Transferência de Calor em Cilindro de Alumínio - Determinação de Coeficientes e Análise

PROBLEM 538 KNOWN Diameter of highly polished aluminum rod Temperature of rod initially and at two later times Room air temperature FIND Values of constants C and n in Equation 526 Plot rod temperature vs time for varying and constant heat transfer coefficients SCHEMATIC ASSUMPTIONS 1 Constant properties 2 Radiation negligible because rod is highly polished 3 Lumped capacitance approximation is valid PROPERTIES Table A1 Aluminum T 328 K c 916 JkgK 2702 kgm3 k 238 WmK ANALYSIS If the heat transfer coefficient is given by Equation 526 then the temperature as a function of time is given by Equation 528 1 1 n n s c i i nCA t Vc 1 where T T and Asc is the area exposed to convection Asc πDL Since the rod temperature is known at two different times Equation 1 can be evaluated at these two times making it possible to solve for the two unknowns C and n The two equations are 1 1 1 1 n n s c i i nCA t Vc 1 2 2 1 n n s c i i nCA t Vc 2ab These equations cannot be explicitly solved for C and n They can be numerically solved in this form using IHT or some other software or they can be further manipulated to solve for the times 1 1 1 4 n n i i cD t nC 2 2 1 4 n n i i cD t nC 3ab where we have used VAsc D4 Taking the ratio of Equations 3a and 3b yields 2 2 1 1 1 1 n i n i t t 30 C 20 C 1 6700 s 90 C 20 C 1250 s 65 C 20 C 1 90 C 20 C n n 0143 1 536 0643 1 n n 4abc Continued Ti 90C T 20C D 35 mm t1 1250s T1 65C t2 6700s T2 30C Ti 90C T 20C D 35 mm t1 1250s T1 65C t2 6700s T2 30C PROBLEM 538 Cont This can be iteratively or numerical solved for n to find n 025 Then C can be determined from Equation 3a or 3b 025 3 2 125 1 025 1 45 C 2702 kgm 916 Jkg K 0035 m 1 1 28 Wm K 4 70 C 4 025 70 C 1250 s n n i i cD C n t 2 125 28 Wm K C n 025 Now that these constants are known the validity of the lumped capacitance approximation can checked The maximum heat transfer coefficient occurs at the initial time 025 2 125 2 28 Wm K 90 20K 81 Wm K n h C T T Thus using the conservative definition Bi hD2k 6 104 The lumped capacitance approximation is valid The heat transfer coefficient corresponding to a rod temperature of 2 i T T T 55C is 025 2 125 2 28 Wm K 55 20K 68 Wm K n h C T T The plot below shows the rod temperature as a function of time using Equation 1 above for variable heat transfer coefficient as well as the rod temperature assuming the constant value of h 68 Wm2K using text Equation 56 0 01 02 03 04 05 06 07 08 09 1 0 2000 4000 6000 8000 10000 i t s constant h variable h COMMENTS 1 Since the heat transfer coefficient is temperature differencedependent variable h the initial cooling rates are larger when this dependence is accounted for As the temperature difference decreases the variable h case cools slower relative to the constant h case 2 The discrepancy between the variable and constant heat transfer coefficient cases is not large under these conditions The difference would be greater if n were larger