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Lecture 8

Lecture 8 - Lecture 8 Dr Srinath V.Ekkad Transient...

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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. Crank-Nicolson implicit scheme - Centered differences Methods to derive finite difference equations include: a. Energy balance b. Taylor series (direct discretization of heat conduction equation) Consider one-dimensional, 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 x-direction only. Area A x L Convection T ∞ , h Insulated sides (heat flow in x-direction only) Insulated end Material with properties k, ρ , C p x m ∆ x N ∆ x=L Area A ∆ x T m p T m+1 p T m-1 p T p T 1 p T N p T N-1 p ∆ x 2 ∆ x 2 Typical interior node Finite-Difference Method The Finite-Difference 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 finite-difference equation for each node of unknown temperature. ─ Solve the resulting set of equations for the nodal temperatures at t = ∆t, 2∆t, 3∆t , …, until steady-state 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 = p∆t , where p assumes integer values, ( p = 0, 1, 2,…). Storage Term Energy Balance and Finite-Difference 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: • Finite-difference form of the storage term: ( 29 1 , , , p p m n m n st m n T T E c t ρ +- = 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 forward-difference approximation . • Example: Two-dimensional 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 finite-difference form o Four ier f number t Fo x α ∆ = ∆ • Unknown nodal temperatures at the new time, t = (p+ 1 )∆t, are determined exclusively by known nodal temperatures at the preceding time, t = p∆t , hence the term explicit solution . Explicit Method (cont.)Explicit Method (cont....
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Lecture 8 - Lecture 8 Dr Srinath V.Ekkad Transient...

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