Page 371 - Modelling in Transport Phenomena A Conceptual Approach

9.2. ENERGY TRANSPORT WITHOUT CONVECTION 351

at
The temperature, TI, T = R1 is given by

Substitution of the numerical values into Eq. (2) gives

280 = T2 + (5 x 106)(1.8 x 10-2)2
4 (0.5)
+ (5 x 106)(1.5 x 10-2)2
2 (0.5)

9.2.2.2 Solid cylinder
Consider a solid cylinder of radius R with a constant surface temperature of TR.
The solution obtained for a hollow cylinder, Eq. (9.2-28) is also valid for this case.
However, since the temperature must have a finite value at the center, i.e., T = 0,
then Cl must be zero and the temperature distribution becomes
I' k(T)dT= -1'; [I'%(u)udu] dr+Cz (9.2-41)

The use of the boundary condition
at r=R T=TR (9.2-42)
gives the solution in the form
6 (9.243)
k(T) dT = 1" f [Jd' %(u) udu] dr

N Case (i) k = constant
Simplification of E&. (9.2-43) gives

(9.2-44)

Case (ii) k = constant; % = constant

In this case Eq. (9.2-43) simplifies to

(9.2-45)

which implies that the variation of temperature with respect to the radial position
is parabolic with the maximum temperature at the center of the pipe.