Question:
Water is discharged from the vertical side of a tank through a circular orifice of diameter 2 cm under laminar flow conditions. The water surface in the tank is maintained at a constant level above the center of the orifice. If the fluid jet has a diameter of 1.6 cm at its vena contracta, then the coefficient of contraction is ______ (rounded off to two decimal places).
Water is discharged from the vertical side of a tank through a circular orifice of diameter 2 cm under laminar flow conditions. The water surface in the tank is maintained at a constant level above the center of the orifice. If the fluid jet has a diameter of 1.6 cm at its vena contracta, then the coefficient of contraction is ______ (rounded off to two decimal places).
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Coefficient of contraction is simply the square of the ratio of the jet diameter at the vena contracta to the orifice diameter.
Updated On: Aug 10, 2026
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Correct Answer: 0.64
Solution and Explanation
Step 1: Recall the definition of the coefficient of contraction.
The coefficient of contraction, \(C_c\), is defined as the ratio of the cross-sectional area of the jet at the vena contracta to the area of the orifice opening:
\[ C_c = \frac{A_{jet}}{A_{orifice}} \]
Step 2: Express the areas in terms of diameters.
Both the jet at the vena contracta and the orifice are circular, so:
\(A_{orifice} = \dfrac{\pi}{4}d_o^2\), \(A_{jet} = \dfrac{\pi}{4}d_j^2\)
\[ C_c = \frac{\frac{\pi}{4}d_j^2}{\frac{\pi}{4}d_o^2} = \left(\frac{d_j}{d_o}\right)^2 \]
Step 3: Substitute the given values.
Orifice diameter \(d_o = 2\) cm, jet diameter at vena contracta \(d_j = 1.6\) cm.
\[ C_c = \left(\frac{1.6}{2}\right)^2 = (0.8)^2 = 0.64 \]
Step 4: Sanity check.
Since the jet always contracts to an area smaller than the orifice, \(C_c\) must be less than 1, and 0.64 is consistent with that physical expectation.
\[ \boxed{C_c = 0.64} \]
The coefficient of contraction, \(C_c\), is defined as the ratio of the cross-sectional area of the jet at the vena contracta to the area of the orifice opening:
\[ C_c = \frac{A_{jet}}{A_{orifice}} \]
Step 2: Express the areas in terms of diameters.
Both the jet at the vena contracta and the orifice are circular, so:
\(A_{orifice} = \dfrac{\pi}{4}d_o^2\), \(A_{jet} = \dfrac{\pi}{4}d_j^2\)
\[ C_c = \frac{\frac{\pi}{4}d_j^2}{\frac{\pi}{4}d_o^2} = \left(\frac{d_j}{d_o}\right)^2 \]
Step 3: Substitute the given values.
Orifice diameter \(d_o = 2\) cm, jet diameter at vena contracta \(d_j = 1.6\) cm.
\[ C_c = \left(\frac{1.6}{2}\right)^2 = (0.8)^2 = 0.64 \]
Step 4: Sanity check.
Since the jet always contracts to an area smaller than the orifice, \(C_c\) must be less than 1, and 0.64 is consistent with that physical expectation.
\[ \boxed{C_c = 0.64} \]
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