ORNL CSD TM-48

Diffusion into a Hollow Cylinder

J. 5. Toiliver M. Reeves

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v)RNI. CSD IAI-4K

Coniraci No. W-7405 cng 26

COMPLTER SCIENCES DIVISION

DIFFUSION INTO A HOLLOW CYLINDER

J. S. Tollivcr

M. Reeves i -°1,a 1 Din i*p»rt *»* pnpartd j *n K -̂>unt >f ».»*k j

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Date Published - October 1978

I'NION C ARBIDF: CORPORATION. M'( I I AR DIVISION 'jpcraiing the

Oak Kidttc(ia»coiJN DitiuMon Plan! . O.ik Ridge National Iaboratim 0;;k Ridge V-l.! Plant Paducah (iaseous Dillusinn Plant

!>>r tnc DIPAR I MTA I Of- FMRCiY

i i i

CONTENTS

List of Figures v

Acknowledgment v i i

Abstract I

I. Introduction and Statement of the Problem I

II. General Solution 5

III. Alternate Soiutbn Useful for the Case I. 5> b 7

IV. Alternate Solution Useful for the Case b > 1 9

V Reduction to Special Cases 13 Specialization of the General Solution 13 Specialization of the First Alternate Solution (I. > b) 14 Specialisation of the Second Alternate Solution (b >̂ I.) 15

VI. Examples 17

VII. Conclusion 23

VIII. Notation 25

IX. References 27

V

f

LIST OF FIGURES

The hollow cylinder of length L. inner radius a. a ad outer radius b 2

Fractional uptake as a function of time for four values of the 'ength-to-radius ratio R IN

Fractional uptake calculated from the general solution compared to that calculated from the first alternate solution with the longitudinal summation truncated O

Comparison between the general solution and the first alternate solution with both summations truncated 20

Comparison between the general solution and the converged second alternate solution 21

Comparison between the general solution and the truncated second alternate solution --

v i i

Acknowledgment

The authors uould like to thank G. I.. Powell of the Y-12 Development Division for biinging this problem to their attention and f«r providing the necessary financial support.

Thanks also are due the secretarial staff of the Computing Applications Department loi their assistance in the preparation of this report.

DIFFUSION INTO A HOLLOW CYLINDER J. S. Tolliver and M. Reeves

ABSTRACT

We have obtained a completely general solution to the problem of diffusion into a hollow cylinder. When one of the dimensions of the cylinder, cither length or radius, is large compared to the other, one of the two infinite series of this solution converges slowly and is not suitable for practical application. For these situations two alternate solutions, utilizing rapidly converging series, have been obtained. Both the concentration distribution and the total uptake as functions of time are considered. A number of specializations are used to show that the solutions are reducible to the cases of a finite solid cylinder, an infinite solid cylinder, an infinite plate, and an infinite hollow cylinder. Finally several examples are given to show the eiTk?cv of our solutions.

I. INTRODUCTION AND STATEMENT OF THE PROBLEM

Figure I shows a hollow cylinder of length L. inner radius a. and outer radius b cylinder is immersed in a gas of constant concentration c, and lias diffusivity I). concentration c obeys the equation of diffusion

a^ = D7>c<r, .«> dl

on (he region

a < r < b and 0 < / < l -

subject to the boundary conditions

c(a,/..t)= L (b / . t ) = c 0

t(r.0.i)=c(r,L.t) = c 0

and (he initial condition

I

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2

0RNL-DW6 77-10296

Z=0

I'H*. I. I he hollow lAlinricr ol length I . inner radius a. and ouler radius b.

3

c(r.z.Ol=0 (2c)

We shall first obtain a general solution to Eq. ( I ) by a treatment similar to that of Carslaw and Jaeger [1959; Section 7.10]. Then we shall employ a method using the Laplace transformation to obtain two aitetnate solutions which are more useful for cases where values of the length and the outer radius differ substantially.

II. GENERAL SOLUTION

In order to obtain a purely homogeneous problem, we first make a change of variables:

c(r^,t)=u(r^,t> + c. (3a)

Here, u satisfies the same partial differential equation as c, namely

* • - - n i dt = DV*u - (3b)

but has the homogeneous boundary conditions

u(a.z.i) = u(b,z.t) = 0 (4a)

and

u(r,0,t) = u(r.L.t) = 0 (4b)

and the initial condition

u(r/,0) = - c 0 (4c)

Upon expressing the Laplacian in cylindrical coordinates. Eq. (3b) becomes

du dt -D I l i Urn) £1H

r d r V d r T dz* (5)

Equation (S) may be solved by separation of variables with the .solution being

u(r.,.«)= l l A m n U 0 ( e n , ) s i n ( ( 2 m

L ' ) f f / ) e ' > ^ n t

m i n i (6a)

where

\

|Jt4J»>vfs>-^..»»spag^..,; - , , ... . - . . . „ . . , . _ ,. ... ,. ._ „̂ ..._.,,.., - ... .- .-..- -:.-.. ^ •-. ,.•--:.=• > . -w° .ca« ! M«yi^ / -

B

: /7nV /'Cm UrrV

<n = 7n b (6d)

and tfc -. are the roots I' . The normalization constant, found by applying th'* initrai condition, fc |. (4c). is given by

4«.., J 0U nb» A, " 2m i J 0(e na) + J0(fnb> «6e)

Thus the solution to Eu. (i) is

•;(r./.i|-c 0 { I 2. L , ; i ~ . r U0Unr>sin ' r — e ' 1 ' -"" 1 .(7)

Jo find the uptake as a function of time,we integrate c(». z. i) over the volume:

M(t)= ( " i i " voy..t,'ruru/ut> •

f:\aluaiini; the integral, we find

Mt'> - l __J1 f f « " " " " ' Jo(ena> J 0(e nb) M(«) * W a J ) m ^ ( n « Cm !)*«„* J 0(e na) + J 0(f nb)

,,-A-' ,tl.wa, ,„«, . , , (8)

Here. M(t) is the amount of gas having diffused into the cylindrical shell after time t. and M(*) is the corresponding quantity after an infinite time.

111. ALTERNATE SOLUTION USEFUL FOR THE CASE L ̂ b

It th- Lngth I is large, the exponent in I q. (6a) is small due to the longitudinal component ol f-.q. (6c) I hi* >mali exponent leads to "low -onvergcr.ee. a dtVticuhv similar to that arising from small value ot time in the exponent. Such di|.icultic> mav he alle\iated h\ application ol a t.apLiee transtornia.ion technique a> follows. Wc begin with F.q. (3b) above and consider a solution having the li*rm

u( r,M) = L:,(eBr»e D ' s ' 11 vl/.t)| ,9)

Ouaitity v represents the variation of the solution from that of an infinite hollow cylinder. The differential equation for v is found to be

with boundary conditions

v(0.t) = v(L.t)= I (|0b)

and initial condition

v(z,0) = 0 MOO

Applying the Laplace transformation to Eqs. (I0), we have the subsidiary equation

g p - - k 2 v = 0 (Ma)

and the boundary conditions

where

The solution to Eq. (Ila) is

vfO)=v(L)=IM (| |b)

k = x/A/D (lie)

v* Aco»h(kz)+Bsinh(kx). (12a)

Imposing the boundary conditions and expanding ihi hyperbolic t-tncti m% in a manner appropuaic for lirge value* of k yields the solution­

i s j , . m e - k » n i t * H + V , , ( m e - k l i m * l » L / !> (I2h)

m l m 0

lading the inverse I.aplace transform and collecting constants. we have

c(ry..t» l I n=l ! I m=0 - v " 1 '

J) I I ) m erfc m = o

/ t m * PL z\ \ \ 2VDt } \ j

(13a)

where A; is found from the initial condition:

jrJ 0(c na) n J 0 (f n a)*Jo(Cnb) (13b)

Integrating to find the fractional uptake, we have

Mi) s , 4 M(~> b 2 a

( y e 1 > { " ' ' Jo(t„a> JoU nt»l

{' L m - 0 icrlc

/ niL \ / lm» l|L\ \2>/Dt) '" 1 :>/Dt/

(I4>

It is apparent from the definition of *„, Eq. (6d). and the properties of the complimentary error function that Eqs. (13a) and (14) will converge rapidly when the dimensions of the cylinder and the time values are such that

b 3 < Dt and I>t < I3 (15)

Equations (13) and (14) are therefore the desired solutions.

')

IV. AITERNATF. SOI .11 ION I SIM I FDR I HI CASH b I

lp contract to the pre\u>u> section, we begin here by putting the radi.it Jencnd»nci- in a torm suitable tor application ot the I.aplace transformation technique. I'onsicjr

irtr./.i| = i:n«tm/le 1 > t "'" t l l o<t.tl| , | f , a )

vi here . C m Or

Here, o satisties

dt " r hi V df /

with boundary conditions

and initial condition

(I6h>

117a)

<Xa,W = 0<b.t)= I (17b)

0(r.0) = 0 (17c)

Solving in Laplace transform space, we find

__l_ |l0(kr)K0(kb) l0(kb)K0(kr)l + |l0(ka)K0(kr) l0(kr)K0(ka)| 0 < r > ~ * l0(ka)K0«kb) l0(kb)K0(ka) ll*>

where

k = v^7D .

l o /croth order, the asymptotic expansions of ihe modified Bessel functions. I., and K . yield

^/ v ' f A ? L-k|<2n*l)p-<b-r)| k |<2i»* I »p*«b r)|l

A f L k|l2«*l)»-(a r)| c M'2n* \)»+(» r ) | | )

w here

(19a)

II)

p = b (Nh>

I he summations .n the above expression result trom an additional expansion ot the denominator ot I q. (IK) Inverting 6(r). applying the initial condiutn. and collecting terms Vicids

n - 0

n - 0

(20a)

where

0 n = ( ? n * l ) p (20b)

To obiain an approximation which is correct to zero:h order- we take only the terms which have the smallest arguments for the complimentary error functions. (Function erfcix) decreases rapidly as argument x increases.) Thus.

c ~ «-o < 4

I — «nU m ' - ) c -Pt , (2m I)

- I \ m l

/Tc r f c/i_l\ /?erfc(V h . (21)

We integrate to find

MID * I

16 M(~> * J<b i /kJ , i )| ;j6r(2m i ) 3 J | : v [VF U^/ j f (22)

It is apparent from the definition of fm in Eq. (16b) that the /eroth order expression. Eq. (22). will be valid whenever

LJ «Dt and D K | b a) 1 (23)

The validity of Eq. (21). however, depends on (he radii considered independently-.

L* « D t . D K b 1 and DKa7 (24)

II

In *:.iv> »hc : t on!> the tir>t :w«> mctiiulilio -t la I 2 4 I arc vui>tieu. lOiu'cnttationN d U uLticti tr.«rr. I a till wtil br iiKoruvt >nlv ;n the rei»:«»r. r -• a

!

!

13

v. REDUCTION TO SPECIAL CASES

In this MX;ton we show how our solutions Irom the p'eviou-. three sections may he spcciali/cd [o ihc cases ol a finite solid c.lindcr. an infinite Mil id cylinder, ap ml mile plale. and an k.itinile hollow cylinder.

Specialization of the General Solution.

Kcginnini' with the fractional uptake lor a finite hollow cylinder. i"q. «X). we reduce it to that lor a finite >olid cylinder by allowing the inner radius a to approach /cr >. I he term

J..«< n .» J„ l t I tb» I t, .-.,-. •! _ "L - , — j . i .-.-•»

J„ l t n . t » r J „ U n l | | t >,

In this limit, then.

Mm Ml ' i ' •-" C J Z j t„-r> : i:m I r

iu I n I

lor a finite solid cylinder. We may further specialize l-q. (2(>) to the expression lor an infinite solid cylinder h\

allowing I. to approach infinity, from t:q. foe), we observe that the term «„. depends on I such that

>n " f £.»> [£ " >o' _ ; h I. • ' b : " ' " t27)

I hits.

Mt-i -ZJ »:,„" irZj ,n-b-m l n I "

or

Ml'-1 Z f , -' h- I2X)

lor an infinite solid cylinder. Here, the summation over m has been replaced b> its closed lorm expression

y ' -. '•'• LJ y Jin I r x

'S

r

! , t-.

•

m

14

rrom IHtight [1961: Eq. (48.12)). Equation (28) agrees with Crank's expression for an infinite solid cylinder [Crank. 1975: Eq. (5.23)].

The next specialization begins again with Eq. (8) for a finite hollow cylinder and allow* b to approach infinity, a to approach zero, and L to remain finite, with the result being the fractional uptake for an infinite slab. From Eqs. (6c) and (lob).

z m

so that

or

^H^H^)'-< •MtQ _ 32 ^% l_ Y 1 t : ' " > * '

\fell K V \ .» "*n»"' M i l _ . » X > •- ( 2 o a ) Mm = J ^ ^ c "*m ' Ml~l~ c 1 Li < 2m i r

provided

I f_ t „ - V = 4

129b)

We have verified Eq. (29b) by numerical means only. Equation (29a) agrees with Crank's expression for an infinite slab [Crank, 1975: Eq. (4.23)].

Specialization of the First Alterrate Solution (L > b).

Here, it is a simple task to reduce the fractional uptake for a long hollow cylinder. Eq. (14). to that of an infinite hollow cylinder by letting I. be not only large, but in fact approaching infinity. Unfortunately, we do not have a readily available expression from Crank or others with which to compare our results. We nevertheless quote the expression so obtained with the comment that an identical expression may be obtained from Eq. (8). the general solution, by letting I. approach infinity-

Mi D 4

M ( - » " l b 2 *2)

\u 3

" J " ( t n J ) J o U n h )

I <n h((»*)* !f><e„h)

(30)

for an infinite hollow cylinder. Proceeding from Eq. (30), let us now allow a to approach zero and recover the fractional

uptake for an infinite solid cylinder. Prom Eq, (25) fh. term involving the Bcssel functions

15

approaches unity. The solution is. then.

M7»I t& (31)

Equation (31) « identical to Eq. (28> which was deduced from the general solution, and both agree with Crank's expression for an infinite solid cylinder. Observing thai E,»v (30) and (31) may both be deduced from the first alternate solution as well as the general solution lends credence to the alternate formulation.

Specialization of the Second Alternate Solution (b > L) -

Beginning with Eq. (22). we deduce the fractional uptake for an infinite slab. Setting a equal to zero produces

Mill Ml«i = I : V 1 Z J cm i>: J I : v J v» VDi' v

(32)

Letting b approach infinity reduces this expression to

Ml-)" ' ir 2j Cm l» : 133) m I

for an infinite slab. Equation (33). deduced from the second alternate solution. »s identical to Eq. (29a). which was obtained from the general solution. This fart serves to validate the formulation of the second alternate solution since both expressions agree with Crank's result.

As a final specialization we once again consider an infinite solid cylinder. Equation (21). which gives the concentration distribution, was formulated to be rapidly convergent whenever the radius b was large compared to I.. It should be noted that the approximations inherent in this equation pertain to the radial variable only. Theret'orc, if wc now allow I. to approach infinity, we do not destroy the correctness of the formulation: at worst, we shall have a slowly converging scries. We do. however, retain the condition Dt < b~ so that our zeroth order approximations will remain valid. Using the fact that the exponential term in Eq. (21) approaches unity for small times and large values of L and setting a equal to zero yield

c(r/,t) * Co %hm\-M^ (34)

The summation over m may be identified with the Fourier series expansion of a function which has the value unity on the region 0 < z < \„ Thus,

16

Equation (35) is just the firs! term of the small time so'ution for an inkinite solid cylinder given by Carsbw and Jaeger [1959; Eq. (3) of Section I3JJ.

B

17

VI. EXAMPLES

In order 10 demonstrate some of the characteristics of our solutions, we plot fractional uptakes as functions of time for several notlow cylinders. Each cylinder has an inner radius a = 0.125 cm. an outer radius b = 0.75 cm. and a diffusivity D = 1.0 cm' sec. The ratio of length to radius. R = L b. is allowed to vaiy from cylinder to cylinder.

Figure 2 illustrates the effect of varying R. As R decreases, which implies that the diffusion from the ends of the cylinder becomes more important, 'he fractional uptake rises more rapidly in time. For the four R values indicated on the graph, the general solution. Eq. (8). and the first alternate solution, Eq. (14). are indistinguishable provided that sufficiently many terms are used in the summations. (In each case we used 100 terms in both the longitudinal summation over m and the radial sum over n. Such equivalence is to be expected since both Eq. (8) and Eq. (14) are exact solurions containing no approximations.

It has been indicated earlier that the first alternate solution will converge rapidly for cylinders having large length-to-radius ratios R provided the time is within the limits specified by Eq. ( 15). Figures 3 and 4 demonstrate this characteristic for three different values of R. Here^as in all figures to follow, the solid curves represent the converged general solution of Eq. (8) and may be considered correct. The dashed curves of Fig. 3 result from truncating the longitudinal sum of the first alternate solution to just one term (m = 0. only, in Eq. (.14)). I he number of ter.ns in the radial summation is kept sufficiently large to insure convergence for this part of the solution. A» expected from the second inequality of Eq. (15). the solutions diverge in the large-time regime with the divergence being less severe for large R. The dashed curves of Fig. 4 result from truncating both summations of the first alternate solution to one term (n = I, only, and m — 0. only, in Eq. (14)). For large times the curves here show the same divergence from the general solution as in Fig. 3. As is to be expected from the first inequality of Eq (t5). they also diverge in the small-time regime, and furthermore, the severity of this divergence is apparently independent of R. The logarithmic vertical axis is included in Fig. 4 to show the approximately exponential nature of the converged regions of the curves for R = I and R = 4.

in contrast to the first a'.cmait solution, the second alternate solution is most appropriate for cylinders having small k.-ngth-to-radius ratios R (cf. Eq. (23)). Also by way of contrast. Eq. (22) is not general in its radial formulation. In effect, the radial summation has been truncated analytically through our approximations to the one term shown in Fq. (22). Figure 5 provides a comparison between the converged general solution (solid curves) and a converged second alternate solution (dashed curves). It shows that our approximations become more and more valid as the ratio R becomes small. As is ak> apparent from the graph, in second alternate curve closely resembles the general-solution curve for R = 0.04. The remaining longitudinal summation of Eq. (22) is truncated to one term to produce the dashed curves of Fig. 6. These may be compared with the solid general solution curves. Obviously there is a large error for the larger values of R. However, for R equal to 0.01. the agreement becomes quite good.

IS

o

Ok e

s >

c . 3

si.

3HYidn 'Hn/Wii

QRifc DWG 77-10655 1 O i

0 • -

0 6

0 2

0 4 -

0 • 10Q

\\\i. .V I nu'iionul upuikc ciilculaicU from ihc gcncnil solution compared in thai ciileuliitcil I row lite I I IM alU'iiiiiic soluiion with ihc lonyiuuiinul summation truncated.

i

10"

ORNL DWG 77-10656

GENERAL SOLUTION FIRST ALTERNATE SOLUTION

i 0 '075 0 - 000

"I 0 026 0 000

t , TIME

l i j». 4. Comparison between the jseneuil solution and the first alternate solution with both summations truncated.

?l

% •

as

V, si

3»vxdn 'WnAim

•c

axvxdn 'Wn/Wn

23

VII. CONCLUSION

Three solutions to the problem ol diffusion into a hollou cylinder have been derived. The first is a general solution and is correct for all values of the spatial dimensions and of the time. The second solution is equally general but more useful for the case of a long cylinder. The third, a /eroih order tpproximation in the radial variable, is rapidly convergent lor the case of a cylinder uith large radius. Each solution has been reduced t«> a number of different specializations for the purpose of validating our results by comparison with accepted work by Crank [1975J and Carslaw and Jaege' [1959). Graphs have been presented to demonstrate the characteristics of the various solutions in several special cases.

25

VIII. NOTATION

A.. Normalisation constant defined by Eq. (6e).

A; Normalization constant defined by Eq. (13b).

a Inner radius of the hollow cylinder ... 1..

b Outer radius o!" the hollow cylinder ... L.

c Concentration ... ML' .

c. Concentration of gas surrounding the cylinder ... M L}.

D pifTusivity of cylinder .. LJ T.

k Parameter defined by Eq. flic).

L Length of cylinder... L.

M(i) Uptake as a function of time ... M.

M(t) M(~) Fractional uptake.

R Length-to-radius ratio L b.

r Radial coordinate... L.

t Time coordinate ... T.

u Function related to concentration by Eq. (3a).

U.. A combination of the Bessel functions J., and Y, defined by Eq. (6b).

V Function carrying z-dependence defined by Eq. (9).

V 1 .aplace transform of v.

7. Longitudinal coordinate ... L.

Gtmn Parameter defined by Eq. (6c).

&» Parameter defined by Eq. (20b).

y» Roots of l'.,

tn Vn b

K 1 .aplace transform variable ... T '.

.v^OWWea

i .i

BLANK PA

26

£.. Parameter dehned bv I:q. 116b).

p Parameter defined bv Eq. I !9b).

£ Function earning radial-dependence defined by Eq. (16a).

6 l.aplace tram'orm ol <J.

IX. REFERENCES

Can.km. H. S.. and Jaeger. J. C . Comiuct.on of Ileal in Soiuh. Clarendon Prcv.. Oxford. 1959.

Crank. John. Tlte Mathematics of l)iffa*ion. Clarendon Pres>. Oxford. 1975. I)vncht. Herbert Bristol. Tables of Integrals ami Other Mathemarival Dflta. Macmillan. New

York. 1961.

ORM. CSI) IM-4K

INTERN \L DISTRIBUTION

1. A. A. Brooks "> I I . P. Carter CSD Library 3. J. S. Crowcll 4. R. I Hibbs 5. J. T . Holdeman 6. C. S. Morgan 7. C. W. Nestor 8. J. E. Park 9. S. K. Pennv

10. H. Posima 11-15. G.I.Powell Ifr40. J. S. Tolliver

16. P. R. Vanstrum 42-43. Central Research Library

44. Document Reference Section - Y-12 45-47. Laboratory Records

48. Laboratory Records - Record Copy 49. ORNL Patent Office

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