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Unformatted text preview: Chapter 2 Heat Conduction Equation 2126 A spherical liquid nitrogen container is subjected to specified temperature on the inner surface and convection on the outer surface. The mathematical formulation, the variation of temperature, and the rate of evaporation of nitrogen are to be determined for steady onedimensional heat transfer. Assumptions 1 Heat conduction is steady and onedimensional since there is no change with time and there is thermal symmetry about the midpoint. 2 Thermal conductivity is constant. 3 There is no heat generation. Properties The thermal conductivity of the tank is given to be k = 18 W/m C. Also, h fg = 198 kJ/kg for nitrogen. Analysis ( a ) Noting that heat transfer is onedimensional in the radial r direction, the mathematical formulation of this problem can be expressed as 2 = dr dT r dr d and T r T ( ) 1 1 196 = =  C = k dT r dr h T r T ( ) [ ( ) ] 2 2 ( b ) Integrating the differential equation once with respect to r gives r dT dr C 2 1 = Dividing both sides of the equation above by r to bring it to a readily integrable form and then integrating, dT dr C r = 1 2 T r C r C ( ) =  + 1 2 where C 1 and C 2 are arbitrary constants. Applying the boundary conditions give r = r 1 : T r C r C T ( ) 1 1 1 2 1 =  + = r = r 2 :  + = T C r C h r C k 2 2 1 2 2 1 Solving for C C 1 2 and simultaneously gives C r T T r r k hr C T C r T T T r r k hr r r 1 2 1 2 1 2 2 1 1 1 1 1 2 1 2 2 1 1 1 = = + = + ( ) and Substituting C C 1 2 and into the general solution, the variation of temperature is determined to be 196 ) / 1 . 2 05 . 1 ( 8 . 549 C ) 196 ( 1 . 2 2 1 . 2 ) m 1 . 2 )( C W/m 25 ( C W/m 18 2 1 . 2 1 C ) 20 196 ( 1 1 1 ) ( 2 1 2 1 2 2 1 2 1 1 1 1 1 1 1 1 =  +    = +  = +  = + + = r r T r r r r hr k r r T T T r r C r C T r C r T ( c ) The rate of heat transfer through the wall and the rate of evaporation of nitrogen are determined from ( ) ( ) ( ) ( . ( ) . ( )( . ) Q kA dT dx k r C r kC k r T T r r k hr =  =  =  =  =    =  4 4 4 1 4 18 21 196 20 1 21 2 18 25 21 2 1 2 1 2 1 2 1 2 W / m C m) C W / m C W / m C m 261,200 W (to the tank since negative) 2 , , m Q h fg = = = 261 200 198 000 J / s J / kg 1.32 kg / s 270 r 1 r 2 h T r196C N 2 Chapter 2 Heat Conduction Equation 2127 A spherical liquid oxygen container is subjected to specified temperature on the inner surface and convection on the outer surface. The mathematical formulation, the variation of temperature, and the rate of evaporation of oxygen are to be determined for steady onedimensional heat transfer....
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This homework help 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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