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- ●TTIVE 10.1. For a fin with a variable cross section, the governing equation is given by (KA) - hpe = 0 (E1) dx with = T-Too. (Fig. 10.16.) (a) Divide the fin into n equal parts and derive the finite-difference form of Eq. (E1) that relates the temperature at node i, Ti, to the temperatures at the neighboring nodes using the forward-finite-difference formula. h h h i-1 1 i i+1 4 Figure 10.16 A fin with a variable cross section.
792 Chapter 10 4"X3" 4 in Ordinary Differential Equations: Boundary-Value Problems 3"x2" 3 in トーリーに Figure 10.17 A stepped beam. + 2"X1" 4 2 in 43 1"X1" 1 in - 14 (b) Express the forward-difference approximations for a fin with (i) insulation at x = 1 and (ii) convection at x = l. (c) Formulate the forward-finite-difference equations for a linearly tapered fin that is insulated at x = / with the following data: A(x) = Ao (1-), p(x) = po (1-). Ao = 1 cm², po = 4 cm, W W 1 = 10 cm, k = 40- ,h= 10- m-K m² K (d) Solve the equations formulated in part (c) and find the temperatures at nodes i, i 1, 2, ..., 5. ,00 100° C, and n = 5.