Mathematics of 

Dispersion With Linear 

Adsorption Isotherm

GEOLOGICAL SURVEY PROFESSIONAL PAPER 411-H
Mathematics of 

Dispersion With Linear 

Adsorption Isotherm

By AKIO OGATA

FLUID MOVEMENT IN EARTH MATERIALS

GEOLOGICAL SURVEY PROFESSIONAL PAPER 411-H

A theoretical consideration of the effect of a reacting 

media on the dispersion of fluids in porous media

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

STEWART L. UDALL, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

The U.S. Geological Survey Library has cataloged this publication as follows:

Ogata, Akio 1927-

Mathematics of dispersion with linear adsorption isotherm. 

Washington, U.S. Govt. Print. Off., 1964.

iii, 9 p. diagrs. 30 cm. ,(U.S. Geological Survey. Professional 

paper 411-H)

Fluid movement in earth materials. 

Bibliography: p. 9.

1. Dispersion. 2. Adsorption. I. Title. (Series)

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

Washington, D.C. 20402
CONTENTS

Page

Abstract._ _________________________________________ HI

Introduction _______________________________________ 1

Adsorption isotherms..._____________________________ 1

Field equation____________________________________ 2

Review of some previous analyses___________________ 3

Mathematical analysis_____________________________ 4

Special situations-____________________________-_-___ 7

Discussion _________________________________________ 8

References _________________________________________ 9

ILLUSTRATIONS

Page 

FIGURE 1. Concentration distribution for system with irreversible linear adsorption isotherm.__________--_--__---------_ H4

2. Plot of /-function____________________________________________________________------------_----------- 7

SYMBOLS

Di Dispersion coefficient in the direction i; i=x, y, z.

O Concentration of the liquid phase.

(70 Concentration of the liquid at a given point, say, o?=0.

N Concentration of the solid phase.

NQ Concentration of the solid phase at saturation.

k, m Constants associated with adsorption isotherm.

u Average velocity of the liquid.

t Time variable.

x Space variable.

A Integration constant.

I0 (x) Modified Bessels function of the first kind of order zero.

J0 (x) Bessels function of the first kind of order zero.

erfc (a;) Complementary error function.

J(x, y) Goldsteins J-function.

L<t>(£) Laplace transform of function

L~l<j>(p) Inverse Laplace transform.

U(t h} Unit step function.

a a2/32&2m/4£2 or a2/32&2m/4.

b ap/2.

a UX/2D.

Dummy variable.

in

FLUID MOVEMENT IN EARTH MATERIALS

MATHEMATICS OF DISPERSION WITH 

LINEAR ADSORPTION ISOTHERM

By AKIO OGATA

ABSTRACT

The simultaneous occurrence of dispersion and adsorption is 

noted frequently in laboratory models. In most investigation, 

especially that of the exchange process, the adsorption phase 

dominates; thus, prediction is based on a reduced system. 

The discussion presented in this paper, however, is based on the 

concept that the orders of magnitude of the adsorptive and the 

dispersive processes are the same. In addition, because of the 

meager knowledge of the macroscopic adsorption, the isotherm 

is assumed to be linear. This leads to a system which may be 

described by the equations

n

da;2 U do;

The paper deals entirely with the solution of the above equations 

for flow in a porous column.

INTRODUCTION

In the transport of two miscible fluids through isotropic porous media, the mixing and the dilution of any 

contained contaminant is dependent on two mechanisms, 

namely, mechanical dispersion and adsorption. The 

mechanical dispersion is defined to include all 

mechanisms that tend to cause spreading of the fluid 

stream which generally arises because of variations in 

fluid velocity throughout microscopic segments of the 

porous medium. Although variations in the magnitude 

and direction of the fluid velocity is noted throughout 

macroscopic segments of heterogeneous materials, the 

theory of dispersion cannot be extended to describe 

this system. The adsorption mechanism is defined as 

the removal or addition of the contained contaminant 

by the solid matrix through which the fluid flows. 

That is, the solid material may be thought to act as a 

mathematical source or sink depending on the concentration of the liquid flowing through the porous medium.

The physicochemical aspects of the problem of 

adsorption are generally approached through thermodynamic principles based on analysis of the total free

energy available at any given state. However, because 

of the complexities of these phenomena and the porous 

medium, such analysis has as yet produced no general 

conclusions on how to make use of the adsorption 

isotherms that characterize naturally occurring porous 

media. In other words, because of the extreme range 

of conditions met in aquifers, adsorption study of small 

segments of the flow system microscopic analysis 

has not produced results directly applicable to the entire 

flow regime. This indicates that for each soil complex 

or aquifer material, the adsorption characteristics of 

various candidate tracer materials must be predetermined so that the results can be given macroscopic 

application.

To circumvent this difficulty, most quantitative 

analyses of laboratory experiments are generally based 

on an assumed adsorption isotherm. In most situations the tracer or contaminant concentration is sufficiently small that a linear isotherm can be assumed. 

This is the condition on which the following analysis is 

based. Adamson (1960), however, discussed various 

findings of recent years in which the linear assumption 

does not hold, especially where high concentrations are 

reached. Because in both laboratory tracer studies and 

actual disposal systems the concentrations are kept at 

a minimum, it is believed that the adoption of a linear 

isotherm is a reasonable approximation.

Linear isotherms have been utilized by chemical 

engineers in the study of exchange processes. There is, 

of course, no proof that the same type of isotherm would 

be applicable for adsorption in a nonactivated bed. 

However, for want of better or more detailed information, the assumption of a nonlinear isotherm would 

only increase the mathematical complexity without 

furthering the systematic progress of the investigations.

ADSORPTION ISOTHERMS

A linear isotherm under isotropic exchange conditions 

may be generally classified as representing either a

HI
H2 FLUID MOVEMENT IN EARTH MATERIALS

reversible or irreversible reaction. The simplest situation is the linear isotherm which represents an irreversible system, as described by the expression

(1)

where N is the solid-phase concentration defined as the 

mass per unit volume of solid material, C is the liquidphase concentration defined as the mass per unit volume 

of liquid, t is the time variable, and k is the proportionality constant. Equation 1 is essentially Langmuir's 

isotherm for low concentrations and for equilibrium 

conditions. For complete discussion of Langmuir's 

isotherm, the reader is referred to Adamson (1960). 

It is noted that equation 1 describes a system in which 

the solid acts as a mathematical sink. That is, the 

amount transferred or the rate of change of solid 

concentration with time depends only on the liquid 

concentration. Physically this system cannot exist 

unless some method is available to remove all material 

transferred from the liquid to the solid phase. In 

terms of the overall mathematical treatment, however, 

inclusion of the terms represented in equation 1 does 

not add to the complexity of the problem.

More generally, isotherms of this nature may be 

expressed as

. -^r=F(O (2) * ot

where F(C] is a function of liquid concentration only 

and must be determined for each system considered.

In all situations, the medium tends to show a fixed 

capacity to adsorb a given substance which in turn is 

set by the level of the liquid concentration. That is, 

dN/dt becomes zero, indicating establishment of equilibrium between the liquid- and the solid-phase concentrations. An example of this type of isotherm is 

that used by Amundson (1948), mathematically represented by the equation

(3)

In equation 3, a is a constant and NQ is the saturation 

capacity of a unit volume of granular material. It is 

noted that equation 3, like equation 1, shows a straightline relation within the range of the porous medium's 

capacity to adsorb. The saturation capacity N0 is 

dependent on the concentration of the liquid phase and 

thus is assumed to be a known constant in any given 

analytical study.

Merriam and others (1952) and Hougen and Marshall (1947) indicated that the adsorption isotherm 

can be depicted by the expression

bN=k(C-mN) (4)

where m is a constant. Note that unlike equation 3, 

equation 4 indicates a possible negative rate of change 

of solid concentration. This indicates that the adsorption process is reversible in the sense that when mN 

^> C the rate of change is negative, hence providing for 

a reverse effect or diminution of the solid-phase concentration. In other words, in a continuous-injection 

system, a continuous adsorption and desorption process 

is possible.

The problems in which adsorption has been investigated extensively are those involving gas or liquid 

flow through activated porous media. There is general 

agreement that the dispersion under these conditions 

is so small that it may be neglected in describing the 

outflow-concentration profile. This assumption eliminates the second-order term in equation 5 and the 

mathematical treatment is thereby simplified to some 

degree. In a liquid-solid system, for the conditions 

imposed in waste-disposal problems, the relative 

magnitudes of the two processes are not known. 

It appears that, for the initial passage of the contaminant, the condition of dN/dt»D(&C/dx2) would be 

valid. However, under conditions of repeated injection, the available free energy may be reduced to a 

great extent by the preceding injection; hence, the 

relative magnitudes may be of the same order. The 

immediate significance of this relates to the predicting 

of the usefulness of any given analysis, in that the 

historical events may figure importantly in evaluating 

the fate of the contaminant.

In the following analysis the isotherm described by 

equation 4 is utilized. This assumption admittedly 

restricts the applicability of the resulting analytical 

solution, but our knowledge is too meager to specify 

these limitations. It is believed that, as more data 

become available, the applicability of the solution will 

become more clearly defined.

FIELD EQUATION

The actual description of mass transport through 

porous media is a matter of conjecture since there is no 

method available to measure all energy states of the 

system. The differential equation presented is thus 

based on a macroscopic concept so that the primary 

result gives a description of the liquid-concentration 

distribution at a given point in space for a given time. 

The mathematical model considered is that of flow 

through a column of semi-infinite extent. It is, thus, 

assumed that

(a) porous medium is homogeneous and isotropic, 

(6) flow field is parallel to the x axis (say), and
MATHEMATICS OF DISPERSION WITH LINEAR ADSORPTION ISOTHERM H3

(c) mass-transfer rate from liquid to solid is linear 

and can be expressed by equation 4 that is,

Assuming that the three components of mass transport are dispersion, convection, and adsorption, the law 

of conservation of mass may be stated as

(5)

where the relationship between 2V and C is given by 

equation 4 and the transport due to dispersion is described by an equation similar to Fick's law in a moving 

medium that is, the total transport is

.

x ~ x bz

The symbols appearing or inherent in equation 5 are 

defined as follows:

D: dispersion coefficient, 

u: average velocity or interstitial velocity, 

k, m: constants relating the adsorption isotherm, 

C: concentration of the liquid phase, 

N: concentration of the solid phase, 

x: space variable, and 

t: time.

The boundary conditions chosen are parallel to 

those of other investigators. Say, at x=0, the liquid 

concentration is maintained at some constant value, 

0 C0 . This condition depicts dispersion phenomena 

from a plane source discharging into a semi-infinite 

medium. Mathematically, these boundary conditions 

can be stated:

and lim C(x, 0 = 0for

z *a>

N(x, 0) = 0for z>

(6)

Before developing an analysis of equation 5, subject to 

conditions 6, it will be of interest to review other solutions obtained in the analysis of exchange phenomena.

REVIEW OF SOME PREVIOUS ANALYSES

Amundson (1948), in describing the mass transport, 

assumed that dispersion is small; hence, equation 5 

becomes

where a is the volume of voids in the medium. The 

assumed adsorption isotherm is given by equation 3. 

The boundary conditions are as specified in 6, except 

for the condition N(x,Q)=Ni(x). For this system, 

the solution obtained is

u ^x> l> ~

t

pO-z/u) -| 

ka I Co(ij)dij I

(if r* ^ j= f(t-x /u) =j ' 

f [No-Ni(&] d£ + exp ka\ C0 (rfdi, \-l 

l»Jo J L Jo J

exp

which is valid for Z> x/u. Because dispersion is 

neglected, it is noted that N(x,t)=Ni(x) for t < x/u, 

which means that no transfer from liquid to solid 

occurs until mass is transported by convection to the 

point of interest. Because the solution is given in 

terms of functions of the coordinates, there is no 

immediate applicability unless these functions are 

defined.

A special circumstance considered by Amundson is 

for the conditions dCy&«dN/& and #,(») = 

N(x, 0)=0. These assumptions may be interpreted to 

mean that the convective transport is large enough 

to balance the amount adsorbed; thus ~bCj~bt remains 

small. For this special circumstance, the solution 

reduces to

exp

Of specific interest is the analysis of Hougen and 

Marshall (1947), who considered the isotherm expressed by equation 4. However, the authors considered only the special circumstance where dispersion 

is negligible and dCydi«d2V/di. Thus, equation 

5 can be written, simply,

bC 1 bJV

"u at

The solution of this system subjected to the boundary 

conditions specified by 6 is

(7 fkx/u

7r = l exp ( kmt) exp ( 

Oo Jo

where J0 (z) is the Bessel function of the first kind of 

order zero. This equation has been calculated 

extensively and was discussed by Marshall and Pigford (1947).

Because exchange phenomena are of primary interest 

in chemical processes and the significant aspect of this 

application is limited to the condition where adsorption 

is of much greater magnitude than dispersion, scientific 

interest has tended to remain centered around the 

previously discussed situations. However, there appears to be one situation for which a solution of equation 

5 has been obtained. This is the most simple condition 

of a linear irreversible isotherm, which is given by 

equation 1. Analysis of this system is analogous to 

solving the heat-conduction equation with radiation 

at the boundary. The solution, as determined for 

example by the writer ("Dispersion in Porous Media,"
H4 FLUID MOVEMENT IN EARTH MATERIALS

£ = ut/x 

FIGURE 1. Concentration distribution for system with irreversible linear adsorption isotherm.

doctoral dissertation, Northwestern Univ., 1958), has 

the form

C 1 ,.. ,

-M) erfc

exp

l-.*j/) * ~| * /2VZ>U

ux~\ , r/,. ut 9n erfc I 1+ 2£>J L\ a;

where M= CJj V

The adsorption term ^ introot

duces a "holdup time" and sets an upper limit to C/CQ 

depending on the value of k. For example,

does not correspond to the point t as indicated by

u

solution of the dispersion equation when Ar=0. In 

addition, the slope of the effluent curve with respect to 

time is smaller and an upper limit other than <7/(70=l 

is reached because of the continual removal of substance from the system. The effect of the adsorption

4y/y»

term L> shown in figure 1 for the special case of 17=-=-=1.

Crank (1956), on the other hand, gave a solution of 

equation 5 for u=Q, using the isotherm depicted by 

equation 4. In his discussion he cited situations of 

diffusion in a plane sheet and in a cylinder. The

solutions are, however, too complex to be discussed in 

this paper. Interested persons are referred to Crank's 

book, in which a complete discussion of numerical 

computation and solution in graphical form is presented.

MATHEMATICAL ANALYSIS

Let us consider now the dispersion and adsorption 

phenomena utilizing equations 4 and 5, subject to conditions 6. There are various analytical methods that 

can be employed to obtain the solution of the differential 

equation. However, for this type of problem perhaps 

the simplest and most straightforward method is that 

employing the operational device called Laplace transforms. Because of its usefulness in analysis, various 

texts have been written on the theory and application 

of the Laplace transform. For discussion and application of complex variables and transform calculus, the 

reader is referred to McLachlan (1955), Churchill 

(1958), or Doetsch (1961). _

The Laplace transform C(x, p) is defined by the 

expression

/ CO

V(x,p)=\ eJo

Hence, substitution of the integral transform into equa-
MATHEMATICS OF DISPERSION WITH LINEAR ADSORPTION ISOTHERM H5

tions 4 and 5 gives the expressions

and

Hx*~u dx = (7)

(8)

It is noted that the transformation reduces the secondorder partial differential equation into a second-order 

ordinary differential equation and reduce^ the first-order 

equation to an algebraic equation.

In establishing equations 7 and 8, two of the boundary conditions, C(x, 0)=0 and N(x, 0)=0, have been 

utilized. The remaining two boundary conditions 

written in terms of G are

and

lim 17=0.

The simultaneous equations 7 and 8 may be readily 

solved^ by solving for N in equation 8 and substituting 

for N in equation 7. This process gives a single 

equation,

D n d*V dV A , * \-r-^ u T~=P ( 1+ rr~ )

dx2 dx * \ p+km/

The resulting equation is a linear second-order ordinary differential equation with constant coefficient for 

which the solution can be readily obtained. The complete solution is

Application of the boundary conditions specifies that 

since C( <» , p) =0, only the negative part of the argument 

is retained, and since C(Q,p~) = CQ/p} A=C0/p. Thus, 

the required solution of the transformed system is

r -p {& [- (9)

The initial step hi determining the inverse transform 

is to refer to any table of inverse Laplace transforms 

for example, Erdelyi and others (1954). If the required 

V does not appear in the tables, the inversion theorem is 

utilized which states that

C(x, f)=~ (10)

where r specifies a given path in the complex plane z. 

The choice of r is dependent on the singular points of 

the function given in equation 9 (McLachlan, 1955). 

The function involved does not, however, lend itself for 

ready determination of the type of singularity involved 

in the computation. Hence, the following mathematical 

manipulation is a process of simplification so that

732-758 64 2

various theorems in conjunction with a table of inverse 

transforms may be utilized to obtain the desired results. 

From any table of integration, it is noted that

A r v

» J°

Hence, equation 9 can be written in terms of an infinite 

integral to eliminate the radical term that is,

where a=ux/2D and ^2== _ 

Accordingly assuming C is continuous, substituting C 

into equation 10 leads to the expression

where p is replaced by z.

Let us consider only the complex part and denote this 

by the symbol F. Letting (z+km)=\, the complex 

part can be written

El __

g-kmt 

2W L exp [A*-

Rearranging the argument of the exponential term, the 

expression can be written

r a3ai ~\ i 

F=exp I kmt -TTJ (k krn) 5 .

f P\ (t «2 /32 \ i c?fPk*m~\ d\ 

Jr eXP L V "4FV + 4£*X J \-krn

Thus, the complex part of the function F is

dx

X km (11)

The nature of equation 11 makes further simplification possible by the use of the shift theorem (McLachlan, 

1955). The shift theorem specifies that if there exists a 

function defined as

where X(£) is continuous or piecewise continuous for 

t^>h, its transform is obtainable by the standard 

method. The value of the integral, however, is zero 

when £< h, which indicates physically a moving system 

or in this circumstance signifies that the concentration 

is zero until the contaminant arrives at the point hit.
H6 FLUID MOVEMENT IN EARTH MATERIALS

It is again noted that the Laplace transform is defined by

Keplacing (t) in the above expression by (t h) gives 

= f e-*'-v<t>(t-h)dt=ert f°

Jh Jk

= f°° e~^<}>(t

v 0

where U(t h) is a unit step function with properties 

U=0 for t<^h and C7= 1 for t^>h. The inversion of this 

function can be written symbolically as

L-le-rt$(p)=<t>(t-h} U(t-h).

Thus, if !j>(p) exists, the inverse transform of ^(t h) 

can readily be obtained since L~l ^>(p) = 4>(t) .

In this specific problem, the function for which the 

inverse must be determined is

exp

As previously stated, L~1 ^(p)~^(t); hence, the required computation can be written in operational form,

where a2=a2/32A:2m/4f .

Because the exact expression of $ does not appear in 

tables of inverse Laplace transforms, the function is 

rewritten

= exp (o2/p) + exp (o'/p).

Each term in the function 0 can then be directly obtained from the table of inverse Laplace transforms 

that is,

Ir1 exp (o2/p)/p=/o(2aVO 

and

L-U/(p+5)=exp(50.

Thus, the first term of 0(£) is I0 (2a-Ji). The second 

term, on the other hand, can be evaluated from the convolution or superposition theorem which states that if 

Lf(t)=J(p) and Lg(t) = H(p), then

= P/(r)<7(«-r)dT= P

Jo Jo

Hence, the second term can be shown to be

p km

Q2 /P p 

P ~"Jfl

The function <$>(t) thus can be written

t P e-* 

Jo

Further, letting a=a/%, where a=a2^2k2m/4: } the shift 

theorem requires that the inverse of equation 11 be 

written

n

] Jo

(12)

equation 12 is valid for £>62/£2 where &2=a2j82/4. 

Since the solution gives zero values for i<a2jS2/4^2, or

/#2 \ / £<j j^ j /^2, the particular solution of equations 4 and 5,

subject to conditions 6, becomes

n Opa /*

^=^=

Co YOT- JZ

jexp [- 

+exp (

exp

(13)

where a=ux/2D and 72 =/32a2/4=z2/4#.

Equation 13 cannot be simplified; however, it can be 

written in terms of a tabulated function J introduced by 

Goldstein (1953). A complete discussion of the various 

properties of the J-i unction was given by Goldstein. 

The function is defined by

=e-« fV< 

Jo

The function J(x, y) appears in a wide variety of problems and therefore no attempt is made to list its properties. The reader is referred to Luke (1962), who 

listed the more significant properties, or to the original 

work of Goldstein. Luke also listed some of the computed tables of the J-function appearing in various 

publications. To illustrate J(x, y), the function computed by the writer is presented in figure 2.

Noting that one of the elementary properties of this 

function is

J(r, y~)+J(y, T)-l=e-<'+»>/0 (2V^)

and utilizing the definition of the J-function, equation 

13 can be written

C_ = 2e« r 

Co vV J

(14)

where y=ky2/£2 and T=km(t y2/%2').

To determine whether the boundary conditions are 

satisfied, it is noted that

/(O, i/) = /(r,0)=e-'
MATHEMATICS OF DISPERSION WITH LINEAR ADSORPTION ISOTHERM

i. Gi

FIGUBE 2. Plot of J-function.

and

lim lim J(T, y) =

Substituting these known conditions, it can be seen 

that the boundary conditions for x 0 and t=0 are 

satisfied. Also when x ><*, C/C0 >0 and, for large 

times that is, r >Q° J(r, y) = l', hence, substituting 

leads to C/C0=1. Thus, all conditions are satisfied 

and by direct substitution, equations 4 and 5 can be 

shown to be satisfied.

Equation 14, although highly complex, cannot be 

reduced any further; however, numerical integration 

can be carried out if k and m can be approximated. No 

attempt is made to calculate this function in this paper, 

although it would be of interest to note two special 

situations for k=Q, m^Q and k^Q, m Q. First consider k=0, m^O. Equation 14 simply becomes

_cCo"

2v/5i

which is the solution for the circumstance of no adsorption that is, for

For m=0 and

7 _"

_/ 4Z)fc\ "

equation 14 reduces to 

C 2e« f 00 T a^v^ J ^_exp

2Vci

which is the solution for 

Furthermore, since

= e-^ erfc *- + e20 erfc

both expressions can be written in terms of the error 

function.

SPECIAL SITUATIONS

As mentioned previously, the orders of magnitude of 

the terms in equation 5 are not known in most adsorption systems. Thus, two circumstances are considered 

in the following pages, the first assuming that dispersion 

is small that is, Z>d2 (7/dz2 is small. Here equation 5 

reduces to

dC bC, c)2V_ ~ ~~ (15)
H8 FLUID MOVEMENT IN EARTH MATERIALS

Second is the circumstance where d(?/d£<CdAYd£, thus 

reducing equation 5 to

_ U _ ~ (16)

The mass transfer to the solid phase is again assumed to 

be expressed by equation 4 and the boundary conditions 

are those given in 6. The conditions under which these 

assumptions are valid are not known; however, these 

solutions may be useful in analyzing some laboratory 

models.

For both circumstances expressed by equations 15 and 

16, the method utilized is the Laplace transformation. 

Hence, applying the Laplace transform to equation 15, 

the subsidiary equations obtained are

u -r--\-pC-}-pN=

and

PN=k(C-mN)

#=(n

V) J

(17)

Equations 17 are now first-order differential equations 

with constant coefficient; the solution can be shown to be

u p -\-krn

From the initial condition C(x, 0) = <70 f or £>0, the 

constant A is again determined to be

A=Co/p.

Further, letting s=p+km and utilizing the inversion 

theorem, the solution can be expressed

-kmt-(k-km)- dz

Or,

dz

Accordingly, utilizing the shift theorem, the function 

for which the inverse is to be determined is

_k*mx , / 

e u ''/ (s-km).

The transformation is carried out in the same manner 

as that used in obtaining equation 14. The required 

solution is, thus,

m f ,-x\ kr t r IJfZ / ^\T I * I IT I n /'"' *' 1 / j ** 1 I V u) u iI0 2-\ km [ t ) 

I L V u V «/ J

-{-km exp j km (t ^

c

7T Co

for t^>x/u. The above relation written in terms of the 

previously described J-fimction is simply

7r=J(y, (18)

kx / x\ where y= and r=km (t ). Equation 18, in

y u \ «/ 

terms of a definite integral, can be written

r -km(t-~* 

*-> -i \

It is noted that the function obtained above is the same 

as that derived by Hougen and Marshall with the

replacement of f t J for t. This function has been 

computed and is presented in graphical form as figure 2.

3/ Further, ' note that for £«-, u' equation n 19 reduces to

~ = e-kx/u 

rt K > Co

which indicates that measurable traces of solute will 

appear at t=x/u provided k is small. In most experimental setups, the holdup time is much greater; thus 

the expression is generally not useful in the analysis of 

experimental data.

Let us consider now the second special circumstance 

in which dN/<)t^>dC/dt. Here the transformed or the 

subsidiary equations become

dC «

and

Solving for U gives

exp -

The inverse obtained in the same manner described 

previously is

?

+ kme *»"

where @=Dka2/u2.

In terms of the J-function, the preceding relation 

becomes

C_ =2e^ f

Co V^ Jo

exp [-!2 - (20)

where y=P/£2 and r=kmt. However, once again the 

function is not reducible and in its integral form can be 

written

~

Co

exp ^-

r

^o

(21)

DISCUSSION

To effectively predict the fate of a tracer or contaminant as it progresses through porous material, two 

phenomena must be predicted beforehand. These 

consist of dispersion due to departures from the flow
MATHEMATICS OF DISPERSION WITH LINEAR ADSORPTION ISOTHERM H9

predicted by use of Darcy's law and the loss of material 

through adsorption by the porous medium. The first 

effect can generally be predicted for laboratory conditions, because flow is controlled to fit the theoretical 

conditions postulated. The adsorption effect, on the 

other hand, cannot be controlled even under conditions 

met in the laboratory. It can only be said to be 

dependent on the tracer and the soil constituents.

Because of this unknown factor that inevitably 

appears in all types of studies involving tracers, the 

need for finding a near-perfect tracer is apparent. But 

discovery of this near-perfect tracer is yet to be made, 

although there appear to be a few elements that display 

only minor attraction to the silica sands commonly 

used in the laboratory. However, when natural soils 

are used as the porous medium, adsorption may play 

an important role in determining the distribution of 

tracer material at some point downstream. This then 

requires that the laboratory data be analyzed to estimate the rate at which the tracer or contaminant is 

transported from the liquid to the solid. Studies of 

exchange processes show that, in the low-liquid-concentration range, a good approximation is obtained by 

assuming a linear adsorption isotherm.

On the basis of these findings the preceding mathematical analyses were developed utilizing the linear 

isotherm described by equation 4. The solutions 

obtained are extensions of previous expressions which 

were developed and used to represent ion-exchange 

phenomena in fixed columns. However, the final solutions, equations 14, 18, and 20, are such that no real 

purpose would be accomplished by numerical computation unless realistic values of k and m in equation 4 

were available.

It is expected that, as some data indicating the 

range of k and m become available, computation of 

equation 14, 18, or 20 will be attempted. From the 

experimental and analytical standpoint, equation 18 

would be the most appropriate for obtaining an approximation of the constants k and m, provided extremely 

slow flow can be obtained in the experimental model.

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O