Appl. Sci. Res. 29

J a n u a r y 1974

LAMINAR

HEAT

TRANSFER

IN P A R A L L E L PLATE FLOW C. A. D E A V O U R S Dept. of Mathematies Kean College of New Jersey Union (New Jersey 07083) USA Abstract

An exact solution for the fluid temperature due to laminar heat transfer in parallel plate flow is found. The formulas obtained are valid for an arbitrary velocity profile. The basic problem encountered involves finding certain expansion coefficients in a series of nonorthogonal eigenfunctions. This problem is solved by passing to a vector system of equations having orthogonal eigenvectors. The method is applicable to more general problems. Nomenclature A B

cg C

/(Y) k

N~ T T~

T~oo

r~ V

2~ ~,+~ õ

matrix defined in (11) matrix defined in (11) expansion coefficients of T{ expansion coefficients of T~ specific heat \ ay] thermome{ric conductivity normalization factors for Z ~ respectively temperature particutar solution of (1) for x > 0 particular solution of (1) for x < 0 homogenons solution of (1) for x < 0 defined by (6) and (7) same as T{ b n t for x > 0 velocißy profile of fluid

(~~, ~~~ ' 1~.~ )

C+~la+~ CglZ~

(0, ~ T ( e ( y + 1) -- õ(y -- 1)) eigenvalne parameter for (8)

--

69 - -

70

P

¢g

C. A. DEAVOURS positive eigenvMues of (8) negative eigenvalues of (8) density of fluid eigenvectors of (8) corresponding to 2+ eigenvectors of (8) corresponding to 2~

Laminar heat transfer in parallel plat e flow

In this paper, we find t h e exact sohtion for heat transfer between parallel plates occupying the region y = ~= 1, --co < x < oo with the right half of the walls, x > 0, maintained at a constant temperature T = T+ and the left half of the walls maintained at a different constant temperature,' T = T-. Problems of this sort have been treated previously El] and redueed to the solution of two infinite sets of simultaneous algebraic equations which are then solved approximately by truncation. We shall find the exact solution of the problem for an arbitrary velocity profile V(y) across the plate (V(y) >_ 0 for [Yr ~< 1). The governing equation is:

ô2T a2T ax---T + ~y2 -

V~c k

aT ~x

# ( aV \~ I--] k \ ay )

(1)

with boundary conditions

T(x,~I)=T+

for

x>0,

(2)

T(x,~I)=T-

for

x<0

(S)

plus some boundary conditions at infinity which arise later in a natural manner. Let T1 denote the solution of our problem for x < 0 and ler T2 denote the solution for X > 0. We shall find these Solutions separately and then match them at x = 0 by requiring t h a t the solution and its normal derivative (and hence all higher derivatives) shall agree at x = 0, i.e.

Tl(o, y) = T2(O, y),

aT1 (0, y) -ôx

OT2

(0, y),

--1 < y < I, -- 1 < y < 1.

(4) (5)

äx

Since the velocity profile V(y) is assumed known, we denote the

L A M I N A R H E A T T R A N S F E R IN P A R A L L E L P L A T E F L O W

71

inhomogenous term in (1) b y / ( y ) ,

k \ay/

=/(y)

Consider first the solution in the region x > 0. If we assume t h a t T tends uniformly to some function independent of x as x tends to infinity, t h e n all derivatives with respect to x t e n d to zero and the limiting equation for x -+ + o o becomes

82T~ _

_

8y 2

= l(y)

whose general solution is readily seen to be

T ~ = A + Æy + ~.~ (y -- s) l(s) ds. In view of (2) we m u s t chose A =

T + - - Sò (1 - - s) l(s) dt,

B=O. We have t a k e n / ( y ) = / ( - - y ) for simplicity of the algebra. As x --> - - o o, we obtain the equation

8y 2

--/(y)

whose general solution is

T_oo = C + Dy + Ig (Y -- s) l(s) ds so that, using (3), we solve for C and D to find t h a t C = T - - - ..[~ (1 - - s ) / ( s ) ds, D=O.

The solntions T±oo are also particnlar solutions of (1) for the regions x > 0 and x < 0 respectively. L e t

Tl(x, y) = T ,.(y) + TE(x, y) and T2(x, y) = T+~(y) + Zr~(x, y)

72

C.A. DEAVOURS

then T~ taust satisfy the equation - -

ôx 2

+

--

ôy2

k

ôx

(6)

for x < 0 with boundary conditions

T~(x, 4-1) ---- 0.

(7)

The function T~ satisfies the same equation and boundary conditions for the region x > 0. We seek separated solutions of (6) in the form

Tsep(x, y) = ~b(y) e -zx. Substitution in (6) yields the equation for ~b, which is ,d/, + (;t2 + 2 qcV(y)) k ~ = 0

(8)

with boundary conditions ~b(~ 1) = 0 to satisfy (7). Equation (8) and its associated boundary value problem constitute an eigenvalue problem, though not of the classical Sturm-Liouville type since the eigenvalues occur nonlinearly in (8). Equation (8) has two sets of complete eigenfunctions, one corresponding to the positive eigenvalues, 4+~, and one corresponding to the negative set of eigenvalues, 2~. An arbitrary function m a y be expanded in terms of either set although the eigenfunctions are not orthogonal so that the coefficients in the expansion are not calculable exactly. In terms of the separated solutions we have

T~(x, y ) =

E Cg~b~(y) e -a=" Rn-

where the C~ are constants and ~bx is the eigenfunction corresponding to ,~~ (unique to a constant multiple). The summation extends only over the ,~~, which are all less than zero, since T~ solves the problem in x < 0. Similafly T~(x, y) = - - Y, C+ ~,+ (y)

e -~~"

Bh+

which holds in x > 0 (minus sign introduced for convenience).

LAMINAR

HEAT TRANSFER

IN PARALLEL

PLATE

FLOW

73

Our problem is then completed if the matching conditions (4) and (5) can be satisfied. These expressed b y requiring that Tl(O, y) + T_oo(Y) = T~(O, y) + Too(y), oTi (O,y)-- aT~ (O,y), Ox Ox

--1 < y < 1

--1 < y <

1

or, in terms of the series solutions Y~ C~~ß;(y) + Y~ C+¢+(y) = Too -- T_oo = A T 2n-

(9)

~.n+

X C;,~;~b;(y) q- Z C+2+¢+(y) = 0. Bh-

(10)

2n +

Since Too - - T_oo = A T satisfies the equation (d2/dy 2)(AT) = 0 with boundary conditions A T ( + 1) then A T = constant = T + -- T as is readily verified from our previous expressions for Too and T_~o. Since $(q-l) = 0 then the expansion can never hold at y---- ~=1 unless A T = O. Of course at x = 0, y = :kl, the boundary conditions are diseontinuous and so will be the solution. Thus, (9) is required to converge to A T = constant for --1 < y < 1 and to 0 for y = =kl in the manner of a trigonometric Fourier series expansion. Due to the aforementioned nonorthogonality of the eigenfunctions numerical methods seem appropriate at this point. However, the problem is solvable without approximate methods b y converting the nonlinear eigenvalue problem of (8) to an equivalent vector system of differential equations in which the eigenvalue occurs linearly and in which the eigenfunction vectors are orthogonal. To this end ler

¢~'= 4" ¢(n"IZn : z [ ~>. ~n will denote the vector (1 (1) Z~~)) We see that Z(n1)'

;t 7(2)

and ocV

o}.

74

C . A . DEAVOURS

these two equations m a y be combined into the vector equation (11)

Z;~ = ;~nA2n + B 2 ~

with Z(~~)(± 1) ----0.

(o i)

Hefe, A=

--1

k

Equations of type (11) have been extensively studied, see for instance [2]. The eigenvector solutions of (11) are orthogonal and complete. To verify the orthogonality suppose 2m is an eigenvector of (11) corresponding to km, then (12)

2 ~ = Z~A 2 ~ + B 2 ~

Z~)(~:l) = 0. Multiplying (l l) by the matrix A and taking the dot product of the resulting equation by --2m, we arrive at - 2 m . 2;~ = - G 2 ~ .

2~~ - 2 m A B 2 ~ .

A similar procedure for (12) yields --2n" ~'.m = --,lm2m "2n -- 2~ "AB2ra.

Subtraction of these two equations can be written after some cancellations and algebra - - ( 2 m . A 2 ~ ) ' = (Z. -- ~.~) 2 n 2 m

for whence we find (,~, - ,~,~) [21 2 ~ . 2 m dx -= - 2 , ,

A2~121 = o.

Thus Itl 2~.2m dx = 0, n ¢ m . An arbitrary vector (Il(Y),/2(Y)) is expandable in terms of the eigenvectors 2n in the form

('0 where an = I+l P ' 2 n dx/I_+l 2 n ' 2 n dx

LAMINAR HEAT T R A N S F E R IN PARALLEL PLATE FLOW

75

in the usual manner. The convergence, termwise differentiation and integration properties of these eigenvector expansions are almost identical with those corresponding properties of ordinary Fourier series expansions. The reader is referred to [2] for details. We now ready to complete the solution of our problem. Equations (9) and (10) can be written as

E 7~Z~ ) + E rs'+7(1)»~= 0 3,~-

~n+

where = I'~/;t~. The first series above is divergent in the ordinary sense but convergent in the generalized sense to (AT)' = AT[~(y + 1) -- d(y -- I)]

where ô is the usual Dirac delta function. This results from the lump discontinuity of (9) at y = ~ 1 * The two previous equations are equivalent to the one vector equation ~?,~_~~__F~ + + ( 0 ) 7~Z~ = A T

~ù-

~o+

~(y + 1) - õ(y - 1)

This is an expansion of the previous type. Thus, the matehing conditions are satisfied if --

1

f + l ô.~~ dx

where Ö =AT

( d(Y-[- 1 ) -0- d ( y - - 1)),

N,~ = ?-I ~~~" Z~± dx

and if

1 f ÷lô.~~+dx. * Actually, since the discontinuities lie at the end points of the interval of convergence, --1 ~ y _ < + 1 , we could have also replaced ~(y + 1) by ½8(y + 1) and 3(y -- I) by ½3(y -- 1); however, any such notational device is valid if consistently followed, [3].

76

LAMINAR HEAT TRANSFER IN PARALLEL PLATE FLOW

Explicit calculation of the coefficients yields AT 7~ -- ;~~ [~b~'(1) -- ~~(--1)~.

(13)

Hence the complete solution of the problem is given by =-fTI(x'Y) T(x,y) (T2(x,y)

for for

x<0 x > 0

where Tl(x, y) = T_~o(y) + Y~ ),g,~g$g(y) e -~~~ Tg,(x, y)

=

T +oo(y) + Y~ y ~+,~~ + ~,~÷ ( y ) e -~~'~ . $.n+

The above method is applicable to other problems of this type and other types of equations ~4], [5J. Received 21 July 1971 In final form 6 August 1973

REFERENCES El] AGRAWAL,H. C., Appl. Sci. Res. Sect. A, 9 (1959/60) 177. [2] LANGER,R. E., Trans. American Math. Society, Oct. (1929) 868. [3] FRIEDMAN,B., Principles and Techniques of Applied Mathematics, John Wiley and Solls, New York, 1956, 154. [4] DEAVOURS,C. A., J. on Math. Analysis (SIAM), I # 2 (1971) 168. [Õ] DEAVOURS,C. A., J. Appl. Mech. 38 (1971) 708.