A high-temperature gas reactor (HTGR) consists of spherical, uranium oxide fuel elements in which there is uniform volumetric heating (q). Each fuel element is embedded in a graphite spherical shell which is cooled by a helium gas flow at 1 atm.

Consider steady-state conditions for which radiation effects may be neglected, the gas velocity and temperature are V = 20 m/s and T_∞ = 500 K, the pellet and shell diameters are D_i = 10 mm and D_o = 12 mm, and the shell surface temperature is T_s.o = 1300 K. The uranium oxide and graphite each have a thermal conductivity of k_p = k_g = 2 W/m ∙ K.

(a) What is the rate of heat transfer from a single pellet to the gas stream?

(b) What is the volumetric rate of heat generation in the pellet and what is the temperature at the pellet-graphite interface (T_s,i>?

(c) Obtain an expression for the radial temperature distribution T(r), in the pellet, expressing your result in terms of the temperature at the center of the pellet, T (0). Evaluate T (0) for the prescribed conditions.

(d) Determine T_s,o, T_s,i and T(0) as a function of the gas velocity for 5 < V< 20 m/s and q = 1.50 x 10⁸ W/m³.

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Uranium oxide Helium V, T. T50 Graphite - D, -De