A spine in the form of a right circular cylinder, made of a material with thermal conductivity k = 45W/m^◦C, is attached to a base structure at Tb = 120^◦C. It loses heat to ambient air at Tamb = 20^◦C by convection via a heat transfer coefficient h = 15W/m^2^◦C. Determine the proportions of an opt

Question

A spine in the form of a right circular cylinder, made of a material with thermal conductivity k = 45W/m°C, is attached to a base structure at [latex] T_{b} = 120°C[/latex] . It loses heat to ambient air at [latex] T_{amb} = 20°C[/latex] by convection via a heat transfer coefficient [latex] h = 15W/m^{2}°C[/latex] . Determine the proportions of an optimum spine if the volume parameter is [latex] \acute{V} = 10^{-6} m^{3}[/latex] . How much heat does it lose to the ambient?What is the heat loss per unit volume of the material of spine?

Accepted Answer

Step 1 With the data specified in the problem, we calculate the parameters that appear in the optimum spine.

[latex]K_{1} = heta _{b} \frac{\pi \sqrt{hk} }{2} =(120-20)\frac{\pi \sqrt{15 imes 45} }{2} =4081.0486[/latex]

[latex]K_{2} =2\acute{V} \sqrt{\frac{h}{k} } =2 imes 10^{-6} imes \sqrt{\frac{15}{45} } =1.1547 imes 10^{-6}[/latex]

Step 2 The diameter of optimum spine is obtained using Eq. 4.65a as

[latex](a) D^{*} =1.3647 \left(\frac{\acute{V} ^{2} h}{k} ight) ^{1/5} (b) L^{*} =0.5370 \left(\frac{\acute{V} k^{2} }{h^{2}} ight) ^{1/5}[/latex] (4.65)

[latex]D^{*} =1.3647\left(\frac{10^{-12} imes 15}{45} ight) ^{1/5} =0.0044m[/latex]

The length of optimum spine is obtained using Eq. 4.65b as

[latex]L^{*} =0.5370\left(\frac{10^{-6} imes 15^{2}}{45^{2}} ight) ^{1/5}=0.0526m[/latex]

Step 3 Heat loss from the spine is then calculated as

[latex]Q_{b} =K_{1} D^{3/2}tanh\beta ^{*} =4081.0486 imes0.0044^{3/2} tanh0.9193=0.829W[/latex]

Step 4 Spine volume is given by

[latex]V=\pi \frac{D^*{^{2} } L}{4} =\frac{\pi \acute{V} }{4} =\frac{\pi imes 10^{-6}}{4} =7.854 imes 10^{-7}m^{3}[/latex]

Heat transfer per unit volume is then given by

[latex]Q=\frac{Q_{b}}{V} =\frac{0.8529}{7.854 imes 10^{-7}} =108590W/ m^{3}= 1.0859 MW/m^{3}[/latex]

Question 4.8: A spine in the form of a right circular cylinder, made of a ...

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