I
TUNG TSANG Department of Chemistry, University of Chicago, Chicago 37, 111.
Transient State Heat Transfer and Diffusion Problems This method can be used when the diffusion coefficient, thermal conductivity,
Y
T H E general equations for transient state heat conduction in solids and for diffusion (in systems where Fick’s law of diffusion is obeyed) are given by:
cp
aT/bt = div ( k grad T )
bC/bt = div (Dgrad C)
7
(1) (2)
The exact solutions of these equations are well known when k, c p , and D are constants. I n general, however, k and c p are functions of T , and D is a function of C. Because of the nonlinearity, it is then very difficult to solve Equations 1 and 2 analytically. Fortunately, in many transient state heat transfer and diffusion problems, k , c p , and D vary only slightly over the whole temperature and concentration range. Therefore, a method similar to time-dependent perturbation methods in quantum mechanics ( 8 ) may be used to obtain an approximate solution. However, one important modification must be made. I n quantum mechanics, the solution usually begins with a pure state a t t = 0, and then the perturbation is applied. The same method cannot be applied to heat transfer and diffusion problems, because a finite nonzero perturbation, no matter how small, will have a large effect on the solution if the time during which the perturbation is applied is long enough. Hence, an unperturbed system must be chosen such that the perturbation becomes zero as time approaches infinity. This means the form of Equations 1 and 2 at t = a must be used as the unperturbed solution. The solution of transient state heat conduction for an infinite slab, a sphere, and an infinitely long cylinder assuming uniform initial temperature distribution and perfect contact with an environment a t constant temperature is given here, u p to the first order. Generalization to other heat conduction problems is also discussed. The diffusion equation (Equation 2 ) may be taken as a special case of the heat conduction equation (Equation 1) merely by setting c p = 1 and k = D. It is assumed that k , cp, and D approach constant values as time approaches infinity,
and
volume
heat
capacity
One-Dimensional Heat Conduction Consider a slab of thickness L and infinite area, with initial temperature T I and in perfect contact with a uniform environment a t temperature To. The two surfaces of the slab are denoted by x = 0 and x = L. By introducing dimensionless variables = k0co-l X po-1L-2rr2t,z = A X / L and , 4 = ( T - To)/ ( T I - To),Equation 1 may be written as:
The initial and boundary conditions are : Atr=O:
+ = l
Atr=m:+=O At z = 0 and
T:
=0
(4)
are
not constant
+
When E 0, the solution of Equation 3 may be expanded into a series in $,@)’s, as ( 2 / r ) l I 2sin nz with n = 1,2,. . . forms a complete set of orthonormal functions, and any function which vanishes at z = 0 and z = A may be expanded into such a sine series. Thus rp may be written as:
The coefficients A,Is are constants, and B,, C,, . . . are functions of 7 only. Substitution of Equations 7 to 12 into Equation 3 gives :
(5)
(6)
If k and c p vary only slightly over the whole range of temperature, distance, 1, 0 4 2 A) and time (0 5 q5
0
s
5 7):
1
+
Gf
= k/ko
(7)
where e is the parameter of perturbation (E l),f and g are defined by Equations 7 and 8 and are in general functions of z, 7,and 4. Temporarily omit the initial condition (Equation 4). When E = 0, a set of unperturbed solutions of Equation 3 satisfying Equations 5 and 6 is known as :
