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Unformatted text preview: Chapter 2 Heat Conduction Equation Variable Thermal Conductivity 2-94C During steady one-dimensional heat conduction in a plane wall, long cylinder, and sphere with constant thermal conductivity and no heat generation, the temperature in only the plane wall will vary linearly. 2-95C The thermal conductivity of a medium, in general, varies with temperature. 2-96C During steady one-dimensional heat conduction in a plane wall in which the thermal conductivity varies linearly, the error involved in heat transfer calculation by assuming constant thermal conductivity at the average temperature is ( a ) none . 2-97C No, the temperature variation in a plain wall will not be linear when the thermal conductivity varies with temperature. 2-98C Yes, when the thermal conductivity of a medium varies linearly with temperature, the average thermal conductivity is always equivalent to the conductivity value at the average temperature. 2-99 A plate with variable conductivity is subjected to specified temperatures on both sides. The rate of heat transfer through the plate is to be determined. Assumptions 1 Heat transfer is given to be steady and one-dimensional. 2 Thermal conductivity varies quadratically. 3 There is no heat generation. Properties The thermal conductivity is given to be k T k T ( ) ( ) = + 2 1 β . Analysis When the variation of thermal conductivity with temperature k ( T ) is known, the average value of the thermal conductivity in the temperature range between T T 1 2 and can be determined from ( 29 ( 29 ( 29 + + + =- - +- =- + =- + =- = ∫ ∫ 2 1 2 1 2 2 1 2 3 1 3 2 1 2 1 2 3 1 2 2 1 2 ave 3 1 3 3 ) 1 ( ) ( 2 1 2 1 2 1 T T T T k T T T T T T k T T T T k T T dT T k T T dT T k k T T T T T T β β β β This relation is based on the requirement that the rate of heat transfer through a medium with constant average thermal conductivity k ave equals the rate of heat transfer through the same medium with variable conductivity k ( T ). Then the rate of heat conduction through the plate can be determined to be ( 29 L T T A T T T T k L T T A k Q 2 1 2 1 2 1 2 2 2 1 ave 3 1- + + + =- = β Discussion We would obtain the same result if we substituted the given k ( T ) relation into the second part of Eq. 2-76, and performed the indicated integration. 2-58 T 2 x k ( T ) L T 1 Chapter 2 Heat Conduction Equation 2-100 A cylindrical shell with variable conductivity is subjected to specified temperatures on both sides. The variation of temperature and the rate of heat transfer through the shell are to be determined. Assumptions 1 Heat transfer is given to be steady and one-dimensional. 2 Thermal conductivity varies linearly. 3 There is no heat generation....
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This note was uploaded on 08/24/2011 for the course ENGR 3150 taught by Professor Engel during the Spring '11 term at Georgia Southern University .
- Spring '11
- Heat Transfer