Orifice Coefficient of Discharge Calculation

This problem requires calculating the coefficient of discharge ($C_d$) for a water jet discharging from an orifice, given the orifice diameter, vena contracta diameter, and the coefficient of velocity ($C_v$). The coefficient of discharge relates the actual flow rate to the theoretical flow rate.

Key Formulas

• The relationship between the coefficients is:

  $C_d = C_v \times C_c$

• The coefficient of contraction ($C_c$) is the ratio of the area at the vena contracta ($A_c$) to the area of the orifice ($A$):

  $C_c = \frac{A_c}{A}$

• Since area is proportional to the square of the diameter ($\frac{\pi d^2}{4}$), the formula for $C_c$ becomes:

  $C_c = \left(\frac{d_c}{d}\right)^2$

  where $d_c$ is the diameter at the vena contracta and $d$ is the orifice diameter.

Step-by-Step Calculation

1. Identify the given values:

   • Orifice diameter, $d = 4 \ cm$
   • Vena contracta diameter, $d_c = 3.5 \ cm$
   • Coefficient of velocity, $C_v = 0.98$

2. Calculate the coefficient of contraction ($C_c$):

   $C_c = \left(\frac{d_c}{d}\right)^2 = \left(\frac{3.5 \ cm}{4 \ cm}\right)^2 = \left(\frac{7}{8}\right)^2 = \frac{49}{64}$

   $C_c = 0.765625$

3. Calculate the coefficient of discharge ($C_d$):

   $C_d = C_v \times C_c$

   $C_d = 0.98 \times 0.765625$

   $C_d = 0.7503125$

4. Round the result to two decimal places:

   $C_d \approx 0.75$

The calculated coefficient of discharge is approximately $0.75$, which falls within the provided correct answer range of $0.74$ to $0.76$.