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Unformatted text preview: ME 608 Numerical Methods in Heat, Mass, and Momentum Transfer Example Problems Instructor: J. Murthy 1. Consider steady 1D conduction in a 1D domain consisting of 3 equal-sized cells as shown in Fig. 1. The right boundary (x=0) is irradiated with a radiative heat flux given by: q = εσ ( T 4 ∞- T 4 ) while the right boundary ( x = L ) is held at a constant temperature T = T b . The thermal conductivity k is constant. (a) Using the finite volume method, develop nominally-linear discrete equations for the unknown temperatures T , T 1 , T 2 and T 3 . Indicate current iterate values with a * superscript. (b) Develop a discrete equation for the unknown boundary flux at x = L , q L . (c) Describe step by step how you would solve this problem to obtain numerical values of the discrete temperatures and the boundary heat flux. Do not attempt to actually solve the problem. 1 2 3 4 x q T= T b Figure 1: Computational Domain for Problem 1 Linearize q as: q = q * + ∂ q ∂ T * ( T- T * ) = ( εσ T * 4 ∞ + 3 εσ T * 4 )- 4 εσ T * 3 T = S C + S P T where S C = ( εσ T * 4 ∞ + 3 εσ T * 4 ) S P =- 4 εσ T * 3 (a) The discrete equations for the boundary face and cell temperatures are given below. Boundary Face 0 1 q = T- T 1 Δ x = S C + S P T T- S P + 2 k Δ x = 2 k Δ x T 1 + S C Cell 1 q = k Δ x ( T 1- T 2 ) k Δ x T 1 = k Δ x T 2 + S C +...
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- Fall '10
- Thermodynamics, Scarborough, numerical values, discrete equations, ﬁrst-order upwind