Question

MW 18 for Lecture 28 1. One side of a 2-m-high and 3-m-wide vertical plate at 80°C is to be cooled by attaching aluminum fins ($$k=237$$ W/mK) of rectangular profile in an environment at 35°C. The fins are 2 cm long, 0.3 cm thick, and 0.4 cm apart. The heat transfer coefficient between the fins and the surrounding air for combined convection and radiation is estimated to be 30 W/mK. Assuming steady one-dimensional heat transfer along the fin and taking the nodal spacing to be 0.5 cm, determine (a) the finite difference formulation of this problem. (b) the nodal temperatures along the fin by solving these equations, (c) the rate of heat transfer from a single fin, and (d) the rate of heat transfer from the entire finned surface of the plate. [Ans: (b) $$T_1=79.64^\circ C$$; $$T_2=79.30^\circ C$$; $$T_3=79.21^\circ C$$; $$T_4=79.14^\circ C$$; (c) 172 W; (d) 53.8 kW] 2. Consider the Uranium plate shown with various heat generation rate of $$\dot{e}_{gen} = \alpha + \beta T$$. $$\alpha = -2.64 \times 10^6$$ W/m$^3$, $$\beta = 6 \times 10^3$$ W/m$^3$$^\circ C$$. Determine the temperature profile in the plate using finite difference method. [Ans: $$T_0=547.5^\circ C$$; $$T_1=506.4^\circ C$$; $$T_2=542.8^\circ C$$; $$T_3=537.3^\circ C$$; $$T_4=529.6^\circ C$$; $$T_5=519.9^\circ C$$]

          MW 18 for Lecture 28
1. One side of a 2-m-high and 3-m-wide vertical plate at 80°C is to be cooled by attaching
aluminum fins ($$k=237$$ W/mK) of rectangular profile in an environment at 35°C. The
fins are 2 cm long, 0.3 cm thick, and 0.4 cm apart. The heat transfer coefficient between
the fins and the surrounding air for combined convection and radiation is estimated to be
30 W/mK. Assuming steady one-dimensional heat transfer along the fin and taking the
nodal spacing to be 0.5 cm, determine (a) the finite difference formulation of this
problem. (b) the nodal temperatures along the fin by solving these equations, (c) the rate
of heat transfer from a single fin, and (d) the rate of heat transfer from the entire finned
surface of the plate. [Ans: (b) $$T_1=79.64^\circ C$$; $$T_2=79.30^\circ C$$; $$T_3=79.21^\circ C$$; $$T_4=79.14^\circ C$$; (c) 172 W;
(d) 53.8 kW]
2. Consider the Uranium plate shown with various heat generation rate of $$\dot{e}_{gen} = \alpha + \beta T$$.
$$\alpha = -2.64 \times 10^6$$ W/m$^3$, $$\beta = 6 \times 10^3$$ W/m$^3$$^\circ C$$. Determine the temperature profile in the
plate using finite difference method. [Ans: $$T_0=547.5^\circ C$$; $$T_1=506.4^\circ C$$; $$T_2=542.8^\circ C$$;
$$T_3=537.3^\circ C$$; $$T_4=529.6^\circ C$$; $$T_5=519.9^\circ C$$]

Added by Valerie A.

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