Page 416 - Bird R.B. Transport phenomena

398 Chapter 12 Temperature Distributions with More Than One Independent Variable

(b) Show that the energy equation for this situation reduces to

pC^f = fc 0 (12В.4-2)

List all the simplifying assumptions needed to get this result. Combine the preceding two
equations to obtain
у § = ,30 (12B.4-3)

in which /3 = /jLk/p C g8.
2
p
(c) Show that for short contact times we may write as boundary conditions
B.C1: T = T 0 forz = 0 and у >0 (12В.4-4)
B.C. 2: T = T o for у = oo and z finite (12B.4-5)
B.C3: T = T, fory = 0 and z > 0 (12В.4-6)
Note that the true boundary condition at у = 8 is replaced by a fictitious boundary condition
at у = oo. This is possible because the heat is penetrating just a very short distance into the
fluid.
(d) Use the dimensionless variables 6(17) = (T — T^/{T - T ) and 77 = y/^9/3z to rewrite
A
o
the differential equation as (see Eq. C.I-9):
Щ + Ъг) 2Л® = Ъ (12В.4-7)
di) 2 drj
Show that the boundary conditions are в = 0 for 17 = 0° and © = 1 at 17 = 0.
(e) In Eq. 12B.4-7, set d<d/dr\ = p and obtain an equation for p{rj). Solve that equation to get
3
dS/diq = p{rj) = C exp (-17 ). Show that a second integration and application of the bound-
}
ary conditions give
3
I exp(-rj )drj
в = -^ = j - Г exp(-v )dv (12B.4-8)
К
-
3
3
exp(-^ )^ Г ( ^ *
Jo
(f) Show that the average heat flux to the fluid is
<?av|-o = | ^ «T, - T ) (12B.4-9)
gy
o
where use is made of the Leibniz formula in §C3.

12B.5. Temperature in a slab with heat production. The slab of thermal conductivity к in Example
12.1-2 is initially at a temperature T . For time t > 0 there is a uniform volume production of
o
heat S within the slab.
o
(a) Obtain an expression for the dimensionless temperature k(T - T )/S b 2 as a function of
0
0
the dimensionless coordinate 17 = y/b and the dimensionless time by looking up the solution
in the book by Carslaw and Jaeger.
(b) What is the maximum temperature reached at the center of the slab?
(c) How much time elapses before 90% of the temperature rise occurs?
2
Answer: (c) t ~b /a
12B.6. Forced convection in slow flow across a cylinder (Fig. 12B.6). A long cylinder of radius R is
suspended in an infinite fluid of constant properties p, /x, C , and k. The fluid approaches with
p