ANWSER
Question 1a:
a. Enthalpy (h): A thermodynamic property representing the sum of the internal energy of a system plus the product of its pressure and volume.
b. Viscosity (µ): A measure of a fluid’s resistance to deformation or flow, often referred to as its “thickness.”
c. Froude’s Number: A dimensionless number used to determine the flow regime in open channels, defined as the ratio of inertial forces to gravitational forces.
d. Reynold’s Number: A dimensionless number used to predict flow patterns, indicating whether the flow is laminar or turbulent, based on the ratio of inertial forces to viscous forces.
Question 1b:
Classification of properties of fluid:
1. Intensive Properties: Independent of mass (e.g., pressure, temperature, density).
2. Extensive Properties: Depend on mass (e.g., volume, energy, enthalpy).
3. Specific Properties: Extensive properties per unit mass (e.g., specific volume, specific energy).
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Question 2:
Given:
– Width (\(b\)) = 7.5 m
– Depth (\(y\)) = 2.25 m
– Slope (\(S\)) = 1/1000
– Chezy’s constant (\(C\)) = 55
Rate of flow (\(Q\)):
Using Chezy’s formula:
\[ Q = A \cdot C \cdot \sqrt{R \cdot S} \]
Where:
– \(A = b \cdot y = 7.5 \times 2.25 = 16.875 \, \text{m}^2\)
– Hydraulic radius (\(R\)) = \(A/P\), where \(P = b + 2y = 7.5 + 2 \times 2.25 = 12 \, \text{m}\)
– \(R = 16.875 / 12 = 1.406 \, \text{m}\)
Substitute values:
\[ Q = 16.875 \times 55 \times \sqrt{1.406 \times 0.001} \]
\[ Q \approx 16.875 \times 55 \times 0.0375 \approx 34.8 \, \text{m}^3/\text{s} \]
Conveyance (\(K\)):
\[ K = A \cdot C \cdot \sqrt{R} \]
\[ K = 16.875 \times 55 \times \sqrt{1.406} \approx 1100 \, \text{m}^3/\text{s} \]
Flow type:
Calculate Froude’s number (\(Fr\)):
\[ Fr = \frac{v}{\sqrt{g \cdot y}} \]
Where \(v = Q/A = 34.8 / 16.875 \approx 2.06 \, \text{m/s}\)
\[ Fr = \frac{2.06}{\sqrt{9.81 \times 2.25}}} \approx 0.44 \]
Since \(Fr < 1\), the flow is tranquil (subcritical).
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Question 3:
Given:
– Discharge (\(Q\)) = 12 m³/s
– Velocity (\(v\)) = 3 m/s
– Chezy’s constant (\(C\)) = 50
Most economical cross-section:
For a rectangular channel, the most economical cross-section occurs when the width (\(b\)) is twice the depth (\(y\)), i.e., \(b = 2y\).
Find \(y\) and \(b\):
\[ Q = A \cdot v \]
\[ A = Q / v = 12 / 3 = 4 \, \text{m}^2 \]
For economical section:
\[ A = b \cdot y = 2y \cdot y = 2y^2 \]
\[ 2y^2 = 4 \implies y = \sqrt{2} \approx 1.41 \, \text{m} \]
\[ b = 2y \approx 2.83 \, \text{m} \]
Slope (\(S\)):
Using Chezy’s formula:
\[ Q = A \cdot C \cdot \sqrt{R \cdot S} \]
Hydraulic radius (\(R\)) = \(A/P = 4 / (2.83 + 2 \times 1.41) \approx 0.7 \, \text{m}\)
Rearrange for \(S\):
\[ S = \left( \frac{Q}{A \cdot C \cdot \sqrt{R}} \right)^2 \]
\[ S \approx \left( \frac{12}{4 \times 50 \times \sqrt{0.7}} \right)^2 \approx 0.0003 \, \text{(1 in 3333)} \]
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Question 4a:
Bernoulli’s equation:
\[ P_1 + \frac{1}{2} \rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g h_2 \]
Where:
– \(P\) = Pressure
– \(\rho\) = Fluid density
– \(v\) = Velocity
– \(h\) = Elevation head
– \(g\) = Acceleration due to gravity
Question 4b:
Limitations of Bernoulli’s equation:
1. Assumes inviscid (frictionless) flow.
2. Valid only for steady flow.
3. Applicable only along a streamline.
4. Neglects compressibility effects (valid for incompressible fluids).
5. Assumes no energy addition or loss (e.g., pumps, turbines).
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Question 5:
Given:
– \(v_1 = 2 \, \text{m/s}\), \(P_1 = 3000 \, \text{Pa}\)
– \(v_2 = 4 \, \text{m/s}\), \(\rho = 1000 \, \text{kg/m}^3\)
Using Bernoulli’s equation (horizontal pipe, \(h_1 = h_2\)):
\[ P_1 + \frac{1}{2} \rho v_1^2 = P_2 + \frac{1}{2} \rho v_2^2 \]
Rearrange for \(P_2\):
\[ P_2 = P_1 + \frac{1}{2} \rho (v_1^2 – v_2^2) \]
\[ P_2 = 3000 + \frac{1}{2} \times 1000 \times (2^2 – 4^2) \]
\[ P_2 = 3000 + 500 \times (-12) = 3000 – 6000 = -3000 \, \text{Pa} \]
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Question 6a:
Types of open channel flow:
1. Steady vs. Unsteady
2. Uniform vs. Non-uniform
3. Laminar vs. Turbulent
4. Subcritical, Critical, Supercritical
Question 6b:
Pipe networking equations:
1. Continuity equation (\(Q = A \cdot v\)).
2. Energy equation (Bernoulli’s or extended).
3. Darcy-Weisbach equation (head loss due to friction).
Question 6c:
Steps in pipe network design:
1. Determine demand and layout.
2. Apply continuity and energy equations.
3. Calculate head losses (friction, minor losses).
4. Ensure pressure and velocity are within limits.
5. Iterate for optimal design.