Page 374 - Modelling in Transport Phenomena A Conceptual Approach

354 CHAPTER 9. STEADY MICROSCOPIC BALANCES WITH GEN.

The energy flux at r = RF is continuous, i.e.,

dTF
dTc
- kF - - kc - (4)
=
dr dr
Substitution of Eqs. (2) and (3) into Eq. (4) gives

Therefore, the temperature distribution given by Eq. (2) becomes

9.2.3 Conduction in Spherical Coordinates

9.2.3.1 Hollow sphere

Consider one-dimensional transfer of energy in the r-direction through a hollow
sphere of inner and outer radii of R1 and R2, respectively, as shown in Figure 9.8.
Let ?R be the rate of generation per unit volume within the sphere.

Figure 9.8 Onedimensional conduction through a hollow sphere with internal
generation.

Since T = (r), Table C.6 in Appendix C indicates that the only non-zero energy
flux component is e, and it is given by

dT
e, = qr = - k - (9.2-49)
dr