Page 294 - Modelling in Transport Phenomena A Conceptual Approach

274 CHAPTER 8. STEADY MCROSCOPIC BALANCES WTHOUT GEN.

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Figure 8.21 Conduction through an insulated hollow sphere.

Differentiation of X with respect to R3 gives

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

Therefore, the critical thickness of insulation for the spherical geometry is given by

2 kj
R -- (4)
cr - (hB)
A representative graph showing the variation of heat transfer rate with insulation
thickness is given in Figure 8.22.
Another point of interest is to determine the value of R", the point at which
the rate of heat loss is equal to that of the bare pipe. Following the procedure given
an Example 8.7, the result is

R" can be determined from Eq. (5) for the given values of Rz, (hB), and kj.