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Sagot :
Absolutely! Let's walk through the solution step-by-step for the given conditions.
### Given Data:
- Revolutions per minute (RPM) = 400
- Power output = 67.5 W
- Head (H) = 60 meters of water
- Inlet diameter at the base of the nozzle = 15200 mm
- Coefficient of velocity (C_v) = 0.97
- Ratio of wheel velocity to jet velocity = 0.40
- Efficiency = 0.83
First, let's convert some of the units to the appropriate measurements (e.g., diameter from mm to meters).
Conversions:
- [tex]\( \text{Inlet diameter} = 15200 \text{ mm} = 15.2 \text{ m} \)[/tex]
### 1. Volume Flow Rate (Q):
The volume flow rate can be calculated using the velocity of the jet and the area of the inlet pipe.
#### a) Calculate Jet Velocity (V_j):
Jet velocity [tex]\( V_j \)[/tex] can be determined using the formula:
[tex]\[ V_j = C_v \sqrt{2 g H} \][/tex]
Where:
- [tex]\( g = 9.81 \, \text{m/s}^2 \)[/tex] (acceleration due to gravity)
- [tex]\( H = 60 \, \text{m} \)[/tex]
With the given coefficient of velocity [tex]\( C_v = 0.97 \)[/tex]:
[tex]\[ V_j = 0.97 \times \sqrt{2 \times 9.81 \times 60} = 0.97 \times \sqrt{1177.2} \][/tex]
#### b) Area of the Inlet Pipe:
[tex]\[ \text{Area of inlet (A)} = \pi \left(\frac{\text{inlet diameter}}{2}\right)^2 = \pi \left(\frac{15.2}{2}\right)^2 \][/tex]
#### c) Volume Flow Rate:
[tex]\[ Q = A \times V_j \][/tex]
Using these formulas, the volume flow rate [tex]\( Q \)[/tex] is obtained as:
[tex]\[ Q = 6039.12 \, \text{m}^3/\text{s} \][/tex]
### 2. Diameter of the Jet (D_j):
The diameter of the jet can be calculated using the flow rate and jet velocity.
#### a) Area of the Jet:
[tex]\[ \text{Area}_{\text{jet}} = \frac{Q}{V_j} \][/tex]
#### b) Diameter of the Jet:
[tex]\[ D_j = \sqrt{\frac{4 \times \text{Area}_{\text{jet}}}{\pi}} \][/tex]
Based on this, the diameter of the jet [tex]\( D_j \)[/tex] is:
[tex]\[ D_j = 15.2 \, \text{m} \][/tex]
### 3. Wheel Diameter (D_w):
The wheel diameter can be calculated using the relationship between wheel velocity and jet velocity, and then determining the wheel circumference.
#### a) Wheel Velocity (V_w):
[tex]\[ V_w = \text{ratio} \times V_j \][/tex]
[tex]\[ V_w = 0.40 \times V_j \][/tex]
#### b) Wheel's Circumference:
[tex]\[ \text{Circumference of the wheel} = \frac{V_w}{RPM / 60} \][/tex]
#### c) Diameter of the Wheel:
[tex]\[ D_w = \frac{\text{Circumference of the wheel}}{\pi} \][/tex]
From these, the diameter of the wheel [tex]\( D_w \)[/tex] is:
[tex]\[ D_w = 0.63562 \, \text{m} \][/tex]
### 4. Pressure in the Inlet Pipe at the Nozzle Base:
Using Bernoulli's equation, assuming that the velocities change from zero to the jet velocity and using the density of water.
#### a) Inlet Pressure:
[tex]\[ \text{Pressure}_{\text{inlet}} = \frac{1}{2} \times \rho \times V_j^2 \][/tex]
Where [tex]\( \rho = 1000 \, \text{kg/m}^3 \)[/tex].
With the given values, the pressure at the inlet pipe is:
[tex]\[ \text{Pressure}_{\text{inlet}} = 553813.74 \, \text{Pa} \][/tex]
### Summary:
1. Volume Flow Rate (Q): [tex]\( 6039.12 \, \text{m}^3/\text{s} \)[/tex]
2. Diameter of the Jet (D_j): [tex]\( 15.2 \, \text{m} \)[/tex]
3. Wheel Diameter (D_w): [tex]\( 0.63562 \, \text{m} \)[/tex]
4. Pressure in the Inlet Pipe at the Nozzle Base: [tex]\( 553813.74 \, \text{Pa} \)[/tex]
### Given Data:
- Revolutions per minute (RPM) = 400
- Power output = 67.5 W
- Head (H) = 60 meters of water
- Inlet diameter at the base of the nozzle = 15200 mm
- Coefficient of velocity (C_v) = 0.97
- Ratio of wheel velocity to jet velocity = 0.40
- Efficiency = 0.83
First, let's convert some of the units to the appropriate measurements (e.g., diameter from mm to meters).
Conversions:
- [tex]\( \text{Inlet diameter} = 15200 \text{ mm} = 15.2 \text{ m} \)[/tex]
### 1. Volume Flow Rate (Q):
The volume flow rate can be calculated using the velocity of the jet and the area of the inlet pipe.
#### a) Calculate Jet Velocity (V_j):
Jet velocity [tex]\( V_j \)[/tex] can be determined using the formula:
[tex]\[ V_j = C_v \sqrt{2 g H} \][/tex]
Where:
- [tex]\( g = 9.81 \, \text{m/s}^2 \)[/tex] (acceleration due to gravity)
- [tex]\( H = 60 \, \text{m} \)[/tex]
With the given coefficient of velocity [tex]\( C_v = 0.97 \)[/tex]:
[tex]\[ V_j = 0.97 \times \sqrt{2 \times 9.81 \times 60} = 0.97 \times \sqrt{1177.2} \][/tex]
#### b) Area of the Inlet Pipe:
[tex]\[ \text{Area of inlet (A)} = \pi \left(\frac{\text{inlet diameter}}{2}\right)^2 = \pi \left(\frac{15.2}{2}\right)^2 \][/tex]
#### c) Volume Flow Rate:
[tex]\[ Q = A \times V_j \][/tex]
Using these formulas, the volume flow rate [tex]\( Q \)[/tex] is obtained as:
[tex]\[ Q = 6039.12 \, \text{m}^3/\text{s} \][/tex]
### 2. Diameter of the Jet (D_j):
The diameter of the jet can be calculated using the flow rate and jet velocity.
#### a) Area of the Jet:
[tex]\[ \text{Area}_{\text{jet}} = \frac{Q}{V_j} \][/tex]
#### b) Diameter of the Jet:
[tex]\[ D_j = \sqrt{\frac{4 \times \text{Area}_{\text{jet}}}{\pi}} \][/tex]
Based on this, the diameter of the jet [tex]\( D_j \)[/tex] is:
[tex]\[ D_j = 15.2 \, \text{m} \][/tex]
### 3. Wheel Diameter (D_w):
The wheel diameter can be calculated using the relationship between wheel velocity and jet velocity, and then determining the wheel circumference.
#### a) Wheel Velocity (V_w):
[tex]\[ V_w = \text{ratio} \times V_j \][/tex]
[tex]\[ V_w = 0.40 \times V_j \][/tex]
#### b) Wheel's Circumference:
[tex]\[ \text{Circumference of the wheel} = \frac{V_w}{RPM / 60} \][/tex]
#### c) Diameter of the Wheel:
[tex]\[ D_w = \frac{\text{Circumference of the wheel}}{\pi} \][/tex]
From these, the diameter of the wheel [tex]\( D_w \)[/tex] is:
[tex]\[ D_w = 0.63562 \, \text{m} \][/tex]
### 4. Pressure in the Inlet Pipe at the Nozzle Base:
Using Bernoulli's equation, assuming that the velocities change from zero to the jet velocity and using the density of water.
#### a) Inlet Pressure:
[tex]\[ \text{Pressure}_{\text{inlet}} = \frac{1}{2} \times \rho \times V_j^2 \][/tex]
Where [tex]\( \rho = 1000 \, \text{kg/m}^3 \)[/tex].
With the given values, the pressure at the inlet pipe is:
[tex]\[ \text{Pressure}_{\text{inlet}} = 553813.74 \, \text{Pa} \][/tex]
### Summary:
1. Volume Flow Rate (Q): [tex]\( 6039.12 \, \text{m}^3/\text{s} \)[/tex]
2. Diameter of the Jet (D_j): [tex]\( 15.2 \, \text{m} \)[/tex]
3. Wheel Diameter (D_w): [tex]\( 0.63562 \, \text{m} \)[/tex]
4. Pressure in the Inlet Pipe at the Nozzle Base: [tex]\( 553813.74 \, \text{Pa} \)[/tex]
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