Diffusion across a corrugated saw-tooth plate

Pergamon

Mechanics Research Communications, %'ol.22, No. 6, pp. 589-597, 1995 Copyright © 1995 Elsevier Science Ltd Printed in the USA. All fights reserved 0093-6413/95 $9.50 + .00

01193-6413(95)0111166-6

D i f f u s i o n across a c o r r u g a t e d saw-tooth plate

C.Y. Wang Departments of Mathematics and Mechanical Engineering Michigan State University, East Lansing, Michigan, 48824

(Received 15 March 1995; accepted for print 12 August 1995)

1

I n t r o d u c t i o n and f o r m u l a t i o n

Diffusion across a barrier is fundamental in numerous engineering processes involving heat or mass transfer (e.g. [1]). The solution of the temperature or concentration is simple (linear) if the barrier is a flat plate. However in many instances the plate is not flat but corrugated. Such corrugations are advantageous not only due to the increased bending stiffness of the plate but also due to the enhanced transport through the increased surface area [2]. Because the geometry of the corrugated plate is complex, analytic solution of the general diffusion problem does not exist, and numerical integration such as finite differences or finite elements is usually applied. The present Note studies the diffusion across a corrugated saw-tooth plate shown in Fig. 1. The aim is two-fold. Firstly we advocate a domain decomposition and eigenfunction expansion method which is much more efficient than direct numerical integration. Secondly such a basic geometry has not been studied in detail before. We would like to determine its transport properties, especially the theoretical increase in total flux due to corrugations. The resuits would be highly useful in the design of enhanced-transport corrugated

589

590

C.Y. WANG plates.

(a)

(b)

Fig.1 a.Cross section of the corrugated plate, b. Separation into regions. In Fig. la, the plate is composed of rectangular sections of thickness h and length (2a + 1)h joined periodically at right angles. Let the temperature T ~ (or concentration) on the top surface be fixed at Tc and the temperature on the bottom surface be T~. We normalize all lengths by the thickness h and let T-

T I - Ti - -

T~- Ti

(1)

The governing equation is then the Laplace's equation

02T 02T c9x2 + ~ =0

(2)

with the boundary conditions T = 1 on top surface and T = 0 on bottom surface. We shall present some analytic studies.

DIFFUSION ACROSS CORRUGATED PLATES

2

591

A p p r o x i m a t i o n for t h e small t h i c k n e s s case

When a >> 1 the thickness is small compared to the period of the corrugations. One may be able to estimate the total heat (mass) transfer by considering the corrugated plate as a series of fiat plates. Using the linear temperature distribution, the heat transfer per period, normalized by (conduetivity)(T,- T~)(depth) is between 4a and 4a + 2. Here the lower limit ignores the comer region while the upper limit included the corners as part of the fiat plate. The error is 0(1).

A

F"

[]

X

A

I B

0 CD

I E

F

Fig.2 Complex transform for the single comer. A better approximation is ~o study the effect of a single corner whose solution can be obtained from complex transforms. Fig.2 shows the complex plane t related to the physical complex plane z by (adapted from [3]) 2i . I /1 1 z=z+iy=l+i-~smV~-~+

cosh ~it

(3)

The solution in the t plane is

U(z)= T ( z , y ) +

i S(z,y) = 1 + i h a t

(4)

592

C.Y. WANG where T(x, y) is the temperature distribution. The temperature gradient along the surface AB is

aT

dU

~/t_ + '1 ' a-3~/dt' = ~ v t - ~

~(dU/dt~

=~(~)=

(5)

Since along AB the variable t is real and negative, let t = - s where s ranges from 1 to oo. Thus

OT V~- 1 Ox +-1

(6)

on the otherhand we find from Eq.(3) y = 1--sin

-1

dy_ d~

+

+

ln(s + x / ~ - 1)

1 ~s+l

~

(7)

(s)

;-1

For large s, Y"~

1 1 ln8+2 +-ln2+O(s-2)r

(9)

The total transfer from the corner is thus (10) The extra heat transfer solely due to the corner region is then

Q - 2(y-1) = 2-1ns l°~ -2(l lns ]°° +~ +11n2-1) + 2 = 1 - - - l n 2 + O ( s -2) 7r

= 0.5587 +

O(s-2)

(11)

DIFFUSION ACROSS CORRUGATEDPLATES

593

Thus a better estimation for the transfer per period of the corrugated plate is

Q = 4a + 2(1 -- 21n2) + O(e -2"a)

(12)

7r

A more significant measure is the heat transfer per nominal width of the corrugated plate Q - x/'2[1 - V~(2a + 1)

3

2 In 2 ~r(2a + 1) + O(e-2"~')]

(13)

Domain decomposition and eigenfunction expansions

For small a the above analysis fails. We shall use an eigenfunction expansion method for the solution. Decompose the plate into two regions with their own Cartesian axes as shown in Fig lb. Region 1 is the interior region of thickness 1 and length 2a. Its solution can be represented by oo

Tl(x, y) = x + y ~ sin(nrx)[Ane "~(u-a) + B,~e-"'~(~'+a)]

(14)

rt:l

where An and Bn are coefficients to be determined. Eq.(14) satisfies the boundary conditions that TI(1, y) = 1 and that Tl(0, y) = 0. However, we also note from geometry that T1 - 1/2 has polar anti-symmetry about the point (1/2,0). Thus 1

1

Ta(x,y) - ~ = -[TI(1 - x , - y ) - ~]

(15)

Substituting Eq.(14) we find the relationship B. = (-1)'~A,~

(16)

594

C.Y. WANG Now for Region 2, the corner region of unit square, the symmetry properties dictate T2(x,y) = T2(y,x). The most general representation, satisfying the boundary conditions T~(x,O) = T~(O,y) = 0 and T~(1, 1) = 1 is o~ T2(x, y) = xy + ~ Cn{sin(nTrx)[e n'~{u-~) - e -'~(y+,)] + n=l sin(nTry) [e nit(x-l) -- e-n~(x+')] } (17) Note that T2 is complete since there is an arbitrary Fourier series along the unprescribed boundaries. The two regions are then matched along their common boundary

T l ( x , - a ) -= T2(x, 1) Tiy(x,--a) T~y(x, 1)

(18) (19)

=

The Fourier components of Eq.(18) show (1 - e -2n'~) An -- (e_Tg~Tg_t- ( _ l ) . ) C n

(20)

Eq.(19) yields

k A"n~rsiu(nrcx)[ e-z'~`~ - (-1)"] = n----1

x + kCnnTr{sin(n~rx)(1 + e -2"~) + (-1)"[e "'(~-') - e-"'(~+a)]}

(21)

1 Using orthigonality of the eigenfunctions, Eq.(21) is Fourier inverted

A n~rre-2n'~-(-1) h I (__])n+i - - + n2t nTr

~.~n-~-( nTr.1 + c_2nr) +

mn(--1)m+" (~ -~m" - 1)

m=l

Cm ~ 7 - ~

(22)

The above infinite series is truncated to N terms, yielding the N linear algebraic equations < ( _ a ) . + ~-2..(o+1)] N 'n(-1)m+n -:m" 1) + C. ). Cm ~--gT-~- (e e -2n'~ + (-1 --(-1)~ 7/271- '

,~ = 1 to X

(23)

DIFFUSION ACROSS CORRUGATEDPLATES

The unkowns C~ are solved by standard methods. The accuracy can be improved by increasingN.

4

R e s u l t s and discussion

The series solution converged fairly fast. In generM 4 terms would give an error of less than 1% and for 10 terms the error decreases to 0.2%.

Fig.3 Isotherms for the case a = 0.5.

Typical isotherms for one period are shown in Fig.3. We see that the largest temperature gradient (he,st flux) occurs at the inside corner. The heat transfer per period is 1 07'2 "0,

595

596

C.Y. WANG N

4~

C.{[1 - ( - 1 ) " e - ' ] 2 +

(1 - e-2")(1

- e ....

) ff'-D"

[e- 2 ' ' " + ( - 1 ) " ]

'

"

+ e-"~"~]}(24)

The heat transfer per nominal width is

(2. + 1),/2

(25)

~

~

~

~

~

~

~

.

.

.

.

.

.

.

.

I

I

05

°O

I

I

i

I

025

05

025

I

Fig.4 Transfer per nominal width Q as a function of a. Dashed curve is from Eq.(13).

Fig. 4 shows Q as a function of the geometric parameter a. The value of decreases from 1.78 as the length a is increased from zero until it reaches a minimum of 1.183 near a = 0.54. Then Q approaches v/2 as a ~ oo. Our approximate formula Eq.(13) is highly accurate for a > 1. In comparison to the value of Q = I for the fiat plate of same thickness, corrugated plates conduct at least 18% more flux. For maximal transfer the value of a should be less than 0.1 where the increase is over 40%. We look forward to experimental confirmation of these results.

DIFFUSIONACROSSCORRUGATEDPLATES For the present problem our series solution is very efficient. The number of computations is less than the square root of that required by direct numerical integration by finite differences or finite elements. Further advantage of the method is that the derivatives and integrals such as those occuring in Eq.(24) can be analytically performed without introducing additional errors.

References [1] J. Crank, The Mathematics of Diffusion, Clarendon Press, Oxford. (1956). [2] H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids, 2~d Ed, Oxford Univ. Press.(1986). [3] R.L. Webb, Enhancement of single-phase heat transfer, in Handbook of Single-Phase Convective Heat Transfer, Eds. S. Kakac, R.K. Shah and W. Aung. Wiley, New York, Chap.17.(1987).

597