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Heat Chap02-126

# Heat Chap02-126 - Chapter 2 Heat Conduction Equation 2-126...

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Chapter 2 Heat Conduction Equation 2-126 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 one-dimensional heat transfer. Assumptions 1 Heat conduction is steady and one-dimensional 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 one-dimensional in the radial r direction, the mathematical formulation of this problem can be expressed as 0 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 ) / 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 2-70 r 1 r 2 h T r -196°C N 2

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Chapter 2 Heat Conduction Equation 2-127 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 one-dimensional heat transfer.
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Heat Chap02-126 - Chapter 2 Heat Conduction Equation 2-126...

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