Dispersion of Soluble Matter in Flow through Granular MediaMark Jay Beran Citation: The Journal of Chemical Physics 27, 270 (1957); doi: 10.1063/1.1743685 View online: http://dx.doi.org/10.1063/1.1743685 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/27/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Dense annular flows of granular media AIP Conf. Proc. 1542, 1198 (2013); 10.1063/1.4812152 Unstable Kolmogorov flow in granular matter Chaos 19, 041106 (2009); 10.1063/1.3202616 Flow and jamming of sheared granular media AIP Conf. Proc. 1145, 31 (2009); 10.1063/1.3179923 Development of a general approach for predicting the pressure fields of unsteady gas flows throughgranular media J. Appl. Phys. 99, 093519 (2006); 10.1063/1.2197028 Effects of an External Circuit on the Dispersion of Soluble Matter in a Magnetohydrodynamic Channel Flow Phys. Fluids 14, 2776 (1971); 10.1063/1.1693408

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270 Y. NISHIJIMA AND G. OSTER

where

or

x In= f X nexp(-X2j2)dX,

-00

J.l3 -x 2 D(c)

-=1+----Dm J.l3 J.l3

1--X+-X3 2 6

(5)

Equation (5) is applicable to all concentrations where the third and higher terms of Eq. (1) are negligible. If, however, higher moments are important as for poly­dispersed polymer systems or for extremely skewed curves, e.g., for concentrated glycerol solutions,2 further terms must be included in these expressions. In practice, the calculations of higher moments than the third from experimental data is subject to considerable error.

THE LINEAR CASE

For sufficiently dilute solutions the concentration de­pendence of the diffusion coefficient can be represented by a linear relation in concentration. The slope, ko, can

THE JOURNAL OF CHEMICAL PHYSICS

be evaluated from Eq. (5). Differentiating Eq. (5) with respect to X, we obtain

J.l3 (J.l3)2 ----X3

d [D(e)] 2 6

dX Dm = [ J.l3 J.l3]2 1--X+-X3

2 6

(6)

But

d:[:~t~o :3 (7)

Hence

ko= [dD(e)]/Do=_1 Dm~, (8) de 2Co Do Ho

where Co is the original concentration, Ho is the height of the normalized curve at X = 0, and Do = Dm(1- J.l3/ 4H 0). Therefore, the linear dependence is obtained from the following three parameters: (1) the third moment of the normalized curve, (2) the height of the curve at the position of the original boundary, and (3) the average diffusion coefficient Dm obtained by the moment method.

VOLUME 27. NUMBER 1 JULY. 1957

Dispersion of Soluble Matter in Flow through Granular Media*

MARK JAY BERANt Harvard University, Cambridge, Massachusetts

(Received November 26, 1956)

This paper treats the problem of dispersion of soluble matter within a one dimensional tube filled with granular media. It is supposed that the solvent is flowing slowly through the tube with mean velocity I), and it is desired to find the future concentration pattern of the soluble matter. It is found that after a long time an initially thin slug of soluble matter is normally distributed about the position vt and that its standard devia­tion is proportional to tl. It is further shown when this effect need not be considered in Ion Exchange prob­lems.

SECTION 1

T HIS paper treats the problem of dispersion of soluble matter within a solvent flowing slowly

through a tube filled with granular media. The assumption is often made in such a situation (pictured in Fig. 1) that if there is no adsorption by the granular media, all soluble matter put into the tube at position x=O at time t=O will be at the same downstream position Xl, time tl later, V= XI/t! being the average velocity of the solvent flowing through the tube. Such an assumption is of limited validity, and it is the purpose here to show that the concentration pattern

* The majority of the work in this article was submitted to Harvard University in partial fulfillment of the degree of Doctor of Philosophy in Engineering.

t Now at Radio Corporation of America, Waltham, Mass­acllUsetts.

of a slug of soluble matter put into the tube initially at x=O becomes instead, when t is large, normally distrib­uted about the point Xl, time tllater, and has a standard deviation proportional to the square root of time. In addition, however, since this problem is of interest in ion exchange problems, a rough criterion will be presented to show when the assumption of constant velocity may be used in the theory of ion exchange.

The first thing that one must realize when considering flow through granular media is that the velocity distribution within the solvent is far from constant; the velocity at the grain boundaries being zero and between the grains often at least twice the average velocity. Further, the grains are so irregularly placed and shaped, since granular media such as common sand is considered here, that the velocity distribution about one set of grains is essentially uncorrelated with

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FLOW THROUGH GRANULAR MEDIA 271

the distribution about grains just a few grain diameters away. Such a velocity distribution can cause consider­able dispersion of the soluble matter within a solvent . ' In much the same way as a gambler can loose a consider-able amount of money tossing coins despite the fact that the probability of winning or losing on each coin toss is 0.5.

SECTION 2

Let us suppose that initially a thin slug of soluble matter is put into a tube (the diameter of which is many, many times the average grain diameter) which is filled with granular material and through which the solvent is flowing at a constant average velocity ii. Initially the slug is assumed to have negligible thickness in the flow direction and to be equally distributed over the tube cross-section. It is noted again that the grains have only statistical properties in common, their individual shapes being as different as different grains of beach sand. One now asks what is the concen­tration of soluble matter, averaged over the tube cross-section, a distance x downstream, at time t later. It is assumed for the moment that molecular diffusion is negligible.

A molecule of soluble matter beginning at some radius r, angle (} will follow a tortuous path through the granular media, its velocity at any instant dependent upon the particular grain configuration through which the solvent containing it is passing. In general as it traverses the tube, it will, at one time or another, travel longitudinally (in the direction of the tube axis) with almost every velocity that is present within the tube. In fact, if one examines the record of longitudinal velocity along any particular stream path versus time, he should find the velocity pattern so irregular and so lacking in correlation between times large compared to d/v (where d is an average grain diameter) that it would be expected that the pattern could profitably be

FIG. 1. Coordinate system used in de­scribing :flow of sol­uble matter through an infinite tube filled with granular media.

~----x.o

x

analyzed on a statistical basis. Taken as a mathematical curve devoid of its physical context this record has all the attributes of a random function.

On a closer look, however, a statistical approach would appear ill founded since there should be no lack of information about a molecule's stream path through granular media no matter how random the path may appear. In any.given situation when the dispersion is measured, the grain packing is assumed to be fixed, even though it is random in the sense that if the tube were filled again the grain configuration would be different. To solve deterministically the problem it is only necessary to be clever enough to arrive at a mathematical expression for the complicated grain boundary condition and with this knowledge solve the equation ac/at+v·vc=O (it is remembered that molecular diffusion has been neglected) where C is the soluble matter concentration at any point (x, y, z), V is the vector fluid velocity and V is the gradient operator. V of course is first determined by using the same grain boundary conditions in conjunction with the Navier-Stokes Equations. Naturally, no one has proven clever enough to solve the problem in this manner, but in principle the concentration pattern that is sought is completely deterministic.

Yet despite the above argument, a statistical approach may be used to solve the problem. It is first necessary to remember that we are dealing with not one, but an enormous number of soluble matter molecules and that though the stream path of every molecule must theoret­ically be determined by a knowledge of the entire grain structure within the tube, the fluid velocity of a molecule is determined principally by the grain con­figuration in the immediate neighborhood of its stream path and is not affected appreciably by the grain configuration a number of grains away. The grains are so irregularly shaped and placed that the stream paths of molecules beginning at time t = 0 just a grain diameter apart are very different in a short time. So different in fact that it is a very reasonable assumption to suppose that one molecule proceeds through the tube with a velocity distribution essentially uncorrelated with the other molecule.

The only realistic way that the velocity patterns of two molecules, beginning initially say a grain diameter apart at t=O, can be compared is by comparing the average properties associated with the patterns. Since the distribution is considered after a long time has passed, each molecule has passed by a sufficient number of grains to allow for homogeneity, and such quantities as the average velocity and the autocorrelation function of velocity are the same for each molecule. One molecule may travel quickly at first and then slowly, another slowly and then quickly, but the average or statistical quantities are the same.

Thus we may envision in our mind an enormous number of soluble matter molecules traveling down the

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272 MARK JAY BERAN

tube, each traveling with fluid velocities essentially uncorrelated to each other except in a statistical sense. To be sure, if two molecules begin their motion just a few molecular diameters apart their velocity patterns will be very closely correlated even after a long time, but if one chooses a single molecule's velocity pattern and asks for the ratio of strongly correlated molecules to essentially uncorrelated molecules (using the words in a loose sense), he should find the ratio «1, so that the above analogy is a sound one.

Using this picture of single molecules traveling down­stream one may theoretically find the concentration distribution of soluble matter along the tube axis after a time t by simply counting the number of molecules occurring at every x position at the time. It is more convenient, however, to change the emphasis from many molecules to a single molecule. To do this it is noted that if the future position of a single molecule is asked and one is ignorant of its initial r, () coordinates, its having equal probability of being anywhere in the tube cross-section, the probability of its being at a position x downstream time t later is the same as the ratio of actual molecules found at position x to the total number of molecules in the initial thin slug. The probability distribution of the single molecule and the concentration distribution of an enormous number of molecules thus differ by only a constant factor.

If then one considers a single molecule which has equal probability of being initially anywhere in the tube cross-section, what can be said about its future downstream travel? It has been assumed that no matter which path it chooses the average (statistical) properties of its velocity pattern will be the same, and it will now be shown that only average properties are necessary to predict its probable downstream position. It is only necessary to know these average properties and suppose the molecule to be following a path along which no more is known than these properties.

There are many functions which describe useful properties of the pattern, but here only one function is necessary. This function, which is denoted as P(v,t/vo,to) Xdv is defined as the probability that the molecule has a velocity v at time t if it had a velocity Vo at time to. Two facts are of prime importance about P(v,t/vo,to).

1. P(v,t/vo,to) depends only upon t-to and not upon t and to individually since the granular media in the tube is uniformly packed.

2. P(v,t/vo,to)d~P(v)dv in a comparatively short time due to the irregularity of the grain structure. That is, after a short time the molecule's velocity at time t is essentially independent of its velocity at time to. Theoretically P(v,t/vo,to)dv~P(v)dv only as t~oo, but practically it is virtually true in a finite time. For simplicity it will be assumed that there is some time T, after which P(v,t/vo,to)dv=P(v)dv.1

1 The mathematical complications that arise here if this as­sumption is not made are prodigious and as yet not fully worked

From the function P(v) it is also possible to obtain the velocity moments such as ii, the average velocity. They are defined as

(vm) = foo vmP(v)dv. -00

Now, oddly enough, knowing only that

1. P(v,t/vo,to) is dependent only upon t-to, 2. P(v,t/vo,to)dv=P(v)dv for t-to"?:.r, 3. ('IfJ)< 00,

it is possible to predict the probability distribution of the molecule as a function of x (the distance along the tube axis) for large t. (It is remembered that this distribution differs from the future concentration distribution of the soluble matter slug by only a constant.)

A central limit theorem in probability theory states that if one seeks the probability distribution of the variable x= fotv(t)dt where vet) has associated with it a probability function P(v,t/vo,to) as described in the foregoing, then P(x,t)dx the probability that x be at a position between x and x+dx at time t is:

1 [- (x- vt)2] P(x,t)=--exp ,

(41rkt)! 4kt

where ii is the average velocity and k is some constant. The central limit theorem used is a simple extension to the continuous case of a theorem proved by Hoeffding and Robbins in 1948.2

It has thus been shown that if a thin slug of soluble matter is placed in a tube filled with granular media, after a long time the concentration of soluble matter averaged over the tube cross-section has the form

const [- (x- iit)2] C(x,t)=--exp .

(41rkt)! 4kt

SECTION 3

The preceding result was obtained in the absence of molecular diffusion, and it is next asked what effect has molecular diffusion.

In a word, though it may be quantitatively very important, it has only one effect on the foregoing solution-it changes the proportionality constant in the standard deviation. The distribution is still normal with mean vt. To see this, let us review for a moment the mechanism by which the soluble matter in the solvent is carried through the tube. The principal mechanism for most average solvent velocities is usually by convection, the mechanism discussed above.

out by the author; although it is believed the same result can be proven with a weaker assumption such as exponential decay of the correlation function.

2 W. Hoeffding and H. Robbins, Duke Math. J. 15, 773-780 (1948).

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FLOW THROUGH GRANULAR MEDIA 273

That is, the soluble matter dispersion is due only to the fluid motion of the solvent. In addition, however, superimposed on this is the molecular diffusion of the soluble matter resulting from the random heat motion of the molecules.

In flow through granular media, molecular diffusion has two effects, which, fortunately, may be super­imposed because of the irregular pattern of grains. First it has the dispersing effect it would have if the solvent remained stationary within the tube, and secondly it has the effect of changing a soluble matter molecule's downstream velocity by moving it to a different stream path. The former effect may be separated from the latter since the grains present so irregular a boundary condition that there is essentially no correlation between the molecular diffusion and the particular position in the tube that the soluble matter happens to be in; on the average the boundary con­ditions affecting molecular diffusion are the same despite the fluid motion. Further it can be shown that due to this effect alone the soluble matter would be normally distributed with zero mean, and standard deviation equal to (2Dt)i, D being the molecular diffusion coefficient of the soluble matter in the solvent.

The later effect presents a much more difficult problem quantitatively but can also be easily seen conceptually. It is remembered that eventually the above problem was reduced to one of considering the distribution P(v,t/vo,to)dv since the only real information known was the probability a molecule would have velocity v at time t if it had velocity Vo at time to. Molecular diffusion does not change this. One may still associate a distribution with the molecule's motion say P' (v,t/vo,to)dv which has the same property as P(v,t/vo,to). Now the particle may change its velocity because it may change its streampath in addition to its changing velocity along a given streampath, but the overall effect is the same. The above central limit theorem may still be applied, only the standard deviation is different.

Thus the effect of molecular diffusion may be simply included, since alone it yields a normal distribu­tion with the standard deviation equal to (2Dt)t and in combination with convection it yields a normal distribution with mean vt and standard deviation proportional to tl. The sum of two independent normal distributions is itself normal, and therefore the concen­tration distribution of a slug of soluble matter initially put into a tube at position x=o is normally distributed about the point vt, time t later, with standard deviation (]" having the form (]"2= 2k't where k' is a constant although different from k.

SECTION 4

The remaining job is now to determine k or k'. In this article principal emphasis will be on the determina­tion of k, since it is only in this case that experimental evidence is available to confirm the theory. However,

the form of k' will be derived when the parameter H=vd/D«1, that is when molecular diffusion dom­inates the dispersion. It can be shown by mathematical manipulation too detailed to reproduce hereS that the standard deviation (]"= (2kt)l has the form:

(]"2=2t i'" dt{ i: J UOUtP(u"t/uo,to)

XP(Uo,to)dUOdUt}

where

The double integral expression is the autocorrelation function and for simplicity will be denoted by R(uo,ut). By an ergodic theorem this function may be obtained from a time record of the velocity of a particle as it traverses the tube; an equivalent form being

lim 1 T

R(uo,Ut)=---f u(s)u(s+t-to)ds.4

T-J.oo T ° When t= to, R(uo,uo) =(]".2 the variance of the velocity

distribution and R(uo,ut) may be written as (]".2p(UO,Ut). Further the expression for fo"'p(uo,ut)dt may be nondimensionalized by letting t=ds/v where d is an average particle diameter, v the solvent velocity and hence s the nondimensionalized time. Thus,

2td(]"V2 f'" (]"2=___ p(uo,u.)ds.

v ° It is next noted that since throughout the paper slow

laminar flow has been assumed the Navier-Stokes equation may be approximated by the equations 1/ eiJp/iJxi= IIV2Vi which are linear. In terms of the variables Yi=Xi/d and Ui=V;jV the solutions are independent of d and V. Thus, if the geometry remains fixed and one only increases or decreases the average grain size, then (]".2/v2=a2, a being a constant, and a and fo"'p(uo,u.)ds are independent of v and d.

Simple algebra then yields (]"2= 2vdtc where c=a2 fo"'p X (uo,u.)ds and hence k= vdc.

The equation k=vdc has been verified experimentally using 20-30 Ottawa sand with the Reynolds number from 0.1 to 2.0. c was found to be approximately 15.5

3 "Dispersion of soluble matter in slowly moving fluids," a thesis presented by Mark Jay Beran to The Division of Applied Science in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the subject of Engineering, Harvard University, Cambridge, Massachusetts, May 1955.

4 See, for example, Norbert Wiener, Extrapolation, Interpolation, and Smoothing oj Stationary Time Series (Technology Press, MIT, Cambridge, Massachusetts, 1950), p. 15, and subsequent pages.

6 See reference 3.

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274 MARK JAY BERAN

In the case when H«l,

U2=2tU.2 f'" p'(uo,ut)dt+2Dt, o

the former term being derived by an identical argument as above. However, here the nondimensionalized time s' =D/d2t is used since when H«l, the magnitude of the convection should have negligible influence on the correlation function p' (uo,u 8). Again .nondimensionaliz­ing the linearized Navier-Stokes equations

where

is a constant independent of V, d, and D and must be determined experimentally.

When H"" 1 and both convection and molecular diffusion influence P'(uo,Ut), no simple procedure appears available and for want of anything better to do U2+U'2 may be added harmonically.

SECTION 5

To show how the previous work affects ion exchange studies, two simple examples will be presented. First, suppose that soluble matter is continually introduced at the point x= 0 (instead of the previous boundary condition of a thin slug of soluble matter at time t=O) and as it flows through the tube it is adsorbed by the granular media at a rate proportional to the amount of material remaining in solution, i.e., ac/at= -khC where kh is an adsorption constant. Thus if the solvent were flowing with a constant velocity at all points in the granular media the concentration at distance x downstream would be C=Coe-khtz, where tx=x/v.

If the soluble matter now disperses according to a normal distribution

1 [-(Vtx -Vt)2] ---exp , (4dt)! 4kt

the concentration at the point vtx downstream now has the value

'" 1 - (Vtx - Vt)2 c=f e-kht-exp[ J

o (47rkt)t 4kt'

where the upper value of 00 has been put on the integral by assuming the process has been going on for some time. This integral may be evaluated and yields

2V2t", exp(-)

Cov 4k C=------

(4k)! (ij2 )! -+kh 4k

When khk/fl<.<l this equation reduces to C=Coe-khtx

while if kh =ij2/4k the formula becomes

C= {~!o exp(O.17khtx) }exp ( -khtx)

which may often be of a different order of magnitude. Thus it is seen that for the effects of velocity disper­

sion to be unimportant khk/v2 must be «1. While the result is certainly oversimplified it does suggest that if in any adsorption problem a time constant similar to l/kh can be determined and if k can be estimated, a reasonable criterion is available for knowing when the effects studied in this paper are unimportant in ion exchange problems.

As a second example suppose the breakthrough curve in an ion exchange column is available and it is desired to know whether the dispersion discussed in this paper has caused serious distortion.

To solve this problem it is only necessary to compare this curve with the concentration curve that would occur if soluble matter were continually introduced at x= 0 and the bed was considered inert. In this case the concentration curve would have the form6

C [(X-vt)] -=!-!erf m --Co kit!

(x-Vt<O)

C [(X-vt)] -= !+!erf m --Co kit!

(x-vt>O)

and the distance between the point where C/Co=O.9 and C/Co=O.l is 3.62k!t!.

If instead of distance one asks the time between these two concentrations, which will be denoted by te, as one would in a breakthrough curve, it is just this expression divided by v, Le., te= 3.62k!t! Iv.

Next measure the breakthrough curve for the time between C/Co=O.1 and C/Co=O.9. Denote this by tB. If tcltB«l, the breakthrough curve should be undistorted by dispersion.

ACKNOWLEDGMENT

The author wishes to thank Professor Harold Thomas of Harvard University for his help in formulat­ing and solving the problem treated in this paper.

6 G. I. Taylor, Proc. Roy. Soc. (London) A219, 186-203 (1953).

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