An orifice having a diameter of 10cm discharging water and the venacontracta formed has a diameter of 9cm. If the coefficient of velocity is 0.96, then what is the coefficient of discharge?a)0.666b)0.777c)0.888d)NoneCorrect answer is option 'B'. Can you explain this answer?

Question

Answer

Understanding Orifice Flow
An orifice is an opening through which fluid can flow. In this case, we need to determine the coefficient of discharge (Cd) using the given parameters.

Given Parameters
- Diameter of orifice (D) = 10 cm
- Diameter of vena contracta (d) = 9 cm
- Coefficient of velocity (Cv) = 0.96

Key Formulas
1. **Area of Orifice (A)**:
$$ A = \frac{\pi D^2}{4} = \frac{\pi (0.1)^2}{4} $$
2. **Area of Vena Contracta (Av)**:
$$ Av = \frac{\pi d^2}{4} = \frac{\pi (0.09)^2}{4} $$
3. **Coefficient of Discharge (Cd)**:
$$ Cd = Cv \times \frac{Av}{A} $$

Calculating Areas
- **Area of Orifice (A)**:
$$ A = \frac{\pi (0.1)^2}{4} = 0.007854 \, m^2 $$
- **Area of Vena Contracta (Av)**:
$$ Av = \frac{\pi (0.09)^2}{4} = 0.006362 \, m^2 $$

Calculating Cd
Now substituting the areas into the Cd formula:
$$ Cd = 0.96 \times \frac{0.006362}{0.007854} $$
Calculating the ratio of areas:
$$ \frac{Av}{A} = \frac{0.006362}{0.007854} \approx 0.809 $$
Now, substituting back into the Cd formula:
$$ Cd = 0.96 \times 0.809 \approx 0.776 $$

Final Result
Rounding to three decimal places, we find:
$$ Cd \approx 0.777 $$
Thus, the coefficient of discharge is approximately **0.777**, which corresponds to option **b**.

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