A venturimeter has its axis vertical, the inlet

and throat diameters being 150 mm and 75 mm

respectively. The throat is 225 mm above inlet

and venturimeter constant = 0.96. Petrol of sp.

gravity 0.78 Àows up through the meter at a

rate of 0.029 m3

/s. Find the pressure difference

between inlet and throat.

To find the pressure difference between the inlet and throat in the venturimeter, we can use Bernoulli’s equation and continuity equation.

### Given dataTo find the pressure difference between the inlet and throat in the venturimeter, we can use Bernoulli’s equation and continuity equation.

### Given 

- **Inlet diameter (D₁)** = 150 mm = 0.15 m
- **Throat diameter (D₂)** = 75 mm = 0.075 m
- **Height difference (\( h \))** = 225 mm = 0.225 m
- **Venturimeter constant (Cₓ)** = 0.96
- **Specific gravity of petrol** = 0.78 → **Density of petrol** \( \rho \) = 0.78 \(\times\) 1000 = 780 kg/m³
- **Flow rate (\( Q \))** = 0.029 m³/s

We need to find the **pressure difference (ΔP) between the inlet and the throat**.

### Step 1: Calculate the velocities at the inlet and throat using the continuity equation.

The continuity equation states:
\[
Q = A_1 V_1 = A_2 V_2
\]
where \( A_1 \) and \( A_2 \) are the cross-sectional areas at the inlet and throat, respectively, and \( V_1 \) and \( V_2 \) are the velocities at the inlet and throat.

The area of a circular section is given by:
\[
A = \frac{\pi D^2}{4}
\]

1. **Calculate \( A_1 \) and \( A_2 \):**

   \[
   A_1 = \frac{\pi \times (0.15)^2}{4} = 0.01767 \, \text{m}^2
   \]
   \[
   A_2 = \frac{\pi \times (0.075)^2}{4} = 0.00442 \, \text{m}^2
   \]

2. **Calculate \( V_1 \) and \( V_2 \):**

   \[
   V_1 = \frac{Q}{A_1} = \frac{0.029}{0.01767} = 1.641 \, \text{m/s}
   \]
   \[
   V_2 = \frac{Q}{A_2} = \frac{0.029}{0.00442} = 6.56 \, \text{m/s}
   \]

### Step 2: Apply Bernoulli's equation between the inlet and throat

Using Bernoulli's equation, including the height difference, we have:
\[
P_1 + \frac{1}{2} \rho V_1^2 + \rho g z_1 = P_2 + \frac{1}{2} \rho V_2^2 + \rho g z_2
\]
where:
- \( P_1 \) and \( P_2 \) are the pressures at the inlet and throat,
- \( V_1 \) and \( V_2 \) are the velocities at the inlet and throat,
- \( z_1 \) and \( z_2 \) are the heights of the inlet and throat, with \( z_2 - z_1 = h \),
- \( g \) is the gravitational acceleration (9.81 m/s²).

Rearrange to find \( \Delta P = P_1 - P_2 \):
\[
\Delta P = \frac{1}{2} \rho (V_2^2 - V_1^2) + \rho g h
\]

Substituting the values:

1. **Calculate the dynamic pressure term**:
   \[
   \frac{1}{2} \rho (V_2^2 - V_1^2) = \frac{1}{2} \times 780 \times (6.56^2 - 1.641^2) = 14809.4 \, \text{Pa}
   \]

2. **Calculate the hydrostatic pressure term**:
   \[
   \rho g h = 780 \times 9.81 \times 0.225 = 1715.4 \, \text{Pa}
   \]

3. **Total pressure difference** \( \Delta P \) (considering the Venturimeter constant \( C_x = 0.96 \)):
   \[
   \Delta P = C_x \left(14809.4 + 1715.4\right) = 0.96 \times 16524.8 = 15844.6 \, \text{Pa}
   \]

### Answer:
The pressure difference between the inlet and throat is approximately **15,844.6 Pa**.

---

Would you like further explanation, or have any questions?

### Related Questions
1. How does the height difference affect the pressure drop in a Venturimeter?
2. What are the limitations of using the Venturimeter constant (Cₓ) in real-life applications?
3. How would the pressure difference change if a denser fluid were used?
4. Can a Venturimeter be used horizontally, and how would this affect the calculations?
5. Why is the throat diameter smaller than the inlet in a Venturimeter?

### Tip
Remember, specific gravity relates the density of a fluid to water, which is useful in converting it directly into density (kg/m³) by multiplying by 1000.

A venturimeter has its axis vertical, the inlet and throat diameters being 150 mm and 75 mm respectively. The throat is 225 mm above inlet and venturimeter constant = 0.96. Petrol of specific gravity 0.78 flows up through the meter at a rate of 0.029 m³/s. Find the pressure difference between inlet and throat.

Find the pressure difference between the inlet and throat of a vertical venturimeter with specific dimensions, fluid properties, and flow rate. This problem involves applying Bernoulli’s equation, continuity equation, and fluid density calculation. Suitable for undergraduate engineering students.

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