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 ncan be expressed as xTTdxdTxTTdxdTnnnnnnΔ−≅Δ−≅++−−121121andNoting 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 ΔxΔxn-1nTn-1 TnTn+1 T(x)x211112121222xTTTxxTTxTTxdxdTdxdTdxTdnnnnnnnnnnΔ+−=ΔΔ−−Δ−=Δ−≅+−−+−+which is the finite difference representationof the second derivativeat 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 nand 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.