Page 322 - Bird R.B. Transport phenomena

306 Chapter 10 Shell Energy Balances and Temperature Distributions in Solids and Laminar Flow

Fig. 10.6-2. Heat conduction through a lami-
nated tube with a fluid at temperature T n in-
side the tube and temperature T h outside.
Fluid at
temperature T t
outside tube

which can also be written as
Region 01: (2irrLq r)\ r - (2mLq r)\ r+£ir = О (10.6-18)
Dividing by 2ITL Ar and taking the limit as Ar goes to zero gives

Region 01: — (rq ) = 0 (10.6-19)
r
Integration of this equation gives

(10.6-20)
in which r is the inner radius of region 01, and q is the heat flux there. In regions 12 and 23,
0
0
rq is equal to the same constant. Application of Fourier's law to the three regions gives
r
dT
Region 01: (10.6-21)
Region 12: (10.6-22)

Region 23: = r q (10.6-23)
o o
If we assume that the thermal conductivities in the three annular regions are constants, then
each of the above three equations can be integrated across its region to give

Region 10: (10.6-24)
In (r /r )
Region 12: 12 = Wo - 2 } (10.6-25)