Problem 1 (43 points) The figure below illustrates a proposed mechanism to provide cooling for a...

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Problem 1 (43 points) The figure below illustrates a proposed mechanism to provide cooling for a nuclear fuel rod. The fuel rod, which has a radius?, and thermal conductivity k,, is surrounded by a highly conductive, metallic casing (cladding) whose outer radius is? and whose thermal conductivity is k. Attached to the cladding are three slender, metallic sheets of thick- ness t and length L. The thermal conductivity of these sheets is k3. The device is immersed in a water bath maintained at a constant temperature T. The heat transfer coefficient between the water and the fuel rod device is h. The fuel rod has a length? . Both ends of the device at z = 0 and z = ? are perfectly insulated. The total heat generation rate is ?gen Watts. For the following questions, assume k (ht) = 100 / L = 0.02, and the system is at steady state. Ex- press your answer in terms of the quantities illustrated in the figure. a) (5 points) Describe all relevant modes of heat transfer between from the center of the fuel rod to the surrounding fluid. b) (3 points) What is the total heat transfer to the water? c) (14 points) Construct a thermal circuit that models the heat transfer from the inner surface of the cladding (? = ?) to the surrounding fluid. Identify all the resistance elements and write down their expressions in terms of the parameters of the problem d) (7 points) What is the temperature, T(?), at the outer surface of the cladding? e) (6 points) What is the temperature, T(?), at the boundary between the fuel rod and the cladding? gen f) (8 points) If the specific heat generation rate in the fuel rod is ?, [W/m], and the thermal diffusivity and specif- ic heat of the fuel rod are cand c, what is the temperature 7(0), at the center of the fuel rod? ht. T Ru Qen W h. T Problem 1 (43 points) The figure below illustrates a proposed mechanism to provide cooling for a nuclear fuel rod. The fuel rod, which has a radius?, and thermal conductivity k,, is surrounded by a highly conductive, metallic casing (cladding) whose outer radius is? and whose thermal conductivity is k. Attached to the cladding are three slender, metallic sheets of thick- ness t and length L. The thermal conductivity of these sheets is k3. The device is immersed in a water bath maintained at a constant temperature T. The heat transfer coefficient between the water and the fuel rod device is h. The fuel rod has a length? . Both ends of the device at z = 0 and z = ? are perfectly insulated. The total heat generation rate is ?gen Watts. For the following questions, assume k (ht) = 100 / L = 0.02, and the system is at steady state. Ex- press your answer in terms of the quantities illustrated in the figure. a) (5 points) Describe all relevant modes of heat transfer between from the center of the fuel rod to the surrounding fluid. b) (3 points) What is the total heat transfer to the water? c) (14 points) Construct a thermal circuit that models the heat transfer from the inner surface of the cladding (? = ?) to the surrounding fluid. Identify all the resistance elements and write down their expressions in terms of the parameters of the problem d) (7 points) What is the temperature, T(?), at the outer surface of the cladding? e) (6 points) What is the temperature, T(?), at the boundary between the fuel rod and the cladding? gen f) (8 points) If the specific heat generation rate in the fuel rod is ?, [W/m], and the thermal diffusivity and specif- ic heat of the fuel rod are cand c, what is the temperature 7(0), at the center of the fuel rod? ht. T Ru Qen W h. T