Question

A mixture of petroleum and natural gas is being transported in a pipeline with a diameter of \(102 \mathrm{~mm}\). The pipeline is located in a terrain that caused it to have an average inclination angle of \(\theta=10^{\circ}\). The liquid phase consists of petroleum with dynamic viscosity of $\mu_{l}=297.5 \times 10^{-4} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}$, density of \(\rho_{l}=853 \mathrm{~kg} / \mathrm{m}^{3}\), thermal conductivity of \(k_{l}=0.163\) \(\mathrm{W} / \mathrm{m} \cdot \mathrm{K}\), surface tension of \(\sigma=0.020 \mathrm{~N} / \mathrm{m}\), and Prandtl number of \(\operatorname{Pr}_{l}=405\). The gas phase consists of natural gas with dynamic viscosity of $\mu_{g}=9.225 \times 10^{-6} \mathrm{~kg} / \mathrm{m}-\mathrm{s}\(, density of \)\rho_{g}=9.0 \mathrm{~kg} / \mathrm{m}^{3}\(, and Prandtl number of \)\operatorname{Pr}_{\mathrm{g}}=0.80$. The liquid is flowing at a flow rate of \(16 \mathrm{~kg} / \mathrm{s}\), while the gas is flowing at \(0.055 \mathrm{~kg} / \mathrm{s}\). If the void fraction is \(\alpha=0.22\), determine the two-phase heat transfer coefficient \(h_{t p}\). Assume the dynamic viscosity of liquid petroleum evaluated at the tube surface temperature to be $238 \times 10^{-4} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}$.

Short answer

Question: Calculate the two-phase heat transfer coefficient for a mixture of petroleum and natural gas flowing in a pipeline with a diameter of 102 mm, flow rates of 16 kg/s for the petroleum and 0.055 kg/s for the natural gas, and a given void fraction. Provide the step by step solution to determine the two-phase heat transfer coefficient. Answer: To calculate the two-phase heat transfer coefficient, follow these steps: 1. Calculate the flow velocities of the liquid and gas phases using the given flow rates and pipeline diameter. 2. Calculate the Reynolds numbers for the liquid and gas phases using the calculated flow velocities and given properties of the liquid and gas. 3. Determine the two-phase flow pattern using the calculated Reynolds numbers and given void fraction. 4. Calculate the Nusselt number for the two-phase flow using an appropriate correlation based on the identified flow pattern, Reynolds numbers, and other given parameters. 5. Calculate the two-phase heat transfer coefficient using the calculated Nusselt number and given properties of the liquid and gas phases.

Step-by-step solution

Calculate the velocities of liquid and gas phases

To calculate the flow velocities of the liquid and gas phases, we will first need the flow rates and the cross-sectional area of the pipeline. Cross-sectional area of the pipeline (\(A\)) can be calculated as follows: \(A = \frac{\pi D^2}{4}\) where \(D\) is the diameter of the pipeline, which is given as 102 mm. Now, let's calculate the flow velocities of the liquid (\(u_l\)) and gas (\(u_g\)) phases: \(\displaystyle u_l = \frac{Q_l}{A}, u_g = \frac{Q_g}{A}\) where \(Q_l\) and \(Q_g\) are the flow rates of liquid and gas phases, given as 16 kg/s and 0.055 kg/s, respectively.

Calculate the Reynolds numbers for liquid and gas phases

Using the calculated flow velocities from Step 1, we can now determine the Reynolds numbers for the liquid (\(Re_l\)) and gas (\(Re_g\)) phases. The Reynolds numbers for each phase can be calculated as follows: \(Re_l = \frac{D \rho_l u_l}{\mu_l}, Re_g = \frac{D \rho_g u_g}{\mu_g}\) Remember to convert the diameter to meters before calculating the Reynolds numbers.

Determine the two-phase flow pattern

In order to determine the appropriate correlations for the Nusselt number, we need to identify the specific two-phase flow pattern in the pipeline. This can be done by using the calculated Reynolds numbers and the given void fraction (\(\alpha\)). Standard flow pattern maps and correlations available in the literature can be referred to for this purpose.

Calculate the Nusselt number for the two-phase flow

Once the flow pattern is identified, we can choose an appropriate correlation to calculate the Nusselt number (\(Nu\)) for the two-phase flow. The specific correlation will depend on the identified flow pattern, the values of the Reynolds numbers, the dynamic viscosities, and other given parameters.

Calculate the two-phase heat transfer coefficient

Finally, we can calculate the two-phase heat transfer coefficient (\(h_{tp}\)) using the Nusselt number calculated in Step 4, and the given properties of the liquid and gas phases: \(h_{tp} = \frac{Nu \cdot k_l}{D}\) where \(k_l\) is the thermal conductivity of the liquid phase and D is the diameter of the pipeline. Do not forget to use appropriate units for the thermal conductivity and diameter while calculating the two-phase heat transfer coefficient.