Problem 1. Consider steady one-dimensional heat flow through the composite structure shown below (i.e. with the top, bottom, front, and back faces perfectly insulated). The thermal conductivities of A, B, C, and D are 150, 10, 180, and 90 Wm^-1K^-1, respectively.

a. Determine the effective thermal conductivity of the composite for the arrangement shown above.

b. To minimize the thermal resistance associated with this structure, you are allowed to replace only one of the four components with an alternative material whose thermal conductivity is twice as large. Which component would you replace with the higher thermal conductivity alternative to produce the largest reduction in the overall thermal resistance of the composite?

c. Which component would you replace in the case of steady one-dimensional heat flow from the top to the bottom of the composite (i.e. with the sides, front, and back perfectly insulated and the temperature difference applied vertically)?

Problem 2. The wind chill that is experienced on a cold windy day is due to increased heat transfer from exposed human skin to the surrounding air.

The human skin can be modeled as a 3 mm-thick layer of fatty tissue (with k = 0.2 Wm^-1K^-1) whose interior surface is maintained at a controlled temperature of 37°C. The average human body may be approximated as a cylinder with an inner core diameter of 40 cm surrounded by a skin layer. The temperature of the outside environment can reach -15°C on a very cold day in State College, and the convection heat transfer coefficient at the outer surface of the skin is 25 Wm^-2K^-1 on a calm day, but increases to 65 Wm^-2K^-1 on a day with 30 kmh^-1 winds.

a. What is the ratio of the skins heat loss per unit area for the windy day to that for the calm day?

b. What is the temperature at the skin's outer surface on the calm day? On the windy day?

c. What surrounding air temperature on a calm day would produce the same heat loss as that on the windy day with an air temperature of -15°C (i.e. how cold does it feel on the windy day)?

d. Estimate the thickness of clothing (k = 0.07 Wm^-1K^-1) that will be required to reduce the heat loss from the skin by 75% on a calm day.

Problem 3. To reduce energy loss from a 6 m-long copper (k = 385 Wm^-1K^-1) refrigerant line of 0.5 cm inside diameter and 0.6 cm outside diameter, a plumber recommends that you insulate the line with a 1 cm- thick layer of Styrofoam insulation (k = 0.033 Wm^-1K^-1). Refrigerant fluid at - 40°C flows inside the copper line with inside convective heat transfer coefficient of 600 Wm^-2K^-1. The line is exposed to ambient air at 20°C with outside convective heat transfer coefficient of 4 Wm^-2K^-1.

a. Would you accept the plumber's recommendation? Justify your answer.

b. Plot the rate of heat loss from the line as a function of Styrofoam thickness.

c. Specify the thickness of Styrofoam insulation needed to reduce the heat loss from the bare refrigerant line by 10 percent. What is the overall heat transfer coefficient for the insulated refrigerant line?