Chapter 5 Numerical Methods in Heat Conduction where k = 0.84 W/m.°C, , Tαρ==×−kC/.039 106m/s2i= To= -3°C hi= 6 W/m2.°C, ho= 20 W/m2.°C, Δx = 0.002 m, and Δy = 0.01 m. The upper limit of the time step Δtis determined from the stability criteria that requires the coefficient of in the Texpression (the primary coefficient) be greater than or equal to zero for all nodes. The smallest primary coefficient in the 9 equations above is the coefficient of Tin the expression since it is exposed to most convection per unit volume (this can be verified). The equation for node 6 can be rearranged as Tmimi+1i9Ti61+⎟⎟⎠⎞⎜⎜⎝⎛Δ+Δ++ΔΔ+⎥⎥⎦⎤⎢⎢⎣⎡⎟⎟⎠⎞⎜⎜⎝⎛Δ+Δ+ΔΔ−=+2529306221621121xTyTTTxkhtTxyxkhtTiiioioiαTherefore, the stability criteria for this problem can be expressed as ⎟⎟⎠⎞⎜⎜⎝⎛Δ+Δ+Δ≤Δ→≥⎟⎟⎠⎞⎜⎜⎝⎛Δ+Δ+ΔΔ−22221121t01121xyxkhxyxkhtooSubstituting the given quantities, the maximum allowable value of the time step is determined to be or, s8.4m)01.0(1)m002.0(1m)002.0)(CW/m
This is the end of the preview. Sign up
access the rest of the document.
This note was uploaded on 01/22/2012 for the course PHY 4803 taught by Professor Dr.danielarenas during the Fall '10 term at UNF.