Page 368 - Modelling in Transport Phenomena A Conceptual Approach

348 CHAPTER 9. STEADY MICROSCOPIC BALANCES WITH GEN.

Substitution of Eq. (9.2-22) into Eq. (9.2-25) gives the governing equation for
- 1 r dr (9.2-26)
temperature as

Integration of Eq. (9.2-26) gives
IC- dT = -ilrR(u)udu+- Cl (9.2-27)
dr r
where is a dummy variable of integration and C1 is an integration constant.
Integration of &. (9.2-27) once more leads to

Evaluation of the constants CI and CZ requires the boundary conditions to be
specified.
Type I boundary condition

The solution of Eq. (9.2-28) subject to the boundary conditions
at r=R1 T=T1
(9.2-29)
at r= R2 T =Tz
is given by

+I"' : [lr R(+du] dr (9.2-30)

Note that when R = 0, Eq. (9.2-30) reduces to Eq. (C) in Table 8.3. Equation
(9.2-30) may be further simplified depending on whether the thermal conductivity
and/or energy generation per unit volume are constant.

Case (i) k = constant
In this case Eq. (9.2-30) reduces to

+ 1" [Jd' R(u) udu] dr (9.2-31)

When ?R = 0, Eq. (9.2-31) simplifies to Eq. (D) in Table 8.3.