5. Consider a wall of a home which consists of three layers of materials of identical thickness, L, as shown in the figures below. The outer layer has a conductivity k W/m-K, the middle layer B has a conductivity kg W/m-K, and the inner layer has a conductivity kc W/m-K. The material conductivities are such that k < kc < kg.
The air inside the home is conditioned to be at Tinf,i through a forced convective system. The convective heat transfer coefficient on the inside of the home is maintained by this system at a value of hi. The outside air is at a temperature of Tinf,o and a steady wind maintains a convective heat transfer coefficient at ho. Assume that Tinf,i > Tinf,o.
Consider two scenarios: Scenario (i), as shown in figure (i), corresponds to nighttime. Scenario (ii) corresponds to the morning time, wherein, in addition to the external convective flow (which remains unchanged from the night conditions), solar irradiation Gsun (W/m^2) is incident on the outer wall. The incident radiation causes a reversal in the flow of heat through the wall as compared to nighttime conditions.
The absorptivity of the wall to solar irradiation is a. The emissivity of the surface is e. Neglect irradiation from the surrounding sky.
A KA L
B KB L
C kc L
Gsun
A KA L
B KB L
C kc L
Tinf,i hi - 1
Tinf,o ho
Tinf,i hi
Tinf,o ho
(i) Night
(ii) Morning
(a) Provide analytical expressions that describe the steady-state heat transfer rate per unit wall area (W/m^2) in terms of the known material properties, dimensions, and boundary conditions for
i. Scenario (i) and
ii. Scenario (ii)
(b) Sketch the steady-state temperature distributions in the three materials that comprise the wall for
i. Scenario (i) and
ii. Scenario (ii)
Provide adequate explanation for the sketched temperature distributions. For example, how do the temperature gradients in each material compare? How do the profiles compare between Scenarios (i) and (ii)?
(c) Consider the transient between nighttime and morning. At t < 0, we have Scenario (i). At t = 0, sunlight with irradiation G is suddenly incident on the outer surface.
Provide differential equations, simplified to the specific problem, which govern the temperature distribution in the materials as a function of location (x) and time (t) along with the initial and boundary conditions. There is no need to solve the equations.
