Page 379 - Modelling in Transport Phenomena A Conceptual Approach
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9.2. ENERGY TRANSPORT WITHOUT CONVECTION 359
Equation (9.2-73) is the macroscopic energy balance under steady conditions by
considering the solid sphere as a system. It is also possible to make use of Newton's
law of cooling to express the rate of heat loss from the system to the surroundings
at T, with an average heat transfer coefficient (h). In this case Eq. (9.2-73)
reduces to
R
R2(h) (TR - T,) = Jd !Rr2 dr (9.274)
Example 9.5 Consider Example 3.2 in which energy generation as a result of
fission within a spherical reactor of radius R is given as
% = !Ro [1- Q2]
Cooling fluid at a temperature of T, flows over a reactor vith an average heat
transfer coeficient of (h). Determine the temperature distribution and the rate of
heat loss from the reactor surface.
Solution
The temperature distribution within the reactor can be calculated from Eq. (9.2-70).
Note that
R(U) u2 du = So Jd' [1- (#)'I u2 du
Substitution of Eq. (1) into Eq. (9.2-70) gives
Evaluation of the integration gives the temperature distribution as
2k [?(E) -i6(d]
7!R0R2 X0R2 1 r 2
T = T' f -- - - 1 r 4 (3)
60 k
This result, however, contains an unknown quantity TR. Therefore, it is necessary
to express TR in terms of the known quantities, i.e., T, and (h).
One way of calculating the surface temperature, TR, is to use the macroscopic
energy balance given by Eq. (9.2-74), Le.,