Question

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?

Short answer

Question: Calculate (a) the average heat transfer coefficient, (b) the rate of heat transfer, and (c) the rate of condensation of ammonia on the outer surface of a vertical tube, given that saturated ammonia vapor at \(25^{\circ} \mathrm{C}\) condenses on the tube with dimensions \(L = 2\,\mathrm{m}\) and \(D_o = 3.2\,\mathrm{cm}\), and temperature \(T_{tube} = 15^{\circ} \mathrm{C}\). Answer: (a) The average heat transfer coefficient, \(h = 1742.03\,\mathrm{W/(m^2\cdot K)}\) (b) The rate of heat transfer, \(q = 3499.89\,\mathrm{W}\) (c) The rate of condensation, \(\dot{m}_c = 2.55\times10^{-3}\,\mathrm{kg/s}\)

Step-by-step solution

Calculate the temperature difference and the condensation temperature

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, \(T_{cond} = 25^{\circ} \mathrm{C}\)

Calculate the Nusselt Number and average heat transfer coefficient

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, \(h = 1742.03\,\mathrm{W/(m^2\cdot K)}\)

Calculate the rate of heat transfer

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, \(q = 3499.89\,\mathrm{W}\)

Calculate the rate of condensation

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.

Key concepts

Condensation

Condensation is a process where a gas transforms into a liquid. This occurs when the gas is cooled to its condensation temperature, causing the gas molecules to lose energy and come together as a liquid. It's an essential mechanism in heat transfer, as it often involves releasing latent heat, a significant component in energy systems.
For instance, in the exercise, saturated ammonia vapor condenses on a vertical tube. During this, latent heat is released, contributing to the process's overall heat transfer. Understanding condensation aids in comprehending how heat is transferred from a gaseous state to a surface, and why certain designs are more efficient at capturing this heat.

• Gas transforms into liquid
• Occurs at condensation temperature
• Releases latent heat

Condensation is especially pivotal in HVAC systems, power plants, and closely related to the efficiency of heat exchangers.

Saturated Vapor

Saturated vapor is a vapor that is in equilibrium with its liquid phase at a given temperature and pressure. This means that the vapor cannot hold any additional moisture, and any reduction in temperature or increase in pressure will cause condensation.
In our exercise, the saturated ammonia vapor at 25°C is in equilibrium condition, ready to condense onto the cooler tube surface at 15°C. By utilizing the properties of saturated vapor, engineers can predict and manage condensation efficiently.

• Vapor in equilibrium with its liquid
• Temperature or pressure change induces condensation
• Used to manage and predict condensation

This concept is crucial as it underpins the principles of phase changes and latent heat exchange.

Thermal Conductivity

Thermal conductivity is a material property that measures the ability of a substance to conduct heat. It is a crucial factor in heat transfer calculations, including situations involving condensation.
In the provided solution, the thermal conductivity of the liquid ammonia is essential for calculating the Nusselt number and subsequently the heat transfer coefficient. The higher the thermal conductivity, the better the material's ability to transfer heat.

• Measures heat conduction ability
• Influences heat transfer calculations
• Essential for Nusselt number calculation

Understanding thermal conductivity allows for the selection and use of materials in systems to ensure effective heat exchange.

Nusselt Number

The Nusselt number is a dimensionless quantity that provides a measure of the convection heat transfer relative to conduction heat transfer across a fluid boundary. It indicates the effectiveness of convective heat transfer during processes like condensation.
In the solution, the Nusselt number is obtained using a specific correlation for film condensation on vertical tubes. This number allows the calculation of the average heat transfer coefficient, showing the efficiency of heat transfer.

• Measures convective vs. conductive heat transfer
• Indicates efficiency of heat exchange
• Crucial for calculating heat transfer coefficient

The Nusselt number is key for understanding heat transfer performance in different systems and conditions.

Latent Heat

Latent heat refers to the energy absorbed or released by a substance during a phase change, without changing its temperature. It plays a vital role in condensation, where heat is released when vapor turns into liquid.
In the example, the latent heat of vaporization for ammonia is used to calculate the condensation rate. This value is pivotal for understanding energy exchanges during phase changes.

• Energy released or absorbed during phase change
• Critical for calculating condensation rate
• Influences energy exchanges in systems

Familiarity with latent heat helps in designing systems that leverage phase changes for efficient thermal management.