Question

If Cv, Cc, and Cd are coefficients of velocity, contraction and discharge respectively, then one of the following equations will hold good

• A. \(C_d = C_v \times C_c\)
• B. \(C_d = C_v / C_c\)
• C. \(C_d = C_v + C_c\)
• D. \(C_d = C_v - C_c\)

Hint:

Fluid Mechanics Orifice Coefficients: Coefficient of Discharge = Coefficient of Contraction $\times$ Coefficient of Velocity ($C_d = C_c \times C_v$).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Fluid dynamics of orifice flow: the coefficient of discharge ($C_d$) is the product of the coefficient of velocity ($C_v$) and the coefficient of contraction ($C_c$).
Key Formula or Approach:
\[ \mathbf{C_d} = \frac{Q_{\text{actual}}}{Q_{\text{theoretical}}} = \left( \frac{A_{\text{actual}}}{A_{\text{theoretical}}} \right) \times \left( \frac{v_{\text{actual}}}{v_{\text{theoretical}}} \right) = \mathbf{C_c \times C_v} \]

Step 2: Detailed Explanation:
In fluid mechanics, orifice flow, and discharge measurement in dairy pipelines:
1. Coefficient of Contraction ($C_c$): The ratio of the area of the jet at the vena contracta ($a_c$) to the area of the orifice ($a$), $C_c = \frac{a_c}{a}$.
2. Coefficient of Velocity ($C_v$): The ratio of actual velocity of jet at vena contracta ($v$) to theoretical velocity ($v_{\text{th}} = \sqrt{2gh}$), $C_v = \frac{v}{\sqrt{2gh}}$.
3. Coefficient of Discharge ($C_d$) (A): The ratio of actual discharge ($Q_{\text{act}} = a_c \cdot v$) to theoretical discharge ($Q_{\text{th}} = a \cdot \sqrt{2gh}$):
\[ C_d = \frac{Q_{\text{act}}}{Q_{\text{th}}} = \frac{a_c \cdot v}{a \cdot \sqrt{2gh}} = \left(\frac{a_c}{a}\right) \times \left(\frac{v}{\sqrt{2gh}}\right) = \mathbf{C_c \times C_v} \]
- (Typical values for a sharp-edged orifice: $C_c \approx 0.62$, $C_v \approx 0.97 \implies C_d \approx 0.60$).

Step 3: Final Answer:
Thus, the equation that holds good is \(C_d = C_v \times C_c\), matching option (A).