Question

Saturated steam at \(30^{\circ} \mathrm{C}\) condenses on the outside of a 4-cm- outer-diameter, 2-m-long vertical tube. The temperature of the tube is maintained at \(20^{\circ} \mathrm{C}\) by the cooling water. Determine \((a)\) the rate of heat transfer from the steam to the cooling water, \((b)\) the rate of condensation of steam, and \((c)\) the approximate thickness of the liquid film at the bottom of the tube. Assume wavy-laminar 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 the rate of heat transfer, the rate of condensation, and the approximate thickness of the liquid film at the bottom of a vertical tube with the given parameters and assumptions. Parameters: - Temperature of saturated steam (T_s) = \(30^\circ \mathrm{C}\) - Temperature of tube (T_t) = \(20^\circ \mathrm{C}\) - Outer diameter of the tube (D) = 4 cm - Length of the tube (L) = 2 m

Step-by-step solution

Identify known parameters and properties

The given parameters are: temperature of saturated steam T_s = \(30^\circ \mathrm{C}\), temperature of tube T_t = \(20^\circ \mathrm{C}\), outer diameter of the tube D = 4 cm, and length of the tube L = 2 m. We need to find other properties relevant to the problem, such as thermodynamic properties of the steam, using steam tables.

Calculate heat transfer coefficient and heat transfer rate

Since we know the temperature of steam and the tube, we can calculate the heat transfer coefficient (h) using Nusselt's equation. Additionally, we also need the steam pressure (P) and the latent heat of vaporization (L), which can be found from the steam table. Use Nusselt's equation to calculate the heat transfer coefficient and heat transfer rate: \(h = \frac {kL}{D}\left[\frac{4g\beta(ρ_v - ρ_l)L^3\Delta T}{Dv_f}\right]^\frac{1}{4}\) where k is the thermal conductivity of steam, g is the acceleration due to gravity, β is the coefficient of volumetric expansion, ρ_v and ρ_l are the densities of the vapor and liquid phases, respectively, L is the latent heat of vaporization, ΔT is the temperature difference, D is the tube diameter, and v_f is the kinematic viscosity of the liquid phase.

Calculate the rate of condensation of steam

With the heat transfer rate (q) found in the previous step, we can calculate the rate of condensation of steam (m_dot) using the following formula: \(m_{dot} = \frac{q}{L}\)

Calculate the approximate thickness of the liquid film at the bottom of the tube

To find the thickness of the liquid film, we need to consider the wavy-laminar flow. The approximate thickness can be obtained using the following formula: \(f_b=\left[12\left(\frac{L^2ρ_l^2gx^3}{wD}\right)\left(\frac{m_{dot}}{\pi D}\frac{1}{ρ_l-ρ_v}\right)^2\right]^{\frac{1}{5}}\) where f_b is the thickness of the liquid film, x is the distance from the top of the tube, and w is the width of the surface wave.

Verify the given assumptions

We need to verify if the assumptions made about the tube diameter being large compared to the thickness of the liquid film are valid. We can simply compare the magnitude of the tube diameter (D) and the film thickness (f_b) calculated in the previous steps. If the tube diameter is considerably larger than the film thickness, the assumptions are considered valid. Combine all the calculations to find the rate of heat transfer, the rate of condensation, and the liquid film thickness. Use the outcomes to validate the assumptions made in the problem.

Key concepts

Nusselt's Equation

To understand the heat transfer during the condensation process, Nusselt's equation serves as a powerful tool. This equation is particularly used to calculate the heat transfer coefficient in condensation and boiling scenarios. Nusselt's equation combines various factors such as physical properties of the fluid (such as thermal conductivity), system geometry, and temperature differences to provide valuable insights into the heat transfer process.

Specifically, for condensation, the Nusselt's equation takes into account the characteristics of both the vapor and the liquid phase, including their densities, and incorporates the gravitational force effect on the liquid film flow. Understanding and applying Nusselt's equation allows one to efficiently evaluate the condensation heat transfer coefficient, a critical step in determining the rate of heat transfer from steam to the surroundings as seen in our example problem.

Rate of Condensation

The rate of condensation plays a crucial role when evaluating the performance of heat exchangers and condensers. It essentially measures the amount of vapor turning into liquid over a certain period. In practical exercises like the one we studied, this rate is pivotal for engineers to design systems that ensure maximum efficiency.

By knowing the rate of condensation, we can calculate the amount of heat being transferred, since each gram of condensing steam releases a specific amount of energy, known as the latent heat of vaporization. Thus, this rate is directly tied to the energy exchange in condensation processes.

Laminar Flow

Understanding the nature of the flow is fundamental in heat transfer calculations. Laminar flow, where fluid flows in smooth paths or layers, has a significant effect on the heat transfer rate. In our exercise example, the assumption of wavy-laminar flow is made. This refers to a flow regime in which the fluid maintains a general laminar motion, yet with some surface disturbance, like waves.

In condensation, the heat transfer is directly affected by the flow of the condensed liquid film. A laminar film promotes steady heat transfer and makes it possible to use steady-state equations to approximate the condensation rate. In the case of a tube, the laminar film flow ensures the validity of Nusselt's theory, which is designed for laminar film condensation.

Steam Properties

The behavior of steam during heat transfer processes is determined by its thermodynamic properties, which include temperature, pressure, specific volume, internal energy, enthalpy, and entropy. For solving problems like the given exercise, properties like the thermal conductivity, density, latent heat of vaporization, and kinematic viscosity of steam are required.

These properties are often available in steam tables and are crucial for various calculations. For instance, the density difference between the vapor and liquid phase, along with the latent heat of vaporization, is essential for calculating the heat transfer coefficient using Nusselt's equation.

Heat Transfer Coefficient

The heat transfer coefficient is a measure that characterizes how well a particular mode of heat transfer occurs across a surface. It essentially quantifies the rate of heat transfer between a solid surface and a fluid per unit area per unit temperature difference.

In our problem applying Nusselt's equation, we use it to relate the rate of heat transfer from steam to the surrounding area. With a higher heat transfer coefficient, you would have a more effective transfer of energy from the steam to the condensing surface. Therefore, accurately determining this coefficient is essential for thermal system design and analysis.

Latent Heat of Vaporization

Latent heat of vaporization is a thermodynamic property that plays a vital role in phase change processes like condensation and evaporation. It is the amount of heat needed to convert a unit mass of a substance from the liquid to the gaseous state or vice versa without a change in temperature.

In the case of our exercise, the latent heat of vaporization is essential for quantifying the amount of thermal energy released as steam condenses on the tube's surface. It is also crucial in calculating the rate of condensation, as mentioned in the problem solution steps. This property directly relates to the amount of heat that must be removed to condense a certain amount of steam, hence its significance in calculations involving heat transfer in condensation.