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Chapter 11: Problem 62

Waste gas \(\left(\mathrm{CO}_{2}\right)\) is vented to outer space from a spacecraft through a circular pipe \(0.2 \mathrm{m}\) long. The pressure and temperature in the spacecraft are \(35 \mathrm{kPa}\) and \(25^{\circ} \mathrm{C}\). The gas must be vented at the rate of \(0.01 \mathrm{kg} / \mathrm{s}\). The friction factor for the flow in the pipe is given by \\[\begin{array}{c}f=64 / \mathrm{Re}, \quad \mathrm{Re}<5000 \\\f=0.013, \quad \mathrm{Re}>5000, \quad \mathrm{Re}=\frac{ \dot{m}}{\pi \mu D}\end{array}\\] The viscosity \((\mu)\) is \(4 \times 10^{-4} \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}\). Determine the required pipe diameter.

Short Answer

Expert verified
The required pipe diameter can be found by iterating calculations based on Reynolds number and flow conditions, starting with assuming laminar flow. The answer will be validated when the resultant diameter gives a Reynolds number consistent with our flow condition assumption.

Step by step solution

01

Calculate initial Reynolds number

First, we need to estimate the Reynolds number to choose the correct friction factor.\[\mathrm{Re}=\frac{ \dot{m}}{\pi \mu D} \]But we don't know the diameter \(D\) of the pipe yet, so let's assume an initial Reynolds number value which is less than 5000 to start with. Let's assume that for that condition, the flow is laminar, so Re<5000.
02

Use the friction factor for laminar flow

Apply the friction factor formula for laminar flow (Re<5000) as follows: \[f=64 / \mathrm{Re}\]As we have assumed Re<5000, we can use this equation. This equation will be used in later step to solve for diameter.
03

Apply Bernoulli’s equation

In order to calculate the diameter of the pipe, we will apply Bernoulli’s equation with some assumptions. The equation is as follows:\[\frac{v^{2}}{2} + gz +\frac{p}{\rho}=\text{constant}\]Assume that the gravitational potential energy and the kinetic energy of the gas are negligible compared to the pressure energy, simplifying Bernoulli’s equation to:\[p_{1} + \frac{1}{2} \rho v_{1}^{2} + \rho g z_{1} = p_{2} + \frac{1}{2} \rho v_{2}^{2} + \rho g z_{2}\]Assuming the gas is ideal: \(p_1 = \rho R T\). Also within a pipe \(v_1 = \dot{m} / (\rho \pi D^{2}/4)\) . Since the gas is vented to outer space where \(p_2=0\), \(z_2=0\) and \(v_2=0\) , we get these equations to calculate unknowns.
04

Calculate Diameter

Substitute the equations from Step 3 and friction factor from Step 2 to derive a quadratic equation for the diameter \(D\). Solve this equation to get the value of \(D\). If the Reynolds number with this diameter is greater than 5000, it means the initial assumption was not correct. In this case, use the turbulent flow friction factor instead of laminar one and solve the equations again. Do not forget that Reynolds number changes with the change in \(D\).
05

Validate the Answer

Recalculate the Reynolds number using the obtained diameter and make sure it matches the assumed flow conditions (laminar or turbulent). If Reynolds number doesn’t match, one should go back to follow the process for the other flow conditions, iterating until you have a consistent solution.

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