Heat Chap05-043 - Chapter 5 Numerical Methods in Heat...

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Unformatted text preview: Chapter 5 Numerical Methods in Heat Conduction Two-Dimensional Steady Heat Conduction 5-43C 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 two-dimensional, ( c ) there is heat generation in the medium, ( d ) the nodal spacing is constant, and ( e ) the thermal conductivity of the medium is constant. 5-44C 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 two-dimensional, ( 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. 5-45C A region that cannot be filled with simple volume elements such as strips for a plane wall, and rectangular elements for two-dimensional 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. 5-38 Chapter 5 Numerical Methods in Heat Conduction 5-46 A long solid body is subjected to steady two-dimensional heat transfer. The unknown nodal temperatures and the rate of heat loss from the bottom surface through a 1-m long section are to be determined. Assumptions 1 Heat transfer through the body is given to be steady and two-dimensional. 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 two-dimensional 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 = 280.9C , T 2 = 397.1C , T 3 = 330.8C , ( b ) The rate of heat loss from the bottom surface through a 1-m 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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Heat Chap05-043 - Chapter 5 Numerical Methods in Heat...

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