Heat Chap05-063 - Chapter 5 Numerical Methods in Heat...

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Unformatted text preview: Chapter 5 Numerical Methods in Heat Conduction Transient Heat Conduction 5-63C The formulation of a transient heat conduction problem differs from that of a steady heat conduction problem in that the transient problem involves an additional term that represents the change in the energy content of the medium with time. This additional term A xC T T t m i m i ( ) / +- 1 represent the change in the internal energy content during t in the transient finite difference formulation. 5-64C The two basic methods of solution of transient problems based on finite differencing are the explicit and the implicit methods . The heat transfer terms are expressed at time step i in the explicit method, and at the future time step i + 1 in the implicit method as Explicit method: Q G V C T T t i m i m i All sides element i element + =- + 1 Implicit method: Q G V C T T t i m i m i + + + =- 1 1 All sides element i+1 element 5-65C The explicit finite difference formulation of a general interior node for transient heat conduction in a plane wall is given by T T T g x k T T m i m i m i m i m i m i- + +- + + =- 1 1 2 1 2 . The finite difference formulation for the steady case is obtained by simply setting i m i m T T = + 1 and disregarding the time index i. It yields 2 2 1 1 = + +- +- k x g T T T m m m m 5-66C The explicit finite difference formulation of a general interior node for transient two-dimensional heat conduction is given by k l g T T T T T T i 2 i node i node i bottom i right i top i left 1 node ) 4 1 ( ) ( +- + + + + = + . The finite difference formulation for the steady case is obtained by simply setting i m i m T T = + 1 and disregarding the time index i. It yields 4 2 node node bottom right top left = +- + + + k l g T T T T T 5-67C There is a limitation on the size of the time step t in the solution of transient heat conduction problems using the explicit method, but there is no such limitation in the implicit method. 5-68C The general stability criteria for the explicit method of solution of transient heat conduction problems is expressed as follows: The coefficients of all T m i in the T m i + 1 expressions ( called the primary coefficient ) in the simplified expressions must be greater than or equal to zero for all nodes m. 5-66 Chapter 5 Numerical Methods in Heat Conduction 5-69C For transient one-dimensional heat conduction in a plane wall with both sides of the wall at specified temperatures, the stability criteria for the explicit method can be expressed in its simplest form as = t x ( ) 2 1 2 5-70C For transient one-dimensional heat conduction in a plane wall with specified heat flux on both sides, the stability criteria for the explicit method can be expressed in its simplest form as = t x ( ) 2 1 2 which is identical to the one for the interior nodes. This is because the heat flux boundary conditions have no effect on the stability criteria. have no effect on the stability criteria....
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Heat Chap05-063 - Chapter 5 Numerical Methods in Heat...

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