Question

Air water slug flows through a 25.4-mm diameter horizontal tube in microgravity condition (less than \(1 \%\) of earth's normal gravity). The liquid phase consists of water with dynamic viscosity of \(\mu_{l}=85.5 \times 10^{-5} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}\), density of \(\rho_{l}=997 \mathrm{~kg} / \mathrm{m}^{3}\), thermal conductivity of \(k_{l}=0.613 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), and Prandtl number of \(\operatorname{Pr}_{l}=5.0\). The gas phase consists of air with dynamic viscosity of \(\mu_{g}=18.5 \times 10^{-6} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}\), density \(\rho_{g}=1.16 \mathrm{~kg} / \mathrm{m}^{3}\), and Prandtl number \(\operatorname{Pr}_{g}=0.71\). At superficial gas velocity of \(V_{s g}=0.3 \mathrm{~m} / \mathrm{s}\), superficial liquid velocity \(V_{s l}=0.544 \mathrm{~m} / \mathrm{s}\), and void fraction of \(\alpha=0.27\), estimate the two-phase heat transfer coefficient \(h_{t p}\). Assume the dynamic viscosity of water evaluated at the tube surface temperature to be \(73.9 \times 10^{-5} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}\).

Short answer

Question: Estimate the two-phase heat transfer coefficient (h_tp) in a slug flow through a horizontal tube in microgravity condition using the Modified Lockhart-Martinelli Correlation. Answer: To estimate the two-phase heat transfer coefficient (h_tp) in a slug flow through a horizontal tube in microgravity condition, follow these steps: 1. Calculate the Reynolds number for both water and air using the given values for superficial gas velocity, superficial liquid velocity, densities, and dynamic viscosities. 2. Calculate the modified Martinelli parameter (X_m) using the void fraction, liquid and gas properties, and the tube diameter. 3. Calculate the heat transfer coefficient for single-phase flow (h_l-μ_0.8) for water using the Nusselt number equation and accounting for the surface temperature dependence of the dynamic viscosity of the liquid phase. 4. Calculate the two-phase heat transfer coefficient (h_tp) using the Modified Lockhart-Martinelli Correlation and the calculated values.

Step-by-step solution

Calculate Reynolds number for both water and air

Using the given values for superficial gas velocity, superficial liquid velocity, densities, and dynamic viscosities, we can calculate the Reynolds numbers for liquid and gas phases: \(Re_{l} = \frac{\rho_{l} V_{sl} D}{\mu_{l}}\) \(Re_{g} = \frac{\rho_{g} V_{sg} D}{\mu_{g}}\) where \(D\) is the diameter of the tube.

Calculate the modified Martinelli parameter (X_m)

The modified Martinelli parameter, which accounts for the effect of void fraction, can be calculated using the void fraction, liquid and gas properties, and the tube diameter: \(X_m = \frac{\rho_{l} \alpha^{1.5}(V_{sl} V_{sg})^{0.5}}{\rho_{g}(1-\alpha)^{1.5}(V_{sl} + V_{sg})^{0.5}}\)

Calculate the heat transfer coefficient for single-phase flow (h_l-μ_0.8) for water

To account for the surface temperature dependence of the dynamic viscosity of the liquid phase, we can use the Nusselt number in the following equation: \(Nu_l = \frac{h_l D}{k_l} = q Re_l Pr_l^{0.8}\) Where q can be calculated as: \(q = \left(\frac{\mu_{l}}{\mu_{0}}\right)^{0.8}\), \(\mu_{0}\) is the dynamic viscosity of water evaluated at the tube surface temperature. Solve for \(h_l\): \(h_l = \frac{q k_l Re_l Pr_l^{0.8}}{D}\)

Calculate the two-phase heat transfer coefficient (h_tp)

Using the Modified Lockhart-Martinelli Correlation and the calculated values, the two-phase heat transfer coefficient can be found: \(h_{tp} = h_l \left[\frac{1 + X_m}{\left(X_m\right)^{0.7}}\right]\) By following these steps using the given values, the two-phase heat transfer coefficient (\(h_{tp}\)) can be obtained which characterizes the heat transfer in slug flow through a horizontal tube in microgravity condition.

Key concepts

Microgravity Condition

Microgravity is a condition where the force of gravity is significantly less than what we experience on Earth. It's often found in space environments, such as on the International Space Station. This unique condition affects fluid dynamics, including how liquids and gases flow and transfer heat.
When dealing with two-phase heat transfer—like in slug flows of air and water—microgravity can significantly impact the way bubbles and slugs of fluid move through a tube. In our exercise, where gravity is less than 1% of Earth's normal gravity, the buoyancy effects are negligible. Instead, surface tension and fluid properties become more dominant in determining flow behavior. This requires special consideration when calculating flow parameters like the heat transfer coefficient.

Reynolds Number Calculation

The Reynolds number is crucial for understanding fluid flow in a tube. It indicates whether the flow is laminar or turbulent. Calculating it separately for both liquid and gas phases helps in assessing the flow regime of each phase within the tube.
The formula for the Reynolds number is given by:

• For the liquid phase: \(Re_{l} = \frac{\rho_{l} V_{sl} D}{\mu_{l}}\)
• For the gas phase: \(Re_{g} = \frac{\rho_{g} V_{sg} D}{\mu_{g}}\)

Where:

• \(\rho\) is the density
• \(V\) represents velocity
• \(D\) is the tube diameter
• \(\mu\) stands for dynamic viscosity

Reynolds number helps determine if the phase flow is smooth (laminar) or chaotic (turbulent), affecting the heat transfer calculations.

Martinelli Parameter

The Martinelli parameter is a dimensionless number used in two-phase flow to relate the pressure drops of the liquid and gas phases. In our context, we use the modified Martinelli parameter \(X_m\) to account for the effect of void fraction in microgravity.
The calculation is done using:
\[X_m = \frac{\rho_{l} \alpha^{1.5}(V_{sl} V_{sg})^{0.5}}{\rho_{g}(1-\alpha)^{1.5}(V_{sl} + V_{sg})^{0.5}}\]
Where:

• \(\alpha\) is the void fraction—representing the volume of the gas phase within the mixture
• \(V_{sl}\) and \(V_{sg}\) are the superficial velocities for the liquid and gas phases

The Martinelli parameter helps measure the ratio of influence between the two phases when calculating the two-phase heat transfer coefficient.

Nusselt Number

The Nusselt number is a dimensionless number representing the efficiency of convective heat transfer relative to conductive heat transfer. It is an essential part of calculating the single-phase heat transfer coefficient in fluids.
In the formula:

• \(Nu_l = \frac{h_l D}{k_l} = q Re_l Pr_l^{0.8}\)

Here:

• \(h_l\) is the heat transfer coefficient of the liquid
• \(D\) is the tube diameter
• \(k_l\) is the thermal conductivity
• \(Pr_l\) represents the Prandtl number of the liquid
• \(q\) adjusts for the viscosity change with temperature, calculated as \(q = \left(\frac{\mu_{l}}{\mu_{0}}\right)^{0.8}\)\

The Nusselt number helps bridge the gap between physical properties and the geometry of the system to calculate heat transfer in a fluid system.