Page 468 - Modelling in Transport Phenomena A Conceptual Approach

448 CHAPTER IO. UNSTEADY MICROSCOPIC BAL. WTHOUT GEN.

Table 10.3 The roots of Eq. (10.2-79).

BiH A1 A2 A3 x, A5
0 0.000 3.142 6.283 9.425 12.566
0.1 0.311 3.173 6.299 9.435 12.574
0.5 0.653 3.292 6.362 9.477 12.606
1.0 0.860 3.426 6.437 9.529 12.645
2.0 1.077 3.644 6.578 9.630 12.722
10.0 1.429 4.306 7.228 10.200 13.214

The unknown coefficients C, can be determined by using the initial condition given
by Eq. (10.2-70). The result is

- 2 sin A, (10.281)
-
A, + sin A,, cos A,
Therefore, the solution becomes
I e=2C A, + sin A, cos A, e- X:T cos(^,<> (10.2-82)
m
sin A,
n=l

When r 1 0.2, the series solution given by Eq. (10.2-82) can be approximated by
the first term of the series.
The rate of energy entering into the slab, Q, is given by
( :IZ=,>
Q=2WH k-
L %I,=,
- 2 WHk (T, - To) 80 (10.283)
--

Substitution of Eq. (10.2-82) into Eq. (10.2-83) gives

4 WHk (T, - To) An sin2 A,
Q= L An + sin A, cos exp (- A2,T) (10.2-84)
n=l
The mount of heat transferred can be calculated from

(10.285)