Page 397 - Bird R.B. Transport phenomena

§12.1 Unsteady Heat Conduction in Solids 379

EXAMPLE 12.1-3 A solid body occupying the space from у = 0 to у = oo i s initially at temperature T . Beginning
o
at time t = 0, a periodic heat flux given by
Unsteady Heat
Conduction near a q v = q 0 cos a>t = (12.1-32)
Wall with Sinusoidal is imposed at у = 0. Here q is the amplitude of the heat flux oscillations, and ш is the (circu-
Heat Flux lar) frequency. It is desired 0 to find the temperature in this system, T(y, t), in the "periodic
steady state" (see Problem 4.1-3).

SOLUTION For one-dimensional heat conduction, Eq. 12.1-2 is
дТ д Т (12.1-33)
~М =СХ 2
М ду
Multiplying by -A: and operating on the entire equation with д/ду gives
d_ dT (12.2-34)
dt ¥

or, by making use of q XJ = -к(дТ/ду),
д\
(12.1-35)
dt
Hence q satisfies the same differential equation as T. The boundary conditions are
y
B.C.I: at у = 0, (12.1-36)
B.C. 2: at у = oo, (12.1-37)
This problem is formally exactly the same as that given in Eqs. 4.1-44, 46, and 47. Hence the
solution in Eq. 4.1-57 may be taken over with appropriate notational changes:

COS ( Q)t - (12.1-38)

Then by integrating Fourier's law

-k\ dT=\ q {y, t) dy (12.1-39)
y
J j J у
Substitution of the heat flux distribution into the right side of this equation gives after inte-
gration

(12.1-40)

Thus, at the surface у = 0, the temperature oscillations lag behind the heat flux oscillations by
тг/4.
This problem illustrates a standard procedure for obtaining the "periodic steady state" in
heat conduction systems. It also shows how one can use the heat conduction equation in
terms of the heat flux, when boundary conditions on the heat flux are known.

EXAMPLE 12.1=4 A homogeneous solid sphere of radius R, initially at a uniform temperature T u is suddenly
immersed at time t = 0 in a volume Vj of well-stirred fluid of temperature T in an insulated
o
Cooling of a Sphere tank. It is desired to find the thermal diffusivity a = k /p C of the solid by observing the
ps
s
s
s
in Contact with a change of the fluid temperature Tf with time. We use the following dimensionless variables:
Well-Stirred Fluid T — T
, T) = ^r—^r = dimensionless solid temperature (12.1-41)