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Unformatted text preview: Chapter 2 Heat Conduction Equation Solution of Steady One-Dimensional Heat Conduction Problems 2-52C Yes, this claim is reasonable since in the absence of any heat generation the rate of heat transfer through a plain wall in steady operation must be constant. But the value of this constant must be zero since one side of the wall is perfectly insulated. Therefore, there can be no temperature difference between different parts of the wall; that is, the temperature in a plane wall must be uniform in steady operation. 2-53C Yes, the temperature in a plane wall with constant thermal conductivity and no heat generation will vary linearly during steady one-dimensional heat conduction even when the wall loses heat by radiation from its surfaces. This is because the steady heat conduction equation in a plane wall is d T dx 2 2 / = 0 whose solution is T x C x C ( ) = + 1 2 regardless of the boundary conditions. The solution function represents a straight line whose slope is C 1 . 2-54C Yes, in the case of constant thermal conductivity and no heat generation, the temperature in a solid cylindrical rod whose ends are maintained at constant but different temperatures while the side surface is perfectly insulated will vary linearly during steady one-dimensional heat conduction. This is because the steady heat conduction equation in this case is d T dx 2 2 / = 0 whose solution is T x C x C ( ) = + 1 2 which represents a straight line whose slope is C 1 . 2-55C Yes, this claim is reasonable since no heat is entering the cylinder and thus there can be no heat transfer from the cylinder in steady operation. This condition will be satisfied only when there are no temperature differences within the cylinder and the outer surface temperature of the cylinder is the equal to the temperature of the surrounding medium. 2-19 Chapter 2 Heat Conduction Equation 2-56 A large plane wall is subjected to specified temperature on the left surface and convection on the right surface. The mathematical formulation, the variation of temperature, and the rate of heat transfer are to be determined for steady one-dimensional heat transfer. Assumptions 1 Heat conduction is steady and one-dimensional. 2 Thermal conductivity is constant. 3 There is no heat generation. Properties The thermal conductivity is given to be k = 2.3 W/m C. Analysis ( a ) Taking the direction normal to the surface of the wall to be the x direction with x = 0 at the left surface, the mathematical formulation of this problem can be expressed as d T dx 2 2 = and T T ( ) 80 1 = = C- =- k dT L dx h T L T ( ) [ ( ) ] ( b ) Integrating the differential equation twice with respect to x yields dT dx C = 1 T x C x C ( ) = + 1 2 where C 1 and C 2 are arbitrary constants. Applying the boundary conditions give x = 0: T C C C T ( ) 1 2 2 1 = + = x = L : - = +- = -- + = -- + kC h C L C T C h C T k hL C h T T k hL 1 1 2 1 2 1 1 [( ( ) ( ) ) ] Substituting C C 1 2 and into the general solution, the variation of temperature is determined to be...
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- Spring '11
- Heat Transfer