Question

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 \(T_{\infty \text { l }}\), while the ambient air temperature outside remains at \(T_{c e 2}\). 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.

Short answer

Answer: The mathematical formulation of the heat conduction problem consists of the heat conduction equation and boundary conditions. Heat Conduction Equation: \(\frac{d^2 T}{dx^2} = 0\) Boundary Conditions: 1. Inner Surface (x = 0): \(-k \frac{dT}{dx} \Big|_{x=0} = h_1 (T_\text{l} - T(0))\) 2. Outer Surface (x = L): \(-k \frac{dT}{dx} \Big|_{x=L} = h_2 (T_\text{e2} - T(L)) + \varepsilon_2 \sigma (T_\text{sky}^4 - T(L)^4)\)

Step-by-step solution

- Define the temperature distribution function

Let \(T(x)\) represent the temperature distribution within the wall at distance \(x\) from the inner surface. The wall thickness is \(L\), and the temperature at \(x = 0\) is \(T_\text{l}\) (interior temperature), while the temperature at \(x = L\) is the outer surface temperature \(T_\text{o}\).

- Write down the heat conduction equation

We can write Fourier's law of heat conduction for the wall: \(\frac{d^2 T}{ dx^2} = 0\) As the problem assumes steady-state and one-dimensional heat transfer, the second-order derivative of temperature with respect to \(x\) equals zero.

- Determine the boundary conditions at x = 0 (inner surface)

At the inner surface (x = 0), the boundary condition is determined by the heat transfer between the wall surface and the interior air through convection: \(-k \frac{dT}{dx} \Big|_{x=0} = h_1 (T_\text{l} - T(0))\)

- Determine the boundary conditions at x = L (outer surface)

At the outer surface (x = L), the boundary condition is determined by the heat transfer with the ambient air through convection and radiation: \(-k \frac{dT}{dx} \Big|_{x=L} = h_2 (T_\text{e2} - T(L)) + \varepsilon_2 \sigma (T_\text{sky}^4 - T(L)^4)\) Where \(\sigma\) is the Stefan-Boltzmann constant.

- Present the final mathematical formulation

With the heat conduction equation and the boundary conditions identified, we can summarize the mathematical formulation of this problem as follows: Heat Conduction Equation: \(\frac{d^2 T}{dx^2} = 0\) Boundary Conditions: 1. Inner Surface (x = 0): \(-k \frac{dT}{dx} \Big|_{x=0} = h_1 (T_\text{l} - T(0))\) 2. Outer Surface (x = L): \(-k \frac{dT}{dx} \Big|_{x=L} = h_2 (T_\text{e2} - T(L)) + \varepsilon_2 \sigma (T_\text{sky}^4 - T(L)^4)\) This is the final mathematical formulation for this heat conduction problem, as asked in the exercise.