Explanation Given information: The diameter of cylindrical rod is 1 cm , thermal conductivity of the rod is 29 ⋅ 5 W / m ⋅ K , heat generated per unit volume is is 4 × 10 7 W / m 3 , and outer surface temperature is 220 ° C . The below figure shows the heat generation and surface temperature of the cylindrical rod. Figure-(1) Write the expression of the differential equation for steady one dimensional heat conduction in cylindrical coordinates. 1 r d d r ( r d T d r ) + q k = 0 d d r ( r d T d r ) = − q r k …… (I) Here, temperature is T , radius is r , heat generated per unit volume is q , and thermal conductivity of the material is k . Integrate the Equation (I) with respect to r . ∫ d d r ( r d T d r ) d r = ∫ − q r k d r ( r d T d r ) = − q k ∫ r d r ( r d T d r ) = − q k [ r 2 2 ] + C 1 ( r d T d r ) = − q r 2 2 k + C 1 d T d r = − q r 2 k + C 1 r …… (II) Here, arbitrary constant is C 1 . Integrate the Equation (II) with respect to r . ∫ ( d T d r ) d r = ∫ ( − q r 2 k + C 1 r ) d r T = − q 2 k ∫ r d r + C 1 ∫ 1 r d r = − q 2 k [ r 2 2 ] + C 1 ln r + C 2 …… (III) Calculation: Substitute 0 for, r and 0 for d T d r in Equation (I) 0 = − ( q ) ( 0 ) 2 k + C 1 0 0 = 0 + C 1 0 C 1 = 0 Substitute r o for r , 0 for C 1 , and T s for T in Equation (III). T s = − q r o 2 4 k + ( 0 ) ( ln r 0 ) + C 2 C 2 = T s + q r o 2 4 k Here, surface temperature of cylinder is T s , and outer radius of cylinder is r o Substitute 0 for C 1 , and T s + q r o 2 4 k for C 2 in Equation (III)

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ISBN: 9781260699326

Author: CENGEL

Publisher: MCG

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Chapter 2, Problem 94P

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HEAT+MASS TRANSFER:FUND..(LL)-W/CONNECT

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