A Laboratory Investigation 

of Open-Channel Dispersion 

Processes for Dissolved, 

Suspended, and Floating 

Dispersants

GEOLOGICAL SURVEY PROFESSIONAL PAPER 433-E

Prepared in cooperation with the 

U.S. Atomic Energy Commission
A Laboratory Investigation 

of Open-Channel Dispersion 

Processes for Dissolved, 

Suspended, and Floating 

Dispersants

By W. W. SAYRE and F. M. CHANG

TRANSPORT OF RADIONUCLIDES BY STREAMS

GEOLOGICAL SURVEY PROFESSIONAL PAPER 433-E

Prepared in cooperation with the 

U.S. Atomic Energy Commission

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1968
UNITED STATES DEPARTMENT OF THE INTERIOR 

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

William T. Pecora, Director

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

Washington, D.C. 20402 - Price 60 cents (paper cover)
CONTENTS

Symbols..-----_-_-.------_-_____________________

Abstract_ ________________________________________

Introduction and acknowledgments.__________________

Theory of dispersion in open-channel flow____________

Fickian diffusion theory.________________________

Diffusion by continuous movements.______________

Analogy with theory of local similarity ____________

Longitudinal dispersion by differential convection 

due to a velocity gradient____________________

Applications of diffusion theory in suspendedsediment transport.._--__-__-__-______________

Description of the experiments _______________________

Hydraulic data.__.-__---_-_______-__________-__

Longitudinal dispersion experiments_____________

Lateral dispersion experiments__________________

Vertical dispersion experiments ___________________

Presentation and analysis of data.____________________

Hydraulic data.________________________________

Page

v

El

1

2

3

6

9

11

11

12

15

16

18

18

Presentation and analysis of data Continued Page

Longitudinal dispersion_________________________ E18

Dye____________ ________ 18

Polyethylene particles-__------_-_-__________ 23

Suspended silt-size particles._________________ 26

Lateral dispersion________-__-_-_-___-___________ 37

Dye___ _ _ __- 37

Polyethylene particles.______________________ 47

Vertical dispersion._____________________________ 51

Discussion of results________________________________ 51

Applicability of Fickian diffusion theory ___________ 51

Relationship of dispersion coefficients to flow

parameters_ ________________________________ 55

Definition of dispersion coefficients,_______________ 59

Longitudinal dispersion of suspended silt-size particles. ---__-.----___--___--_--_-_-_------___- 61

Treatment of boundaries as reflecting barriers._____ 63

Conclusions.-__-_____-_-_-__--_-_--__________-_____ 63

Experimental evaluation of response characteristics of

the concentration-measuring systems ________________ 65

Literature cited_____________________________________ 70

ILLUSTRATIONS

Page 

FIGURE 1. Schematic diagram of the flume and circulation system-_-_______-__--_-_________---__-_-----------_- Ell

2. Photograph and sketch showing arrangement of roughness cleats on the bed of the flume.___--------__-_ 12

3-5. Photographs showing:

3. Point velocity measuring equipment__________-__-____-_____-______-_-___---_-------------__ 13

4. Improved dumping trough for dye and suspended sediment in longitudinal dispersion experiments. _ 14

5. Continuous and discrete sampling systems used for determining dye and sediment concentrations in

longitudinal dispersion experiments..____________-______-________-__------_------------- 14

6. Graph showing fall-velocity distribution curves for the sediments used in longitudinal dispersion experiments. 16

7. Photograph of point source used in polyethylene-particle dispersion experiments..___________-_----_---_ 17

8. Circuit diagram of time-measuring system used to determine traveltimes of polyethylene particles________ 17

9-11. Photographs showing:

9. Point-source arrangement used in lateral dispersion experiments with fluorescent dye_____________ 17

10. Continuous-sampling system and traversing mechanism used in lateral dispersion experiments with

fluorescent dye.-_-____________________________________--__________----_--_----------- 17

11. Compartmented sieve collector used in lateral dispersion experiments with floating polyethylene

particles--__________________________________________-_____________-_---___-_-------- 18

12-57. Graphs showing:

12. Variation of resistance function with depth of flow_______________________--_-_-__--_------_- 18

13. Velocity profiles for flow conditions in dispersion experiments.______________-_--__--_-------_-- 19

14-16. Relative concentration of dye as a function of time at various dispersion distances in:

14. Run LO-D-1.___________________________________________________________________ 19

15. Run LO-D-2_____________________________-_______---___-_-_----------_-__------- 20

16. Run LO-D-3.________________________________________.________________ 21

17. Mean traveltime as a function of dispersion distance, longitudinal dispersion of dye_____________ 22

18. Variance of traveltime as a function of dispersion distance, longitudinal dispersion of dye _______ 22

19. Dimensionless peak relative concentration as a function of dimensionless dispersion distance, longitudinal dispersion of dye.-_______________--______-__---_-_-____--------__----_-------- 23

m
IV CONTENTS

FIGURES 12-57. Graphs showing Continued Page

20. Recovery ratio for area under concentration versus time curves as a function of dispersion distance, E24 

longitudinal dispersion of dye._________________________________________________________ 25

21. Typical traveltime distribution, longitudinal dispersion of polyethylene particles. ________________

22. Mean traveltime as a function of dispersion distance, longitudinal dispersion of polyethylene par- 26 

ticles_ ______________________________________________________________________________

23. Variance of traveltime as a function of dispersion distance, longitudinal dispersion of polyethylene

particles.___________________________________________________________________________ 27

24. Typical traveltime distribution data, longitudinal dispersion of natural silt and glass beads._______ 28

25-32. Relative concentration of sediment as a function of time at various depths and dispersion distances:

25. Run LO-FS-1, 15-30,* silt_______________________________________________ 29

26. Run LO-FS-2, 15-30,* silt -----------------------__-__-__________________ 30

27. Run LO-CS-1, 53-62M sUt. 31

28. Run LO-FG-1, <44fi glass beads___-________________________________________ 32

29. Run LO-FG-3, <44/* glass beads______________._____-_________________________ 33

30. Run LO-CG-1, 53-62M glass beads_______________________________________________ 34

31. Run LO-CG-2, 53-62M glass beads__-____._.___________________________________ 35

32. Run LO-CG-3, 53-62M glass beads_______________--_-___-________________________ 36

33, 34. Mean traveltime as a function of dispersion distance, longitudinal dispersion of:

33. Natural silt______________________________________________________________________ 37

34. Silt-size glass beads.______________________________________________________________ 38

35, 36. Variance of traveltime as a function of dispersion distance, longitudinal dispersion of:

35. Natural silt______________________________________________________________ 39

36. Silt-size glass beads.______________________________________________________________ 40

37, 38. Dimensionless peak relative concentration as a function of dimensionless dispersion distance, longitudinal dispersion of:

37. Natural silt____________________________________________________..___.__ 42

38. Silt-size glass beads.______________________________________________________________ 42

39, 40. Recovery ratio for area under concentration versus time curves as a function of dispersion distance, 

longitudinal dispersion of:

39. Naturalsilt______________________________________________________________________ 43

40. Silt-size glass beads.______________________________________________________________ 44

41-43. Lateral distribution of dye at various depths and dispersion distances:

41. Run LA-D-1______________________________________________________________ 45

42. Run LA-D-2______________________________________________________________.._____ 45

43. Run LA-D-3______._____________________________________________________________ 46

44. Variance of lateral displacement as a function of dispersion distance, lateral dispersion of dye ______ 46

45. Recovery ratio for area under lateral-distribution curves as a function of dispersion distance, lateral

dispersion of dye_____________________________________________________________________ 48

46. Lateral distribution of dye for eccentrically located source.____________________________________ 49

47. Typical lateral distribution of polyethylene particles.__________________________________________ 51

48. Variance of lateral displacement as a function of dispersion distance, lateral dispersion of polyethylene particles ____________________________________________________________________ 52

49. Coefficient of variation for vertical distribution of dye as a function of relative dispersion distance. _ 53

50. Coefficient of variation for vertical distribution of dye as a function of relative dispersion distance

divided by the resistance function._____________________________________________________ 54

51. Longitudinal and lateral dispersion coefficients as functions of ynUT ---------------------------- 55

52. Longitudinal and lateral dispersion coefficients as functions of ynUT from Krenkel's and (Mob's data. 57

53. Variation of resistance function with flow depth for Krenkel's data_____________________________ 58

54. Comparison of Taylor's and logarithmic velocity profiles.____________.______________----_---__ 59

55. Integral length scale of large lateral turbulence components at the water surface as a function of

flow conditions ______________________________________________________________________ 60

56. Ratio of silt and dye longitudinal dispersion coefficients as a function of/3_______________________ 62

57. Attenuation of peak concentration with dispersion distance in a rectangular channel of width B having

a continuous point source at the centerline_______________________________________________ 64

58. Schematic diagram of experimental setup for determining response characteristics of sampling systems- 66

59. Graphs showing three typical system-response functions ____________-____________--_---_-----_ 68

60. Graphs showing comparisons between hypothetical input and output curves illustrating the effect of

system-response characteristics on longitudinal dispersion data.__________________-_-_-_--- 69
CONTENTS

TABLES

TABLE 1. Hydraulic conditions and types of dispersion experiments._________________________________

2. Longitudinal dispersion of dye: summary of data____________-________---_----__----------

3. Longitudinal dispersion of floating polyethylene particles: summary of data ______________

4. Longitudinal dispersion of silt-size particles: summary of data._____________-______-------__-

5. Lateral dispersion of dye: summary of data_________________-___-__------_-_--------_----

6. Lateral dispersion of floating polyethylene particles: summary of data._______________________

7. Lengths of initial increments of dispersion distance for experimental flow conditions__ ___________

8. Lateral dispersion coefficients, von Karman coefficients, and absolute roughness-______________.

9. Values of the parameter ft and the median particle fall velocity in the silt dispersion experiments.

10. Summary of results of system-response experiments _____________________---_---_--__------.

11. Corrections for system-response lag which were applied to flume data ________________________

Page 

E12 

25 

28 

41 

50

53

54

59

62

69

69

Symbol

SYMBOLS

Definition 

Am Area under experimental concentration-distribution curve, determined by

planimetering.

A t Theoretical area under concentration-distribution curve assuming perfect 

mixing and sampling.

B Width of a rectangular channel.

C Concentration, by weight, of dispersant in water.

Cv Coefficient of variation, ratio of standard deviation to mean.

E Mean rate of energy dissipation per unit mass of fluid, given by UgS.

/( ; ) A probability-density function. The symbol(s), in the parentheses, preceding 

the semicolon designates the variable (s) with respect to which probability is 

distributed. The symbol following the semicolon designates the specific point 

in space or time to which the probability applies.

F Sensitivity of fluorometer, expressed in fluorometer units per unit of concentration.

g Acceleration of gravity.

I Random variable denoting the time required for a particle of dispersant to 

travel from the source at x=Q to the sampling nozzle.

Kx, Ky, Kz Coefficients of dispersion in the x, y, and 2 directions.

Kc Component of Kx due to differential convection.

KT Component of Kx due to turbulence.

L Length scale of smallest eddies whose energy is converted directly into heat by 

viscous dissipation.

Lt Lagrangian integral time scale of turbulence.

Lx, Lz Length scale for large-scale turbulence components in the x and z directions.

Lm Length of initial increment of dispersion distance within which the onedimensional Fickian diffusion theory does not apply.

0 Random variable denoting the duration of the time interval between the release 

of a particle of dispersant from the source and its registration by the detection 

system.

3 Volumetric rate of release of dispersant from a continuous point source.

Q Volumetric rate of discharge of water in the channel.

R Random variable denoting the time lag between the arrival of a particle of 

dispersant at the sampling nozzle and its registration by the detection system.

R Reynolds number, defined as Uyjv in a wide, open channel and as UD/v in 

a pipe.

RU'^T") Lagrangian correlation coefficient, which correlates the fluctuating component 

u'i of the velocity of a fluid particle at the times t and t-\-r.

S Slope of the energy gradient in a uniform open-channel flow.

t Time.

t Mean traveltime of a dispersing substance over a prescribed distance.

T Water temperature.

1 Special symbols, subscripts, superscripts, and overscores if not given here, are defined where they first appear in the text.

Dimensions Units

T~l or L-i

L/272 

T

UIT 

T

ft

ppb or ppm

ft2 per sec3 

see"1 or ft"1

ft per sec2 

sec

ft2 per sec 

ft2 per sec 

ft2 per sec 

ft

sec

ft

ft

sec

ml per sec 

ft3 per sec 

sec

sec 

sec

°C
VI CONTENTS

U

V

u*

u.

Ur

vv 

w

X

tv

f

TO

Symbol Definition

Time-averaged local velocity of flow in the x direction.

Average velocity of flow in the cross section.

Difference _between the local velocity U and the cross-sectional average

velocity U.

Time-averaged local velocity of a suspended sediment particle. 

Time-averaged velocity of a particle floating on the water surface. 

Shear velocity in a wide, open channel, defined as VTO/P or vWn*SSettling velocity of a sediment particle in quiescent water. 

Weight of dispersing substance. 

Longitudinal distance. 

Vertical distance.

Normal depth; depth of flow in a channel with uniform flow. 

Lateral distance.

Sediment suspension parameter, defined as VP!KUT. 

Specific weight of water.

Local eddy diffusivity, sometimes called the kinematic eddy viscosity. 

Local eddy diffusivity for sediment particles. 

Local vertical eddy diffusivity.

Vertical eddy diffusivity averaged over the depth of flow. 

Eccentricity of source, the lateral displacement of a point source from the

centerline of the channel. 

Von Karman turbulence coefficient. 

Kinematic viscosity. 

Longitudinal displacement from a yz plane moving in the x direction at velocity

T7, defined as a; Ut.

Variance of a concentration distribution with respect to time. 

r \ Variances of longitudinal and lateral concentration distributions. 

Time. 

Shear stress at the bed, defined as yynS.

Dimensions

UT

LIT

LIT

LIT

LIT

LIT

LIT

F

L

L

L

L

F/L?

l?IT

l?\T

l?IT

I? IT

L

UIT

L

y2

I?

T

F/L*

Units

ft per sec

ft per sec

ft per sec

ft per sec

ft per sec

ft per sec

ft per sec

gm

ft

ft

ft

ft

Ib per f t»

ft2 per sec

ft2 per sec

ft2 per sec

ft2 per sec

ft

ft2 per sec 

ft

sec2

ft2

sec

Ib per ft2
TRANSPORT OF RADIONUCLIDES BY STREAMS

A LABORATORY INVESTIGATION OF OPEN-CHANNEL DISPERSION PROCESSES FOR 

DISSOLVED, SUSPENDED, AND FLOATING DISPERSANTS

By W. W. SAYRE and F. M. CHANG

ABSTRACT

The establishment of more reliable criteria for regulating the 

discharge of contaminants into streams and rivers requires an 

improved understanding of fluvial transport and dispersion 

processes. This requirement applies particularly to contaminants 

which are transported as part of the suspended or bed-material 

sediment load.

A series of experiments conducted in a rigid-boundary laboratory flume having an artificially roughened bed compared 

longitudinal dispersion of suspended silt-size sediment particles 

with longitudinal dispersion of a fluorescent dye solution. 

Additional information on dispersion in open channels was 

provided by lateral dispersion experiments with dye and by 

longitudinal and lateral dispersion experiments with floating 

polyethylene particles. Several dispersion theories for openchannel flows are reviewed, and the experimental data are 

analyzed in the light of these theories.

The experimental results indicate that in a channel with a 

rough boundary the longitudinal and lateral dispersion processes 

for dissolved dispersants and floating particles converge to a 

Fickian-type diffusion process quite rapidly with increasing 

dispersion distance. The experimentally determined values of 

the longitudinal dispersion coefficient for the dye closely correspond to values calculated by J. W. Elder's theory of longitudinal 

dispersion in open channels. Except in the sediment-dispersion 

experiments, the dispersion coefficients in each set of experiments were found to be approximately proportional to the 

product of the depth and the shear velocity. The longitudinal 

dispersion process for suspended silt-size particles, although it 

resembles the process for the dye, is modified by the tendency 

of the particles to settle toward the slower moving flow near 

the bed and eventually to be deposited on the bed. The longitudinal distribution of particle deposition can be calculated 

with satisfactory accuracy by a procedure based on T. R. 

Camp's theory of the influence of turbulence on sedimentation in settling tanks. The confining effect of the sidewalls 

on the lateral distribution of dye can be accounted for satisfactorily by the reflection-superposition principle in which 

channel boundaries are treated as reflecting barriers.

To compare the response characteristics of the different concentration-measuring systems used in the experiments, the 

response of the systems to rapidly changing concentrations was 

investigated both experimentally and analytically. The results 

indicate that the system output can be expressed as the convolution of the response function of the system to a unit impulse 

and the input to the system.

INTRODUCTION AND ACKNOWLEDGMENTS

With the growing demands on the Nation's water 

resources, it becomes increasingly imperative that 

rational criteria be applied to the problem of allocating 

available water supplies among all the various competing uses. As requirements grow, the allowable margin 

for error decreases. All too frequently the information 

necessary for making rational decisions is lacking.

One example of this problem is the use of streams 

and rivers as channels for the disposal of industrial, 

agricultural, and domestic wastes. Waterways have 

traditionally performed this function. If pollution is 

not controlled, however, the availability of water for 

other uses may be sharply curtailed. To control pollution, the release of potentially harmful contaminants 

into waterways must be regulated so that the capacity 

of the flow to maintain the concentration of contaminants within permissible limits is not exceeded. This 

regulation requires knowledge of the rates at which 

stream systems are capable of transporting and dispersing contaminants. In general, transport and dispersion rates depend on the physical and chemical 

nature of the contaminant, and on the physiographic 

characteristics of, and the flow discharge in, the stream 

system. However, the relationships between these 

factors and the transport and dispersion processes are 

extremely complex. Due to inadequate understanding 

of both the relationships and the processes, criteria for 

predicting transport and dispersion rates are often 

unreliable.

With the introduction since World War II of radioactive wastes and the new generation of agricultural 

pesticides and herbicides, the need for improved 

criteria for predicting transport and dispersion rates 

has become more acute. This is largely because many 

of these contaminants (1) have a tolerance level 

several orders of magnitude lower than that for most 

other pollutants, (2) are chemically very stable and

El
E2 TRANSPORT OF RAIXIONIU'CLII>EiS BY STREAMS

retain their toxicity for long periods of time before 

yielding to natural decay processes, and (3) cannot be 

removed by conventional water-treatment practices.

For a number of years the Water Resources Division 

of the U.S. Geological Survey in cooperation with the 

Division of Reactor Development and Technology of 

the U.S. Atomic Energy Commission has been engaged 

in investigations of dispersion processes in streams. As 

either a direct or indirect result of this program, the 

dispersion of dissolved contaminants has been investigated by Godfrey and Frederick (1963), Glover (1964), 

Yotsukura, Smoot, and Cahal (1964), and Yotsukura 

and Fiering (1964); available information on the uptake 

and transport of radionuclides by stream sediments was 

assembled by Sayre, Guy, and Chamberlain (1963); 

and Sayre and Hubbell (1965) investigated the transport and dispersion of contaminants which have 

become attached to bed-material sediment particles 

in an alluvial channel.

The principal objective of this investigation is to 

bridge the gap between what is known about the 

dispersion process for dissolved contaminants and 

what is known about the dispersion process for contaminants that are transported as part of the bedmaterial load. Dissolved contaminants are transported 

at the mean velocity of the stream and are dispersed by 

the mechanism of turbulent diffusion and by the 

differential convection currents associated with velocity 

gradients. Contaminants that are transported in 

colloidal suspension have been shown by Sayre and 

Hubbell (1963) to disperse in the same manner as 

dissolved contaminants and at essentially the same 

rate. On the other hand, contaminants which are 

transported as part of the bed-material load move in a 

sequence of steps of random length between which 

particles may remain buried for considerable periods. 

Hence, these contaminants are transported and dispersed at rates that are much less frequently by 

several orders of magnitude than the rates for dissolved or colloidally suspended contaminants. Between 

these two extremes are the contaminants that are 

transported mainly with the suspended-sediment load. 

These contaminants are either adsorbed by, or behave 

as, sediment particles in the silt- and fine-sand-size 

ranges. These particles may travel for long distances 

in suspension and, while in suspension, behave very 

much like fluid or colloidally suspended particles. 

However, they have a tendency to settle and eventually 

be deposited on the bed, where they behave as bedmaterial particles until they are reentrained in the 

flow.

This report describes a series of experiments conducted in a rigid-boundary laboratory flume having an 

artificially roughened bed. In the experiments the

longitudinal dispersion of suspended silt-size particles 

was compared with the longitudinal dispersion of a 

fluorescent dye solution, under nearly identical conditions. Because of inconsistencies between the results 

of previous experiments on the longitudinal dispersion 

of dissolved dispersants and Elder's (1959) theory 

of longitudinal dispersion in an open channel, the 

artificial roughness on the flume bed was designed to 

create flow conditions that would satisfy as nearly 

as possible the requirement, specified in the theory, 

of no transverse velocity gradient. To provide additional information on dispersion processes in open 

channels, lateral dispersion experiments with fluorescent dye and longitudinal and lateral dispersion 

experiments with small polyethylene particles floating 

on the water surface were also performed. The application of several dispersion theories to open-channel 

flow situations is reviewed, and the experimental data 

are analyzed within the framework of these theories.

The experiments were conducted in the hydraulic 

laboratory of Colorado State University at Fort Collins. 

The basic plan of the experimental program was 

conceived, and the first experiments were conducted, 

under the supervision of D. W. Hubbell, hydraulic 

engineer, of the U.S. Geological Survey. Tsung Yang, 

R. S. McQuivey, and J. A. Danielson, graduate students at Colorado State University, participated 

extensively in the design of experimental equipment 

and in the collection and analysis of data.

THEORY OF DISPERSION IN OPEN-CHANNEL FLOW

The processes by which dissolved and suspended 

particulate matter are dispersed in open-channel flow 

have been subjected to considerable theoretical analysis 

in recent years. Owing to a number of complex factors 

that are almost invariably associated with open-channel 

flow turbulence, velocity gradients in the vertical 

and horizontal directions, and secondary flows originating from bends or obstructions theoretical treatments 

of the problem have been only partially successful. 

Not even for the relatively simple conditions of uniform flow in a straight channel of constant cross 

section has a theory been formulated that accounts 

adequately for all of the above factors. Nevertheless, 

a considerable amount of useful information is contained in the technical literature. However, this 

information is scattered among many publications 

and much of it is not oriented toward open-channel 

dispersion processes. In this section an attempt is 

made to synthesize the most pertinent existing theories 

into a coherent theoretical background for the convenience of the general user in interpreting the experimental results presented and analyzed in the 

following sections.
OPEN-CHANNEL DISPERSION PROCESSES E3

The dispersion process can be described rather 

rigorously in differential-equation form by the diffusion 

equation for turbulent flow, which is based on the principle of conservation of mass. Stated in tensor form for 

dispersion in a steady flow of incompressible fluid, and 

with the assumption that the fluid properties of the 

dispersant are identical with those of the transporting 

medium, this equation is

where

C=CJrc' = C(xi) £2, £3, t) is the local concentration 

of dispersant expressed as the sum of a slowly 

varying part, C, and a rapidly fluctuating 

part, c',

<=time,

Ui=Ui+u'i=Ui(xi, Xz, #3) is the local velocity of 

flow expressed as the sum of the time-averaged 

velocity Ut and a turbulent component, u'i}

xt=distance and the index i=l, 2, 3 indicates 

direction in a rectangular coordinate system, and

CM coefficient of molecular diffusivity.

A coefficient of turbulent diffusion, e r =er?.. (xi, x2 , 

z3), may be defined as

(2)

Assuming that the processes of molecular and turbulent 

diffusion are independent and therefore additive, 

Mickelsen (1960), the turbulent and molecular diffusion 

coefficients can be combined by addition to give

(3)

Incorporation of equations 2 and 3 and elimination of 

the averaging bars permits equation 1 to be written in 

the form

In equation 3 the question of whether the processes of 

molecular and turbulent diffusion actually are independent is mainly of academic interest, because in 

ordinary open-channel flows eT is several orders of 

magnitude larger than eM . If, in addition, the coordinate 

axes are defined so that they coincide with the principal 

axes of the diffusion tensor, e 0=0 for i^j. If we now 

define ffj= i for i=j, equation 4 simplifies further to

d<7 =5 MI ,. ,

(4a)

Consider a uniform flow in a straight channel of 

constant cross section so that the basic flow and 

turbulence structure does not change with respect to 

the direction of flow. If a coordinate system is chosen 

in which the subscripts 1, 2, and 3 indicate respectively 

the direction of flow, the direction normal to the channel bed, and the horizontal direction normal to the 

flow direction, it is evident that U2 =Us=Q. Furthermore, under these conditions it is to be expected that 

ti is a function of a?_ and xs, but not of x\. Equation 4 

may now be written

?4i"-. I LM-3_J

or, after switching to the more conventional form of 

Cartesian notation in which xi} Xz, £3 are replaced by 

x, y, z; Ui, Uz, U3 by U, V, W; and c1} e2 , e3 by ez, cv , e f,

>_7, v -xTT H~>T7 z >kT I (^a)

This is the basic Eulerian diffusion equation which 

applies to dispersion in uniform turbulent flow. In 

equation 5 a U, fx, e v> and e z are generally functions of 

y and z.

Theoretical work relating to dispersion in openchannel flow has been devoted mainly to (1) analysis of 

the nature of ez , ey , and e z from the standpoint of 

fluid mechanics, (2) the analysis of certain limiting 

cases for which solutions to equation 5 can be obtained, 

and (3) the transformation of equation 5 into forms 

which are more amenable to solution.

Approaches yielding useful results have been based 

on (1) the semiempirical Fickian diffusion theory,

(2) the theory of diffusion by continuous movements,

(3) Kolmogoroff's theory of local similarity in turbulence, (4) the theory of longitudinal dispersion by 

differential convection due to a velocity gradient, and 

(5) diffusion theory as applied to the transport of suspended sediment. Because all these theories have a 

direct bearing on the experiments described in this 

paper, a review of the pertinent results and limitations 

is appropriate here.

FICKIAN DIFFUSION THEORY

In applying the Fickian diffusion theory to the process of dispersion in a turbulent flow, an exact analogy 

with the process of molecular diffusion is assumed. 

Thus, the coefficients ex, ev , and e z are assumed to be 

constants, which we shall call Kx , Kv, and Kz, and_ the 

time-average velocity, U, which we shall call U, is

281-661 O - 68 - 2
E4 TRANSPORT OF RADIONIUCUBES BY STREAMS

assumed to be the same everywhere in the flow field. 

In a real flow U is defined as the average velocity in 

a cross section. With these very drastic assumptions, 

equation 5a becomes

(6)

The convection term, U ^ , means that dispersion is

occurring within a frame of reference which is moving 

at a velocity U in the x direction.

Two types of dispersion in open-channel flow are 

of particular interest: (1) longitudinal dispersion from 

an instantaneous plane source which is distributed 

uniformly over the cross section, and (2) lateral dispersion from a continuous point source.

The longitudinal dispersion reduces to a one-dimensional problem because, owing to the restrictions on 

equation 6 and the initial source conditions, C is no 

longer dependent on y and z. Equation 6 then becomes

(7)

for which a solution satisfying the initial condition is

(8)

where

(70 =source strength of dispersant, which in a 

rectangular channel of flow depth yn and 

width B is equal to W/yByn, where W is 

the total weight of dispersant, and

(8a)

The function j(x; t) is the probability-density function 

of the normal probability law with mean ~x=Ut, and 

variance ozx=2lKjt,. In comparing equation 8 with 

experimental results it is usually more convenient to 

consider C as a function of t with a; as a parameter 

rather than as a function of x with t as a parameter. 

Then equation 8 becomes

C(t, x)=j(t; x}

where

(x-Uffl

(9)

(9a)

in which f(t; x) is the probability-density function for 

the distribution of dispersant flux with respect to t at a 

fixed value of x. The mean and variance of J(t; x) as

given by Yotsukura (1963) are

and

tj(t;z)dt==+ (10)

(U)

Experimental evidence of Yotsukura, Smoot, and 

Cahal (1964), Glover (1964), and Godfrey and Frederick (1963) indicates that the Fickian diffusion theory 

provides at best a crude representation of the longitudinal dispersion process in open channels. In general, the agreement between the Fickian theory and 

experimental observations is poor in the early stages 

of dispersion, but tends to improve with increasing 

dispersion time or distance from the source. Thus, 

under some conditions, for example dispersion in a 

uniform rectangular channel with straight alinement, 

equations 8 and 9 are useful as asymptotic solutions 

for large values of t and x respectively.

Let us turn now to lateral dispersion from a point 

source. Since it is assumed implicitly in equation 6 that 

the dispersion process in any one of the three coordinate 

directions is independent of the process in either or 

both of the other two directions, the joint probabilitydensity function j(x, y, z; f) is equal to the product of 

the one-dimensional probability-density functions, 

/(z; 0 > J(y't 0> and/(s; 0- The solution of equation 6 

for the initial condition of an instantaneous point 

source at the origin can then be written in the form

W C(x, y, z, t}= W 

, y, z; t)=J(x; t)f(y; t}j(z; t) (12)

where

/(*; 0=

and

; t}=-

(12a) 

(12b)

(12c)

The three functions/(.; f), where the dot represents an 

arbitrary variable with respect to which probability is 

distributed, are, like equation 8a, probability-density 

functions of the normal probability law. In a threedimensional domain they are one-dimensional only in 

the sense of marginal probability-density functions. 

For example, the function

/»« fB/2

f(y, t)=\ f(x, y, z; f)dzdx

JO J-BIZ
OPEN-CHANNEL DISPERSION PROCESSES E5

describes the vertical distribution of all of the dispersant irrespective of position with respect to x and z. 

It does not apply to the distribution in any particular 

cross section.

Assuming that yn/B is small and that there is sufficient turbulence to rapidly distribute the dispersant 

throughout the depth of flow, the concentration with 

respect to y soon tends to become uniform that is, 

f(y; t)-*l/yn. Then equation 12 becomes

~

W (13)

The solution of equation 6 for a point source discharging continuously at a constant volumetric rate q 

so that W=q'yCodt, where Co is the concentration of 

dispersant at the source, can be obtained by applying 

the superposition principle to equation 13 whereby

C(z, x)=\ O(z, x, t tjdti

*J oo

yt 00

 I C(z, X, r}dr 

Jo

J(x; r)j(z; r)dr.

In the above operation the dispersion time r=t ti of a 

particle of dispersant is the difference between the 

observation time, t, and the release time, t\. Making 

the appropriate substitutions and performing the 

integration,

Ux U

where K0 ( ) is a modified Bessel function of the second 

kind, of order zero. Values of K0 ( ) are tabulated in 

mathematical reference books such as Korn and Korn 

(1961). For

equation 14 converges to

(15)

( x\ in which/ (z; ff)i$ the probability-density function of

the normal probability law given in equation 12c, wherein t is replaced by x/U. Except close to the source, the 

conditions for convergence are usually easily obtainable 

in open-channel flow so that from a practical standpoint 

the difference between the two functions is generally

scarcely distinguishable. The variance corresponding to 

equation 14,

(16)

is larger than the variance corresponding to equation 15,

(17)

" U

by the constant amount, =1 - C72

In a channel of finite width, equations 14 and 15 

apply only in the region extending downstream from the

section where j(y; i) = to the sectioji where a significant amount of dispersant reaches the sidewalls at 

x~Q.QlUB2IKz. In a rectangular channel of width B, 

if the sidewalls are assumed to behave as reflecting 

barriers, equations 14 and 15 may be extended to 

account for the confining effects of the walls. The solution 

for the general case in which the source is displaced a 

lateral distance, f, from the origin at the center of the 

channel is

n=l

C(nB-t

-(-DK 2, z), (18)

in which

-B/2<z<B/2 and

n=the number of reflection cycles.

The C(-, x) terms on the right side of equation 18 are as 

defined by equations 14 or 15, but with z replaced by 

the first expression for example, nB f+( I) w 2^-inside 

the parentheses. Downstream from x~0.15UB2/KZ) 

where probably not more than two reflection cycles 

are contributing significantly to the value of Ci(z, x), the 

dispersant becomes distributed uniformly across the 

channel and

oC Ci(z, x)==^ 5-=constant. (19)

Experimental evidence of Orlob (1961), Sayre and 

Chamberlain (1964), and Patterson and Gloyna (1963) 

indicates that the Fickian diffusion model represents 

lateral dispersion considerably better than it represents 

longitudinal dispersion.

In summary, the Fickian diffusion theory gives, at 

best, an approximate kinematic description of dispersion in open channels. It provides little insight into the
E6 TRANSPORT OP RADIONU'CLIDEiS BY STREAMS

actual mechanics of the dispersion process and no insight into how the value of the dispersion coefficients 

are related to flow and channel characteristics. Dispersion coefficients for a particular set of conditions 

are evaluated empirically from a set of observed concentration-distribution data by means of the definition

(20)

or other relationships derived therefrom.

DIFFUSION BY CONTINUOUS MOVEMENTS

The theory of diffusion by continuous movements 

(Taylor, 1921), like the Fickian theory, is restricted to 

giving a kinematic description of dispersion, but the 

description is much more realistic because it is based 

on the turbulence properties of the flow. Taylor's equation for dispersion in one direction in a turbulence field 

that is spatially homogeneous and stationary in time is

where

(21)

= the variance at time t of the distribution in 

the i direction of a group of fluid particles that 

were located at the origin at time £=0,

Wj'2=the mean of the squared instantaneous turbulent velocity components in the i direction,

t= dispersion time, and

T} ., T . , ,. 

- is the Lagrangian correlation R r> / \

,(T)= v /

*

coefficient which correlates values of u't for fluid 

particles at the times t and £+T.

Equation 21 describes the dispersion of the fluid particles about the mean position of the group. Thus, if the 

turbulence field is being convected at the time-averaged 

velocity Ui, the dispersion refers to instantaneous displacement in the i direction from a point moving with 

velocity Ut- In general, a\(t) depends on the functional 

form of RU '_(T) which, like all Lagrangian turbulence

properties, is difficult to determine and is therefore 

usually not known. However, useful information can be 

obtained from equation 21 for the limiting conditions 

of (1) very small dispersion times where Lim RU >(T) =

and equation 21 reduces to

(22)

and (2) large dispersion times where Lim R »(r) = 0 and

T-X)

equation 21 reduces to

where

-2u? 

Jo (23)

Lt = I R '(r)dr is the Lagrangian integral time 

1 Jo * 

scale of turbulence.

Given the homogeneity and stationarity of the turbulence, the second term on the right of equation 23 is a 

constant, so that as t becomes very large,

which is equivalent to the variance given by the Fickian 

theory when the substitution Ki U{ iLt i is made.

Equation 21 was originally derived to describe dispersion in a homogeneous turbulence field one in 

which the statistical properties of the turbulence are 

the same at every point. Thus, at first glance it seems 

inapplicable to dispersion in open channels where the 

statistical properties of the turbulence generally vary 

with the distance from the boundary. However, Orlob 

(1958, 1961) showed that planes which are equidistant 

from the boundary in wide channels having uniform 

flow, for example the water surface, do indeed satisfy 

the criteria for a homogeneous turbulence field. Also, 

Batchelor and Townsend (1956) pointed out that, for 

steady uniform flows confined by rigid boundaries, the 

turbulence structure is homogeneous with respect to 

the longitudinal direction, and the instantaneous 

velocity of a fluid particle is necessarily a stationary 

random function of time as soon as the influence of the 

particle's initial position in the cross section has been 

lost. This says that the identity Us = U*-{~ux, where U* 

is the discharge velocity, (Q/A), is independent of x and 

becomes a stationary random function of time even 

though u'x is a function of position in the cross section 

also. Therefore, under conditions of uniform flow, 

equation 21 applies to lateral and longitudinal dispersion 

in planes which are parallel to the bed in wide channels, 

and equation 23 applies to longitudinal dispersion in 

any uniform channel. A very significant aspect of this 

conclusion is that it gives theoretical support, if not 

proof, to the applicability of the Fickian diffusion theory 

to dispersion in open channels for large dispersion 

times.

ANALOGY WITH THEORY OF LOCAL SIMILARITY

One of the basic hypotheses in Kolmogoroff's (1941) 

theory of local similarity in turbulence is that at 

sufficiently large Reynolds numbers the small-scale 

turbulence is locally isotropic, and that within the
OPEN-CHANNEL DISPERSION PROCESSES E7

portion of the turbulence energy spectrum characterized 

by the local isotropy the statistical characteristics of 

the turbulent motion are uniquely determined by the 

mean rate of energy dissipation per unit mass of fluid, 

E, and the kinematic viscosity, v.. A major result of this 

hypothesis is the definition of the length scale

 _/"8Y/4

'~W ' (25)

which characterizes the size of the smallest eddies whose 

energy is converted directly into heat by viscous 

dissipation. Batchelor (1953) gave the term "universal 

equilibrium range" to the portion of the turbulence 

energy spectrum in which the similarity principle defined 

by equation 25 is applicable. In this range, which 

covers the high frequency portion of the spectrum, the 

removal of energy by viscous dissipation chiefly at the 

upper end of the range is exactly compensated by the 

insertion of energy due to inertia transfer from the 

lower frequency components at the lower end of the 

range.

Orlob (1958, 1961) assumed an analogy between 

molecular and turbulent diffusion in which the viscosity, v, is replaced by a coefficient of turbulent diffusion, K. From dimensional considerations and the 

analysis of the lateral dispersion of small polyethylene 

particles on the water surface of an open channel, he 

arrived at the formula for the lateral dispersion coefficient at the surface,

(26)

which is similar in form to equation 25. In equation 26,

E=UgS=me&n rate of energy dissipation per 

unit mass of fluid in a broad open channel where 

U is the mean flow velocity, g is gravitational 

acceleration, and S is the slope of the energy 

gradient, and

Lz=UsLtz=length scale for large-scale lateral turbulence components, defined as the product of 

the mean velocity at the water surface, Us, 

and the Lagrangian integral time scale, Ltz, for 

the lateral component of turbulence.

Hino (1961) carried the analysis a step further by 

deriving a modified form of equation 26,

(26a)

in which

U

./ \UJ

accounts for the influence of boundary roughness and 

UT=^gynS is the shear velocity. Krenkel (1960) applied 

a similar model to that of Orlob in analyzing the longitudinal dispersion of a tracer solution in open-channel 

flow.

The physical justification for Orlob's analogy remains obscure despite (1) the fact that Orlob's and 

Krenkel's data are reasonably well represented by 

equations based on Kolmogoroff's similarity concept 

and (2) the resemblance of equation 26 to the empirical 

"Four-Thirds Law" which has found application in 

large-scale dispersion phenomena in the atmosphere 

and in the ocean (Richardson, 1920; Richardson and 

Stommel, 1948) . As pointed out by von Karman and Lin 

(1949), turbulent diffusion is associated with the lowfrequency end of the turbulence energy spectrum where 

the relationships among the turbulence characteristics 

and the energy transfer mechanisms are quite different 

from those in the universal equilibrium range. However, 

this observation does not necessarily rule out the validity of Orlob's analogy.

From a practical standpoint, equation 26 is not 

readily applicable to predicting dispersion in open 

channels because the length scale Lt, although necessarily related to the dimensions of the flow field, is 

difficult to determine. On the basis of his data, Orlob 

(1958) proposed the empirical formula

Yotsukura, Smoot, and Cahal (1964) suggested that 

in a broad open channel

Lx ~yn,

which does not seem to be compatible with Orlob's 

formula.

In summary, there is not yet sufficient evidence to 

either prove or disprove the validity of extending 

Kolmogoroff's similarity hypothesis to include dispersion by turbulence. If the extension is justified, an 

adequate method for predicting Lt in open-channel 

flows remains to be determined.

LONGITUDINAL DISPERSION BY DIFFERENTIAL 

CONVECTION DUE TO A VELOCITY GRADIENT

In this class of theories the basic Eulerian dispersion 

equation,

. ,_ , f»-+<*' (5a)

is converted by assumptions and transformations into 

simpler forms that permit an evaluation of the longitudinal dispersion coefficient, Kx, in terms of param-
E8 TRANSPORT OF RADIONUCLIDEiS BY STREAMS

eters which describe the flow. This value of Kx can 

then be used in equation 8 to predict the dispersion 

pattern. Considering longitudinal dispersion from an 

instantaneous plane source, which extends across the

channel in a two-dimensional flow so that all ^ terms ftz

equal zero, and assuming that the rate of spread of 

dispersant in the longitudinal direction due to turbulence is very small in comparison to the rate of spread 

due to differential convection imposed by a velocity 

gradient, equation 5a becomes

Elder (1959), following the method derived by Taylor 

(1954) for longitudinal dispersion in a turbulent axisymmetric pipe flow, introduced the transformations

which is the difference between the local time-averaged 

velocity U at y and the mean velocity in the cross 

section U, and

which is the longitudinal displacement from the plane 

moving at velocity U about which dispersion is occurring. Taylor and Elder furthermore assumed that at 

large dispersion times in the transformed system

3 «0, and that

where =constant. With these assumptions, equation 27 becomes

(28)

Upon defining a coefficient of longitudinal dispersion 

due to convection, Kc, in terms of the rate of transfer 

of dispersant across a section at £,

and substituting this definition into equation 28, equation 28 can be integrated twice to obtain

1 cvn ry i ry tfc=~ U* - U*dydydy. (29)

tfn Jo Jo fy Jo

Ellison (1960) integrated equation 29 by parts and 

obtained

1 Cvn i / Cv Kc==^- HI u*

tfn Jo fv \Jo

which is a more convenient form for computational 

purposes. Taking the origin at the water surface and 

employing the von Karman-Prandtl logarithmic velocity-distribution equation,

(30)

and also the Reynolds analogy, which states the 

equivalence of mass and momentum transfer,

-vc uv (31)

Elder (1959) integrated equation 29 and obtained

Kc=0.404 ^p- (32)

In equations 30 through 32, UT= Vro/p=V^n'S' is the 

shear velocity and K is the von Karman turbulence 

coefficient. The experimental results of Al-Saffar (1964), 

and the results of Kalinske and Pien (1944), who 

obtained fairly good agreement between ty evaluated 

by means of equation 31 and

fy=~2~dx

from dispersion experiments in a flume, provide 

empirical justification for applying Reynolds' analogy. 

If it is assumed that the turbulence is isotropic, the 

coefficient of longitudinal dispersion due to turbulence, 

which was neglected in the development leading to 

equation 32, can also be estimated. Using equations 30 

and 31,

Vn Jo 6

Following Taylor, Elder assumed that the convection 

and turbulence components of dispersion are additive, 

whereupon he obtained for the total coefficient of 

longitudinal dispersion in a rectangular channel having 

uniform two-dimensional flow,

(34)

If the parabolic velocity-distribution function

obtained by integrating equation 31 with
OPEN-CHANNEL DISPERSION PROCESSES E9

is used instead of the logarithmic distribution function 

in equation 29, then

#c=0.457 ^- (32a) K3

In view of the generally accepted observation that the 

value of Kc is extremely sensitive to small changes in 

velocity distribution, the agreement with equation 32 

is remarkably close.

The assumptions made in going from equation 27 to 

equation 28 are rather gross; however, they seem to be 

approximately true at large dispersion times. Therefore, 

equations 29 and 34 should be considered applicable 

only in the limiting case of large dispersion times.

Aris (1956) approached the problem of evaluating Kx 

in a more rigorous fashion. By taking the p'th moment 

of equation 5a with respect to £, where £=x Ut as 

before, so that

*, 0= «/ ( r (35)

and averaging the resulting equation over the crosssectional area, A, of the channel so that

C,(y, z, t)dydz, (36)

Aris derived an equation in mp for which he was able 

to obtain solutions for p=Q, p=l, and p=2. Employing 

the definition of the dispersion coefficient he was then 

able to obtain

IT- d°i « v T d imi =^ Lim ~jf=~ Lim -=- ( 

2 t->m dt 2 (_> dt \m0 \m0

Fischer (1964a) showed that this result for a rectangular 

channel with two-dimensional flow and isotropic turbulence becomes

Kx=lv+Kc

where Kc and ? are identical with the functions obtained by Elder which are given in equations 29 and 33. 

The result of Aris' analysis strengthens considerably 

the credibility of the results obtained by Taylor and 

Elder. Aris' method, in its more general form, also appears to be applicable to three-dimensional uniform 

flow where U=U(y, z), even though the mathematical 

difficulties are apt to be quite formidable.

Yotsukura and Fiering (1964, 1966) obtained some 

particular solutions of equation 27 by means of a numerical methods technique with the aid of a digital 

computer. The longitudinal concentration distribution 

curves generated in the solutions resembled experimental curves in that they were highly skewed in the 

upstream direction at small dispersion times and tended

to approach a normal distribution as an asymptotic 

limit at large dispersion times. Because Yotsukura and 

Fiering used a different velocity distribution function, 

their results cannot be compared quantitatively with 

Elder's.

Perhaps the most significant contribution of the 

differential convection theories has been to identify 

ynUT as an important parameter which relates both Kx 

and Tv to flow conditions. However, experimentally 

determined values of Kx/yn Ur ranging from 3 to 24 in 

laboratory flumes and from 15 to 800 in canals and 

natural streams have been reported in the literature by 

Glover (1964), Yotsukura, Smoot, and Cahal (1964), 

and Godfrey and Frederick (1963). Such a wide variation indicates the existence of other major influences 

such as velocity variation with respect to z, variation in 

K, boundary roughness, scale effect, and nonuniformity 

of the channel. Fischer (1964a) discusses several possible 

causes for these disparities.

If, as is suggested by the results of some experimental 

investigations, Kc/ynUT is a function of U/UT, dimensional considerations suggest that ynU is perhaps a more 

significant parameter than yn Ur. Indeed, some investigators (Orlob, 1958; Yotsukura and others, 1964) have 

found that for given sets of data a significant correlation 

exists between the Schmidt number &=Ki/v and the 

Reynolds number R=ynU/v. However, as with the 

experimentally determined Kx/yn Ur values, here also 

the disparities among various sets of data encompass 

approximately two orders of magnitude.

APPLICATIONS OF DIFFUSION THEORY IN 

SUSPENDED-SEDIMENT TRANSPORT

The basic differential equation for the dispersion of 

suspended sediment, when the concentration by volume 

is small, is

d<7, TT d<7 d f 5CH , ^

which is the same as equation 4 except that the subscript p here indicates reference to sediment particles. 

Upon introduction of the same simplifying assumptions 

as before, for a wide channel with uniform two-dimensional flow, and a uniform distribution of dispersant in 

the z direction, equation 37 becomes

itf 52 <7, <

With the exception of the term Vp ^ > where Vp is the

settling velocity of the sediment particles, equation 38 

has the same form as the corresponding equation for a 

fluid dispersant. Additional assumptions implicit in 

equation 38 are that all the sediment particles have
E10 TRANSPORT OF RADHONUCLIDEiS BY STREAMS

identical transport characteristics, that Vp is the same 

in turbulent and quiescent fluid, and that UP =U at a 

point. With respect to the last assumption, however, it 

should be noted that the cross-sectional average velocity 

of sediment in the x direction is

P" UCdy

TJ =^2______, 

p C*n

Cdy

Jo

(39)

which is different from the cross-sectional average 

velocity of flow, UAlthough equation 38 in its complete form has never 

been solved, certain special cases which permit drastic 

simplifications have been solved. For example, for 

the steady-state equilibrium conditions where both

 and equal zero, equation 38 reduces to the 

d£ dx

well-known equation

which for many years has served as the basis for computing the vertical distribution of suspended sediment 

in alluvial channels. For a turbulence tank in which

5(7 =0 and fp = constant, Dobbins (1944) obtained a 

ox

formal solution to the equation

(41)

which agreed well with the results of experiments in 

which the transient response to the shift from one 

steady state to another was investigated.

Experience in comparing predictions based on equations 40 and 41 with experimental results (Task Committee on Preparation of Sedimentation Manual, 1963) 

indicates that the theory is basically sound. Experiments by Brush (1962) on the dispersion of sediment 

particles in a submerged jet show, furthermore, that 

ePi tends to approach e f as the particle diameter, d, 

decreases, becoming approximately equal when d = 0.2 

mm. This result is in essential agreement with an 

analysis based on the equation of motion of a particle 

in a turbulent flow field (Tchen, 1947; Hinze, 1959), 

which shows that ePi ~ti when vLti/d2 > about 3.5. 

Therefore, in most instances ePv , like ev, can be evaluated 

by means of Reynolds' analogy (eq 31). Some comparisons of fv computed from Reynolds' analogy and ePv 

computed from equation 40 using experimental data 

were given by Vanoni (1946).

A very rough approximation of KXp for the longitudinal dispersion of suspended sediment was obtained 

by Elder (1959) by means of equation 29 with U*=

U Up, where Up is defined by equation 39. Assuming 

a parabolic velocity distribution, where fv is a constant, 

according to Reynolds' analogy, Elder obtained

Kx = "- (64+21/3-308/32 -210/33 -35/34), (42)

where fi=Vp/icUT. In the derivation of equation 42 it 

was also assumed that no deposit of sediment on the 

bed occurs and that the dispersing particles are distributed vertically according to the probability-density 

function

/J y i- fr?*^ where 0<,K1. (43)

Equation 43 is obtained by solving equation 40 for 

steady-state equilibrium where the diffusivity term,

is obtained by solving equation 31 for a logarithmic 

velocity distribution. Equation 42 has not been verified 

experimentally.

If it is assumed instead that the particles are distributed vertically according to the probability-density 

function

S N /"» f\ RfitJ 111

(44)

y

-i

which results from solving equation 40 for a constant 

diffusivity

then equation 29 leads to

(45)

where

60V*- 1 )

Equations 42 and 45 give nearly identical results for 

the ratio KXJKX . As shown in figure 56, KtJKt has 

an asymptotic value of unity as /3 approaches zero, 

rises to a weak maximum of 1.0055 at /3 = 0.033, and 

decreases rapidly to zero near #=0.5.

Although diffusion theory has not as yet (1965) been 

successfully applied to predicting longitudinal and 

lateral dispersion of suspended sediment particles, 

successful application of diffusion theory in some other 

areas of sediment transport suggest that such an 

application is possible.

DESCRIPTION OF THE EXPERIMENTS

All the experiments were conducted in a recirculating 

flume 150 feet long having a rectangular cross section
OPEN-CHANNEL DISPERSION PROCESSES Ell

FIGURE 1. Schematic diagram of the flume and circulation system, a, pumping units; hi, bj, orifices; c, head-box and diffuser; d, baffles and screens; e, flume (8 ft

X 2 ft X 150 ft); f, tailgate; h, tail-box; j, jacks supporting the flume.

7.83 feet wide by 2 feet deep. The interior of the flume 

was surfaced with tarred plywood except for a section 

of the right sidewall 40 feet long which was made of 

transparent plexiglass. Figure 1 shows a schematic 

diagram of the flume.

In order to satisfy approximately the restrictions on 

Elder's (1959) longitudinal dispersion theory that (1) 

the velocity varies only with distance above the bed 

and (2) the turbulence is homogeneous with respect to 

the x and z directions, a uniform arrangement of 

artificial roughness was placed on the bed of the flume. 

The roughness elements consisted of wooden cleats and 

were arranged in the pattern shown in figure 2.

HYDRAULIC DATA

Hydraulic data were obtained to provide a base of 

reference for the dispersion experiments. These data 

included water discharge, water-surface and bed slopes, 

depth, water temperature, and velocity profiles. A 

side-contracted orifice in the pipeline was used to 

measure water discharges. Bed- and water-surface 

elevations and slopes were determined using an engineer's level and rod and Lory point gages. Depths 

of flow were obtained by determining the difference 

in elevation between the mean water surface and bed 

profiles. No correction for the roughness cleats was 

applied to the bed level. However, the volume occupied by the roughness cleats, if spread evenly over 

the floor of the flume, would have added only 0.004 

foot to the bed level. Water temperatures were measured with a standard laboratory mercury thermometer. 

Because a period of two or three days each was required for completion of many of the runs, temperature fluctuations of up to 2°C occurred during several 

runs. Point velocity measurements were obtained with 

a 15 mm-diameter Delft propellor meter and the electronic pulse-shaping and counting system shown in 

figure 3.

Uniform flow conditions for all runs were established 

by manipulating the setting of the vertical-slat tailgate until the water-surface slope was parallel to the 

bed slope. This could be done quite accurately by 

intentionally setting the tailgate to obtain slight MI 

and Mz backwater curves and then finding the correct 

tailgate setting by interpolation.

Each type of dispersion experiment was performed 

using one or more of three different flow conditions. 

The hydraulic data describing these flow conditions 

and information characterizing the various types of 

dispersion experiments are listed in table 1.

Velocity profiles for each of the three flow conditions were obtained at cross sections approximately 

100 feet downstream from the flume entrance. In 

order to check the applicability of the two-dimensional 

flow assumption used in the dispersion analysis, velocity profiles were obtained at several verticals across 

the flume. For the intermediate flow condition (Q ~ 

7 cfs), sets of velocity profiles were also obtained at 

several cross sections along the flume in order to ascertain the distance downstream from the flume entrance required for the establishment of uniform flow.

281-661 O - 68 - 3
E12 TRANSPORT OF RADIONUCLIDES BY STREAMS

FIGURE 2. Arrangement of roughness cleats on the bed of the flume.

These measurements indicated that a distance of 30 

feet was sufficient to establish a pattern of velocity 

distribution which did not change very much in the 

x direction. The velocity measured at each point was 

an average during a period of 100 seconds.

LONGITUDINAL DISPERSION EXPERIMENTS

Longitudinal dispersion experiments were conducted 

with three different types of dispersants fluorescent 

dye, suspended silt-size particles, and floating polyethylene particles. A different set of equipment and 

techniques was required for each type of dispersant.

In both the dye and the suspended-sediment experiments the same basic procedure was employed. The 

dispersant was dumped into the flume from an elevated 

trough which extended across the flume a short distance 

above the water surface in such a way that the initial 

distribution of dispersant approximated an instantaneous plane source with the dispersant uniformly distributed over the entire flow cross section. The concenTABLE 1. Hydraulic conditions and types of dispersion experiments

Dispersion Run Q(cfs) J/n (ft) 17 (ft per sec) T(°C) Source Dispersant

Longitudinal LO-D-1

2

3

FS-1

2

CS-1

FG-1

3

CG-1

2

3

P-l

2 

3

Lateral LA-D-1

2

3

P-l

2

3

Vertical V-D-1

2

3

2.88

7.17

14.9

2.88

7.17

2.88

2.93

14.9

2.93

7.12

14.9

2.93

7.12 

14.9

2.93

7.12 

7.12

14.9

2.93

7.12

14.9

2.93

7.12

14.9

0.493

.798

1.217

.493

.798

.493

.486

1.217

.486

.814

1.217

.486

.814 

1.217

.486

.803 

.814

1.217

.486

.803

1.217

.486

.814

1.217

0.747

1.15

1.56

.747

1.15

.747

.770

1.56

.770

1.12

1.56

.770

1.12 

1.56

.770

1.13 

1.12

1.56

.770

1.13

1.56

.770

1.12

1.56

0.001

.001

.001

.001

.001

.001

.001

.001

.001

.001

.001

.001

.001 

.001

.001

.001 

.001

.001

.001

.001

.001

.001

.001

.001

20 Rhodamine B dye.

20

5

20

20

20

7

6

5

4

6 .

Instanteous plane source.

Do.

Pontacyl brilliant pink B dye.

15-SO/i natural silt.

Do.

53-62/1 natural silt.

<44/i glass beads.

Do.

53-62/1 glass beads.

Do.

Do.

6 ] (Polyethylene particles.

4 ^Intermittent point source. ! Do. 

10 J I Do.

7 1 f Pontacyl brilliant pink B dye.

20 l/i I Rhodamine B dye. 4 ^Continuous point source. 1 Pontacyl brilliant pink B dye.

7 J I Do.

6 1 (Polyethylene particles.

20 ^Intermittent point source. < Do.

9 J I Do.

6 ] (Pontacyl brilliant pink B dye.

4 ^Continuous point source. < Do.

8 J I Do.
OPEN-CHANNEL DISPERSION PROCESSES E13

FIGURE 3. Point velocity measuring equipment. Above: electronic counting and 

recording system. Right: propeller meter.

tration of dispersant was then measured as a function 

of time at each of four downstream sampling stations 

as the slug of dispersing material was carried by. The 

dispersion distances, defined as the distances from the 

source to the sampling stations, were 32.8, 65.6, 88.5, 

and 115.0 feet. The source was always located at least 

30 feet downstream from the flume entrance.

In runs LO-D-1 and -2, LO-CG-2, and the LO-FS 

and -CS series of runs, a simple dumping trough was 

used in which mixing and suspension of sediment was 

done with manual stirring rods. The dispersant was 

introduced into the flow simply by tipping the trough 

on edge. Because an even distribution of dispersant 

across the flume was difficult to obtain with this 

trough, and because some of the sediment remained 

deposited in the trough after dumping, an improved 

trough, shown in figure 4, was used in the remaining 

experiments. The improved trough contained a rotating

mechanical agitator for keeping the sediment in suspension, compartments for maintaining an even distribution of dispersant along the length of the trough, and 

a spring-loaded hinge mechanism which allowed one 

side of the trough to swing open from the top so that 

all of the sediment was instantaneously flushed into the 

flow. Visual inspection through the transparent sidewall 

indicated that a satisfactory distribution of dispersant 

throughout the depth of flow was obtained from both 

the troughs.

A continuous sampling and concentration-measuring 

system was used to measure the dye concentrations, 

and a discrete sampling system was used to collect 

suspended-sediment samples. Both systems are shown 

in figure 5.

The continuous system consisted of a %-inch ID 

sampling nozzle and polyethylene tube having a combined length of 7.3 feet feeding into a Model 111
E14 TRANSPORT OF RADTONTJCLJDE.S BY STREAMS

FIGURE 4. Improved dumping trough for dye and suspended sediment in 

longitudinal dispersion experiments.

FIGURE 5. Continuous and discrete sampling systems used for determining dye 

and sediment concentrations in longitudinal dispersion experiments. The continuous system is on the left, and the discrete system on the right.

Turner Fluorometer equipped with a continuous-flow 

door having a 5-cc continuous-flow square cuvette and 

%-inch I.P.S. fittings. The response speed of the fluorometer was increased by installing a lamp adaptor 

with a green T-5 envelope lamp. In some of the experiments the response was improved further by removing the fine mesh wire screen from the lamp 

adaptor. A strip-chart recorder connected to the 

fluorometer gave a continuous record of dye concentration in the solution flowing through the fluorometer. 

The sampling nozzle was positioned at middepth on 

the centerline of the flume. The velocity in the sampling nozzle and feed tube was maintained at approximately 2.5 feet per second for ah1 runs. The response 

characteristics of this system are discussed in the 

section on experimental evaluation beginning on page 

E65.

The Turner Fluorometer is capable of measuring 

concentrations of fluoroescent dyes such as Rhodamine 

B and Pontacyl Brilliant Pink B which are as small 

as 0.1 ppb. The theory and practice of fluorometry as 

applied to dispersion measurements in surface waters 

has been reviewed by Feuerstein and Selleck (1963).

Because there was only one continuous sampling 

system, it was necessary to repeat the dumping and 

the concentration-measurement operation for each 

dispersion distance. Sufficient time between successive 

dumpings was allowed to permit the dye to become 

uniformly distributed throughout the circulation 

system.

The discrete sampling system used in the sedimentdispersion experiments consisted of four separate sampling units with one unit serving each dispersion 

distance. Each unit consisted of three %-inch ID 

sampling nozzles feeding continuously through 4-foot 

lengths of %-inch ID mayon tubing into an enclosed 

plexiglass box in which negative pressure was maintained by means of a siphon. Inside the sampling box 

there was a rotatable circular rack with spaces for 

holding 25-cc glass vials at equal intervals around the 

perimeters of three concentric circles. There were 60 

spaces on each circle. The three sampling nozzles were 

positioned on the centerline of the flume at depths 

beneath the water surface of one-fourth, one-half and 

three-fourths of the flow depth. Each feed tube terminated at a point on top of the box immediately 

above a point on the circumference of one of the three 

circles, so that by rotating the rack the vials could be 

filled in rapid sequence. Thus, a sequence of 60 discrete 

samples from each sampling nozzle could be obtained 

during a single revolution of the rack. A ratchetoperated microswitch and an event recorder automatically provided a record of the times at which each 

group of three vials was being filled. In the experiments, 

the rack was rotated manually by means of a handle 

on top of the box according to a predetermined time 

schedule so that the elapsed time between the first 

and the last samples would be longer than the time 

required for the dispersing sediment cloud to pass the
OPEN-CHANNEL DISPERSION PROCESSES E15

sampling station. The sampling velocity in the nozzles 

and feed tubes was maintained at 1.35 feet per second 

in all of the sediment-dispersion experiments. As explained in the section beginning on page E65, the response characteristics of the sampling system would 

have been improved considerably if the velocity in the 

feed tubes had been increased to about 2.5 feet per 

second.

After the samples had been collected, the concentration of sediment in each sample was determined 

by using the Turner Model 111 Fluorometer as a nephelometer an instrument which measures the amount 

of light scattered by dilute suspensions. For this 

operation the fluorometer was equipped with a standard 

cuvette door. Of the various light-filter combinations 

which were tried, a color spec. No. 2A-12 primary 

filter together with a color spec. ND 1 percent secondary 

filter seemed to be the most satisfactory. With this 

arrangement, a linear calibration of fluorometer readings 

as functions of sediment concentration was obtained, 

and concentrations of silt-size particles down to less 

than 10 ppm could be measured. Each concentration 

measurement entailed transferring a part of the sample 

into a 5-cc square cuvette, placing the cuvette in the 

fluorometer, and reading the fluorometer dial. Owing to 

the high sensitivity of the fluorometer to any one of a 

number of possible sources of contamination, considerable care had to be observed with respect to 

cleanliness of hands and glassware. However, with a 

reasonable degree of care and practice, concentration 

measurements of different portions from the same 

sample could usually be repeated with deviations not 

exceeding about 5 percent.

In the sediment-dispersion experiments, since the 

discrete sampling systems at each of the four sampling 

stations could be operated simultaneously, only one 

dumping was required for each run. In addition, the 

continuous system was operated at one of the four 

stations during each run, which provided a direct 

comparison between the two systems.

Four kinds of silt-size particles were used in the 

sediment-dispersion experiments. These were natural 

silt in the 15-30^ size range, natural silt in the 53-62ju 

size range, glass beads in the <44jt size range, and glass 

beads in the 53-62ju size range. The natural silt particles 

had a specific gravity of about 2.65 and were irregular 

in shape. The glass beads had a specific gravity of about 

2.6 and were approximately spherical. Fall-velocity 

distribution curves for the four kinds of particles are 

shown in figure 6.

In the floating polyethylene particle experiments, the 

particles were released intermittently from a point 

source located on the centerline of the flume, and the 

travel time of each particle to each of four timing

stations along the flume was measured. The stations 

were located at distances of 16.4, 32.8, 49.2, and 65.6 

feet downstream from the source. The particles were 

disk shaped, having a diameter of one-eighth inch and 

a thickness of one-sixteenth inch, and had a specific 

gravity of about 0.96. The particles were the same kind 

as were used in lateral dispersion experiments by 

Sayre and Chamberlain (1964) and similar to those 

used by Orlob (1958, 1961). The source, which is 

shown in figure 7, consisted of a manually operated 

particle dispenser and a funnel, with a Ke-inch-diameter 

tip, positioned just above the water surface. Each 

timing station consisted of a string stretched across the 

flume immediately above the water surface and a unit 

consisting of a telegraph key and one or more flashlight 

batteries connected to a strip-chart recorder as shown 

in figure 8.

The release time of each particle was marked on the 

recorder chart by the event-marking pen, and the 

arrival time of a particle at each station was marked 

by a pulse on the recorder chart caused by depressing 

the telegraph key. The station at which a particular 

arrival had occurred was identified by the pulse height 

which was proportional to the number of batteries in 

the branch of the circuit for that station. With a chart 

speed of 8 inches per minute, the traveltime of each 

particle could be estimated to the nearest 0.1 second 

by measuring the distance on the chart between release 

and arrival pulses.

For each flow condition two sets of data were obtained. They differed only in the time interval between 

particle releases. For each set of data at least 100 

particles were released in sequence.

LATERAL DISPERSION EXPERIMENTS

Lateral dispersion experiments were conducted with 

both fluorescent dye and floating polyethylene particles as dispersants. In both kinds of experiments the 

dispersant was released from a point source in the 

center of the flume and the distribution of dispersant 

across the flume at each of four or more stations downstream from the source was determined. The four 

primary stations were located at dispersion distances 

of 16.4, 32.8, 49.2, and 65.6 feet from the source. In 

some of the experiments lateral distributions were 

obtained at dispersion distances of 88.5 and 115.0 

feet also.

In the dye experiments a concentrated dye solution 

was released continuously at a constant rate from a 

%-inch brass tube. The horizontal leg at the discharge 

end of the tube was oriented in the direction of flow, 

and the rate of release of dye solution was set so that 

the discharge velocity from the tip was equal to the
E16 TRANSPORT OP RADIONUCLIDEiS BY STREAMS

1.0 

0.8

0.6 

0.4

0.2

0.1

0.08

0.06

0.04

0.02

0.01

J&,

0.0 ,'V

T=20°C

0.01 0.1 0.2 0.5 1 2 5 10 20 30 40 50 60 70 80 90 95 98 99 99.5

PERCENTAGE OF PARTICLES HAVING FALL VELOCITY LESS THAN THAT INDICATED 

FIQUBE 6. Fall-velocity distribution curves for the sediments used in longitudinal dispersion experiments.

99.9 99.99

local ambient flow velocity. The source arrangement 

is shown in figure 9.

Lateral dye concentration profiles were obtained by 

slowly moving the sampling nozzle of the continuoussampling system, which was described in the previous 

section, across the flume. The traversing mechanism, 

shown with the continuous sampling system in figure 

10, was mounted on the flume carriage. It consisted 

principally of a chain drive operated by a hand crank 

and a series of regularly spaced electrical contacts, 

which, together with the recorder event-marking 

system, recorded the lateral position of the sampling 

nozzle as it was being moved across the flume. Approximately 2-3 minutes was required for each traverse. 

As with the longitudinal dye dispersion experiments 

sufficient time between traverses was allowed to permit 

the dye to become uniformly distributed throughout 

the flume circulation system. At each dispersion distance three traverses were made, with both the source 

and the sampling nozzle positioned respectively at 

one fourth, one-half, and three-fourths of the flow depth.

The equipment and techniques employed in the 

lateral dispersion experiments with floating polyethylene particles were virtually the same as those described 

by Orlob (1958, 1961) and Sayre and Chamberlain 

(1964). The intermittent point source arrangement was 

the same as that used in the longitudinal dispersion 

experiments. At a given dispersion distance the particles 

were caught in a sieve having 1-centimeter-wide compartments. The sieve was placed across the flume as 

shown in figure 11. The lateral distribution of particle 

paths was determined by counting and recording the 

number of particles trapped in each compartment. This 

operation was repeated with lots of 100 particles at 

each dispersion distance for time intervals between 

particle releases of 1, 2, and 5 seconds.

VERTICAL DISPERSION EXPERIMENTS

The vertical dispersion experiments with fluorescent 

dye were performed with the limited objective of 

determining the dispersion distance required in order 

to obtain a uniform vertical distribution of dye solution
OPEN-CHANNEL DISPERSION PROCESSES E17

FIGTTKE 7. Point source used in polyethylene-particle dispersion experiments.

Station 1 Station 2 Station 3 Station 4

Source

FIGUBE 8. Circuit diagram of time-measuring system used to determine traveltimes of polyethylene particles.

originating from a continuous point source positioned 

at middepth on the centerline of the flume. No attempt 

was made to define the vertical dispersion pattern. The 

source and sampling systems were the same as those 

used in the lateral dye dispersion experiments. Beginning at a dispersion distance of about 3 feet, several 

vertical concentration profiles were taken along the 

centerline of the flume at increasing dispersion distances 

until the concentration in the vertical was found to be 

essentially uniform. A concentration profile was obtained by slowly moving the sampling nozzle, which 

was mounted on a point gage assembly, from the bed

FIGUBE 9. Point-source arrangement used in lateral dispersion experiments with 

fluorescent dye.

FIGTJKE 10. Continuous-sampling system and traversing mechanism used in lateral 

dispersion experiments with fluorescent dye.
E18 TRANSPORT OF RADIONTJCLIDEiS BY STREAMS

14 

12

3'°

fe

Iz

LU g

o

u.

LU

O 

O

LU 6

<

(/)

V)

% 4

2

0

x5

tw\V<

VV^

^

<

/

^

x = QA2. X^Q.^

SYMBOL FLOW

 1

* 2

. 3

42ft

FIGURE 11. Compartmented sieve collector used in lateral dispersion experiments 

with floating polyethylene particles.

to the water surface or vice versa. The time span for 

each vertical profile was about 2-4 minutes.

PRESENTATION AND ANALYSIS OF DATA 

HYDRAULIC DATA

The characteristics of the flows in which the dispersion experiments were conducted are defined by the 

hydraulic data. The mean flow characteristics of discharge, depth, velocity, slope, and temperature are 

listed in table 1. In figure 12 the resistance coefficient, 

U/UT, is shown as a function of the normal depth, yn . 

The data follows the trend of the von Karman-Prandtl 

resistance law for turbulent flow in a wide channel with 

a rough bottom,

f^- -

Ur K

(46)

In figure 13 the measured velocity profiles are seen to 

compare well with the von Karman-Prandtl law for 

velocity distribution near a rough boundary,

C7 

C7/ (47)

For the curves in both figures 12 and 13, the values of 

the von Karman turbulence coefficient and the roughness measure are respectively «=0.42 and x=0.042 

foot. The application of equations 46 and 47 in roughboundary open-channel flow has been discussed in 

detail by Sayre and Albertson (1963).

0.2 0.3 0.4 0.5 0.6 0.8 1.0

NORMAL DEPTH (yn), IN FEET

2.0 3.0

FIGURE 12. Variation of resistance function with depth of flow.

The velocity distribution measurements, which 

were obtained at several verticals across the width of 

the flume, indicated that there was no significant 

variation of velocity with respect to lateral position 

except in the immediate vicinity of the sidewalls. The 

thickness of the boundary layer within which the 

velocity distribution was influenced by the proximity 

of the sidewalls was not greater than 0.5-1.0 foot. Thus, 

the condition of two-dimensional flow was essentially 

satisfied.

The mean flow data in table 1 and figure 12 show 

some scatter. For some undetermined reason the conditions of flows 1 and 2 sometimes could not be duplicated. This tended to occur when a period of several 

weeks or months was allowed to elapse between different sets of runs for the same flow condition. The 

most likely explanation is that alternate periods of 

wetting and drying, coupled with loading changes, 

caused the plywood bed of the flume to assume slightly 

different configurations which could not be duplicated.

LONGITUDINAL DISPERSION

DYE

The basic data in the LO-D series of experiments 

consisted of sets of distribution curves in which fluorometer readings were recorded on strip charts as a continuous function of time. For convenience in analyzing 

the data, each distribution curve was normalized by
OPEN-CHANNEL DISPERSION PROCESSES E19

dividing the fluorometer readings by the area under the 

curve, which gave all the normalized distribution curves 

an area of 1. The normalized distribution curves are 

shown as the dashline curves in figures 14,15, and 16 for

flows 1, 2, and 3 respectively. The time base for these 

curves was corrected for response lag in the concentration-measuring system, using the appropriate T values 

from table 11 in the section beginning on page E5. The

0.80 

0.60

0.40

UJ 

UJ

u_ 

- 0.20

O

UJ

m 0.10

£ 0.08 

O 

2 0.06

UJ

z 0.04 

fc

Q

0.02 

0.01

X

/

i

r

UT =

^ /

7

u.

L(iri

J 

Fk 

= 0.12

|^ + 1

w 1 

5 ft pe

)

sec

0 2 4 6 8 10 12 14 

 Average velocity in middle 6 ft of flume

l.UU

0.80 

0.60

0.40 

0.20

0.10 

0.08

0.06 

0.04

0.02 

0 01

^

 /

!

JW

 <

7

UT-

^=-MlnUr *

yy

Flow 

3.160

I + 1 *

2 

ft per sec

10.00 

8.00 

6.00

4.00 

2.00

1.00 

0.80 

0.60

0.40 

0.20

0.10 

0.08 

0.06

0.02 

0.01

/

y

UT*

/

k[/

Ur

FlowS 

= 0.198 ft pe

^

 

 ^

^

I (In

H"

 &-r

 u ,< J+l)

' sec

0 2 4 6 8 10 12 14 8

DIMENSIONLESS VELOCITY (U/UT)

 Average velocity near side wall; 0.65 ft from wall in flow 2, 0.33 ft from wall in flow 3 

x = 0.42, X =0.042 ft for all flows

10 12 14

FIGUBE 13. Velocity profiles for flow conditions in dispersion experiments.

0.06

0.05

- 0.04 i

CO

Q

O 0.03

UJ 

CO

z. 0.02 

T?

0.01

10

x = 32 8 feet

./s

20 30 40 50

x= 65.6 ft

60 70 80 90 100 50 60 70 80 90 100 110 120 130 140 150

0.05

0.04,

0.03

0.02

0.01

x= 88.5 ft x= 115.0 ft

80 90 100 110 120 130 140 150 160 170 180 110 120 130 140 150 160 170 180 190 200 210

DISPERSION TIME (t), IN SECONDS 

 Equation 9a Observed at y/yn = 0.50

FIGURE 14. Relative concentration of dye as a function of time at various dispersion distances in run LO-D-1.

281-661 O -
E20 TRANSPORT OF RADKQMJCLIDEiS BY STREAMS

0.09

0.08

0.07

0.06

0.05

0.04

0.03 

Q.02 

0.01

0

I

0.05

0.04

0.03

0.02

0.01

0

x =32.8 ft

0 10 20 30 40 50 60 70 80 90 100 20 30 40 50 60 70 80 90 100 110 120

40 50 60

x= 88.5 ft

70 80 90 100 110 120 130 140 60 '0

x= 11,5.0 ft

80 90 100 110 120 130 140 150 160

DISPERSION TIME (t), IN SECONDS 

 Equation 9a Observed at V/Vn= 0.50

FIGURE 15. Relative concentration of dye as a function of time at vario is dispersion distances in run LO-D-2.

solid-line curves which were obtained from equation 

9a, using Kx values determined from the data, provide a 

comparison between the experimental distribution 

curves and theoretical curves given by the one-dimensional Fickian diffusion model. Characteristically, the 

experimental curves tend to have a steeper rising limb 

and a somewhat longer tail on the receding limb than do 

the theoretical curves. However, the shape of the 

experimental curves tends to approach the form of the 

more symmetrical theoretical curves as the dispersion 

distance increases, which suggests that equation 9a 

is valid as an asymptotic limit. Even at relatively 

short dispersion distances there is fairly good agreement 

between the theoretical and experimental curves. The 

differences between individual realizations of the experimental curves, when repeated, are in several cases 

as great as the differences between the experimental 

and theoretical curves. This is an indication of the 

randomness of the dispersion process. Irregularities hi 

the experimental curves such as the kinks in figure 16 

were presumably caused by large eddies. As a rule, such

irregularities tended to even out with increasing dispersion time.

Mean traveltimes, calculated from the experimental 

distribution curves by the method of moments using 

the formula

(48)

are shown as a function of dispersion distance in figure 

17. In ec uation 48 the width of the increments A2* was 

varied between 1 and 4 seconds, depending on the rate 

of chango of concentration with time; tt is the dispersion 

time at :he midpoint of the increment, and /(<*; x) is 

the avenge value of the relative concentration in the 

increment. For comparison, the relationship between 

mean tri.veltime and dispersion distance according to 

the Ficki an theory (eq 10) is also shown. Except for a 

3-second displacement, agreement between the experimental ind theoretical relationships is good. The 

reciprocal of the slopes of the curves, which indicates 

the mear rate of movement of the dye, is equal to the
RELATIVE CONCENTRATION f(t; x), IN SECONDS

p ppppppp

b bbbbbbb 

W o >- Mooji yiOi-vi

0| | | f r
E22

180

TRANSPORT OF RADOONTJCLIDES BY STREAMS

180

20 40 60 80 100

DISPERSION'DISTANCE (x), IN FEET

120

FIGURE 17. Mean traveltime as a function of dispersion distance, longitudinal 

dispersion of dye.

average flow velocities determined from the hydraulic 

measurements, as required by the Fickian theory. The 

3-second displacement could have been caused either 

by an initial vertical distribution of dye which was 

weighted toward the water surface or an overcorrection 

for response lag.

Variances of the experimental distribution curves 

were calculated, also by the method of moments, using 

the formula

*?=§ %f(tt; z)A*,-?. (49)

These variances, like the time base, were corrected for 

system-response lag by subtracting the appropriate a\ 

values taken from table 11 in the section beginning on 

page E65. The corrected variances are shown as a 

function of dispersion distance in figure 18. With the 

exception of the region near the origin or at short 

dispersion distances, the data show that the relationship 

between the variance and the dispersion distance is 

linear. The linearity of this relationship, which is 

implicit in the Fickian diffusion theory, permits the

20 40 60 80 100 

DISPERSION DISTANCE (x), IN FEET

120

FIGURE 18. Variance of traveltime as a function of dispersion distance, longitudinal dispersion of dye.

longitudinal dispersion coefficient, Kx, to be determined 

from the relationship

Z7" da* t=~iT Lim ~T^' 

2 *_> , dx

(50)

Equation 50 was used for evaluating Kx from the data.

The fact that the experimentally determined o^ versus 

x curves fall consistently below the curves given by 

equation 11 indicates that the dispersion rate during 

the initial phase of the dispersion process was somewhat less than that specified by the one-dimensional 

Fickian theory. Going from flow 1 to flow 3, the differences between the actual and theoretical initial 

dispersion rates tends to increase. This suggests that 

the duration of the initial phase within which the 

Fickian theory does not apply is a function of the flow 

conditions.

In figure 19 the peak relative concentrations of the 

experimental distribution curves are shown as a function of the dimensionless distance. Comparison of the 

data with the maxima given by equation 9a indicates 

that the rate of attenuation of the observed peak con-
OPEN-CHANNEL DISPERSION PROCESSES E23

It)

0.07

0.06

0.05

0.04

0.03

0.02

0.01

Run LO-D-1 

Run LO-D-2 

Run LO-D-3

From eq 9 a

50 100 150 200 250

DIMENSIONLESS DISPERSION DISTANCE (xU/KJ

300 350 400

FIGTJEE 19. Dimensionless peak relative concentration as a function of dimensionless dispersion distance, longitudinal dispersion of dye.

centrations with increasing dispersion distance is in 

reasonably close accord with that given by the Fickian 

theory. The fact that the experimentally determined 

peak concentrations tend to exceed the theoretical 

concentrations at small dispersion distances is consistent 

with the previous observation that actual initial dispersion rates tend to be less rapid than the rates specified by the theory.

In figure 20 the recovery ratio, Am/At, for each of the 

distribution curves is plotted against dispersion distance. The recovery ratio is an index of the amount of 

dispersant actually recovered by the sampling apparatus 

relative to that which would be recovered if the sampling 

were truly representative and the calibration perfect. 

It is conveniently defined as the ratio of the area, Am, 

under the experimental distribution curve to the area, 

At} under the theoretical distribution curve. 

Numerically,

A __ A'JF

At 35,300 W/Q (51)

where A'm is the area under the experimental curve in 

fluorometer unit-seconds, F is the sensitivity of the 

fluorometer, determined by calibration, in fluorometer

units per parts per billion of dye, and W is the weight 

of dye, in grams, used in the experiment. The numerical 

conversion factor changes the units of At from gramseconds per cubic foot to parts per billion-seconds.

The data in figure 20 show considerable scatter. 

Probably the most significant factor contributing to the 

scatter was that the concentration of dye at the sampling point was not representative of the mean concentration in the cross section. This would be consistent 

with the randomness of the dispersion process which is 

apparently more pronounced in the initial phases. In 

addition, some of the more likely sources of scatter are 

errors in the operations of weighing, pipetting, and 

dilution in the preparation of the dosing and calibration solutions.

The information obtained from the longitu dinal 

dispersion experiments with dye is summarized in 

table 2.

POLYETHYLJ3NE PARTICLES

The basic data from the LO-P series of experiments 

consisted of traveltime for known distances of single 

floating polyethylene particles. The particles were 

released sequentially in lots of approximately 100, with
E24 TRANSPORT OF RAmONHCTCLIDES BY STREAMS

1.3 

1.2 

1.1 

1.0 

0.9 

0.8

1.5

-^ 1-4 

5

£ 

S

0 1.3

< 

oc

g 1-2

o 

o

LJ

o: 1.1 

1.0

1.4 

1.3 

1.2 

1.1 

1.0 

0.9

0.8

0.7 

C

^

Average

Run LO-D-1

I

 A 

X

Average A

A

Run LO-D-2

A

A

 Continue 

@ Discrete <

us system 

>ystem

 *

/

m

®

Average

Run LO-D-3

) 20 40 60 80 100 120 14

DISPERSION DISTANCE (x), IN FEET 

FIGURE 20. Recovery ratio for area under concentration versus time curves as a function of dispersion distance, longitudinal dispersion of dye.
OPEN-CHANNEL DISPERSION PROCESSES E25

TABLE 2. Longitudinal dispersion of dye: summary of data

Run (LO-D-) x (ft)

F (fluorom- At Am W(gta) eter units (ppb-sec) (ppb-sec) Usec)

<rt* f(f, Z)max

(sec*) (sec-i)

per ppb)

I....... .... - 32.8

65.6

88.5

115.0

2................... 32.8

65.6

88.5

115.0

3. ............... 32.8

65.6

88.5

115.0

32.8

(i) 32. 8

0.0330

.0495

.0667

.0667

.0667

.0525

.110

.110

.110

.110

.0515

.0412

.0690

.0768

.0706

.0706

.0808

.101

.0412

.0412

2.96

2.72

2.80

2.88

2.84

2.76

2.64

2.72

2.72

2.68

6.75

6.80

6.85

6.80

6.80

6.80

6.70

6.80

6.75

4.84

405

607

819

819

819

268

542

642

542

642

122

97.5

140

177

167

167

191

237

97.5

97.5

405

691

907

792

803

362

675

666

599

627

123

86.6

114

227

192

176

202

234

76.5

71.3

41.1

85.5

116.9

163.6

150.4

26.5

54.2

66.9

78.1

98.6

19.9

19.4

39.6

38.3

54.1

53.5

70.2

71.2

19.1

20.6

47

96

133

174

144

28

66

52

77

104

11.8

12.6

34

28

41

48

65

74

8.0

9.1

0.0545

.0408

.0365

.0303

.0343

.0820

.0470

.0545

.0440

.0382

.119

.118

.0835

.0705

.0652

.0601

.0480

.0515

.132

.163

1 Discrete system.

a time interval, A£, between successive releases. In 

analyzing the data it was assumed that the flow 

velocity at the water surface was a statistically stationary function of time so that the statistical characteristics 

of the distribution of traveltimes would be the same as 

if the entire lot of particles had been released 

simultaneously.

A typical set of traveltime distribution data is shown 

in the form of a histogram in figure 21. The ordinate 

of the histogram, which represents the relative frequency with which the time required for a particle

to travel from 0 to a; is in the range t} ^-< Afc

2

s

s)= (52)

In equation 52,

tj = time at the midpoint of the j'th time interval,

A^ =0.3 sec = duration of the j'th time interval,

n} = number of particles counted in the/th time

interval, and 

N=total number of particles counted.

For comparison, the curve representing equation 9a 

with U = US and Kx =KXs evaluated from the data 

is also shown in figure 21. The subscript s denotes

Run LO-P-2 

x = 49.2 ft

A t 15 sec

30 31 32 33 34 35 

TRAVELTIME (t), IN SECONDS

36 37 38 39

FIGURE 21. Typical traveltime distribution, longitudinal dispersion of polyethylene particles.
E26 TRANSPORT OF RADIONUCLIDES BY STREAMS

10 20 30 40 

DISPERSION DISTANCE I

50 60 

,IN FEET

FIGURE 22. Mean traveltime as a function of dispersion distance, longitudinal 

dispersion of polyethylene particles.

reference to the water surface. The irregularity of 

the histogram, which was typical of all sets of data, 

indicates that the number of particles in each lot, 

^=100, was too small to permit a detailed comparison between the shapes of the histograms and the distribution curves generated by equation 9a. However, 

N was sufficiently large to permit reliable estimates 

to be made of the mean and variance of the traveltimes. 

In figure 22 the mean traveltime

(53)

where tt is the traveltime of the i'th particle, is shown 

as a function of dispersion distance. The average 

velocities of the particles, indicated by the slopes 

dx 

-=, are US = 1.1Q, 1.49, and 2.16 feet per second. These

agree closely with the water-surface velocities of 1.03, 

1.53, and 2.06 feet per second calculated by equation 

47 for the flow conditions in runs LO-P-1, LO-P-2, 

and LO-P-3. Changing the time interval, At, between 

particle releases had no apparent eftect on the mean 

traveltimes.

In figure 23 the unbiased estimate of the variance of 

the traveltimes, calculated according to the formula

(54)

is shown as a function of dispersion distance. Longitudinal dispersion coefficients, KXg, for the particles 

were evaluated from this data using equation 50 with 

U replaced by Tjs. The results for flow 3 differ markedly 

from those for flows 1 and 2 in two important respects. 

The first difference is the apparent dependence of the 

<r\ versus x relationship on At in run LO-P-3. No 

satisfactory explanation for this has been found. 

The argument that the periodicity of the turbulence 

was sufficient to produce so pronounced an effect is 

inconsistent with the notion that turbulence is basically 

a random phenomenon. The second difference is the 

obvious nonlinearity of the d\ versus x relationships 

in run LO-P-3 for z<20 feet. The results from run 

LO-D-3 in figure 18 are similar in this respect. The 

initial nonlinearity may be associated with a Lagrangian velocity correlation as suggested by the 

theory of diffusion by continuous movements. However, 

the existence in flow 3 of a scale of turbulence sufficiently large to indicate a persistence of correlation 

over a distance of 20 feet would be somewhat surprising. 

It is quite possible that for small x the relationship 

between tr? and x would be nonlinear for flows 1 

and 2 also, but that the region of nonlinearity does 

not extend to sufficiently large x to be apparent from 

the data in figure 23.

A summary of the longitudinal dispersion data 

obtained in the polyethylene particle experiments is 

given in table 3.

SUSPENDED 8LLT-8IZE PARTICLES

In contrast to the dye dispersion experiments where 

the basic data for the concentration-distribution curves 

consisted of continuous recordings of the fluorometer 

reading, the basic data for most of the concentrationdistribution curves in the silt dispersion experiments 

consisted of a set of discrete fluorometer readings for a 

group of 25-30 small water-sediment samples. These 

samples were obtained in rapid sequence as the silt 

cloud passed the sampling station. Also, whereas 

sampling was restricted to one point at middepth in 

the dye dispersion experiments, samples were collected 

simultaneously from three sampling depths (y/yn=Q.25, 

0.50, and 0.75, where y is the distance above the flume 

bed) in the silt dispersion experiments.

The distribution data shown in figure 24, where 

each plotted point represents the fluorometer reading 

and dispersion time for a single sample, is quite typical. 

Figure 24 illustrates quite well the degree of definition 

which is obtainable by nephelometric tracing techniques. The degree of definition is limited not only by 

sample reading techniques, which are complicated by 

the tendency of the particles to settle while a reading 

is being made, but also by sampling statistics as related 

to the number of particles contained in the sample.
OPEN-CHANNEL DISPERSION PROCESSES E27

10 20 30 40 

DISPERSION DISTANCE (x),\M FEET

50 60 70

FIGUBE 23. Variance of traveltime as a function of dispersion distance, longitudinal dispersion of polyethylene particles.

Both of these restrictions tend to become stronger 

with increasing particle size.

The data for all the experiments were plotted in the 

manner shown in figure 24. Smooth curves were drawn 

by eye through the data and a line representing the

background fluorometer reading was drawn. The time 

base was corrected for response lag using the appropriate r values from table 11 in the section beginning 

on page E65. The curves were normalized by dividing 

the net fluorometer readings (fluorometer reading minus

281-661 O - 68 - 5
E28

60 

50 

40 

30 

20

CD 10

z

Q 

< UJ n

o: ° 

ca

UJ

uJ 60

O

o:

§ 50

_j

LL.

40 

30 

20

10

0

5

TRANSPORT OF RADM

_____ ft,

._.__ ._

 /

^

i/

Batj

r

L

ikgroun

V

 \

t

\

d = 1.5

<

/

/

Bac

/A

kgrourx

x

V

<

J = 8.5

RunL<

k

 ^v

._ .

RunLC

V

)-FS-l

^-- 

-FG-1

<^

~ -

 j ~

0 60 70 80 90 100 110 120 130 140

l°MUCL

60

50 

40 

30 

20 

10 

0 

24 

20 

16 

12 

8

4

0

5

JDES BY STREAMS

 1

 

^

/

 / < ' 

V

/

ackgr

./

/

/

E

fr

\l

ackgr

0 60 70 80 90 100 110 120 130 140 150

DISPERSION TIME (t), IN SECONDS

FIGURE 24. Typical traveltime distribution data, longitudinal dispersion of natural silt and glass beads, flow 1, z=65.6 feet, y/yn =0.50.

TABLE 3. Longitudinal dispersion of floating polyethylene 

particles: summary of data

Run (LO-P-) At (sec)

1....... .... 10

20 

2.. . .. 15

40 

3-.- ..-..... 10

30

* (ft)

16.4

32.8 

49.2 

65.6 

16.4 

32.8 

49.2 

65.6

16.4

32.8 

49.2 

65.6 

16.4 

32.8 

49.2 

65.6

32.8 

49.2 

65.6 

16.4 

32.8 

49.2 

65.6

Number of 

particles 

released

116

115

-ion

98 

115

110

Number of 

particles 

counted '

105

99 

93 

99 

111 

114 

110 

109

11Q

96 

105 

112 

97 

97 

95 

95

105 

91 

84 

110 

107 

105 

106

T(sec)

15.1

29.9 

44.8 

58.8 

15.4 

30.2 

45.0 

59.9

11.1

22.1 

33.2 

44.1 

10.9 

21.5 

32.8 

44.0

14.9 

22.8 

31.0 

7.6 

14.8 

22.4 

30.0

<r (sec 2 ) t

0.74

1.28 

2.48 

3.19 

.75 

1.71 

2.39 

3.27

.45

1.35 

1.97 

2.00 

.42 

1.14 

1.64 

3.07

.27

1.12 

2.22 

3.63 

.22 

.65 

1.29 

2.06

1 Particles which were not counted were not seen by the observers.

background) by the area enclosed by the distribution 

curve and the background line.

In each run, one distribution curve was obtained 

with the continuous sampling and recording system. 

This data was processed to obtain normalized distribution curves in exactly the same way as the dye dispersion data.

The normalized distribution curves for all the silt 

dispersion experiments are shown in figures 25 through 

32. Given in these figures for the indicated dispersion 

distances are the distribution curves obtained at each 

of the three sampling depths, the distribution curve 

(for one dispersion distance only) obtained with the 

continuous sampling and recording system, and, for 

comparison, the distribution function according to 

equation 9a. The different curves are identified by the 

symbols defined in the figures.

The values of Kx used in obtaining the curves from 

equation 9a are either the same as those determined 

from the dye dispersion experiments for the same flow 

conditions, or they are adjusted values of Kx, used 

where the flow conditions as listed in table 1 differed 

slightly from the conditions for the corresponding dye 

experiments. The adjustment was made by assuming, 

in accordance with equation 34, that Kx is directly 

proportional to ynUr so that Kx can be adjusted in 

proportion to the change in ynUrThe original intent was to compare the sediment 

dispersion curves directly with the dye curves. However, 

because of the inability to repeat flow conditions more
OPEN-CHANNEL DISPERSION PROCESSES E29

0.07

0.06

0.05

0.04

g 0.03

z 

o

LJ 0.02

z

-< 0.01

«

* O°i'

= 32.8ft z=65.6ft

10 20 30 40 50 60 70 80 90 100 50 60 70 80 90 100 110 120 130 140 150

80 90 100 110 120 130 140 150 160 170 180 110 120 130 140 150 160 170 180 190 200 210

DISPERSION TIME (t), IN SECONDS 

 Equation 9a - Observed at *//# =0.25 Observed at y/yn =0.50 Observed at V/yn =0.75 - Continuous system

FIGURE 25. Relative concentration of 15-30/t silt as a function of time at various depths and dispersion distances in run LO-FS-1.

closely it was considered preferable to compare both to 

a common standard, equation 9a.

The distribution curves for the fine particles for the 

most part agree fairly closely with the curves derived 

from equation 9a. Indeed, in several instances the fine 

particle curves seem to follow the theoretical curves 

more closely than do the dye curves. In comparison, 

the curves for the coarse particles characteristically 

tend to have lower peaks and longer tails on the receding 

limb. This results from the tendency of the coarse 

particles to settle more rapidly, which is conducive to 

the establishment of a concentration gradient that is 

more heavily weighted toward the bottom. Therefore, 

there is greater probability that the coarse particles 

will either be retarded in the slower moving region of 

flow near the bed or be deposited on the bed. In opposition to the settling process, however, turbulence is 

continually tending to redistribute the particles more 

uniformly in the vertical, so that some of the particles 

which have been temporarily retarded or deposited are

reentrained in the main stream of the flow. These are 

the particles which contribute to the tails on the curves.

A comparison of the curves for runs LO-CG-1, 

LO-CG-2, and LO-CG-3 in figures 30, 31, and 32 

shows that the tails tend to become progressively more 

pronounced going from flow 1 to flow 3. This suggests 

that the capacity of the turbulence to reentrain the 

retarded and deposited particles tends to increase with 

the depth and velocity of the flow. The lack of any 

pronounced tails in the curves for the fine particles 

suggests that the level of turbulence was sufficient to 

largely inhibit the establishment of a negative concentration gradient. These observations are consistent 

with the recovery ratio data in figures 39 and 40, which 

give an indication of the settling rates, and the mean 

traveltime data in figure 34, which indicates the mean 

velocity of the particles.

A comparison of the distribution curves obtained at 

the different sampling depths indicates a two-dimensional relative concentration-distribution pattern,
E30 TRANSPORT OF RADIONTJCLIDES BY STREAMS

10 20 30 40 50 60 70 80 90 100 20 30 40 50 60 70 80 90 100 110 120

40 50 60 70 80 90 100 110 120 130 140 60 70 80 90 100 110 120 130 140 150 160

DISPERSION TIME (t), IN SECONDS 

 Equation 9a Observed at y/yn =0.25 Observed at y/yn =0.50 Observed at y/yn =0.75 Continuous system

FIGURE 26. Relative concentration of 15-30^ silt as a function of time at various depths and dispersion distances in run LO-FS-2.

/(£> y> x), which is repeated quite consistently in most 

of the observations. As the observation level is raised 

from the bed toward the water surface, the position of 

the distribution curve, f(t; y, x), is shifted slightly along 

the time axis in the direction of decreasing time, and 

the peak relative concentration tends to increase 

slightly, reflecting a small decrease in the degree of 

dispersion. This suggests that a longitudinal cross section taken through the cloud of dispersing particles 

would be somewhat trapezoidal in shape with both the 

leading and trailing ends inclined in the direction of 

flow, but with the leading end less inclined. This result 

is entirely consistent with the distribution pat'terns, 

C(x> y> t), developed by Yotsukura and Fiering (1964) 

from their numerical solution of equation 27. The data 

shows, however, that the degree of dependence of 

/(£) U) x) on y in the present experiments is quite small 

and for most purposes can safely be ignored.

The normalized relative concentration-distribution

data for the silt-size particles were processed in exactly 

the same way as the data from the dye experiments to 

obtain the graphical relationships in which the mean 

traveltime, the variance of traveltime, the dimensionless 

peak relative concentration, and the recovery ratio are 

shown as functions of dispersion distance. The only 

difference was the substitution of parts per million for 

parts per billion as concentration units in calculating 

the recovery ratio and the consequent reduction of the 

constant in equation 51 by a factor of 1,000. Many of 

the observations pertaining to these relationships which 

were made for the dye data apply equally well to the 

sediment. Only those which differ significantly are 

discussed.

The data in figures 33 and 34 show that in most runs 

the mean velocity of the suspended particles agrees very 

closely with the mean flow velocity. The exceptions are 

runs LO-CG-2 and LO-CG-3, where the mean velocity 

of the particles was somewhat retarded. This was
OPEN-CHANNEL DISPERSION PROCESSES E31

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0

0.07

0.06

0.05

0.04

0 10 20 30 40 50 60 70 80 90 100 50 60 70 80 90 100 110 120 130 140 150

80 90 100 110 120 130 140 150 160 170 180 110 120 130 140 150 160 170 180 190 200 210

DISPERSION TIME (t), IN SECONDS 

 Equation9a Observed at V/Vn =0.25 Observed at y/yn =0.50 Observed at V/Vn =0.75 - Continuous system

FIGURE 27. Relative concentration of 63-62p silt as a function of time at various depths and dispersion distances in run LO-CS-1.

probably caused by the development of a negative 

vertical concentration gradient and perhaps also by 

temporary deposition on the bed.

The relationship between a\ and x, shown in figures 

35 and 36 for the fine particles, follows equation 11 

quite closely. However, the relationship for the coarse 

particles differs markedly. In flow 1, a* for the coarse 

particle data is two to three tunes the value given by 

equation 11 at x=32.8 feet, but as x increases the 

relationships tend to converge. In flows 2 and 3, the 

trend is exactly opposite. No satisfactory explanation 

for this reversal in trend has been found.

The trends of the peak relative concentration data in 

figures 37 and 38 are consistent with those of the variance data but are considerably less exaggerated. This 

is reasonable because the peak relative concentration 

is approximately inversely proportional to the square 

root of the variance.

The conclusions concerning the general properties of

the two-dimensional relative concentration distribution, 

J(t, y; x), which were inferred from a comparison of the 

distribution curves obtained at different sampling 

depths, are reinforced by the data on mean traveltime, 

variance, and peak relative concentration. According 

to the conclusions, with increasing yfyn, values of t 

and 0-2 should occur in descending order, and values of 

J(t',x)max should occur in ascending order. Out of a 

total of 31 sets of observations, the expected order 

occurs 17 times for t, 9 times for <r*t , and 14 tunes for 

J(t', z)max- If all possible orders were equally likely, the 

expected orders would have occurred, on the average, 

only one out of six tunes.

Applying a similar analysis to the recovery ratio 

data in figures 39 and 40, for a concentration gradient 

of sediment which is weighted toward the bottom, 

values of AmjA t should occur in descending order with 

increasing y/yn . For the finer particles, the expected 

order occurs 6 out of 15 times and for the coarser
E32 TRANSPORT OF RAmOMJCLIDES BY STREAMS

z = 65.6ft

20 30 40 50 60 70 80 90 100 110 120 50 60 70 80 90 100 110 120 130 140 150

A

x = 115.0 ft

70 80 90 100 110 120 130 140 150 160 170 110 120 130 140 150 160 170 180 190 200 210

DISPERSION TIME (t), IN SECONDS 

 Equation 9a Observed at V/yn =0.25 Observed at y/Vn =0.50 Observed at y/yn =0.75 Continuous system

FIGURE 28. Relative concentration of <44n glass beads as a function of time at various depths and dispersion distances hi run LO-PG-1.

particles, 8 out of 16 times. This indicates a tendency, 

although not a very consistent one, for the concentration gradient in the vertical to be negative, as was 

expected.

The recovery ratio data for the sediment differs 

fundamentally from the data for dye because of the 

tendency for the particles to settle and deposit on the 

bed of the flume. Since At is the theoretical area which 

would have been under the concentration-distribution 

curve had no settling occurred, the ratio Am\At represents the relative amount of sediment remaining in 

suspension at the sampling level, y/yn, integrated over 

the time of passage of the sediment cloud past the 

sampling nozzle. If Am is averaged over the depth of 

flow,

1 /»vre

= 

Vn Jo

then the ratio Am/At represents the relative amount of 

sediment in the entire vertical which remains in suspension as the cloud passes the sampling location, x. 

The curves in figures 39 and 40, which show Am/At as 

a function of x, were determined from a theory given 

by Camp (1944, 1946) for sedimentation in settling 

tanks with turbulent flow. According to Camp's theory,

~__ 79 2 02 .,30

where fj=Vp/KUT as in equation 42, and a* (*=1> 2,
OPEN-CHANNEL DISPERSION PROCESSES E33

0.13r

0.12

0.11

10 20 30 40 50 60 70 80 90 20 30 40 50 60 70 80 90 100 110

0.07

0.06

0.01

30 40 50 60 70 80 90 100 110 120 40 50 60 

DISPERSION TIME ft), IN SECONDS 

.Equation 9a _____Observed at V/yn =0.25 ___Observed at y/y =0.50 __-_ Observed at y/yn =0.75

80 90 100 110 120 130

-Continuous system

FIGUBE 29. Relative concentration of <44jtt glass beads as a function of time at various depths and dispersion distances in run LO-FQ-3.
E34 TRANSPORT OF RADIONIUCLIDES BY STREAMS

20 30 40 50 60 70 80 90 100 110 120 50 60 70 80 90 100 110 120 130 140 150

70 80 90 100 110 120 130 140 150 160 170 110 120 130 140 150 160 170 180 190 200 210

DISPERSION TIME (t), IN SECONDS 

 Equation 9a -----Observed at y/yn =0.25 Observed at y/yn =0.50 Observed at y/yn =0.75 Continuous system

FIGURE 30. Relative concentration of 53-62^ glass beads as a function of time at various depths and dispersion distances in run LO-C G-1.

3, . . .,) are the successive real positive roots of the 

transcendental equation

n , a 3j8 2 cot a= Z" 

3p a

Equation 55 is a solution of equation 41 for the initial 

condition of a vertically uniform distribution of sediment at the entrance of the tank, and the boundary 

condition of no entrainment of sediment from the bed. 

In order to obtain this solution it was necessary to

ftQ __ ftQ

assume that: (1) -^7=U^ , (2) the velocity distribuot ox

tion, U(y), is parabolic so that ey is a constant, and 

(3) fpv=fy=Kyn UT/Q. In_performing the calculations 

required for plotting the Am/At versus x curves, Camp's

graphical representation of equation 55 was used. Also 

the fall-velocity distribution data given in figure 6 

were corrected for temperature by means of Stoke's 

law. Finally, the fall-velocity distribution data were 

separated into five ranges, using 20-percent increments. 

The calculations were done separately for each range, 

using the average fall velocity within the range, and 

the final values of Am/At were obtained by averaging 

the values obtained for the five different ranges.

The theoretical Am/At versus x relationships agree 

reasonably well with the experimental data for all 

runs except for run LO-CS-1, where the data were 

probably subject to a large systematic error of undetermined origin. Likely sources of experimental 

error in the remaining runs were an uneven initial
OPEN-CHANNEL DISPERSION PROCESSES E35

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0

x=328ft '

10 20 30 40 50 60 70 80 90 100 30 40 50 60 70 80 90 100 110 120 130

x = li5.0ft

50 60 70 80 90 100 110 120 130 140 150 70 80 90 100 110 120 130 140 150 160 170

DISPERSION TIME (t), IN SECONDS 

 Equation9a Observed at y/yn =0.25 Observed at y/yn = 0.50 Observed at y/yn =0.75 Continuous system

FIGURE 31. Relative concentration of 53-62/u glass beads as a function of time at various depths and dispersion distances in run LO-C G-2.

distribution of sediment across the flume, calibration 

errors, and an abnormally high rate of deposition near 

the source when initial local concentrations of sediment 

exceeded the transport capacity of the flow. The initial 

distribution of sediment across the flume tended to be 

more even in those runs in which the improved dumping trough was used. Calibrating the fluorometer for 

sediment concentration was one of the most difficult 

operations in the entire experimental program. Because 

the calibrations required such small quantities of sediment, the extent to which the calibration samples were 

truly representative is open to question. Another possible source of error was a shift in the size distribution of 

the suspended sediment with increasing dispersion distance caused by the tendency of the coarser particles to 

be deposited sooner. Because the sensitivity of the fluorometer increases as particle size decreases, this would 

cause the fluorometer to register higher concentrations 

than actually existed at large dispersion distances. A 

spot check consisting of a visual examination with a 

microscope indicated that the average size of particles 

in a sample collected at £=115 feet did not differ 

significantly from that of a sample obtained at the 

source. However, spot checks with small samples can 

be misleading.

A summary of the data obtained in the longitudinal 

dispersion experiments with silt-size particles is given 

in table 4.
TRANSPORT OF RADIOMJOLIDES BY STREAMS

0.07 

0-06

0.05

0.04

0.03

0.02

0.01

0

\

x =65.6 ft

10 20 30 40 50 60 70 80 90 20 30 40 50 60 70 80 90 100 110

= 88.5 ft

30 40 50 60 70 80 90 100 110 120 50 60 70 80 90 100 110 120 130 140

DISPERSION J\ME(t), IN SECONDS 

.Equation 9a Observed at y/^ =0.25 ___Observed at j//j^ =0.50 Observed at y/yn =0.75 Continuous system

FIGURE 32. Relative concentration of 53-62/t glass beads as a function of time at various depths and dispersion distances in run LO-CG-3.
OPEN-CHANNEL DISPERSION PROCESSES E37

180

80 100 120 0 20 

DISPERSION DISTANCE (x), IN FEET

40 120

FIGUBE 33. Mean traveltime as a function of dispersion distance, longitudinal dispersion of natural silt.

LATERAL DISPERSION

DYE

The basic data in the LA-D series of experiments 

consisted of strip-chart recordings of fluorometer 

readings, which were obtained as the sampling nozzle 

was moved slowly across the flume. Event marks on the 

chart provided a record of the lateral position of the 

sampling nozzle. The data were corrected for the 4.8- 

second lag time of the sampling system, and lateral 

distribution curves were obtained by plotting fluorometer readings as a function of lateral displacement 

from the flume centerline. The traverse speed was 

slow enough so that no appreciable distortion of the 

curves was caused by mixing in the sampling tubes. 

The curves were smoothed somewhat to reduce the

time fluctuating component of concentration, and they 

were normalized by dividing the ordinates by the 

area under the curve. The normalized lateral distribution curves are shown in figures 41, 42, and 43. 

Each figure shows all of the lateral dye distribution 

curves obtained for a particular flow condition. At each 

of the indicated dispersion distances, curves are given, 

in most cases, for the three sampling depths y/yn=Q.25, 

0.50, and 0.75. The sampling depths are identified by 

the symbols defined in figure 41. For comparison, the 

theoretical lateral distribution function according to 

equations 14 and 18 is also shown. Use of equation 18 

was necessary only when a significant amount of dye 

reached the flume walls and was reflected back toward 

the center. The values of Kz and Kx used in equation 14
ESS TRANSPORT OF RADIONTJCLIDES BY STREAMS

180

160

140

120

~ 100

= 0.25 

0.50 

o y/yn =0.75 

@ Continuous system

100 120 0 20 

DISPERSION DISTANCE (*), IN FEET

FIGURE 34. Mean traveltime as a function of dispersion distance, longitudinal dispersion of silt-size glass beads.

100 120

were obtained respectively from the data in figure 44 and 

the results of the longitudinal dispersion experiments.

The data for the three sampling depths from runs 

LA-D-1 and LA-D-2 in general agree quite closely 

with one another and with equation 14, the agreement 

tending to improve with increasing dispersion distance. 

The data show that the degree of dispersion tends to 

increase somewhat with decreasing y/yn - However, dispersion in the vertical direction tends to equalize the 

concentration in the vertical, thus largely masking the 

effect of variation in lateral diffusivity with depth. 

The data for run LA-D-3 show a consistent drift to the 

left at the relative depths y/yn 0.25 and 0.50 and a 

drift to the right at y/yn=0.75.

In figure 44 the estimated variances of the lateral distribution curves are shown as functions of dispersion 

distance. Because the main purpose for determining the 

variances was to evaluate the lateral dispersion coefficient, Kz, as a function of flow conditions in a twodimensional flow field, it was necessary to determine

the variances which would have occurred without the 

confining influence of the sidewalls. Therefore of was 

not calculated by the method of moments but from the 

relationship

s :j2> (56) 2

which follows directly from the normal probability law. 

In equation 5Q, f(z; z) max is the peak observed relative 

concentration. Whereas values of o\ computed by the 

method of moments are affected by the sidewalls as 

soon as an appreciable amount of dispersant reaches 

the walls, values of a\ computed by equation 56 are 

not affected until a significant quantity of the dispersant, which is reflected back from the walls, actually 

reaches the centerline.

Lateral dispersion coefficients were evaluated from 

the data in figure 44 by means of the relationship

2 dx (57)
OPEN-CHANNEL DISPERSION PROCESSES E39

~ Q

SQN003S Nl '(> a) 30NVIMVA
300

tM

V)

Q

0 250

0 

LlJ 

C/>

- 200

tM

b 

" " 150

LLJ

o 

< 

H 10°

50 

0

Flow 1

/

A

9

A

Ok \S

 y

«ft £r

 y 

* c

o 2/ 

A y

>Vn =0.25

/yn =0.50

fy* =0.75

antinuous 

system > 

i'y n = 0.25

>yn =0.50

o y/yn =0.75

@ Continuous 

system

4?

^TJ

 X «

1

 glass beads

53-62 M 

> glass beadsA ^/' 

7

/

r

' 8

Flow 2

0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140

DISPERSION DISTANCE (*), IN FEET

FlowS

20 40 60 80 1

FIGURE 36. Variance of traveltime as a function of dispersion distance, longitudinal dispersion of silt-size glass beads.
OPEN-CHANNEL DISPERSION PROCESSES

TABLE 4. Longitudinal dispersion of silt-size particles: summary of data 

[Boldface indicates continuous system]

E41

Run (LO-) x (ft)

FS-1.. -----.....-.......... 32.8

65.6

88.5

115.0

2..... ............... 32.8

65.6

88.5

115.0

CS-1....... _ .............. 32.8

65.6

88.5

115.0

FG V-* 1 A-_ ---.____-_-_--______ O.4. °.*> Oft

65.6

88.5

115.0

3 ...................... 32.8

65.6

88.5

115.0

CG-1.. .................. 32.8

65.6

88.5

115.0

2....................... 32.8

65.6

88.5

115.0

y/y0.25

.60

.75

.25

.50

.75

.25

.50

.75

.25

.50

.75

.50

.25

.50

.75

.25

.50

.75

.25

.50

.75

.50

.25

.50

.75

.25

.50

.75

.50

.25

.50

.75

.25

.50

.75

.25

.50

.75

.50

.25

.50

.75

.25

.50

.75

.25

.50

.75

.25

.50

.75

.50

.25

.50

.75

.25

.50

.75

.25

.50

.75

.25

.50

.75

.66

.25

.50

.75

.25

.50

.75

.25

.50

.75

.25

.50

.75

.25

.50

.75

.25

.50

.75

.50

.25

.60

.75

W (gm) F (fluorometer At (ppm-sec) Am 

unit per ppm)

283

283

283

283

283

283

283

283

283

283

283

283

283

830

830

830

830

830

830

830

830

830

830

ftdft irtu

646

646

646

646

646

646

646

646

646

646

646

646

750

750

750

750

760

750

750

750

750

750

750

750

750

2,500

2,600

2,500

2,500

2,500

2,500

2,500

2,500

2,500

2,500

2,500

2,500

2,500

1,533

1,533

1,533

1,533

1,533

1,533

1,533

1,533

1,533

1,533

1,533

1,533

1,533

2

A7H I Uf U

2,070

2,070

2,070

2,070

2,070

2,070

2,070

2,070

2,070

2,070

2,070

2,070

0.24

.24

.24

.24

.24

.24

.24

.24

.24

.24

.24

.24

.24

.24

.24

.24

.24

.24

.24

.24

.24

.24

.24

.069

.069

.069

.069

.069

.069

.069

.069

.069

.069

.069

.069

.069

.150

.150

.160

.170

.150

.150

.160

.150

.150

.150

.405

.405

.405

.150

.150

.150

.170

.150

.160

.150

.160

.150

.150

.150

.150

.150

.074

.074

.074

.085

.074

.074

.074

.174

.174

.174

.174

.174

.174

.059

.069

.069

.069

.059

.059

.205

.205

.205

.082

.205

.205

.205

3,460

3,460

3,460

3,460

3,460

3,460

3,460

3,460

3,460

3,460

3,460

3,460

3,460

4,080

4,080

4,080

4,080

4,080

4,080

4,080

4,080

4,080

4,080

7,920

7,920

7,920

7,920

7,920

7,920

7,920

7,920

7,920

7,920

7,920

7,920

7,920

9 040

9' 040

9,040

9,040

9,040

9,040

9,040

9,040

9,040

9,040

9,040

9,040

9,040

5,930

5,930

5,930

5,930

5,930

5,930

5,930

5,930

5,930

5,930

5,930

5,930

5,930

18,500

18,500

18,500

18,500

18,500

18,500

18,500

18,500

18,500

18,500

18,500

18,500

18,500

10,300

10,300

10,300

10,300

10,300

10,300

10,300

10,300

10,300

10,300

10,300

10,300

10,300

, (ppm-sec) ~t (sec) <r

4,880

4,900

4,430

4,230

3,570

3,930

3,940

3,910

3,760

2,920

3,020

3,330

3,160

4,640

4,680

4,940

4,610

4,350

3,980

3,910

3,960

3,860

4,580

11,400

11,700

10,600

7,840

7,390

7,910

7,930

8,100

7,130

7,880

6,200

6,520

6,000

6,870

6,470

6,640

5,930

7,000

6,190

6,320

4,860

4,920

5,000

4,940

4,940

4,970

4,700

4,400

3,860

5.270

4,520

4,270

4,130

4,380

4,470

3,970

4,640

4,400

4,200

8,960

8,610

8,420

10,400

5,430

5,410

4,600

4,070

4,080

4,050

3,360

3,310

3,250

11,100

13,700

10,700

5,430

5,660

5,700

5,630

5,170

5,000

7,030

6,540

6,650

6,380

49.6

48.3

48.9

89.7

86.8

87.2

115.4

114.5

114.1

166.0

155.8

155.2

159.7

31.3

30.4

30.2

69.4

67.8

57.0

79.1

77.9

77.9

107.5

51.1

51.2

50.9

93.0

89.1

91.0

93.1

125.4

123.8

124.1

160.5

160.9

159.2

45.6

46.3

43.9

43.2

88.5

86.3

84.7

117.5

116.4

116.0

155.2

153.7

153.1

21.3

21.1

20.6

20.8

45.4

43.9

42.3

59.2

61.0

61.3

76.0

77.1

75.9

49.1

49.1

46.7

46.1

88.0

86.6

83.7

118.7

116.2

115.6

160.9

151.3

149.8

31.0

33.0

32.2

63.1

64.1

60.5

86.3

84.6

81.9

88.4

104.4

111.0

103.3

>-t (sec") f(t;x)max (sec-i)

107

90

116

92

68

78

154

144

148

165

168

183

150

31

44

34

55

50

45

68

68

63

125

118

122

156

110

92

109

81

179

147

176

253

253

226

79

143

81

45

104

83

66

105

96

95

167

153

142

21

26

24

27

59

44

38

82

106

126

76

99

85

149

215

188

189

99

115

93

113

96

103

170

178

142

43

67

99

106

173

75

135

128

92

286

320

404

313

0.0555

.0595

.0610

.0440

.0515

.0500

.0310

.0320

.0305

.0295

.0320

.0290

.0305

.0730

.0700

.0860

.0520

.0530

.0685

.0455

.0445

.0475

.0385

.0445

.0405

.0445

.0365

.0395

.0370

.0410

.0345

.0325

.0290

.0255

.0255

.0280

.0558

.0667

.0632

.0614

.0444

.0447

.0458

.0392

.0407

.0427

.0325

.0329

.0332

.108

.113

.122

.0977

.0646

.0602

.0646

.0511

.0471

.0497

.0426

.0402

.0452

.0529

.0557

.0646

.0520

.0437

.0404

.0434

.0436

.0396

.0418

.0308

.0321

.0349

.0660

.0650

.0500

.0530

.0495

.0530

.0425

.0460

.0500

.0390

.0325

.0215

.0315

See footnote at end of table.
E42 TRANSPORT OF RADIONTJCLIDES BY STREAMS

TABLE 4. Longitudinal dispersion of siU-sise particles: summary of data Continued

Bun (LO-) x (ft) y/yB W (gm) F (fluorometer At (ppm-sec) Am (ppm-sec) ~t (sec) <r2< (sec2) f(t;x) max (sec-') 

unit per ppm)

CG-3 

2 KjUCONCENTRATION PEAK f(t;x) DIMENSIONLESS RELATIVE mm 

ooppoo oooooo O bbbbbb 2

g. oi-'MwJ^aiOiOi-'MCoj^aioi 

32.8 0.25 5,000 0.074 11,850 12,700 26.5 79 0.0595 

.50 5,000 .074 11,850 8,970 25.8 84 .0617 

.75 5,000 .074 11,850 6,980 25.0 68 .0813 

.50 5,000 .085 11,850 12,600 26.3 96 .0766 

65.6 .25 5,000 .074 11,850 11,406 49.4 155 .0427 

.50 5,000 .074 11,850 8,830 48.0 98 .0459 

.75 5,000 .074 11,850 8,490 43.2 55 .0525 

88.5 .25 5,000 .074 11,850 9,080 59.9 106 .0402 

.50 5,000 .074 11,850 8,210 60.4 105 .0437 

.75 5,000 .074 11,850 6,890 62.6 167 .0442 

115.0 .25 5,000 .174 11,850 10,700 87.0 321 .0279 

.50 5,000 .174 11,850 10,600 86.1 290 .0276 

.75 5,000 .174 11,850 10,400 84.1 281 .0318

Inuous system.

\

c

\\

»

^

4

/From eq

r -

15-30A

9a

-± 

silt

 ___

H

\

A ^

t^ ^ From

A^^ ^

53-62,

eq 9a

~^~ir~~

/silt

 

D 50 100 150 200 250 300 350 

DIMENSIONLESS DISPERSION DISTANCE (xU/KJ

Flow 1 Flow 2

. y/vn =o.25 « v/yn =o.25

*y/yn = 0.50 A V/Vn =0.50 

 y/Vn =0.75 y/yn =0.75 

® Continuous system » Continuous system

0.07 

0.06 

0.05

Ib

^ 0.04

(D

^ 0.03

^

| 0.02

<

cc

JE 0.01

O 

0

s °

d 0.06 

cc

*:

£ 0.05

</) </)

^ 0.04

O 

(/)

JE 0.03

S 

Q

0.02 

0.01

°

o

_1 A

\ffl

V 

^XT

N<^ From 

^*-^

<44Agla

eq 9a 

1

 - -.

ss beads

~- -* 

 

A

A

A \D Of \

<#

a

t v

?^< \ Q?

o

£a

^- From

A'^ ~-_

ff

t.

53 -62^

eq 9a

~~ - 

glass be:

~ 1^-

ds

 

3 50 100 150 200 250 300 350

DIMENSIONLESS DISPERSION DISTANCE (xU/Kx) 

Flow 1 Flow 2 Flow 3 

 y/yn =°- 25 my/yn =0.25 ny/yn =0.25

* y/yn = o. 50 A y/yn =0.50 A y/yn = o. 50

. y/yn = 0.75 cy/yn =0.75 oy/yn = 0.75 

®Continuous system ©Continuous system @ Continuous system

FIGURE 37. Dimensionless peak relative concentration as a function of dimensionless dispersion distance, longitudinal dispersion of natural silt.

FIGURE 38. Dimensionless peak relative concentration as a function of dimensionless dispersion distance, longitudinal dispersion of silt-size glass beads.
OPEN-CHANNEL DISPERSION PROCESSES E43

1.0

\

£

0 0.5 i-

-

8 g

Run LO-FS-2

 8 

-

cc

> 2.0 

o

1.0

Run LO-CS-1

0.5

0.2

0.1

20 40 60 80

DISPERSION DISTANCE (x), IN FEET 

n y/yn =0.25 A y/yn =0.50 o y/yn =0.75

 Continuous system From equation 55

100 120

FIGURE 39. Recovery ratio for area under concentration versus time curves as a function of dispersion distance, longitudinal dispersion of natural silt.
E44 TRANSPORT OF RAmONUCLIDES BY STREAMS

1.5

1.0

0.5

Run LO-FG-3

0.15

0.5

DISPERSION DISTANCE (x), IN FEET

a y/yn =0.25 A 2//2/re =0.50 o y/yn =0.75 

 Continuous system From equation 55

100 120

FIGURE 40. Recovery ratio for area under concentration versus time curves as a function of dispersion distance, longitudinal dispersion of silt-size

glass beads.
RELATIVE CONCENTRATION f(z;x), IN FEET~

§ a

« o M p B'o

RELATIVE CONCENTRATION /(*;*,), IN FEET' 1

_ w 8s ! 

§ I

OL <l
E46 TRANSPORT OF RADIONTJCLIDES BY STREAMS

0.5

- 0-4

0.3

|0.2

g 0.1

< 

cr

z 0

0.4 

0.3 

0.2 

0.1'

x ='32.8 ft

x = 49.2 ft

u.d

0.2

0.1

0

^

^

^

7

^^i;=^^

^5^

x=

^

65.6 ft

ss

0.1

-4.0 -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0

LATERAL DISPLACEMENT FROM CENTER

OF FLUME (z), IN FEET

From equations 14 and 18 Observed at y/yn=0.25 

-Observed at y/yn = 0.50 Observed at y/yn = 0.75

FIGURE 43. Lateral distribution of dye at various depths and dispersion 

distances in run LA-D-3.

Run

A,=

-^r

LA-D 

0.30S 

0.01C

^-^"

-1

Ht 2 p 

)3ft 2

-r

er se( 

per s

^t

3C

^--^ -K

_--- 

to.

*~_->^

^

Run

Kz =

^

LA-D

0.70;

0.02,

-2 

3ft 2 p 

Jft'p

Jf

er se 

er se

-""" \^-

m

\^C ^

*^\

<^ ^

^

Run

Kz =

^

LA-D-3 

1.33 ft 2 pe 

0.0385 ft 2

^^

per sc

t

>C

^

X^^

^

\<

^

^

A

^

) 10 20 30 40 50 60 70 80 90 100 110 12C

DISPERSION DISTANCE (x), IN FEET 

 y/yn =0.25 *y/yn =0.50 y/yn =0.75

FIGURE 44. Estimated variance of lateral displacement as a function of dispersion 

distance, lateral dispersion of dye.

The range of x within which equation 57 gives the 

desired lateral dispersion coefficient is limited at the 

lower end by the distance required to obtain uniform 

vertical distribution and at the upper end by the distance at which reflections from the channel sides begin 

to have an appreciable effect. The a\ versus x relationships are quite well represented by equation 16 from 

Fickian diffusion theory for the indicated values of Kz 

and Kx . As with the silt dispersion experiments, values 

of Kx were adjusted where necessary to compensate for 

the slight differences in yn UT that existed between the
OPEN-CHANNEL DISPERSION PROCESSES E47

flow conditions for the lateral dispersion experiments 

and the corresponding longitudinal dispersion experiments.

The fact that the linearity of the v\ versus x relationship, as defined by the data, extends nearly back 

to the source indicates that convergence of the lateral 

dispersion process to the Fickian model is very rapid. 

This means that the region within which the process 

depends on the Lagrangian correlation coefficient is 

confined to very short dispersion distances.

Although equation 16 applies reasonably well to the 

data obtained at all three sampling depths, the data 

in figure 44 indicate a correlation between al and y/yn 

that is consistent with the observations made in discussing the characteristics of the lateral distribution 

curves. According to these observations, the values of 

0-2 obtained at a particular dispersion distance should 

be arranged in descending order with increasing y/yn . 

The anticipated order occurs six times out of 14 sets of 

observations, whereas with random ordering it would 

occur only one out of six times.

Recovery ratio data for the lateral dispersion experiments is given in figure 45. For lateral dispersion from 

a continuous point source the recovery ratio is

Am= (Am/F)a 

At 3

(58)

where B is the width of the flume, in feet, q is the discharge rate of dye at the source, in milliliters per 

second, C0 is the initial concentration of dye at the 

source, in grams per liter, a is a correction factor for 

nonuniform distribution of flow across the channel, 

and the other variables are as defined in equation 51. 

The correction factor a is given by the formula

where C(z) is the local concentration, C is the average 

concentration across the width of the channel, and 

q(z) is the local water discharge per unit of channel 

width. From the velocity distribution measurements 

it was determined that a should have remained within 

the range !<«<!.03 in the experiments. In the calculations for figure 45 the correction was neglected, and 

it was assumed that a=l.

In addition to the random error in figure 45, there 

appears to be a systematic error of approximately plus 

10 percent. The systematic error is almost certainly 

due to a temperature difference between the dye 

solution flowing from the source and the flume water 

which was used for establishing the discharge rate of 

the source prior to each run. The temperature of the 

dye solution, which was not recorded, varied from that

of the flume water up to room temperature, depending 

on how long the dye solution was permitted to stand 

before use. In most of the experiments the room temperature was on the order of 10°C higher than that of 

the flume water. Because the flow in the tube leading 

from the dye reservoir to the source was laminar, the 

temperature difference was sufficient to cause a significant increase in the actual discharge rate of the dye 

solution over the discharge rate of the flume water 

which had been used for measuring q. Therefore, the 

values of q used in equation 58 were too low. Inclusion 

of the correction factor a, for nonuniform distribution 

of flow, would have tended to increase the values of 

AmfAt still more.

In order to verify further the applicability of the 

reflecting barrier concept, two experimental lateral 

distribution curves were obtained with the source 

displaced 2 feet to the right of the centerline. These are 

shown as the dashed curves in figure 46. The solid 

curves, which are shown for comparison, were determined from the reflecting barrier equation (18) together with equation 14.

The data from the lateral dispersion experiments 

with dye are summarized in table 5.

POLYETHYLENE PAKTICLE8

The lateral dispersion data for polyethylene particles 

was analyzed in much the same manner, and with 

essentially the same assumptions, as the longitudinal 

dispersion data for polyethylene particles. The width 

of the flume was divided into 9-centimeter increments; 

the number of particles in each increment was counted, 

and the relative frequency of occupancy,

(59)

was calculated for each increment. Equation 59 is 

entirely analagous to equation 52. The relative frequencies were plotted in the form of histograms as 

shown in the example given in figure 47. Also shown in 

figure 47 for comparison is the lateral distribution function according to equations 14 and 18 with the values 

of Kz and Kx taken from the results of the dye dispersion experiments for the same flow condition.

The example given in figure 47 is quite typical in 

that the dispersion pattern of the polyethylene particles 

is only very roughly approximated by the theoretical 

function. Also, the multimodal pattern of the histogram 

is fairly typical as is the tendency of the particles to 

keep away from the sidewalls. Both these characteristics 

suggest the existence of the multicellular type of 

secondary circulation described by Nemenyi (1946), in 

which there are one or more vortex pairs with longitudinal axes distributed across the channel. The members
E48 TRANSPORT OF RADIONTJOLIDES BY STREAMS

1.3 

1.2 

1.1

1.0 

0.9

1.3

-v L2

^ ^1

^ 

o i-l

<

cc.

g 1-0

So

LU

oc 

0.9

0.8 

1.4 

1.3 

1.2 

1.1 

1.0

0.9 

C

£

 t 

A

A

 

Average^

i

Run LA-D-1

A

 

A

A

 A 

A

 

A

Average^

Run LA-D-2

:

 A 

 

A

A

 

A

Average^

A

Run LA-D-3

D 20 40 60 80 100 120 14

DISPERSION DISTANCE (x), IN FEET 

 y/yn =0.25 A y/yn =0.50 y/yn =0.75

FIGUEE 45. Recovery ratio for area under lateral-distribution curves as a function of dispersion distance, lateral dispersion of dye.
OPEN-CHANNEL DISPERSION PROCESSES E49

0.5 r

0.4

0.3

0.2

0.1

2 0 

£ 0.5

^0.4

0.3

0.2

0.1

x =88.5 ft

r =2.0 ft

= 115.0ft

=2.0 ft

-3.0 -2.0 -1.0 0 1.0 2.0 

LATERAL DISPLACEMENT FROM CENTER OF FLUME (z), IN FEET 

 From experiment From equations 14 and 18

3.0 4.0

FIGUEE 46. Lateral distribution of dye for eccentrically located source. Run LA-D-2,1^=0.719 ft* per sec, K,=O.Q23 ft2 per sec.
E50 TRANSPORT OF RADIONTJOLIDEiS BY STREAMS

TABLE 5. Lateral dispersion of dye: summary of data

Run x 

(LA-D-) (ft)

1................... ........ 16.4

32.8

49.2 

65.6 

115.0 

21......................... 16.4

(f). ....... ........... 32.8

(^). .................. 49.2

(I)................... 65.6

(2) 88. 5 

(2). .................. 115.0

3.......................... 16.4

32.8

49.2 

65.6 

115.0

Co q F 

y/yn (gm per liter) (ml per see) (fluorometer 

units per ppb)

0.25 

.50 

.75 

.25 

.50 

.75

.25 

.50 

.75 

.25 

.50 

.75 

.50 

.50

.25 

.50 

.75 

.25 

.50 

.75

.25 

.50 

.75 

.25 

.50 

.75

.25 

.50 

.75 

.25 

.50 

.75

.25 

.50 

.75 

.25 

.50 

.75

.25 

.50 

.75 

.25 

.50 

.75 

.50

0.10 

.10 

.10 

.10 

.10 

.10

.15 

.15 

.15 

.15 

.15 

.15 

.25 

.25

.66 

.38 

.38 

.90 

.57 

.57

.90 

.67 

.67 

1.30 

.67 

.67

.333 

.333 

.333 

.333 

.333 

.333

.250 

.250 

.250 

.350 

.350 

.350

.500 

.500 

.500 

.600 

.600 

.600 

.500

1.31 

1.91 

1.91 

1.31 

1.91 

1.91

1.44 

1.93 

2.03 

1.44 

1.93 

2.03 

2.29 

1.95

2.30 

3.02 

3.44 

2.30 

3.02 

3.44

2.30 

3.02 

3.44 

2.30 

3.02 

3.44

2.40 

3.05 

3.35 

2.40 

3.05 

3.35

3.21 

4.13 

4.60 

3.24 

4.11 

4.61

3.24 

4.11 

4.50 

3.10 

4.08 

4.60 

4.15

6.60 

6.60 

6.60 

6.50 

6.65 

6.65

6.50 

6.45 

6.60 

6.55 

6.55 

6.80 

6.75 

6.85

2.80 

2.68 

2.68 

2.74 

2.66 

2.64

2.68 

2.56 

2.68 

2.76 

2.60 

2.72

7.90 

8.20 

8.50 

7.80 

8.40 

8.10

6.60 

6.55 

6.75 

6.50 

6.70 

6.95

6.95 

6.80 

6.70 

6.80 

6.85 

6.75 

6.90

At 

(ppb-ft)

12.4 

18.0 

18.0 

12.4 

18.0 

18.0

20.4 

27.3 

28.6 

20.3 

27.3 

28.6 

53.8 

45.8

58.9 

44.8 

50.8 

80.3 

66. 6 

76.2

80.6 

78.1 

89.2 

116.0 

78.5 

89.0

31.1 

39.6 

43.5 

31.1 

39.6 

43.5

14.8 

19.1 

21.3 

20.9 

26.7 

29.8

30.0 

38.2 

41.7 

34.4 

45.4 

51.2 

38.6

Am } 

(ppb-ft)

13.6 

19.6 

20.7 

13.8 

20.2 

19.6

21.6 

33.2 

35.2 

21.1 

32.0 

32.1 

60.2 

51.5

50.0 

47.4 

45.2 

81.3 

86.0 

87.5

85.5 

76.2 

101.2 

115.6 

70.8 

91.0

38.8 

46.0 

44.1 

39.2 

48.1 

56.0

18.3 

21.2 

26.2 

22.1 

27.8 

32.6

30.5 

42.6 

43.1 

37.3 

53.4 

56.7 

44.4

rtex)m*x (ft-i)

0.575 

.642 

.692 

.404 

.415 

.455

.330 

.339 

.382 

.289 

.295 

.304 

.222 

.229

.560 

.455 

.520 

.354 

.347 

.346

.267 

.276

.277 

.225 

.245 

.270

.202 

.209 

.200 

.188 

.182 

.187

.467 

.505 

.532 

.302 

.310 

.317

.250 

.218 

.275 

.207 

.199 

.227 

.155

<ft\

0.47 

.37 

.34 

.85 

.91 

.72

1.32 

1.37 

1.12 

1.75 

1.80 

1.72 

3.05 

3.16

.52 

.77 

.51 

1.39 

1.52 

1.36

2.22 

2.06 

1.94 

3.23 

2.64 

2.28

3.50 

3.34 

3.99 

4.43 

4.98 

4.66

.87 

.71 

.64 

1.67 

1.55 

1.21

2.38 

3.33 

2.02 

3.70 

3.55 

2.76 

6.35

1 yB =0.803 ft. 

3 yB =0.814 ft.

of each pair rotate in opposite directions. The sense of 

rotation is such that the direction of secondary flow 

is away from the sidewalls at the water surface. At 

those regions across the channel toward which the 

secondary flows converge at the water surface, the 

particles tend to become concentrated, accounting for 

the multiple modes. No doubt this type of secondary 

circulation existed also in the lateral dye dispersion 

experiments; however, its effects would have been 

masked by the mass exchange, due to turbulence, 

occurring between different levels in the flow field.

The variances of the lateral distributions of particles 

are plotted as a function of dispersion distance in figure 

48. Because of the irregularity of the distribution histograms, the variances were calculated by the method of 

moments rather than by the peak relative concentration method used in analyzing the results of the 

lateral dispersion experiments with dye or the cumulative probability paper method introduced by Orlob 

(1958) for polyethylene particles. The time intervals 

between successive particle releases for each set of 

data are identified by the symbols as defined in figure

48. No consistent effect of time interval duration is 

detectable. Shown for the sake of comparison with the 

data are the theoretical <r2 2 versus x relationships corresponding to the lateral distribution functions, with 

and without sidewall effects, for the values of the dispersion coefficients obtained from the dye dispersion 

experiments. The dashed curves were taken directly 

from figure 44. The solid curves were determined by 

calculating the variances of the theoretical lateral 

distribution curves in figures 41, 42, and 43 that had 

been corrected for sidewall effect. Since the values of 

0^2 determined from the data were calculated by the 

method of moments, which is immediately sensitive 

to sidewall effect, they tend to follow the solid curve. 

It is significant that the lateral dispersion at the water 

surface, as indicated by the variances of the polyethylene particle distributions, tends to follow quite 

closely the pattern within the flow as established from 

the dye experiments. The tendency of the particle data 

to lie slightly below the curve may be due to a tendency 

for the effect of the Lagrangian correlation coefficient 

at the surface to persist over a longer distance. The
OPEN-CHANNEL DISPERSION PROCESSES E51

\ From equations 14 and 18

-3 -2 -1 0 1 

LATERAL DISPLACEMENT (z), IN FEET

FIGUEE 47. Typical lateral distribution of polyethylene particles.

fact that the particle data lies still further below the 

curve at large dispersion distances is probably due to 

the effect of secondary circulation tending to keep the 

particles away from the sidewalls.

The results of the lateral dispersion experiments with 

polyethylene particles are summarized in table 6.

VERTICAL DISPERSION

The vertical dispersion experiments were performed 

to obtain an estimate of the dispersion distance required for dye released continuously from a point 

source at middepth to become uniformly mixed throughout the depth of flow. Because the concentration profiles were obtained close to the source and only a 

short period of time was allowed for each, they contained a relatively large component which fluctuated 

randomly with time. For this reason, the observed 

concentration profiles did not provide a suitable basis 

for determining the time-averaged concentration distribution in the vertical. An indication of the uniformity of the concentration in the vertical was obtained, however, by calculating for each profile the 

coefficient of variation

where

- (60)

n-1 tt

in which d is the average concentration in the i'th 

increment of depth, and

is the average concentration in the vertical. In figure 

49, C, is shown as a function of x/yn, the relative 

dispersion distance at which the profiles were obtained. 

The data indicate that for the conditions of the experiments a dispersion distance of x=2Qyn is sufficient 

to achieve a virtually uniform vertical concentration 

distribution.

DISCUSSION OF RESULTS 

APPLICABILITY OF FICKIAN DIFFUSION THEORY

In the analysis of the data the Fickian diffusion 

theory was assumed to be applicable. No particular 

justification was given for this assumption other than 

that it seemed to work. Fischer (1966) has shown that 

equation 7, which describes longitudinal dispersion as a 

one-dimensional diffusion process, is applicable when 

the concentration of dispersant in the cross section is 

nearly uniform. In terms of the notation used in deriving 

equation 29, a sufficient, although perhaps not necessary, condition is that C2 (y) «Ci(£), where Ci(£) 

is understood to be the mean concentration in
E52 TRANSPORT OF RADIONUCLIDES BY STREAMS

Run LA-P-1

? 3

o 2

Run LA-P-2

10 20 30 40 50 60 70 80 

DISPERSION DISTANCE (x), IN FEET

 A£=l second At=2 seconds * Ai = 5 seconds x.others _____ From figure 44

90 100 110 120

Taking sidewall effect 

_ into account using 

equation 18

FIGURE 48. Variance of lateral displacement as a function of dispersion distance, lateral dispersion of polyethylene particles.
OPEN-CHANNEL DISPERSION PROCESSES E53

TABLE 6. Lateral dispersion of floating polyethylene particles: 

summary of data

[Minus sign Indicates left of centerline of flume]

Run Number of 

(LA-P-) At (sec) x (ft) particles 2~(ft) *, (« 2)

counted

1... ----- 1 16.4

32.8

49.2

65.6

2 16.4

32.8

49.2

65.6

5 16.4

32.8

49.2

65.6

10 115. 0

20 115. 0

2...... --- 1 16.4

32.8

49.2

65.6

115.0

2 16.4

32.8

49.2

65.6

115.0

5 16.4

32.8

49.2

65.6

115.0

S. ........... 1 16.4

32.8

49.2

65.6

2 16.4

32.8

49.2

65.6

5 16.4

32.8

49.2

65.6

10 or 30 115. 0

101

102

105

99

103

99

100

98

107

100

99

100

»119

U22

97

100

94

100

120

97

104

100

99

130

100

99

101

102

124

100

101

100

100

103

100

99

100

101

100

100

102 2256

0.03

0

.26

.13

.06

.13

.23

.11

.10

.13

.40

.05

.02

.32

-.10

-.16

-.30

.04

.34 -.11

-.15

-.11

-.30

-.26

-.01

-.16

-.12

-.19

.06

.09

.58

.03 -.10

.28

.14 -.08

-.23

.13

.11 -.06

-.24

-.32

0.36

.64

1.26

1.65

.41

.66

1.45

2.24

.36

.73

1.39

1.45

2.64

2.39

.63

1.23

2.22

2.35

2.97

.58

1.38

1.62

2.32

2.65

.56

1.07

1.35

2.15

3.08

.41

1.14

2.08

3.00

.59

1.62

2.24

2.94

.44

1.54

2.44

3.02

2.72

» Collected during run LO-P-1. 

2 Collected during run LO-P-3.

the cross section. Considering longitudinal dispersion 

in a two-dimensional flow with a logarithmic velocity 

gradient for the initial condition of a uniformly distributed plane source, it is apparent that the initial 

uniformity is immediately destroyed by the convective 

effect of the velocity gradient. Thus, at first, the distribution of dispersant in the flow assumes the general 

shape of the mean velocity profile producing the highly 

skewed longitudinal concentration distribution characteristically associated with short dispersion times. 

Turbulence, however, tends to promote mixing in 

the vertical direction, leading to an eventual recovery 

of the condition of nearly uniform concentration of 

dispersant with respect to depth over most of the length 

of the dispersing cloud, at which time the Fickian 

model becomes applicable. Dispersant which is initially 

located near either the bed or the water surface does 

not become uniformly mixed over the depth of flow 

as rapidly as dispersant which is initially located near 

middepth. It is, therefore, logical to suppose that the 

length of the initial increment of dispersion distance

0.5

To

'1 0.4

2

O

1-

>

LL0

H

2

LLJ

0 0.2

LLLL.

LLJ

0

O

0.1 

0

A

\

\

0

m

. ^^

 Run V-D -1

A Run V-D-2

 Run V-D-3

t

~- ._

* *

k>

10 20 30 40 5C

RELATIVE DISPERSION DISTANCE (x/yj

FIGURE 49. Coefficient of variation for vertical distribution of dye as a 

function of relative dispersion distance.

in which the one-dimensional Fickian model does not 

necessarily apply, is the same as the dispersion distance 

required for the dispersant released from a continuous 

cross-channel line source located either at the bed or 

the water surface to become uniformly distributed with 

respect to depth.

The required distance can be estimated if it is 

assumed that the coefficient of vertical diffusion can be 

approximated by its average value, given in equation 

33, and that the bed and water surface can be considered as reflecting barriers. With these assumptions, 

and the substitution of y for z, equation 18 can be 

used to calculate the concentration distribution in the 

vertical. Let it be supposed that a sufficient condition 

for the applicability of the one-dimensional diffusion 

model is thatThe further assumption that the vertical dispersion 

process follows the normal probability law with 

variance <r2v=2t yx/U then leads to the realization of 

this condition when <ry=yn/1.3, for which the required 

length of the initial increment is

(61)
E54 TRANSPORT OF RAmONTJCLIDEfi BY STREAMS

The assumptions leading to equation 61 are indeed 

somewhat rough; however, the functional relationship 

which states that the length of the initial increment is 

directly proportional to depth and velocity and inversely proportional to the shear velocity and the 

degree of uniformity of the velocity distribution seems 

quite reasonable.

Equation 61, although it was developed for longitudinal dispersion from an instantaneous plane source, 

should apply equally well to lateral dispersion from a 

continuous point source located either at the water 

surface or the bed. For a source located at middepth, 

the criterion corresponding to equation 61 gives

£.=0.078 =

fy UT

(62)

as the length of the initial increment of dispersion 

distance downstream from which the one-dimensional 

Fickian model for lateral dispersion should apply. 

For other initial conditions where the source is located 

either within the upper half or the lower half of the 

flow, the values of the numerical coefficients lie between the extreme values specified in equations 61 and 

62.

Conceptually, Lm is related to the Lagrangian integral 

length scale of turbulence in that it is a measure of the 

dispersion distance over which the influence of the 

initial position of a fluid particle in the cross section 

persists. Indeed, it would not be surprising to find that 

the relationship is more than superficial.

A rough experimental confirmation of the analysis 

leading to equations 61 and 62 can be obtained from 

the vertical dispersion data in figure 49. These data are 

replotted in figure 50 in which the coefficient of vari-

^7*/?y ation, Cn is shown as a function of d~ Also given in

figure 50 are two calculated curves which show C, as a

x/y 

function of =~ for (1) a source located either at the

bed or the water surface and (2) a source located at 

middepth. For sources located in intermediate positions, 

C, should lie in the crosshatched area between the 

curves. The calculated concentration profiles used for 

computing C, were determined from Jhe normal probability law with variance a*=2evz/U, and from the 

assumption that the bed and water surface behave as 

reflecting barriers. The experimental data in figure 50 

should follow the curve for the source at y/yn=Q.5. 

The fact that the experimental C, values are substantially larger than the calculated ones is largely due to 

the random-in-time concentration fluctuations which 

contributed significantly to the experimental Cv values. 

The approximate relationship

0(x,y)-C(x)

C(x) (63)

which applies to both sets of concentration profiles on 

which the curves in figure 50 are based, permits figure 50 

to be compared directly with equations 61 and 62. 

Thus, the criterion <7S <0.07 is equivalent to the 

criterion

C(x, y)- 

0(x)

used in the derivations of equations 61 and 62.

The lengths of the initial increments, according to the 

criteria given in equations 61 and 62, are listed in table 7 

for the experimental flow conditions.

TABLE 7. Lengths of initial increments of dispersion distance for 

experimental flow conditions

Values o/Lm, in feet Flow

From eq 61 

12.5 

24.6 

41.2

From eq 62

3. 1

6.1

10.3

The dispersion distance exceeded Lm by a safe 

margin in all longitudinal dispersion experiments, 

except those for flow 3 at #=32.8 feet, and for all 

lateral dispersion experiments, except those for flows 

2 and 3 at £=16.4 feet. It is assumed here that the 

appropriate values of Lm for the lateral dispersion

 RunV-D-l 

A Run V-D-2 

. Run V-D-3

FIGURE 50. Coefficient of variation for vertical distribution of dye as a function 

of relative dispersion distance divided by the resistance function.
OPEN-CHANNEL DISPERSION PROCESSES E55

experiments, with the source located at y/yn=Q.25 or 

0.75, lie about halfway between the two sets of values 

given in table 7. Note also in figure 18 that the dispersion distance at which the experimental at versus x 

relationship for run LO-D-3 becomes linear, as is 

required by the Fickian model, corresponds closely to 

the Lm value from equation 61 given in table 7 for flow 

3. It is also likely that the magnitude of the average 

displacements of the data in figure 18, from the curves 

defined by equation 11, is a function of Lm. On the 

basis of the analysis given here, it is concluded that 

use of the one-dimensional Fickian diffusion model in 

analyzing the data contained in this report is fully 

justified. A possible exception is the coarse-silt longitudinal dispersion data for which justification is 

questionable on other grounds.

RELATIONSHIP OF DISPERSION COEFFICIENTS 

TO FLOW PARAMETERS

The dispersion coefficients determined from the data 

for the longitudinal dispersion of dye, the longitudinal 

dispersion of polyethylene particles, and the lateral 

dispersion of dye are shown in figure 51 as functions 

of ynUT. The longitudinal dispersion coefficient for 

dye, which may be expressed by the formula

Kx=5.ZynUT, (64)

is in excellent agreement with equation 34 of Elder 

(1959) for which the numerical coefficient, evaluated 

for /c=0.42, is 5.5.

The longitudinal dispersion coefficient for polyethylene particles floating on the water surface is 

expressed approximately by

^=0.690.17,. (65)

On the water surface the dispersion process is due 

mainly to turbulence, and there is no direct contribution from differential convection due to the velocity 

gradient. Therefore, the coefficients in equations 64 

and 65 provide a basis for estimating roughly the 

relative contributions of convection and turbulence to 

the total longitudinal dispersion. The estimate is good 

only for an order of magnitude comparison, because 

the contribution due to turbulence, which is no doubt 

a function of depth, is specified by equation 65 only 

at the water surface. If it is assumed that the rate 

of longitudinal spread observed at the water surface 

is representative of the direct contribution by turbulence to the rate of longitudinal spread within the 

body of the flow, then the turbulence component of 

the longitudinal dispersion coefficient, KXT, can be 

estimated by making the appropriate correction for 

reference velocity. Using the ratio derived from the

O.OlL

0.01

yn UT ,W FEET 2 PER SECOND

FIGTIEE 51. Longitudinal and lateral dispersion coefficients as functions of »nUT.

von Karman-Prandtl velocity distribution equation,

u *u

which has the average value 1.34 throughout the range 

of experimental flow conditions, the corrected coefficient is _

r. (66)

This is from three to four times as large as the coefficient given in equation 33, which was based on the 

assumption of isotropic turbulence. This result implies, 

therefore, that the turbulence in open channels is 

not isotropic.

The lateral dispersion coefficient for dye, expressed 

by the formula

K.=0.17y»Uv , (67)

is the same for the three depths ?//2/«=0.25, 0.50,_and 

0.75, provided that the same reference velocity U, in 

equation 57, is used in all three cases. The experimental 

results of Batchelor, Binnie and Phillips (1955), in
E56 TRANSPORT OF RADIONUCLIDES BY STREAMS

which the mean velocity of neutrally buoyant particles 

in a sufficiently long reach of pipe was found to be 

equal to the mean flow velocity, regardless of the 

initial and final positions of the particles in the pipe 

cross section, indicate that U is the correct reference 

velocity for fluid particles.

Although the of versus x relationship for the lateral 

dispersion of polyethylene particles on the surface is 

shown by figure 48 to agree well with that for the dye, 

the lateral dispersion coefficient for the polyethylene 

particles should be US/U times as large to account for 

the difference in reference velocity. Thus, the equation 

for the lateral dispersion coefficient for particles floating 

on the surface is

=l (68)

The coefficient in equation 68 agrees well with the 

coefficient 0.24 obtained by Sayre and Chamberlain 

(1964) for lateral dispersion of polyethylene particles 

in the same flume, but with a sand bed. The ratio of 

the longitudinal to the lateral dispersion coefficients 

for the polyethylene particles is 2.6, which agrees well 

with the ratio 2.5 obtained by Orlob (1961).

The results shown in figure 51 support the TaylorElder differential convection theory according to which 

the dispersion coefficients are proportional to the 

product yn UT . In order to test this proportionality 

further, the longitudinal dispersion coefficients obtained 

by Krenkel (1960) and the lateral dispersion coefficients 

for polyethylene particles obtained by Orlob (1958) for 

runs with uniform flow are shown as functions of ynUT 

in figure 52. These data, which were originally analyzed 

according to the local similarity theory, and later by 

Yotsukura, Smoot, and Cahal (1964), in terms of a 

functional relationship between the Schmidt number 

and the Reynolds number, also indicate a proportionality between the dispersion coefficients and ynUr. 

However, the values of the coefficient are different 

from those in figure 51.

Considering first the longitudinal dispersion coefficient, Elder's analytically derived equation 34 

states that Kxlyn Ur varies approximately inversely 

with the cube of the von Karman coefficient, K, or 

in other words, that KxKzlynUT ~QAQ^. An estimated 

value of K=0.36 for Krenkel's data w^as obtained by 

plotting the resistance function against normal flow 

depth as shown in figure 53. The value of KxK3/yn UT 

for Krenkel's data is 0.43 when K =0.36. For the 

LO-D series in figure 51, where *=0.42, KxK3/yn UT= 

0.39. Both of these are in good agreement with the 

theoretical value in equation 34. Experimental evidence 

apparently bears out the theoretical prediction that

Kx is extremely sensitive to small changes in K. Additional confirmation of this is obtained if the velocity 

distribution function used by Taylor (1954) in deriving 

the equation

Kx=W.Qr0 UT , (69)

where r0 is the pipe radius, for turbulent flow in pipes, 

is approximated with a logarithmic velocity distribution as shown in figure 54. If K=0.32, from figure 

54, is substituted into the equation

(70)

which is the counterpart of equation 34 for longitudinal 

dispersion in pipes, equation 70 becomes equal to equation 69. The coefficient 0.328 in equation 70 was 

determined by integrating the polar counterpart of 

equation 29, using the logarithmic velocity distribution 

function in figure 54.

Few experiments have been performed under conditions which conform to the requirements, set forth in 

the derivation of equation 34, of uniform two-dimensional flow with a logarithmic velocity distribution. 

When this requirement has been met, agreement has 

been good. If this requirement is not met, wide variations between predictions based on equation 34 and 

experimental results should not be surprising. Fischer 

(1964a) has shown, for example, that relatively slight 

deviations from a uniform lateral velocity distribution 

are capable of overwhelming all other effects which 

contribute to longitudinal dispersion.

Although it has not been established theoretically 

that the lateral dispersion coefficient Kz is proportional 

to yn UT, the fact that the differential convection and 

eddy diffusivity theories indicate that both Kx and 

y are proportional to yn UT suggests the same kind of 

relationship for Kz. Experimental evidence has borne 

this out to some extent; however, the proportionality 

coefficient has been found to vary somewhat from channel to channel. According to Yotsukura (written 

commun., 1965), values of Kg/yn UT ranging from 0.1 

to 0.4 for laboratory flumes, and up to 0.7 for natural 

channels, have been reported in the literature. Yotsukura's tabulation shows an apparent tendency for 

KzfynUr to increase with boundary roughness. However, this tendency is not clearly defined, and it could 

depend mainly on other conditions under which the 

data were obtained, such as nonuniformity of flow, 

nonhomogeneity of the flow and turbulence structure 

in the lateral direction, differences in channel width, 

and whether the dispersant was floating on the surface 

(as in the LA-P series) or entrained in the flow (as in 

the LA-D series). It is likely that some of the small 

KzlynUr values are due to a suppression of Kz caused
p

b

po

Or^- 

o

LONGITUDINAL DISPERSION COEFFICIENT (KJ, IN FEET 2 PER SECOND 

§ P

2. Hp

g N. O

5f "0

I 8 

I S «s

Cj

tf

p_

^p 

b 

o 

o

I I I I I I__________|_____I I I I I I I

P P § '°

LATERAL DISPERSION COEFFICIENT (Kz ), IN FEET 2 PER SECOND
E58 TRANSPORT OF RADIONTJCLIDES BY STREAMS

o 

o

UJ

O

I

- 5 in °

LJ

cc

0.02

X

*=0.36,x =0.016 ft

0.04 0.06 0.08 0.1 0.2

NORMAL DEPTH fig,IN FEET

FIGUEE 53. Variation of resistance function with flow depth for Krenkel's data.

0.4 0.6 0.8

by a small width-to-depth ratio. However, except 

when B/yn is very small, there does not appear to be 

any systematic correlation between Kz/yn UT and 

B/yn . The relationship between the lateral dispersion 

coefficients for floating and entrained dispersants 

given in equation 68 clearly depends on the resistance

function > and hence on boundary roughness. HowU T

ever, this does not indicate the nature of the relationship of either Kz/yn UT or Ktl/yn Ur to boundary 

roughness.

Hino (1961), in an analysis based on the local 

similarity theory and a turbulence energy balance, 

showed that

Kz =const ynUr. (71)

According to this result and equation 68, Ks/yn UT should 

decrease with boundary roughness. This contradicts 

the apparent trend of the experimental results cited by 

Yotsukura. However, if it is assumed that equation 71 

is correct, setting equation 71 equal to equation 26a 

results in the formula

U. Lz=const yn

=const yn 77- (72)

for the integral length scale of lateral turbulence components at the water surface. In figure 55 Orlob's data 

for runs with uniform flow and data from the alluvialchannel experiment of Sayre and Chamberlain (1964)
OPEN-CHANNEL DISPERSION PROCESSES E59

1.0, 

0.8 \

0.6 

0.4

9: 0.2

0.

o

I 0.1

I 0.08

y o.oe I 

§

^ 0.04

Q

0.02 r

O.Oll__/- 

~9 -8

Taylor's velocity profile

with x =0.32

-7 -6 -5 -4 -3 -2 -1

DIMENSIONLESS VELOCITY ( U~ U) 

UT

FIGURE 54. Comparison of Taylor's and logarithmic velocity profiles.

are compared with equation 72. These data are described at least as well by equation 72 as by Orlob's 

(1958) empirical formula

Although the question of whether or not K2/yn UT 

varies with boundary roughness remains unsettled, 

equation 72 provides a useful link between the local 

similarity and eddy diffusivity theories of lateral 

dispersion.

Attempts at relating the variation in Kz/yn UT to the 

von Karman turbulence coefficient, K, and the absolute 

roughness, % (as defined in eq 46), were also unsuccessful. 

The results are summarized in table 8.

DEFINITION OF DISPERSION COEFFICIENTS

The classical definition of the dispersion coefficient for 

a given quantity of dispersant, for which the spatial 

distribution is fixed at some initial instant, is that given 

in equation 20,

Fischer (1966) has shown that this definition is valid, 

regardless of the initial concentration distribution of 

dispersant, provided only that the dispersion process 

obeys a gradient-type diffusion law, for example, equation 6. Due to experimental considerations, it is difficult

TABLE 8. Lateral dispersion coefficients, von Karman coefficients, 

and absolute roughness

Source of data K,/Vn Ur K x

(ft)

Remarks

Series LA-D. 

Series LA-P.

Sayre-Chamberlain.... 

Orlob . - ..--..-.

0.17 0.42 0.042 Dispersant entrained in flow.

Flume with rough bed. 

.23 .42 .042 Dispersant floating on water

surface. [/,/[/«1.3-1.4. Flume

with rough bed. 

.24 .38 .0012 Dispersant floating on water

surface. LT,/[7«1.2. Flume with

sand bed. 

.17 .40 .0058 Dispersant floating on water

surface. U,jU^ 1.3-1.5. Flume

with rough bed.
E60

10

i.o

TRANSPORT OF RADIONTJCLIDES BY STREAMS

I I T

 Orlob

0 Sayre and Chamberlain

100

M. -^-, IN FEET 

UT

FIGUEE 55. Integral length scale of large lateral turbulence components at the water surface as a function of flow conditions.

to obtain instantaneous observations of the spatial 

concentration distributions as required by the classical 

definition. Therefore, relationships for which data are 

relatively easy to obtain such as

K

v U*. dtf x=-^ Lim '

2 x^m ax

for longitudinal dispersion from an instantaneous source 

(eq 50), and

for lateral dispersion from a continuous source (eq 57), 

are generally used instead. It has been shown, for 

example, by Fischer (1964b), Levenspiel and Smith 

(1957), and Orlob (1961), that the values of the dispersion coefficients given by equations 50 and 57 are the 

same as those given by equation 20.

In principle, the most straightforward way of evaluating the variances in the above equations is by the 

method of moments for example, eq 49. However, in 

practice, difficulties are sometimes encountered. A common difficulty is that of, accurately fixing the points 

where the tails of the concentration-distribution curves

become tangent to the background level. This difficulty 

is due to the extreme sensitivity of the second moment 

of a distribution curve to the length of the tails. If there 

is any appreciable fluctuation in background concentration, the difficulty is compounded. In the determination of lateral dispersion coefficients, complications 

arise in the method of moments when dispersant reaches 

the side of the channel and is reflected back toward the 

center.

Owing to difficulties such as these, indirect methods 

based on the properties of the normal probability law 

are sometimes used for estimating either the variance 

or the dispersion coefficient. Among these are methods 

based on the rate of attenuation of the peak relative 

concentration (eq 56), the width of the concentrationdistribution curve at one-half the peak value (Taylor, 

1954 and Elder, 1959), and the method based on the 

slope of a curve obtained by plotting log (tl/2 j(t;x)} 

against (x-Ut) 2/t (Krenkel and Orlob, 1963). All of 

these methods give the same results if the concentration 

distribution follows closely the normal probability law. 

However, if there is any appreciable deviation from the 

normal probability law, results obtained by the different 

methods in general do not agree with one another.
OPEN-CHANNEL DISPERSION PROCESSES E61

Because the normal probability law gives, at best, only 

an approximate representation of actual concentration 

distributions, there has in fact been considerable difficulty in comparing the results of different investigators. 

The values of the dispersion coefficients obtained by the 

indirect methods are partially dependent on the initial 

concentration distribution and the early phases of the 

mixing process. The advantage of the direct methods, 

based on the rate of change of variance, is that the value 

of the dispersion coefficients so determined are independent of these effects after the initial mixing period. 

Facilitating the comparison of data is, by itself, 

sufficient grounds for standardizing methods of evaluating dispersion coefficients from experimental data. 

It is recommended here that equation 50, with <r] 

determined by the method of moments (eq. 49), be used 

for evaluating the longitudinal dispersion coefficient. 

If there is a problem in locating the points where the 

concentration-distribution curve becomes tangent to 

the background level, both a minimum and a maximum 

probable variance for the distribution curve in question 

can be estimated by terminating the distribution curve 

at points which give, respectively, the minimum and 

maximum reasonable tail lengths. If this procedure 

is repeated for a series of distribution curves obtained 

at various dispersion distances, plotting both the 

minimum and maximum estimates of variance as a 

function of dispersion distance defines a band within 

which the correct a2 versus x relationship should lie. 

Increasing the range of dispersion distances with respect 

to the width of the band tends to minimize the error in 

the dispersion coefficient. Essentially the same method 

with equation 57 is recommended for evaluating lateral 

dispersion coefficients, except when reflection from the 

sides is a factor, in which case the peak attenuation 

method, with a\ calculated from equation 56, is 

recommended.

LONGITUDINAL DISPERSION OF SUSPENDED 

SILT-SIZE PARTICLES

No general analytical framework for the results of 

the longitudinal silt dispersion experiments has been 

found, because, as yet (1965), it has not been possible 

to either solve or simplify equation 38 in a manner 

that permits expression of the concentration distribution function, C(z,f), or the moments thereof in terms 

of hydraulic and sediment parameters. A solution for 

the moments, mp (t), using Aris' (1956) eigenvalue 

method, was attempted; however this approach led 

to a system of ordinary differential equations with 

variable coefficients, for which there is no standard 

method of obtaining a formal solution. A numerical 

methods-type solution of these equations does, however, 

appear to be feasible and will be tried in the future.

Assuming that equation 38 in combination with 

appropriate boundary and initial conditions is capable 

of representing with good approximation the longitudinal dispersion of silt-size particles, the solution 

should indicate the combined effect of longitudinal 

dispersion owing to the velocity gradient and the 

tendency of the particles to settle. The settling of the 

particles should cause the development of a vertical 

concentration gradient which is weighted toward the 

slower moving flow near the bottom. In comparison 

with the dye, this should cause a retardation of the 

mean velocity and a tendency toward more rapid 

longitudinal spreading of the suspended particles. A 

comparison of the dye and silt data in general bears 

out these trends. The tendency for the particles to 

settle is clearly indicated in the recovery-ratio data, 

figures 39 and 40. The development of a vertical concentration gradient, although not clearly evident from 

the recovery-ratio data, is reflected in figures 33 and 

34 where the mean velocity of the suspended coarse 

particles is seen to be slightly less than that of the flow. 

Figures 35 and 36 indicate that the longitudinal spread 

of the suspended coarse particles is characteristically 

greater than that of either the fine particles or the dye. 

Evidently with the fine particles there was not enough 

of a concentration gradient to cause a significant 

difference between the fine particle and dye data in 

figures 33 through 38.

From dimensional and physical considerations it 

would seem that the parameter p=Vp/i<UT} which is 

an index of the vertical concentration distribution of 

suspended sediment under equilibrium conditions, 

should also be important in describing the longitudinal 

dispersion of suspended sediment particles under nonequilibrium conditions. The j8 values corresponding to 

the median fall velocities of the particles for the conditions of the experiments are given in table 9. The 

dispersion of the suspended fine particles agrees closely 

with the dispersion of the dye and is essentially independent of j8. However, the 0 values for the coarseparticle experiments apparently do not correlate systematically with any of the index parameters for 

longitudinal dispersion, for example,

This is seen in figure 56, where the data for the coarseparticle experiments consists of the four points farthest 

to the right. The curve in figure 56 is from Elder's 

equation 42. It is clear that the curve and the coarseparticle data do not follow the same trend.
E62 TRANSPORT OF RADTONUCLIDES BY STREAMS

10 TTT

0.1 

0.01

n r i i i u

From eq 42

0.1

*UT

1.0

FIGURE 56. Ratio of silt to dye longitudinal dispersion coefficients, Ktp/Kz, 

as a function of ft.

TABLE 9. Values of the parameter /3 and the median particle fall 

velocity in the silt dispersion experiments

Run (LO-)

FS-1- ...............

CS-1-..... .............

FG-1... ................

3... ------------

CG-1..... ...... ........

3... .............

Particle size 1 

GO

15-30

15-30

53-62

<"44

53-62

53-62

53-62

"CO)

20

20

20

7

6

4

6

(ft per sec)

0.00069

.00069

.0071

.0017

.0016

fifij.7

.0045

.0048

ft

0.013

.010

.13

.031

.019

.090

.066

.058

In the calculations for plotting the curve in figure 56, 

the legitimacy of the definition

was assumed. Strictly speaking, in view of the fact that 

the dispersion data for the coarse particles deviates 

significantly from the Fickian model, both in the skewness of the relative concentration versus time curves 

and in the nonlinearity of the a-] versus x relationship, 

there is no justification for supposing that the dispersion process for these particles can actually be characterized by means of a gradient-type diffusion model 

with a constant coefficient.

It would seem that the effect of particle size, as 

indicated by /3, on the longitudinal dispersion of the 

coarse particles should become less going from flow I 

to flow 3, but the coarse-particle data lead to the 

opposite conclusion. Anomalies such as this one led 

to the investigation of the response characteristics of

the concentration measuring systems, which is discussed 

at length hi the section beginning on page E65. However, 

the corrections for system-response lag, even though 

relatively larger for the coarse particles than for the 

fine particles or the dye, did not appreciably change 

the pattern of the results.

It is noteworthy that whereas Camp's (1944, 1946) 

settling tank theory, in the form of equation 55 for 

the recovery ratio, agrees reasonably well with the trend 

of the experimental data, equations 42 and 45 for the 

longitudinal dispersion coefficient for suspended sediment particles do not. This is true, despite the fact that 

the implications introduced by assuming a parabolic 

velocity distribution are essentially the same in both 

cases. However, a fundamental difference, which 

probably accounts for this lack of agreement, is the 

assumption in the derivation of equations 42 and 45 

of an equilibrium vertical distribution of suspended 

particles that is independent of x and t. A comparison 

of the relative concentration data in figures 25 through 

32 for the different sampling depths shows clearly that 

the data contradict this assumption.

In considering some of the broader aspects of the 

transport and dispersion of a group of marked fluvial 

sediment particles, the significance of the recovery 

ratio data and Camp's settling tank theory extends 

beyond a material balance analysis. Let it be recalled 

that Am/At is the ratio of the amount of sediment 

remaining in suspension as the dispersing cloud passes 

the sampling location x, to the amount which was 

originally in suspension at the source. Consequently,

^ T

Jo

(73)

represents the proportional amount of sediment deposited on the bed between the source and location x. 

In equation 73 Jx(x) is a probability-density function 

which defines the longitudinal distribution of deposited 

particles. In a channel with an alluvial bed the deposited particles would become part of the bed-material 

load. The transport and dispersion of these particles 

would then proceed according to the manner postulated 

by Hubbell and Sayre (1964) and Sayre and Hubbell 

(1965) in which particle motion is described as an 

alternating sequence of steps and rest periods of 

random length and duration. The longitudinal deposition-distribution function, /^(z), would then become 

the initial source distribution function for the same 

group of particles in the bed-material dispersion process. 

Furthermore, even though the mechanics of entrainment are ignored, it is reasonable to suppose that the 

step length distribution function for those bed-material 

particles that are transported mainly in suspension 

should be closely related to /*(*). Indeed, if it is
OPEN-CHANNEL DISPERSION PROCESSES E63

assumed that they are the same, and the recovery 

ratio function is approximated by the exponential 

function

A,

At

 =e-*i*t

(74)

then the corresponding probability-density function for 

the distribution of step lengths will be

where \jk\ is the mean step length. Equation 75 is 

identical to the step-length distribution function used 

by Hubbell and Sayre (1964) and Sayre and Hubbell 

(1965) in the development of their bed-material dispersion model. For small values of /3 (/3<0.6), Camp's 

settling tank equation (eq 55) can be closely approximated by equation 74 by making the substitution

i=-(0.80+1). 

Uyn

(76)

The value of k\ given by equation 76 differs by only 

7 percent from the value determined experimentally for 

tracer particles in the North Loup River by Sayre and 

Hubbell (1965), under conditions where suspension was 

an important factor in the transport of the tracer 

particles. However, the value of k\ determined by 

Hubbell and Sayre (1964) from a laboratory flume 

experiment in which there was no suspended load is 

approximately five times as large as that given by 

equation 76.

TREATMENT OF BOUNDARIES AS REFLECTING 

BARRIERS

In confined channels it has been assumed by some 

investigators (Diachishin, 1963; Nobuhiro Yotsukura, 

written commun., 1965) that surfaces such as the 

channel sides and bottom and the water surface behave 

as reflecting barriers for dispersing substances. Implications of this assumption are that the concentration 

gradient at any one of these surfaces, taken normal to 

the surface, is zero, and that a mirror image of that 

part of the concentration profile which would extend 

beyond the barrier, if the barrier were not present, is 

reflected back into the flow field. This is analogous to 

the concept of a perfectly insulated surface, adjacent to 

a conducting medium, which is commonly employed in 

heat diffusion theory.

Comparisons of the experimental lateral distribution 

curves for dye with the theoretical curves based on 

the Fickian model, equation 14, and the reflection 

principle, equation 18, which are given in figures 41-

43, and in particular figure 46, indicate that application 

of the reflection principle here is indeed valid. These 

results are very encouraging because they support the 

acceptability of the reflecting barrier concept which 

greatly facilitates the solution of dispersion problems 

in confined channels. The vertical dispersion data are 

not sufficiently detailed to evaluate the applicability of 

the reflecting barrier concept at the water surface and 

channel bed. However, as seen in figure 50, the reflection principle seems to be reasonably adequate for 

predicting the dispersion distance required for uniform 

mixing in the vertical.

The utility of the reflecting barrier concept in the 

solution of dispersion problems is demonstrated in 

figure 57, where the maximum concentration in the 

cross section of a dispersant released continuously at 

the centerline is given as a function of mean flow 

velocity, channel width, Kz, and Kx. The curves were 

obtained from equations 14 and 15 using the reflection 

technique, equation 18. For comparison, experimental 

data are shown in figure 57 also. According to the curves 

dispersant reflected from the sides noticeably affects "US*

the centerline concentration at about x=0.04 -= > Ks

and uniform concentration across the channel is

UB2

achieved at about x=0.15 ., 

&z

CONCLUSIONS

The following conclusions are restricted to conditions 

of uniform, two-dimensional, turbulent flow in an open 

channel with a rough bed:

1. The longitudinal dispersion process for a dissolved 

dispersant can be represented with good approximation, except in the initial stages, by the onedimensional Fickian diffusion equation.

2. The value of the longitudinal dispersion coefficient 

for a dissolved dispersant can be calculated with 

good accuracy by Elder's (1959) equation,

0.404 K

which is based on a coupling of turbulent dispersion with the differential convection rates due 

to a velocity gradient.

3. The length of the initial increment of dispersion 

distance, downstream from which the onedimensional Fickian diffusion equation for the 

longitudinal dispersion of a dissolved dispersant 

should become applicable, is approximately
E64 TRANSPORT OF RADIONUCLIDEiS BY STREAMS

Symbol Flow KZ /UB

 1 0.00171

A 2 0.00260

 3 0.00315

0.001

100 1000 10,000 

DIMENSIONSLESS DISPERSION DISTANCE (^)

100,000

FIGURE 57. Attenuation of peak concentration with dispersion distance in a rectangular channel of width B having a continuous point source at the centerline.

4. The longitudinal dispersion process for particles 

floating on the water surface can also be represented by the one-dimensional Fickian diffusion 

equation. The dispersion rate for floating particles 

is a function of turbulence at the water surface 

only. The value of the longitudinal dispersion 

coefficient for floating particles is approximately 

one-tenth the value of the coefficient for a dissolved dispersant that is distributed throughout 

the depth of flow.

5. The longitudinal dispersion process for suspended 

silt-size sediment particles differs from the process for a dissolved dispersant in that the particles 

tend to settle toward the slower moving flow 

near the bed and eventually deposit on the bed.

6. The longitudinal distribution of particles that are 

deposited along the channel can be calculated 

with satisfactory accuracy by a procedure based 

on Camp's (1944, 1946) theory of the influence 

of turbulence on sedimentation in settling tanks.

7. The lateral dispersion process for a dissolved dispersant released from a continuous point source 

can be represented with good approximation by 

the two-dimensional Fickian diffusion equation,

10.

provided that the dispersant is evenly distributed 

with respect to depth. The required minimum 

dispersion distance for uniform vertical distribution, if the source is at middepth, is approximately

KUr

The value of the lateral dispersion coefficient for a 

dissolved dispersant is approximately one-thirtieth that of the longitudinal dispersion coefficient.

The_ confining effect of the sidewalls on the lateral 

distribution of a dissolved dispersant can be 

satisfactorily accounted for by the reflectionsuperposition principle in which boundaries are 

treated analytically as reflecting barriers.

The lateral dispersion of floating particles released 

from an intermittent point source at the water 

surface can also be represented as a Fickian 

diffusion process. However, the dispersion pattern 

is somewhat distorted by the effects of secondary 

circulation. The rate of lateral dispersion per unit 

of dispersion distance is approximately the same 

for floating particles as for a dissolved dispersant
OPEN-CHANNEL DISPERSION PROCESSES E65

that is distributed evenly throughout the depth 

of flow. 

The following conclusions are general insofar as they

are not restricted to any particular type of channel or

flow conditions :

11. If a dispersion process does not follow the Fickian 

diffusion law, different methods of evaluating 

dispersion coefficients may lead to significantly 

different results. In order to facilitate the comparison of results it is recommended that, insofar 

as possible, evaluations be based on equations of 

the type

~ Ida2

where <r2 is evaluated by the method of 

moments.

12. The Turner fluorometer, used as a nephelometer, 

provides a rapid and convenient means for 

measuring concentrations of suspended fine sediments. However, because the method is sensitive 

to surface characteristics and size of particles, 

a separate calibration must be performed for 

each type of sediment.

13. The response of concentration-measuring systems 

to rapidly changing concentrations can be evaluated by means of the convolution principle that 

permits the output of the system to be expressed 

as the convolution of the response function of 

the system and the input to the system. The 

response function of the system can be determined by experiment. From a knowledge of the 

response function, corrections for lag and distortion due to slow response can be made.

EXPERIMENTAL EVALUATION OF RESPONSE

CHARACTERISTICS OF THE 

CONCENTRATION-MEASURING SYSTEMS

When the longitudinal dispersion data were being 

analyzed, some questions concerning the response 

characteristics of both the continuous and the discrete 

sampling systems were raised. With what degree of 

fidelity were the combined sampling and concentrationmeasuring systems indicating the rate of change of 

concentration actually occurring at the sampling 

nozzle? Was the response lag sufficient to cause an 

appreciable amount of distortion in the experimentalconcentration versus time curves? Could a method be 

found for correcting the experimental data for the 

effects of response lag? These questions were investigated both analytically and experimentally. Because 

the problem of system response is common to many 

aspects of experimental hydraulics and because of the 

general applicability of the investigational methods, the

results of these investigations are considered worth 

reporting here.

The investigations were based on the convolution 

principle, which according to Lee (1960), is universally 

applicable for determining the output of a linear 

system in terms of the system unit-impulse response 

and the input. For our problem, the principle may be 

stated in the form of a convolution integral,

C0(t, = r/*(*- 

Jo

(77)

where

C0(t, x)= output of combined sampling and concentration-measuring system with the sampling

nozzle located at dispersion distance x, 

/«(£ £i)= response function of the sampling system to an

instantaneous unit impulse occurring at the

sampling nozzle at time ti, 

Gr(ti, x)= input to the system; that is, the concentration

versus time relationship actually occurring

at the sampling nozzle, 

£=time registered by the output end of the

system, and 

£i = time of input; that is, the time of arrival at

the sampling nozzle.

The convolution integral possesses various useful 

properties. The most important for this particular 

application is that it expresses the relationship between 

the probability-density functions of two independent 

random variables and the probability-density function 

of the sum of these random variables. Let / be a random 

variable that expresses the time required for a particle 

of dispersant to travel from an instantaneous plane 

source at x=0 to the sampling nozzle, and let I be 

distributed according to the probability-density function

Let R be a random variable, independent of 7, which 

expresses the elapsed time between the arrival of a 

particle of dispersant at the sampling nozzle and the 

registration of the particle by the system, and let R be 

distributed according to the probability-density function JR(T) where r=t ti. Finally, let 0 be a random 

variable which expresses the total elapsed time from 

the release of a particle at the source until its registration by the system, and let 0 be. distributed according 

to the probability-density function

) yp C0(t,
E66 TRANSPORT OF RAmONUOLIDEiS BY STREAMS

According to the above interpretations, it is apparent 

that

0=R+I (78) 

and that

Mb x)=fR+I (t; x)= r/*(*-*i)//(*i; x)dti. (79) Jo

It is possible to determine /0 (-; x) and/B (-) by means 

of experiment. However,//( ; z), which is the function 

of primary concern, cannot be so determined. Therefore, 

a means of expressing the unknown input function in 

terms of the known output and response functions would 

be highly desirable. Unfortunately, an expression of 

this type cannot readily be obtained unless the natures 

of the functions are specified mathematically, in which 

case //( ; x) can be determined from a knowledge of 

fo(", %), /B(')> and their Fourier transforms. If, as is 

usually the case, a mathematical description of these 

functions is lacking, it nevertheless follows from equation 78 and elementary probability theory that the 

mean value of the output time is equal to the sum of 

the means of the response and the input times,

and likewise for the variances,

(80)

(81)

Thus, T and o\ can be considered, respectively, as 

indices of the lag time and distortion imposed on the 

input by the combined sampling and concentration 

measuring system. A particularly significant result of 

equation 81 is that, subject to possible restrictions on 

the assumptions in the foregoing analysis, values of the 

longitudinal dispersion coefficients obtained in the flume 

experiments were not influenced by the response characteristics of the system. This is because

K7^7- T dffj v dffn x ccLim -7jL =Lim -r*

z-»co ax i->oo ax

since a2R is not a function of x.

The interpretation of /, 0, and, for the discrete 

sampling system, R as random variables is straightforward. These variables are associated entirely with 

dispersion, either in the flume or in the feed tubes of 

the sampling system, and dispersion is essentially a 

random phenomenon. However, in the continuous 

system where the form of JR(T) is due to the combined 

effects of dispersion in the feed tube and the electronic 

characteristics of the fluorometer and recorder, the 

latter of which are essentially deterministic, the justification for interpreting R as a random variable is 

less clear. A partial justification is that/B(r) was shown

by experiments to possess the necessary mathematical 

attributes of a probability-density function.

In order to obtain a quantitative measure of the 

effect of system response on the experimental longitudinal dispersion data, a series of experiments was 

performed in which JR(T) was determined under conditions closely approximating those in the flume 

experiments. Conditions which were duplicated are 

sampling velocity, length and diameter of sampling 

nozzles and feed tubes, and water temperature. The 

experimental setup, shown in schematic form in figure 

58, consisted essentially of two 22-inch-diameter plexiglass tanks, each having a capacity of approximately 

100 liters, a 4-way stopcock, the feed tubes, and either 

a continuous or a discrete sampling system. One of 

the tanks contained water with a known concentration 

of dispersant, and the other tank contained clean 

water. The dispersant tank was equipped with a handoperated agitator, which when given an oscillating 

motion was capable of maintaining a steady concentration of suspended silt-size particles in the tank. By 

means of the 4-way stopcock the source of flow to the 

sampling system could be switched instantaneously

Waste

Tank containing 

clean water

Waste

FIGURE 58. Experimental setup for determining response characteristics of 

sampling systems.
OPEN-CHANNEL DISPERSION PROCESSES E67

from the clean-water lank to the dispersant tank or 

vice versa. The rate of flow in the feed tubes was 

controlled with tube clamps. The same rate of flow 

was maintained in both the sampling system and the 

waste feed tubes. The continuous sampling system 

consisted of the same Turner Model 111 Fluorometer 

and strip chart recorder that were used in the flume 

experiments. The discrete sampling system was a 

simplified version of that used in the flume experiments 

and consisted merely of a movable rack, containing 

30 glass vials, which was passed under the outlet end 

of the feed tube so as to obtain discrete samples at 

one-second intervals.

In theory, jR (T) can be obtained either as a direct observation of the response of the system to an instantaneous unit pulse input or as the derivative with respect to 

time of a unit step function input. In the experiments, 

an instantaneous pulse was approximated by switching 

the source of flow to the sampling system from the 

clean-water tank to the dispersant tank for a period of 

1 second and then switching it back. A step function 

input was obtained by switching the source from the 

clean-water tank to the dispersant tank until the response reached equilibrium with the concentration in 

the dispersant tank. In both cases jR (T) was put in the 

form of a probability-density function by normalizing

/»00

the data to make I /B (r)^r=l. Good experimental

Jo

agreement between the two methods was obtained. 

The pulse input method, despite the 1-second duration 

of the pulse, is considered to be the more reliable due to 

the inaccuracy inherent in graphically differentiating the 

response to the step function input, particularly for the 

first 1 or 2 seconds where the slope is very steep. Most 

of the data reported here were obtained by the pulse 

input method.

For some conditions, response experiments were 

repeated for different concentrations of dispersant in 

the dispersant tank. The results showed that a slight 

degree of dependence on concentration did in fact 

exist. For both the discrete system, with sediment as a 

dispersant, and the continuous system, with either 

dispersant, the tendency was toward a more sluggish 

response as the concentration of dispersant was in 

creased. The causes, however, were not the same. In 

the discrete system with sediment, the increase in 

sluggishness was due to a decreased capacity of the 

flow in the feed tube to transport all of the particles 

simultaneously in suspension at high concentrations. 

In the case of the continouus system with dye, the cause 

was associated with the electronics of the fluorometer 

and the recorder rather than intrinsically with concentration. That is, a response requiring a large instrument deflection was accomplished less rapidly than

a response requiring a small deflection. In the continuous 

system with sediment, both causes were involved. As 

indicated previously, however, the degree of dependence 

of JR(T) on either the concentration of sediment or the 

required instrument deflection was found to be quite 

small and is therefore not considered sufficient to 

appreciably upset the assumption of system linearity 

or the contention that R and / are independently distributed, random variables.

A summary of the results obtained in the response 

experiments and associated supplementary data are 

given in table 10. The values of ~ and o\ were computed 

by the method of moments using the /R(r) curves 

and equations 48 and 49. It is immediately apparent 

that (TR 2 for the sediment system-response curves is 

strongly dependent on the velocity in the feed tubes, 

but essentially independent of the Reynolds number, 

R, of the flow in the feed tubes despite the fact that 

the R values indicate flow conditions ranging from 

laminar to turbulent. This result contradicts estimates 

of the expected dispersion in the feed tubes, which were 

made prior to the flume experiments. Calculations 

based on Taylor's (1953, 1954) theories of longitudinal 

dispersion in laminar and turbulent pipe flow indicated 

that dispersion rates of dye in the feed tubes would 

be negligible in comparison to dispersion rates in the 

flume. As fall velocities of the silt-size sediment particles are on the order of 0.1 percent of the flow velocities 

in the feed tubes, it was assumed that this would be 

so for low concentrations of particles also. This assumption was borne out by the response experiments 

with a feed tube velocity of 2.5 feet per second, but 

not by those with a velocity of 1.3 feet per second. 

In retrospect, it seems likely that, at the lower velocity, 

forces such as those associated with molecular diffusivity, secondary circulation, and turbulence were 

capable of distributing the dye, but not the particles, 

over the cross section of the tube. This explanation is 

compatible with the long tails, which were typical of 

the response functions for sediment, obtained with the 

discrete system.

The results of the system-response experiments were 

used as a basis for estimating corrections to be subtracted from the values of t and v\ found in the flume 

experiments. Values of ~T and v\ used in making these 

corrections are listed in table 11.

Three typical system-response functions are shown 

in figure 59. These response functions were obtained 

under conditions closely approximating those in the 

flume experiments LO-D-2, LO-FS-2, and LO-CG-2. 

In order to illustrate more clearly the effect of the 

system-response characteristics on the longitudinal 

dispersion data, output curves are compared with 

hypothetical input curves for the same three runs in
E68 TRANSPORT OF RAWONUCJLIDES BY STREAMS

LO

0.8

0.6

0.4

0.2 \-

0 

1.0

T 0.8

CO 

Q

8

% 0.6

0.4 

0.2

0 

1.0

0.8 

0.6 

0.4 

0.2

Input

Response

Run LO-D-2

Continuous system

Dye, lamp adapter screen installed

T* 20°C

r =6.7 sec 

= 4.3 sec2

Input

Response

Run LO-FS-2 

Discrete system 

< 44M glass beads 

T~ 20°C

f=4.6 sec 

o =9.6 sec 2

Input

Response

Run LO-CG-2 

Discrete system 

53-62 M glass beads 

T~ 5°C

f =5.5 sec

6 8 10 12 14 

r , IN SECONDS

FIGURE 59. Three typical system-response functions.

16 18 20

figure 60. The input curves were obtained from equation 9a with t=t\, z=65.6 feet and U and Kx for the 

flow conditions in runs LO-D-2, LO-FS-2, and LOCG-2. The output curves were determined by means 

of a numerical integration of equation 79 using the 

input curves, and the response functions in figure 59 

with r=t ti and dtl=Mi=l second. The results indicate that the distortion imposed on the longitudinal

dispersion data by the response characteristics of the 

measuring systems is not great. Also, the values of 

the means and variances listed in figures 59 and 60 

are in good agreement with equations 80 and 81, considering the fact that the moments for the response 

and output curves were calculated by the approximate 

method of numerical integration using equations 48 

and 49.
OPEN-CHANNEL DISPERSION PROCESSES E69

TABLE 10. Summary of results of system-response experiments

Dispersant

Pontacyl Brilliant Pink B...

System

. _ , _ ... Continuous. _____

.... do ....... .......

....do.i. ..... .....

....do.' ....... ...... .

....do.'. - ... ...

.......... Continuous... ......... .

....do..... .. ........... .

.__.do . ...............

Velocity in Temp, 

sampling tube (°C) 

(ft per sec)

2.46 

2.54

2.46

1.28

1.35

1.28

2.46

1.32

2.46

2.46

1.37

1.32

21.5 

4.0 

20.0 

4.0 

21.5 

4.0

23.0 

23.0

24.5 

5.0 

25.0 

5.2

R

4730 

2980 

4560 

1510 

2600 

1510

4870 

2620

5020 

3000 

2820 

1610

Approx. F 

Approximate (fluorometer 

initial units per 

concentration unit of 

concentration)

18, 38 ppb 

12 ppb 

18 ppb 

12 ppb 

11 ppb 

7.5 ppb

500 ppm 

500 ppm

1,000 ppm 

1,000 ppm 

1, 000 ppm 

1, 000 ppm

4.9 

3.8 

15.0 

4.7 

4.4 

4.7

.147 

.101

.070 

.070 

.032 

.032

r (sec) 9\

6.5 

5.1 

6.7 

3.2 

3.0 

3.4

6.5 

4.6

6.0 

6.2 

5.4 

5.5

^(secJ)

1.7 

2.1 

4.3

.8 

.5

.7

1.7 

9.6

2.6 

3.1 

15.7 

16.3

i Lamp adaptor screen in place.

' From derivative of step function response.

TABLE 11. Corrections for system-response lag which were applied to flume data

Flume run (LO-) System f (sec) "K (sets') Remarks

D-l... .................

2......-.. .......

3.. .............. ....

3.. - -- - -

F8-1.. ........ ......._. 

I....... ............ 

2.. ........ ........ .

2... ......-__.....

CS-1.. ..............

1.. .................

FG-1-. ........... .....

1... ...............

3----...-.. ....

3-..-.. .........

CO-1. - 

1... ...............

2..................

2.... ..............

3.. ............. ...

3.. ............... .

................. Continuous...... _ ...........

-.- ...- .. ....do... ...... ...... ........... .

................ ... .do...... ........... ..........

......-..-..--.. Discrete-----.----------- ---.

................ ....do .......................

.. ___ ... _ ... Continuous.. .................. 

..... ___ ..... Discrete _ ....................

. ______ ... Continuous..- --... ----------

................ Discrete. ------ --.

................. Continuous....................

... _ ........... Discrete _ . ____ . __ .

....... __ ...... Continuous... ....... .

................. Discrete . ..... .... .

. ___ . ____ Continuous.. _ ... ___ . _

. _ . __ ...... Discrete __ __ .- _ ...

. _____ ...... Continuous.. _ .... ......

................. Discrete ..-.... . ....

.... _ ....... _ Continuous... _ . _ ... _ ...

.............. _ Discrete _ _ . _ ......

... _ . _ ....... Continuous.. _ .. __ ........

.......... 6.4

......_... 6.4

.......... 5.1

.......... 3.3

.......... 4.6 

.......... 6.4 

.......... 4.6

.......... 6.4

.......... 5.4

.......... 6.9

.......... 4.8

.......... 5.5

.......... 4.8

.......... 5.5

.......... 5.5

.......... 6.2

.......... 5.5

.......... 7.2

.......... 5.5

.......... 6.2

4.3

4.3

2.1

.7

9.6 

4.3 

9.6

4.3

15.7

5.1

10.1

2.1

10.1

2.1

16.3

3.1

16.3

5.6

16.3

3.1

Lamp adaptor screen installed in fluorometer.

Do.

15-3Qft silt assumed to behave as <44j» glass beads. 

Lamp adaptor screen installed in Suorometer. 

15-3Qft silt assumed to behave as <44p glass beads.

Lamp adaptor screen installed in fluorometer.

53-62/1 silt assumed to behave as 53-62p glass beads.

Lamp adaptor screen effect estimated.

Temperature effect estimated.

Do.

Do.

Do.

Lamp adaptor screen effect estimated.

0.10

R jn LO-FS-2

I 

I

K\

40 50 60 70 80 90 100

£i=58.1 sec 

<r £= 63.0 sec 2 

f =64.9 sec 

<rn2= 66.9 sec 2

n LCKCG-2

X)

30 40 50 60 70 80 90 100 

tj.t.lN SECONDS

?!=58.1 sec

<rf= 63.0 sec 2

i= 62.7 sec

<r 2=71.2 sec 2

30 40 50 60 70 80 90 100

£1 = 59.7 sec 

° 1=70.2 sec 2

£=65.1 sec 

<r 2= 86.3 sec 2

FIGURE 60. Comparisons between hypothetical input and output curves illustrating the effect of system-response characteristics on longitudinal dispersion data.
E70 TRANSPORT OF RADIONUCLIDEiS BY STREAMS

LITERATURE CITED

Al-Saffar, A. M., 1964, Eddy diffusion and mass transfer in 

open channel flow: California Univ., Ph.D. dissert., 138 p.

Aris, R., 1956, On the dispersion of a solute in a fluid flowing 

through a tube: Royal Soc. (London) Proc., Ser. A, v. 235, 

p. 67-77.

Batchelor, G. K., 1953, The theory of homogeneous turbulence: 

London, Cambridge Univ. Press, 197 p.

Batchelor, G. K., Binnie, A. M., and Phillips, 0. M., 1955, The 

mean velocity of discrete particles in turbulent flow in a 

pipe: Physical Soc. [London] Proc., v. 68, Sect. B, p. 

1095-1104.

Batchelor, G. K., and Townsend, A. A., 1956, Turbulent diffusion, in G. K. Batchelor, and R. M. Davies, eds., Surveys 

in mechanics: London, Cambridge Univ. Press, p. 352-399.

Brush, L. M., Jr., 1962, Exploratory study of sediment diffusion: 

Jour. Geophys. Research, v. 67, no. 4, p. 1427-1435.

Camp, T. R., 1944, Discussion of "Effect of turbulence on 

sedimentation": Am. Soc. Civil Engineers Trans., v. 109, 

p. 660-667.

 1946, Sedimentation and the design of settling tanks: 

Am. Soc. Civil Engineers Trans., v. Ill, p. 895-958.

Diachishin, A. N., 1963, Waste disposal in tidal waters: Am. Soc. 

Civil Engineers Proc., v. 89, no. SA4, Paper no. 3602, p. 

23-43.

Dobbins, W. E., 1944, Effect of turbulence on sedimentation: 

Am. Soc. Civil Engineers Trans., v. 109, p. 629-678.

Elder, J. W., 1959, The dispersion of marked fluid in turbulent 

shear flow: Jour. Fluid Mechanics, v. 5, pt. 4, p. 544-560.

Ellison, T. H., 1960, A note on the velocity profile and longitudinal mixing in a broad open channel: Jour. Fluid Mechanics, v. 8, pt. 1, p. 33-40.

Feuerstein, D. L., and Selleck, R. E., 1963, Tracers for dispersion 

measurements in surface waters: California Univ., Berkeley, 

Sanitary Engineering Research Lab. Rept. 63-1, 69 p.

Fischer, H. B., 1964a, Longitudinal dispersion by velocity 

gradients in open channel flow: Pasadena, California Inst. 

Technology, W. M. Keck Lab. Hydraulics and Water 

Resources Tech. Memo. 64-4, 12 p.

 1964b, Determination of dispersion coefficients by the 

change in moment method: Pasadena, California Inst. 

Technology, W. M. Keck Lab. Hydraulics and Water 

Resources Tech. Memo. 64-6, 22 p.

 1966, A note on the one-dimensional dispersion model: 

Internat. Jour. Air and Water Pollution, v. 10, no. 6/7, p. 

443-452.

Glover, R. E., 1964, Dispersion of dissolved or suspended materials in flowing streams: U.S. Geol. Survey Prof. Paper 

433-B, 32 p.

Godfrey, R. G., and Frederick, B. J., 1963, Dispersion in natural 

streams: U.S. Geol. Survey open-file rept., 75 p.

Hino, Mikio, 1961, Discussion of "Eddy diffusion in homogeneous 

turbulence": Am. Soc. Civil Engineers Trans., v. 126, pt. 1, 

p. 417-419.

Hinze, J. 0., 1959, Turbulence: New York, McGraw-Hill Book 

Co., 586 p.

Hubbell, D. W., and Sayre, W. W., 1964, Sand transport studies 

with radioactive tracers: Am. Soc. Civil Engineers Proc., 

v. 90, no. HY4, Paper no. 3900, p. 39-68.

Kalinske, A. A., and Pien, C. L., 1944, Eddy diffusion: Industrial 

and Engineering Chemistry, v. 36, p. 220-223.

Karman, T. von, and Lin, C. C., 1949, On the concept of similarity in the theory of turbulence: Rev. Modern Physics, 

v. 21, p. 516-519.

Kolmogoroff, A. N., 1941, The local structure of turbulence in 

incompressible viscous fluid for very large Reynolds numbers: Acad. Sci. URSS Comptes rendus (Doklady), v. 30, 

no. 4, p. 301-305.

Korn, G. A., and Korn, T. M., 1961, Mathematical handbook 

for scientists and engineers: New York, McGraw-Hill Book 

Co., 943 p.

Krenkel, P. A., 1960, Turbulent diffusion and the kinetics of 

oxygen absorption: California Univ., Ph. D. dissert., 142 p.

Krenkel, P. A., and Orlob, G. T., 1963, Turbulent diffusion and 

the reaeration coefficient: Am. Soc. Civil Engineers Trans., 

v. 128, pt. Ill, p. 293-334.

Lee, Y. W., 1960, Statistical theory of communication: New 

York, John Wiley & Sons, 509 p.

Levenspiel, Octave, and Smith, W. K., 1957, Notes on the 

diffusion-type model for the longitudinal mixing of fluids in 

flow: Chemical Engineering Science, v. 6, p. 227-233.

Mickelsen, W. R., 1960, Measurements of the effect of the 

molecular diffusivity turbulent diffusion: Jour. Fluid 

Mechanics, v. 7, pt. 3, p. 397-400.

Nemenyi, P. F., 1946, Discussion of "Transportation of suspended sediment by water": Am. Soc. Civil Engineers 

Trans., v. Ill, p. 116-125.

Orlob, G. T., 1958, Eddy diffusion in open channel flow: California Univ., Sanitary Engineering Research Lab., Contr. 

19, 144 p.

 1961, Eddy diffusion in homogeneous turbulence: Am. 

Soc. Civil Engineers Trans., v. 126, pt. 1, p. 397-438.

Patterson, C. C., and Gloyna, E. F., 1963, Radioactivity transport in water the dispersion of radionuclides in open 

channel flow: Univ. Texas, Environmental Health Engineering Research Lab. Tech. Rept. to U.S. Atomic Energy 

Comm., 87 p.

Richardson, L. F., 1920, Some measurements of atmospheric 

turbulence: Royal Soc. London Philos. Trans., v. 221A, p. 

1-28.

Richardson, L. F., and Stommel, H., 1948, Note on eddy diffusion in the sea: Jour. Meteorology, v. 5, p. 238-240.

Sayre, W. W., and Albertson, M. L., 1963, Roughness spacing 

in rigid open channels: Am. Soc. Civil Engineers Trans., v. 

128, pt. 1, p. 343-427.

Sayre, W. W., and Chamberlain, A. R., 1964, Exploratory 

laboratory study of lateral turbulent diffusion at the surface 

of an alluvial channel: U.S. Geol. Survey Circ. 484, 18 p.

Sayre, W. W., Guy, H. P., and Chamberlain, A. R., 1963, Uptake 

and transport of radionuclides by stream sediments: U.S. 

Geol. Survey Prof. Paper 433-A, 35 p.

Sayre, W. W., and Hubbell, D. W., 1963, Transport of radionuclides in fresh water Dispersal of bed sediments, in 

U.S. Atomic Energy Comm., Transport of radionuclides by 

fresh water systems: U.S. Atomic Energy Comm. TID7664, p. 327-352.

 1965, Transport and dispersion of labeled bed material, 

North Loup River, Nebraska: U.S. Geol. Survey Prof. 

Paper 433-C, 48 p.

Task Committee on Preparation of Sedimentation Manual, 

1963, Sediment transportation mechanics suspension of 

sediment: Am. Soc. Civil Engineers Proc., v. 89, no. 

HY5, Paper no. 3636, p. 45-87.

Taylor, G. I., 1921, Diffusion by continuous movements: 

London Math. Soc. Proc., Ser. 2, v. 20, p. 196-211.
OPEN-CHANNEL DISPERSION PROCESSES E71

Taylor, G. T., 1953, Dispersion of soluble matter in solvent flowing 

slowly through a tube: Royal Soc. (London) Proc., Ser. A, 

v. 219, p. 186-203.

 1954, The dispersion of matter in turbulent flow through 

a pipe: Royal Soc. (London) Proc., Ser. A, v. 223, p. 

446-468.

Tchen, C. M., 1947, Mean value and correlation problems connected with the motion of small particles suspended in a 

turbulent fluid: Delft Tech. Univ. Lab. Aero-and Hydrodynamics Rept. 51.

Vanoni, V. A., 1946, Transportation of suspended sediment by 

water: Am. Soc. Civil Engineers Trails., v. Ill, p. 67-133.

Yotsukura, Nobuhiro, 1963, Turbulent dispersion of miscible 

materials in open channels: in U.S. Atomic Energy Comm., 

Transport of radionuclides in fresh water systems: U.S. 

Atomic Energy Comm. TID-7664, p. 311-326.

Yotsukura, Nobuhiro, and Fiering, M. B., 1964, Numerical 

solution to a dispersion equation: Am. Soc. Civil Engineers 

Proc., v. 90, no. HY5, Paper no. 4046, p. 83-104.

 1966, Closure to "Numerical solution to a dispersion 

equation": Am. Soc. Civil Engineers Proc., v. 92, no. 

HY3, p. 67-72.

Yotsukura, Nobuhiro, Smoot, G. F., and Cahal, D. I., 1964, 

Dispersion in open channel flow: U.S. Geol. Survey openfile report, 19 p.

U.S. GOVERNMENT PRINTING OFFICE: 1968 O-28I-66I