91Ó°ÊÓ

The pressure drop in a section of pipe can be calculated as \\[\Delta p=f \frac{L \rho V^{2}}{2 D}\\] where \(\Delta p=\) the pressure drop (Pa), \(f=\) the friction factor, \(L=\) the length of pipe \([\mathrm{m}], \rho=\) density \(\left(\mathrm{kg} / \mathrm{m}^{3}\right), V=\) velocity \((\mathrm{m} / \mathrm{s})\) and \(D=\) diameter (m). For turbulent flow, the Colebrook equation provides a means to calculate the friction factor, \\[\frac{1}{\sqrt{f}}=-2.0 \log \left(\frac{\varepsilon}{3.7 D}+\frac{2.51}{\operatorname{Re} \sqrt{f}}\right)\\] where \(\varepsilon=\) the roughness \((\mathrm{m}),\) and \(\mathrm{Re}=\) the Reynolds number \\[\mathrm{Re}=\frac{\rho V D}{\mu}\\] where \(\mu=\) dynamic viscosity \(\left(\mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}\right)\) (a) Determine \(\Delta p\) for a 0.2 -m-long horizontal stretch of smooth drawn tubing given \(\rho=1.23 \mathrm{kg} / \mathrm{m}^{3}, \mu=1.79 \times 10^{-5} \mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}\) \(D=0.005 \mathrm{m}, V=40 \mathrm{m} / \mathrm{s},\) and \(\varepsilon=0.0015 \mathrm{mm} .\) Use a numer- ical method to determine the friction factor. Note that smooth pipes with \(\mathrm{Re}<10^{5},\) a good initial guess can be obtained using the Blasitus formula, \(f=0.316 / \mathrm{Re}^{0.25}\) (b) Repeat the computation but for a rougher commercial steel pipe \((\varepsilon=0.045 \mathrm{mm})\)

Step by step solution

01

Calculate the Reynolds number Re

To calculate the Reynolds number, use the formula: \[\mathrm{Re}=\frac{\rho V D}{\mu}\] where \(\rho = 1.23 \ \mathrm{kg / m^3}\), \(V = 40 \ \mathrm{m / s}\), \(D = 0.005\ \mathrm{m}\), and \(\mu = 1.79 \times 10^{-5} \ \mathrm{N \cdot s / m^2}\). Re = \(\frac{(1.23)(40)(0.005)}{1.79\times10^{-5}} \)

02

Estimate the friction factor (f) using the Blasius formula

The Blasius formula gives a good initial guess for the friction factor when the Reynolds number is less than \(10^5\). \(f= \frac{0.316}{\mathrm{Re}^{0.25}}\) Now, calculate f using the Reynolds number calculated in Step 1.

03

Find the friction factor (f) using the Colebrook equation and numerical method

We will use a numerical method, such as the Newton-Raphson method, to iteratively solve for the friction factor (f) using the initial guess obtained from Step 2 and the Colebrook equation. \[ \frac{1}{\sqrt{f}} = -2.0 \log \left(\frac{\varepsilon}{3.7D} + \frac{2.51}{\mathrm{Re} \sqrt{f}}\right) \] Note that \(\varepsilon=0.0015\) mm, which needs to be converted to m for calculations.

04

Calculate the pressure drop \(\Delta p\)

Now that we know the friction factor (f), we can calculate the pressure drop \(\Delta p\) using the formula: \[ \Delta p=f \frac{L \rho V^{2}}{2 D} \] where the length of the pipe L = 0.2 m. ## Problem (b): Determine the pressure drop for the rougher commercial steel pipe ##

05

Calculate the Reynolds number Re

The Reynolds number is the same as in Part (a), as the relevant parameters have not changed.

06

Estimate the friction factor (f) using the Blasius formula

Again, use Blasius formula to estimate the friction factor (f) with the same Reynolds number.

07

Find the friction factor (f) using the Colebrook equation and numerical method

Use the same numerical method for finding the friction factor (f), but update the roughness value. Note that the new roughness value for the commercial steel pipe is \(\varepsilon=0.045\) mm, which needs to be converted to m for calculations.

08

Calculate the pressure drop \(\Delta p\)

Finally, calculate the pressure drop \(\Delta p\) for the rougher commercial pipe using the updated friction factor (f) and same pipe length L = 0.2 m as in the formula: \[ \Delta p=f \frac{L \rho V^{2}}{2 D} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Pressure Drop Calculation

The pressure drop in a pipe is a crucial concept in fluid mechanics, often needed to understand how fluids behave in real-world systems like water pipes and air conditioning. To calculate this drop, we rely on the formula: \[\Delta p = f \frac{L \rho V^2}{2D}\]Each variable in this equation represents:

• \(\Delta p\): The pressure drop across a section of pipe, measured in Pascals (Pa).
• \(f\): Friction factor, depicting how rough the pipe's surface is.
• \(L\): Length of the pipe in meters (m).
• \(\rho\): Fluid density in kilograms per cubic meter (kg/m³).
• \(V\): Fluid velocity in meters per second (m/s).
• \(D\): Pipe diameter in meters (m).

These factors all contribute to the loss of pressure as fluid flows through a pipe. Calculating pressure drop is essential for designing efficient piping systems and ensuring they can handle the demands placed upon them.

Friction Factor Determination

The friction factor \(f\) is critical in determining the pressure drop of a fluid flowing through a pipe. It accounts for the influence of a pipe's surface roughness on the flow. There are several ways to estimate \(f\), often depending on whether the flow is laminar or turbulent.For turbulent flow, a popular initial estimation method is the Blasius formula, expressed as:\[f = \frac{0.316}{\text{Re}^{0.25}}\]This formula is effective for smooth pipes where the Reynolds number \(\text{Re}\) is less than \(10^5\).The friction factor is then refined using the Colebrook equation, integrating surface roughness more precisely.This iterative process helps engineers and students understand how different parameters affect fluid resistance and can optimize designs accordingly.

Colebrook Equation

The Colebrook equation is a key step in accurately determining the friction factor for turbulent flow within pipes. It accounts for both the pipe roughness and the Reynolds number, establishing a more accurate friction factor:\[\frac{1}{\sqrt{f}} = -2.0 \log \left(\frac{\varepsilon}{3.7D} + \frac{2.51}{\text{Re}\sqrt{f}}\right)\]In this equation:

• \(\varepsilon\): Roughness of the pipe's interior surface, measured in meters.

The Colebrook equation is implicit, meaning \(f\) appears on both sides. Solving it requires numerical methods like Newton-Raphson, as it is not easily solved algebraically. This method provides a better depiction of how real-world conditions influence the flow and is essential for designing efficient piping systems.

Reynolds Number

The Reynolds number (\text{Re}) is a dimensionless quantity crucial in fluid mechanics, describing the flow regime in a piping system. It's calculated as:\[\text{Re} = \frac{\rho V D}{\mu}\]Where:

• \(\rho\): Fluid density (kg/m³).
• \(V\): Fluid velocity (m/s).
• \(D\): Pipe diameter (m).
• \(\mu\): Dynamic viscosity (Nâ‹…s/m²).

A high Reynolds number indicates turbulent flow, whereas a low number suggests laminar flow. Understanding \(\text{Re}\)is pivotal for engineers to predict how fluids move, helping to design appropriate systems and select suitable methods for friction factor determination.

Numerical Methods in Engineering

In fluid mechanics, many equations cannot be solved directly, like the Colebrook equation, necessitating numerical methods. These methods provide approximate solutions through iterations, allowing us to find answers to complex problems that don't have straightforward formulas. Commonly used numerical methods include:

• Newton-Raphson Method: An iterative process to find successively better approximations of the roots (solutions).
• Bisection Method: A simple yet effective method that repeatedly bisects an interval and then selects a subinterval preserving the sign of the function.

These tools are essential in engineering, providing accuracy and reliability in computations, especially in determining values like the friction factor in fluid flow scenarios. Without them, engineers would struggle to predict flow behavior in real-world applications.