Calculate the salient features in the boiling curve when water at a pressure of 1 atmosphere absolute boils over a polished copper surface. Make use of the various correlations presented above.

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Step 1 Properties of saturated water and saturated steam at [latex]T_{s}=100{ }^{\circ} C[/latex] are

Saturated water at [latex]T_{s}=100{ }^{\circ} C[/latex]

Density: [latex] ho_{f}=958.4 kg / m ^{3}[/latex]

Thermal conductivity: [latex]k_{f}=0.680 W / m ^{\circ} C[/latex]

Dynamic viscosity: [latex]\mu_{f}=2.79 imes 10^{-4} kg / m s[/latex]

Specific heat: [latex]C_{p f}=4216 J / kg ^{\circ} C[/latex]

Prandtl number: [latex]\operatorname{Pr}_{f}=1.73[/latex]

Surface tension: [latex]\sigma=0.059 N / m[/latex]

Saturated steam at [latex]T_{s}=100{ }^{\circ} C[/latex]

Density: [latex] ho_{g}=0.5981 kg / m ^{3}[/latex]

Thermal conductivity: [latex]k_{g}=0.0248 W / m ^{\circ} C[/latex]

Latent heat of vaporization: [latex]h_{f g}=2.257 imes 10^{6} J / kg[/latex]

Specific heat: [latex]C_{p g}=2020 J / kg ^{\circ} C[/latex]

Dynamic viscosity: [latex]\mu_{g}=1.2 imes 10^{-5} kg / m s[/latex]

Step 2 The critical heat flux is calculated using Eq. 17.83 as

[latex]q_{c r}=0.18 ho_{g} h_{f g}\left[\frac{\sigma g\left( ho_{f}- ho_{g} ight)}{ ho_{g}^{2}} ight]^{\frac{1}{4}} \approx 0.18 ho_{g} h_{f g}\left[\frac{\sigma g ho_{f}}{ ho_{g}^{2}} ight]^{\frac{1}{4}}[/latex] (17.83)

[latex]q_{c r}=0.18 imes 0.5981 imes 2.257 imes 10^{6}\left[\frac{0.059 imes 9.81(958.4-0.5981)}{0.5981^{2}} ight]^{\frac{1}{4}}= 1.524 × 10^6 W/m^2[/latex]

Step 3 Using Rohsenow correlation (17.88) we calculate the wall superheat parameter under DNB as

[latex]\Delta T=\frac{C h_{f g}}{C_{p f}}\left[\left(\frac{q_{n b}}{\mu_{f} h_{f g}} ight)\left(\frac{\sigma}{g\left( ho_{f}- ho_{g} ight)} ight)^{\frac{1}{2}} ight]^{0.33} \operatorname{Pr}_{f}^{m+1}[/latex] (17.88)

[latex]\Delta T_{D N B}=\frac{0.0182 imes 2.257 imes 10^{6}}{4216}\left(\frac{1.524 imes 10^{6}}{2.79 imes 10^{-4} imes 2.257 imes 10^{6}} ight)^{0.33} imes[/latex] [latex]\left(\frac{0.059}{9.81(958.4-0.5981)} ight)^{0.165} imes 1.73=17.66{ }^{\circ} C[/latex]

where the constant C has been chosen as 0.0182 for water boiling over polished copper.

Step 4 Assuming that nucleate boiling initiates when [latex]\Delta T=6.5 K[/latex], the corresponding heat flux is given by

[latex]q_{I B}=q_{c r}\left(\frac{\Delta T_{I B}}{\Delta T_{D N B}} ight)^{\frac{1}{0.33}}=1.524 imes 10^{6}\left(\frac{6.5}{\Delta 17.66} ight)^{\frac{1}{0.33}}=73722 W / m ^{2}[/latex]

Step 5 The minimum heat flux under film boiling condition is calculated based on Eq. 17.91

[latex]q_{M F B}=\frac{q_{c r}}{2 \sqrt{\frac{ ho_{f}+ ho_{g}}{ ho_{g}}}}[/latex] (17.91)

[latex]q_{M F B}=\frac{1.524 imes 10^{6}}{2 \sqrt{\frac{958.4+0.5981}{0.5981}}}=19029.8 W / m ^{2}[/latex]

We shall assume that the heat transfer coefficient in film boiling is given by Eq. 17.92. The heat flux is then given by [latex]q_{M F B}=h_{f} \Delta T_{M F B}[/latex]. We may then solve for [latex]\Delta T_{M F B}[/latex] as

[latex]\Delta T_{M F B}=\frac{\left(\frac{q_{M F B}}{0.425} ight)^{\frac{4}{3}}}{\left(\frac{k_{g}^{3} ho_{g}\left( ho_{f}- ho_{g} ight) g h_{e}}{\mu_{g} \frac{L}{2 \pi}} ight)^{\frac{1}{3}}}[/latex]

To begin with let us assume that [latex]h_{e}=h_{e}^{0} \approx h_{f g}=2.257 imes 10^{6} J / kg[/latex]. The characteristic length [latex]L_{c}[/latex] is given by

[latex]L_{c}=2 \pi \sqrt{\frac{0.059}{9.81(958.4-0.5981)}}=0.1157 m[/latex]

Then we get

[latex]\Delta T_{M F B}^{0}=\frac{\left(\frac{19029.8}{0.425} ight)^{\frac{4}{3}}}{\left(\frac{0.0248^{3} imes 0.5981(958.4-0.5981) imes 9.81 imes 2.257 imes 10^{6}}{1.2 imes 10^{-5} imes \frac{0.1157}{2 \pi}} ight)^{\frac{1}{3}}}=166.3[/latex]

We shall now correct the result by correcting [latex]h_{e}[/latex] as

[latex]h_{e}^{1}=h_{f g}+0.4 C_{p g} \Delta T=2.257 imes 10^{6}+0.4 imes 2020 imes 166.3=2.3913 imes 10^{6}[/latex]

The corrected value of superheat is then obtained as

[latex]\Delta T_{M F B}^{1}=\frac{\Delta T_{M F B}^{0}}{\left(\frac{h_{e}^{1}}{h_{e}^{0}} ight)^{0.33}}=\frac{\Delta 166.3}{\left(\frac{2.391 imes 10^{6}}{2.257 imes 10^{6}} ight)^{0.33}}=163.1{ }^{\circ} C[/latex]

Step 6 We may also calculate the film boiling heat flux with another superheat value of [latex]\Delta T=500^{\circ} C[/latex]. Then we have

[latex]h_{e}^{1}=2.257 imes 10^{6}+0.4 imes 2020 imes 500=2.661 imes 10^{6}[/latex]

The corresponding heat flux is then obtained using Eq. 17.92 as

[latex]h_{f}=0.425\left[\frac{k_{g}^{3} ho_{g}\left( ho_{f}- ho_{g} ight) g h_{e}}{\mu_{g} \Delta T\left(\frac{L_{c}}{2 \pi} ight)} ight]^{\frac{1}{4}}[/latex] (17.92)

[latex]q_{f}=0.425 imes 500^{\frac{3}{4}}\left[\frac{0.0248^{3} \cdot 0.5981(958.4-0.5981) \cdot 9.81 \cdot 2.661 imes 10^{6}}{1.2 imes 10^{-5}\left(\frac{0.1157}{2 \pi} ight)} ight]^{\frac{1}{4}}= 45296.3 W/m^2[/latex]

The calculations made in this example are used to make a plot of the boiling curve as shown in Fig. 17.19. We join the points by straight lines on the log-log plot as shown in the figure. Experimental boiling curves indicate fair agreement with the values calculated here.

Question 17.11: Calculate the salient features in the boiling curve when wat...

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