitted, and for the flume equations when the slope, 5, is omitted.

Finally, the partial correlation coefficients 

listed in table 11 also indicate a difference between 

field and flume prediction equations. The field 

equations have partial correlation coefficients decreasing in the order of H, U, S, and W, with the 

coefficient for W negligible. On the other hand, the 

flume equations have partial correlation coefficients decreasing in the order of 5, W, RH, and U.

If a regression analysis is performed on all of the 

data used in preparing table 11, the resulting prediction equation is

TABLE 12. Values of Student-t for comparison of the exponents in equation 162 with those for the two corresponding field and flume equations of table 11

= 721

50.

f/0.064 (162)

which has a standard error of estimate of logarithms of kz of 0.15572, or 30.1 percent. Results of 

Student-f tests comparing the exponents of the 

variables in equation 162 with the exponents in 

the two corresponding equations of table 11 are 

given in table 12. From table 12, it must be concluded that equation 162 is significantly different 

at some level less than the 0.005 level from the 

corresponding flume and field A2 prediction equations of table 11. Not only are the flume and field 

prediction equations significantly different from 

each other, but also the equation using the combined data is different from both of them.

Variable

U 

W

RH

S

Student-t values

Field

 11.49 

- 3.04 

16.50 

12.00

Flume

- 4.06 

11.12 

4.58 

5.15

Discrepancies similar to those pointed out here 

between field and flume kz prediction equations 

have been observed in other areas of hydraulic 

engineering research. For example, in the field of 

alluvial boundary hydraulics, Maddock (1969, 

p. A67) concluded:

The variability of natural channels cannot be studied in flumes 

because of constraints placed upon the possible combinations 

of width, depth, velocity, slope, and load that are inherent in 

field operations.

And Colby (1964, p. A45) stated:

Flume investigations can provide much helpful information on 

sediment transportation; but, until scale effects are understood more completely, flume investigations of the discharge 

of sands are not model studies of the discharge of sands in field 

streams.

As another example of the apparent difference 

between field and flume phenomena, the present 

authors attempted to relate to the form of kz 

regression equations the form of the regression 

equations for the turbulent intensity at one-tenth 

the flow depth below the water surface. The 

turbulent intensity, u', is the root-mean-square of 

the deviations of the velocity fluctuations at a 

point around the mean value of the velocity at the 

point. It is equal to the square root of twice the 

kinetic energy per unit of mass of the fluid turbulence at the point, and thus the near-surface value 

of u' might reasonably be expected to be related to 

the reaeration coefficient.

The results of the regression analysis of data 

obtained from R. S. McQuivey (written commun., 

1970) are listed in table 13. A general model analogous to equation 161 was assumed for the turbulent intensity. Because simultaneous A2 and u' 

measurements are not available, no conclusions 

could be made about the functional relation, if any, 

between kz and u'.

On the other hand, in terms of the dependence of 

u on S and U, the results of this study are similar 

to the results of the reaeration study in that the 

flume prediction equations show the highest 

partial correlation with S whereas the field equations show the highest partial correlation with U. 

However, the results of this study differ from the 

results of the reaeration study in that the 

dependency of u' on W is significant both in the 

flume and in the field. The results of the Student-/
54 REAERATION IN OPEN-CHANNEL FLOW 

TABLE 13. Dependence of turbulent intensity on the mean hydraulic parameters

Equation

21 jpo.43» 50.399

//0. 396 50.50:1

f /0.143 IfO 798 eo.544 1 nnn <-> " j

0.890 //O 158 [70.422 50.313

23.750 //°-°» [/°-764 ff 0.75751.183 

0.121 //0.042 [71.050 50.090

Partial correlation coefficients

H

.779 

-.390

.219 

.234 

.176

U

O.Q85

.121

.315 

.718 

.871

W S

0.302 0.560 

.700 

.527 .590

.381 

.490 .516 

.271

ESL

0.115 

.117 

.146

.168 

.087 

.098

Number 

of data 

points

20

20 

27

27 

43 

43

Type of data

Flume, 

Do. 

Flume,

Do. 

Field. 

Do.

rigid, 

alluvial.

tests comparing the exponents on the variables 

in the two flume situations with those of the field 

situation of table 13 are listed in table 14. These 

Student-? values show that at least one exponent 

in each flume equation differs from the corresponding exponent in the field equation at significance levels less than 0.005. It must be concluded 

that the flume regression equations for the turbulent intensity at one-tenth the flow depth below 

the water surface are significantly different from 

the field equation.

TABLE 14. Results of the Student-t test of the exponents in 

the flume equations and in the field equations of table 13

Equation

Flume, rigid....... .......

Do.......................

Flume, alluvial.........

Do.......................

Student-t values

H

 0.30

.79

 2.13

.79

V

 1.75 

 2.33 

 2.50 

 2.33

W

-0.89 

.15

s

 5.16 

1.54 

 4.02 

1.54

As a final example of the difference between 

regression equations for flume and field data, consider equations for predicting mean velocity from 

W, //, and S obtained from the same data as the 

k2 prediction equations of table 11. These equations 

along with the partial correlation coefficients and 

the standard errors of estimate are listed in 

table 15.

TABLE 15. Prediction equations for mean velocity, obtained 

from the data used in table 11

Equation

[/ =

37.55 ffo.187^.168 50.742

98.80 /? > "" 5°- 4

29.82 fp0.33Sfl^.608 50 .423

25.20 R^> 54050.400

Partial 

correlation 

coefficients

W RH

0.256 0.676

.937

.340 .468

.409

S

0.861

.891

.755

.721

ESL Ep 

(percent)

0.166

.170

.138

.146

31.8

32.4

27.3

28.6

Type 

of

Field.

Do.

Flume.

Do.

The partial correlation coefficients and the small 

change in standard error of estimate on omitting 

W from the equations indicate considerably less 

dependency of U than kz on this parameter. The

partial correlation coefficients further indicate 

that U in the flume is less dependent on depth than 

in the field. The difference between the two situations is, however, best illustrated by the Student-? 

test. Comparing the exponents of the third equation with those of the first in column one of table 

15, the Student-? values for W. RH, and S are 1.65,

-5.02, and -9.94, whereas in the second and fourth 

equations, for RH and 5, the Student-? values are

-8.05 and -10.58. This indicates that the prediction equations for the two situations are different 

at some significance level less than 0.005.

Possible reasons for the difference between the 

regression equations for flume and field reaeration 

coefficient data are (1) differences in the basic 

physical processes, or misinterpretation of them, 

(2) statistical quirks, and (3) deficiencies or bias 

in the available data.

Possible physical differences between the two 

systems are easy to visualize. For example, the 

straight, uniform flume flow may have entirely 

different turbulent transfer properties than the 

tortuous, nonuniform field stream flows. R. S. 

McQuivey (written commun., 1970) indicated that 

the turbulence profile in a 200-foot-long by 8-footwide flume at depths of about 1 foot does not fully 

develop at any point in the flume. This indicates 

some question about the similarity between the 

turbulent transfer properties near the surface of 

field and flume flows. In addition, the high partial 

correlation coefficient between /c2 and W in the 

flume as compared with u and U indicates a 

possible secondary circulation effect on k2 which is 

not present in the other phenomena. If such an 

effect is present, simple WjH scaling may not 

account for it. As a further example, the comments 

of Haddock (1969) and Colby (1964) quoted previously could be applied equally as well to reaeration as to sediment transport. That is, the different 

constraints on the two physical systems might prevent similar mass-transfer regimes from developing at the water surface. Finally, the difference
PREDICTION OF THE REAERATION COEFFICIENT 55

between the two sets of equations may be due to a 

basic misunderstanding of the turbulent-masstransfer process near the surface that is, it may 

be due to a lack of knowledge of the pertinent 

scaling parameters, or to the fact that the correct 

parameters have not been measured.

Certain quirks are inherent in regression analysis which might cause field and flume regression 

equations for &2 to differ. In the first place, the 

differences between the two systems cause the 

experimental variables to be selected differently 

that is, in the flume, slope and discharge vary and 

determine velocity, whereas in the field, depth and 

discharge vary to determine velocity. This difference in selection of "independent" variables for 

analysis can be reflected in the regression equations, as can the differences in the adjustments of 

the so-called independent variables to each other. 

In the second place, if a wide enough range of 

data is not available for each of the independent 

variables, the significance of the variable may be 

missed in the regression analysis, or the variable 

may be represented incorrectly in the regression 

equation. In this sense, dependency of the flume 

equations on W is particularly questionable because only two values were available in the 

literature.

Secondly, if the form of the regression equation 

chosen is incorrect, the equations for the two systems of different scale might be expected to differ 

radically. This is, in a different sense, a statement 

of the scaling problem mentioned above.

Finally, no regression equation is better than the 

data used in its derivation, because if the data contain errors or bias, this will be reflected in the 

prediction equations. In the field data, especially, 

the accuracy of the data can be questioned. First,

because in none of the studies reported, except 

possibly that of Churchill, Elmore, and Buckingham (1962), have sedimentation, bottom deposits, 

and photosynthesis been accounted for adequately. 

Second, in the two sets of flume data, there is a 

possibility that one or both of the data sets were influenced by an oxygen-consuming reaction in the 

flume channel, and this would have a tendency to 

bias the resulting data. (See appendix A.)

In addition to the differences pointed out above 

between regression equations derived from different data sets, it can also be shown that the 

regression equations (listed in table 11) for the field 

data are significantly different from most 

comparable equations found in the literature.

The Student-r test was used to compare exponents on various variables of the equations in 

table 11 with the corresponding exponents from 

the equations in the literature, and the results are 

presented in table 16. The Student-f values show 

that all but the two equations of Owens, Edwards, 

and Gibbs (1964) have at least one exponent which 

differs from the exponent in the corresponding 

field equation at a level less than the 0.005 significance level. Because the second equation of Owens, 

Edwards, and Gibbs (1964) was derived, with the 

exception of the six data sets of Gameson, Truesdale, and Downing (1955), from the same data as 

the corresponding field regression equation, it is 

not surprising that these equations do not differ 

significantly.

The previous discussion suggests at least two 

reasons for pursuing the search for a new physical 

model describing reaeration. The first is the difference observed between the regression equations 

for the field and flume situations. It would seem 

that equations describing the physical process of

TABLE 16. Results of the Student-t test of exponents in the regression equations of this study and corresponding equations

in the literature

Regression equations Comparable equations 

(this study) in the literature

/yO.413 CO. 273

n in v" 6"

9 - 59 //1.665

o

24.55(t/S£)°-*>8 

5.026 {/»-«»//- '6

10.90 L"> -73//-1.75

9.41 (/°«//- 185

(

f A0.85 

^)

18.58 Vff(j^)"2

(Chezy C>17) 

73 (7-=20°C)

(7"=20°C) 

(Chezy C< 17)

Student-fo.oos

// t/

-3.89 0.04 

-5.60 2.98 

-2.95 -5.04

-.23 -.07 

-15.58 -3.01

-6.00 ........ 

-3.64 ........

5 (percent) Source

op K O'Connor and 

" ' Dobbins (1958).

Churchill and 

 " " others(1962).

Owens and

 " others (1964).

........ ........ Do.

Negulescu and 

 ' Rojanski (1969).

K qc qq Q O'Connor and 

Dobbins (1958).

1.92 ....... Thackston (1966).

{Equations by Fortescue and Pearson (1967) and Isaacs and Gaudy (1968) are also of this form.
56 REAERATION IN OPEN-CHANNEL FLOW

reaeration should be applicable no matter what 

the scale of the system. The second is the fact that 

none of the available theoretical models can be 

related to the regression equation for field data. 

This is because either the theoretical model is not 

expressed in terms of the mean hydraulic parameters, or because for the ones which are expressed 

in this fashion, it has been shown that the 

exponents in the field regression equation differ 

significantly from those in the theoretical models. 

Furthermore, the commonly measured mean hydraulic parameters U, H, W, and S may not be 

sufficient to model the phenomenon. Perhaps some 

other parameters must also be measured, such as 

those characteristic of the near-surface turbulence, or of the velocity in the air above the flow. 

Using all the available data and using the standard error of estimate of logarithms as the basis 

for judgment, one can see from table 11 that the 

best prediction equation for reaeration coefficients 

of natural streams is

r/0.413 C0.273

*2 = 46.05 ^Jog . (163)

However, considering the percentage variation 

(table 11), the equation

r/0.674

fe = 9-&977T56 < 164> 

has a percentage variation only 1.1 percent greater 

than that for equation 163. Considering the difficulty and expense of measuring accurately the 

slope of a field stream (Yao, 1969), an equation that 

does not contain the slope is preferred for routine 

use as a prediction equation for k2 in natural 

streams.

In appendix B, in addition to the field data used 

in deriving equation 164, there are other field data 

which report only U and //. From the 121 sets of 

field data which contain U and //, the least-squares 

regression equation for k2 is

r/0.607

^ re^ISSr (165) 

Equation 165 has a standard error of estimate of 

logs of 0.203 (37.5 percent). The Student-/ test was 

used to compare the exponents on U and //; comparing equation 165 with equation 164 (119 degrees 

of freedom) for U and //, the Student-/ values are 

-1.42 and 4.22, whereas comparing equation 164 

and 165 (60 degrees of freedom) gives Student-/ 

values of 1.14 and -2.70. Thus equations 164 and 

165 have one exponent which differs from its 

counterpart at a significance level less than 0.005. 

The standard error of estimate of equation 165 

has increased markedly over that of 164. This is 

probably due to a decrease in overall quality of the 

data. However, because equation 165 is based on

the widest range of field data presently available, 

this equation is probably the best available for 

prediction of reaeration coefficients for natural 

streams.

EFFECT OF DETERGENTS AND WIND ON THE 

REAERATION COEFFICIENT

Up to this point, the reaeration process has been 

considered as occurring under ideal conditions 

that is, in relatively pure water with no wind 

effects. The literature, however, provides evidence 

that the presence of sewage and surfactants in the 

water tends to reduce KL, the liquid-film coefficient 

for gas transfer, compared with the pure-water 

value, whereas winds tend to increase 'KL with 

respect to the value existing for quiescent air conditions. Recall that KL = K2H, where K* is the 

reaeration coefficient and H is the mean depth of 

flow. The results of some of these studies are summarized in the following paragraphs.

Mancy and Okun (1965) discussed two theories 

concerning the effect of surfactants on the rate 

of oxygen absorption. On the one hand, they cited 

four references suggesting that surfactants 

reduce the absorption rate only by decreasing the 

hydrodynamic activity at the interface. On the 

other hand, they cited six references suggesting 

that surfactants reduce the absorption rate both 

by increasing the diffusional resistance at the 

interface and by decreasing the hydrodynamic 

activity at the interface.

Eckenfelder, Raymond, and Lauria (1956) hypothesized that surfactants would reduce KL in 

two ways: first, by increasing the diffusional 

resistance through a reduction in the surface tension, and second, by decreasing the surface 

renewal rate through an increase in viscosity. 

Metzger (1968) studied the absorption of helium 

and nitrogen under identical mixing conditions in 

a stirred tank containing 0.5 mg/1 of ABS (alkyl 

benzene sulfonate), a surfactant. He found that 

the film thickness and surface-renewal parameters of the Dobbins (1956) film-penetration model 

were larger and smaller, respectively, than the 

values obtained for the same mixing conditions 

with distilled water. Downing, Melbourne, and 

Bruce (1957) studied the effects of household 

detergents and sewage on oxygen absorption in a 

stirred tank and in a wave tank. They found that 

mixed household detergents reduced KL from the 

clean-water value by various amounts depending 

on the concentration of detergent, the initial KL 

value, and the way in which the water was 

agitated. The effects of sewage effluents and 

settled sewage containing detergents were similar
EFFECT OF DETERGENTS AND WIND ON THE REAERATION COEFFICIENT 57

to the effects observed for mixed detergents, but 

the effects of sewage effluents and settled sewage 

were less when the detergents were not present. 

Gameson, Truesdale, and Varley (1956) studied 

the effect of about 1 mg/1 ABS on oxygen absorption by flowing water in a laboratory open channel. 

They found that the percentage reduction in KL 

depended on the degree of turbulence, expressed 

in terms of KL. The maximum reduction occurred 

for intermediate KL values, equivalent to those of 

lowland English streams. Lower and higher turbulence conditions resulted in smaller percentage 

reductions. Similarly, Metzger (1968) and Mancy 

and Okun (1965) found in stirred-tank studies that 

the largest reduction in KL occurred for intermediate mixing intensities.

Gameson, Truesdale, and Downing (1955) added 

detergents to a small English stream and found 

that there was no apparent relation between the 

reduction in KL and the detergent concentration, 

in the range from 0.5 to 2 mg/1. The reduction in 

KL decreased with increasing KL for values ranging 

from about 30 to 120 cm/hr (centimeters per hour). 

No maximum was observed in the natural streams, 

apparently because KL values were all larger than 

5 to 10 cm/hr, the level at which the maximum reduction occurred in the laboratory studies.

Ogden, Gibbs, and Gameson (1959) studied the 

effect of salinity and the combined effect of 

salinity, detergents, and sewage on the reaeration rate of flowing water in a flume. They found 

that the percentage reduction in KL was equal to 

six times the square root of the salinity expressed 

as grams of salt per 1,000 grams of mixture, but 

that the reductions caused by detergents and 

sewage did not appear to depend on the presence 

of salts in the water. Kehr (1938) studied the effect 

of sewage on reaeration in a laboratory flume and 

found that the percentage reduction increased 

with the percentage of sewage and the flow velocity; however, the maximum velocity considered 

was only about 0.5 fps.

Lynch and Sawyer (1954) studied the effect on 

oxygen absorption of 11 synthetic detergents available on .the market in 1954. They found with one 

exception that all reduced the oxygen-transfer 

coefficient, and they pointed out that because 

these detergents were not biodegradable, they 

eventually entered the streams and rivers, where 

they could significantly reduce the reaeration 

coefficients. However, since 1954, biodegradable 

detergents have been developed, and these should 

reduce the effect that detergents have on reaeration coefficients.

Downing and Truesdale (1955) studied the effect 

of oil films on oxygen absorption in a stirred tank. 

They found no significant effect until the film 

thickness exceeded 10~ 4 cm. Thus, they concluded 

that oil films were not a problem because films 

thicker than 10~ 4 cm were not likely to persist very 

long in natural streams.

Poon and Campbell (1967) found that the 

presence of small concentrations of biologically 

inert suspended particles increased the rate of 

oxygen absorption in an aerator. This finding is in 

agreement with the observation of R. S. McQuivey 

(oral commun., 1970) that suspended sediment increases the intensity of turbulence over that 

observed in a clear water flow.

Whereas surfactants and sewage generally reduce KL, the presence of winds and surface disturbances produced by the winds generally increase 

KL. Imhoff and Fair (1956, p. 291) discussed wind 

effects in general terms. They suggested that 

winds passing over the water surface may double 

the absorption rate, that wind-induced waves may 

increase the absorption rate 10 times, and that 

whitecaps may, by droplet aeration, increase it 

100 times.

Eloubaidy (1969) studied reaeration both with 

and without wind waves in a laboratory wind-water 

tunnel. He concluded that the absorption rate was 

significantly increased when waves developed on 

the water surface, and this increase was considerably more than could be accounted for by the increase in surface area. He presented a prediction 

equation containing the shear velocity, the 

kinematic viscosity, and the wind velocity. This 

equation, however, was verified only with a very 

limited amount of data.

Downing and Truesdale (1955) studied the effect 

of surface wind on oxygen absorption in a laboratory ripple tank. They found that there was little 

change in KL until a wind velocity between 3.0 and 

3.5 meters per second was reached. For larger 

velocities, KL increased rapidly. Similar results 

were obtained in tank experiments by Kanwishe; 

(1963), who found little change in the gas-exchange 

coefficient up to a velocity of about 3 meters per 

second but a coefficient that increased approximately as the square of the velocity for wind 

velocities between 3 and 10 meters per second. 

Downing and Truesdale (1955) also studied the 

effect of wave height and frequency on KL. They 

found that the rate of absorption of oxygen increased almost linearly with increase in wave 

height, increasing approximately fourfold for a 

fourfold increase in height. For irregular choppy
58 REAERATION IN OPEN-CHANNEL FLOW

waves of the same frequency, KL increased less 

rapidly. They also found that KL increased with an 

increase in the frequency of the waves, although 

there was more scatter in the results. Finally, 

Juliano (1969) found that absorption-rate constants 

measured in the Sacramento-San Joaquin estuary 

system varied directly with wind velocity. There 

was considerable scatter in the relation; however, 

this scatter was attributed to the fact that wind 

effects vary greatly from point to point in the 

reach of interest.

CONCLUSIONS AND RECOMMENDATIONS 

CONCLUSIONS

From the review of reaeration in open-channel 

flow presented in previous sections, we have 

arrived at several general conclusions.

1. The approximations that can be made in the 

general DO balance equation (see pages 22-29, 

33-39) vary radically from situation to situation. 

Therefore, careful consideration of all the circumstances must be made before any terms of the 

equation are disregarded.

2. The most general and realistic conceptual 

model for the reaeration process of open-channel 

flows is the surface-renewal-damped-eddy diffusivity model of King (1966). This model merits 

further development in relating its parameters to 

the mean flow parameters that are easily 

measured.

3. In theory, the tracer technique is far superior 

to the other two techniques commonly used for 

determining the reaeration coefficient of openchannel flows. In practice, however, the use of 

radioactive tracers involves problems from the 

standpoint of obtaining permission for use in 

natural streams, and from the standpoint of possible hazard to the personnel involved in the study.

4. Sensitivity analysis of the DO balance technique shows that if photosynthetic-oxygen production is present, its neglect is probably the most 

serious source of error in the calculation of the 

reaeration coefficient. Following this in decreasing order of effect are errors in the upstream and 

downstream DO deficit, ki, and La.

5. Sensitivity analysis of the disturbed-equilibrium technique shows that the most serious 

sources of error in the calculation of the reaeration 

coefficient are, in decreasing order of effect on the 

calculated k2, errors in the downstream DO deficit, 

net photosynthetic production, upstream DO 

deficit, and time of travel.

6. The expected root-mean-square error of 15 

percent for the tracer technique is much less than 

the 65-percent error expected from the DO balance 

technique and the 115-percent error expected from 

the disturbed-equilibrium technique. Thus, it 

would seem well worthwhile to expend further time 

and effort on developing the tracer technique for 

use in streams of all sizes.

7. The best available reaeration-coefficient data 

for natural streams is that of Churchill, Elmore, 

and Buckingham (1962). With these data and those 

of Owens, Edwards, and Gibbs (1964), 62 field data 

sets can be considered complete; that is, they contain all the mean hydraulic parameters commonly 

measured: velocity, depth, width, and slope.

8. Analysis of available reaeration-coefficient 

data from natural streams and from laboratory 

flumes indicates that there is a significant difference between the k2 regression equations obtained from the two groups of data. The most likely 

reasons for this are (a) an inability to scale correctly the turbulent mass-transfer process at the 

water surface, (b) statistical quirks involved in 

regression analysis using the complexly interdependent hydraulic parameters of mean velocity, 

depth, width, and slope, and (c) experimental bias 

in one or more of the individual groups of data.

9. For the prediction of k2 in natural streams, 

regression analysis indicates that the equation 

with the smallest standard error of estimate consistent with the ease of obtaining the mean flow

f/0.607

parameters is £2 = 8.76 -,1 689, where k-2 is in reciprocal days.

RECOMMENDATIONS

A more detailed and reliable theoretical model 

than is presently available for the transfer of 

oxygen through the surface of an open-channel 

flow is needed. The model should account for the 

effect of impurities in the flow such as sewage, 

sediment, dissolved solids, and surfactants. It 

should consider the effect of wind blowing over 

the water surface, and it should be related directly 

to hydraulic parameters such as mean velocity, 

depth, width, slope, and possibly near-surface vertical and (or) longitudinal turbulence. The model 

should, furthermore, scale the interface masstransfer phenomenon so that field and flume-scale 

data fit one general equation. This would make 

possible (with adequate independent verification 

of the equation) the use of experimental results 

from laboratory flumes to predict field-scale 

reaeration coefficients.
REFERENCES 59

For verification of the model in large-scale field 

streams in which some degree of pollution, photosynthesis, and other interferences may be present, 

or for obtaining more reliable k2 values than are 

presently available, the tracer technique should 

be modified to avoid the use of the radioactive 

tracers. This could be done, for example, by using 

the fluorescent dye as the conservative tracer and 

gas chromatography for determining the concentration of some inert tracer gas.

If the DO balance technique and the disturbedequilibrium technique are to be used for calculating reaeration coefficients, then conclusions 

4 and 5 above indicate that better methods must 

be found for accounting for photosyntheticoxygen production and for the respiration of the 

suspended and attached aquatic plants. In fact, all 

the source-sink terms should be more carefully 

considered in reaeration-coefficient studies than 

they are presently.

Other factors concerning DO in streams require 

immediate consideration. For example, consider 

the lateral distribution of DO in a stream cross 

section. The measurements by Churchill, Elmore, 

and Buckingham (1962) indicate very large DO 

variations in a stream cross section. The implications of this observation in terms of average DO 

transferred into the stream and in terms of the 

cross-section variation of the surface masstransfer coefficient are very important. A further 

field study in terms of lateral DO distribution 

might be a study of the effect of sewage outfall 

placement on dispersion, or concentration of pollution by the stream.

The probabilistic behavior of DO variations at a 

particular cross section also needs further study. 

Studies should be aimed at determining the hazard 

level associated with a particular pollution input 

at some point upstream of the cross section of 

interest.

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60 REAERATION IN OPEN-CHANNEL FLOW

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REFERENCES 61

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Thackston, E. L., and Krenkel, P. A., 1969a, Reaeration prediction in natural streams: Am. Soc. Civil Engineers 

Jour., v. 95, no. SA-1, p. 65-94.

_____1969b, Discussion of atmospheric oxygenation in a 

simulated stream: Am. Soc. Civil Engineers Jour., v. 95, 

no. SA-2, p. 354-356.

Thayer, R. P., and Krutchkoff, R. G., 1967, Stochastic model 

for BOD and DO in streams: Am. Soc. Civil Engineers 

Jour., v. 93, no. SA-3, p. 59-72.

Theriault, E. J., 1926, The rate of deoxygenation of polluted 

waters, in Stream pollution a symposium: Am. Soc. 

Civil Engineers Trans., v. 89, p. 1341-1350.

_____1927, The oxygen demand of polluted waters: Washington, U.S. Public Health Service, Public Health Bull. 

173.

Thomann, R. V., 1963, Mathematical model for dissolved oxygen: Am. Soc. Civil Engineers Jour., v. 89, no. SA-5, 

p. 1-30.

_____1965, Recent results from a mathematical model of 

water pollution control in the Delaware estuary: Water 

Resources Research, v. 1, no. 3, p. 349-359.

Thomas, H. A., Jr., 1937, The "slope" method of evaluating 

the constants of the first-stage biochemical oxygen-demand 

curve: Sewage Works Jour., v. 9, no. 3, p. 425-430.

Toor, H. L., and Marchello, J. M., 1958, Film penetration 

model for mass and heat transfer: Am. Inst. Chem. Engineers Jour., v. 4, no. 1, p. 97-102.

Truesdale, G. A., Downing, A. L., and Lowden, G. F., 1955, 

The solubility of oxygen in pure water and sea water: 

Jour. Appl. Chemistry, v. 5, p. 53-62.

Tsivoglou, E. C., 1967, Tracer measurement of stream reaeration: Federal Water Pollution Control Adm. Rept., 

June, 86 p.
REFERENCES 63

Tsivoglou, E. C., Cohen, J. B., Shearer, S. D., and Godsil, 

P. J., 1968, Field studies, part II of Tracer measurement 

of stream reaeration: Water Pollution Control Federation 

Jour., v. 40, no. 2, pt. 1, p. 285-305.

Tsivoglou, E. C., O'Connell, R. L., Walter, C. M., Godsil, P. J., 

and Logsdon, G. S., 1965, Laboratory studies, part I of 

Tracer measurements of atmospheric reaeration: Water 

Pollution Control Federation Jour., v. 37, no. 10, p. 1343- 

1362.

Velz, C. J., 1958, Significance of organic sludge deposits: The 

Robert A. Taft Sanitary Eng. Center Technology Rept. 

W58-2, p. 47-57.

Velz, C. J., and Gannon, J. J., 1963, Biological extraction and 

accumulation in stream self-purification: Internat. Jour. 

Air and Water Pollution, v. 7, p. 587-595.

Wezernak, C. T., and Gannon, J. J., 1968, Evaluation of

nitrification in streams: Am. Soc. Civil Engineers Jour.,

v. 94, no. SA-5, no. 6159, p. 883-895. 

Whipple, W., Jr., Hunter, J. V., Davidson, B., Dittman, F.,

and Yu, S., 1969, Instream aeration of polluted rivers:

Water Resources Research Inst., Rutgers Univ., New

Brunswick, N. J., 196 p. 

Williams, W. E., 1951, Relation of BOD to stream deoxygenation: Sewage and Indus. Wastes, v. 23, no. 9, p. 1161-1175. 

Wolman, Nathaniel, 1960, U.S. Senate Committee Pre-print

32, 86th Congress, 2d sess., Aug. 

Yao, K. M., 1969, Discussion of reaeration prediction in

natural streams: Am. Soc. Civil Engineers Jour., v. 95,

no. SA-6, p. 1189-1190. 

Zanoni, A. E., 1967, Waste water deoxygenation at different

temperatures: Water Research, v. 1, no. 8/9, p. 543-566.

APPENDIXES

APPENDIX A: EFFECT OF A CORROSION REACTION ON REAERATION-COEFFICIENT 

MEASUREMENTS IN A LABORATORY FLUME

[Symbols used in appendix A are given in a list at the end of the appendix]

A recirculating-flume system generally consists 

of a headbox, a section of open channel, a tailbox, 

and a closed return-flow conduit. The system may 

be conceptualized as a flow loop consisting of an 

open channel and a pipe in series. If first-order 

oxidation reactions proceed in the channel and in 

the return pipe simultaneously with reaeration in 

the channel, the characteristic differential equations are:

and

(channel)

and

(pipe)

(A-l)

dt - dx2 " " (A-2)

where b is the first-order rate constant for the 

oxidation reaction, D is the DO deficit, C is the DO 

concentration, K2 is the reaeration coefficient, U 

is the mean water velocity, / is time, x is longitudinal position, and the 1 and 2 subscripts refer 

to the channel and pipe, respectively. These equations assume that the only source of oxygen is 

reaeration, the only sink is the oxidation reaction, 

and longitudinal dispersion effects are negligible. 

The oxidation reaction is assumed to follow the 

basic law of chemical kinetics for a first-order 

heterogeneous reaction; that is, the rate of 

reaction is proportional to the concentration of the 

reactant, the dissolved oxygen.

Note that D\=D\ (x\, t) and £>2 = Z)2 fe, t) and also

that Jti = 0, JC2 = L2 designates the beginning of the

channel, and XI = LI, ;t2 = 0 the end of the channel.

The initial and final conditions in the system are:

Di(0,0)=D2 (L2 ,0) = D0 (A-3)

where L2 is the length of the pipe. The solutions to 

equations A-l and A-2 (derived by the authors) 

under these conditions are:

D, l-e

D2 (jt2 ,0 = Cs [l-e "2 ]

b 9

+ (D0 -Dx )e- at e 

where from

and

D,(L,,0)=D2 (0,0):

, (A-6) 

(A-7) 

(A-8)

+

b2 L.2

b,

. e

_</r2 +&i> 

(A-9)

and

in which V is the volume of the water in the system, 

Q is the discharge of the system, Cs is the saturation concentration, and Li is the length of the 

channel. Because of the presence of the headbox, 

tailbox, and pump, the dimensions L\ and L2 in a 

real flume are equivalent dimensions.

Equation A-9 gives the deficit at the station 

*i=0, x2 L2 as /-*oo. This non-zero value of the 

deficit is the value at which the oxygen consumed 

by the reactions is just replaced by that supplied 

by surface reaeration. Equation A-5 shows that if 

there is an oxygen-consuming reaction in the 

channel or return pipe of a flume, then because of 

the first term on the right of that equation and because of the non-zero D^ value, the logarithm D 

versus / relation at a station jc t will not be a straight 

line as shown in figure 2. Thus the system does not 

follow the first-order model usually assumed valid 

for flume systems. Also, if b\ and (or) b-> are not 

equal to zero, the curves of logarithm D versus t at 

two different stations in the channel will not be 

parallel, as they would be were b\ and 62 = 0. 

Furthermore, it is expected, even discounting the 

b l -a)^l curvature of the lines and their deviation from

parallel, that the quantity M= , as determined

x. \ (j

(A-5) by the method of figure 2 will be some weighted

67
68 REAERATION IN OPEN-CHANNEL FLOW

average of Kz and b\ and will not equal the reaeration coefficient, K2. Examples presented later show 

that if bi=0, the error due to the curvature and 

deviation from the parallel is negligible. If bi^O, 

equations A-5, A-9, and A-10 can be used to obtain 

an estimate of the true values of K2 and b\ from 

values of M measured by the procedure of figure 2. 

If

and

+ D' e

)fl2. 

"l

ul

(A-ll)

+D x e

Ul

+ (D'0 -D'x)e e , - 

where in the double subscripts the first subscript, 

1, indicates the channel and the second, 1 or 2, 

indicates one of two longitudinal locations in the

channel, Af=

*12 Xll

=-rr, and the deficits have been

nondimensionalized by dividing by Cs, which is indicated by the primed notation, then from equations A-ll and A-12,

Ax.

Equation A-13 may be written

Y where 17= - (K2 + bi) and S= h* Because rj is

small, e~v in the first term on the right of equation A-14 is adequately represented by the first 

two terms in its Taylor series expansion, and equation A-14 becomes

Taking the natural logarithm of equation A-15, 

expressing the right-hand side in its power series 

form, and neglecting all powers of 8 rj greater 

than 1, it may be shown that

(A- 16)

Because e~^ =

/

Ay- 

 ,

Lf\

(A-17)

(A-18)

Equation A-18 states that, assuming one has 

available DO-deficit measurements of two samples, 

one from the upstream station, and one from the 

downstream station at a lag time r=Ajc/f/ later, 

then the K2 value as determined from A/ = Aln D/r 

differs from the true value only if bi^Q. The 

amount of the deviation of M from K2 increases 

as the deficit at the upstream station decreases. 

If the method of figure 2 is used, that is, if M is 

determined not from two instantaneous point 

samples but from experimental results which were 

forced to fit two parallel straight lines, the resulting curve-fitting errors will be added to those 

resulting; from the non-zero b\ value.

To determine Kz and bi from experimental data, 

a plot of In D'i versus t at two different stations, 

as in figure 2, should be made without forcing the 

data to fit two parallel straight lines. For two 

sufficiently different values of D'n, one determines 

M, as in figure 2, and computes bi and K2 from two 

applications of equation A-18. If the data are 

forced to fit two parallel straight lines, M will be 

the same for all D'n, and b\ and K2 will be impossible to determine. Other methods for determining 

Kz, bi, and b2 may be derived from equations A-9, 

A-10, and A-18. These methods require only a 

single determination of M, but they involve conducting the experiment until f ><» to determine 

experimentally Dx . These methods, in addition, 

require measurements of Li and L2 , indeterminate 

quantities in real flume systems. Thus the procedure outlined at the start of this paragraph 

appears to be the best one for determining K2 in 

a flume system in which a first-order oxygenconsuming reaction proceeds in the channel in 

conjunction with reaeration.

Equation A-18 may be rewritten as

^M = |L(1 ~ Dn)> (A-19)

A.2 A.2 Lf 11

from which it is seen that if b\ and D'n are constants, the relative error in the measured Kz increases as Kz decreases. Thus if b\ is a constant in 

a set of flume experiments, the resulting errors 

will be more severe at the lower end of the experimental Kz range, with the possibility of severely 

biasing any resulting least-squares prediction 

equations. This could be a reason for the widely 

different values of the multiplicative constants 

of the regression equations obtained by Thackston 

and Krenkel (1969a) from the analysis of their data 

(Thackston, 1966; Krenkel, 1960).

The discussion above suggests a procedure for 

conducting a series of flume experiments for 

measuring K2 which would determine if oxygenconsuming reactions are influencing the K2 mea-
APPENDIX A 69

surements. The procedure involves running a 

special experiment prior to the start of a test 

series. In this experiment, test conditions are 

selected which are expected to yield a K2 value that 

would rank among the lowest expected in the test 

series. The experiment is conducted under conditions typical for the entire set, but over a larger 

DO-deficit range than normal-that is, for a long 

time. If there is no oxygen-consuming reaction in 

process in the system, the curves of logarithm D 

versus t observed at the two measurement stations 

will be straight and parallel, and Kz may be computed by the method of figure 2. If a reaction is in 

progress in the system, this will be indicated, as t 

becomes large, by curvature and nonparallelism 

of the curves of logarithm D versus t at the two 

measuring stations. If the reaction is in the return 

conduit, b\ as computed by the method suggested 

above will be zero, and as will be shown in the 

examples to follow, the analysis method of 

figure 2 will be accurate enough. If &i^0, K2 must 

be calculated by the method given above. The trial 

experiment described above will point the way for 

conducting the other experiments in the series. If 

the reaction in the channel is insignificant in the 

test case, it will most likely also be insignificant in 

the others. If it were significant, its effects should 

be considered in the remainder of the experiments.

EXAMPLES

The usefulness of the K2 determination procedure described above is best illustrated by 

example. The examples which follow utilize, with 

the exception of the reaction-rate constants b\ 

and b2 , values of flume and flow parameters which 

are typical for small laboratory flumes. The values 

of hi and b2 must necessarily be highly speculative, 

because available flume data are not sufficient to 

allow determination of typical values for these 

parameters. For a mixing tank, however, Tsivoglou, O'Connell, Walter, Godsil, and Logsdon (1965) 

reported data which indicate that the reactionrate constants in a largely metal system can be on 

the order of 2xlO~5 per second. The reaction is 

most likely due to corrosion of the walls of the system and would be expected to vary widely from 

one system to another; consequently, slightly 

larger values for b\ and b2 have been chosen for 

use in the examples described. The relative error 

in a measured K2 due to a particular value of bi is 

dependent on the ratio bi/K2 , so that for the values 

of b\ used in the examples, the relative errors will 

be more or less severe for respectively larger or 

smaller K2 values.

In the examples, DO deficits at two sampling 

stations were calculated using equation A-5 and 

the parameters listed in table 17, where a and Dx 

were calculated from equations A-9 and A-10.

TABLE 17. Parameters used in calculating the synthetic 

flume data and the resulting calculated reaeration coefficient

Example

No.

1

2

3

4

5

Ki

_L

sec xlO4

8.00

8.00

8.00

8.00

4.00

b,

1

sec xlO5

0

0

4.00

4.00

4.00

b.,

1

sec xlO5

8.00

8.00

8.00

8.00

4.00

a

1

sec xlO4

3.50

3.50

3.65

3.65

1.90

D,,

0.800

.400

.800

.400

.400

Q

0.144

.144

.179

.179

.211

M

1

sec xlO4

8.00

8.00

7.81

7.27

3.34

NOTE.- Parameters which were the same for all examples are: 

K=16 ft3, 0 = 0.3 cfs, t/,= 1.8 fps, t/ 2 =1.2 fps,

tt = 36 ft, t2 = 40 ft, AT,i = 8 ft at upstream, and x, 2 = 28 ft at downstream 

sampling stations.

From the synthetic flume data, M was calculated 

for each of the examples by the method of figure 

2. Least-squares straight lines were fit to 20- 

minute increments of the calculated values of the 

logarithm deficit and t for each of the two sampling 

stations. The 20-minute sampling period was 

chosen because this was the period used by both 

Thackston (1966) and Krenkel (1960). The slopes of 

the two lines were averaged, and new intercepts 

were computed for parallel least-squares lines 

through the two sets of points. The quantity M was 

then computed from the new intercepts and the 

average slope. These calculated M values are listed 

in table 17. In this procedure, M is not a function 

of time as it is if the lines are not parallel.

Figure 18 shows plots of logarithm D versus t 

for the upstream sampling stations in examples 3, 

4, and 5; the lines are the least-squares fits to the 

synthetic data points. The curvature due to the 

non-zero Dx value is virtually unnoticeable over 

the 20-minute time period. In fact, if experimental 

data with its scatter were used, the curvature 

would probably be impossible to detect. Thus, over 

a 20-minute experimental time, it is virtually 

impossible to detect a non-zero Dx , and it is 

impossible to conclude whether or not an oxygen 

consuming reaction is proceeding in the system. 

As can be seen from table 17, for examples 3, 4, 

and 5, the deviation of M, the calculated K>, from 

the true value is significant. Thus it is possible 

for oxygen-consuming reactions to cause significant errors in calculated K2 values without their 

presence being noticed in the experimental results.

The values of M for examples 1 and 2 are not 

significantly different from 8.0xlO~* seconds-'. 

This shows that, at least for short time periods, if
70 REAERATION IN OPEN-CHANNEL FLOW

-.20

-1.20

0 6 8 10 12 14 

TIME, IN MINUTES

16 18 20

FIGURE 18. Synthetic flume data from the upstream sampling station.

b\ is zero, the deviation of M from the true K2 , due 

to the curvature induced by a non-zero Dx , is 

negligible. Thus, when Cs is known, the K-2 calculation procedure illustrated in figure 2 will contain significant errors only if b\ is not zero.

The upstream deficit values at f=0 minutes for 

examples 3, 4, and 5, respectively, are 0.676, 0.356, 

and 0.379. The corresponding values of M calculated from equation A-18 are 7.82, 7.28, and 

3.34xlO~4 seconds" 1 . These are very nearly equal 

to the M values calculated by the procedure of 

figure 2 listed in column 8 of table 17. Even better 

correspondence between the measured M and the 

M calculated from equation A-18 is obtained if 

instantaneous values of the upstream and downstream deficits are used to calculate M. For 

instance, in example 5 at t = lQ minutes the upstream deficit is 0.368, and the M value calculated 

from the deficits is 3.31xlO~4 seconds" 1 , whereas 

the M calculated from equation A-18 is 3.31 x 10~4 

seconds^ 1 . The close correspondence of the M

values obtained in the two ways indicates that 

despite the approximations made in its derivation, 

equation A-18 provides an accurate means for 

obtaining K-2 and b\ from two measured values of M.

The procedure for determining K2 and b\ from 

experimental data is illustrated in figure 19. 

Synthetic data parameters shown in figure 19 are 

the same as those for example 4 in table 17. Deficits 

at the upstream and downstream sampling 

stations are plotted for two periods: from 0 to 10 

minutes and from 50 to 60 minutes. The procedure 

consists of plotting the natural logarithm of deficit 

versus time, determining M graphically (as in fig. 

2) for each of the 10 minute periods, and solving for 

bi and K2 using equation A-18 and the resulting 

two sets of D'n and M values. For the example 

given in figure 18, D'n and M for the initial 10- 

minute period are, respectively, 0.376 and 7.20xlO~4 

seconds" 1 , whereas for the period from 50 to 60 

minutes, they are 0.245 and 6.56 xlO"4 seconds" 1 . 

Using equation A-18, the calculated bi and K> 

values are, respectively, 4.51xlO" 4 seconds" 1 and 

7.95xlO" 4 seconds" 1 . The agreement of these 

values with the bi and K2 values used in generating the synthetic data is very good, considering 

that the two M values were determined graphically.

If in a flume study there is an error in the value 

of Cs being used, there will be a corresponding 

error in all the computed deficits. In such a case, 

the true deficit, DT is related to the plotted deficit 

by D=DT +d. The resulting equation for the plotted 

deficit, assuming no oxygen-consuming reactions 

in either the channel or the return conduit, is

_ _ X

-c* e 2 ulf (A-21)

Equation A-21 is similar to equation A-5, with 

bi=Q. It has been shown that in the case of bi = Q, 

the error in the K-2 computed using the method of 

figure 2 is negligible. Thus in a flume study, a 

negligible error is introduced by using a value of 

Cs which is incorrect by a small amount Ad. This is 

in contrast to the nature of errors in analysis of 

mixing-tank data where K2 is determined from the 

slope of a single line (Isaacs and Gaudy, 1968; 

Tsivoglou and others, 1965).

SYMBOLS USED IN APPENDIX A

Symbol Definition

bi Rate constant for the first-order oxidation reaction in the channel of the flume, in reciprocal seconds.

b-2 Rate constant for the first-order oxidation reaction in the return pipe of the flume, in reciprocal seconds.

Ci Dissolved-oxygen concentration in the water in the channel of the flume, in milligrams per liter.

C2 Dissolved-oxygen concentration in the water in the return pipe of the flume, in milligrams per liter.

Cs Dissolved-oxygen concentration at saturation, in milligrams per liter.

D Computed deficit of dissolved oxygen, in milligrams per liter.
APPENDIX A 71

-0.91

-1.43 -

-1.450

3000

100

3100

200 300 400

3200 3300 3400

TIME, IN SECONDS

500

3500

600

3600

FIGURE 19. Determination of 61 and K? from M values calculated at two different

DO deficits.
72 REAERATION IN OPEN-CHANNEL FLOW

Symbol Definition

Di Equivalent to Di(jci,/)f the dissolved-oxygen deficit at longitudinal position x in the channel of the flume at time t, in

milligrams per liter. (Subscript 1 indicates the channel of the flume.) 

D2 Equivalent to D2 (jt2 ,0, the dissolved-oxygen deficit at longitudinal position x in the return pipe of the flume at time t, in

milligrams per liter. (Subscript 2 indicates the return pipe of the flume.) 

Do Initial dissolved-oxygen deficit in the system at time /=0, in milligrams per liter. 

D* Final dissolved-oxygen deficit in the system at time / = °°, in milligrams per liter. 

Do Nondimensional deficit of dissolved oxygen in the system at time (=0, equal to D0/CS . 

D'» Nondimensional deficit of dissolved oxygen in the system at time ?=°°, equal to DX /CS .

Dr True deficit of dissolved oxygen, equal to the computed deficit minus an error, or D-d, in milligrams per liter. 

Dr» Initial true deficit of dissolved oxygen, equal to the initial computed deficit minus an error, or D0 -d, in milligrams per

liter. 

D'n Equivalent to D\(xn,t), the nondimensional deficit of dissolved oxygen in the channel at longitudinal position xt at time t.

(Deficit nondimensionalized by dividing by Cs ; first subscript in the double subscript (1) indicates the channel of the

flume, and the second subscript (1) indicates upstream measurement point in the channel.) 

D'i2 Equivalent to D'i(jti 2 ,/ + A/), the nondimensional deficit of dissolved oxygen in the channel at longitudinal position x» at

time t+&t. (Deficit nondimensionalized by dividing by Cs ; first subscript in the double subscript (1) indicates the

channel of the flume, and the second subscript (2) indicates downstream measurement point in the channel; A? is the

flow time or (xiz x\\)IU\, where Ui is the mean water velocity in the channel.) 

d Error in the saturation concentration of dissolved oxygen, in milligrams per liter. 

Kz Reaeration coefficient or rate constant for oxygen absorption from the atmosphere, natural logarithm base, in reciprocal

days.

Li Length of the flume channel, in feet. (Equivalent dimension because of the presence of the headbox and tailbox.) 

Lz Length of the return pipe, in feet. (Equivalent dimension because of the presence of the headbox, tailbox, and pump.)

X AJC M Equal to A In Dj (nomenclature of fig. 2 and the main part of the report) or A In D'/ (nomenclature of appendix A); U U\

A" = Ajt=the distance, in feet, between the upstream and downstream measurement points in the flume channel; 

U=Ui = the mean water velocity, in feet per second, in the flume channel; A In D = A In D =the difference in the 

natural logarithms of the dissolved-oxygen deficits at the upstream and downstream measurement points with the 

downstream measurement lagging the upstream by the flow time, AJC/(/I. See figure 2.

Q Water discharge, in cubic feet per second.

t Time, in seconds.

(/i Mean water velocity in the channel of the flume, in feet per second.

Uz Mean water velocity in the return pipe of the flume, in feet per second.

Xi Longitudinal position in the channel of the flume, in feet.

xn Upstream measurement point in the channel of the flume, in feet.

Xiz Downstream measurement point in the channel of the flume, in feet.

Equal to -77 -77-?+(K2 +61)77- » in reciprocal seconds. 

v I Uz c/jj

8 Equal to bi/(Kz + bi). 

Equal to (Kz + bi). 

T Lag time or flow time, equal to AJC/(/I, in seconds.
APPENDIX B: COLLECTED DATA

TABLE 18. Reaeration coefficients and hydraulic data from 

Churchill, Elmore, and Buckingham (1962)

TABLE 19. Reaeration coefficients and hydraulic data from 

Owens, Edwards, and Gibbs (1964)

fcst

(day-')

2.272

1.440

.981

.496

.743

1.129

.281

3.361

2.794

1.568

.455

.389

.270

.550

.544

.604

1.251

.273

.225

1.881

.842

.883

.915

.995

.547

.881

.252

5.558

1.712

3.222

tAdjusted to

H

(feet)

3.27

5.09

4.42

6.14

5.66

7.17

11.41

2.12

2.93

4.54

9.50

6.29

7.52

7.07

5.44

8.06

3.88

9.28

10.19

3.29

4.74

5.72

6.95

4.29

6.01

7.16

8.49

3.42

3.02

2.83

(fc2)20% US

V

(fps)

3.07

3.69

2.10

2.68

2.78

2.64

2.92

2.47

3.44

4.65

2.94

2.51

3.15

3.30

3.11

4.28

2.73

2.41

3.06

2.40

3.46

4.02

4.52

1.85

2.75

3.23

3.71

5.00

3.05

3.51

iing equation 104.

S

(X10 1 )

10.8502

8.6426

2.7058

2.9014

3.6970

1.6448

1.7994

14.2342

14.5225

13.9099

1.6893

2.9038

3.0443

3.7775

5.9961

5.7667

6.0706

1.2571

1.7789

15.5378

13.5458

11.7032

10.1594

3.9709

3.7523

3.6794

3.6430

23.5142

17.4603

18.8692

w

(feet)

229

242

344

358

376

313

312

617

635

667

374

414

443

450

330

346

90

537

549

520

535

542

550

517

531

539

548

182

124

115

tot (day-1 )

2.35

2.06

3.20

2.37

4.57

2.09

1.18

3.18

6.18

.90

1.66

.75

2.12

1.41

.31

31.80

24.53

24.57

25.59

28.34

22.80

49.17

30.77

18.46

21.05

16.06

12.04

20.32

18.90

20.25

17.09 

19.16

H

(feet)

1.21

1.15

1.09

1.50

1.08

.98

1.12

1.46

1.31

2.44

2.03

1.74

1.47

1.82

2.41

.72

.87

.85

.40

.40

.39

.60

.69

1.00

.82

.78

.64

.48

.72

.66

.67 

.69

u

(fps)

0.14

.14

.13

.37

.16

.15

.13

.38

.47

.23

.22

.28

.37

.50

.43

1.37

1.19

1.07

.44

.56

.63

1.83

1.81

1.54

1.07

1.27

.46

.60

1.16

1.31

1.30 

1.25

s

(X10<)

3.94

3.94

3.94

3.94

4.24

4.24

4.24

4.24

11.65

8.95

4.08

2.69

8.72

1.56

1.56

106.00

84.00

61.00

92.00

61.50

60.50

92.00

61.50

60.50

101.60

83.00

61.20

65.70

61.20

65.70

65.70 

65.70

w

(feet)

28.7

25.8

27.3

27.7

28.1

28.2

26.6

27.8

25.0

17.9

22.5

22.5

20.1

39.8

34.9

21.9

20.9

23.7

15.3

12.1

11.0

16.1

14.2

11.5

21.8

19.3

17.3

17.7

20.7

20.0

19.9 

20.0

tAdjusted to (fcs)20°, using equation 104.

73
74 REAERATION IN OPEN-CHANNEL FLOW

TABLE 20. Reaeration coefficients and hydraulic data from 

Gameson, Truesdale, and Downing (1955)

fczt (day : H

(feet)

U 

(fps)

W 

(feet)

57.7

48.7

55.6

46.2

34.3

35.2

0.41 

.41 

.42 

.42 

.42 

.34

1.04

1.12

.99

1.02

.73

.74

5.7 

5.3 

5.6 

5.4 

6.0 

6.0

t Ad justed to (ki)io°, using equation 104.

TABLE 21. Reaeration coefficients and hydraulic data from 

O'Connor and Dobbins (1958) (Streeter and Phelps, 1925)

fc2 t

(day-1 )

0.45

.23

.15

.34

.17

.26

.20

.14

.19

.37

.22

.71

.15

.20

.26

.22

.31

.21

.23

.86

.92

.45

.19

.24

.17

.19

.16

.14

H

(feet)

9.2

8.8

9.3

6.5

7.0

5.8

5.3

4.8

6.4

4.0

8.6

7.6

7.1

5.5

13.8

15.0

8.6

7.2

6.6

14.3

9.3

9.0

18.7

8.8

8.2

7.2

7.3

24.2

u

(fps)

2.32

1.98

2.18

1.41

.59

.32

.22

.19

.43

.23

.40

.40

.22

.51

2.50

4.20

1.36

.86

.53

3.68

1.01

.62

3.72

1.48

1.18

.91

.96

3.90

s

(X104 )

0.68

.68

.68

.68

1.97

1.97

1.97

1.97

1.23

1.23

.95

.95

.95

1.67

1.97

1.23

1.23

1.23

1.23

.95

.95

.95

1.67

1.67

1.67

1.67

1.67

.68

t Adjusted to (fc2)20°, using equation 104.

TABLE 22. Reaeration coefficients and hydraulic data from 

O'Connor and Dobbins (1958) (various sources)

fc:t 

(day-1 )

5.80

3.07

1.22

.92

.23

.28

.30

.048

.018

.026

H

(feet)

0.9

1.9

4.0

3.1

9.2

9.0

8.9

12

32

37

u

(fps)

0.97

.55

.73

.65

1.37

1.57

1.63

.32

.50

.88

s

(X101 )

36

14

4.9

6.9

.27

.27

.27

t Adjusted to (£2)20°, using equation 104.

TABLE 23. Reaeration coefficients and hydraulic data from 

Tsivoglou and others (1967,1968)

fc2t

(day-1 )

1.273

1.142

.977

.646

1.653

.872

1.251

1.170

1.261

3.373

1.163

1.491

2.344

1.420

.977

H

(feet)

1.76

1.94

2.09

3.11

1.67

2.36

1.99

2.26

2.83

2.30

1.95

2.42

2.13

2.06

1.66

u

(fps)

0.636

.419

.359

.458

.570

.346

.335

.436

.286

.563

.529

.441

.539

.425

.457

W 

(feet)

73.7

101.5

110.0

57.9

86.7

101.0

123.8

83.72

101.9

63.7

80

77.3

71.9

94.2

108.8

t Adjusted to (fa) 20% using equation 104.

TABLE 26. Reaeration coefficients and hydraulic data from 

Negulescu and Rojanski (1969)

fc2t

(day-1 )

15.90

11.50

17.25

18.70

8.64

11.50

8.64

14.45

H

(feet)

0.492

.492

.164

.164

.492

.492

.492

.361

u

(fps)

1.902

1.640

.951

.853

1.050

.656

.886

1.082

W 

(feet)

0.66

.66

.66

.66

.66

.66

.66

.66

tAdjusted to (£2)20°, using equation 104.
APPENDIX B 75

TABLE 24. Reaeration coefficients and hydraulic data from 

Thackston (1966)

TABLE 25. Reaeration coefficients and hydraulic data from 

Krenkel (1960)

tot (day-1 )

18.72

20.75

25.45

34.85

40.6

44.9

52.5

24.35

19.57

18.38

21.1

15.39

16.69

9.98

13.19

26.15

39.55

40.55

43.8

31.90

39.45

16.74

16.32

36.45

14.62

21.95

25.87

24.25

24.72

64.05

10.88

11.38

5.78

44.55

31.55

39.3

23.80

18.45

28.80

14.36

29.80

19.08

22.45

24.30

23.35

16.82

22.45

32.08

19.32

18.82

25.55

18.60

t Ad justed to

H

(feet)

0.125

.109

.124

.092

.098

.122

.117

.170

.112

.104

.107

.094

.107

.113

.099

.065

.065

.063

.072

.052

.078

.082

.085

.070

.101

.119

.146

.082

.129

.057

.141

.175

.232

.154

.150

.189

.089

.067

.037

.080

.078

.111

.096

.121

.088

.083

.073

.082

.085

.080

.072

.148

(£2)20°, using

U

(fps)

1.323

1.449

1.878

1.707

1.858

2.320

2.029

2.079

1.225

.994

.921

.802

.940

.773

.776

.754

1.116

.961

1.069

.678

.848

.802

.365

.389

.645

.865

1.100

.636

1.110

.558

.476

.955

.484

1.471

1.743

2.015

1.333

.978

.592

.843

.762

.900

.764

.968

.738

.688

.595

.666

.855

.782

.753

.741

equation 104.

S

(X1<M)

59.6

89.4

116.1

144.6

162.3

178.5

141.8

84.3

55.7

43.0

38.0

23.6

42.3

24.5

28.2

57.1

112.4

89.4

89.7

67.4

52.4

43.8

31.5

60.0

65.9

79.4

79.1

107.6

107.4

201.2

14.4

42.6

6.5

133.2

203.8

168.8

24.1

17.4

18.2

10.3

44.1

32.4

32.4

32.4

32.4

32.4

32.4

32.4

46.5

40.6

40.6

30.9

w

(feet)

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

2.0

feit (day-i)

34.37

69.91

37.26

24.66

64.32

82.39

99.38

50.93

75.93

40.06

104.3

115.3

34.51

26.57

100.7

111.7

59.22

78.99

77.25

40.80

91.04

67.25

29.11

66.73

14.31

49.39

102.5

114.9

109.2

43.70

62.70

71.81

64.31

96.17

19.65

32.64

41.50

95.67

82.67

13.26

53.02

69.10

53.23

96.13

97.18

108.7

13.04

38.05

52.01

53.31

32.32

10.63

16.46 

39.18

28.85

37.62

42.99

50.54

H

(feet)

0.0802

.0823

.0820

.0824

.0861

.0852

.0857

.0844

.0871

.0894

.0930

.0901

.0878

.0932

.0876

.0882

.0964

.0971

.0974

.0974

.0963

.1091

.1115

.0993

.1001

.1048

.1045

.1090

.1043

.1232

.1182

.1161

.1189

.1360

.1416

.1422

.1415

.1427

.1410

.1447

.1463

.1451

.1478

.1521

.1507

.1481

.1650

.1651

.1650

.1662

.1651

.2014

.1983 

.1985

.1989

.1995

.1968

.1989

U

(fps)

0.473

.739

.385

.295

.788

.897

1.020

.634

.823

.461

1.330

1.205

.553

.364

1.024

1.063

.859

1.020

1.150

.605

1.185

1.200

.662

.898

.243

.793

1.272

1.590

1.438

.830

.967

1.085

1.108

1.590

.539

.733

.904

1.450

1.510

.359

1.180

1.340

1.030

1.810

2.030

2.140

.408

1.050

1.350

1.560

.828

.441

.743 

1.186

1.020

1.315

1.478

1.790

s

(X104 )

44.38

100.00

28.25

15.75

100.00

145.88

162.86

70.33

106.75

32.00

239.90

212.50

44.00

17.50

158.75

177.75

93.75

127.50

164.00

45.25

137.50

137.50

411.80

91.56

8.00

63.44

175.00

236.00

212.60

51.75

76.25

104.00

100.45

156.25

17.50

31.75

49.25

118.25

138.50

7.75

77.25

106.00

59.00

180.00

212.60

234.30

7.50

49.50

78.50

105.75

33.25

7.50

18.00 

49.25

32.75

59.00

77.00

113.60

w

(feet)

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0

1.0 

1.0

1.0

1.0

1.0

1.0

tAdjusted to (£2)20°, using equation 104.

U. S. GOVERNMENT PRINTING OFFICE : 1972 O - 466-751