Question

Saturated steam at 1 atm condenses on a \(2-\mathrm{m}\)-high and 10 -m-wide vertical plate that is maintained at \(90^{\circ} \mathrm{C}\) by circulating cooling water through the other side. Determine (a) the rate of heat transfer by condensation to the plate, and (b) the rate at which the condensate drips off the plate at the bottom. Assume wavy-laminar flow. Is this a good assumption?

Short answer

Answer: The rate of heat transfer by condensation to the plate is approximately 2410 W. 2) What is the rate at which the condensate drips off the plate at the bottom? Answer: The rate at which the condensate drips off the plate at the bottom is approximately 1.068 x 10^-3 kg/s.

Step-by-step solution

Obtain the properties for saturated steam at 1 atm

First, the properties of saturated steam at 1 atm should be obtained. From steam tables, we find: Saturation temperature: \(T_s = 100^{\circ}\mathrm{C}\) Latent heat of vaporization: \(L_v = 2.257 \times 10^6 \, \mathrm{J/kg}\) Density of vapor: \(\rho_v = 0.598 \, \mathrm{kg/m^3}\) Density of liquid: \(\rho_l = 958 \, \mathrm{kg/m^3}\)

Write the dimensional analysis relationships for condensation heat transfer in wavy-laminar flow

The Nusselt number, \(Nu = \frac{hL}{k}\), in wavy-laminar condensate flow can be expressed as: \(Nu = 0.943\left(\mathrm{Gr}_L\mathrm{Pr}\right)^{1/4}\) where \(Gr_L\) is the Grashof number based on the liquid film thickness, \(L\), and Pr is the Prandtl number. The Grashof number can be estimated as: \(\mathrm{Gr}_L = \frac{g\beta \rho_l (\rho_l - \rho_v)L^3}{\mu_l^2}\) The Prandtl number is given by: \(\mathrm{Pr} = \frac{\mu_l c_{p_l}}{k_l}\)

Calculate the Grashof number, Prandtl number, and Nusselt number

Next, we calculate those dimensionless quantities: First, let's identify the remaining properties needed for the calculations: Coefficient of volume expansion, \(\beta = \frac{1}{T_s+273.15} \approx 3.24 \times 10^{-3}\, \mathrm{K^{-1}}\) Viscosity, \(\mu_l = 2.82 \times 10^{-4} \, \mathrm{kg \cdot m^{-1}s^{-1}}\) Heat capacity, \(c_{p_l} = 4.22 \times 10^3 \, \mathrm{J \cdot kg^{-1}K^{-1}}\) Thermal conductivity, \(k_l = 0.679 \, \mathrm{W \cdot m^{-1}K^{-1}}\) Now we can calculate Grashof and Prandtl numbers: \(\mathrm{Gr}_L = \frac{9.81 \times 3.24 \times 10^{-3} \times 958 \times (958 - 0.598) \times (2)^3}{(2.82 \times 10^{-4})^2} \approx 5.56 \times 10^9\) \(\mathrm{Pr} = \frac{2.82 \times 10^{-4} \times 4.22 \times 10^3}{0.679} \approx 1.749\) Now, we can determine the Nusselt number: \(Nu = 0.943\left(5.56 \times 10^9 \times 1.749\right)^{1/4} \approx 356\)

Calculate the heat transfer coefficient

Using the Nusselt number, we can now calculate the heat transfer coefficient: \(h = \frac{Nu \cdot k_l}{L} = \frac{356 \cdot 0.679}{2} \approx 120.5\, \mathrm{W\cdot m^{-2}K^{-1}}\)

Determine the rate of heat transfer by condensation

With the heat transfer coefficient, we can now calculate the rate of heat transfer, \(q\), by condensation: \(q = hA(T_s - T) = 120.5 \times 2 \times 10 \times (100 - 90) \approx 2410\, \mathrm{W}\) This is the rate of heat transfer by condensation to the plate (a).

Calculate the rate at which condensate drips off the plate

To find the rate at which the condensate drips off the plate, we need to use the heat transfer rate we just found: \(\dot{m} = \frac{q}{L_v} = \frac{2410}{2.257 \times 10^6} \approx 1.068 \times 10^{-3}\, \mathrm{kg/s}\) This is the rate at which the condensate drips off the plate at the bottom (b).

Check the validity of the wavy-laminar flow assumption

Finally, we should check if the assumption of wavy-laminar flow is valid. For wavy-laminar flow, the Reynolds number, \(Re_L\), should be less than \(1800\). The Reynolds number can be estimated as: \(Re_L = \frac{4\dot{m}}{\mu_lW} = \frac{4 \times 1.068 \times 10^{-3}}{2.82 \times 10^{-4} \times 10} \approx 15.1\) Since \(Re_L \approx 15.1 < 1800\), the assumption of wavy-laminar flow is valid. In conclusion, (a) the rate of heat transfer by condensation to the plate is approximately \(2410\, \mathrm{W}\), (b) the rate at which the condensate drips off the plate at the bottom is approximately \(1.068 \times 10^{-3}\, \mathrm{kg/s}\), and the assumption of wavy-laminar flow is valid.