Page 395 - Bird R.B. Transport phenomena

§12.1 Unsteady Heat Conduction in Solids 377

Hence Eq. 12.1-14 can be satisfied by
2
2
©„ = A C n expKn + l) 7T r] cos (n + 5)7777 (12.1-25)
n
The subscripts n remind us that A and С may be different for each value of n. Because of the
linearity of the differential equation, we may now superpose all the solutions of the form of
Eq. 12.1-25. In doing this we note that the exponentials and cosines for n have the same values
as those for -(n + 1), so that the terms with negative indices combine with those with posi-
tive indices. The superposition then gives

2
2
В = 2 Д, ехр[-(и + ^) 7г т] cos (n + 5)7777 (12.1-26)
J7 = 0
in which D n = А„С„ + Л_ „ С_ (н + 1) .
(
+1)
The D are now determined by using the initial condition, which gives
n
1 = f D n cos (n + l)irri (12.1-27)
/; = 0
Multiplication by cos(m + 5)7777 and integration from ту = - 1 to ту = +1 gives

r +1 •> с + I
cos (m + 5)7777 dr] = 2 Ц, cos (m + 5)7777 cos (w + 5)7777 d-ц (12.1-28)
•'-l и = 0 • ' - I
When the integrations are performed, all integrals on the right side are identically zero, ex-
cept for the term in which n = m. Hence we get
sin (m + 5)7777 \{m + 5)7777 + J sin l(m + 5)777?
(12.1-29)
(m + 5)77 (m + 5)^
After inserting the limits, we may solve for D m to get

D m = ^ " T (12.1-30)
(m + 5)77
Substitution of this expression into Eq. 12.1-26 gives the temperature profiles, which we now
rewrite in terms of the original variables 2

2
^ ^ - 2 2 ^ ^ - ехр[-(и + ^)Vaf/b ] cos (w + ^) ^ (12.1-31)
ii - io »=o (и + 5)77 t?
The solutions to many unsteady-state heat conduction problems come out as infinite series,
such as that just obtained here. These series converge rapidly for large values 2 of the dimen-
sionless time, at/b . For very short times the convergence is very slow, and in the limit as
2
at/b 2 approaches zero, the solution in Eq. 12.1-31 may be shown to approach that given in Eq.
12.1-8 (see Problem 12D.1). Although Eq. 12.1-31 is unwieldy for some practical calculations,
a graphical presentation, such as that in Fig. 12.1-1, is easy to use (see Problem 12A.3). From
the figure it is clear that when the dimensionless time т = at/b 2 is 0.1, the heat has "pene-
trated" measurably to the center plane of the slab, and that at т = 1.0 the heating is 90% com-
plete at the center plane.
Results analogous to Fig. 12.1-1 are given for infinite cylinders and for spheres in Figs.
12.1-2 and 3. These charts can also be used to build up the solutions for the analogous heat
conduction problems in rectangular parallelepipeds and cylinders of finite length (see Prob.
12C.1).

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd edition, Oxford University Press
2
(1959), p. 97, Eq. (8); the alternate solution in Eq. (9) converges rapidly for small times.