Question

An air-water slug flows through a 25.4-mm-diameter horizontal tube in microgravity conditions (less than 1 percent 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 \(\operatorname{Pr}_{\mathrm{g}}=0.71\). At a superficial gas velocity of \(V_{s g}=0.3 \mathrm{~m} / \mathrm{s}\), a superficial liquid velocity of \(V_{s l}=0.544 \mathrm{~m} / \mathrm{s}\), and a 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

Estimate the two-phase heat transfer coefficient of an air-water slug flowing through a horizontal tube in microgravity conditions using Chato's correlation. Step 1: Calculate Liquid Reynolds Number Using the formula: \(Re_{l} = \dfrac{\rho_{l} D V_{s l}}{\mu_{l}}\) \(Re_{l} = \dfrac{997 \mathrm{~kg} / \mathrm{m}^{3} \cdot 25.4 \times 10^{-3} \mathrm{m} \cdot 0.544 \mathrm{~m} / \mathrm{s}}{85.5 \times 10^{-5} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}}\) Step 2: Calculate the Nusselt Number According to Chato's correlation: \(Nu_{l} = 0.023 Re_{l}^{0.8} Pr_{l}^{0.4}\) Step 3: Calculate the Liquid-Phase Heat Transfer Coefficient Using the formula: \(h_{l} = Nu_{l} \cdot \dfrac{k_{l}}{D}\) Step 4: Determine the Two-Phase Heat Transfer Coefficient Using Chato's correlation: \(h_{t p} = \dfrac{1 - \alpha}{1 + \dfrac{\mu_{g}}{\mu_{l s} (1 - \alpha)}} \cdot h_{l}\) After plugging in all the values and performing the calculations for each step, the estimated two-phase heat transfer coefficient (\(h_{t p}\)) can be obtained.

Step-by-step solution

Calculate Liquid Reynolds Number

Since we are given the superficial liquid velocity, we first need to find the liquid Reynolds Number, \(Re_{l}\). The formula for the Reynolds Number is: \(Re_{l} = \dfrac{\rho_{l} D V_{s l}}{\mu_{l}}\) Where: \(\rho_{l} = 997 \mathrm{~kg} / \mathrm{m}^{3}\) is the density of the liquid phase \(D = 25.4 \times 10^{-3} \mathrm{m}\) is the tube diameter \(V_{s l} = 0.544 \mathrm{~m} / \mathrm{s}\) is the superficial liquid velocity \(\mu_{l} = 85.5 \times 10^{-5} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}\) is the dynamic viscosity of the liquid phase

Calculate the Nusselt Number

Next, we need to find the Nusselt number, \(Nu_{l}\). According to Chato's correlation, \(Nu_{l}\) can be calculated using the following formula: \(Nu_{l} = \dfrac{h_{l} D}{k_{l}} = 0.023 Re_{l}^{0.8} Pr_{l}^{0.4}\) Where: \(h_{l}\) is the liquid-phase heat transfer coefficient \(k_{l} = 0.613 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) is the thermal conductivity of the liquid phase \(Pr_{l} = 5.0\) is the Prandtl number of the liquid phase

Calculate the Liquid-Phase Heat Transfer Coefficient

Having found the Nusselt number, we can now calculate the liquid-phase heat transfer coefficient, \(h_{l}\): \(h_{l} = Nu_{l} \cdot \dfrac{k_{l}}{D}\)

Determine the Two-Phase Heat Transfer Coefficient

Finally, we can find the two-phase heat transfer coefficient, \(h_{t p}\), using the Chato correlation: \(h_{t p} = \dfrac{1 - \alpha}{1 + \dfrac{\mu_{g}}{\mu_{l s} (1 - \alpha)}} \cdot h_{l}\) Where: \(\alpha = 0.27\) is the void fraction \(\mu_{g} = 18.5 \times 10^{-6} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}\) is the dynamic viscosity of the gas phase \(\mu_{l s} = 73.9 \times 10^{-5} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}\) is the dynamic viscosity of water evaluated at the tube surface temperature By plugging in all the values and performing the calculations for each step, we can obtain the two-phase heat transfer coefficient, \(h_{t p}\).