A long steel bar has the cross section shown in Fig. P5–149. The bar is removed from a heat treatment oven at T_i = 700°C and placed on the bottom of a tank filled with water at 10°C. To intensify the heat transfer, the water is vigorously circulated, which creates a virtually constant temperature T_s = 10°C on all sides of the bar, except for the bottom side, which is adiabatic. The properties of the bar are c_p = 430 J/kg · K, k = 40 W/m · K, and ρ = 8000 kg/m³.
(a) Write the finite difference equations for the unknown temperatures in the grid using the explicit method. Group all constant quantities in one term. Identify dimensionless parameters such as Bi and Fo if applicable.
(b) Determine the range of time steps for which the explicit scheme is numerically stable.
(c) For Δt = 10 s, determine the temperature field at t = 10 s and t = 20 s. Fill in the table below.

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Node 1 2 3 4 5 6 7 T(10 s) T,= 10°C: 6 2 -5 cm- T(20 s)