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 \(73.9 \times 10^{-5} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}\).

Step by step solution

01

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.

02

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: \(X_m = \frac{\rho_{l} \alpha^{1.5}(V_{sl} V_{sg})^{0.5}}{\rho_{g}(1-\alpha)^{1.5}(V_{sl} + V_{sg})^{0.5}}\)

03

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

04

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!