Question

A vertical 0.5- \(\mathrm{m} \times 0.5-\mathrm{m}\) square plate is used in a process to condense saturated water vapor. If the desired rate of condensation is \(0.016 \mathrm{~kg} / \mathrm{s}\), determine the necessary surface temperature of the plate at atmospheric pressure. For this problem, as a first approximation, assume a film temperature of \(90^{\circ} \mathrm{C}\) for the evaluation of the liquid properties and a surface temperature of $80^{\circ} \mathrm{C}$ for the evaluation of modified latent heat of vaporization. Are these good assumptions?

Short answer

Based on the given information and calculations, the necessary surface temperature of the plate at atmospheric pressure to achieve the desired rate of condensation is approximately 280°C. It is important to note that the initial assumption for the surface temperature (80°C) may not be accurate, and a more accurate evaluation should consider the actual surface temperature during the calculations. However, the assumption of film temperature at 90°C seems reasonable for fluid properties evaluation.

Step-by-step solution

Gather necessary information

We are given the following information: - Area of square plate: \(A = (0.5 \mathrm{m}) \times (0.5 \mathrm{m}) = 0.25 \mathrm{m}^2\) - Desired rate of condensation: \(m' = 0.016 \mathrm{kg/s}\) - Film temperature: \(T_f = 90^{\circ}\mathrm{C}\) - Initial surface temperature to evaluate modified latent heat of vaporization: \(T_s = 80^{\circ}\mathrm{C}\)

Calculate the heat transfer rate

The heat transfer rate can be calculated using the relation: \(Q = m' \times h_{fg}\), where \(h_{fg}\) is the modified latent heat of vaporization. We have \(m' = 0.016 \mathrm{kg/s}\) and \(h_{fg}\) can be obtained from the steam tables at given initial surface temperature (\(T_s = 80^{\circ}\mathrm{C}\)), which is approximately \(2435 \mathrm{kJ/kg}\). Now, calculating the heat transfer rate: \(Q = (0.016 \mathrm{kg/s}) \times (2435 \times 10^3 \mathrm{J/kg}) = 38.96 \times 10^3 \mathrm{W}\)

Apply Nusselt number relation

We can use Nusselt number to find the necessary surface temperature at atmospheric pressure. The Nusselt number is given by the relation: \(Nu = \frac{hL}{k}\), where \(h\) is the heat transfer coefficient, \(L\) is the characteristic length, and \(k\) is the thermal conductivity of the fluid. We are given that the film temperature (\(T_f\)) is \(90^{\circ}\mathrm{C}\) for the evaluation of liquid properties. We can obtain the following values from the water property tables at this temperature: - Dynamic viscosity (\(\mu\)): \(0.281 \times 10^{-3} \mathrm{Pa\,s}\) - Thermal conductivity (\(k\)): \(0.685 \mathrm{W/m\,K}\) - Prandtl number (\(Pr\)): \(1.76\) - Density (\(\rho\)): \(965 \mathrm{kg/m^3}\) The convective heat transfer coefficient (\(h\)) can be obtained using the Nusselt number correlation for laminar natural convection over vertical surfaces: \(Nu = 0.943 \times (Gr \times Pr)^{1/4}\), where \(Gr\) is the Grashof number. The Grashof number can be calculated using the relation: \(Gr = \frac{g \beta (\Delta T) L^3}{\nu^2}\), where \(g\) is the acceleration due to gravity, \(\beta\) is the thermal expansion coefficient (approximated as \(\frac{1}{T_f}\)), \(\Delta T\) is the temperature difference between the surface and the fluid, \(L\) is the characteristic length, and \(\nu\) is the kinematic viscosity. For water, the kinematic viscosity (\(\nu\)) can be calculated as: \(\nu = \frac{\mu}{\rho} = \frac{0.281 \times 10^{-3} \mathrm{Pa\,s}}{965 \mathrm{kg/m^3}} = 2.91 \times 10^{-7} \mathrm{m^2/s}\). Now, substituting the given values in the Grashof number expression: \(Gr = \frac{9.81 \frac{1}{(90+273)} (\Delta T) (0.5)^3}{(2.91 \times 10^{-7})^2}\) We can now substitute the expressions for \(Nu\) and \(Gr\) into the Nusselt number relation and solve for \(\Delta T\): \(0.943 \times (\frac{9.81 \frac{1}{(90+273)} (\Delta T) (0.5)^3}{(2.91 \times 10^{-7})^2} \times 1.76)^{1/4} = \frac{h(0.5)}{0.685}\) We also know that \(Q = hA\Delta T\), so we can substitute the expression for \(Q\): \(38.96 \times 10^3 \mathrm{W} = h(0.25 \mathrm{m}^2) \times \Delta T\) Now, we can solve these two equations together to find the values of \(h\) and \(\Delta T\). \(h = 816.7 \mathrm{W/m^2\,K}\) and \(\Delta T = 190 \mathrm{K}\)

Calculate the necessary surface temperature

Now, we can calculate the necessary surface temperature at atmospheric pressure by adding the calculated temperature difference (\(\Delta T\)) to the given film temperature: \(T_s = T_f + \Delta T = 90^{\circ} \mathrm{C} + 190 \mathrm{K} = 280^{\circ} \mathrm{C}\)

Check if the initial assumptions were good

Initially, we assumed a film temperature (\(T_f\)) of \(90^{\circ}\mathrm{C}\) for the evaluation of liquid properties and a surface temperature (\(T_s\)) of \(80^{\circ}\mathrm{C}\) for the evaluation of modified latent heat of vaporization. Now that we have calculated the necessary surface temperature, we can compare it to the initial assumptions. The calculated surface temperature (\(T_s = 280^{\circ}\mathrm{C}\)) is much higher than the initially assumed value of \(80^{\circ}\mathrm{C}\). Therefore, the initial assumption for the surface temperature may not be accurate, and a more accurate evaluation should consider the actual surface temperature during the calculations. However, the film temperature assumption (\(T_f = 90^{\circ}\mathrm{C}\)) seems reasonable since it is close to the actual conditions and was used for fluid properties evaluation. This assumption should not have a significant impact on the results.