Saturated ammonia vapor at \(25^{\circ} \mathrm{C}\) condenses on the outside of a 2 -m-long, 3.2-cm-outer-diameter vertical tube maintained at \(15^{\circ} \mathrm{C}\). Determine \((a)\) the average heat transfer coefficient, \((b)\) the rate of heat transfer, and \((c)\) the rate of condensation of ammonia. Assume turbulent 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?

First, we need to calculate the temperature difference between the ammonia vapor and the tube, and also find the condensation temperature of ammonia. The temperature difference is given by \(\Delta T = T_{vapor} - T_{tube}\). The condensation temperature can be found using the saturation temperature corresponding to the saturation pressure for ammonia at \(25^{\circ} \mathrm{C}\). \(\Delta T = 25^{\circ} \mathrm{C} - 15^{\circ} \mathrm{C} = 10^{\circ} \mathrm{C}\) From the ammonia property tables, at \(25^{\circ} \mathrm{C}\): - Saturation pressure, \(P_{sat} = 1628\,\mathrm{Pa}\) - Condensation temperature, \(T_{cond} = 25^{\circ} \mathrm{C}\)

Next, we need to calculate the Nusselt number using the condensation Nusselt number correlation for turbulent film condensation on a vertical cylinder, which is given by: \(Nu = \frac{hD_o}{k_l} = 0.729\left(\frac{g \rho_l (\rho_l - \rho_v) k_l^3 L\Delta T}{\mu_l \mathrm{Pr}_l}\right)^{1/4}\) Where: - \(h\) is the average heat transfer coefficient - \(Nu\) is the Nusselt number - \(D_o\) is the tube outer diameter - \(k_l\) is the thermal conductivity of the liquid - \(g\) is the acceleration due to gravity - \(\rho_l\) is the density of the liquid - \(\rho_v\) is the density of the vapor - \(L\) is the length of the tube - \(\Delta T\) is the temperature difference - \(\mu_l\) is the dynamic viscosity of the liquid - \(\mathrm{Pr}_l\) is the Prandtl number of the liquid The average heat transfer coefficient can then be calculated as follows: \(h = \frac{k_l\,Nu}{D_o}\) Using the ammonia property tables at \(25^{\circ} \mathrm{C}\): - \(k_l = 0.685\,\mathrm{W/(m\cdot K)}\) - \(\rho_l = 602.3\,\mathrm{kg/m^3}\) - \(\rho_v = 0.619\,\mathrm{kg/m^3}\) - \(\mu_l = 306.8 * 10^{-6}\,\mathrm{Pa\cdot s}\) - \(\mathrm{Pr}_l = 1.098\) - \(g = 9.81\,\mathrm{m/s^2}\) Substitute the values into the Nusselt number correlation and calculate the heat transfer coefficient: \(Nu = 0.729\left(\frac{9.81\times 602.3\times (602.3-0.619)\times 0.685^3\times 2\times 10}{306.8\times10^{-6}\times 1.098}\right)^{1/4} = 81.24\) \(h = \frac{0.685\times 81.24}{0.032} = 1742.03\,\mathrm{W/(m^2\cdot K)}\) (a) The average heat transfer coefficient, \(h = 1742.03\,\mathrm{W/(m^2\cdot K)}\)

The rate of heat transfer can be calculated using the formula: \(q = h A_s \Delta T\) Where: - \(q\) is the rate of heat transfer - \(A_s\) is the outer surface area of the tube The outer surface area of the tube can be calculated as: \(A_s = \pi D_o L = \pi \times 0.032 \times 2 = 0.201\,\mathrm{m^2}\) Substituting the values, we get the rate of heat transfer: \(q = 1742.03\times 0.201 \times 10 = 3499.89\,\mathrm{W}\) (b) The rate of heat transfer, \(q = 3499.89\,\mathrm{W}\)

The rate of condensation can be calculated by dividing the rate of heat transfer by the latent heat of vaporization: \(\dot{m}_c = \frac{q}{h_{fg}}\) Where: - \(\dot{m}_c\) is the rate of condensation - \(h_{fg}\) is the latent heat of vaporization From the ammonia property tables at \(25^{\circ} \mathrm{C}\): - \(h_{fg} = 1370\,\mathrm{kJ/kg}\) Substituting the values and converting the units: \(\dot{m}_c = \frac{3499.89\,\mathrm{W}}{1370\times 10^3\,\mathrm{J/kg}} = 2.55\times10^{-3}\,\mathrm{kg/s}\) (c) The rate of condensation, \(\dot{m}_c = 2.55\times10^{-3}\,\mathrm{kg/s}\) Finally, the assumptions in the problem are good because the derived heat transfer coefficient is well within the range found in other cases for film condensation. Furthermore, the tube diameter being large relative to the thickness of the liquid film at the bottom of the tube allows for a more concise and accurate calculation of the heat transfer coefficient.