Page 286 - Modelling in Transport Phenomena A Conceptual Approach

266 CHAPTER 8. STEADY MICROSCOPIC BALANCES WITHOUT GEN.

This conclusion is true only for planar surfaces. In the case of curved surfaces,
however, close examination of Eq. (8.2-32) indicates that while addition of insu-
lation increases the thickness, i.e., R2 - R1, it also increases the heat transfer
area, i.e., ALM. Hence, both numerator and denominator of Eq. (8.2-32) increase
when the insulation thickness increases. If the increase in the heat transfer area is
greater than the increase in thickness, then resistance decreases with a concomitant
increase in the rate of heat loss.
For the geometry shown in Figure 8.17, the rate of heat loss is given by

X

where k, and ki are the thermal conductivities of the wall and the insulating
material, respectively.

Figure 8.17 Conduction through an insulated cylindrical pipe.

Note that the term X in the denominator of Eq. (1) is dependent on the insulation
thickness. Differentiation of X with respect to R3 gives

To determine whether this point corresponds to a minimum or a maximum value,
it is necessary to calculate the second derivative, i.e.,