The development of safe nuclear fuel pellets is critical to future reactor designs. One project the engineers have is to look into what happens when the reactor cooling system fails and the heat-transfer mechanism changes from forced convective cooling of the pellets to free convective cooling (coefficient \(-h_{f c}\) ). The reactor cooling fluid is essentially infinite in extent and remains at a uniform temperature of \(T_{o}\). The pellet generates heat at ate that varies with the pellet temperature.

a. Assuming the lumped capacitance approach is appropriate, derive the transient differential heat balance and solve the resulting differential equation (pellet temperature at \(t=0\) is \(T_{1}\) ).

b. How would you define the Biot number for this situation?

c. From the following information, what will be the temperature of the pellet at \(t=10 \mathrm{sec}\). following cooling system failure?

d. Using the above information, how large must a pellet be before the lumped capacitance approach fails?

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p = 1500 kg/m h = 55 W/mK 9. = 55,000 W/m K q q. (T-T) W/m k = 50 W/m K T = 125C pellet = 4 mm C = 450 J/kg K T = 20C