Answered: Water flows at an average velocity of…

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Water flows at an average velocity of 10 m/s through a horizontal tube of diameter 5 cm. The pressure is measured at 1 m intervals as follows:

(a) Estimate the total head loss.
(b) Estimate the entrance length based on the data and based on the accepted formula.
(c) Estimate the wall shear stress in the fully developed region.
(d) Estimate the overall friction factor based on the data. Estimate the friction factor in the fully developed region only.

### Fluid Mechanics Problem: Estimating Pressure Drop in a Horizontal Tube

**Problem Statement:**

Water flows at an average velocity of 10 m/s through a horizontal tube with a diameter of 5 cm. The pressure is measured at 1-meter intervals as follows:

\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
x \, (\text{m}) & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
p \, (\text{kPa}) & 304 & 273 & 255 & 240 & 226 & 213 & 200 \\
\hline
\end{array}
\]

**Tasks:**

(a) Estimate the total head loss.

(b) Estimate the entrance length based on the data and accepted formulas.

(c) Estimate the wall shear stress in the fully developed region.

(d) Estimate the overall friction factor based on the data. Estimate the friction factor in the fully developed region only.

**Graph Explanation:**

- **Graph/Data Table:** The provided data table shows the pressure (p in kPa) measured at several points (x in meters) along the length of the tube.

- At \(x = 0 \, m\), the pressure is 304 kPa.
- At \(x = 1 \, m\), the pressure is 273 kPa.
- At \(x = 2 \, m\), the pressure is 255 kPa.
- At \(x = 3 \, m\), the pressure is 240 kPa.
- At \(x = 4 \, m\), the pressure is 226 kPa.
- At \(x = 5 \, m\), the pressure is 213 kPa.
- At \(x = 6 \, m\), the pressure is 200 kPa.

**Detailed Steps for Solutions:**

1. **Total Head Loss:**
- Use the pressure difference between the two endpoints of the tube to determine the total head loss.

2. **Entrance Length Estimation:**
- Calculate the entrance length using empirical correlations for turbulent or laminar flow as applicable.

3. **Wall Shear Stress:**
- Derive the wall shear stress from the pressure gradient in the fully developed region by applying the

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