open-channel model is A 30-meter-long constructed to meet the Froude number law. What is the flow rate in the model for a 700 m/s prototype flood if the scale is 1:20? Additionally, calculate the force ratio.
open-channel model is A 30-meter-long constructed to meet the Froude number law. What is the flow rate in the model for a 700 m/s prototype flood if the scale is 1:20? Additionally, calculate the force ratio.
Chapter2: Loads On Structures
Section: Chapter Questions
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![### Fluid Mechanics: Open-Channel Flow Modeling
**Problem Statement:**
A 30-meter-long open-channel model is constructed to meet the Froude number law. What is the flow rate in the model for a 700 m/s prototype flood if the scale is 1:20? Additionally, calculate the force ratio.
**Solution:**
1. **Understanding the Froude Number Law:**
The Froude number (Fr) is a dimensionless number that is used to compare the flow types in different systems to ensure dynamic similarity. The Froude number is given by:
\[
\text{Fr} = \frac{V}{\sqrt{gL}}
\]
where:
- \( V \) is the velocity of the fluid,
- \( g \) is the acceleration due to gravity,
- \( L \) is a characteristic length.
2. **Scale Ratios:**
The given scale is 1:20. This means that the model is 1/20th the size of the prototype. Given that:
- Length scale ratio, \(\lambda_L = \frac{L_m}{L_p} = \frac{1}{20}\)
- Velocity scale ratio, \(\lambda_V = \sqrt{\lambda_L} = \sqrt{\frac{1}{20}}\)
3. **Calculating Model Velocity:**
If the prototype velocity (\(V_p\)) is 700 m/s, the model velocity (\(V_m\)) can be calculated as:
\[
V_m = V_p \times \sqrt{\frac{1}{20}} = 700 \times \frac{1}{\sqrt{20}} \approx 700 \times 0.2236 = 156.52 \, \text{m/s}
\]
4. **Flow Rate Calculation:**
Assuming the model and prototype have the same cross-sectional area (\(A\)), the flow rate (\(Q\)) is given by:
\[
Q = A \times V
\]
Since \(A\) scales as \(L^2\),
- Area scale ratio, \(\lambda_A = \lambda_L^2 = \left(\frac{1}{20}\right)^2 = \frac{1}{400}\)
So, the scaled flow rate for the model,
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Transcribed Image Text:### Fluid Mechanics: Open-Channel Flow Modeling
**Problem Statement:**
A 30-meter-long open-channel model is constructed to meet the Froude number law. What is the flow rate in the model for a 700 m/s prototype flood if the scale is 1:20? Additionally, calculate the force ratio.
**Solution:**
1. **Understanding the Froude Number Law:**
The Froude number (Fr) is a dimensionless number that is used to compare the flow types in different systems to ensure dynamic similarity. The Froude number is given by:
\[
\text{Fr} = \frac{V}{\sqrt{gL}}
\]
where:
- \( V \) is the velocity of the fluid,
- \( g \) is the acceleration due to gravity,
- \( L \) is a characteristic length.
2. **Scale Ratios:**
The given scale is 1:20. This means that the model is 1/20th the size of the prototype. Given that:
- Length scale ratio, \(\lambda_L = \frac{L_m}{L_p} = \frac{1}{20}\)
- Velocity scale ratio, \(\lambda_V = \sqrt{\lambda_L} = \sqrt{\frac{1}{20}}\)
3. **Calculating Model Velocity:**
If the prototype velocity (\(V_p\)) is 700 m/s, the model velocity (\(V_m\)) can be calculated as:
\[
V_m = V_p \times \sqrt{\frac{1}{20}} = 700 \times \frac{1}{\sqrt{20}} \approx 700 \times 0.2236 = 156.52 \, \text{m/s}
\]
4. **Flow Rate Calculation:**
Assuming the model and prototype have the same cross-sectional area (\(A\)), the flow rate (\(Q\)) is given by:
\[
Q = A \times V
\]
Since \(A\) scales as \(L^2\),
- Area scale ratio, \(\lambda_A = \lambda_L^2 = \left(\frac{1}{20}\right)^2 = \frac{1}{400}\)
So, the scaled flow rate for the model,
\[
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