Consider the East wall of a house that has a thickness of \(L\). The outer surface of the wall exchanges heat by both convection and radiation. The interior of the house is maintained at \(T_{\infty 1}\), while the ambient air temperature outside remains at \(T_{\infty 2}\). The sky, the ground, and the surfaces of the surrounding structures at this location can be modeled as a surface at an effective temperature of \(T_{\text {sky }}\) for radiation exchange on the outer surface. The radiation exchange between the inner surface of the wall and the surfaces of the walls, floor, and ceiling it faces is negligible. The convection heat transfer coefficients on the inner and outer surfaces of the wall are \(h_{1}\) and \(h_{2}\), respectively. The thermal conductivity of the wall material is \(k\) and the emissivity of the outer surface is \(\varepsilon_{2}\). Assuming the heat transfer through the wall to be steady and one-dimensional, express the mathematical formulation (the differential equation and the boundary and initial conditions) of this heat conduction problem. Do not solve.

Step by step solution

01

Heat transfer processes

The given problem involves three different heat transfer mechanisms: convection on the inner surface, convection on the outer surface, and radiation on the outer surface. We can write the heat fluxes for each mechanism as follows: - Inner surface convection: \(q_{c1} = h_{1}(T_{1} - T_{\infty 1})\) - Outer surface convection: \(q_{c2} = h_{2}(T_{2} - T_{\infty 2})\) - Outer surface radiation: \(q_{r2} = \varepsilon_{2}\sigma(T_{2}^4 - T_{\text{sky}}^4)\) where \(T_1\) and \(T_2\) are the inner and outer surface temperatures of the wall, respectively, and \(\sigma\) is the Stefan-Boltzmann constant.

02

Conservation principle

The heat transfer through the wall is assumed to be one-dimensional and steady, meaning that there is no heat accumulation within the wall. Hence, the heat transferred through the wall by conduction must equal the total heat fluxes on the inner and outer surfaces, due to convection and radiation: \(q_{cond} = -k \frac{dT}{dx} = q_{c1} + q_{c2} + q_{r2}\)

03

Governing differential equation

To express the governing differential equation for heat conduction in the wall, we employ Fourier's law of heat conduction along with the given thermal conductivity \(k\): \(\frac{d^2 T}{dx^2} = 0\)

04

Boundary conditions

To complete the mathematical formulation of the problem, we need to specify the boundary conditions on the inner and outer surfaces of the wall (\(x = 0\) and \(x = L\), respectively). At \(x = 0\), we know the heat flux due to convection: \(-k\frac{dT}{dx}\Big\vert_{x=0} = h_{1}(T_1 - T_{\infty 1})\) At \(x = L\), the total heat flux due to convection and radiation is given by: \(-k\frac{dT}{dx}\Big\vert_{x=L} = h_{2}(T_2 - T_{\infty 2}) + \varepsilon_{2}\sigma(T_{2}^4 - T_{\text{sky}}^4)\) The resulting mathematical formulation is: - Differential equation: \(\frac{d^2 T}{dx^2} = 0\) - Boundary conditions: \(-k\frac{dT}{dx}\Big\vert_{x=0} = h_{1}(T_1 - T_{\infty 1})\) and \(-k\frac{dT}{dx}\Big\vert_{x=L} = h_{2}(T_2 - T_{\infty 2}) + \varepsilon_{2}\sigma(T_{2}^4 - T_{\text{sky}}^4)\)

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