Page 346 - Bird R.B. Transport phenomena

330 Chapter 10 Shell Energy Balances and Temperature Distributions in Solids and Laminar Flow

2
2
Now introduce the dimensionless quantities £ = r/R and В = k R E /k LT 0 and show that Eq.
0
e0
10C.1-3 then becomes
( ) Вг- П0С.1-4)
k e0
When the power series expressions for the conductivities are inserted into this equation we get

2
- т 4 ( d - a,® - a ® 2 + • • • ) £ ? ! ) = B(l - 0,0 - )3 20 + • • •) (10C.1-5)
2
This is the equation that is to be solved for the dimensionless temperature distribution.
(b) Begin by noting that if all the a, and /3, were zero (that is, both conductivities constant),
then Eq. IOC. 1-5 would simplify to

When this is solved with the boundary conditions that 0 = finite at f = 0, and 0 = 0 at £ = 1,
we get
0 = \B(\ - f) (10C.1-7)
This is Eq. 10.2-13 in dimensionless notation.
Note that Eq. 10C-5 will have the solution in Eq. 10C.1-7 for small values of B—that is,
for weak heat sources. For stronger heat sources, postulate that the temperature distribution
can be expressed as a power series in the dimensionless heat source strength B:
2
2
0 = \B(1 -f )(l + B® } + B 0 2 + • • •) (10C.1-8)
Here the 0„ are functions of f but not of B. Substitute Eq. IOC. 1-8 into Eq. IOC. 1-5, and equate
the coefficients of like powers of В to get a set of ordinary differential equations for the 0,,, with
n = 1,2,3, These may be solved with the boundary conditions that 0,, = finite at f = 0 and
0 = 0 at £ = 1. In this way obtain
И
2
2
2
0 = \B{\ - £ )[l + 5 ( ^ ( 1 - ?) - TsfaO ~ f )) + O(B )] (lOC.1-9)
2
2
where O(B ) means "terms of the order of B and higher/'
(c) For materials that are described by the Wiedemann-Franz-Lorenz law (see §9.5), the ratio
k/k eT is a constant (independent of temperature). Hence

Combine this with Eqs. lOC.1-1 and 2 to get
1 - ttl® - a 2 0 2 + • • • = (1 - fa® - p 2® 2 + • • -)(1 + 0) (lOC.1-11)
Equate coefficients of equal powers of the dimensionless temperature to get relations among
the a, and the Д: a = y8 = 1, a = /3 + j8 , and so on. Use these relations to get
x T 2 } 2
2
2
2
0 = \B(\ - f )[l - ^8(03, + 2) + {(fa - 2)f ) + O(B )] (10СЛ-12)
10C.2. Viscous heating with temperature-dependent viscosity and thermal conductivity (Figs. 9.4-
1 and 2). Consider the flow situation shown in Fig. 10.4-2. Both the stationary surface and the
moving surface are maintained at a constant temperature T . The temperature dependences
o
of к and /x are given by
j- = 1 + a^® + a ® 2 + • • • (10C.2-1)
2
2
^ = ^ = 1 + 0,6 + /3 0 + • • • (10C.2-2)
2
in which the a, and Д are constants, <p = \/1± is the fluidity, and the subscript "0" means
"evaluated at T = T ." The dimensionless temperature is defined as 0 = (T - T )/T .
o 0 Q