A cylinder of diameter D = 0.1m, length L = 0.15m is maintained at a uniform temperature of Tw = 60 °C in still air at T∞ = 20 °C. The two flat ends of the cylinder are perfectly insulated. Compare the heat loss from the cylinder if its axis is (a) horizontal and (b) vertical.

Question

A cylinder of diameter D = 0.1m, length L = 0.15m is maintained at a uniform temperature of [latex]T_{w}=60^{\circ} C[/latex] in still air at [latex]T_{\infty}=20^{\circ} C[/latex]. The two flat ends of the cylinder are perfectly insulated. Compare the heat loss from the cylinder if its axis is (a) horizontal and (b) vertical.

Accepted Answer

We base the thermo-physical properties of air at [latex]T_{m}=\frac{T_{w}+T_{\infty}}{2}=\frac{60+20}{2}=40^{\circ} C =273 + 40 = 313 K[/latex].

Kinematic viscosity: [latex]v_{m}=17.07 imes 10^{-6} m ^{2} / s[/latex]

Thermal conductivity: [latex]k_{m}=0.0274 W / m ^{\circ} C[/latex]

Prandtl number: [latex]P r_{m}=0.699[/latex]

Isobaric expansion coefficient: [latex]\beta_{m}=\frac{1}{313}=0.0032 K ^{-1}[/latex]

Case (a) Cylinder axis vertical

The characteristic dimension in this case is [latex]L_{c h}=L=0.15 m[/latex]. We have for the cylinder [latex]\frac{D}{L}=\frac{0.1}{0.15}=0.667[/latex] . The Rayleigh number is

[latex]R a_{L}=\frac{9.81 imes 0.0032(60-20) 0.15^{3}}{\left(17.07 imes 10^{-6} ight)^{2}} 0.699=1.017 imes 10^{7}[/latex]

The Grashof number is given by [latex]G r_{L}=\frac{R a_{L}}{P r_{m}}=\frac{1.017 imes 10^{7}}{0.699}=1.454 imes 10^{7}[/latex] . We then have

[latex]\frac{35}{G r_{L}^{\frac{1}{4}}}=\frac{35}{\left(1.454 imes 10^{7} ight)^{\frac{1}{4}}}=0.567<0.667[/latex]

Hence, the correlation for a vertical plate may be made use of. The flow is laminar and we make use of Eq. 16.51 to get

[latex]\overline{N u}_{L}=0.55 R a_{L}^{\frac{1}{4}}[/latex] (16.51)

[latex]\overline{N u}_{L}=0.55 imes\left(1.017 imes 10^{7} ight)^{\frac{1}{4}}=31.06[/latex]

The average heat transfer coefficient is given by

[latex]\bar{h}=\frac{31.06 imes 0.0274}{0.15}=5.67 W / m ^{2 \circ} C[/latex]

The heat loss from the cylinder is then given by

[latex]Q_{ ext {vert }}=\bar{h} \pi D L\left(T_{w}-T_{\infty} ight)=5.67 imes \pi imes 0.1 imes 0.15 imes 40=10.7 W[/latex]

Case (b) Cylinder axis horizontal

The characteristic dimension in this case is [latex]L_{c h}=D=0.1m[/latex]. The Rayleigh number is

[latex]R a_{D}=\frac{9.81 imes 0.0032(60-20) 0.1^{3}}{\left(17.07 imes 10^{-6} ight)^{2}} 0.699=3.013 imes 10^{6}[/latex]

The flow is laminar and we make use of Eq. 16.59 to get

[latex]\overline{N u}_{D}=0.53 R a_{D}^{\frac{1}{4}}, 10^{4}

[latex]\overline{N u}_{L}=0.53 imes\left(3.013 imes 10^{6} ight)^{\frac{1}{4}}=22.08[/latex]

The average heat transfer coefficient is given by

[latex]\bar{h}=\frac{22.08 imes 0.0274}{0.1}=6.05 W / m ^{2{ }^{\circ} C }[/latex]

The heat loss from the cylinder is then given by

[latex]Q_{h o r}=\bar{h} \pi D L\left(T_{w}-T_{\infty} ight)=6.05 imes \pi imes 0.1 imes 0.15 imes 40=11.4 W[/latex]

The cylinder with its axis horizontal loses marginally more heat than when the cylinder axis is vertical.

Question 16.8: A cylinder of diameter D = 0.1m, length L = 0.15m is maintai...

This Question has been Answered.

Already have an account?

Login

Related Answered Questions