Page 367 - Modelling in Transport Phenomena A Conceptual Approach

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9.2. ENERGY TRANSPORT WITHOUT CONVECTION 347

9.2.2 Conduction in Cylindrical Coordinates

9.2.2.1 Hollow cylinder

Consider onedimensional transfer of energy in the r -direction through a hollow
cylinder of inner and outer radii of R1 and Rz, respectively, as shown in Figure
9.7. Let W be the rate of energy generation per unit volume within the cylinder.

T

E
Figure 9.7 Onedimensional conduction through a hollow cylinder with internal
generation.

Since T = T(r), Table C.5 in Appendix C indicates that the only non-zero
energy flux component is e, and it is given by

(9.2-22)

For a cylindrical differential volume element of thickness AT as shown in Figure
9.7, the inventory rate equation for energy, Eq. (9.2-l), is expressed as
W
2TL (TQr)I, - 2TL (TQT)lr+A, ~TTATL = O (9.223)
Dividing each term by 27rL AT and taking the limit as AT --+ 0 gives

(9.2-24)

Id
- - (rqr) = W (9.2-25)
r dr