Question

Steam is being condensed at \(60^{\circ} \mathrm{C}\) by a 15 -m-long horizontal copper tube with a diameter of \(25 \mathrm{~mm}\). The tube surface temperature is maintained at \(40^{\circ} \mathrm{C}\). Determine the condensation rate of the steam during \((a)\) film condensation, and (b) dropwise condensation. Compare and discuss the results.

Short answer

Answer: In this problem, the dropwise condensation rate is significantly higher than the film condensation rate. This is because dropwise condensation minimizes thermal resistance and maintains a higher heat transfer coefficient, resulting in a more efficient heat exchange process.

Step-by-step solution

Assumptions

We assume that the condensation process is steady-state, meaning the rate of the steam condensing does not change over time. We also assume a uniform temperature profile along the tube.

Film Condensation

Step 1: Calculate the temperature difference

The temperature difference between the steam and the tube's surface is given as \(\Delta T = T_{steam} - T_{surface} = 60 - 40 = 20^{\circ}\mathrm{C}\).

Step 2: Obtain properties of the steam

The properties of the steam at \(60^{\circ}\mathrm{C}\) can be found in steam tables. For this problem, we will focus on the following properties: - Latent heat of vaporization: \(L = 2400 \mathrm{~kJ/kg}\) - Density of condensate: \(\rho_c = 983 \mathrm{~kg/m^3}\) - Viscosity of condensate: \(\mu = 4.8 \times 10^{-4} \mathrm{~kg/m \cdot s}\) - Thermal conductivity of condensate: \(k_c = 0.64 \mathrm{~W/m \cdot K}\)

Step 3: Calculate the Nusselt number

For film condensation on a horizontal tube, the Nusselt number is given by: \(Nu = (\frac{hD}{k_c}) = 0.725 \cdot [\frac{g(\rho_c - \rho_v)L\rho_v D^3}{\mu k_c \Delta T}]^{1/4}\) where \(h\) is the heat transfer coefficient, \(D\) is the diameter of the tube, \(g\) is the acceleration due to gravity, \(\rho_v\) is the density of the vapor, and the other properties have been previously defined. The vapor density can be approximated as \(\rho_v = 0.6\mathrm{~kg/m^3}\). Plug in the given values and parameters to get the Nusselt number: \(Nu \approx 0.725\cdot [(\frac{(9.81)(983-0.6)(2400)(0.6)(0.025)^3}{(4.8\times 10^{-4})(40)(20)})]^{\frac{1}{4}} \approx 81.96\)

Step 4: Calculate the heat transfer coefficient and condensation rate

From the Nusselt number, the heat transfer coefficient can be obtained as: \(h = (\frac{Nu\cdot k_c}{D}) \approx (\frac{81.96 \cdot 0.64}{0.025}) \approx 2100\mathrm{~W/m^2 \cdot K}\) For film condensation, the condensation rate (mass flow rate of steam) can be calculated as: \(\dot{m} = \frac{hA\Delta T}{L}\), where \(A\) is the surface area of the tube. \(A = \pi DL\), where \(L\) is the length of the tube. Using the given values for surface area and tube length, the condensation rate is calculated as: \(\dot{m} = (\frac{2100 \cdot \pi(0.025)(15)(20)}{2400\times 10^3}) \approx 3.28 \times 10^{-3}\mathrm{~kg/s}\)

Dropwise Condensation

Calculate the heat transfer coefficient

For dropwise condensation, we will use the Rohsenow method. The heat transfer coefficient is usually several times higher than that for film condensation (\(h_{dropwise} \approx 4h_{film}\)). Using the calculated heat transfer coefficient for film condensation, obtain the heat transfer coefficient for dropwise condensation: \(h_{dropwise} = 4h_{film} \approx 4 \times 2100 \approx 8400\mathrm{~W/m^2 \cdot K}\)

Calculate the condensation rate

Calculate the condensation rate for dropwise condensation using the same formula as before, but with the updated heat transfer coefficient: \(\dot{m}_{dropwise} = (\frac{8400 \cdot \pi(0.025)(15)(20)}{2400\times 10^3}) \approx 1.31 \times 10^{-2}\mathrm{~kg/s}\)

Comparison and Discussion

The condensation rates for both methods are as follows: - Film condensation: \(\dot{m} = 3.28 \times 10^{-3}\mathrm{~kg/s}\) - Dropwise condensation: \(\dot{m}_{dropwise} = 1.31 \times 10^{-2}\mathrm{~kg/s}\) Dropwise condensation rate is significantly higher than the film condensation rate. This is due to the dropwise condensation's ability to minimize the thermal resistance and maintain a high heat transfer coefficient. In practical applications, techniques that promote dropwise condensation can greatly enhance the efficiency of heat exchange processes involving steam condensation.