Dispersion of 

Dissolved or Su

Materials in Flowipg Streams

GEOLOGICAL SURVEY PROF E S SIGNAL PAPER 433-B

Prepared in cooperation with the 

lf,S. Atotnif Energy Coifnmisstoffi
Dispersion of 

Dissolved or Suspended 

Materials in Flowing Streams

By ROBERT E. GLOVER

TRANSPORT OF RADIONUCLIDES BY STREAMS

GEOLOGICAL SURVEY PROFESSIONAL PAPER 433-B

Prepared in cooperation with the 

U.S. Atomic Energy Commission

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1964
UNITED STATES DEPARTMENT OF THE INTERIOR 

CECIL D. ANDRUS, Secretary

GEOLOGICAL SURVEY

H. William Menard, Director

First printing 1964 

Second printing 1979

For sale by the Branch of Distribution, U.S. Geological Survey, 

1200 South Eads Street, Arlington, VA 22202
CONTENTS

Abstract--__-_-_____-_---________________

Introduction. __-__-_-_----_-___________--

Acknowledgments.-.-- _ -_-_____-_-___-__-_ 

Notation.-_-____-_________--_-_--_--____

Definitions of terms used in this report.-----

Basic differential equation _________________

Solutions-_________-____-__-______---_.__

Product law__._-__________.__-._______.

Use of the product law._-_____-_-_-__-____

Velocity variation as a dispersing mechanismContinuous introduction of salinity______-_-

Specialchart_ _-___-_-----___________--___

Laboratory and field investigations _________

Purposes.----------------------------

Planning of tests____________________

Electrode resistance.--______.___-_-___

Details of methods and equipment------

Fort Collins tests_________________________

Description of tests.-_-______-_-__-__-

Calibrations.

Bl

1

2

3

4

4

5

6

6

6

7

8

9

9

10

10

10

13

15

16

Comparison of diffusion coefficients, Fort Collins tests. __ 

Allenspark tests__________-_-__----________---______

Longitudinal diffusion coefficients obtained from the 

Allenspark tests ------------------------------

The South Platte dispersion test..____________________

Mohawk test___________-______--__--__-----_----_

Lateral dispersion -__________----____-----__---_--

Columbia River test._______-_____-_-___-_---------_

Integral relationships,-.. ----------------------------

Evaluation of integrals. _______------------------__--

Determination of diffusion coefficients by analytical procedures. .___________-____--_---_----_--_-----_---

Summary of longitudinal and lateral diffusion coefficients _ 

Examples..----------------------------------------

Longitudinal dispersion.-_-----------------------

Transverse dispersion.__________________________

Outline of computation procedure.____________________

Selected references._____-_______--______-_----------

ILLUSTRATIONS

FIGURE 1. Special chart for estimating salinity changes 

as modified by boundaries._____________

2. Longitudinal diffusion test record Intermittent operation of recorder, Fort Collkis 

tests. -_---_______________.-_______j__

3. Longitudinal diffusion test record Continuous operation of recorder, Fort Collins 

tests.________________________________

4. Transverse diffusion test with continuous 

feed at side, Fort Collins tests..---------

5. Test flume 8 feet wide "with trough and point 

gages in place, Fort Collins tests_______

6. Electrode numbers and positions, Fort Collins tests ___ _________________________

Page

B9

11

12

12

13

13

Page 

B16

17

18

18

20

20

21

22

23

24

26

26

26

28

31

31

Page

FIGURE 7. Test flume 8 feet wide showing electrode assembly and bar roughness, Fort Collins 

tests. _____._-_-- _- --__ -_- B14

8. View of Allenspark flume showing the triangular section.__ __________-_-------_--_ 17

9. Allenspark electrode arrangement _______ 17

10. Comparison of observed and computed salinities at points 3 and 4, South Platte test - _ 28

11. Comparison of computed and observed distributions at 3,590 feet from the source 

Lateral dispersion, Colombia River test__ 29

12. Comparison of observed and computed salinity distributions at 29,900 feet from the 

source Lateral dispersion, Columbia 

River test-__________.____----_------_ 30

TABLES

Page

TABLE 1. Flow data, Fort Collins tests________________________-________-------_------_---------------------- B15

2. Correlations among electrode, galvanometer, and trace numbers, Fort Collins tests-__-___-______-_---------- 15

3. Comparison of experimentally and analytically determined longitudinal diffusion coefficients, Fort Collins tests. _ 16

4. Longitudinal diffusion coefficients obtained from the Allenspark tests._______________-_-__----------------- 19

5. Analyses of samples for potassium ion, South Platte test----_-__-----.-------------------------- ------ 19

6. Discharge measurements and stream channel data, South Platte test____._-______-_-__-_--_-------------- 20

7. Lateral diffusion coefficients obtained from selected Fort Collins tests. ______-___-___-_-.------------------- 21

8. Comparison of computed and observed values of longitudinal and transverse diffusion coefficients ___________ 26

9. Sample computation for point 3, South Platte test___________-__------------_-------------------------- 27

10. Computation of lateral diffusion from a continuous point source, Columbia River test___--___---------------- 29

11. Estimated salinities 29,900 feet from the source, Columbia River test-------------------------------------- 29

m
TRANSPORT OF RADIONUCLIDES BY STREAMS

DISPERSION OF DISSOLVED OR SUSPENDED MATERIALS IN FLOWING STREAMS

BY ROBERT E. GLOVER

ABSTRACT

The findings on stream dispersion described here were obtained by postulating certain possible types of dispersal mechanism, developing the characteristics of these mechanisms analytically, and comparing the results so obtained with data obtained from tests.

The possibility that the dispersal mechanism was a diffusion 

process was first considered. Analysis of this mechanism led to 

the conclusion that maximum salinities due to a local introduction of salinity at an upstream point should vary inversely as 

the square root of the distance from this point. This conclusion was confirmed by tests made on the 400-foot long flume at 

Allenspark and is in accord with the results of salt velocity tests 

made on long conduits. It is also in agreement with the results 

of theoretical work dealing with the combined effect of velocity 

variations through the cross section of a flowing stream and 

turbulence (Taylor, G. I., 1954 ; and Elder, 1959). The dispersing mechanism is therefore identified as being of the diffusion 

type.

A second possibility considered was that dispersion in the 

downstream direction was predominantly due to velocity variations in the cross section of the stream. This postulate yielded 

a formula which indicated that the maximum salinity observed 

at a downstream point should vary inversely as the distance 

from the point of introduction. This variation is not confirmed 

by the data obtained from tests in spite of the fact that these 

velocity variations are the most potent single factor in the mechanism of longitudinal dispersion. The importance of lateral mixing due to turbulence as a factor in the longitudinal dispersion 

mechanism must therefore be recognized. This finding is in accord with theoretical developments.

Comparisons of theoretical and test data show a very satisfactory agreement between theoretical findings and flume-test results, but natural streams yield diffusion coefficients approximately 10 to 50 times as large as those obtained on theoretical 

grounds.

Lateral-diffusion coefficients are much smaller than those for 

longitudinal diffusion because turbulence is the primary factor 

in lateral diffusion. In the same stream, the ratio of the two 

coefficients is of the order of 300.

INTRODUCTION

Streams have long been used as a means for disposal 

of wastes. It is a common practice, for example, to discharge sewage into flowing streams where oxidation 

and bacterial actions can destroy the organic materials 

it contains. The dangers inherent in such practices

have long been realized, but with the growing accumulation of nuclear wastes a new danger from contamination by radioactive material may be added. It is difficult to insure that dangerous contaminations cannot 

happen in some way because, even though the radioactive material is buried, ground water may carry some 

of it in solution and discharge it into a stream. It is the 

purpose of this investigation to study the characteristics 

of the process of dispersal so that, if it is discovered 

that dangerous materials have escaped, estimates may 

be made of their location and concentration.

At the beginning of this investigation certain possible 

dispersion mechanisms were selected, and the consequences of these were worked out analytically. These 

mechanisms included the concept of dispersion as a diffusion process and the concept of longitudinal dispersion by variation of the stream velocity through the 

cross section. Laboratory and field investigations were 

then made, and the observed and theoretical results were 

compared to determine whether an explanation of the 

observed facts could be obtained from either of the postulated mechanisms. It will be seen later that the diffusion mechanism did provide a complete explanation.

The characteristic feature of a diffusion process is a 

rate of propagation of the material being dispersed that 

is proportional to the gradient of the concentration of 

the material. The propagation is in the direction of 

decreasing concentration. The factor of proportionality is called a diffusion coefficient. In this investigation, diffusion coefficients were determined experimentally. Where appropriate, the diffusion equations 

presented here contain a term to account for the possibility that the added materials may be carried along 

bodily by the mean stream velocity.

Recent analytical studies based upon measured or derived velocity variations through an idealized stream 

cross section and the mechanism of turbulence have 

shown that longitudinal stream dispersion can be descrijbed by a virtual coefficient of diffusion (Taylor, 

G. I., 1954; and Elder, 1959). Such behavior is in

Bl
B2 TRANSPORT OF RADIONUCLIDES BY STREAMS

accord with results obtained from this investigation, 

and a digest of the work is included. A surprising result comes out of the comparison; very good correlations between theoretical work and test results were 

obtained from the flume experiments, but dispersion 

rates that are many times those indicated by the analytical work were obtained for natural streams.

One method of velocity determination utilizes the 

time interval required for a salt cloud to traverse a 

known distance between electrode positions. This is 

known as the salt-velocity method. Effective use has 

been made of the recorded experience in this field in developing the methods described here. In return, these 

developments provide a new theoretical basis for saltvelocity practices.

In preparing this presentation, it has been kept in 

mind that the methods developed here might have to be 

used at a tune of threatened disaster when simple and 

reliable methods would be much needed, .and unnecessary complications would become a dangerous nuisance. 

An investigation such as this one is definitely in the 

realm of mathematical physics and requires at many 

points the use of judgment and the making of decisions 

so that the end product will be at the same time as reliable and as simple to use as possible. The formulas 

presented here have been carefully tied in with hydraulics concepts so that the wealth of empirical data 

accumulated in this field can be brought to bear in case 

of need. Some effort has also been applied to the task 

of bringing out relationships that will contribute to a 

sound and useful understanding of the characteristics 

of the dispersion mechanism. In places, some concessions have been made in the interest of simplicity. 

These will be mentioned at the appropriate place in the 

text.

In an investigation of the type described here, there 

is always a possibility that the time and funds available 

may be inadequate to settle al] the questions which arise 

during the progress of the work. It will be pertinent, 

therefore, to comment briefly upon some factors of this 

kind which appeared during these studies.

The surprising difference in the dispersive power of 

streams running in flumes and natural channels indicates that more work is needed to determine factors, 

such as those contained in table 8, that govern a wider 

range of conditions.

A persistent phenomenon in dispersion tests on natural streams indicates a loss of tracer with time. The 

limited data now available indicate that the added materials may upset a previously stable chemical regimen 

with the result that some of the tracer material is removed by chemical reactions between the tracer material

and the materials of the bed and banks of the stream. 

More study is needed to clarify the nature and behavior 

of this factor.

Throughout the report the term "salinity" is used 

to describe not only the dissolved salts that may be present in a stream but also those suspended materials that 

respond to the diffusion process in the same manner as 

dissolved salts. Such suspended materials include not 

only the very fine sediments such as fine clay and colloids, but also include micro-organisms such as plankton, which may collect and carry radioactive material.

For the benefit of those who may have occasion to 

apply these methods, it will be helpful to identify the 

formulas to be used in typical situations. Formula 3 

applies where a quantity of contaminating material is 

suddenly released in a cross section of a stream and is 

dispersed in the direction of flow. If conditions are 

similar except that the contaminating material is originally distributed through the volume of the stream 

over a length Z, formula 4 applies. When a finite 

amount of a contaminant is suddenly released at a point 

in the stream and may be dispersed longitudinally and 

also transversely and vertically, formulas to apply can 

be made up of formulas 3,13, and 14, combined according to the product law described in a later paragraph of 

this report. If the contaminating material is supplied 

continuously at a point on the bank of the stream, 

formula 30 will apply. Diffusion coefficients for use 

with these formulas can be obtained from table 10. 

Formulas 13, 14, and 30 are appropriate when the contaminating material is released at the bank of the stream 

where lateral dispersion can act in only one lateral 

direction, this being the direction toward the other bank 

of the stream. If the point of release is away from the 

bank, where lateral dispersion can move the salinity 

toward both banks, then one-half of the material introduced can be assumed to move each way.

ACKNOWLEDGMENTS

The technical advice and assistance of Mr. C. K. Daum 

in designing, fabricating, and operating the equipment 

used in the Fort Collins and Allenspark experiments 

is gratefully acknowledged. A note of thanks is due 

the Bureau of Reclamation for loan of the oscillograph 

and timer equipment used in making the Fort Collins 

and Allenspark tests. Acknowledgment is also made 

of the General Electric Co. who supplied patterns of 

lateral dispersion for the Columbia River, expressed as 

percentages of an original concentration.
DISPERSION OF DISSOLVED OR SUSPENDED MATERIALS IN FLOWING STREAMS B3

This report was prepared under the administrative 

supervision of G. Earl Harbeck, Jr., Project Hydrologist, and C. C. McDonald, Chief, General Hydrology 

Branch, U.S. Geological Survey. This study is one of 

several made by the Geological Survey in cooperation 

with the U.S. Atomic Energy Commission.

NOTATION

The following notation will be used:

A

A,

B

b

C' 

C

D

d 

e

E

E(u)

77? / f

rll J2j JS

9 

I

Ji

K' 

K"

Kn

K-x> Kv ,. 

L

L'

n

Qs

area (sq ft or ft2) 

a constant

- (sec)

radius of a circular pipe (ft)

the average width of the stream (ft)

y2

=7W (I Per sec)

=BDV Stdt

Chezy's coefficient (Vft per sec2 )

concentration

_Kx

V

the average depth of the stream (ft) 

diameter of an electrode bar (ft) 

the base of the natural system of logarithms, 2.71828+ (dimensionless) 

electrical potential in volts (v). Alternating voltages are given as root mean

square values. 

the probability integral corresponding to

the argument u (dimensionless) 

functions defined in the text 

acceleration of gravity (ft per sec2 ) 

electrical current (amperes)

r Sdt J2=BDV

Jo Jo 

Von Karman's constant (dimensionless)

conductivity (Ohmsj(feet)

a constant defined in the text

modified Bessel functions of the second

kind of order n. (This definition conforms to that of the British Association

Mathematical tables, vol. 6.) 

diffusion coefficients for the directions x,

y, and z, respectively (ft2 per sec) 

a distance measured downstream from the

point x=0 (ft)

the length of an electrode (ft) 

a dimensionless number 

a factor specifying hydraulic roughness 

a quantity of salt (Ib) 

a quantity of salt introduced per unit of

time (Ib per sec)

u

v*

V

V

Wn

w 

x,y, z

TT 

P

TO

hydraulic radius (ft)

slope of a stream (dimensionless)

salinity, the amount of salt in unit volume

of water (Ib per ft3 ) 

maximum salinity observed at a station

(Ib per ft3)

electrode spacing (ft) 

time coordinate (sec)

a time defined by the relation (

z> :)=:

the mean transit time corresponding to the 

center of gravity of the salinity-time 

curve obtained at a point on a stream 

(sec)

a generalized variable whose value becomes

determinate in any specific case I 

-*/ the shear velocity (ft per sec)

a substitution variable

the mean stream velocity (ft per sec)

=BD f Sdx

J 

integrals described in the test 

weight per unit of volume (lb/ft3 ) 

coordinates measured, respectively, in the 

direction of streamflow, across the 

stream and vertically downward (ft). 

These quantities are measured from a 

point on the bank of the stream which 

is assumed to be the point at which the 

salinity is introduced. These quantities 

are considered positive in the directions 

stated. 

xV

distance measured upward from the bed

of a stream (ft) 

a substitution variable 

a time variable 

=3.14159 +

/it*

= the density of the fluid

9 

the friction stress exerted by the stream

on its bed (Ib per ft2)

Although the notation is expressed here in terms of 

English units, any consistent system of units may be 

used with the formulas given here, except for the expressions derived from conventional hydraulic formulas which are traditionally tied to specific unit systems. 

These formulas are identified by a dagger (|).
B4 TRANSPORT OF RADIONUCLIDES BY STREAMS

DEFINITIONS OF TERMS USED IN THIS REPORT

Density current: A current moving along the bottom 

or top of a stream because of a difference in densities. Such differences of density may be due to 

temperature differences or differences of salinity between the water of a stream and waters being discharged into it. Suspended sediment may also 

cause density differences.

Dispersion: A general term used here to describe a scattering of dissolved or suspended material through 

the volume of a stream without reference to the 

nature of the mechanism which brings it about.

Diffusion: A specific term used to indicate a dispersion 

mechanism of the type commonly associated with 

diffusion phenomena. These phenomena are expressed by differential equations of a characteristic 

type. Equations 1, 2, 7, 9, and 12 are of this type. 

They imply a salinity propagation rate proportional to the salinity gradient.

Diffusion coefficient: A coefficient which represents the 

rate of salinity propagation across a unit area under 

a unit salinity gradient. It has the dimensions 

ft2 per sec.

Salinity: As used here this term designates any kind of 

a dissolved or suspended material which may be 

dispersed through the volume of a stream.

Salt-velocity test: A test made to determine the mean 

velocity of a stream by observing the progress of 

a salt cloud injected into it.

Velocity variations: Velocity variations in the cross section of a stream. The term velocity, as used here, 

indicates a temporal mean at a point. It is such a 

velocity as would be determined by a current meter.

Suspended material: Any material held continuously in 

suspension, such as clay particles, plankton organisms, and the like which may carry radioactive material and which move freely with the water.

ppm: Parts per million, by weight.

BASIC DIFFERENTIAL EQUATION

Analytical treatments of diffusion phenomena began 

with the work of Fourier in the year 1812. He was 

principally concerned with the conduction of heat in 

solids, although he recognized that the same basic relationships would apply in other fields also. Since his 

time, phenomena of the diffusion type have been observed in many other fields of which chemical processes, 

osmosis, ground-water flow, evaporation, atmospheric 

pollution, sediment transport, combustion processes, 

propagation of ocean salinity into estuarine channels, 

biologic changes, flow of water through unsaturated

soils, and the propagation of electricity may be mentioned.

The diffusion of momentum by turbulence is of fundamental importance in streamflow. The processes by 

which water evaporates from lakes, is picked up into the 

atmosphere and carried away by the wind, and the 

transport of sediment by streams are related to the 

process here under study because turbulence is an important factor in each. From our standpoint, it will 

be important to realize that in the processes to be studied, the differences from other fields are enough, however, so that developments in them cannot be taken 

over and applied to the process of dispersion in streams. 

Much help may, nevertheless, be obtained from the 

pioneer work of Fourier.

The analytical and experimental developments to be 

described relate to dispersion in three dimensions. 

These are in the downstream, cross-stream, and vertical 

directions. The rate of diffusion in the downstream 

direction has been investigated by Taylor (1954) and 

Elder (1959) by a fluid mechanics approach. A development for vertical dispersion has been described 

by Vanoni (1953). In a later chapter, it will be shown 

how these developments can be used to calculate values 

for the dispersion coefficients introduced into the present 

development.

For the benefit of those readers who may be unfamiliar with differential equations and their solutions, it 

may be explained that a differential equation is a mathematical formula which expresses some fundamental requirement. Equation 1 is such a formula. It not only 

expresses quantitatively how salt is propagated, but it 

also expresses a requirement that once the salt has entered the stream its amount will remain unchanged. A 

solution of a differential equation is an expression, such 

as formula 3, which satisfies the requirements imposed 

by the differential equation and, in addition, satisfies 

certain special conditions called boundary conditions. 

The task of finding a solution of a differential equation 

which meets an appropriate set of boundary conditions 

is basically one of great difficulty. The labors of mathematicians over the last two centuries have, however, 

provided solutions of many basic types. The ones used 

here come principally from the work of Fourier. His 

work was presented to the French Academy in the 

year 1812.

The concepts which form the basis of the analyses 

may be described in the following way: 

1. Salinity can be carried along by the flow of the 

stream.
DISPERSION OF DISSOLVED OR SUSPENDED MATERIALS IN FLOWING STREAMS B5

2. The variations of velocity through the cross section 

of a stream disperses any existing salinity concentration in the direction of flow.

3. Turbulence and, to a minor extent, molecular diffusion contribute to the dispersion of a salinity concentration in the vertical, horizontal, and downstream directions.

To put these concepts into a form suitable for analytical

use, we may assume that

4. Where a salinity gradient exists, salinity will be 

propagated in the direction of decreasing salinity.

5. The rate of propagation of salinity across a unit of 

area is proportional to the salinity gradient along 

a line normal to the area.

Some of these concepts are certainly true and all are 

reasonable, but with respect to item 5 we may entertain 

some reservations concerning the linearity of the relation, as assumed. Here a choice must be made. We 

may note, first, that if a nonlinear relationship is to be 

used instead of the linear type assumed, we will be 

forced to deal with a nonlinear partial differential equation, and this will involve us in mathematical difficulties 

of the most serious kind. As a matter of tactics, it 

seems much better to assume a linear relation, to deal 

with the linear differential equation that results, and to 

reserve judgment until we can compare the results of 

this analysis with the results of tests. It may be worthwhile to state here that both test data and the independent analyses to be described later justify this course 

of action.

The effect of velocity variations as described in item 

2 is very important when dealing with a mass of salinity 

introduced into a stream at a given time and place, such 

as could occur if a cloud of radioactive material were 

released into a stream. It has a lesser significance 

where the salinity is supplied continuously to a stream 

so that the salinity gradient in the longitudinal direction is low.

Our analyses may be begun by noting that the linear 

relationship of item 5 permits the rate of transport of 

salinity across unit areas normal to the coordinate direction a?, y, and 2 to be expressed in the form (see p. 

for explanation of symbols used) :

~ t>S v d£ , v 5S Kx ^ > Ky ^ > and Kz ^- 

ox * dy dz

If attention is now fixed upon an elemental volume 

dx dy da through which the fluid flows, the rate of 

change of its salt content will be determined by the rate 

salinity is propagated into the volume across three of 

the faces and the rate it is propagated out of the volume

across the three remaining faces. The expression of 

this relationship for the x direction is typical:

, \ , , r, dx) dydz=K*

Since our coordinate system is fixed to the ground, 

we note that at the point x the combination of a salinity

p\ o

gradient > and a mean velocity V will produce a 

rate of salinity change in the element of volume of

T7 5S , , , V ?r- dxdydz.

The condition that the rate of change of salinity in the 

elemental volume will be consistent with the rates at 

which salinity is entering and leaving it is

.r o- }dxdydi 

oz /

V ^ dxdydz. dx

When this expression is simplified by dividing out tlie 

common factor dx dy dz, it takes the form

 r=r J\_T T- I X\.y ^o~~r~ -t^-2 ^T 9" r ->T* v-*"/

di dx dy ds ox

Although in this equation F appears as a constant, the 

variations of velocity within the stream cross section 

are not ignored but are accounted for in the magnitudes 

of the diffusion coefficients Kx, Ky, and Kz.

SOLUTIONS

A solution will first be obtained for the one dimensional case defined by the differential equation:

Qo

dx (2)

Extension to the two and three dimensional cases will 

be made later by use of a product law.

Let us assume that a quantity of salt Qs is dispersed 

throughout a cross section of a stream having a breadth 

B and average depth D at x=O when £=O. Then a 

solution satisfying equation 2 is:

/x-Vt

S=

BD (3)

^^Kxt

If a quantity of sale Qs is dispersed through the 

volume BDL at the time £=O, a solution of equation

2 is:

Q (4)

695-721 O - 63 - 2
B6 TRANSPORT OF RADIONUCLIDES BY STREAMS

PRODUCT LAW

The term "Product law" as used here applies to a 

method of constructing solutions of partial differential 

equations of diffusion with two or three space variables 

from solutions of a differential equation of the same 

type with one space variable. These solutions are 

formed as products of solutions having a single space 

variable, and this relation is indicated in the name.

The writer has used this method of constructing solutions to the diffusion equation (Glover, 1934; and Tuthill and others, 1951) over a period of years and has 

found it to be an effective means for simplifying the 

treatment of problems which involve diffusion in more 

than one direction.

The process to be described here should not be confused with the older method which seeks a solution as 

a product of functions which have only one independent 

variable. The functions used here can have two 01 

more independent variables. The following developments will clarify the details of this procedure. 

Suppose we have a solution

Sl =F1 (x) Kx, t, V} 

which satisfies the differential equation

ax2

(5)

(2)

a solution

fi=Ft(y, Ky, 0 (6) 

which satisfies the differential equation

a* y a?/2

and a solution

A=F»(e, Kz, t) 

which satisfies the differential equation

then the product

$ = $1/2/3

is a solution of the differential equation

(7)

(8)

(9)

(10)

as _

a* * ax2 » a?/2

^ V 

>i~2 V > (*-)

m

Similarly, the product

S=SJ2 

is a solution of the differential equation

a* ax2 * a?/2

(ii)

(12)

A proof that the product law holds may be obtained 

by performing on equations 10 or 11 the differentiations 

required by equations 1 or 12 and making use of the 

relations 2, 7, and 9.

USE OF THE PRODUCT LAW

While the product law affirms that the product 10 

will be a solution of equation 1 and the product 11 will 

be a solution of equation 12, it does not assure that the 

desired initial condition or the boundary conditions will 

be met. If we take

(3)

and

(13)

(14)

the initial condition will be properly met for the products 10 and 11 if the quantity of salt Qs is introduced at 

the origin at a?=0. The boundary conditions will be 

met so long as the salinity does not reach a barrier which 

prevents further diffusion in some direction. A chart 

showing the progress of diffusion, when barriers are 

present, is included and will be described later. A similar set of solutions can be based on equation 4. If we set

~*~BDL (4)

then a solution of equation 1 is

S=Stf2 J3 (15) 

and a solution of equation 12 is

S=SJ2 . (16) 

VELOCITY VARIATION AS A DISPERSING MECHANISM

The formulas of the preceding paragraphs are based 

upon the assumption that dispersion is a diffusion phenomena, and it may be noted here that the experiments
DISPERSION OF DISSOLVED OR SUSPENDED MATERIALS IN FLOWING STREAMS B7

made have confirmed the validity of this assumption. 

In the early phases of this investigation, however, another type of formula was developed on the basis that 

variations of velocity x in the cross section of the stream 

were the dominating factor in longitudinal dispersion. 

Even though experiments have shown that this basis 

is inadequate, an account of this development may be 

warranted because the reasons for its failure throw so 

much light on the nature of the dispersion mechanism. 

In developing this formula, it was supposed that the 

velocity at the bed and banks of the stream was zero 

and that velocities in the remainder of the section 

ranged from this minimum value to a maximum near 

the middle of the cross section near or at the surface. 

If a quantity of salt were to be distributed uniformly 

throughout a cross section of the stream at some instant it would be dispersed longitudinally, at later 

times, by the variations of velocity through the section. 

All the salt would, however, be found between the point 

of introduction and the point reached by the thread of 

maximum velocity. On the basis that the stream has a 

uniform cross section, it can be inferred that both the 

average salinity and the maximum salinity in this reach 

would be inversely proportional to time and that a modicum of salt would always remain at the point of introduction. Because the investigations of Taylor 

(1954) and Elder (1959) show that these velocity variations are the dominating factor in longitudinal dispersion, as was assumed, it may well be asked why it is that 

the experiments fail to confirm an inverse first power 

law as would be implied by the above considerations. 

The answer seems to be found in the lateral dispersion 

due, principally, to turbulence. The modicum of salt 

can not remain at the point of introduction because it 

is picked up by the turbulence and thrown into the moving portions of the stream where it can be carried along. 

The salinity remains more closely bunched, and the decay of maximum salinity follows an inverse square-root 

law rather than an inverse first power law. This process is described in detail by Taylor and Elder. It must 

be conceded, then, that transverse dispersion due to turbulence is an indispensable factor in the mechanism of 

longitudinal dispersion. Some other important characteristics of this longitudinal dispersion mechanism 

will be discussed in later paragraphs.

CONTINUOUS INTRODUCTION OF SALINITY

The difficulties encountered in the laboratory tests, 

when an attempt was made to study lateral diffusion 

rates by using salinity samples introduced locally, made

1 The term "velocity" is used here in the sense of a temporal mean at a point. It is 

a velocity that would be measured with a current meter.

it expedient to introduce the salt solution continuously. 

By this means, it was possible to minimize the tendency 

of the salt solutions to cause density currents which 

produced an excessive rate of lateral spreading. A special solution of the diffusion differential equation is 

needed in order to reduce these test data. Such a solution also is needed because continuous introductions of 

salinity must be expected in problems involving actual 

rivers, and it will be essential to have formulas available 

for estimating their dispersion. The following development is based upon the idea of dispersion by diffusion. 

If a quantity of salinity Qs is introduced instantaneously at a?=0, y=0, and ^=0 and allowed to disperse 

both in the downstream direction and laterally, the distribution of salinity is given by an expression of the 

type:

(17)

This can be put into the form

xV

o ___

Q,e* (18)

where *-(£> < 19)

Suppose salinity is introduced continuously, at aj=0, at 

the rate q8 , then the effect of a quantity of salt qtd£ introduced at the time £ on the salinity at the point a, y 

at the time t will be given by an integral of the type:

2KX

cr O 

ft

I

<2^rD^[KMy Jo Jo (*~^

Let

u= (t-Q

then, by substitution

(20)

(21)

(22)

U

By use of the relations developed in the chapter on 

integral relation's this expression can be written

(23)
B8 TRANSPORT OF RADIONUCLIDES BY STREAMS

The method used to obtain this expression should insure that it is an exact solution of equation 12, and an 

independent check will confirm this. An attenipt to 

use it for computation purposes will show, however, 

that it is not well adapted to these uses.

A simplification can be made if the quantity \74/2 is 

large enough to justify the use of the first term of the 

asymptotic series :

/ n-V * <*>=(£)

as an approximation. This will be permissible if l/8z 

is small compared to unity. Then as an approximation

( approximate if «1 ). (25)

\ oS /

and this relation takes the

 fe:\ 1/2

In the present case z = 

form

K, 

Since

"V4/3 '* (^valid if L«l\ 

\ 8V4/8 /

(26)

then

< 28)

K and, if the quantity -^ ^ is also small compared to 

unity,

1+^2~1 +

lKx y2 (

2 Kv x*

. , .txy proximate if

(29)

If both of these conditions are met, the exact solution 

of equation 23 can be approximated by the relation

Vy*

I approximate if «1

\ 8V4/3

^l^1 (30)

It is fortunate that if the limitations given here are 

exceeded, the salinity will generally be very small, and 

this makes it possible to use the formula almost without 

restriction. Doubtful cases can be expected to fall in 

the zone close to the point of introduction of the salt, 

and in this region equation 23 could probably be used 

without much difficulty. This expression applies to a 

stream of unlimited width, but the method of images

may be used to adapt it to use with streams of finite 

width.

SPECIAL CHART

Solutions of the form of equation 13 are appropriate 

if the medium is of infinite extent in the direction of y. 

When the width of the stream is limited, as is always 

true, these solutions are valid until the parameter

reaches approximately 2. This value of the parameter 

indicates that the diffusing salinity has reached the far 

bank of the stream. The distribution of salinity across 

the stream at this time is indicated by the curve of 

figure 1 identified by the value

o

For later times, it will be necessary to substitute for the 

original expression 13 an equation 31 of another type 

which conforms to the boundary conditions imposed by 

the banks. A suitable form for this use is a Fourier 

series of cosine terms which can be adjusted to make 

the salinity gradients zero at the banks. The first three 

terms of such a series are given in equation 31 and 

graphed on figure 1. The terms in this expression have 

been adjusted to give, very nearly, the distribution 

shown by the curve for

B

on figure 1. Then for times longer than t2 , figure 1 is 

used instead of the original expressions.

A similar situation exists in vertical diffusion when 

the bed of the stream halts the downward diffusion of 

salt. The curves of figure 1 may be used for this problem also. To adapt them for this use replace y by 2, 

Ky by Ks, /2 by /3 , and B by D.

The treatment described involves a slight approximation because the time when the salinity reaches the far 

bank is poorly defined and a somewhat arbitrary choice 

of a changeover time must be made. Such a choice will 

be acceptable if the total amount of salt, as indicated by 

the equations 13 and 31, remains essentially unchanged. 

The amount of salinity which is lost track of by converting from equation 13 to 31 may be judged from the 

height of the ordinate at y/B = l.QQ for the curve

B

on figure 1.
DISPERSION OF DISSOLVED OR SUSPENDED MATERIALS IN FLOWING STREAMS B9

This chart was computed from the Fourier series:

A

CD=OA820e cos ^+0.0722,

+0.4432. (31)

LABORATORY AND FIELD INVESTIGATIONS

PURPOSES

The equations as described in previous paragraphs 

contain certain coefficients which must be determined 

experimentally, and this requirement alone would justify a program of experimentation. There are, however, other ways in which the results of a carefully 

planned series of experiments can be turned to advantage, and the experiments described here were arranged 

to gain these additional advantages.

If turbulence is a factor in the dispersion (and there 

are strong reasons for believing that it is) and the turbulence velocities are proportional to the mean stream 

velocity, then it should be expected that so long as the 

cross section of the stream remains unchanged, quantities of the type of KA , Kj,, and K», should be proportional to the mean velocity V. Variations of velocity in

an unchanging cross section can be arranged in the laboratory. In natural streams, a change of flow will generally change the stage so that these relations will be 

upset, but where dams maintain water levels these conditions may be approximated. If this expectation is 

confirmed, it would follow that for a quantity of salt Qs, 

introduced at the origin at time zero, the maximum concentration observed at a downstream point would be independent of the velocity, although the time required 

for the maximum to appear at the downstream point 

would grow greater as the velocity V is reduced. The 

maximum salinity observed at any downstream point 

would vary inversely as the three halves power of the 

distance x from the origin to the point, as long as the 

diffusion proceeds in the three directions: downstream, 

across stream, and vertically. If diffusion takes place 

only in the downstream and cross-stream directions, the 

maximum salinity will be inversely proportional to the 

distance from the origin. After the salinity has become 

uniformly distributed throughout the stream cross section, the maximum salinities should be inversely proportional to the square root of the distance from the 

origin.

2.0

1.0 0.000

0.1 0.2 0.3 0.4 0.5

(I)

0.6 0.7 0.8 0.9 1.0

FIOUKE 1. Special chart for estimating salinity changes as modified by boundaries.
BIO TRANSPORT OF RADIONUCLIDES BY STREAMS

With these considerations in mind, the purposes to 

be served by an experimental program may be outlined 

in the following way:

1. To evaluate the coefficients of equations 3 and 13 for 

several stream cross sections having known geometrical and hydraulic properties.

2. To discover the ways in which these coefficients vary 

with stream geometry and hydraulics.

3. To discover what factors are of primary importance

in producing dispersion in streams. 

Records. The original data collected from the Fort 

Collins, Allenspark and South Platte tests are filed at 

the Federal Records Center, Denver 25, Colo., where 

they are available for reference.

PLANNING OF TESTS

In the Fort Collins, Allenspark and South Platte 

tests, the test program was based upon computations 

prepared beforehand. Salt concentrations, dispersion 

patterns, salt quantities, time schedules, and electrode 

resistances were all computed. The design of circuitry 

was based upon these computed results. The effectiveness of this method is shown by the test record. Of 77 

tests run, only one run was lost. This was due to a poor 

choice of a stepping relay interval made by the writer. 

This test was the No. 1 run at Fort Collins. In 1 day at 

Fort Collins 15 tests were made; the average was 8 per 

day. The early tests were planned with the aid of the 

formula based on velocity variations which required no 

knowledge of a diffusion coefficient. Sampling schedules were prepared for the South Platte test in a similar 

way.

ELECTRODE RESISTANCE

If two parallel wires of diameter d and length L' 

are spaced a distance s' apart in a fluid of conductance 

K1', the current / due to an electromotive force E is 

approximately

j_

rK'L'E

(32)

For dilute solutions, the conductance has almost a linear 

relation to concentration, and equation 32 may be 

rewritten,

I=K"C' (33)

in which K" is a constant and C' is the concentration in 

pounds of salt per cubic foot of water.

DETAILS OF METHODS AND EQUIPMENT

A solution of common salt was used as a tracer material in the flume tests at Fort Collins and at Allenspark.

In these tests, the amount of tracer present was measured electrically through its influence on the conductivity of the flowing water. The electrodes used to 

measure the salt concentration were round brass bars. 

Electrode arrangements are shown in figures 7 and 9. 

An alternating electrical pressure of about 1 volt was 

applied to these electrodes. The frequency used was 

60 cycles per second. In order to avoid danger from 

electrical shock, the 110-volt supply was stepped down 

in two stages to about 1 volt by use of an autotransformer which energized bell-ringing transformers. The 

secondary insulation of the bell-ringing transformers 

isolated the electrode circuit from the supply. The 

electrode circuit included the secondary of the transformer, a resistor and the electrode, connected in series. 

A separate bell-ringing transformer was used on each 

electrode circuit.

An oscillograph was used to make the actual recordings. The oscillograph galvanometers were connected 

across the series resistor. The resistance of this series 

resistor was always kept less than 1 percent of the 

electrode resistance. Because the electrode resistance 

varied with the depth of water in the flume, the amount 

of salt originally present in the water and the amount 

of salt added, series resistances of 0.27, 0.47, and 1.00 

ohms were used at various times to maintain this relationship. The series resistor, a scale setting resistor, 

and the galvanometer were connected in series in the 

galvanometer circuit. The galvanometers were electrically damped and had an undamped natural frequency 

of about 120 cycles per second. Very slow paper speeds 

were used for' these tests, and each galvanometer therefore produced a record in the form of a band whose 

width indicated the amount of salt present at the 

electrode. Records of this type are shown on figures 

2, 3 and 4. Its initial width indicated the amount of 

salt naturally present in the flume water. This initial 

width was used for calibration purposes in a manner 

to be explained later.

The time scale on all the records was imposed by an 

electrically operated timer that closed a contact at 1- 

second intervals. For the high-velocity runs the timer 

was connected to a spare galvanometer which made a 

trace near the edge of the record, but for the tests with 

low velocities the tinier was connected to a stepping 

relay with 40 contacts. When so connected, it advanced 

the relay one contact every second. By energizing the 

oscillograph from selected contacts, an intermittent operation of the oscillograph could be arranged. Energized in this way, it would operate for 1 second at prearranged intervals. This device was used to hold the 

record to a convenient length when the test required a,
DISPERSION OF DISSOLVED OR SUSPENDED MATERIALS IN FLOWING STREAMS Bll

m fc

TRACES 

Starting time

Calibration

20ft

III 40ft

60ft

Time

lllllllllll

No 18

Longitudinal diffusion

0.882 Ib salt introduced from trough.

Flume 7.94 ft wide Flow 2 CFS.

Intermittent record with 5-sec. time intervals.

/?,=0.025

25 50

I 

75 100 125 150

FIGURE 2. Longitudinal diffusion test record intermittent operation of recorder. Oscillograph recorder connected to each fifth contact of a stepping relay actuated by a

1-second timer to shorten the test record. Fort Collins test.

long time for completion. A record obtained in this 

way is shown on figure 2.

Some of the tests required the current through each 

pair of electrodes to be recorded individually while other tests required a response as a group representing a 

section. To facilitate the rearrangement of electrical 

connections, a control panel was arranged so that the 

desired changes could be made by using switches. This 

control panel was used for both the Fort Collins and 

Allenspark tests.

A spare galvanometer was used to indicate the start 

of a run, and another galvanometer was used for calibration purposes. To make the calibration, a special 

electrode was transferred from a standard to the flume 

water during the test. At Fort Collins the city water 

was used as a standard, but at Allenspark a solution 

containing 100 ppm of salt in distilled water was used 

as the standard.

These markings are shown clearly on the record of 

test 23. This record is reproduced as figure 3. The 

trace indicating the start of the test is at the top, and 

the calibration trace is the next one below it. Because

the flume water was pumped from a sump filled from 

the city mains, the calibration trace does not change 

width in many of the records. After a number of runs 

were made, the salt used in the test began to show on 

the records. The initial width of the trace indicates 

the amount of salt originally present in the flume water. 

The 1-second time marks show at the bottom of the 

record. Run 18, reproduced as figure 2, is an intermittent record made by using eight contacts on the stepping relay. Each bar of the record therefore represents a 5-second time interval. The starting time and 

calibration records appear at the top of the record, as 

described above.

A transverse diffusion run is shown on record 25. 

This is reproduced as figure 4. The starting time and 

calibration marks show as before. The electrode numbers are shown on this record also. The position of 

these electrodes is shown on figure 6. The electrodes, 

galvanometer, and trace correlation is given in table 3.

Salt was introduced from a trough as shown in figure 5. The salt was put in solution, and the solution 

was placed in the trough. At the start of the test, the
B12 TRANSPORT OF RADIONUCLIDES BY STREAMS

TRACES 

Starting time

60ft

Time 0

No. 23 

Longitudinal diffusion 

0.882 Ib salt introduced from trough 

Flume 7.94 ft wide Flow 8 CFS. 

Continuous record with l-sec time marks 

/?!= 0.025

5 10 15 20 25 30 35 40 45 50

FIGURE 3. Longitudinal diffusion test record continuous operation of recorder. Starting time trace at top. Calibration trace next below starting time trace. One-second 

time marks at bottom. Width of the three electrode traces indicates amount of salt present at the electrode. Fort Collins test.

TRACES 

Starting time 

Calibration

1 

5 

9

C/5 o

tY £.

Ld 

CQ .-

1 10

a 3

o

0

y 11

Ld 

4

8

12 

Time

^V^V^v^^^^.y^^-.^^,^^ __ n-^* Ajfc --~.n **-. -«*-^ ,-, |.

^ " "" - ^ ____ ^ ( ___ s «-^r _ »^^ -r T- M- -4UU. _ - "" ^ <r "":"jiwi"-" 1"- iTmi^^iPpwcr-v^^^^^^ia^^^

 -,->. . - - - - -^a§ -»IM-I _ ~ ~i*iT - - J ^

Traverse diffusion

ft^., - .. ^ flump Zjiidt/LJlfiiitn HJ.J. CF^ ... a -- - * - -

Continuous record"wrm !-sec Iime marks

" ' ~J '' W~- " "^~ r~~~ "" ' "VJ ~ T ... T ^ sr ~T

"""~"!P -»v -» » Tff -^ffl* ^-

0 5 10 15 20 25 30 35 40 45 50 55 60

FIGURE 4. Transverse diffusion test with continuous feed at side. Starting time trace at top. Timer trace at bottom. Trace identification at side. Fort Collins test.

handle was rotated to drop the salt solution into the 

water. The trough was coated with parafin wax to 

insure that all of the solution would be released. It 

was arranged to just clear the water surface at the 

maximum flow depth so that the salt charge could be 

released with the minimum amount of fall.

A darkroom was available at Fort Collins for development of the records. At Alleiispark a light-tight box 

with a shelf was used for development. This was fitted 

with a door and arm holes protected by light-tight 

sleeves. In use, the record holder from the Oscillograph 

was placed on the upper shelf and the door closed. The
DISPERSION OF DISSOLVED OR SUSPENDED MATERIALS IN FLOWING STREAMS B13

developing trays were kept in the bottom of the box. 

The operator worked through the arm holes to unload 

the record holder and develop the record. The box was 

of plywood about 1.5 by 2 by 3 feet. Because commerical power was not available at Allenspark, the 60- 

cycle per second power supply was obtained from a 

storage-battery-operated vibrator of the type used in 

car radios.

The South Platte test was made by a different procedure. In this test potassium carbonate was used as a 

tracer material, and samples were taken in plastic 

bottles and evaluated by use of a flame photometer. 

This technique proved to be both expeditious and 

effective.

Flow measurements were made by a calibrated orifice 

at Fort Collins, by rectangular and triangular weirs at 

Allenspark, and by current meter on the South Platte 

Test.

FORT COLLINS TESTS

In order to provide experimental data which could 

be used for determination of constants and the effects 

of stream geometry and hydraulics, a series of experiments were performed in a flume in the Hydraulic 

Laboratory of the Colorado State University at Fort 

Collins, Colo. This flume is constructed of plywood 

sheets and is nominally 8 feet wide and 2 feet deep. A 

view of the interior of the flume is shown in figure 5.

FIGURE 5. Test flume 8-feet wide with trough and point gages in place, Fort Collins

tests.

Its overall length was 72 feet. The usable length was 

somewhat less than this because of a stilling-well arrangement at the inlet end which served as a distributor. 

Of the total length, 60 feet was used in the experiments 

by introducing salt solution at a point near the inlet end 

of the flume and measuring the salt concentrations electrically at intervals of 20 feet. A plan view showing the 

stilling well and electrode stations is shown on figure 6. 

The flume was supplied with water by a pump which 

has a capacity in excess of 8 cubic feet per second.

7.94 ft.- h 4.0 ft. H

Well

<t_TroughK

i 

i

(

Upstream 

point ' 

gage )

m1 .2

Electrodes

<

Downstream ) 

point 

gage

9 10

2 

3

4

5 

3 .4

6

7

8 

^Manometers

9

10

,11

13

11 12

14

T

i

Well

"t Trough

2 3

\ 

Electrodes

10 11 12

- -

Well

1 3

2 4

5 7

6 8

9 11

10 12

FIGURE 6. Electrode numbers and positions, Fort Collins tests.

Water-quantity measurements are made by means of a 

calibrated orifice in the pump discharge line. The 

Kutter's "wi" value for the bare flume is about 0.010, but 

this can be raised to about 0.025 by laying grids of reinforcing bars on the floor. A view of the flume with 

the bars installed is shown on figure 7.

Measurements of salt concentration were made electrically by imposing an alternating voltage of about 1 

volt across electrodes mounted vertically on the floor 

of the flume and recording the current flow by means 

of an oscillograph. In dilute concentrations, the current flow is nearly proportional to the amount of salt 

present. Three sets of four electrodes each were used. 

These consisted of 3/16-inch-diameter brass rods spaced 

1 inch apart and mounted in a plastic base arranged 

for attachment to the floor of the flume as shown in 

figure 7. Switches on a control panel permitted a recording of the current flow at each electrode separately,

695-721 O - 63 - 3
B14 TRANSPORT OF RADIONUCLIDES BY STREAMS

FIGURE 7. Test flume 8-feet wide showing electrode assembly and bar roughness, 

Fort Collins tests. The electrode bars are attached to plastic plates mounted on 

the floor and are supported by a crossbeam at the top. The bar grids on the floor 

of the flume raise the roughness n\ =0.025.

or a simultaneous record of the four arranged in a 

series-parallel arrangement which had the same overall resistance as- a single electrode. For the longitudinal diffusion test, the salt solution was introduced 

from a trough which spanned the flume 20 feet upstream from the first electrode, as shown in figure 5, 

but for the lateral diffusion tests the salt solution was 

introduced continuously, at this same location, by gravity flow through a rubber tube connected to a tank. 

Water levels were read from a set of piezometer tubes 

connected by plastic tubing to short sections of metal 

tube which were flush-mounted in the floor of the flume 

near the middle. Checks of water level were made at 

two points by means of point gages which were arranged to take readings on the floor of the flume and 

at the water surface. At the downstream end of the 

flume, water levels were controlled by an adjustable 

slatwork gate.

In order to obtain data for a range of cross sections 

and hydraulic conditions, experiments were run on sections 8 feet wride and 0.5 foot deep, 4 feet wide and 1 

foot deep, and 1.667 feet wide and 1.667 feet deep.

The following sequence of flow rates was planned: 1

Flume width 

(feet)

Flow rates 

(cubic feet per second)

8, 

8, 

5.6,

4, 

4,

1.667__ _______ _ . 2.8,

: This is the planned sequence. For test value, see table 1.

2,

2,

1.4,

1, 0. 5

These planned water depths and flow rates were only 

approximately met in the tests and the actual conditions obtained were recorded in each test. The test 

conditions are summarized in table 1. The bar roughness was used to vary the hydraulic conditions in several tests. In all, 65 runs were made.

To insure an accurate recording of salinity by electrical means, the resistance of the current measuring 

instrument must be small compared to the resistance of 

the electrode-electrolyte. Computations and tests 

showed that the minimum resistance of 3/16-inch brass 

rods with a 6-inch immersion would approximate 300 

ohms, and therefore 1 ohm in the current measuring circuit would meet our requirements. Arrangements were 

made so that for greater depths of immersion, a correspondingly smaller value of resistance could be inserted in the circuit.

The recorder used was a 24-element recording oscillograph. The galvanometers, when properly damped, 

have a flat characteristic response to 120 cycles per second with a sensitivity in excess of 2,000 millimeters per 

milliampere at a distance of 1 meter. The galvanometer was connected, with suitable attenuator circuits, 

across the 1 ohm series resistor.

The paper speed in the oscillograph could be varied 

from about 1/4 to 30 inches per second. The one-quarter inch per second minimum speed was used throughout the tests. A description of the tests run is given in 

the table on page B15.

When the velocity of the water was high, it was 

possible to make a continuous recording of the conductivity without making an excessively long record. 

In these runs, second-time-marks were put on the record 

by using one galvanometer connected to a battery 

through a switch operated by a motor. The motor is 

a d-c type, but it is controlled by an escapement mechanism for constant speed.

For runs in which the velocity of the water was low, 

intermittent recordings of 1 second duration were made 

periodically with the periods varying from 5 seconds 

to 20 seconds. Intermittent operation was obtained 

by use of the second-timer with a stepping relay. The 

relay lias 40 contacts. By advancing the relay one step 

per second, cycles of 5,10, and 20 seconds each are available by connecting to the proper terminals. The oscillograph, timer motor, and stepping relay were all operated 

from two 6-volt storage batteries.

The electrodes were aligned in three sets of four pairs 

each as shown in figure 6. Switching made it possible 

to operate the individual pairs of electrodes into their 

corresponding galvanometer in the oscillograph or to 

connect each set of four pairs in a series-parallel combination for recording on one galvanometer. The individual arrangement was used in the transverse diffusion 

study, and the series-parallel connections were used for 

the longitudinal studies.
DISPERSION OF DISSOLVED OR SUSPENDED MATERIALS IN FLOWING STREAMS B15

TABLE 1. Flow data, Fort Collins tests 

[L, longitudinal dispersion. TS, transverse dispersion from a spot source. TC, transverse dispersion form a continuous source.]

Test Number

2.................... ......

3. ........................... .

4, 5_......... ........................ .

6-12...........

13-15..................................

16................ . ......

17.................... ........

18-20 .........................

21,22............

23-26................. .........

27-30........ .......................

31-34..................................

35-38........ ...... .......

39-41............... . .....

42,43... ...............................

44.............. . . ....

45............................ .

46-49.... ..................

50-53. ...................

54-56.............................

57-60 ....

61, 62..... ....... ...............

63,64. ....

65............ ..

Flume 

width (feet)

7.94 

7.94 

7.94 

7.94

7.94 

7.94 

7.94 

7.94 

7.94 

7.94 

7.94 

7.94 

7.94

7.94 

7.94 

7.94 

7.94 

7.94

4.00

4.00 

4.00

1.67 

1.67 

1.67

Flow (cubic 

feet per 

second)

8.00 

6.00 

4.00 

2.00

1.74 

2.08 

1.40 

2.00 

4.00 

8.00 

8.00 

4.00 

1.00

.50 

1.00 

1.00 

.50 

.50

7.75

4.00 

2.05

1.50 

2.80 

5.80

Roughness 

coefficient 

n\

0.010 

.010 

.010 

.010

.010 

.010 

.025 

.025 

.025 

.025 

.010 

.010 

.010

.010 

.025 

.025 

.025 

.010

.010

.010 

.010

.010 

.010 

.010

Water depths at indicated manometer 

(feet)

2

0.312 

.448 

.414 

.434

.429 

.458

.409 

.448 

.514 

.339 

.384 

.411

.368 

.405 

.389 

.359 

.392

.859

.862 

.887

8

0.447 

.544 

.506 

.525

.521 

.553

.493 

.514 

.492 

.461 

.482 

.506

.461 

.499 

.481 

.452 

.486

.951

.954 

.979

14

0.549 

.638 

.601 

.620

.616 

.650

.581 

.596 

.479 

.566 

.579 

.601

.555 

.592 

.576 

.546 

.583

1.044

1.046 

1.071

Indicated water depth 

at point gage 

(feet)

Upstream

0.336 

.492 

.457 

.480

.446 

.473 

.479 

.410 

.429 

.457

.413

.455 

.439 

.402 

.439

.910

.905 

.928

1.628 

1.607 

1.639

Downstream

0.508 

.599 

.566 

.590

.551 

.569 

.474 

.542 

.545 

.568

.521 

.559 

.551 

.507 

.543

1.014

1.014 

1.040

1.733 

1.710 

1.708

Test No. 

and (or) type

L. 

L. 

L.

6TS 7L 8-11TS 

12TC. 

TC. 

TC. 

L. 

18L 19-20TC. 

21L 22TC. 

23L24-26TC. 

TC. 

TC. 

35-36L 

37-38TC. 

39L 40-41TC. 

42L 43TC. 

TC. 

L. 

46-47L 

48-49TC. 

50L 51-52TC

53L.

54L 55-56TC 

57-58L 

59-60TC. 

61L 62TC. 

63L 64TC 

L.

DESCRIPTION OF TESTS

Since the electrodes were energized from a 60-cycleper-second alternating voltage source, the resulting current alternations caused the oscillograph galvanometer 

beam to swing over a width determined by the peak 

values of the current. The galvanometers had a sufficiently high undamped natural frequency to be able to 

follow faithfully these 60-cycle-per-second variations. 

One of the galvanometers was connected to a special 

electrode which was used for standardization. Fort 

Collins city water was used as a standard, and a reading 

was taken in a beaker of this water at the start of each 

test. During the test, this electrode was transferred to 

another beaker filled with water from the flume just 

before the start of the test. Since each electrode recorded the salinity of the flume water, these readings in 

the Fort Collins water and the flume water provide a 

means for setting the scale for each of the electrode 

records. The time scale was set by the 1-second timer 

in all cases as the paper speed was too slow to permit 

use. of the timer built into the oscillograph. The correlation among electrode numbers, galvanometer numbers, 

and trace numbers is shown in table 2.

Between tests the water ran continuously to keep the 

salt content of the water in the sump uniform. The 

amount of salt used in each longitudinal diffusion test 

was 400 grams or 0.882 pounds. The dry salt was 

weighed out on a balance and was then put into solution and poured into the trough. At the beginning of

TABLE 2. Correlations among electrode, galvanometer, 

and trace numbers, Fort Collins tests

Trace'

1 Time---.. . .__-_-. - .__

3 

4-._ __ . -_. _ _ -_.-_- _

5 ------------------------

6 ------------------------

7

8.. 

q

10 

11 

19

13 

14 

Electrode

1

5

9

2

6

10

3

7

11

4

8

12

Galvanometer

5

6

7

8

9

10

11

12

13

14

15

16

17

18

i Traces are read from the top of the page down. Galvanometer 5 

gives time of injection. Galvanometer 4, trace 15, is used for the 1-second 

timer record. Galvanometer 6 is used for Fort Collins water and flume 

water recordings. These trace assignments were used to avoid a confusing overlap of the records.

these tests, the salt solution was dumped uniformly 

across the width of the flume onto the surface of the 

running water. The trough consisted of a 120° sector 

of a cylinder concentric with the axis upon which it was 

mounted. A paraffin coating on the trough prevented 

adherence of any appreciable amount of salt solution to 

the trough surface. The mounting of the trough was 

such that the salt solution was released very close to the 

surface of the running water in the flume. In all tests, 

it is believed that the bottom of the trough was within 

an inch of the water surface. In the lateral diffusion
B16 TRANSPORT OF RADIONUCLIDES BY STREAMS

tests, the continuous salt solution injection was sometimes made near the bottom in the tests with an 8-footwide flume and a 0.5-foot water depth, but in the remainder of the tests the salt solution was introduced at 

the surface.

A few lateral diffusion tests were made by placing a 

6-inch-diameter tube against a rubber mat on the bottom 

of the flume and filling it with salt solution to approximately the level of the water outside and pulling the 

tube away quickly at the start of the test. This procedure gave trouble, however, because of the density 

currents set up by the greater density of the salt solution. Use of permanganate color disclosed that these 

density currents persisted even though a siphon was 

used to bring the two levels to equilibrium before the 

tube was withdrawn. This type of test was therefore 

abandoned. These tests were Nos. 6, 8, 9, 10, and 11. 

In test 11, the salt content was reduced to 0.441 pound 

to reduce the difference in density between the salt solution and the water, but the dye disclosed that the density 

differential was still sufficient to cause flow.

In another test, an ammonia-salt solution was used 

to investigate the effects of density on the results of the 

tests. This test run is No. 47. Test 46 is identical except that it was run with salt. The ammonia-salt solution was made by mixing concentrated ammonia solution, distilled water, and salt in approximately the following proportions: 500 cubic centimeters ammonia solution, 300 cubic centimeters distilled water, and 90 

grams of dry salt. This solution can be adjusted to have 

the same density as water. This solution is not recommended for continuous use because of its acrid odor and 

the nonlinearity of its conductance-concentration characteristics at high dilutions. However, in this test it 

served to disclose the presence of density currents in the 

tests conducted with salt solutions introduced into low 

velocity flows. Approximately 1 gallon of ammoniasalt solution used in this one test so contaminated the 

sump that it was necessary to drain the sump and refill 

it with fresh water. The sump was drained and refilled 

with fresh water subsequent to tests 41 and 49.

The weight of salt stated in connection with the tests

represents the weight of dry crystalline salt used. The 

salt was always put into solution before being used in 

the tests.

CALIBRATIONS

After the tests were completed, a calibration was made 

by placing a test electrode in beakers containing 

samples. The series was read four times and the test 

electrode was rinsed in distilled water after each reading. The following table shows the result of the 

calibration.

Average

trace

Sample width

(inches')

Fort Collins water, new sample___________________ 1. 024

Distilled water______^-______------------------- 057

Distilled water+100 ppm of c.p. NaCl__________ 3. 182

Distilled water+100 ppm of Salt 2______________ 3. 102

Distilled water + 100 ppm of Salt l__________.-_-_- 3. 115

Fort Collins old sample+100 ppm of Salt !________ 4. 085

Salt 1 was used in tests up to and including test 42. 

Salt 2 was used in the remainder of the tests with the 

exception of test 47, which was run with the ammoniasalt solution.

COMPARISON OF DIFFUSION COEFFICIENTS, FORT 

COLLINS TESTS

The longitudinal diffusion coefficients obtained from 

selected Fort Collins tests are shown in table 3. Some 

transverse diffusion coefficients were also obtained from 

these tests and will be described later. Of the tests 

listed in the table, the test conditions approached the 

ideal more closely in test 23 than in any of the others. 

Because of the high flow rate and the roughness, the 

flow was strongly stable and had a surface nearly parallel to the bed of the flume. The gradients were also 

high enough to permit a good slope determination. The 

diffusion coefficients, identified as from test, are obtained 

from the observed salt concentrations by the use of 

equation 3. The computed values are obtained by using 

ing a formula adapted from the work of G. I. Taylor 

(1954) as described later under the heading "Determination of diffusion coefficients by analytical procedures." 

The comparative values are shown in the following 

table:

TABLE 3. Comparison of experimentally and analytically determined longitudinal diffusion coefficients, Fort Collins tests

Test

2........

3.. ...........

18.... ............. .

23.........

58.............

65... .

Flow 

(cubic feet 

per second)

8.0 

6.0 

2.0 

8.0 

2.05 

5.80

Flume 

width 

(feet)

7.94 

7.94 

7.94 

7.94 

4.0 

1.67

Flume 

depth 

(feet)

0.447 

.544 

.493 

.492 

.984 

1.674

Mean 

velocity 

(feet per 

second)

2.254 

1.389 

.511 

2.048 

.533 

2.072

ni

0.010 

.010 

.025 

.025 

.010 

.010

n (feet)

0.402 

.478 

.438 

.438 

.660 

.557

c,

128 

131 

52 

52 

138 

134

V*

(feet per 

second)

0.100 

.060 

.056 

.225 

.0217 

.0884

Diffusion coefficients 

(square feet per second)

From test

0.806 

.703 

.461 

1.285 

.421 

.877

Computed 1

0.810 

.580 

.493 

1.985 

.286 

.985

1 Computed from the formula -K*=20.2 n »».
B17

ALLENSPARK TESTS

The experimental flume at Allenspark, Colo., is 400 

feet long. Because a long conduit permits a more nearly 

complete development of the pattern of salinity distribution with length, this length is favorable for experimentally observing the variation of maximum salinity 

with distance. A view of this flume is shown in figure 

8. The flume, being triangular in cross section, also

FIGURE 8. View of Allenspark flume showing the triangular section. Point gages 

for measuring water-surface elevations were mounted on spacers crossing the 

flume. Readings were made with a level.

affords an opportunity to make tests to learn whether 

the shape of the cross section exerts any influence upon 

the rate of dispersion. A series of 11 tests were made 

on this flume. They confirm the inverse square root 

law and indicate that the shape of the cross section does 

not greatly alter the rate of dispersion. The Alienspark flume is supplied from North St. Vrain Creek. 

The water has only a small content of dissolved solids 

which favors the use of salt as a tracer material. The 

amount of tracer present was measured through its

effect oh the conductivity of the water, as at Fort 

Collins.

The flume used for these tests is constructed of plywood sheets which are 4-feet wide and 8-feet long. 

These are assembled to form a V-shaped trough with 

an angle of 90° in the bottom. The cross-section of a 

stream flowing in this flume is, therefore, triangular in 

shape. The slope of this flume can be varied by use of 

jacks which slide it up or down on ways on one side. 

At the time of these tests, the flume was set to a slope 

of 0.00048. The interior of the flume is coated with an 

aluminum base paint, and great care was used by the 

builders to insure a smooth and uniform interior and a 

good alinement. Elevations of the bottom of the flume 

were read by means of a graduated bar. A similar bar 

was lowered until a needle point at the end made contact with the water surface, and the graduations were 

then read with a level. For the dispersion tests, a measured quantity of chemically pure sodium chloride, hereafter called salt, was dissolved in water and placed in a 

trough. This trough was the shortest of the three used 

at Fort Collins. At the start of each test, this salt solution was dropped into the water in the flume by rotating 

the trough around the axis by means of a handle. Passage of the salt cloud was recorded electrically by connecting electrodes in series with a source of alternating 

voltage and an oscillograph galvanometer element, as at 

Fort Collins. The electrodes were of the form shown 

in figure 9. They were made of %6 -mch-diameter rods,

FIGURE 9. Allenspark electrode arrangement. Holes in electrode support permit 

adjustment to depth of flow. Plastic sleeves prevent spurious readings due to 

wave motion.

spaced at 1.5-inch centers and clamped to a plywood 

frame. This assembly was supported from a brace 

which spanned the flume at the top. Holes in the plywood frame permitted an adjustment of the electrode 

position so that at the center of the flume the electrode
B18 TRANSPORT OF RADIONUCLIDES BY STREAMS

was at 0.6 depth of the stream cross section. Small plastic shields were placed around the electrodes near the 

waterline to prevent spurious variations of conductivity, due to wave motion, as shown in figure 9. The five 

electrodes used in these tests were spaced To feet apart. 

The salt trough was 75 feet upstream from the first electrode and 11 feet downstream from the upper end of the 

flume. The downstream electrode was 14 feet upstream 

from the lower end of the flume. Since the conductivity 

of a solution of salt is closely proportional to the concentration, in weak solutions, the amplitude of the galvanometer swing is a measure of the salt present. A 

shunt was used to adjust the galvanometer sensitivity. 

Use of an alternating voltage, obtained from a batterypowered vibrator, causes the galvanometer spot to sweep 

over a band width which is proportional to the amount 

of salt present. Galvanometer sensitivities were adjusted to give a band width of about 0.5 inch with the 

salinity naturally present in the creek water, and the 

amount of salt introduced was adjusted to increase this 

double amplitude to about 6 inches at the first electrode. 

Calibration was obtained by means of a supplementary 

test electrode which was placed alternately in a beaker 

of creek water and a beaker containing a 100 ppm salt 

solution. A separate galvanometer was used to indicate 

the time when the salt solution was dropped into the 

flume. This galvanometer was actuated by a switch 

which was thrown at the instant the salt solution struck 

the water. Another galvanometer was used with a 

1-second interval timer to put time marks on the record. 

On tests 9. 10. and 11 the start indicating circuit was 

used to provide an independent time check where periods of 2.5 and 2.0 minutes were indicated on the record as timed by a stop watch. Communication facilities 

were provided between the oscillograph and salt injection stations by voice-powered phones.

Water flows were measured with rectangular and triangular weirs. The recording equipment was housed 

in a station wagon. Wires from the electrodes were 

brought in overhead and distributed to the same control 

panel that was used in the Fort Collins tests. Power 

supply for the galvanometers came from the 6-volt d-c 

to 110-volt a-c converter equipment, and storage batteries supplied the power for operating the oscillograph 

and the converter. To insure that a proper record was 

obtained before the flow in the flume was changed, a 

portable darkroom was used to develop the records as' 

soon as they were taken.

LONGITUDINAL DIFFUSION COEFFICIENTS

OBTAINED FROM ALLENSFABK TESTS 

The diffusion coefficients obtained from the Alienspark tests are shown in table 4. The test value was

obtained by analyzing the recorded salinity patterns 

with the aid of equation 3. The analytically determined values are computed from the flume properties 

by use of the formula adapted from the work of G. I. 

Taylor (1954), as described later under the heading 

"Determination of diffusion coefficients by analytical 

procedures." The mean velocities shown were obtained 

by applying salt-velocity procedures. The flow rates 

were obtained by applying the observed velocities to 

the area of the cross section. The table on page B19 

shows the coefficients as determined from the Alienspark test data.

When the peak salinities observed at the five electrode 

stations are plotted against distance on logarithmic 

cross-section paper, it is found that the relation of 

salinity versus length is of the form

The values of n obtained from these plots are shown on 

the following tabulation :

Values of n as obtained from logarithmic plots of the test 

data

Test

9 _____ __

11

!___ __ ____ ____

2 ._ ____ _ __

4

5

g

7_________ 

Averaee _

AfomtVmJ 

water 

depth r 

(/erf) 

.___ ---__ 1.0 0

1.0

.-___---__ 1.0

_. _ 1-0

.____----_ 1.5

__________ 1.5

________ 1.5

2.0

2.0

2.0

2.0

i 

331

354

318

316

552

523

544

433

485

338

463

423

The developments described indicate that the value 

of n should be one-half. The tabular values vary from 

0.316 to 0.552 and average 0.423. These are considered 

to be in substantial agreement with the ideal value. 

Tests on very long conduits yield a value close to the 

ideal (Taylor, 1924).

THE SOUTH PLATTE DISPERSION TEST

A dispersion test was made on the South Platte River 

in October 1958. The reach chosen for this test extends 

from the Kersey bridge to the Kuner bridge. In this 

reach the river runs southeasterly, the Kuner bridge 

being 4 miles east and about 2 miles south of the Kersey 

bridge. The length of river between the two bridges is 

about 5 miles. The South Platte runs over a sandy bed 

in this area and generally follows more than one channel. At the Kersey bridge, for example, there were five 

channels. This stream is, therefore, of the braided type.

On October 22, the flow at the Kersey bridge was 600 

cubic feet per'second. Based upon this flow, detailed
DISPERSION OF DISSOLVED OR SUSPENDED MATERIALS IN FLOWING STREAMS 

TABLE 4. Longitudinal diffusion coefficients obtained from the Allenspark tests

B19

Test

8,9, 10, !!_____ _______________________

1, 2, 3_ __________________________

4, 5,6, 7_ _____________________________

Flow 

(cubic feet 

per second)

1. 50 

4.99 

10.22

Maximum 

depth 

(feet)

1.005 

1.473 

1.990

Mean 

velocity 

(feet per 

second)

1. 49 

2.30

2. 58

Hydraulic 

radius 

(feet)

0.355 

. 522 

.704

Slope

0. 000480 

. 000480 

. 000480

ni

0. 0092 

.0092 

.0092

»* 

(feet per 

second)

0. 0622 

.0901 

.0961

Kx (square feet per 

second)

From 

testi

0. 55

.88 

1. 32

Computed 

20.2 n»*

0.45 

.95 

1.37

1 Based on trace amplitudes corrected to conform to the known amounts of salt used in the tests.

plans were worked out for the test. A diffusion coefficient was selected, based upon the best data available, 

and the dispersion pattern to be expected at the points 

where the north-south section lines crossed the stream 

was computed. The Kersey bridge was designated point 

0. Points 1, 2 and 3 are between the Kersey and Kuner 

bridges. The Kuner bridge was at point 4. A detailed 

program of sampling and stream gaging was then prepared and entered in the five field notebooks to be used 

by the observers at the five observation stations. Because the observers would be out of touch with each 

other during the test period, the program was based 

upon a time schedule. In addition to the samples and 

stream gagings, both air and water temperatures were 

read.

The tracer material chosen for this test was potassium 

carbonate. This material was chosen because it is nonpoisonous and highly soluble in water and because 

potassium yields a brilliant flame color. This latter 

consideration was important since a flame photometer 

was to be used to analyze the samples. A preliminary 

test of the sample collected on October 22 showed 9 

ppm of potassium ion as a natural constituent of the 

stream. To produce sufficient contrast to this background it was estimated that 1,000 pounds of potassium 

carbonate would be required. In preparation for the 

test, six 50-gallon drums were fitted with a valve and 

a nipple.

Six men participated in the test. The drums were 

placed on the downstream walkway of the bridge with 

the valve and nipple directed downstream, and about 

40 gallons of solution was made up in each drum.

At 11:00 o'clock the men simultaneously opened a 

valve on each of the six drums to release the tracer 

solution into the stream. When the drums were nearly 

empty, the valve was closed and the remaining tracer 

material was dumped over the bridge rail. This operation took 80 seconds. The observers then proceeded to 

the downstream observation points. Samples were collected in 4-ounce size plastic bottles. The readings and 

samplings were complete at 3:35 p.m. The data 

obtained are shown in tables 5 and 6.

TABLE 5. Analysis of samples for potassium ion, 

South Platte test

Location

Point 0 __ ___ . ..

Point 1- __

Point 2.- __ _

Points. .

Point 4 _ _ .....

Laboratory No.

29191.... __ ... 

2927A.. ..... .

2927B... _______

29270 

2927D. _

2927ETime of

collection

12: 50 p.m.

10: 00 a.m.

10:30

11:30

12:00 m.

12:30 p.m.

1:00

1:30

2:00

2:30

11:00 a.m.

11:10

11:20

11:30

11:40

11:45

11:47

11:49

11:51

11:54

12:00 m.

12: 10p.m.

12:20

12:30

11:30 a.m.

11:40

11:50

12:00 m.

12: 10 p.m.

12:15(341)

12:15 (621)

12:15(900

12:15 (1181)

12:20

12:30

12:40

12:50

1:00

1:10

1:20

1:30

12:00 m.

12:15 p.m. 

12:30

12:45

1:00

1:10

1:12^

1:14

1:18

l:22^i

1:30

1:45

2:00

2:15

2:30

12: 30p.m.

12:45

1:00

1:15

1:30

1:45

1:52

1:55

1:59

2:04

2:12

2:16

2:30

2:45

3:00

3:15

3:30

(ppm) K+

9.2

8.4

8.0

8.0

8.4

8.0

8.0

ao

8.0

8.0

7.8

7.8

7.8

7.8

9.4

27.0

22.8

1&6

16.8

17.8

12.2

8.8

8.2

8.0

8.0

8.0

7.8

7.8

7.8

7.8

12.0

14.4

14.6

16.4

16.6

13.4

11.2

9.2

8.4

8.0

9.0

8.2

8.2 

8.2

8.4

9.6

13.6

14.8

14.8

14.6

13.2

12.8

10.0

9.2

8.2

8.6

8.0

8.0

8.0

8.0

8.0

8.0

8.0

8.0

9.6

12.4

12.6

11.8

10.2

8.6

8.0

7.8

7.8

i Large sample collected in a glass bottle. This sample contained no 

tracer.
B20 TRANSPORT OF RADIONUCLIDES BY STREAMS

A large sample collected at the Kersey bridge during 

the test was found, on analysis, to contain 1,420 ppm 

of dissolved solids. The day of the test was overcast 

with a slight wind blowing. Water temperatures remained at about 50° F while air temperatures remained 

at about 49 ° F. A traverse of the river at point 2 showed 

a maximum water temperature differential of only 

0.27° F.

The potassium carbonate used was of technical grade 

having a purity of 98 to 100 percent. It is of the dry 

anhydrous form and visually is a white, free-running 

powder. The 1,000 pounds of tracer material are estimated to have contained 566 pounds of potassium ion.

Levels run subsequent to the test gave the following 

figures:

Watersurface 

Station elevation

Kersey Bridge________________________________ 4579. 17

Kuner Bridge._______________________________ 4550. 76

Fall in reach. 28. 41

A diffusion coefficient of 174 ftYper sec was obtained 

for the conditions of this test. Some loss of tracer was 

accounted for in making the determination. Results 

of stream gagings made during the test are shown in 

the table 6.

MOHAWK TEST

A test of stream dispersal characteristics was made in 

August 1954 by introducing 4.53 curies of radioactive 

phosphorus into the Mohawk River at the Knolls 

Atomic Power Laboratory site (Simpson, Beetem, 

Ruggles, 1958). This laboratory is located on the right 

bank of the stream downstream from Schenectady and 

about 1.7 miles upstream from the Vischers Ferry dam. 

The progress of the tracer was watched for about 2 

weeks. The observations covered the approximately 13 

miles of stream from the Knolls Atomic Power Laboratory site to the junction of the Mohawk and Hudson 

Rivers at Cohoes, N.Y. In this reach, the Vischers 

Ferry dam and the Crescent dam below it create pools 

which are a part of the Erie Canal system. Traffic 

moves from the Hudson River at Waterford, N.Y.,

through locks 1 to 6 into the pool above the Crescent 

dam. At the Vischers Ferry dam, lock 7 lifts the traffic to the pool above this dam. The reach between the 

Crescent dam and the Hudson River has two more dams 

with the Cohoes Falls between them. The river makes 

several sweeping bends in its passage through the test 

reach.

The test period beginning August 9, 1954, was one 

of low flow, the average flow being around 1,100 cubic 

feet per second (Thomas, 1956). Hydroelectric power 

plants at the Vischers Ferry and Crescent dams utilized 

this flow intermittently. The tracer was released into a 

sewer outlet. The water being discharged from the 

sewer was colder than the water in the river, and the 

density difference caused the formation of a density current which carried the tracer along the bottom to the 

deepest part of the channel before it became dispersed 

through the cross section of the stream.

This test has been described by Thomas (1956), 

Simpson, Beetem, and Ruggles (1958), and Parker 

(1958). Diffusion coefficients were obtained by Thomas 

(1956) and Parker (1958). The data of table 10 for 

this test are adapted from the analysis of Thomas who 

obtained 67 ft2 per sec for the first 7.5 miles of the 

test reach. Parker obtained a coefficient of 100 ft2 per 

sec for the river to mile 13 at the Cohoes waterplant.

Plankton samples collected during the test showed a 

high activity (New York State Department of Health, 

1954), indicating that the phosphorus was being collected by these organisms. There was no evidence however that dispersion rates were altered by their activity 

since they are very small and float freely in the water. 

Agencies cooperating in this test were the U.S. Geological Survey, New York State Department of Health, 

The Knolls Atomic Power Laboratory, and Harvard 

University.

LATERAL DISPERSION

Fort Collins test flume. When the Fort Collins tests 

were made, no data were available to indicate how rapid 

the lateral dispersion might be. The test equipment 

was therefore arranged to permit the salt concentration

TABLE 6. Discharge measurements and stream channel data, South Platte test

Location

Point 0 (Kersey Bridge) ____ ____

Point !____ _ _ _ __ _ _____

Point _!___________ ___ _ _

Point 3______ __ _ _ _ __

Point 4 (Kuner Bridge) _____ _ __ _ _

Time of flow 

measurement 

(p.m.)

1:50 to 3:00

12:40 to 1:10

1:40 to 2:43

1:45 to 2:30

12:05 to 12:50

Discharge 

(cubic feet 

per second)

588

553

551

559

553

Channels

5

1

1

2

2

Total top 

width

(feet)

213

160

141

180

189

Area 

(square 

feet)

242

247

232

271

272

Velocity 

(feet per 

second)

2.43

2. 24

2. 38

2.06

2. 03

Hydraulic 

radius 

(feet)

1. 14

1. 54

1. 64

1. 51

1.44

Eiver distance from 

Kersey 

Bridge 

(feet)

0

6, 100

12, 400

19, 900

27, 000
DISPERSION OF DISSOLVED OR SUSPENDED MATERIALS IN FLOWING STREAMS B21

at each electrode to be recorded separately so that tests 

could be made 'from which the lateral dispersion rates 

could be determined. In the first lateral diffusion tests, 

the salt solution was placed in a tube pressed against a 

sponge rubber pad on the bottom of the flume, and the 

levels outside and inside brought to equilibrium by 

means of a syphon. The salt solution was then released by pulling the cylinder upward. Some potassium permanganate color was also included with the 

salt, and observation showed that the lateral spreading 

being obtained was due to a difference in density because the salt solution went to the bottom of the flume 

where it spread out. To overcome this difficulty, the 

salt was put in solution in a container and supplied continously to a selected point through a rubber tube. 

Analyses of the records of these tests indicate that the 

effects of density were still present to some degree since 

the tests made at low velocities show relatively high dispersion rates. However, by selecting tests in which 

the velocity was high, some lateral diffusion coefficients 

can be obtained. These are shown in the table 7.

COLUMBIA RIVER TEST

Data for this test were supplied through the courtesy 

of the General Electric Co. The reactor cooling water 

discharge used in the test is released at essentially a 

single point in the bottom of the river about 300 feet 

from the plant shore. Approximately 38 percent of 

the river flowed between the point of discharge and the 

plant shore. A mixture of short-lived radioisotopes 

in the reactor effluent was used as a tracer. The effluent 

is warmer than the water in the river.

The ten traverses sampled were selected to be as nearly normal to the direction of flow of the main current 

as could be judged visually. They were selected at distances of 950, 1,750, 2,850, 3,590, 6,230, 8,870, 14,150, 

14,430, 24,710 and 29,990 feet from the point of discharge. The river makes a very gradual S-curve in 

the reach encompassed by these traverses. Velocities 

were measured with a cup-type velocity meter, and 

enough measurements were obtained to permit velocity 

contours to be plotted for each section.

The data relate to concentrations based on a value 

of 100 units per second arbitrarily assigned to the reactor effluent at the time of discharge to the river. The 

data were all corrected for decay. The samples were 

collected from a uniform depth of 3 feet below the 

surface of the river, as it had been previously determined that vertical concentration variations are negligible on the scales used for plotting. The test was performed near the middle of the annual low-water period 

of the Columbia, and no major changes of flow rate had 

occurred during the preceding three months. The flow 

rate of the river as reported by an adjacent USGS gaging station during the day of the test was 62,000 cubic 

feet per second.

The lateral diffusion coefficient, as obtained by the 

writer, is based upon the data from the first four stations. In this 3,590-foot reach, the river has a nearly 

uniform width of 1,000 feet. The maximum depth 

varies from about 16 to 24 feet. The concentration 

curve for station 4 is quite regular.

The measured flows in the reach including traverses 

1 to 4, inclusive, range from 40,400 to 45,900 cubic feet 

per second and average 43,625. The flow of the river 

in this reach is affected by the operation of powerplants. 

The flow reported by the U.S. Geological Survey represents a mean for the day while the measured flows 

represent the conditions at the time the test was being 

made. The measured flows were used in making the 

analyses described here.

Several difficulties were cited in the original report. 

Because the river is large and swift, it is difficult to 

locate sampling points with exactitude. Because the 

time of transit was somewhat uncertain, it was difficult 

to allow precisely for the decay of the short-lived isotopes. There was some variation at the source. A 

standard deviation of 10 percent was noted in hourly 

samples. There was a marked temperature difference 

between the effluent and the river water. A small rapids is present just downstream from traverse 5. This 

is 6,230 feet downstream from the discharge point.

In spite of these difficulties the data are correlated 

well by a diffusion coefficient Ky=2.l ft2 per sec. If the

TABLE 7. Lateral diffusion coefficients obtained from selected Fort Collins tests

Test

25- __ _--___.___..________.

27______-_____________ _ .

31 __-_-_________...._.__.____

51____._. _._.___.._._._.____....

Ml

0. 025 

. 010 

.010 

. 010

Flow (cubic 

feet per 

second)

8. 00 

8. 00 

4.00

7. 75

Flume

Width (feet)

7. 94 

7. 94 

7.94 

4.00

Depth (feet)

0.496 

.461 

.482 

. 951

Mean 

velocity (feet 

per second)

2. 04 

2. 19 

1. 05 

2.04

Diffusion coefficient (square 

feet per second)

From test 1

0. 035 

.0066 

.044 

.0050

0.067 n »*

0. 015 

.0065 

.0031 

. 0053

Ratio of 

coefficients

2.3 

1.0 

14.2 

1.0

1 These analyses were made by means of equation 30.
B22 TRANSPORT OF RADIONUCLIDES BY STREAMS

diffusion coefficients for transverse and vertical diffusion are comparable, an estimated diffusion coefficient 

would be, as will be explained later:

Ky= 0.067 Dv.*

With Z> = 10.03 feet and ^=0.025, Manning's formula 

gives d = 87.29 and ^*= 0.287 ft per sec. The mean 

velocity is V=4.418 ft per sec. These values yield the 

product 2.88 and a diffusion coefficient of 0.193 ft2 per 

sec. In this case, therefore, the value observed in an 

actual stream is approximately 10 times the value obtained from investigations of vertical diffusion rates.

INTEGRAL RELATIONSHIPS

Oscillograph charts resembling those obtained from 

the experiments described here have often been obtained 

during salt velocity determinations of the flow passing 

through penstocks. This method has been widely used, 

and considerable literature is available on developments 

in this art. If it can be shown that the relationships 

developed in the practice of this art are in accord with 

those derived from the concept of stream dispersion as 

a diffusion process, then the belief that stream dispersion is of this nature would be strongly confirmed. It 

seems worthwhile to do the small amount of work 

needed to make this test. It will be seen later that this 

concept is confirmed and that some important relations 

are also obtained as a by-product.

The relations of interest for the present purpose are:

1. For two stations where a salinity-time pattern is 

recorded, the time interval between the centers of 

gravity of the two patterns represents the time required for the mean velocity to traverse the distance between the two stations.

2. The maximum salinity observed varies inversely as 

the square root of the distance travelled by the salt 

cloud.

To investigate the possibility of correlating the relationship of item 1 we need two integrals. These are:

and

or

and

f

Jo

J2=BDV f~Stdt

Jo

Jl

_QSV C c

V» Jo

J: __& r c

: ~p I 

VT Jo

d«

dt.

(34)

(35)

(36)

(37)

The integrals of equations 34 and 35 represent, respectively, the area under the salinity time curve and the 

moment of this area about the origin of the time scale. 

The ratio e/2 :e/i represents the time to the center of 

gravity of the salinity-time area. This interpretation 

is appropriate when, as is usually the case, the salinity 

time curve is represented by a graph in rectangular coordinates with salinity as ordinate and with time represented by a distance measured along the abscissa.

The substitution of variables

u=- xV

/4Kxt 

reduces these integrals to the form

(38)

a (* e

:-Jo du

and

a»

(39)

(40)

The two integrals which appear here are evaluated in 

the following chapter. By using the relations developed there, it can be shown that

Ji=Q,

and

(41)

(42)

where C (Kx/V] is a constant. The relation is confirmed if the velocity is determined by the use of two 

electrode stations and the time of transit from one to 

the other is determined by subtraction. The time of 

transit represented by the center of gravity is

-r -IT

2C There term -= is a constant which will disappear if

the difference between two values of tm is taken. This 

is the customary procedure in salt-velocity work. It 

is clear, then, that the concept of dispersion in streams 

as a diffusion process is in accord with salt velocity 

experience.

We may now consider item 2. The condition that the 

mean velocity will have reached the point x is expressed 

by the relation x Vt = 0. This relation will reduce the 

exponential term of equation (3) to unity. At this time 

also the salinity will be very near its maximum. Then 

when x F£=0, the salinity varies inversely as the 

square root of the time. This relation is closely con-
DISPERSION OF DISSOLVED OR SUSPENDED MATERIALS IN FLOWING STREAMS B23

firmed by tests in long pipes (Taylor, 1924). It is clear, 

therefore, that the concept of stream dispersion as a 

diffusion process is in complete accord with salt-velocity 

experience.

Having established that the concept of stream dispersion is in accord with salt velocity practice and experience, our attention may now be turned to some relationships which involve these integrals. At the outset we 

may note that the salinity introduced is certainly represented by the expression

+ 

Sdx

or

dx.

The substitution of variable

x-Vt

l±Kxt 

reduces this integral to the form

e-»du.

The integral involved here has the value 

result of this computation is then

(43)

(44)

(45)

(46)

and the

(47)

This is the expected result.

If equations 34, 41, 43, and 47 are now compared, 

it will be seen that we have the integral of the salinity 

time curve for a given point x and the integral of the 

salinity-space curve at a given time both representing 

the original amount of salt Qs introduced into the 

stream. In this connection, it is of interest to note that 

a similar development, based upon the concept of dispersion by velocity differentials, yields Qs from the 

salinity-space integral, as it should, but that the 

salinity-time integral does not yield Qs. If the integration is carried over a finite time, the result is often 

close to Qs ; but if the integration is carried from zero 

to infinity, an infinite result is obtained. The data from 

an actual test would, of course, involve only a finite 

time interval. This difference suggests that, while the 

integrals described above show that the salinity-time 

integral should yield Qs, if the dispersion mechanism 

is of the diffusion type, a determination of this quantity 

from test data may be sensitive to experimental errors 

due to the manner of making the salinity measurements. 

A comparison of the results of the Fort Collins and

Allenspark tests indicates that this is true. The Fort 

Collins records yield salinity-time integrals which indicate a lesser amount of salt than was known to have 

been used, while the salinity-time curves from the Allenspark tests yield total salinities in excess of the 

amount of salt introduced. The discrepancy reaches 56 

percent in the Fort Collins tests and 30 percent in the 

Allenspark tests. In both tests the sampling was done 

by electrodes distributed through the stream cross section. The salinity-time curves should have been well 

defined in these tests, but, as indicated above, the area 

of the salinity-time curve did not determine closely the 

amount of salt present. This possibility should be taken 

into account when making interpretations of test data. 

In return for the assistance received from saltvelocity experience, it may be pointed out that the combination of analytical and experimental identification 

of stream dispersion as a diffusion process and the integral relations described above can be used to provide 

a new theoretical basis for the salt-velocity method of 

velocity determination.

EVALUATION OF INTEGRALS

To evaluate integrals of the type

. a"

U* du (48)

and

W^r^ ^-du, (49) 

Jo u

consider the integral:

(50)

/» CO .Of2

I 11? 

*= e u*du. 

Jo

By differentiation with respect to a (Osgood, 1933, 

p. 486),

da Jo

du. (51)

Let

a a , , a , u=-> «=-' and du= 5- dij.

By substitution

Then

 n u

dWp_ _£ j 

da Jo

da

5=-2W0 .

(52)

(53)
B24 TRANSPORT OF RADIONUCLIDES BY STREAMS

This is a differential equation whose solution is:

(54)

The arbitrary constant of the solution was determined 

from the known value of the probability integral to 

which the integral 50 reduces when a=0.

By the use of equations 51 and 54 the value of integral 

48 can be determined as

(55)

By equating the second derivatives of W0 with respect 

to u, as obtained from equations 50 and 54, equation 49 

can be evaluated as

(56)

To evaluate an integral of the type

11 _E.

/» 00

1 Jo ~~

du (57)

consider the integral

-i= r

Jo

udu. (58)

By differentiation with respect to ft (Osgood, 1933)

.. 0

and

If

then

and

'--/; du

dp -I du.

v=- 

u

C" 

I

Jo u

, du= -

(59)

(60)

(61)

(62)

(63)

A solution of this differential equation which satisfies 

the requirement that, when ft = Q (McLachlan, 1955, 

p.133),

= e- udu=-e~u = 1

is

(64)

(65)

where K± represents the modified Bessel function of the 

second kind of order 1. (British Association for the 

Advancement of Science, 1937.) By differentiation 

with respect to /?, use of the relation

(66)

together with equation 59, it can be shown that

rJo u

(67)

The substitution of variable

v*=u (68) 

in the integral of equation 67 will yield the result

/'

Jo

(69)

This transformation shows that the integrals evaluated by the two developments described above are closely related. The first group has the even powers of the 

variable in the denominator, while those of the second 

group have the odd powers, when they are put into a 

comparable form.

DETERMINATION OF DIFFUSION COEFFICIENTS BY 

ANALYTICAL PROCEDURES

Some noteworthy progress has been made in the 

analytical determination of diffusion constants for flowing fluids. Diffusion in pipes has been treated in two 

papers by Taylor. (Taylor, 1953, p. 186-203; 1954, p. 

446-468.) The latter of the two relates to the disperson 

of matter in turbulent flow through a pipe. For this 

flow he determines a virtual coefficient of diffusion:

#=10. low*, (70)

where a represents the radius of the pipe and (Prandtl, 

1952, p.126)

(71)

is the shear velocity. The symbol TO represents the friction stress exerted by the fluid, of density p, on the wall. 

This virtual coefficient represents the combined action o| 

variation of velocity over the cross section of the tube
DISPERSION OF DISSOLVED OR SUSPENDED MATERIALS IN FLOWING STREAMS B25

and turbulent diffusion in the radial direction. It is 

therefore of a type comparable to the longitudinal diffusion constant Kx as used here.

A later development (Elder, 1959) applies to flow in 

a wide channel. His analysis is based upon a logarithmic form of the velocity profile between the bed of the 

stream and the free surface. The result he obtains, if 

expressed in the present notation, is

(72)

In writing this expression, Elder's stream depth h has 

been replaced by the hydraulic radius TV Of the factor 

5.93, the part contributed by the combination of velocity 

variations and lateral dispersion due to turbulence is 

5.86, while the part contributed by longitudinal dispersion due to turbulence is 0.07. The above expression 

has the same form as that found by Taylor (1954), but 

the factor is only about one-third as large. Elder has 

made experiments on a small flume in which a dispersing dye concentration was recorded photographically. 

He found a part of the dye being carried by the turbulent portion of the stream and a part by the laminar 

sublayer. For the turbulent portion he obtained a factor 

6.3 which is about 8 percent above the theoretical value 

5.9.

Elder (1959) has also discussed the dispersion of particles of finite size and has noted that, for nonbuoyant 

particles previous experiments indicate that very small 

discrete particles are dispersed at the same rate as the 

fluid particles.

The quantities involved in the diffusion coefficient are 

clearly identified by these investigations. They include 

a dimension descriptive of the cross section of the 

stream, here represented by the hydraulic radius n 

and the shear velocity. It therefore seems appropriate 

to express the results of experiments by the ratio

The virtual coefficient of longitudinal diffusion is also 

given by Taylor (1954) in the form in the present 

notation :

= 14. (73)

where y is a friction factor of the Darcy-Wiesbach type. 

The results obtained by Taylor (1954) may be adapted 

to our present purposes by expressing the pipe radius 

a in terms of the hydraulic radius TV On this basis 

we obtain

(74)

Some further relations will facilitate the use of this 

expression. If we write equation 71 in the form:

W

(75)

where w represents the weight of a unit volume of water 

and g the acceleration of gravity and note that in a 

stream of hydraulic radius n and slope s^

(76)

we can obtain by substitution (see also Boyer, 1954)

t>*=V0ri*i- (77)

If we compare this with the well-known Chezy formula 2 (King, 1954, p. 7-15)

(78) t

we may write, by introducing the quantity g:

(79) t

Then the shear velocity is given by relations of the type: 

y=^U, or »* =^f- V. (80) f

In the cases under consideration here the ratio v^.: V 

would range, approximately, between the limits 0.03 

to 0.2. For some purposes it will be advantageous to 

express equation 74 in the form

Kx=20.2rlV 4f (81) t

This equation shows that, for a given cross section, the 

diffusion coefficient should be proportional to the mean 

velocity. It is interesting to note that Taylor's researches confirm the inverse square root law for longitudinal diffusion phenomena.

We may now turn our attention to the transverse dispersion rates. These are represented by constants of 

the type Ky and Ke. Because they depend upon turbulence only, instead of a combination of velocity variations and turbulence, they may be expected to be much 

smaller than the longitudinal dispersion constant Kx 

The case of vertical dispersion has been considered by

* Tabulations of Ci values are generally dimensioned quantities. The quotient -y=

is dimensionless. Care should be taken to have C\ and g in the same units when 

computing the quotient. 

t See note at end of notation list.
B26 TRANSPORT OF RADIONUCLIDES BY STREAMS

Vanoni (1953, pp. 141-144) in connection with the 

phenomenon of sediment suspension. He finds that 

the dispersion rate is given by an expression of the type

(82)

where &!=0.4 is von Karman's constant and em of 

Vanoni's notation corresponds to our Kz. This expression has been used as a basis for derivation of formulas 

for the distribution of sediment in streams, and it is 

stated that these have been confirmed by numerous laboratory and field measurements.

For our purposes we may use an average value

or

(83)

(84)

A comparison of this form and that for the longitudinal 

dispersion constant Kr will disclose that their magnitudes stand in the ratio of about 300 to 1 with Kx being 

the larger. Because both Ky and Kg are due to turbulence alone, they may be of the same order of magnitude.

SUMMARY OF LONGITUDINAL AND LATERAL 

DIFFUSION COEFFICIENTS

A digest of the analytical and experimental data on 

diffusion coefficients developed in this study are contained in table 8. Some results obtained by other investigators are also listed in this table for purposes of 

comparison, and the sources of these data are indicated. 

Analytical investigations based upon fluid mechanics 

considerations show that the diffusion coefficient Kx and 

Ky should be proportional to the product of the shear 

velocity v* and the hydraulic radius rx . The data are 

presented in terms of a ratio to this product.

EXAMPLES

The following examples have been prepared as a guide 

for applying the equations described here. By making 

the examples apply to streams for which test data are 

available, comparisons of computed and test data can 

be made, and the factor of loss of tracer can be examined. The computations will be made as for a new 

problem, and the values so obtained will then be compared with those obtained from the tests.

LONGITUDINAL DISPERSION

The first of these computations will be made for the 

conditions prevailing on the 27,000-foot reach of the 

South Platte River between Kersey and Kuner, Colo., 

on the date of October 28, 1958. The South Platte is

TABLE 8. Comparison of computed and observed values of longitudinal and transverse diffusion coefficients

Kx

5.86

20.2

18

24

13

500

800

0.36

.22

.14

.72

Source

Analysis.... .......

Analysis. .... .......

Triangular flume

experiments.

Rectangular flume

experiments.

Experiment on a rectangular flume with

bar roughness on

the bottom.

Test on a natural

stream (South

Platte test).

Test on a developed

stream (Mohawk

River test).

Experiment on a rectangular flume with

bar roughness on

the bottom.

Rectangular flume 

experiment.

Rectangular flume

experiment.

Test on a natural

stream (Columbia

River test).

Remarks

Longitudinal Diffusion

Value obtained from the analytical and experimental investigations of Elder (1949) on dispersion in open channels.

V &IUG 8>Q£ipt6Ci from txiG sin&ly tiC£il find Gxpcri~

mental work of Taylor (1954) on dispersion in

pipes.

Value obtained from tests 1, 2, and 3 of the Alienspark series. Flume 400 ft long with plywood

sides at 90°; depth, 1.473 ft; flow, 4.99 ft8 per sec;

average velocity of flow, 2.30 ft per sec; hydraulic

radius, 0.521 ft; slope, 0.000480; Kutters «i value,

0.0092 (Powell and Posey, 1959); diffusion coefficient, 0.825 ft2 per sec.

Value obtained from test 3 of the Fort Collins

series. Plywood flume, 7.94 ft wide and 75 ft

long; average depth, 0.544 ft; flow, 6.0 ft8 per sec; 

mean velocity of flow, 1.39 ft per sec; hydraulic

radius, 0.478 ft; Kutters n\ value 0.010; diffusion

coefficient 0.702 ft« per sec.

Value obtained from test 23 of the Fort Collins

series. Plywood flume, 7.94 ft wide and 75 ft

long with a mesh of welded reinforcing bars laid

on the floor; average depth, 0.492 ft; hydraulic

radius, 0.438 ft; flow, 8.0 ft8 per sec; average velocity of flow, 2.05 ft per sec; Kutters n\ value, 0.025;

diffusion coefficient, Kx=l.26 ft« per sec.

Value obtained from a test on the South Platte

River near Kersey, Colo. Braided stream running over a sandy bed; flow, 554 ft8 per sec;

average velocity of flow, 2.16 ft per sec; hydraulic

radius, 1.52 ft; effective slope. 0.00105; m=0.028; 

diffusion coefficient 1^=174 ft« per sec. (Some

loss of tracer accounted for); test reach 27,000

ft long.

Value obtained from a test on the Mohawk River

near Cohoes, N.Y. Conditions of great complexity with dams, intermittent flow through

powerplants, expanding sections, crookedness,

barge traffic, lockage, and density currents.

Average flow, 1,100 ft8 per sec; diffusion coefficient, 67 ft" per sec; average depth 20 ft (Thomas,

1956.) with n=20 ft, m =0.025, V=0.07525 ft per

sec, ,, =0.00435 ft per sec, ^._m This value

nv«

accounts for loss of tracer.

Transverse Diffusion

Value obtained from test 25 of the Fort Collins

series. Bar roughness laid in bottom of flume to

raise the roughness to n\ =0.025; flume, 7.94 ft

wide, 0.484 ft deep; flow, 8.0 ft8 per sec, Ky =0.036

ft per sec.

Value obtained from test 27 of the Fort Collins 

series. Bare flume with «i =0.010. Flume, 7.94

ft wide and 0.453 ft deep; flow, 8.0 ft8 per sec;

Ky =0.0089 ft2 per sec.

Value obtained from test 51 of the Fort Collins

series. Bare flume with n» =0.010. Flume, 4

ft wide and 0.943 ft deep; flow, 7.75 ft8 per sec;

Ky =0.0077 ft2 per sec.

Value obtained from a test on the Columbia River

near Richland, Wash. Top width about 1,000

ft; maximum depth 16 to 24 ft; diffusion coefficient obtained from observations on a 3,590-ft

reach; flow, 43,600 ft8 per sec; mean velocity, 4.42

ft per sec; hydraulic radius, 10.03 ft; Ky=2.0

ft2 per sec.

a braided stream that runs over a sandy bed. On the 

day of the test the flows averaged 554 ft3 per sec, the 

areas, A, of the cross sections 256 sequare feet, the top 

widths 168 feet, the hydraulic radii, n, 1.532 feet, and 

the mean velocity, F, 2.164 ft per sec. These are averages obtained from four discharge measurements made 

in the reach during the test. A reference to table 6 will 

indicate that there was not much variation in the flow 

in the test reach.

Potassium carbonate was used as a tracer. Samples 

were taken and measurements were made by means of
DISPERSION OP DISSOLVED OB SUSPENDED MATERIALS IN FLOWING STREAMS B27

a flame photometer. This device measured the potassium ion of which the 1,000 pounds of tracer contained 

566 pounds. The sensitivity of the flame photometer 

is about 0.1 ppm. The tracer was put in solution in 

six drums and was released into the river at the Kersey 

bridge at 11:00 a.m. The time required for the release 

was 1 minute and 20 seconds.

We will compute the salinity pattern at points 3 and 

4. A longitudinal diffusion coefficient must first be 

selected. By reference to table 8, it is seen that for 

a stream of this type a ratio of Kx :riv:lt.=600 is 

appropriate.

A suitable value for the shear velocity v* can be obtained from equation 55:

= 

19

t

For a stream of this kind, a reasonable value for the 

coefficient of roughness would be n±=0.025. 

By use of Manning's formula with nj.=0.025,

n\

r} /6=63.9, t

then

and

__. ~

W*=: V 2.164 =^^=0.193. 11.26 11.26 

The estimated diffusion coefficient is

Kx= (500)r!0, = (500) (1.536) (0.193) = 147 ft2 per sec.

This value is somewhat smaller than the one obtained 

from the analysis of the test data. This was based upon 

a measured gradient. The difference is due to the choice 

of U-L. A value of about ^ = 0.028 would be required to 

reproduce the test value. It will be found, however, 

that a reasonably good correlation can still be obtained 

because the presence of Kx under the square root sign 

in equation 3 makes it somewhat insensitive to variations of the diffusion coefficient.

The distance from the point of introduction of the 

tracer to point 3 was 19,900 feet. The time required 

for the mean velocity to reach this point may be estimated as 19900 * 2.178=9140 seconds, or 2 hours 32 minutes and 20 seconds. This would put the estimated time 

of arrival of the peak of the salinity pattern at about 

1:32 p.m. The observed time of arrival was at about 

1:15 p.m., or 17 minutes earlier. Differences of this

t See note at end of notations list.

magnitude should be expected when dealing with natural streams.

The computations will cover the period from 12:30 

p.m. to 2:30 p.m. at 5-minute intervals. If

Qs =566 pounds of potassium ion,

BD=256 tt*=A, 

and

sec

then

Q. 566

(256) (42.93) =0.0515.

The computation is shown in table 9. It is based on 

the use of equation 3.

TABLE 9. Sample computation for point S, South Platte test

Time 

(hr, 

min)

12:50

12:55

1:00

1:05

1:10

1:15

1:20

1:25

1:30

1:35

1:40

1:45

1:50

1:55

2:00

2:05

2:10

2:15

2:20

2:25

2:30

t (sec)

6600

6900

7200

7500

7800

8100

8400

8700

9000

9300

9600

9900

10200

10500

10800

11100

11400

11700

12000

12300

12600

x- Vt (ft)

5618

4968

4319

3670

3021

2372

1722

1073

424

-225

-874

-1524

-2173

-2822

-3471

-4120

-4770

-5419

-6068

-6717

-7366

(x-Vt)*

±Kzt

8. 1472

6.094

4.414

3.060

1.993

1.183

.601

.225

.034

.009

.136

.400

.789

1.292

1.901

2.605

3.400

4.276

5.228

6.249

7.336

(x-VW

±Kzt

0.00029

. 00226

.01215

. 04689

. 13629

.30636

.54827

.79852

.96658

.99075

.87320

.67052

. 45448

. 27472

.14942

.07390

.03337

.01390

.00537

.00193

.00065

<S (Ib per 

cu ft 103)

0.00018

. 00140

.00737

. 02788

.07947

.17531

.30807

. 44080

. 52471

.52909

. 45897

. 34705

.23175

. 13807

. 07405

.03612

.01610

.00662

. 00252

.00090

.00030

*E> (ppm)

0.003

.022

.119

.447

1.274

2.809

4.937

7.064

8.409

8.479

7.355

5.562

3.714

2.213

1.187

.579

.258

.106

.040

.014

.005

* Equivalent potassium ion concentration.

A comparison of these estimated salinities with the 

observed salinities is shown on figure 10.

The observed salinities are listed in table 5. It will be 

noted that while the observed pattern is about as wide 

as the computed pattern, the observed salinities reach 

only 6.6 ppm as compared to the estimated 8.7 ppm. 

Some loss of tracer is indicated.

A similar condition is found at point 4 where the 

estimated travel time is

27,000 10 . AA , 

-= 12 > 400 seconds

and the estimated maximum salinity is

Sm =

HI-35 =0.000472 Ib per F ft3 

or 7.6 parts per million of potassium, by weight. This
B28 TRANSPORT OF RADIONUCLIDES BY STREAMS

1:00 PM

TIME

FIQUEE 10. Comparison of observed and computed salinities at points 3 and 4, South

Platte test.

conversion is based upon a density of water of 62.4 

pounds per cubic foot. The maximum observed salinity 

at this point was 12.6 8.0=4.6 ppm.

A good correlation of peak amplitudes can be obtained by using a diffusion coefficient of about 320 

square feet per second, but the computed patterns are 

then much too wide. The areas of the observed salinitytime curves in terms of potassium ion at the four observation points are given in the following table.

Indicated 

amount of

Point potassium 

ion (Ib)

Kersey bridge.________ 566

!.._________________._ 501

2______________.__.__ 571

3______________._____ 510

4____________._ 304

If the difficulty of determining the amount of salinity 

from the salinity-time curve, as revealed by the Fort 

Collins and Allenspark tests, is kept in mind, the first 

three stations show a reasonably consistent behavior. 

The amount of potassium ion indicated at point 4 is 

low by comparison. This may be due to the way the

Remarks

Amount introduced.

samples were collected at point 4. The sampling schedule had been worked out for a flow of 600 cubic feet 

per second as determined from a recording gage reading obtained by telephone. However, on the morning 

of the test the flow was down to about 554 cubic feet 

per second. This had an effect on the timing and, at 

point 4, caused the salt cloud to pass while a series of 

readings across the stream were being made. Because 

these samples were taken from a bridge, this series took 

more time than those made at the other stations where 

they were obtained by wading. A similar evidence of 

tracer loss was also present in the Mohawk test.

A bump is present in the descending limb of the 

curves of observed salinity at points 1, 3, and 4. More 

closely spaced readings at point 2 would probably have 

shown its presence there also. The reason for this may 

be found in the way the tracer was introduced. The 

tracer was put in solution in drums placed on a walkway of the Kersey bridge. Initial release was made by 

opening a valve. As soon as the contents of the drum 

was reduced to the point where a man could lift the 

drum, the valve was quickly closed, and the remaining 

contents were dumped over the bridge rail. The bump 

is probably due to the dumped part.

This comparison should indicate the type of behavior 

to be expected in a natural stream. The propriety of 

the analytical formulation used is supported by the work 

of Taylor (1954) and Elder (1959).

TRANSVERSE DISPERSION

A computation of lateral diffusion will be made for 

the conditions on a 3,590-foot reach of the Columbia 

River near Richland. On the day of the test, a tracer 

material was being continuously released at a point 

about 300 feet from the bank of the stream. A rate of 

100 units per second was arbitrarily assigned to this 

release. The observations were reported in terms of this 

arbitrary designation. This is an adequate specification for our purposes. The top width of the stream at 

this point is about 1,000 feet, and approximately 38 

percent of the river flow was passing between the point 

of release and the bank. Over the test reach the flows 

averaged 43,600 cubic feet per second, the areas of the 

cross section 9,870 square feet, the hydraulic radii 10.03 

feet, and the mean velocities 4.418 feet per second.

If a roughness coefficient of 0.025 is appropriate, then 

Manning's formula gives

t

and

~_±_._
DISPERSION OF DISSOLVED OR SUSPENDED MATERIALS IN FLOWING STREAMS B29

from which

4.418 t

From table 8 an appropriate value for Ky can be 

obtained:

Kt^-=0.72

Then Kv= (0.72) (10.03) (0.287) =2.07 ft2 per sec. With

100 Brt / -4. \ v gs =-jr-=50 (units per sec) each way,

z

D= 10.03 feet, 

F=4.418 ft per sec, 

z=3590 feet,

50

10.03 V»(2.07) (4.418) (3590)

=0.015725

4.418 =0.00014863. 4KyX (4) (2.07) (3590)

The computation of salinity is made as shown in the 

following table. Because the tracer can diffuse both 

ways, the qg value is taken as one-half of the total 

supply. The computation is made by use of equation 

30 which was obtained by use of the Product Law.

TABLE 10. Computation of lateral diffusion from a 

continuous point source, Columbia River test

»(feet)

0 _ ______ ______

25...... __ ..... _ ....

SO........ __ .............

75---.- __ -_ . .

100 _______________

125.......... __ ...........

150 -.- .- .

175 _______________

200- _ . .

Vy*

*KyX

0

0.093

.372

.836

1.486

2.32

3.34

4.55

5.95

W* 

e ****

1.0000

Q11O

fiSOd

4335

.2263

.0982

.0354

.0106

.0026

S (units per 

cuft)

0. 015725

01433

AinoA

.00682

00356

.00154

.00056

.00017

.0000

A comparison of the computed and observed values 

is shown on figure 11.

It will be of interest to estimate the dispersion at a 

point much farther downstream. This will also illustrate the use of an image. The last observations were 

made at a point 29,900 feet downstream from the source.

For this point

50

and

10.03 Vx(2.07) (4.42) (29,900) 

V 4.42

=0.00538

-=0.0000178. (4) (2.07) (29,900)" 

Table 11 is computed with the aid of equation 30:

t See note at end of notations list.

Computed 

distribution"

I . Observed distribution 

as scaled from the 

original chart

-100 0 100 

DISTANCE FROM CENTER OF PATTERN, IN FEET

FIQUKE 11. Comparisons of computed and observed distributions at 3,590 feet from 

the source Lateral dispersion, Columbia River test.

TABLE 11. Estimated salinities 29,900 feet from the 

source, Columbia River test

y

(it}Cit\ v.ieet.j

0

50

100

150

200

300

400

500

600

700

QAA OUv

onn *7UU

1,000

V02

4-Kvr

0

.0404

.1616

.3635

.6463

1.454

2.585

4.039

5. 8165

7. 9169

10.340

13. 087

16. 157

-Vy*

e*KtX

1.00000

.961

.850

.695

.524

.234

.104

.0176

.0030

.00036

.000032

S

(units per 

cuft)

0.005378

. 005169

.004571

.003738

.002818

.001258

.000559

.000095

.000016

.000002

Adjusted 

width

(feet)

0

72.5

145

217.5

290

435

580

725

870

1015

1160

1305

1450

The results of this computation and the observed 

salinities are shown on figure 12. The curve a-a' represents the salinity as computed from equation 30 on 

the basis that the width of the stream is unchanged 

and that the bank of the stream does not halt the lateral 

spread of salt. Curve &-&' represents the estimated 

salinity distribution, as described for «-a', if the near 

bank were absent. A reference to the figure will show, 

however, that the near bank does halt the spread of 

salinity in this direction. To account for this, it may 

be assumed that the stream extends beyond the bank and 

that a second plume of tracer material is released 300
B30 TRANSPORT OF RADIONUCLIDES BY STREAMS

0.008

0.006

Position of Point of 

image release

\ 

\

\

\

\

\

\

\

\ 

^ <

\

\ \

b^/

 Near bank

; m s"

 (3

 / y^

// /

>7

/

\

Xc'^

 © 0 Estimated salinity as corrected for change 

in width of stream

 X

a**jOr' *~^. ''\VX''"\\

\ ' 

\

A

\

^.

\

.

(- \e( "V

\ ' :

N^-^_.

'"x.

^^-I'.'^- J

Far bank 

0.004

0.002

0.000

300 300 600 900 

DISTANCE FROM NEAR BANK, IN FEET

1200 1500

FIGURE 12. Comparison of observed and computed salinity distributions at 29,900 feet from the source lateral dispersion, Columbia River test. Curves o-a', 6-6', c-c' 

and d-d' illustrate steps in constructing curve of estimated salinity (see section on "Transverse dispersion").

feet from it to create a condition that is everywhere 

symmetrical about the line of the near bank. Such a 

hypothetical symmetrical source is called an image. 

The real source and its image produce effects which 

nullify each other at the bank, thereby satisfying the 

condition of no flow of salt through the bank. The effect 

of the image is represented by curve c-c'. The sum of 

curves &-&' and c-c' yields curve d-d'. This curve has 

a zero gradient at the bank and accounts for the presence 

of the bank. Then the curves a-of and d-d' represent 

the estimated salinity distribution if the stream had its 

original top width of 1,000 feet. At this station, however, the stream has a top width of 1,450 feet. To 

account for the change of width, the plotted points of 

curves a-a' and dr-d' were displaced toward the far bank, 

the new distance from the near bank being in the ratio 

of 1,450 to 1,000 to the old distance. This correction 

is based upon the reasonable assumption that the lateral 

spreading of the stream carried the salinity along 

with it.

Although the observed and estimated salinity distributions are somewhat different, the maximum values 

of salinity and the amount pf lateral dispersion are 

not greatly different.

These comparisons represent the type of correlations 

obtained in actual tests. They indicate what might be 

expected from an application to a new test.

Use of the chart of figure 1 may be illustrated if it is 

assumed, as an approximation, that the tracer used in 

this test was released near the bank. This chart was 

prepared for use with the product law where equation 3 

is to be used with equation 13 or 14, or both, but it 

may be adapted to the present use by interpreting a? 

as the distance travelled by the mean velocity V in the 

time t and by interpreting the factor /2 on the ordinate 

of the chart as the quantity

S

<L* \ 

BDVj

Based upon a mean velocity of 4.418 feet per second, 

the time required to traverse the distance a? =29,900 

feet would be 6,768 seconds. If the original width 

5 = 1,000 feet is used, the parameter B/^/±Kyt takes 

the value 4.23. A glance at figure 1 will show that the 

salinity has not reached the far bank and will not do 

so short of an a? distance of about 135,000 feet. At this 

distance, the travel time is £=(135,000/4.418) =30,562 

seconds, and the chart parameter has the value

B 1,000

V(4) (2.07) (30,562)

=2.0.
DISPERSION OP DISSOLVED OR SUSPENDED MATERIALS IN FLOWING STREAMS B31

The salinity has then reached the far bank and the 

salinity profile across the stream is as shown on the 

curve having this value of the parameter. At the near 

bank, where y/B=Q.Q, this curve has an ordinate of 

2.25, the salinity at this point will be

(10°)(2- 2567)

(9,874) (4.418)

= 0.00517 unit per cubic foot

At the middle of the stream where y/Z?=0.5 the ordinate is 0.83 and

£=(0.00517) 0.83

(2.2568) =0.00186 unit per cubic foot.

At a distance of 528,000 feet or 100 miles downstream 

from the source where #=119,500 seconds, or 1.37 days, 

the parameter would have the value 1.01 and the crossstream profile would "be as shown on the chart for this 

parameter. The maximum salinity at this point would 

be 0.00264 unit per cubic foot. Ultimately the parameter approaches zero, the ordinate approaches 1.000, 

and the salinity becomes uniformly distributed through 

the cross section with a value of

The ultimate value is

100

BDV (9,874) (4.418) =0.002293 unit per cubic foot.

The tracer used in this test was composed of shortlived isotopes. The above ultimate value does not, 

therefore, represent the observed activity, which might 

be very low, but rather the activity which would be 

observed if the tracer had retained its original strength. 

An increase of stream width could be accounted for in 

the manner described previously. A continuous supply 

of salinity released adjacent to a bank of a stream will 

not disperse as rapidly as it will if released at a point 

away from the bank because, when released near the 

bank, it can diffuse in only one direction. A comparison of values obtained in this example will show that 

the estimated maximum salinity attained at OB =135,000 

feet for material released adjacent to the bank is comparable to the maximum salinity found at 29,900 feet 

when the salinity is released at a point out in the 

stream.

OUTLINE OF COMPUTATION PROCEDURE

For the benefit of those who wish to apply these methods to a new problem it will be helpful to outline the

procedure followed in these examples. The steps to 

be taken are as follows:

1. Obtain the flow of the stream and its cross section.

2. From available hydraulic data select an n^ value 

appropriate for the stream conditions and compute 

the Chezy coefficient Oi from Manning's formula, 

using the hydraulic radius obtained from the data 

of step 1 preceding.

3. Use equation 80 to estimate the shear velocity v* 

using the mean stream velocity obtained from the 

data of step 1 and the C-^ value of step 2 preceding.

4. From table 8 select appropriate factors and use them 

with the values r^ and v% previously obtained to 

estimate values for the Kx, Ky, and Ks diffusion coefficients.

5. For the case of a finite quantity of salinity suddenly 

released into a stream under conditions where 

longitudinal dispersion is important use equation 

3. If lateral as well as longitudinal dispersion 

must be accounted for, use equations 3 and 13 combined according to the product law.

6. For a sustained release use equation 30.

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B32 TRANSPORT OF RADIONUCLIDES BY STREAMS

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O

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