Page 222 - Modelling in Transport Phenomena A Conceptual Approach
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202 CHAPTER 7. UNSTEADY-STATE MACROSCOPIC BALANCES
Assumption # 4
"=[ 397 ]
(lg6) 100 >> 1
D; (8924)(387) 02
=
Comment: Note that Eq. (6) can simply be obtained from the first-law of thenno-
dynamics written for a closed system. Considering the copper sphere as a system,
AU = Qint -I- W =$ Qint = AU = m&AT 21 mCpAT
Example 7.7 A solid sphere at a uniform temperature of TI is suddenly im-
mersed in a well-stirred fluid of temperature To in an insulated tank (TI > To).
a) Determine the temperatures of the sphere and the fluid as a function of time.
b) Determine the steady-state temperatures of the sphere and the fluid.
Solution
Assumptions
1. The physical properties of the sphere and the fluid are independent of tern
perature.
2. The average heat transfer coefficient on the surface of the sphere is constant.
3. The sphere and the fluid have uniform but unequal temperatures at any in-
stant, i.e., Bi << 1 and mixing is perfect.
Analysis
a) Since the fluid and the sphere are at different temperatures at a given instant, it
is necessary to write two differential equations: one for the fluid, and one for the
sphere.
System: Solid sphere
The terms in Eq. (7.5-8) are
min = mo,t = 0
ws = 0
Qint = - (rD;)(h)(Ts - Tj)
