Thermodynamics HW Solutions 410

# Thermodynamics HW Solutions 410 -...

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Chapter 5 Numerical Methods in Heat Conduction 5-26 "!PROBLEM 5-26" "GIVEN" k=180 "[W/m-C]" L=0.05 "[m]" b=0.01 "[m]" w=1 "[m]" "T_0=180 [C], parameter to be varied" T_infinity=25 "[C]" h=25 "[W/m^2-C]" T_surr=290 "[K]" M=6 epsilon=0.9 tan(theta)=(0.5*b)/L sigma=5.67E-8 "[W/m^2-K^4], Stefan-Boltzmann constant" "ANALYSIS" "(a)" DELTAx=L/(M-1) "Using the finite difference method, the five equations for the temperatures at 5 nodes are determined to be" (1-0.5*DELTAx/L)*(T_0-T_1)+(1-1.5*DELTAx/L)*(T_2- T_1)+(h*DELTAx^2)/(k*L*sin(theta))*(T_infinity- T_1)+(epsilon*sigma*DELTAX^2)/(k*L*sin(theta))*(T_surr^4-(T_1+273)^4)=0 "for mode 1" (1-1.5*DELTAx/L)*(T_1-T_2)+(1-2.5*DELTAx/L)*(T_3- T_2)+(h*DELTAx^2)/(k*L*sin(theta))*(T_infinity- T_2)+(epsilon*sigma*DELTAX^2)/(k*L*sin(theta))*(T_surr^4-(T_2+273)^4)=0 "for mode 2" (1-2.5*DELTAx/L)*(T_2-T_3)+(1-3.5*DELTAx/L)*(T_4- T_3)+(h*DELTAx^2)/(k*L*sin(theta))*(T_infinity-
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Unformatted text preview: T_3)+(epsilon*sigma*DELTAX^2)/(k*L*sin(theta))*(T_surr^4-(T_3+273)^4)=0 "for mode 3" (1-3.5*DELTAx/L)*(T_3-T_4)+(1-4.5*DELTAx/L)*(T_5-T_4)+(h*DELTAx^2)/(k*L*sin(theta))*(T_infinity-T_4)+(epsilon*sigma*DELTAX^2)/(k*L*sin(theta))*(T_surr^4-(T_4+273)^4)=0 "for mode 4" 2*k*DELTAx/2*tan(theta)*(T_4-T_5)/DELTAx+2*h*(0.5*DELTAx)/cos(theta)*(T_infinity-T_5)+2*epsilon*sigma*(0.5*DELTAx)/cos(theta)*(T_surr^4-(T_5+273)^4)=0 "for mode 5" T_tip=T_5 "(b)" Q_dot_fin=C+D "where" C=h*(w*DELTAx)/cos(theta)*((T_0-T_infinity)+2*(T_1-T_infinity)+2*(T_2-T_infinity)+2*(T_3-T_infinity)+2*(T_4-T_infinity)+(T_5-T_infinity)) D=epsilon*sigma*(w*DELTAx)/cos(theta)*(((T_0+273)^4-T_surr^4)+2*((T_1+273)^4-T_surr^4)+2*((T_2+273)^4-T_surr^4)+2*((T_3+273)^4-T_surr^4)+2*((T_4+273)^4-T_surr^4)+((T_5+273)^4-T_surr^4)) 5-13...
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