Question

Saturated water vapor at atmospheric pressure condenses on the outer surface of a \(0.1\)-m-diameter vertical pipe. The pipe is \(1 \mathrm{~m}\) long and has a uniform surface temperature of \(80^{\circ} \mathrm{C}\). Determine the rate of condensation and the heat transfer rate by condensation. Discuss whether the pipe can be treated as a vertical plate. 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

Answer: The wavy-laminar flow model resulted in a non-existent film thickness, meaning the initial assumptions were not valid. Alternative methods to determine the rate of condensation and heat transfer rate include using experimental data or different models that consider more factors influencing the condensation and heat transfer processes.

Step-by-step solution

Determine the physical properties of water

At the given temperature of \(80^{\circ} \mathrm{C}\), refer to the Steam Tables or an online steam properties calculator to obtain the following properties of water: - Saturation pressure, \(P_{sat} = 47.36 \mathrm{~kPa}\) - Liquid density, \(\rho_L = 971.97 \mathrm{~kg/m^3}\) - Vapor density, \(\rho_V = 0.4343 \mathrm{~kg/m^3}\) - Dynamic viscosity of liquid, \(\mu = 3.58 \times 10^{-4} \mathrm{~kg/(m \cdot s)}\) - Specific heat of vaporization, \(L_{v} = 2214.5 \mathrm{~kJ/kg}\)

Find the film thickness

According to the wavy-laminar flow model, the thickness of the liquid film at the bottom of the tube, \(\delta\), can be expressed as: \(\delta = 0.943 \left( \frac{\mu L_v (T_s - T_{sat})}{g(\rho_L - \rho_V) x} \right)^{1/4}\) where - \(T_s = 80^{\circ} \mathrm{C}\) (surface temperature) - \(T_{sat} = 80^{\circ} \mathrm{C}\) (saturation temperature) - \(x = 1 \mathrm{~m}\) (vertical height of the tube) - \(g = 9.81 \mathrm{~m/s^2}\) (acceleration due to gravity) Now, plug in the given values and calculate the film thickness: \(\delta = 0.943 \left( \frac{3.58 \times 10^{-4} \cdot 2214.5 \times 10^{3} \cdot (80-80)}{9.81(971.97 - 0.4343) \cdot 1} \right)^{1/4} = 0 \mathrm{~m}\) As the film thickness is zero, the film does not really exist. This means that our assumptions of wavy-laminar flow and the tube diameter being large relative to the film thickness are not valid.

Rate of condensation and heat transfer rate

Since our assumptions in the previous step did not hold, it is not possible to directly calculate the rate of condensation and heat transfer rate from our derived equations. However, these values can still be determined experimentally or using other appropriate models, which consider more factors influencing the condensation and heat transfer processes.

Conclusion

In this exercise, the initial assumptions of wavy-laminar flow and the tube diameter being large relative to the film thickness do not appear to be valid. Therefore, to determine the rate of condensation and heat transfer rate, alternative methods, such as experimental data or different models, should be used.

Key concepts

Laminar Flow

Laminar flow is a key principle in fluid mechanics where fluid particles move in parallel layers without disruption. In the context of condensation heat transfer, especially with smooth surfaces, flow is often assumed to be laminar when the velocity and layer interactions are minimal. This assumption simplifies the analysis of condensation, such as on the surface of a pipe.

Here, the term "wavy-laminar" was used to describe the type of flow expected around the tube during condensation. This flow typically occurs when the Reynolds number (a dimensionless number used to predict flow patterns) is low, indicating that inertial forces are smaller compared to viscous forces. This scenario ensures stable and predictable fluid flow.

However, when analyzing real-world cases, it's essential to verify if this assumption truly fits. If unexpected factors like turbulence or instability appear, adjustments in the model calculations are necessary to accurately predict heat transfer and condensation rates.

Saturation Pressure

Saturation pressure refers to the pressure at which a liquid boils and turns into vapor for a given temperature. It is a critical concept in understanding phase changes, such as the condensation of water vapor into liquid. When a substance reaches its saturation pressure, any additional cooling leads to condensation.

In the exercise, the saturation pressure of water vapor at the pipe's surface temperature of 80°C was noted as 47.36 kPa. This means that at this pressure, the water vapor is at equilibrium with its liquid phase. If the temperature or pressure were altered, it would significantly affect the condensation process and heat transfer rate.

Understanding this concept helps in predicting how changes in environmental conditions such as pressure and temperature differences influence the rate of heat transfer during condensation.

Film Thickness

In condensation heat transfer, film thickness is a crucial parameter that determines how heat is transferred across a surface with liquid films. It is the distance from the surface to the outermost edge of the condensed liquid film.

The problem intended to calculate this thickness using a specific model for wavy-laminar flow. However, as calculated, the film thickness was zero due to the terms canceling out in the expression a \(a= 0.943 \left( \frac{\mu L_v (T_s - T_{sat})}{g(\rho_L - \rho_V) x} \right)^{1/4}\).

This suggested that either the conditions did not allow for film formation, or alternative methodologies needed to be employed. Generally, the film thickness depends on various factors including the properties of the fluid and geometry of the surface. A thicker film may indicate a higher resistance to heat flow, whereas a thinner film could allow for more efficient heat transfer. Understanding and predicting film thickness is crucial for optimizing condensation processes in a variety of applications.