On the dispersion of a solute in a fluid flowing through a tube

8/14/2019 On the dispersion of a solute in a fluid flowing through a tube

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Oxidation of organic szclphides. V IIn this case, such complexing will be enhanced by the ready electron availability

a t the sulphur atoms.The author is indebted to Dr D. Barnard for helpful discussions and to Dr L.

Bateman for assistance in the preparation of this paper. The work described formspart of a program of research undertaken by the Board of the British RubberProducers' Research Association.

Barnard, D. 1956 J. Chem. Soc . p. 489. (part V).Barnard, D. Hargrave, K. R. 1952 A n a l y t . c h i m . a c t a 6 23Barnard, D., Hargrave, K. R. Higgins, G. 1956 J. Chem. Soc . (in the Press).Bateman, L. Cunneen, J. 1955 J . Che m. Soc. p. 1596 (pa rt 111 .Bateman, L. Hargrave, K. R. 1 9 5 4 a Proc . Roy . Soc . A 224 389 (part I .Bateman, L. Hargrave, K. R. 1 g j 4 b Proc . Roy . Soc . A 224 399 (part 11 .Bateman, L. Shipley, F. W. 1955 J . Chem . Soc . p. 1996 (part IV).Boozer, C. E. Hammond, G. S. 1 9 5 4 J . A m e r . C h e m . S o c. 76 3861.

On the dispersion of a solute in a fluid flowingth rough a t u b eDepartment of Chemical Engineering, University of Minnesota

(Communicated by Sir Geoffrey Taylor, F.R.8.-Received 3 September 1955)Sir Geoffrey Taylor has recently discussed the dispersion of a solute under the simultaneousaction of molecular diffusion and variation of the velocity of the solvent. A new basis forhis analysis is presented here which removes the restrictions imposed on some of the para-meters a t the expense of describing the distribution of solute in terms of its moments in thedirection of flow. It is shown that the rate of growth of the variance is proportional to thesum of the molecular diffusion coefficient,D, and the Taylor diffusion coefficient K U ~ U ~ / Dwhere U is the mean velocity and a is a dimension characteristic of the cross-section of thetube. An expression for K is given in the most general case, and it is shown that a finitedistribution of solute tends to become normally distributed.

I n three recent papers, Sir Geoffrey Taylor has discussed the dispersion of solublematter in a fluid flowing in a straight tube (Taylor 1953, 1954a,b . In the first ofthese, viscous flow through a tube of circular cross-section was considered, andit was found that the solute was dispersed about a point moving with the meanvelocity of flow, U with an apparent diffusion coefficient a2U2/48D.Here D is themolecular diffusion coefficient and a the radius of the tube. In the last of thesepapers, Taylor showed that the conditions under which this analysis was validcould be expressed as 4LIA B Ua/D 6.9, where L is the length over which appre-ciable changes in concentration occur.

5 2


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8 R. rist might be hoped that so elegant a result should have some meaning when these

restrictions are removed, and it is possible to obtain this by fixing attention onthe movement of the centre of gravity of the distribution of solute and the growthof its higher moments. These may be studied in some detail and give a useful pictureof the dispersion under the most general conditions.

2 THEGENERAL EQUATIONS O F DIFFUSION AND FLOW I N STRAIGHT TUBEConsider an infinite tube whose axis is parallel to the axis Ox of a rectangular

co-ordinate system Oxyz. Let S denote the domain occupied by the interior of thetube in the plane Oyx, and let s be its area and the curve I its perimeter. I n steadyuniform flow the velocity u is everywhere in the direction Ox and is a function of

and given by U(Y,) = up X(Y,)), (1)where U is the mean velocity and defines the velocity relative to the mean. If

here s no slip at the wall of the tube = 1 on I?.Let C(x,y, x, t denote the concentration of solute a t the point x,y, z and time t

and let D (y,) be its diffusion coefficient. The function defines the variation ofthe diffusion coefficient and D s its mean value over the cross-section of the tube.The equation governing C is thus

t is convenient to take an origin moving with the mean speed of the stream andto reduce the variables to dimensionless form. Let a be a dimension characteristicof the cross-section S and let

= X W l a ,

then the equation for C becomes

The conditions to be applied to the solution areQ( 7, c, 0 = Q O ( E 3 r > c ) ,

where a a v denotes differentiation along the normal to I? and sufficient conditionson the behaviour of C as c a et


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Dispersion of solute n uid owing through tube 9be the pth moment of the distribution of solute in the filament through 7, a t timet and the pth moment of th e distribution of solute in the tube. The condition to beimposed on C as E- -t. o is thus that these moments should exist and be finite,a condition fulfilled if the solute is originally contained in a finite length of the tube.Multiplying equation (4)by .. p and integrating with respect to .. from o o co

and the conditions (5) and (6) become

If equation (9) is averaged over the cross-section, the use of Green s theorem andthe condition (11) reduces i t to

(where, as in equation (S), a bar denotes the average over(10) becomes mp(0)= mpo.

The equations (9) subject to conditions (10) and (11)for p

(12)8 ) and the condition

(13)= 0, 1,2, ... now form

a sequence of inhomogeneous equations which can be solved for the moments c .In principle these might be solved to a sufficiently high value ofp for the distributionto be constructed to any degree of accuracy, but a very useful picture of the progressof the dispersion can be obtained from the first three or four moments. Since i twill be shown that the distribution ultimately tends to normality the first twomoments are ultimately sufficient to describe the distribution.

3. THETUBE OF CIRCULAR CROSS-SECTIONBefore proceeding with the discussion of the general case, it is interesting t o

consider the case of a tube of circular cross-section and to note how the presenttreatment agrees with Taylor s. In this case we take Ox to be the axis and to be theradius of the tube and transform to polar co-ordinates

r=pcos0 , {=ps ino .In the case of viscous flow = (1 2p2), = 1 and D is the molecular diffusioncoefficient.

and of this equation we seek solutions of period 27~n 8 and satisfyingC ~ P > = c p O ~ 7 e), (I5)


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7 R. ArisEquation (12) becomes

I n particular, dm,/dr = 0 so that m, is a constant which we may take as unity. Thisof course merely states that the total quantity of solute is constant.

For p = 0, equations (14) and (16) giveCGco(p,0,7) = 1+ C {A; cos me+B; sin m0) J,(a,,p) exp &,r) ,m=O n l (18)

where a is the nth zero of dJm(p)/dp. he constants A and B are chosen to satisfyequation (15), i.e.

Ln cos me; j0 Ofo sin m s ~ ~ m ( a ~ )o0ip, ) dp, (19)the suffixes m and n to a being understood. (Itwill generally cause no confusion tosuppress the suffixesmn on a, Ap and Bp.)

For = 1 equation (17) gives

In this integral the constant term vanishes since the mean value of (1 2p2) s zeroand the terms with m 0 vanish by integration round the circle. Hence

Thus as a m,/dr -+ 0 and the centre of gravity ultimately moves with themean speed of the stream. Choosing the origin in the original plane of the centreof gravity, m,, = 0 and

ml = Sp A a Jo(ao,)1 xp :, 7)).n = l

(20)Thus the centre of gravity ultimately moves to a position

relative to its original position. The actual distance moved is proportional toa2U/D, the constant of proportionality depending on the initial distribution ofsolute. If the mean speed of the stream is being measured by the time taken forthe centre of gravity of the distribution to move between two points there will bea slight error, though in practice it will usually be negligible.

For a more detailed picture of the variation in positionof centre of gravity acrossthe tube, equation (14) must be solved. For p = 1 this becomes


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Dispersion of a solute n a uid owing through a tube 7 1of which the complete solution consists of three parts: (i) he term in the particularintegral arising from the constant part of c,; (ii) the rest of the particular integralwhich may be written

pZ(AO osm0+B0 in me) (p) xp 2r),with satisfying the equation

and (iii) the complementary functionX A1cosm6+B1sinm0)J,(xp) exp ( %),

with A1 and B chosen to fit the initial value cl0(p,0).The first part is independent of r and 0, being pp2(1 +p2),and choosing

A: so that mlo 0 the expression for c1 isc1 ( 2+4p4)+pZ(AOosm0+B0 inm0) (p)exp ( %)

ZA;, ,, +X A1cos m0+B1sinm0) J,(xp) exp ( ). (23)Thus ultimately the centre of gravity is distributed across the tube according tothe formula c, m,, +&p(& 2 +*p4)* (24)t may readily be seen by averaging the equation for that the form given for

m,, in equation (23) agrees with that previously derived in equation (21). t isinteresting that the ultimate distribution of c is independent of 0; physically thisis because circumferential diffusion brings a molecule into a stream of precisely thesame speed and so contributes nothing to the movement of the centre of gravity.

In Taylor s analysis (1953, 19546) an expression is derived for the variationof C withp when aC,/ax is constant. Equation (6)of Taylor (1954 may be writtenin the notation of this paper

i p ( & p2 + *p4),which with aC,/a[ constant means that the distance from any plane to the pointa t which the concentration C C,,, the mean concentration in the plane, is given bythe expression on the right-hand side of (25), .e. this is the shape of the surfacesof constant concentration. t follows that the centres of gravity would then bedistributed on the same surface. Equation (24)shows that this last statement istrue even though aC,/at is not constant.

Substituting the value of c, in equation (17)with 2 and neglecting termswhich tend to zero we have

Thus the rate of growth of the variance rapidly becomes constant andm2- 1+p2/48) +a constant

which becomes negligible by comparison.


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7 R risIf V is the variance of the distribution of solute about the moving origin, i.e.

V = /Isdy dz/IW s Ut)2C z,y, z, t de,- mthenIt is not unreasonable to use the left-hand side of this equation as the definition ofthe effective diffusion constant K, the more so as it will be shown that any dis-tribution tends to normality. With this definition K is the sum of the moleculardiffusion coefficient, D, and the apparent diffusion coefficient k = a2U2/48D,whichwas discovered by Taylor in his first paper (Taylor 1953, equation (25)). Equation(26),however, is true without any restriction on the value of p, or on the distributionof solute. The constant 1/48 is a function of the profile of flow, and for so-calledpiston flow with ~ this constant is zero and K = D as it should.

In the full expression for c2 the terms which do not vanish as r-tco are m2(r),a constant function of p and a constant depending on the initial value of c,. Theconstant function of p must satisfy the equation

whereThis is precisely the polynomial obtained by Taylor under the assumption thataQ,/ag is not quite constant but that the second derivative a2C,/aE2 is (Taylor1954b equation (24)).

Thus, apart from a constant, c2 s ultimately given byc2 2(1+p2/48) *pnzlm(- 6+p2 p4)+g(p)

and

Neglecting constant terms which become negligible in comparison to the dominantterm proportional to r this equation gives

m3 ml, m2 7p3r/210. (27)Apart from a term m which is a constant, the expression on the left-hand side of(27) s the third moment about the mean m, so that the absolute skewnessP = r n~ /m

which tends to zero as r -+co. Thus any distribution of solute tends to become moresymmetrical.An examination of the equation for c, shows that the dominant terms will beproportional to r and will be m3(r)and the product of r and a function of p. Ingeneral,


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Dispersion o a solute in a uid owing through a tube 73where M,, and .MZn+,are even and odd polynomials in p of degree 2n and 2n+ 1respectively and

C n = X 2 n T n +G z ~ ( P )'-', (%+I M 2 n + l T n +G2n+l(~)n, 29)where the mean values of the functions G2,(p) and G,,+,(p) are zero. This remarkmay be used to show tha t the distribution tends to normality but the proof of thiswill be left for the general case.

The case of a distribution of solute initially constant over any cross-section ofthe tube is particularly simple forA BO = 0 and so by (21)m,, 0. When T 0,c, is independent of p and so A1 B1 0 for all m 0 and

andInserting this value in the equation for m, and integrating

m, m,, +2(1+p2/48) + 128p2Ea01,8{1 xp - a$ ,~ ) }- m,, +2(1+pa/48) +p2/360.The smallness of the last term may be judged by considering the case p2 48. Thenm2-m,, (p2/24)7+ 1/15),so that this last term is equivalent to an additionaltime of a2/15D.

The case of an instantaneous point source a t p p,, 0 = 0 is of interest. All themoments are initially zero and coo 0 for p p,, 0 0 and m, 1. Then

c 1+2 c ( l - )-I -a2? cosmOJ,,(ap) J,(apo)/(Jm(~)}2, (31)The dependence of m,, on the distance of the point source from the axis is the sameas the dependence of the centre of gravity of the solute in any filament on itsdistance from the axis.

5. THEGENERAL CASEThe general case of a tube of arbitrary cross-section flow profile and variation

of diffusion coefficient for which the equations were set up in $1,may be consideredin a similar way. Again m, must be a constant which may be taken as unity and


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7 R. risUnder very general conditions on F and @ there will exist a positive increasing

sequence of eigenvalues A, and a complete set of ortho-normal eigenfunctions v,satisfying the equation V @Vv,) A, vn = (34)in nd the boundary conditions

on I?. Any function satisfying the boundary conditions and with continuousderivatives up to the second order can be expanded in a series of these functions(see, for example, Courant & Hilbert 1937 . If the constants A, are such that

then the solution of the equations (33) is

As before, the density of solute in any streamline rapidly becomes uniform acrossthe tube, the relaxation time being of the order of AT , where A, is the smallest ofthe eigenvalues.

Inserting this value of c in equation (12) with p 1 it is again evident that thecentre of gravity ultimately moves with the mean speed of the stream. Its finalposition relative to a moving origin originally at the centre of gravity is

where the bar denotes an average over the cross-section of the tube.Apart from mlm and terms which vanish as r-tco, c1 will contain a constant

function of 7 and whose mean value is zero. This function arises from the constantterm 1 n c, when this is inserted in equation (9) with p = 1and so must satisfy theequations

a4- 0 (on r).a v iThen cl-m,, +p (r, C for all other terms will vanish at least as rapidly ase xp h , ~ ) .

From equation (12) with = 2

where again the vanishing terms have been neglected. But by definition 1 and= K,a pure number dependent only on the geometry of the cross-section andthe functions - and X. Thus again defining the effective diffusion constant K as

one-half the rate of growth of the variance


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ispersionof a solute in a uid owing through a tube 7 5showing how a Taylor diffusion coefficient can be found in the general case. By- reen's theorem and equations (39)we have = = V(@O$)= @(V'C )~.

Consideration of the dominant terms in the successive moments again shows thatc,, and c,,,, are of the form

where the mean values of the functions G are zero.Substituting c,,+, in equation (9) with p = 2n+ 1 and neglecting all except the

where has already been defined by the equations (39). Substituting this value ofG,,,, in equation (12) with p = (2n+2) and again neglecting all save the dominantterms dn* t12= 2 n + 2 ) 2 n + l ) ~ ~ , ~ r ~ + 2 n + 2 ) 2 n + l ) ~ l ~ l ~ ~ ~ ~ ~ ~drand the dominant term of m,,,, is

It follows that as r - o ,

These are the relations which exist between the moments of the normal distributionand in this sense the mean concentration is ultimately distributed about a pointmoving with the mean speed of the stream according to the normal law of error,the variance being 2 ( 1 + ~ p , ). It should be noted tha t the approach to normalityis as 7-l a very much slower process than the vanishing of terms in the expressionsfor the moments, which is as exp ( lr).

6 TURBULENTLOW I N A TUBE OF CIRCULAR CROSS-SECTIONThe case of turbulent flow and diffusion which was treated in Taylor's second

paper (Taylor 1g54cc clearly comes under the general case. If and are functionsof p = q2+c2 * nly and is the radius of the tube equation (9)becomesand the equation for d is

of which the solution ispUx(p ) p .


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76 R. ArisThis is the function tabulated in column (8) of table 1 (Taylor 1g54a , and the finalcolumn (11) of this table is

In 4of this paper the effect of longitudinal diffusion is estimated and is added todiffusion coefficient K@. Equation (41) of the present paper supplies a rigorousjustification for this.

7. VISCOUS FLOW IN A TUBE OF ARBITRARY CROSS-SECTIONI t is well known th at the velocity u 7,C) for viscous flow in a straight pipe

satisfies the equationsV ~ - a 2 g / px (in s),(

u = 0 (on r ,where in this equation is the pressure and p the viscosity, and it follows thatsatisfies the equations V ~ X a (in S

x = - 1 ( o n I? ,where a = a2 U is a constant.dxICombining this equation with equation 39), and putting = 1, we have theequations for 4 V q5= a (in S ,

which enables 8 and so K to be calculated for any cross-section.As an illustration of this a tube of elliptical cross-section may be considered. If

the major and minor semi-axes are a and b respectively

Taking a as the characteristic length it may be shown, after a certain amount ofalgebraic labour, tha t 1 24 24e2+ 5e4K = 48 24-12e2 'where e = (1 2/a2 *is the eccentricity of the cross-section. For e = 0, K is 1/48as it should be for a circle and when e = 1, K is 5/12 of this value. A very small valueof (1 represents a very narrow elliptical slit and in such a slit diffusion acrossthe slit renders the concentration very nearly constant in this direction, so thereare effectively only two spatial dimensions.*

I am indebted to M r C. H. Bosanquet or pointing this out.


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Dispersion of a solute in a Jluid owing through a tube 7Let 5 be in the direction of the narrow dimension of the slit, 2a be its major

dimension, andP 7) be proportional to the breadth of the slit at any point. Then theequation for C is ac a2c i ac+ - p -px r)-.a at2 gar r3 s

The function ~ 7 )s here the mean value of ~ 7 , across the slit, so th at

In fact all averages must be taken with P 7) as a weighting factor. An equationsimilar to equation 9) obtains for the moment c,, namely,

I n precisely the same way as before we derive the value of K,

whereand

In the case of an elliptical slit

and 5/12 x 48, which confirms th e limiting value in equation 46).It is well known that for a given pressure drop, the flow is greater in a circular

tube than in an elliptical one of the same area, and if in the Taylor diffusion coeffi-cient a is replaced by the area of cross-section na2for the circle and nab for theellipse) the constant is least for e 0. Thus the dispersion in a circular tube isless than in an elliptical tube of the same area.

Courant, R. Hilbert, D. 1937 Methoden, der mathematischen Ph ysi k, 1. Berlin: Springer.Taylor, Sir Geoffrey 1953 PTOC .Roy. Soc. A 219 186.Taylor, Sir Geoffrey 19j4a Proc. Roy. Soc. A 223 446.Taylor, Sir Geoffrey 19546 Proc. Ro y. Soc . A 225, 473

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On the dispersion of a solute in a fluid flowing through a tube

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