https://file.aiquickdraw.com/mathbot/file/1727644852825o0isgjgl.jpg

Let's break down the problem from the image.

### Given dataLet's break down the problem from the image.

### Given 
- The oil flows through a pipe that contracts from:
  - Diameter at point A: 450 mm
  - Diameter at point B: 300 mm
- After point B, the pipe forks:
  - Branch 1 (C): Diameter = 150 mm
  - Branch 2 (D): Diameter = 225 mm
- Velocity at point A (\(V_A\)) = 1.8 m/s
- Velocity at point D (\(V_D\)) = 3.6 m/s

We need to compute:
1. **Discharge at C and D**.
2. **Velocities at B and C**.

### Approach:

#### Step 1: Apply the continuity equation
The continuity equation states that:
\[
Q = A \cdot V
\]
Where:
- \( Q \) is the discharge (flow rate)
- \( A \) is the cross-sectional area of the pipe
- \( V \) is the velocity of the fluid

For any given point:
\[
A = \frac{\pi \cdot d^2}{4}
\]
Where \( d \) is the diameter of the pipe.

### Step 2: Total discharge at A
The discharge at point A can be calculated as:
\[
Q_A = A_A \cdot V_A
\]
Where \( A_A \) is the cross-sectional area at point A.

Let's compute it using these formulas and proceed with solving for discharges and velocities at other points.### Results:
1. **Discharge at points C and D**:
   - Discharge at C (\( Q_C \)): 0.143 m³/s
   - Discharge at D (\( Q_D \)): 0.143 m³/s
   
2. **Velocities at points B and C**:
   - Velocity at B (\( V_B \)): 4.05 m/s
   - Velocity at C (\( V_C \)): 8.1 m/s

These results are based on applying the continuity equation and assuming incompressible flow.

Would you like further clarification or additional details?

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Here are 5 related questions to deepen your understanding:
1. What is the significance of the continuity equation in fluid mechanics?
2. How would the results change if the fluid were compressible?
3. Can you explain how discharge is conserved in a branching pipe?
4. What would happen to the velocity if the diameter at point B were increased?
5. How would you calculate the pressure drop between points A and D?

**Tip**: In fluid flow problems, the continuity equation helps maintain the mass balance across different sections of a pipe system.

Oil flows through a pipe that contracts from 450 mm diameter at A to 300 mm diameter at B and then forks, one branch being 150 mm in diameter discharging at C and the other branch 225 mm diameter discharging at D. If the velocity at A is 1.8 m/s and the velocity at D is 3.6 m/s, compute the discharge at C and D, and the velocities at B and C.

Learn how to compute discharge and velocities in a pipe system with varying diameters using the continuity equation. This problem involves calculating the discharge at C and D and determining the velocities at B and C for an oil flow system. Suitable for students studying fluid mechanics at the undergraduate level.

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