Assume that the inner cylinder in Problem 7.40 is a heat source consisting of an aluminum-clad rod of uranium, 5-cm-OD and 2-m-long. Estimate the heat flux that raises the temperature of the bismuth 40°C and the maximum center and surface temperatures necessary to transfer heat at this rate.
GIVEN
Liquid bismuth flowing through an annulus
Annulus inside diameter (Di) = 5 cm = 0.05 m
Annulus outside diameter (Do) = 6.1 cm = 0.061 m
Bismuth velocity (V) = 4.5 m/s
Bismuth temperature (Tb) = 316°C
Inner cylinder is an aluminum clad uranium heat source
Cylinder length (L) = 2 m
From Problem 7.40: c h
26,800 W/(m2 K)
FIND
(a) The heat flux (QG/At) necessary to raise the bismuth temperature 40°C, and
(b) The maximum center (Tu,o) and surface (Tu,ro) temperatures of the uranium
ASSUMPTIONS
Steady state
The Bismuth temperature given is the bulk Bismuth temperature
Thermal resistance of the aluminum is negligible
Thickness of the aluminum is negligible
SKETCH
(a) The rate of heat transfer required to raise the Bismuth by 40°C is
Therefore, the average heat flux is
The temperature difference between the uranium and bismuth (?Tub) required to transfer this heat can
be calculated from
The maximum uranium surface temperature will occur at the outlet where the bismuth temperature is
The rate of internal heat generation per unit volume is
The maximum temperature at the center of the uranium is given
Therefore, the average uranium temperature is approximately
Repeating the calculation using the thermal conductivity of uranium evaluated at this temperature
yields the following result
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