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Unformatted text preview: Lecture 8 Dr. Srinath V.Ekkad Transient Conduction Finite Difference Method In order to include the effects of the time variable, various finite difference schemes have been devised. These include: 1. Explicit scheme  Forward differences in time 2. Implicit scheme  Backward differences in time 3. CrankNicolson implicit scheme  Centered differences Methods to derive finite difference equations include: a. Energy balance b. Taylor series (direct discretization of heat conduction equation) Consider onedimensional, transient heat conduction with no heat generation in a material exposed to convection on the left face and insulated on the right face. The sides are perfectly insulated so heat can flow in the xdirection only. Area A x L Convection T , h Insulated sides (heat flow in xdirection only) Insulated end Material with properties k, , C p x m x N x=L Area A x T m p T m+1 p T m1 p T p T 1 p T N p T N1 p x 2 x 2 Typical interior node FiniteDifference Method The FiniteDifference Method An approximate method for determining temperatures at discrete (nodal) points of the physical system and at discrete times during the transient process. Procedure: Represent the physical system by a nodal network , with an m, n notation used to designate the location of discrete points in the network, Use the energy balance method to obtain a finitedifference equation for each node of unknown temperature. Solve the resulting set of equations for the nodal temperatures at t = t, 2t, 3t , , until steadystate is reached. What is represented by the temperature, ? , p m n T and discretize the problem in time by designating a time increment t and expressing the time as t = pt , where p assumes integer values, ( p = 0, 1, 2,). Storage Term Energy Balance and FiniteDifference Approximation for the Storage Term For any nodal region, the energy balance is in g st E E E + = g g g (5.81) where, according to convention, all heat flow is assumed to be into the region. Discretization of temperature variation with time: Finitedifference form of the storage term: ( 29 1 , , , p p m n m n st m n T T E c t + = 2200 g Existence of two options for the time at which all other terms in the energy balance are evaluated: p or p+ 1. 1 , , , p p m n m n m n T T T t t + (5.74) Explicit Method The Explicit Method of Solution All other terms in the energy balance are evaluated at the preceding time corresponding to p . Equation (5.74) is then termed a forwarddifference approximation . Example: Twodimensional conduction for an interior node with x=y. ( 29 ( 29 1 , , 1, 1, , 1 , 1 1 4 p p p p p p m n m n m n m n m n m n T Fo T T T T Fo T + + + = + + + + (5.76) ( 29 2 finitedifference form o Four ier f number t Fo x = Unknown nodal temperatures at the new time, t = (p+ 1 )t, are determined exclusively by known nodal temperatures at the preceding time, t = pt , hence the term explicit solution . Explicit Method (cont.)Explicit Method (cont....
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This note was uploaded on 03/30/2008 for the course ME 3304 taught by Professor Stern during the Spring '08 term at Virginia Tech.
 Spring '08
 STERN

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