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 \((\mathrm{Pa}), f=\) 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 numerical method to determine the friction factor. Note that smooth pipes with \(\operatorname{Re}<10^{5}\), a good initial guess can be obtained using the Blasius 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

First, calculate the Reynolds Number: \(\operatorname{Re}=\frac{\rho V D}{\mu}\). Given the values of \(\rho=1.23 \mathrm{kg} / \mathrm{m}^{3}, \mu=1.79 \times 10^{-5}\mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}, D=0.005\ \mathrm{m}, V=40\ \mathrm{m} / \mathrm{s}\), \[\operatorname{Re} = \frac{(1.23)(40)(0.005)}{1.79 \times 10^{-5}} = 13743.58\] #Step 2: Estimate Initial Friction Factor#

Use the Blasius formula to find an initial estimate of the friction factor: \(f=\frac{0.316}{\operatorname{Re}^{0.25}}\) \[f=\frac{0.316}{13743.58^{0.25}}=0.0274\] #Step 3: Apply the Newton's Method to the Colebrook equation#

02

Use Newton's method iteratively to determine a more accurate value of the friction factor: The Colebrook equation is: \(\frac{1}{\sqrt{f}}=-2.0 \log \left(\frac{\varepsilon}{3.7D}+\frac{2.51}{\operatorname{Re} \sqrt{f}}\right)\) Differentiate this equation with respect to f: \[\frac{d}{df}\left(\frac{1}{\sqrt{f}}\right) = -\frac{1}{2 f^{3/2}}\] Apply Newton's method formula: \[f_{n+1} = f_n - \frac{F(f_n)}{F'(f_n)}\] where \(F\left(f_n\right) = \frac{1}{\sqrt{f_n}} + 2.0\log \left(\frac{\varepsilon}{3.7D}+\frac{2.51}{\operatorname{Re} \sqrt{f_n}}\right)\) and \(F'\left(f_n\right) = -\frac{1}{2f_n^{3/2}}\) Iterate until the change in successive estimates is negligible or it reaches a tolerable error limit. #Step 4: Calculate the pressure drop in both cases#

Now that we have the friction factor, we can compute the pressure drop for both cases of tubing using the formula: \(\Delta p = f\frac{L \rho V^{2}}{2 D}\). (a) For the smooth tubing: Using the friction factor, length of the pipe L=\(0.2\ \mathrm{m}\), density \(\rho = 1.23\ \mathrm{kg/m}^3\), velocity \(V=40\ \mathrm{m/s}\), and diameter \(D = 0.005\ \mathrm{m}\), \[\Delta p = (0.0274)\frac{(0.2)(1.23)(40^2)}{2(0.005)}= 22096.8\ \mathrm{Pa}\]. (b) For the rough steel pipe: Follow the same procedure to find the friction factor for the given roughness \(\varepsilon=0.045\ \mathrm{mm}\). After calculating it, use the formula to calculate the pressure drop.

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Key Concepts

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

Pressure Drop Calculation

The calculation of pressure drop in a pipe is crucial in fluid dynamics as it helps in understanding the resistance exerted by the pipe on the fluid flow. The pressure drop can be modeled using the equation: \[ \Delta p = f \frac{L \rho V^{2}}{2 D} \] This equation considers several parameters:- **Friction Factor (\( f \))**: Represents the friction between the fluid and pipe walls. Determining this accurately is essential for precise pressure drop calculations.- **Length of Pipe (\( L \))**: Longer pipes naturally cause a greater pressure drop.- **Density (\( \rho \))**: This is the fluid's mass per unit volume. Heavier fluids experience higher pressure drops.- **Velocity (\( V \))**: The speed of the fluid flow. Higher velocity results in a higher pressure drop.- **Diameter (\( D \))**: A smaller diameter leads to a higher pressure drop due to limited flow space.
Understanding these components allows engineers to design systems with proper pressure levels, ensuring efficient flow and minimized energy usage.

Colebrook Equation

The Colebrook Equation is pivotal in calculating the friction factor for turbulent flow within pipes, especially when dealing with practical engineering applications involving both smooth and rough pipes. The equation is given by:\[ \frac{1}{\sqrt{f}} = -2.0 \log \left(\frac{\varepsilon}{3.7 D} + \frac{2.51}{\operatorname{Re} \sqrt{f}}\right) \]This equation is implicit in nature, meaning that it requires iterative methods, such as Newton's method, to solve for the friction factor \( f \). Key elements include:- **Relative Roughness (\( \varepsilon / D \))**: This ratio affects how flow interacts with pipe walls.- **Reynolds Number**: A unitless number that helps determine whether the flow is laminar or turbulent.
The Colebrook equation is crucial for determining the correct friction factor, which influences the design and optimization of pipe systems in various engineering and industrial applications.

Reynolds Number

The Reynolds Number \( \text{Re} \) is an essential dimensionless quantity in fluid mechanics used to predict flow patterns in different fluid flow situations. Its formula is: \[ \text{Re} = \frac{\rho V D}{\mu} \]This number helps identify whether the fluid flow is laminar or turbulent:- **Laminar Flow**: Occurs at \( \text{Re} < 2000 \), with fluid flowing in parallel layers.- **Turbulent Flow**: Occurs at \( \text{Re} > 4000 \), characterized by chaotic property fluctuations.
In the zone between these values (i.e., around 2000-4000), the flow is in a transitional phase. The Reynolds Number influences various factors like the friction factor and overall flow behavior, making it a vital part of fluid dynamics calculations.

Friction Factor Estimation

Estimating the friction factor is vital for evaluating pressure drops in pipes accurately. For turbulent flows, particularly when \( \text{Re} < 10^5 \), the Blasius formula provides a suitable estimation:\[ f = \frac{0.316}{\text{Re}^{0.25}} \]This serves as an initial estimate before refining using methods such as the Newton’s method with the Colebrook equation. Key points include:- **Initial Estimate**: Helps start the iterative process to reach a more precise solution.- **Iterative Methods**: Techniques such as Newton’s are essential for complex equations like the Colebrook's equation.- **Accuracy**: A precise friction factor is crucial for designing efficient piping systems in terms of energy and cost-efficiency.
Accurate friction factor estimation ensures that systems perform effectively under specified conditions, minimizing energy losses and ensuring safety in operational environments.