9.35. Tubular heat-transfer surfaces are used in many heat exchangers, and the performance of these can be analyzed on the basis of heat con­duction through hollow circular cylinders with convection at both interior and exterior surfaces. The physical system in cross section and the thermal — circuit diagram are shown in Fig. 9.2; it is assumed that all the heat flow is in a radial direction. The inner and outer radii of the cylinder are a and b, respectively; ha and hb are the heat-transfer coefficients for the inner and outer fluids, and ta and tb are the mixed-mean temperatures of the inner and outer fluids, respectively.

9.36. The convection thermal resistances are equal to l/haAa and l/hbAb, where Aa and Ab are the areas of the inner and outer surfaces. The thermal conduction resistance of the cylinder wall remains to be de­termined. If lis the length of the cylinder, then the heat-flow rate, expressed at an arbitrary radius r, due to conduction within the wall is given by equation (9.6) as

q = — k(2irrl) —,

where к is the thermal conductivity and 2wl is equivalent to A, the area through which heat flow occurs. This expression may be rearranged and integrated over the wall thickness, i. e., from r = a to r = b; the result

Fig. 9.2. Heat conduction in a hollow cylinder with convection boundaries.

THERMAL CIRCUIT

is

q ln(b/a)/2irJt/’ (911′)

where M is the difference in temperature across the cylinder wall.

9.37. It is evident from equation (9.11) that the thermal resistance Rw of the wall is given by

= In (b/a) w 2irkl ‘

The total thermal resistance of the system under consideration is conse­quently

___ 1_ In (b/a) _1______ 1__

~ h. A„ + 2тткі + hbAb ~ UbAb’

where Ub is the overall heat-transfer coefficient based on the outer surface of the cylinder. According to the thermal-circuit concept

Я ~ ^ ~ UbAb(ta tb)

or

where

_______ 1_______

b b In (b/a) 1 [4]

Kp + * + к