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Unformatted text preview: Chapter 5 Numerical Methods in Heat Conduction TwoDimensional Steady Heat Conduction 543C For a medium in which the finite difference formulation of a general interior node is given in its simplest form as T T T T T g l k left top right bottom node node + + + + = 4 2 : ( a ) Heat transfer is steady, ( b ) heat transfer is twodimensional, ( c ) there is heat generation in the medium, ( d ) the nodal spacing is constant, and ( e ) the thermal conductivity of the medium is constant. 544C For a medium in which the finite difference formulation of a general interior node is given in its simplest form as T T T T T node left top right bottom = + + + ( ) / 4 : ( a ) Heat transfer is steady, ( b ) heat transfer is twodimensional, ( c ) there is no heat generation in the medium, ( d ) the nodal spacing is constant, and ( e ) the thermal conductivity of the medium is constant. 545C A region that cannot be filled with simple volume elements such as strips for a plane wall, and rectangular elements for twodimensional conduction is said to have irregular boundaries . A practical way of dealing with such geometries in the finite difference method is to replace the elements bordering the irregular geometry by a series of simple volume elements. 538 Chapter 5 Numerical Methods in Heat Conduction 546 A long solid body is subjected to steady twodimensional heat transfer. The unknown nodal temperatures and the rate of heat loss from the bottom surface through a 1m long section are to be determined. Assumptions 1 Heat transfer through the body is given to be steady and twodimensional. 2 Heat is generated uniformly in the body. 3 Radiation heat transfer is negligible. Properties The thermal conductivity is given to be k = 45 W/m ⋅ °C. Analysis The nodal spacing is given to be ∆ x = ∆ x = l =0.05 m, and the general finite difference form of an interior node for steady twodimensional heat conduction is expressed as T T T T T g l k left top right bottom node node + + + + = 4 2 where C 5 . 93 C W/m 214 ) m 05 . )( W/m 10 8 ( 2 3 6 2 2 node ° = ° ⋅ × = = k l g k l g The finite difference equations for boundary nodes are obtained by applying an energy balance on the volume elements and taking the direction of all heat transfers to be towards the node under consideration: 4 200 240 290 260 : (interior) 3 Node 4 290 325 290 350 : (interior) 2 Node 2 ) ( 325 2 290 240 2 : ) convection ( 1 Node 2 3 2 2 2 1 1 1 1 = + + + + = + + + + = + + + + ∞ k l g T k l g T k l g T T hl l T l k l T kl l T l k where C 20 , W/m 10 8 C, . W/m 50 C, W/m. 45 3 6 2 ° = × = ° = ° = ∞ T g h k Substituting, T 1 = 280.9°C , T 2 = 397.1°C , T 3 = 330.8°C , ( b ) The rate of heat loss from the bottom surface through a 1m long section is W 1808 = ° + + + × ° ⋅ = + + + = = = ∞ ∞ ∞ ∞ ∞ ∑ ∑ C 20)/2] (325 20) (280.9 20) (240 20)/2 m)[(200 1 m C)(0.05 W/m 50 ( ) 325 )( 2 / ( ) ( ) 240 ( ) 200 )( 2 / ( ) ( 2...
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This note was uploaded on 03/19/2008 for the course ME 410 taught by Professor Benard during the Spring '08 term at Michigan State University.
 Spring '08
 BENARD
 Heat Transfer

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