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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. T2 x k(T) L T1 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 . )1()( 20 TkTk \u3b2+= 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 can be determined from 21 and TT ( ) ( ) ( )\u23a5\u23a6\u23a4\u23a2\u23a3\u23a1 +++= \u2212 \u23a5\u23a6 \u23a4\u23a2\u23a3 \u23a1 \u2212+\u2212 =\u2212 \u239f\u23a0 \u239e\u239c\u239d \u239b + =\u2212 + =\u2212= \u222b\u222b 2 121 2 20 12 3 1 3 2120 12 3 0 12 2 0 12 avg 3 1 33 )1()( 2 1 2 1 2 1 TTTTk TT TTTTk TT TTk TT dTTk TT dTTk k T T T T T T \u3b2 \u3b2\u3b2\u3b2 This relation is based on the requirement that the rate of heat transfer through a medium with constant average thermal conductivity kavg 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 ( ) L TT ATTTTk L TT AkQ 212121 2 20 21 avg 3 1 \u2212 \u23a5\u23a6 \u23a4\u23a2\u23a3 \u23a1 +++=\u2212= \u3b2& 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. PROPRIETARY MATERIAL. © 2007 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission. 2-53 2-103 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. Properties The thermal conductivity is given to be )1()( 0 TkTk \u3b2+= . r2 T2 r r1 T1 k(T) Solution (a) The rate of heat transfer through the shell is expressed as )/ln( 2 12 21 avgcylinder rr TT LkQ \u2212= \u3c0& where L is the length of the cylinder, r1 is the inner radius, and r2 is the outer radius, and \u239f\u23a0 \u239e\u239c\u239d \u239b ++== 2 1)( 120avgavg TTkTkk \u3b2 is the average thermal conductivity. (b) To determine the temperature distribution in the shell, we begin with the Fourier\u2019s law of heat conduction expressed as dr dTATkQ )(\u2212=& where the rate of conduction heat transfer is constant and the heat conduction area A = 2\u3c0rL is variable. Separating the variables in the above equation and integrating from r = r Q& 1 where to any r where , we get 11 )( TrT = TrT =)( \u222b\u222b \u2212= TTrr dTTkLrdrQ 11 )(2\u3c0& Substituting )1()( 0 TkTk \u3b2+= and performing the integrations gives ]2/)()[(2ln 21 2 10 1 TTTTLk r rQ \u2212+\u2212\u2212= \u3b2\u3c0& Substituting the expression from part (a) and rearranging give &Q 02)( )/ln( )/ln(22 1 2 121 12 1 0 avg2 =\u2212\u2212\u2212++ TTTT rr rr k k TT \u3b2\u3b2\u3b2 which is a quadratic equation in the unknown temperature T. Using the quadratic formula, the temperature distribution T(r) in the cylindrical shell is determined to be 1 2 121 12 1 0 avg 2 2)( )/ln( )/ln(211)( TTTT rr rr k k rT \u3b2\u3b2\u3b2\u3b2 ++\u2212\u2212±\u2212= Discussion The proper sign of the square root term (+ or -) is determined from the requirement that the temperature at any point within the medium must remain between . 21 and TT PROPRIETARY MATERIAL. © 2007 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission. 2-54 2-104 A spherical 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. Properties The thermal conductivity is given to be )1()( 0 TkTk \u3b2+= . r1 r2 T1k(T) r T2Solution (a) The rate of heat transfer through the shell is expressed as 12 21 21avgsphere 4 rr TT rrkQ \u2212 \u2212= \u3c0& where r1 is the inner radius, r2 is the outer radius, and \u239f\u23a0 \u239e\u239c\u239d \u239b ++== 2 1)( 120avgavg TTkTkk \u3b2 is the average thermal conductivity. (b) To determine the temperature distribution in the shell, we begin with the Fourier\u2019s law of heat conduction expressed as dr dTATkQ )(\u2212=& where the rate of conduction heat transfer is constant and the heat conduction area A = 4\u3c0rQ& 2 is variable. Separating the variables in the above equation and integrating from r = r1 where to any r where , we get 11 )( TrT = TrT =)( \u222b\u222b \u2212= TTrr dTTkrdrQ 11 )(42 \u3c0& Substituting )1()( 0 TkTk \u3b2+= and performing the integrations gives ]2/)()[(411 21 2 10 1 TTTTk rr Q \u2212+\u2212\u2212=\u239f\u239f\u23a0 \u239e \u239c\u239c\u239d \u239b \u2212 \u3b2\u3c0& Substituting the expression from part (a) and rearranging give Q& 02)( )( )(22 1 2 121 12 12 0 avg2 =\u2212\u2212\u2212\u2212 \u2212++ TTTT rrr rrr k k TT \u3b2\u3b2\u3b2 which is a quadratic equation in the unknown temperature T. Using the quadratic formula, the temperature distribution T(r) in the cylindrical shell is determined to be 1 2 121 12 12 0 avg 2 2)( )( )(211)( TTTT rrr rrr k k rT \u3b2\u3b2\u3b2\u3b2 ++\u2212\u2212 \u2212\u2212±\u2212= Discussion The proper sign of the square root term (+ or -) is determined from the requirement that the temperature at any point within the medium must remain between . 21 and TT PROPRIETARY MATERIAL. © 2007 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission. 2-55 2-105 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 linearly. 3 There is no heat generation. Properties The thermal conductivity is given to be )1()( 0 TkTk \u3b2+= . T2 k(T) T1 L Analysis The average thermal conductivity of the medium in this case is simply the conductivity value at the average temperature since the thermal conductivity varies linearly with temperature, and is determined to be K W/m24.34 2 K 350)+(500 )K 10(8.7+1K) W/m25( 2 1)( 1-4- 12 0avgave \u22c5= \u239f\u23a0 \u239e\u239c\u239d \u239b ×\u22c5= \u239f\u239f\u23a0 \u239e\u239c\u239c\u239d \u239b ++== TTkTkk \u3b2 Then the rate of heat conduction through the plate becomes kW 30.8==\u2212×\u22c5=\u2212= W30,820 m 15.0 0)K35(500m) 0.6 m K)(1.5 W/m24.34(21avg L TT AkQ& 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. PROPRIETARY MATERIAL. © 2007 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission. 2-56 2-106 EES Prob. 2-105 is reconsidered. The rate of heat conduction through the plate as a function of the temperature of the hot side of the plate is to be plotted. Analysis The problem is solved using EES, and the solution is given below. "GIVEN" A=1.5*0.6 [m^2] L=0.15 [m] T_1=500 [K] T_2=350 [K] k_0=25 [W/m-K] beta=8.7E-4 [1/K] "ANALYSIS" k=k_0*(1+beta*T) T=1/2*(T_1+T_2) Q_dot=k*A*(T_1-T_2)/L T1 [W] Q [W] 400 9947 425 15043 450 20220 475 25479 500 30819 525 36241 550 41745 575 47330 600 52997 625 58745 650 64575 675 70486 700 76479 400 450 500 550 600 650 700 0 10000 20000 30000 40000 50000 60000 70000 80000 T1 [K] Q [ W ] PROPRIETARY MATERIAL. © 2007 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using