Thermodynamics HW Solutions 399

Thermodynamics HW Solutions 399 - Chapter 5 Numerical...

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Chapter 5 Numerical Methods in Heat Conduction 5-7 We consider three consecutive nodes n- 1, n , and n +1 in a plain wall. Using Eq. 5-6, the first derivative of temperature dT dx / at the midpoints n - 1/2 and n + 1/2 of the sections surrounding the node n can be expressed as x T T dx dT x T T dx dT n n n n n n Δ Δ + + 1 2 1 1 2 1 and Noting that second derivative is simply the derivative of the first derivative, the second derivative of temperature at node n can be expressed as n+ 1 Δ x Δ x n -1 n T n- 1 T n T n+ 1 T ( x ) x 2 1 1 1 1 2 1 2 1 2 2 2 x T T T x x T T x T T x dx dT dx dT dx T d n n n n n n n n n n Δ + = Δ Δ Δ = Δ + + + which is the finite difference representation of the second derivative at a general internal node n . Note that the second derivative of temperature at a node n is expressed in terms of the temperatures at node n and its two neighboring nodes
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This note was uploaded on 01/19/2012 for the course PHY 4803 taught by Professor Dr.danielarenas during the Fall '10 term at UNF.

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