Page 268 - Modelling in Transport Phenomena A Conceptual Approach

248 CHAPTER 8. STEADY MICROSCOPIC BALANCES WITHOUT GEN.

Example 8.1 Consider a solid wne of circular cross-section as shown in Figure
8.6. The diameter at z = 0 is 8cm and the diameter at z = L is 10cm. Calculate
the steady rate of heat transfer if the lateral surface is well insulated and the thermal
conductivity of the solid material as a function of temperature is given by

k(T) = 400 - 0.07T

where k is an W1m.K and T is in degrees Celsius.

L = 40 cmd

Figure 8.6 Conduction through a solid cone.

Solution

The diameter increases linearly in the z-direction, i.e.,

D(z) = 0.05 z + 0.08

Therefore, the cross-sectional area perpendicular to the direction of heat flu is
given as a function of position in the form

aD2 T
A(z) = - - (0.05 z + 0.08)2
=
4 4
The use of Eq. (A) in Table 8.1 with To = 80"C, TL = 35 "C and L = 0.4m gives
the heat transfer rate as

/80 (400 - 0.07 T) dT
Q= J35 = 280W
r0.4 dz
I,, 40.05 z + 0.08)2/4

Example 8.2 Consider the problem given in Example 2.2. Determine the tem-
perature distribution within the slab.