A heat exchanger connects two pieces of machinery. The left end of the exchanger (x = 0), is held at a constant temperature, Tref. The right end is insulated. Along the exchanger’s length, heat is lost to the surrounding environment – air at a temperature T0 – by convection. Given Tref, T0, the length L of the heat exchanger, its thermal conductivity κ and the convection heat transfer coefficient η, we seek a model for the steady-state temperature distribution along the length of the exchanger. Assume the problem domain is one-dimensional, with x ∈ [0, L]. The constitutive equations are: • Heat flux in the exchanger q(x) q(x) = −κ dT dx • Heat loss to the environment by convection qconv(x) qconv(x) = ηx T(x) − T0 Formulate the continuous steady-state problem. Recall that the complete problem statement includes boundary conditions. When the governing equation has been obtained, write out the finite difference scheme for the problem, which will include a stencil specifying the equation at each node in terms of the node index i. Make sure to include the approximation for the boundary conditions.
Sample Solution