Question

Saturated water vapor at a pressure of \(12.4 \mathrm{kPa}\) is condensed over an array of 100 horizontal tubes, each with a diameter of \(8 \mathrm{~mm}\) and a length of \(1 \mathrm{~m}\). The tube surfaces are maintained with a uniform temperature of \(30^{\circ} \mathrm{C}\). Determine the condensation rates of the tubes for \((a)\) a rectangular array of 5 tubes high and 20 tubes wide and \((b)\) a square array of 10 tubes high and 10 tubes wide. Compare and discuss the results of (a) and (b).

Short answer

Based on the given information and calculations, the condensation rates of saturated water vapor over the rectangular and square tube arrays are found to be the same. However, the arrangement of tubes in each array may affect the thermal performance and efficiency of the system due to the different geometries.

Step-by-step solution

Identify the given information.

We are given the following information: - Saturation pressure of water vapor: \(p_s = 12.4 \ \text{kPa}\) - Diameter of the tubes: \(d = 8 \ \text{mm}\) - Length of each tube: \(L = 1 \ \text{m}\) - Surface temperature of tubes: \(T_s = 30^\circ \ \text{C}\)

Calculate the mass transfer coefficient for the vapor.

To determine the mass transfer coefficient (\(h_m\)) for the vapor, we will use Nusselt's filmwise condensation theory. According to this theory, the mass transfer coefficient is determined by the following formula: \[h_m = \frac{k_{L}}{d}\]\[-0.948\left(\frac{G_{z}}{\mu_{L}}\right)^{1/24}(Gr_{l})^{1/4}\] where \(k_{L}\) is the liquid thermal conductivity, \(G_{z}\) is the mass flow rate per unit area of vapor along the tubes, \(\mu_{L}\) is the liquid dynamic viscosity, and \(Gr_{l}\) is the liquid Grashof number. For water vapor, \(k_{L} = 0.671 \ \text{W/(m·K)}\), and \(\mu_{L} = 2.82 \times 10^{-4} \ \text{Pa·s}\). We'll use the saturation temperature of water vapor at \(p_s\) and the given tube surface temperature to calculate the required properties in the formula.

Calculate the condensation rate for a single tube.

Once we know the mass transfer coefficient (\(h_m\)), we can calculate the condensation rate (\(m_{cond}\)) for a single tube using the following formula: \[m_{cond} = h_m \cdot A_{cond}\] where \(A_{cond}\) is the condensation area of the tube, given by \(A_{cond} = \pi d L\).

Calculate the condensation rate for the rectangular array.

We know there are 5 tubes high and 20 tubes wide in the rectangular array, which makes a total of 100 tubes. To calculate the total condensation rate (\(M_{rect}\)) for the rectangular array, we simply multiply the condensation rate for a single tube by the number of tubes in the array: \[M_{rect} = 100m_{cond}\]

Calculate the condensation rate for the square array.

Similarly, for the square array (10 tubes high and 10 tubes wide), we again multiply the condensation rate for a single tube by the number of tubes in the array to find the total condensation rate (\(M_{square}\)): \[M_{square} = 100m_{cond}\]

Compare and discuss the results of (a) and (b).

Since both arrays are comprised of 100 tubes each and that the surface temperature of the tubes remains constant in both cases, the condensation rate for both arrays should be identical. However, the arrangement of tubes in the array may influence the thermal performance and efficiency of the system. In summary, both for the rectangular and square tube arrays, the condensation rates will be the same. However, the geometrical distribution of tubes may play a role in influencing their overall efficiency.

Key concepts

Heat Transfer

Understanding heat transfer is essential when investigating condensation over surfaces like tubes. Heat transfer in condensation involves the transformation of vapor into liquid as a result of cooling. When steam comes into contact with a cooler surface, its temperature decreases, causing the vapor to release heat and change into its liquid form. This process is influenced by factors such as temperature gradients, surface material, and the characteristics of the fluid. Effective heat transfer ensures that more vapor condenses on the surface, which is crucial for applications involving heat exchangers or cooling systems.

For the tubes in the given exercise, the uniform surface temperature simplifies the analysis. The difference between the saturation temperature of the vapor at 12.4 kPa and the tube surface temperature of 30°C drives the heat transfer. This gradient is pivotal in determining how efficiently heat can be removed, thereby impacting the rate of condensation across the tube array.

Mass Transfer

Mass transfer refers to the movement of a fluid's mass from one point to another, often as a result of concentration differences. In the context of condensation, mass transfer is the process where vapor molecules move towards a cooler surface and accumulate as a liquid film. This transformation from vapor to liquid involves changes in mass distribution.

Nusselt's filmwise theory provides an insight into how mass is transferred during condensation. As vapor condenses on a surface, a liquid film is formed. The mass transfer rate depends on several factors, such as the properties of the liquid and vapor, and the surface characteristics. In the exercise, the calculation of the mass transfer coefficient using Nusselt's theory helps in estimating how efficiently mass is transferred from vapor to liquid phase on each tube in the array.

Nusselt's Filmwise Condensation Theory

Nusselt's filmwise condensation theory is a fundamental concept in understanding how condensation occurs on surfaces. It describes the behavior of a thin film of liquid forming as vapor cools and condenses on a surface. This theory helps predict the rate at which vapor will condense into liquid by considering factors such as heat and mass transfer coefficients.

In the exercise, the mass transfer coefficient is derived using Nusselt's approach. The theory accounts for the liquid thermal conductivity, the mass flow rate of vapor per unit area, and other properties such as viscosity and Grashof number. By using this theory, engineers and scientists can better predict condensation rates, which is critical for designing efficient systems involving heat exchangers or cooling.

Grashof Number

The Grashof number is a dimensionless number in fluid dynamics and heat transfer, often used to predict the pattern of fluid flow. It plays a crucial role in analyzing natural convection, which occurs when fluid motion is driven by buoyancy forces resulting from density variations due to temperature differences.

In the context of condensation on tubes, the liquid Grashof number is essential for determining the nature of flow in the liquid film formed. It helps understand how convection influences the heat and mass transfer processes during condensation. A higher Grashof number indicates stronger buoyancy effects, which can enhance the convective heat transfer, affecting the overall condensation rate on the surface. The relationship between Grashof number and other thermal properties is vital for optimizing and designing efficient systems in engineering applications.