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?1, while the ambient air temperature outside remains at T?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 Tsky 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 h1 and h2, respectively. The thermal conductivity of the wall material is k and the emissivity of the outer surface is ?2.

Question

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.

Answer 1

The outer surface of the East wall of a house exchanges heat with both convection and radiation., while the interior surface is subjected to convection only. Assuming the heat transfer through the wall to be steady and one-dimensional, the mathematical formulation (the differential equation and the boundary and initial conditions) of this heat conduction problem is to be obtained.

Assumptions 1 Heat transfer is given to be steady and one-dimensional. 2 Thermal conductivity is given to be constant. 3 There is no heat generation in the medium. 4 The outer surface at x = L is subjected to convection and radiation while the inner surface at x = 0 is subjected to convection only.

Analysis Expressing all the temperatures in Kelvin, the differential equation and the boundary conditions for this heat conduction problem can be expressed as