Saturated water vapor at atmospheric pressure condenses on the outer surface of a \(0.1\)-m-diameter vertical pipe. The pipe is \(1 \mathrm{~m}\) long and has a uniform surface temperature of \(80^{\circ} \mathrm{C}\). Determine the rate of condensation and the heat transfer rate by condensation. Discuss whether the pipe can be treated as a vertical plate. Assume wavy-laminar flow and that the tube diameter is large relative to the thickness of the liquid film at the bottom of the tube. Are these good assumptions?

At the given temperature of \(80^{\circ} \mathrm{C}\), refer to the Steam Tables or an online steam properties calculator to obtain the following properties of water: - Saturation pressure, \(P_{sat} = 47.36 \mathrm{~kPa}\) - Liquid density, \(\rho_L = 971.97 \mathrm{~kg/m^3}\) - Vapor density, \(\rho_V = 0.4343 \mathrm{~kg/m^3}\) - Dynamic viscosity of liquid, \(\mu = 3.58 \times 10^{-4} \mathrm{~kg/(m \cdot s)}\) - Specific heat of vaporization, \(L_{v} = 2214.5 \mathrm{~kJ/kg}\)

According to the wavy-laminar flow model, the thickness of the liquid film at the bottom of the tube, \(\delta\), can be expressed as: \(\delta = 0.943 \left( \frac{\mu L_v (T_s - T_{sat})}{g(\rho_L - \rho_V) x} \right)^{1/4}\) where - \(T_s = 80^{\circ} \mathrm{C}\) (surface temperature) - \(T_{sat} = 80^{\circ} \mathrm{C}\) (saturation temperature) - \(x = 1 \mathrm{~m}\) (vertical height of the tube) - \(g = 9.81 \mathrm{~m/s^2}\) (acceleration due to gravity) Now, plug in the given values and calculate the film thickness: \(\delta = 0.943 \left( \frac{3.58 \times 10^{-4} \cdot 2214.5 \times 10^{3} \cdot (80-80)}{9.81(971.97 - 0.4343) \cdot 1} \right)^{1/4} = 0 \mathrm{~m}\) As the film thickness is zero, the film does not really exist. This means that our assumptions of wavy-laminar flow and the tube diameter being large relative to the film thickness are not valid.

Since our assumptions in the previous step did not hold, it is not possible to directly calculate the rate of condensation and heat transfer rate from our derived equations. However, these values can still be determined experimentally or using other appropriate models, which consider more factors influencing the condensation and heat transfer processes.

In this exercise, the initial assumptions of wavy-laminar flow and the tube diameter being large relative to the film thickness do not appear to be valid. Therefore, to determine the rate of condensation and heat transfer rate, alternative methods, such as experimental data or different models, should be used.