Question

Consider a two phase flow of air-water in a vertical upward stainless steel pipe with an inside diameter of \(0.0254 \mathrm{~m}\). The two phase mixture enters the pipe at \(25^{\circ} \mathrm{C}\) at a system pressure of \(201 \mathrm{kPa}\). The superficial velocities of the water and air are \(0.3 \mathrm{~m} / \mathrm{s}\) and \(23 \mathrm{~m} / \mathrm{s}\), respectively. The differential pressure transducer connected across the pressure taps set \(1 \mathrm{~m}\) apart records a pressure drop of \(2700 \mathrm{~Pa}\) and the measured value of void fraction is \(0.86\). Using the concept of Reynolds analogy determine the two phase convective heat transfer coefficient. Hint: Use EES to calculate the properties of water and air at the given temperature and pressure.

Short answer

Based on the step-by-step solution provided, answer the following question: Question: Calculate the two-phase convective heat transfer coefficient for a water-air mixture with a void fraction of 0.86 flowing inside a pipe with an inside diameter of 0.0254 m. The air has a superficial velocity of 23 m/s and the water has a superficial velocity of 0.3 m/s. The mixture is at a temperature of 25°C and a pressure of 201 kPa.

Step-by-step solution

Calculate air and water properties

The first step is to find the properties of air and water at the given temperature and pressure. You can use the EES (Engineering Equation Solver) software or any other software/database that provides the properties of fluids for this purpose. Here's the data we need: For air at \(25^{\circ} \mathrm{C}\) and \(201\mathrm{kPa}\): - Density: \(\rho_{air}\) - Dynamic viscosity: \(\mu_{air}\) For water at \(25^{\circ} \mathrm{C}\) and \(201\mathrm{kPa}\): - Density: \(\rho_{water}\) - Dynamic viscosity: \(\mu_{water}\)

Calculate Reynolds numbers

Now that we have the fluid properties, we can calculate the Reynolds numbers for air and water. The Reynolds number is given by: $$ Re = \frac{\rho vD}{\mu} $$ Where \(\rho\) is the density, \(v\) is the superficial velocity, \(D\) is the inside diameter of the pipe, and \(\mu\) is the dynamic viscosity. For air: $$ Re_{air} = \frac{\rho_{air} (23\frac{\mathrm{m}}{\mathrm{s}}) (0.0254\mathrm{~m})}{\mu_{air}} $$ For water: $$ Re_{water} = \frac{\rho_{water} (0.3\frac{\mathrm{m}}{\mathrm{s}}) (0.0254\mathrm{~m})}{\mu_{water}} $$

Apply the Reynolds analogy

We can now use the Reynolds analogy to find the two-phase convective heat transfer coefficient, \(h_{tp}\). According to the Reynolds analogy: $$ h_{tp} = C\frac{\rho_{tp}v_{tp}C_{p_{tp}}}{\mu_{tp}} $$ Where \(C\) is a constant, \(\rho_{tp}\) is the density of the two-phase mixture, \(v_{tp}\) is the superficial velocity of the two-phase mixture, \(C_{p_{tp}}\) is the specific heat capacity of the two-phase mixture, and \(\mu_{tp}\) is the dynamic viscosity of the two-phase mixture. The density of the two-phase mixture can be found using the void fraction: $$ \rho_{tp} = \alpha \rho_{air} + (1 - \alpha) \rho_{water} $$ With \(\alpha = 0.86\), we can calculate \(\rho_{tp}\). Next, we find the superficial velocity of the two-phase mixture: $$ v_{tp} = v_{air} + v_{water} = 23\frac{\mathrm{~m}}{\mathrm{s}} + 0.3\frac{\mathrm{~m}}{\mathrm{s}} = 23.3\frac{\mathrm{~m}}{\mathrm{s}} $$ The specific heat capacity and dynamic viscosity of the two-phase mixture can be found using the property data obtained in Step 1: $$ C_{p_{tp}} = \alpha C_{p_{air}} + (1 - \alpha) C_{p_{water}} $$ $$ \mu_{tp} = \frac{\alpha \mu_{air} + (1 - \alpha) \mu_{water}}{\alpha + (1 - \alpha)} $$ Finally, we can find the convective heat transfer coefficient: $$ h_{tp} = C\frac{\rho_{tp}v_{tp}C_{p_{tp}}}{\mu_{tp}} $$ Note that the value of the constant \(C\) depends on the flow regime (laminar or turbulent) and various correlations can be used to determine its value. After finding \(C\), you can calculate the two-phase convective heat transfer coefficient \(h_{tp}\).

Key concepts

Reynolds Number

The Reynolds number is a crucial parameter in fluid dynamics that helps determine the flow regime of a fluid through a pipe. It's a dimensionless quantity used to identify whether the flow is laminar or turbulent. The Reynold's number can be calculated using the formula: \[ Re = \frac{\rho vD}{\mu} \] where:

• \( \rho \) is the fluid's density,
• \( v \) is the superficial velocity,
• \( D \) is the pipe diameter,
• \( \mu \) is the dynamic viscosity of the fluid.

A low Reynolds number (typically below 2000) indicates laminar flow, where fluid moves in smooth layers, while a high Reynolds number (above 4000) suggests turbulent flow, characterized by chaotic fluid motion. In two-phase flow with air and water, calculating the Reynolds number for each component (air and water) helps to analyze the flow conditions within the pipe.

Convective Heat Transfer Coefficient

The convective heat transfer coefficient, denoted as \( h \), is a measure of the heat transfer capability of a fluid in motion. It represents how effectively heat is removed or transferred from a surface to a moving fluid, or vice versa. In the context of two-phase flow, you're interested in the two-phase convective heat transfer coefficient \( h_{tp} \). The concept of Reynolds analogy can aid in determining \( h_{tp} \). It establishes a relationship between momentum transfer and heat transfer in turbulent flows, allowing you to derive \( h_{tp} \) from known fluid properties and parameters: \[ h_{tp} = C \frac{\rho_{tp}v_{tp}C_{p_{tp}}}{\mu_{tp}} \]where \( C \) is a constant that depends on the flow regime. This formula highlights the direct roles density, velocity, specific heat capacity (\( C_{p_{tp} \)), and viscosity play in defining how efficiently heat can be transferred in the system. Adjusting any of these parameters will inherently change \( h_{tp} \), affecting the rate of heat transfer.

Void Fraction

Void fraction is a vital concept in two-phase flow, referring to the volume fraction of the gas phase within a liquid-gas mixture. It is defined as the ratio of the cross-sectional area occupied by the gas to the total cross-sectional area of the pipe. This gives a quantitative measure of how much space the gas takes up in the mixture. In formulaic terms, void fraction \( \alpha \) can be expressed as:\[ \alpha = \frac{A_{gas}}{A_{total}} \] where \( A_{gas} \) is the area occupied by the gas, and \( A_{total} \) is the entire cross-sectional area. Void fraction can significantly impact the behavior of the two-phase flow, affecting pressure drop, flow rates, and the overall system's heat transfer characteristics. A higher void fraction indicates more gas in the mixture, which often alters flow dynamics and heat transfer properties.

Superficial Velocity

Superficial velocity is a key parameter when studying multiphase flows such as two-phase flows. It represents the velocity a single phase would have if it solely occupied the pipe volume. This allows for the characterization of each phase's behavior in multi-component flow systems. For each phase, the superficial velocity can be calculated using:\[ v_s = \frac{Q}{A} \]where:

• \( v_s \) is the superficial velocity,
• \( Q \) is the volumetric flow rate,
• \( A \) is the cross-sectional area of the pipe.

In the given exercise, the superficial velocities for air and water help to determine how each phase interacts within the pipe. By summing these velocities, you obtain the total velocity of the two-phase mixture. Understanding superficial velocities helps predict flow patterns and assess the efficiency of mass and energy transfer within the system.