2. One-dimensional heat transfer in a straight fin (see sketch below) is governed by the following heat equation d2T - + c2T + c3 = 0 dX2 C1 = = with C1 = kA, C2 = -hp, C3 = hpTwhere A denotes the cross-sectional area of the fin, is the perimeter of the fin, Ty is the temperature of the surrounding fluid, k is the thermal conductivity, and h is the convective heat transfer coefficient. The equation is subject to essential and natural boundary conditions. Aluminum fins of a rectangular profile, shown in the figure below, are used to remove heat from a surface whose temperature is 100 C. The temperature of the surrounding fluid (ambient air) is 20 C. The thermal conductivity of aluminum is 168 W/m K (W/m C). The natural convective heat transfer coefficient associated with the surrounding air is 30 W/m K (W/m C). The fins are 80 mm long, 5 mm wide and 1 mm thick. Want: Develop the linear finite element solution for the temperature resulting from a three- element partition. Take the three elements to be of equal length. Develop the solution in matrix form and solve for the temperature at the nodes and the heat flux at x = 0 (i.e. ka (x = 0)). dx The solution should be subject to the following essential and natural temperature boundary conditions, respectively: T(x = 0) = = 100 C dT KA (x = L = 80 mm) = 0 dx The latter boundary condition denotes an insulated tip. T Actual temperature profile Tbase Approximate temperature profile Tfluid (2) (3) (1) X 2 base 3 4 L Figure Temperature distribution for a fin of uniform cross section. L 80 m 1000 W W m C1 = kA = 168 = 5 x m2 = 8.4 x 10 10002 m C C W W 12 C2 = -hp = -30 m = -0.36 m2 C 1000 = - m C W W 12 C3 = hpTg = -30 -m x 20 C = 7.2 m2 C 1000 = = - m