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.,
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