STEADY DIFFUSION OR CONDUCTION W I T H A VARIABLE COEFFICIENT Theorems are presented concerning the steady-state concentration or temperature distribution and rate of diffusion of mass or heat in a fixed medium in which the flux is given by Fick’s or Fourier’s law with a coefficient depending on concentration or temperature only.
THIS note presents some theorems
concerning the steadystate concentration or temperature distribution and rate of diffusion of mass or heat in a fixed medium of any shape in which the flux is given by Fick’s or Fourier’s law with a coefficient which depends upon concentration or temperature only. Theorem I. If no sources or sinks are present and on all or part of the boundary of the medium the concentration or temperature is fixed at two different constant values, C1 and Cz, the remaining part being impervious, (A) the concentration or temperature, Cl, at each point in the medium is given by
where eI depends only upon the geometry and (B) the rate of transfer through the medium (taken as positive when the flow is inwards on the SIsurface) is given by
CI
=
Sc,
( X W )
(4)
Equation 3 is reduced to Laplace’s equation V%l = 0
(5)
with
and the solution is
since, from Equation 5, €1 can depend only upon position. This proves Theorem IA. With the aid of Equation 4 el can be related to the concentration which would be obtained if the coefficient were constant where K is a positive constant which depends only upon geometry and is proportional to the characteristic length for geometrically similar systems. Theorem I includes diffusion through the interstices of a porous material whose walls are impervious (in which case the medium is the substance inside the pores) and is a generalization of a result obtained earlier for a porous disk ( 2 ) . I t is also a generalization of the usual result given for unidirectional transfer (7). Equation 2 indicates that a rate expression developed for a system with a constant coefficient may be applied to a system with a varying coefficient by replacing the coefficient in the former expression by the integral mean coefficient. The proof is similar to that given by Toor ( 2 ) . Fick’s or Fourier’s law and the continuity equation yield (3) for every point in the medium. If S, is the surface (which may be made up of unconnected parts) on which the concentration is Cr and So the surface (which also may be made up of unconnected parts) which is impervious, the boundary conditions are
CI(S1) = Cl Cl(S2)
=
cz
(34
(7) The flux is given by j I =
-D(CI)vCl
=
-
l:
D(C)dC Vel
(8)
so the rate of transfer into the medium across St is given by
The term in brackets depends only upon geometry and examination shows that it must be a positive constant on the SI surface, so Theorem IB follows. We also have N,, N12= 0. I t follows simply from dimensional analysis that for geometrically similar media K is proportional to the characteristic length. Theorem 11. When sources or sinks are present which depend upon position only and on all or part of the boundary the concentration or temperature is fixed at the constant value, C1, the remaining part being impervious, (A) the concentration or temperature CII at any point in the medium is given by
+
and everywhere on the Sosurface
With the variable 152
l&EC FUNDAMENTALS
where eII depends only upon the geometry, and intensity and distribution of the sources and sinks, and (B) the distribution
of the mass or heat flow across the boundary is independent of the value of C1. The equation to be satisfied inside the medium is
+
V [D(CII)VC~II G(x,y,~)= 0
(11)
Crr(S1) = CII(S2) = Cl
(11a)
with
____
bCII(Y0)
=
so the rate of transfer through the medium can be considered to be the same as would be obtained in the absence of sources or sinks. The proof of the above is obtained by noting that CIII must satisfy the same equation as CrI (Equation 11) but with the boundary values
(1lb)
3%
CIII(S1) =
c1
(184
The proof of the theorem is obtained from the transformation
s,
CII (x,y,r)
err =
D(C)dC
(12)
which reduces Equation 11 to Poisson's equation,
=o
(13)
and by direct substitution Equation 16 is seen to be the solution. The rate of flow inwards across the surface St from Equation 16 is
with
n
J Si and Theorem IIB follows from Equations 19, 9, and 15 when 611
= erdx,
y,
p in Equation 15 is taken as i.
4
(14)
which yields Theorem HA. Because of the linearity of Equation 13, the integral in Equation 30 may be considered to be the sum of the contributions of the various sources or sinks in the medium; each point source or sink (of differential strength) contributes the quantity D(C)dC to the integral in Equation 10. Equation 12 also relates €11 to the conditions existing when the coefficient is constant EII(X,
y,
4
=
D*[C*II(X, y,
2)
- Cll
(14a)
Since the first term is the flow which would be obtained in the absence of sources and sinks and the second the flow caused by the sources and sinks, the first term may be considered to be the flow through the medium. The above results may be generalized to a medium whose boundary is fixed at more than two different concentrations or temperatures or on all or part of which there is a given continuous variation of concentration or temperature. If there are no sources or sinks, for example, and the boundary is divided into 7 surfaces each at a concentration C,, the remaining surfaces being impervious, the concentration or temperature at any position in the medium is given by
The rate of transfer (inwards) across any part of the boundary is given by
from which Theorem I'[B is obtained. Theorem 111. When sources or sinks are present which depend only upon position and on all or part of the boundary the concentration or temperature is fixed at two different values, C1 and Cz, the remaining part being impervious, (A) the concentration or temperature CIII at any point in the medium is given by
s,
CUI
(X,),Z)
D(C)dC =
€I(X,
y,
4
CI ( x , y , z )
+ ll
+
EIl(X,
y,
2) =
CI1 (x,u,z)
D(C)dC SCl
D(C)dC
where e, solves Laplace's equation, is 1 on surface k, and zero on all other surfaces except the impervious ones, where its normal derivative is zero. Similarly the rate of transfer across any part of the boundary, say surface S,, is given by
where K j depends upon geometry. The result when the concentration or temperature varies continuously is obtained by going to the limit in the above equations.
D(C)dC (16) Nomenclature
where e, and eII are the functions appearing in Theorems I and I1 and CI and CII arc the concentrations or temperatures obtained for the conditions of Theorems I and 11, it being understood that surfaces SI,S2,and So are chosen the same in all three cases, and (B) the rate of transfer across either surface is the sum of the rates obtained in Theorems I and 11, due account being taken of sign,
C
$G j K N n0
= = =
= = = = =
concentration or temperature diffusion coefficient or thermal conductivity integral mean value of D with respect to C rate of generation of mass or heat per unit volume vector flux of mass or heat positive constant inwards rate of transfer of mass or heat distance in direction normal to So surface VOL. 6
NO. 1
FEBRUARY 1967
153
n
= unit vector directed outwards and normal to
surface = surface = distance coordinates = gradient operator = divergence operator = defined variously by Equations 4 and 12 = number of surfaces on which concentration is fixed
S x, y , z
V V2 e
7
SUPERSCRIPTS
*
= value when coefficient is constant
I
=
reference value
literature Cited (1) McAdams, W. H., “Heat Transmission,” 3rd ed., p. 12, McGraw-Hill, New York, 1954. (2) Toor, H. L.,J . Phys. Chem. 64, 1580 (1960).
H. L. TOOR
SUBSCRIPTS I, TI, I11 = conditions of theorem I, 11, or I11 = surfaces 0, 1, 2 0, 1, 2 i = lor2 i,k = indices = any part of boundary P
Carneeie Institute of Technoloey -. PittsLrgh, P a .
RECEIVED for review June 28, 1966 ACCEPTEDSeptember 2; 1966 Work carried out while author was at Alagappa Chettrar College of Technology, University of Madras, Madras, India
FIN EFFICIENCY FOR THIN SPHERICAL SHELLS The fin efficiency, q, for a thin spherical shell heated on a circular edge at uniform temperature, T, and m m-1 i / cooled by convection with a uniform heat transfer coefficient, h, is derived as q = !. = 1 II ( A
+
J
n
+ n2)xom/(m!)2(m+ 1 )
o)
m-1
(A
m=l
+
n=O
+ n + n2)xom/(rn!)zwhere A = hr2/k6, dimensionless; xo =
+
(cos 0 0 1)/2; r, 8, k = radius, thickness, thermal conductivity of the shell; 0 0 = angle from the pole above to the isothermal circle at J; i= average temperature of the shell. The computed results for 0 6 xo 1 and 0,001 A 1000 are summarized in a graph.
