Question

The pressure in a pipeline that transports helium gas at a rate of $7 \mathrm{lbm} / \mathrm{s}\( is maintained at \)14.5$ psia by venting helium to the atmosphere through a \(0.4\)-in-internal-diameter tube that extends $30 \mathrm{ft}$ into the air. Assuming both the helium and the atmospheric air to be at \(80^{\circ} \mathrm{F}\), determine \((a)\) the mass flow rate of helium lost to the atmosphere through the tube, (b) the mass flow rate of air that infiltrates into the pipeline, and (c) the flow velocity at the bottom of the tube where it is attached to the pipeline that will be measured by an anemometer in steady operation.

Short answer

Answer: To find the mass flow rates of helium lost to the atmosphere and air infiltrating into the pipeline, as well as the flow velocity at the bottom of the tube, follow these steps: 1. Use the Ideal Gas Law to calculate the mass flow rate of helium based on the given pressure and temperature. 2. Calculate the mass flow rate of helium lost to the atmosphere using the pressure and volume flow rate. 3. Calculate the mass flow rate of air infiltrating into the pipeline by applying the principle of mass conservation. 4. Calculate the flow velocity at the bottom of the tube using the mass flow rate of helium from step 2. Once the calculations are done based on given data, you will find the mass flow rates of helium lost to the atmosphere and air infiltrating into the pipeline, and the flow velocity at the bottom of the tube.

Step-by-step solution

Ideal Gas Law Calculation for Helium

First, we need to use the Ideal Gas Law to find the mass flow rate of helium based on the given pressure and temperature. The ideal gas law is given by: \(PV = nRT\) where P is the pressure, V is the volume, n is the number of moles, R is the specific gas constant, and T is the temperature in Kelvin. Since we require molar mass of helium (M) to find mass flow rate, R can be related to the universal gas constant (R_u) using the equation: \(R = \frac{R_u}{M}\) The universal gas constant for helium is \(206 \, \mathrm{ft.BundleUp} .\mathrm{lbm.BundleUp^{2}} / \mathrm{s.BundleUp^{2}.R.BundleUp}\) Rewriting the Ideal Gas Law by substituting the value of R: \(PV = \frac{mRT_u}{M}\) Now we need to find the mass flow rate (m') of helium, so we need to differentiate the equation above with respect to time: \(P \frac{dV}{dt} = \frac{1}{M} \frac{dm}{dt} R T_u\) Next, rearrange the equation to find the mass flow rate: \(\frac{dm}{dt} = \frac{PM}{RT_u} \frac{dV}{dt}\)

Calculate mass flow rate of helium lost to the atmosphere

We now know the mass flow rate of helium in the pipeline is given as \(7 \, \mathrm{lbm / s}\). We can now plug in the given pressure to calculate the mass flow rate of helium lost to the atmosphere: \(\frac{dm_{He}}{dt} = \frac{14.5 \, \mathrm{psia}}{R \times 80 + 460} \frac{dV_{He}}{dt}\) To find the volume flow rate, \(\frac{dV_{He}}{dt}\), we can use the relationship: \(A_{He}v_{He} = A_{air}v_{air}\) where \(A\) is the cross-sectional area and \(v\) is the velocity. We can rewrite the equation as follows: \(v_{He} = \frac{A_{air}v_{air}}{A_{He}}\) Using the given properties of the tube, we can now calculate the cross-sectional areas: \(A_{He} = A_{air} = \pi \left( \frac{0.4 \, \mathrm{in}}{2} \right)^2\) Now we can determine \(v_{He}\): \(v_{He} = \frac{A_{air}v_{air}}{A_{He}}\) We now have the information we need to find the mass flow rate of helium lost to the atmosphere from the equation in step 1: \(\frac{dm_{He}}{dt} = \frac{14.5 \, \mathrm{psia}}{R \times 80 + 460} \frac{dV_{He}}{dt} = \frac{PM}{RT_u} A_{He} v_{He}\)

Calculate mass flow rate of air infiltrating into the pipeline

Since we know the mass flow rate of helium in the pipeline and the mass flow rate of helium lost to the atmosphere, we can now calculate the mass flow rate of air infiltrating into the pipeline using the principle of mass conservation: \(\frac{dm_{air}}{dt} = \frac{dm_{pipeline}}{dt} - \frac{dm_{He}}{dt}\) \(\frac{dm_{air}}{dt} = 7 \, \mathrm{lbm / s} - \frac{dm_{He}}{dt}\)

Calculate the flow velocity at the bottom of the tube

Finally, we need to find the flow velocity at the bottom of the tube where it is attached to the pipeline. This will be measured by an anemometer in steady operation. We can find the flow velocity using the mass flow rate of helium from Step 2: \(v_{He} = \frac{\frac{dm_{He}}{dt}}{\rho_{He} A_{He}}\) Now that we have calculated the flow velocity, we have completed the task of finding the mass flow rates of helium lost to the atmosphere and air infiltrating into the pipeline, and the flow velocity at the bottom of the tube.