Oil flows through a pipe that contracts from 450mm diameter at A to 300mm diameter at B and then forks. One branch being 150mm diameter discharging at C and the other branch being 225mm discharging at D. If the velocity at A=1.5m/s and the velocity at B is 3.6m/s,
(i)Find the discharge at C and D, 
(ii) find the velocities at B and C

To solve this problem, we'll use the principle of continuity, which is based on the conservation of mass. The principle states that the mass flow rate remains constant throughout the pipeline, and it can be expressed mathematically as:

\[
Q = A \cdot v
\]

Where:
- \(Q\) is the volumetric flow rate (in cubic meters per second, m³/s),
- \(A\) is the cross-sectional area of the pipe (in square meters, m²),
- \(v\) is the velocity of the fluid (in meters per second, m/s).

### Step 1: Find the cross-sectional areas
We'll begin by calculating the cross-sectional areas of the pipe at each section, using the formula for the area of a circle:

\[
A = \frac{\pi d^2}{4}
\]

Where:
- \(d\) is the diameter of the pipe.

#### At section A (diameter = 450 mm = 0.45 m):

\[
A_A = \frac{\pi (0.45)^2}{4} = 0.159 m^2
\]

#### At section B (diameter = 300 mm = 0.30 m):

\[
A_B = \frac{\pi (0.30)^2}{4} = 0.071 m^2
\]

#### At section C (diameter = 150 mm = 0.15 m):

\[
A_C = \frac{\pi (0.15)^2}{4} = 0.0177 m^2
\]

#### At section D (diameter = 225 mm = 0.225 m):

\[
A_D = \frac{\pi (0.225)^2}{4} = 0.0398 m^2
\]

### Step 2: Apply the continuity equation

The volumetric flow rate \(Q\) is constant at sections A and B, meaning the total flow at A equals the total flow at B. First, calculate the flow rates at A and B.

#### Flow rate at A:

\[
Q_A = A_A \cdot v_A = 0.159 \cdot 1.5 = 0.2385 \, m^3/s
\]

#### Flow rate at B:

\[
Q_B = A_B \cdot v_B = 0.071 \cdot 3.6 = 0.2556 \, m^3/s
\]

The slight difference between \(Q_A\) and \(Q_B\) is due to rounding.

Now, for the forked branches, the total flow rate at B must be equal to the sum of the flow rates at C and D.

\[
Q_B = Q_C + Q_D
\]

We know that the cross-sectional areas at C and D are different, so we need to determine the flow rate at each section.

### Step 3: Find the velocities at C and D

We can calculate the velocities at sections C and D using the continuity equation \(Q = A \cdot v\).

Let’s assume that the flow splits proportionally according to the cross-sectional areas of sections C and D. The total flow rate at B is 0.2556 m³/s, so:

#### Velocity at C:

\[
v_C = \frac{Q_C}{A_C}
\]

#### Velocity at D:

\[
v_D = \frac{Q

Oil flows through a pipe that contracts from 450mm diameter at A to 300mm diameter at B and then forks. One branch being 150mm diameter discharging at C and the other branch being 225mm discharging at D. If the velocity at A=1.5m/s and the velocity at B is 3.6m/s, (i) Find the discharge at C and D, (ii) find the velocities at C and D.

Learn how to calculate the discharge and velocity of oil flowing through a pipe that contracts and forks into two branches. This problem applies fluid mechanics principles, including the continuity equation and volumetric flow rate, and involves solving for the velocities at different pipe sections. Ideal for undergraduate fluid mechanics students.

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