Determination of the Heat Transfer Coefficient An average convective heat transfer coefficient for flow of 90°𝐶 air over a plate is measured by observing the temperature – time history of a 40𝑚𝑚 thick copper slab (𝜌 = 9000𝑘𝑔/m³, 𝑐𝑝 = 0.38𝑘𝑗/𝑘𝑔°𝐶, 𝑘 = 370𝑤/𝑚°𝐶) exposed to 90°𝐶

Question

Determination of the Heat Transfer Coefficient
An average convective heat transfer coefficient for flow of 90°𝐶 air over a plate is
measured by observing the temperature – time history of a 40𝑚𝑚 thick copper slab (𝜌 = 9000𝑘𝑔/m³, [latex]c_{p}[/latex] = 0.38𝑘𝑗/𝑘𝑔°𝐶, 𝑘 = 370𝑤/𝑚°𝐶) exposed to 90°𝐶 air. In one test run, the initial temperature of the plate was 200°𝐶, and in 4.5 minutes the temperature decreased by 35°𝐶. Find the heat transfer coefficient for this case.
Neglect internal thermal resistance.

Accepted Answer

[latex]{\mathrm{Given}}\colon\;T_{\infty}=90^{\circ}C;t=40m m\;o r\;\;0.04m;\; ho=9000k g/m^{3};\;c_{p}=0.38k j/k g^{\circ}C;[/latex]

[latex]T_{o}=200^{\circ}C;\;T(t)=200-35=165^{\circ}C;\; au=4.5m i n=270s[/latex]

Characteristic length of the sphere is,

[latex]L=\frac{t}{2}=\frac{0.04}{2}=0.02m[/latex]

[latex]B i=\frac{h L}{k}=\frac{0.02h}{370}=5.405 imes10^{-5}h[/latex]

[latex]F o=\frac{k}{ ho c_{P}L^{2}}\cdot au=\frac{370}{900 imes0.38 imes10^{3} imes0.02^{2}} imes270=73.03[/latex]

[latex]B i imes F o= 5.405 imes 10^{-5} imes 73.03h=394.73 imes10^{-5}h=0.003947h[/latex]

[latex]\frac{ heta}{ heta_{o}}=\frac{T(t)-T_{\infty}}{T_{o}-T_{\infty}}=e^{-B i imes\!F o}[/latex]

[latex]\frac{ heta}{ heta_{o}}=\frac{165-90}{200-90}=e^{-0.003947h}[/latex]

[latex]0.682\,=\,e^{-0.003947h}[/latex]

[latex]\ln0.682=-0.003947h \ \ln \ e[/latex]

[latex] h=\frac{\ln \ 0.682}{-0.003947 imes 1}=96.97w/m^{2^{\circ}} C[/latex]

Question 7.7: Determination of the Heat Transfer Coefficient An average co......

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