HW3 Solutions

# HW3 Solutions - ME 364 HW 3 Solutions Problem 1 You have...

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ME 364 HW 3 Solutions Problem 1: You have been contracted by ASHRAE (the American Society of Heating, Refrigeration, and Air- Conditioning Engineers) to measure the thermal conductivity of various, new materials for insulating pipes. Your contract specifies that you will measure the thermal conductivity to within 10%. Your initial design for the test setup is shown in Fig. 1. Figure 1: Test facility for measuring pipe insulation The test facility consists of a pipe (with conductivity k pipe = 120 W/m-K) with inner diameter, D i,pipe = 6.0 inch and thickness th pipe = 0.5 inch that carries a flow of chilled water, T water = 10°C. The heat transfer coefficient between the water and the internal surface of the pipe is h water = 300 W/m 2 -K. The pipe is covered by a th ins =2.0 inch thick layer of the insulation (with conductivity k ins ) that is being tested. Two thermocouples are embedded in the insulation, one connected to the outer surface ( T ins , out ) and the other to the inner surface ( T ins,in ). The insulation material is surrounded by a th m =3.0 inch thick layer of a material with a well-known thermal conductivity, k m = 2.0 W/m-K. Two thermocouples are embedded in the material at its inner and outer surface ( T m,in and T m,out , respectively). Finally, a band heater is wrapped around the outer surface of the material. Assume that the thickness of the band heater is negligibly small. The band heater provides band q =3 kW for every m of its length. The outer surface of the band heater is exposed to ambient air at T air = 20°C and has a heat transfer coefficient, h air = 10 W/m 2 -K and emissivity = 0.5. A contact resistance of c R  =1e-4 m 2 -K/W is present at all 3 interfaces in the problem (i.e., between the pipe and the insulation, the insulation and the material, and the material and the band heater). You may assume that the problem is 1-D (i.e., there are no variations along the length of the pipe) and do the problem on a per unit length of pipe ( L =1 m) basis. a.) Draw a resistance network that represents the test facility. Clearly label each resistance and indicate what it represents. Be sure to indicate where in the network the heat input from the band heater will be applied and also the location of the thermocouples mentioned in the problem statement.

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The resistance network is shown in Fig. 2 and includes convection with the water and the air ( R conv,w and R air ), conduction through the pipe, insulation, and material ( R pipe , R cond,ins , and R cond,m ), contact resistances between the pipe and insulation ( R c, 1 ), the insulation and material ( R c, 2 ), and the material and the band heater ( R c ,3 ), and radiation ( R rad ). Figure 2: Resistance network representing the test facility. b.) If the insulating material under test has a conductivity of k ins = 1.0 W/m-K then predict the heater temperature, T htr , and determine how much of the heater power is transferred through the insulation ( ins q ).
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