The cylindrical chamber of a pebble bed nuclear reactor is of length L = 10m and diameter D = 3 m. The chamber is filled with spherical uranium oxide pellets of core diameter Dp = 50 mm. Each pellet generates thermal energy in its core at a rate of E_g and is coated with a layer of non-heat-generating graphite, which is of uniform thickness 8 = 5 mm, to form a pebble. The uranium oxide and graphite each have a thermal conductivity of 2 W/m ∙ K. The packed bed has a porosity of ε = 0.4. Pressurized helium at 40 bars is used to absorb the thermal energy from the pebbles. The helium enters the packed bed at T_i = 450°C with a velocity of 3.2 m/s. The properties of the helium may be assumed to be c_p = 5193 J/kg ∙ K, k = 0.3355 W/m ∙ K, p = 2.1676 kg/m³, μ = 4.214 x 10^-5 kg/s ∙ m, Pr = 0.654.

(a) For a desired overall thermal energy transfer rate of q = 125 MW, determine the mean outlet temperature of the helium leaving the bed, T_o and the amount of thermal energy generated by each pellet, Eg.

(b) The amount of energy generated by the fuel decreases if a maximum operating temperature of approximately 2100°C is exceeded. Determine the maximum internal temperature of the hottest pellet in the packed bed. For Reynolds numbers in the range 4000 < ReD < 10,000, Equation 7.85 may be replaced by μj_H = 2.876 Re_D^-l + 0.3023 Re_D^-0.35.

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