Afl-AlIO 454 11ISCONSIN UN V-MADSSON MATHEMATICS RESEARCH CENTER F/S 20/14NEW DESCRIPTION4S OF DISPERSIN IN FLOW THROUGH TUBES: CONVOLUTI-ETC (U)

NO I J C WANG, W E STEWAR T OAAG29-8O-C-0041

UNCLASSIF IED MRC-TSR-2297 NL-EmhhIlllllllEEEEEEEEEEL

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James C. Wang and Warren E. Stewart

Mathematics Research CenterUnivesFity of Wisconsin- Madison610 Walnut StreetMadison, Wisconsin 53706

November 1981 ~

j9ceived August 5, 1981)

Approved for public releaseDistributien unlimited

Sponsored by

U. S. Army Research Office and National Science FoundationP. 0. Box 12211 Washington, DC 20550Research Triangle ParkNorth Carolina 2770982 0 03 85

UNIVERSITY OF WISCONSIN - MADISONMATHEMATICS RESEARCH CENTER

NEW DESCRIPTIONS OF DISPERSION IN FLOW THROUGH TUBES:

CONVOLUTION AND COLLOCATION METHODS

James C. Wang and Warren E. Stewart

Technical Summary Report #2297

November 1981

ABSTRACT

The convective dispersion of a solute in steady flow through a tube is

analyzed, and the concentration profile for any Peclet number is obtained as a

convolution of the profile for infinite Peclet number. Close approximations

are obtained for the concentration profile and its axial moments, by use of

orthogonal collocation in the radial direction. The moments thus obtained

converge rapidly, and the concentration profile less rapidly, toward exactness

as the number of collocation po-nts is increased. A two-point radial grid

gives results of practical accuracy; analytical solutions are obtained at this

level of approximation.

AMS (MOS) Subject Classifications: 35A35, 41A10, 41A63, 44A30, 65N35, 65N40,

76R05

Key Words: Dispersion, Diffusion, Polynomial approximation, Orthogonal

collocation, Method of lines, Laplace transform, Fourier

transform, Convolution, Superposition, Moment-generating function

Work Unit Number 2 - Physical Mathematics

Sponsored by the United States Army under Contract No. DAAG29-80-C-0041 andthe National Science Foundation under Grant Numbers ENG76-24368 and CPE79-

13162.

SIGNIFICANCE AND EXPLANATION

'Convective dispersion plays an important role in many processes of

chemical reaction and separation. Such systems are commonly analyzed by use

of a radially-averaged diffusion equation, This approach has been popularized

by the simple results of Taylor (1953) and Aris (1956) for the cross-sectional

average concentration, and perhaps also by the belief that the radial

variations of concentration are too small to require detailed consideration.

Subsequent investigators (e.g. Gill and Sankarasubramanian (1970), Chatwin

(1977), De Gance and Johns (1978)) have extended this approach to shorter

times and to other boundary conditions.

'The foregoing approach is not easy to apply to reaction or separation

processes if the fluid properties vary, nor if the kinetics or equilibria are

non-linear. Therefore, in this paper we consider an alternate method: we

solve the full diffusion equation by orthogonal collocation in the radial

direction. Non-reactive systems are emphasized here; reactive ones will be

studied more fully in the sequel to this p

The responsibility for the wording and views expressed in this descriptive..summary lies with MRC, and not with the authors of this report. 0-ti

44 i -. C

C *..D

NEW DESCRIPTIONS OP DISPERSION IN FLOWv THROUGH TrJBES:CONVOLUTION AND COLLOCATION ME.THODS

James C. Wanq and Warren E. Stewart

INTRODUCTION

The collocation procedure described here is a variant of that qiven by

Villadsen and Stewart (1967); see also Finlayson (1972) and Villadsen and

?ichelsen (1978). For axially symmetric states in a tubular reactor, we

approximate the profile of each unknown state variable YMin the form

n 21

Y M a.i (T,Z)Et 0 4 &~ 4 1 *(1)

m i=0 i

This expansion can also be written as a Laqranqep polynomial

- n+1Y M= IW k(9~)Y M(r ,Z,E k 0 4 E 4 1 (2)

m k=1k m k

with radial basis functions

W (E) - R (t-2 2)/ 2 _ 2) *(3)

Thus, each approximate profile Y mis represented by Laqranqe interpolation

in terms of its values Y M(T,Z, F') at a set of chosen radial nodes 'k. We

choose E 2,**. as the zeros of the shifted Legendre polynomial Pn (x)1 nn

(Abramowitz and Stequn, 1972) and choose E as the wall location, =1.n+1

Any needed derivatives or integrals of Y Mcan now be represented as

linear combinations of the nodal values Y (~h this is known as the method

of ordinates (Villadsen and Stewart, 1967). For example, the dimensionless

radial derivatives of interest at the radial node E are:i

dY n+1

n+1I= ik Y mk Y(T,Z)

k-1

Sponsored by the United States Army under Contract No. DAAG29-80-C-0041 andthe National Science Foundation under Grant Numbers ENG76-24368 and CPE79-13162.

LK ---'-----

dY n+1 dw

, - k- C (E d Z(5)

n+1 ~a A- DikYk(T,Z)

k-1

In this way, we reduce all radial derivatives to summations; the same can be

done for differences and integrals with respect to C. For the above choice

of nodes, weighted integrals of Y over the cross-section can be obtainedm

with high accuracy by use of Gauss' quadrature formula.

The boundary conditions at the wall may e linear or non-linear. If they

are linear, then Y can be expressed linearly in terms of the interiorm ,n+1

values Ymk by use of the boundary condition and Eq. (4). This permits

elimination of Y from &T. (5); the resulting new weight coefficientsM ,n+1

will be denoted by B1k. In non-linear cases, on the other hand, one retains

the wall values as working variables and applies the boundary conditions

numerically during the solution process.

CONVOLUTION RELATION

Consider the developed isothermal flow of a binary Newtonian or non-

Newtonian fluid in an infinitely long circular tube. The continuity equation

for either species may be written in the dimensionless form

30 ail 1 aa +t" 1. a2g 1 1T + V(&) z- r (& a2 2 (6)

- + Pe2 az2

with the initial and boundary conditions

At T- 0 f(T,Z,&) = G(Z,t) (7)

At =I : f(T,Z,C) + K"Q(T,Z,F.) = 0 for T > 0 (8)

-2-

At 0 : (T,Z,g) 0 for T > 0 . (9)

Here Ki'' and Ku are the first and second dimensionless numbers of

Damkhler (1936) for homoqeneous and heteroqeneous reactions. The Laplace-

Fourier transform of Es. (6) - (9) from (T,Z,&) into (s,p,g) is

2 + -

(8 + Kill _L'01 + pv(w)o - + G(v,E) (10)Pe

2

At 1 (s,p,) + KuU(s,p,0) = 0 (11)

At o l(s'PA) = 0. (12)

Let 0. be the solution of Egs. (6) - (9) with Pe . Its transform

satisfies eqs. (10) - (12) with the p2 term suppressed. Use of the s-shift

theorem and Fourier convolution (Campbell and Poster 1967) then gives:

2, - exp(! !J)1.('rp, ) (13)

Pe 2

f- il exp[- 2EL (Z-X)2]-(TXpC)dx " (14)

Thus, the complete solution of Bqs. (6) - (9) is obtained by Gaussian

smoothinq of the infinite-Pe solution.

TWO-POINT COLLOCATION1 NON-REACTIVE CASE

Let Ci be the solution of ESo. (6) - (9) with K" K''' 0, Pe *,

and the followinq initial mass fraction profile:

G(ZA) - 8(Z)M(O) . (15)

Collocation with n - 2, and elimination of the wall concentration throuqh

the boundary condition (8), yields an initial-value prohlem for the nodal

-3-

___

functions C.(T,Z, 1 - C,,(T,Z) and 2 C 20(T,Z). The boundary

condition (9), and the symmetry of the true solution, are satisfied throuqh

the use of only even powers of & in the collocation basis functions rsee Eq.

(3)]. In place of Eq. (10) we then obtain

2seic + PVjLic . I j- + 01 i - 1,2 (16)

k=1

in which Vi and 1P are the nodal values of V(&) and O(E). To complete

the solution, we evaluate the coefficients B! from Eqs. (5) and (8), with

the Gaussian nodes

2 1 1

&J1 2

(17)2 1 1F2 = +- 2

and invert the transforms. The solution differs from zero only within the

reqion IJZ 4 IT, and is qiven there by

0 1 lT+ZlI I(X) 89 2 0IX) :

C14 W exp(-8T){. 2 X + Vl2 1) + 1 6(Z-T)) (18.1)

Q2BIT-Zl 11(X) 8Q110 (X) -

C 2- expC-ST). 2X + IV .V + 92 6(Z+T)} (18.2)

in which

Z = Z - (V + V )T/2 (19)1 2

T - (V - V )T/2 (20)1 2

i' V2-2B , 256(V1 V (21)

x./B(T2 - 2 ) . (22)

-4-

" ~ .7. " ,

Interpolating the solution according to Eq. (2), and integrating over the tube

cross-section, we find the mean value to he (CI, + C 2)/2. Gaussian

quadrature of the integral qives the same result.

CENTRAL MOMENTS OF THE TWO-POTNT SOLUTION

rt C be the solution of Eqs. (6) - (9) and (15) with K" = K'' = 0.

The mean mass fraction in a cross-section is defined, for these axisymmetric

problems, by

<C> = 2 f0Cd * (23)

The central axial moments of <C> are

i = - (z - <V>t) i<C>dZ (24)

and have the moment-qeneratinq function

1i = - t i e x p ( < V >T p ) C_(T, p, &) >1 (25)

= dp p=O (5

in which C(r,p,F) is the Fourier transform of C(T,Z,&).

For the collocation solution with n = 2, we thus obtain

=0 1 (01 + Q2 ) (26)

1 = (Ol- O)(Vl" v)['-exp(-16T)U4 = ( ) 27 )

1 2 )(V-V) 2 6

2 (V1 t 2 T 1-exp(-16T) } (28)2 0 2 8 4 64

For a radt.ally uniform pulse of unit strength, Pi - 1; and for developed

laminar flow of a Newtonian fluid, V1 = I - . Eqs. (26) - (28) then give

P0 = 1 (29)0

= 0 (30)

" 1 1 I--exp (-1 6'T)2= ('.2 + - --)2T - 1536 " (31)2 PP 2 2 1536

: -5-

- . ..- . .-

The exact solutions derived by Aria (1956), Gill and Sankarasubramanian

(1970), and Chatwin (1977) for a radially uniform pulse confirm Eqs. (29) and

(30). Their expressions for P2 can he reduced (after a siqn correction in

Aria' solution) to the common form

1 1 32 -8 ~ 2(A + -L-)2T - + 32 X exp(-A T) (32)2 e 2 192 1440

in which A is the jth zero of the Bessel function J1(x). The terms

proportional to T constitute the prediction of the standard Fickian

dispersion model, with the long-time asymptotic dispersion coefficient found

by Taylor (1953) and completed by Aris (1956). For brevity, the latter model

is called the Taylor-Aria model.

Before testinq these results numerically, we qive collocation solutions

of hiqher order for the axial moments of the function C(T,Z,t).

COLLOCATION SOLUTIONS OF HIGHER ORDEP

The moments of the mass fraction profile about the oriqin are defined by

mi(TI) " t. Zi l(TZ,g)dZ . (33)

The qeneratinq function for these moments is

i ,,i ',,m ( --.TIE(',,)]p~ (34)

The moments at finite Pe can be obtained from those at infinite Pe by use

of Eqs. (13) and (34).

A collocation solution for m (T,4) at Pe - 6 is obtained as follows.i

Let mi be the vector of mesh-point values of mi(TE):

TI a -mi(TI ),eo46mi(TEn)1 . (35)

We apply the Fourier transformation and n-point radial orthoqonal collocation

to Eqs. (6) - (9) with K" K'"' - 0 and Pe - . Then by use of Eq. (34)

-6-

- --

7

we obtain

TT.i" m + i .-im 'fi Dii . 36

Eqs. (13), (34), and (36) can then be used to obtain the moments at finite

values of Pe. Finally, the cross-sectional average <mi> can be obtained by

Gaussian quadrature of

Tables 1, 2, and 3 show the collocation solutions and the exact solutions

for lo0, I'. and P 2 for two initial solution distributions. The

collocation solutions converge rapidly toward the exact results for all values

of T, and for both initial distributions. From Table 1 we see that the

collocation solution for n i 2 is particularly accurate in the case of a

uniform initial radial distribution.

TWO-DIMENSIONAL INITIAL DISTRIBUTIONS

Any axially symmetric initial distribution can be expressed in the form

G(ZF j) j lb P MZQk (t) (37)

k k j

with suitable basis functions J(Z) and 0 (E). Let C (T,Z,t) be the

solution of Eqs. (6) - (9) with constant K" and K''', and with the initial

distribution M(,Z,.) k Then the solution correspondinq to the

initial condition (37) is obtainable by superposition,

(T,Z,)- bjk fl i(X)Ck(t,ZX,)dX (38)j k

and has the Fourier transform

= b jkV p0(T,'') .(39)j k

Let tjand be the ith moments of the functions i'd(Z) and Ck(T,Z,E)

respectively with respect to Z. Then from Eqs. (34) and (39) we get the

moments mi in the form

-7-

t -.- --

1-. .: . .. " ... .. - : '

Table 1 Second moment 1000(P 2 _ -) for initial condition Q(M) = 12 Pe2

T Collocation Exact Taylor-Aris

n=2 n=3 n=4

.01 .00791 .00782 .00780 .00779 .1042

.05 .1623 .1575 .1574 .1574 .5208

.10 .5221 .5059 .5059 .5059 1.042

.15 .9705 .9441 .9442 .9442 1.562

.20 1.459 1.425 1.425 1.425 2.083

.25 1.965 1.927 1.927 1.927 2.604

.40 3.517 3.474 3.474 3.474 4.167

1.0 9.766 9.722 9.722 9.722 10.42

Exact values are calculated from Eq. (32). Thefirst two moments aqree for all three methods.

-8-

t]

Table 2 First moment 100 M I for initial condition Q() =2

T Collocation Exact

n=2 n=3 n=4

.01 -.1540 -.1516 -.1511 -.1511

.05 -.5736 -.5540 -.5541 -.5541

.10 -.8314 -.8083 -.8086 -.8086

.15 -.9472 -.9298 -.9299 -.9299

.20 -.9992 -.9880 -.9880 -.9880

.25 -1.0226 -1.0160 -1.0159 -1.0159

.40 -1.0399 -1.0388 -1.0388 -1.0388

1.0 -1.0417 -1.0417 -1.0417 -1.0417

• Exact values are calculated from Chatwin

(1977). The first moment is independent ofPe, in view of Eqs. (13) and (34).

-9-

t44 _ ___

Table 3 Second moment 1000(u 2 - - for initial condition O() 2F2

2 Pe

' Collocation Exact

n=2 n=3 n=4

.01 .00791 .00747 .00738 .00737

.05 .1623 .1390 .1389 .1389

.10 .5221 .4434 .4447 .4447

.15 .9705 .8415 .8431 .8431

.20 1.459 1.294 1.296 1.296

.25 1.965 1.778 1.779 1.779

.40 3.517 3.305 3.305 3.305

1.0 9.766 9.549 9.549 9.549

*Exact values are calculated from Chatwin

(1977).

-10-

--- ..

a(TIC) - b M 1 4 i ,k (TIC) (40)jkjk r r4 i-r rj k r0 r

for any initial distribution of the form in Eq. (37). This result shows the

influence of the initial conditions on the moments of the mass fraction

profile at any later time.

PROFILES FOR A PULSE OF FINITE LENGTH

As a further test of the collocation solutions, we calculate the mass

fraction profiles for an initial pulse of maximum amplitude unity and of

finite lenqth L equal to 0.1 unit of Z. The coordinate

U = (2Z - T)/(L + T) (41)

is introduced here to facilitate finite element calculations of the

solution. The followinq initial condition is used,

G(Zt) = i(U) (42)

in which w(U) is a C1 cubic spline:

V(U) = 1 for UIl 4 0.975

O(U) = 1 - 3 (lUI-0975) 2 + 1(}-9753 for 0.975 4 I 41 (43)0.025 , ,0.025

O(U) 0 for luI > 1

With this initial condition, the solution C. (valid for Pe = ) is

permanently zero outside the reqion -1 4 U 4 1.

Fiqures 1, 2, and 3 show the cross-sectional averaqe mass fraction <w>

as a function of T and z at Pe - 15, 40, and infinity. Three methods of

solution are shown: orthogonal collocation on finite elements of U, two-

point radial collocation, and the Taylor-Aris model. The radial collocation

was done in all cases as described above; the finite elements of U were C1

cubic splines with lenqth AU = 0.025. Each method was first applied with

-11-

Pe m , and Eq. (14) was then used to qet results at finite Peclet numbers.

The two-point collocation profiles (dashed curves) model the true solutions

very well, except for some lack of smoothness at short times with Pe .

The Taylor-Aris model exaqqerates the width of the tails of the pulse in every

case.

, -12-

1.0 I

0

in 0

0 4

a.~" ..

.4 P

*. t:.0N0 0 I

as 0

1.00:2 m. 0 4o- 0 4 0

r4.44 .

1.0G

* .0

0 . -- - I0z 000

1.0

.= O I

-=*.6t .O 1 *6

1.0

40

IDI

A c~0 eq

100-41

40 I .4 0J

06,*, *6 004 0

-4 31' -Au

a 0

6 .4

as 0

41

.4

* 00

-14-.

.64 0•8

ml . .. ' ".. .. .. I~l[r '% m( 0: .. . '' " ' . . . - .. . " '

1.0

.01

4001.0.

*>, . .

0 _.

1.0- "0 1

1. 0

%4 . M14 9.40N 0

d a4

=.40 0

a 0

0. I 14 t1

1.0 -1.0

1 .0 -Ow

NOTATION

Aik coefficients for gradient operator in Eq. (4)

aim, bjk coefficients of basis functions.

Bik coefficients for Laplacian operator in Eq. (5)

B'k coefficients for Laplacian operator with boundary point1ik

eliminated.

S' matrix of order n with elements B!ik

C solution of Eqs. (6) - (9) with constant KO and K'" and

with initial condition 6 (z)O(M)

Ck special form of C for initial condition k(z)Qk(4)

C. limit of C as Pe + a

G(C,z) initial distribution, Eq. (7)

In X) modified Bessel function of the first kind

K'',K" first and second Davk8hler numbers

Mi ith axial moment of 0 defined in Eq. (33)

vector of nodal values of mj, Eq. (35)

n number of collocation points interior to C - I

p Fourier transform variable

Pe - RVmax/VAB , Peclet number

Q(M) radial function in initial condition

0, value of Q() at ith collocation point

kQ (M) kth radial basis function in initial condition

s Laplace transform variable

U transformed Z coordinate in Ea. (41)

V() velocity profile, relative to centerline value

V1 value of V() at ith collocation point

j diagonal matrix with elements Vii - Vi

-16-

Y mth state variable (e.q., temperature or species mass fraction)m

Z - ZDB/(Vmax R2), dimensionless axial coordinate

< > averaqe over flow cross-section

i(r) binomial coefficientr

Greek Letters

Pi ith central moment of C defined in Eq. (24)

= r/R, dimensionless radial coordinate

2-tDA/R1 , dimensionless time

O(U) axial function in initial condition of Eq. (43)

P(Jz) jth axial basis function in initial condition of Eq. (37)

w solute mass fraction

solution for w with qeneral initial condition G(z,&)

-17-

-. F " L ,.

LITERATURE CITED

Abramoitz, N. and I. A. Stecrun, ed., "Handbook of Mathematical Functions",

hover (1972).

Kris, R., "On the Dispersion of a Solute in a Fluid Flowinq Throuqh a Tube",

Proc. Roy Soc. A, 235, 67 (1956).

Campbell, G. A. and R. M. Foster, "Fourier Integrals for Practical

Applications", D. Va '-'grand, Princeton, New Jersey (1967).

Chatwin, P. C., "The Tadd..O '. velopment of Lonqitudinal Dispersion in

Straiqht Tubes".. e; d Mech., 80, 33 (1977).

Damk8hler, G., "Ei4nt . t 4 r Str8munq, Diffusion und W&rmefiberqanqes auf die

Leistunq von Reacktoneffen. I. Allqemeine Gesichtspunkte ffir die

bbertraqunn eih.ia chemischen Prozesses aus dem Kleinen ins Grosse", Z.

Electrochem. 42, 846 (1936).

De Gance, A. E. and L. E. Johns, "On the Dispersion Coefficients for

Poiseuille Flow in a Circular Cylinder", Appl. Sci. Res., 34, 227

(1978).

Finlayson, B. A., "The Method of Weighted Residuals and Variational

Principles", Academic Press, New York (1972).

Gill, W. N. and R. Sankarasubramanian, "Exact Analysis of Unsteady Convective

Diffusion", Proc. Roy. Soc. A, 316, 341 (1970).

Taylor, Sir Geoffrey, "Dispersion of Soluble Matter in Solvent Flowing Slowly

Throuqh a Tube", Proc. Roy. Soc. A, 219, 186 (1953).

Villadsen, J. V. and M. L. Michelsen, "Solutions of Differential Equation

Models by Polynomial Approximation", Prentice-Hall, Enqlewood Cliffs,

New Jersey (1978).

Villadsen, J. V. and W. E. Stewart, "Solution of Boundary-Value Problems by

Orthoqonal Collocation", Chem. Eng. Sci., 22, 1483 (1967), 23, I515

(1968).

JCW/WES/Jvs-18-

I . ... , , :,

REPORT DOCUMENTATION PAGE READ 1NS1.RVT1,)oS

1. REPORT NUMUER 2. GOVT ACCESSION NO 3. RLCIPIENT'S CATALOG NUMUkR

i- I ,4IfjL' _ _ _ __ _ _

4. TITLE (id Subtitle) S. TYPE OF REPORT & PERIOD COVERED

New Descriptions of Dispersion in Flow Through Summary Report - no specific

Tubes: Convolution and Collocation Methods s. PERFORMING ORG. REPORT NUMBER

7. AUTHOR(*) S. CONTRACT OR GRANT NUMDER(a)

ENG76-24368; CPE79-13162James C. Wang and Warren E. Stewart DAAGZ-80-C-0041

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASK

Mathematics Research Center, University of AREA & WORK UNIT NUMBERS

Work Unit Number 2 -610 Walnut Street Wisconsin Physical MathematicsMadison, Wisconsin 53706

It. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

November 1981(see Item 18 below) Is. NUMBER OF PAGES

18

i. MONITORING %GENCY NAME & ADDRESS(If different brom Controling Office) IS. SECURITY CLASS. (of tl report)

UNCLASSIFIED154L' OECLASSIFICATIOWi DOWN GRADING

SCHEDULE

IS. DISTRIBUTION STATEMENT (of this Report)

Approved for public release; distribution unlimited.

17. DISTRIBUTION STATEMENT (of the abstact ewteted in Block 20, it diffleent /atw Report)

14. SUPPLEMENTARY NOTESU. S. Army Research Office and National Science FoundationP. 0. Box 12211 Washington, DC 20550Research Triangle ParkNorth Carolina 27709

IS. KEY WORDS (Continue on revers ide if necessary amd Identify by block number)

Dispersion, Diffusion, Polynomial approximation, Orthogonal collocation, Methodof lines, Laplace transform, Fourier transform, Convolution, Superposition,Moment-generating function

20. ABSTRACT (Continue an reveree side it necesary and Identify by block number)

The convective dispersion of a solute in steady flow through a tube isanalyzed, and the concentration profile for any Peclet number is obtained as a con-volution of the profile for infinite Peclet number. Close approximations are ob-tained for the concentration profile and its axial moments, by use of orthogonalcollocation in the radial direction. The moments thus obtained converge rapidly,and the concentration profile less rapidly, toward exactness as the number ofcollocation points is increased. A two-point radial grid gives results of prac-tical accuracy; analytical solutions are obtained at this level of approximation.

DD , F. 1473 toIrON OF I NOV ,s IS OSSOLETE UNCLASSI PIEDV sCU1gTV CLASSIFICATION OF T"IS PAGE (l"ea Dats nter

I