YOU HAVE 1 HOUR AND 50 MINUTES TO COMPLETE THIS EXAM. PLEASE PLEASE SHOW ALL OF YOUR WORK ON THE PAGE PROVIDED. YOU MAY NOT REFER TO ANY PRINTED OR DIGITAL MATERIALS OR USE ANY ELECTRONIC DEVICE, EXCEPT A CALCULATOR FOR BASIC COMPUTATION AND ANY NOTES PROVIDED BY THE INSTRUCTOR FOR USE DURING THE EXAM. I HAVE READ AND UNDERSTAND THESE EXAMINATION RULES.
1. Carefully draw on the diagram below the profile of temperature along the centerline axis of a solid fin from its point of attachment to a hot surface at x = 0, where T = T , to its tip at x = L, where T = T . Assume that the cross-sectional area of the fin is constant along the axis and that steady-state heat transfer away from the fin and into the surrounding fluid at T occurs. Explain the shape of the temperature profile using heat-transfer principles and mathematical relations (i.e., why it is linear, concave up or concave down). Also, label and explain the values of T and T (i.e., why each is greater than, less than or equal to T). Note that there is no internal heat generation within the fin. (15 points)
T
T
X (fin centerline) x=0 (fin base)
x=L (fin tip)
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2. A perfectly round hot steel ball with uniform temperature T and conductivity k is immersed into a cold bath of water at T T. You must fully justify your answers to the following questions to receive credit. (15 points) a. Explain whether the temperature gradient dT/dr at the center of the ball is positive, negative or zero and how its absolute value varies from the beginning through the end of the cooling process (r is the radial coordinate measured from the center of the ball). b. Now consider the temperature gradient dT/dr in the water bath on the outside surface of the ball and explain whether this gradient is positive, negative or zero and how its absolute value varies from the beginning through the end of the cooling process.
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3. An experimental arrangement for measuring the thermal conductivity of solid materials involves the use of two long rods that are equivalent in every respect, except that one is fabricated from a standard material of known thermal conductivity k A while the other is fabricated from the material whose thermal conductivity k B is desired. Both rods are attached at one end to a heat source of fixed temperature T b, are exposed to a fluid of temperature T , and are instrumented with thermocouples to measure the temperature at a fixed distance x 1 from the heat source. If the standard material is aluminum, with k A = 200 W/m K, and measurements reveal values of T A = 75 C and TB = 60 C at x1 for T b = 100 C and T = 25 C, what is the thermal conductivity k B of the test material? (25 points)
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4. A spherical vessel used to contain a hot liquid has a 5-cm-thick stainless steel wall (k = 17 W/m K) and an 3 inner diameter D i = 1.0 m. The hot liquid inside of the vessel has a density ! = 1100 kg/m and specific heat c = 2400 J/kg K and is initially at a temperature of 100 C. The exterior surface of the vessel is 2 exposed to ambient air at T = 25 C, with h = 6 W/m K. Assuming validity of the lumped-capacitance model, determine (a) the amount of time required for the liquid to cool to 50 C and (b) the temperature of the vessel’s outer wall at this time. (c) How large does the thermal conductivity of the liquid need to be to justify your assumption of validity of the lumped-capacitance model? Note that no heat generation occurs in either the liquid or the vessel wall. (30 points) !
