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Created: 2025/02/25 00:32
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# Experiment no.: 08
# Experiment name: Determination of co-efficient of discharge using orifice discharge apparatus
## Objectives
The objectives of this experiment are:
1. To determine static pressure head in the tank and total pressure head in the free jet.
2. To calculate velocity coefficient ($\phi$), contraction coefficient ($\alpha$), and discharge coefficient ($C_d$).
3. To check consistency between the three coefficients.
## Scopes
The scopes of this experiment includes:
1. Understanding flow characteristics through an orifice by determining velocity, contraction, and discharge coefficients.
2. Analyzing the relation between the three coefficients and determining coefficient of discharge with velocity coefficient and contraction coefficient.
3. Understanding energy losses due to frictions and conversion from pressure head to velocity head, and how these coefficients can be used to calibrate the system for practical flow measurement applications.
## Datasheet
## Calculation
The measured value of:
1. Static pressure head, $h_{static} = 9.4\space cm = 0.094\space m$
2. Total pressure head, $h_{pitot} = 9.1\space cm = 0.091\space m$
3. Orifice diameter, $d_{orifice} = 29\space mm = 0.029\space m$
4. Stream diameter, $d_{stream} = 27.2\space mm = 0.0272\space m$
5. For measurement of actual discharge, duration = $7.81\space s$ and volume = $5\space L$
Hence,
Theoretical velocity,
$$
v_{theo} = \sqrt{2gh_{static}}
$$
$$
or,\space v_{theo} = \sqrt{2\times 9.8\times 0.094} = 1.357\space m/s
$$
Stream velocity,
$$
v_{stream} = \sqrt{2gh_{pitot}}
$$
$$
or,\space \sqrt{2\times 9.8\times 0.091} = 1.336\space m/s
$$
Therefore, velocity coefficient,
$$
\phi = \frac{v_{stream}}{v_{theo}}
$$
$$
or,\space \phi = \frac{1.336}{1.357} = 0.985
$$
Now, contraction coefficient,
$$
\alpha = \frac{A_{stream}}{A_{orifice}} = \frac{d_{stream}^2}{d_{orifice}^2}
$$
$$
or,\space \alpha = \frac{0.0272^2}{0.029^2} = 0.880
$$
Now, actual discharge,
$$
Q_a = \frac{volume}{time} = \frac{5}{7.81} = 6.402\times 10^{-4}\space m^3/s
$$
Theoretical discharge,
$$
Q_t = A_{orifice}v_{theo}
$$
$$
or,\space Q_t = \pi(\frac{0.029}{2})^2\times 1.357 = 8.963\times 10^{-4}\space m^3/s
$$
Therefore, coefficient of discharge,
$$
C_d = \frac{Q_a}{Q_t}
$$
$$
or,\space C_d = \frac{6.402}{8.963} = 0.714
$$
## Discussion
In this experiment, three key dimensionless coefficients were determined. These are:
1. Velocity coefficient, $\phi = 0.985$,
2. Contraction coefficient, $\alpha = 0.880$, and
3. Discharge coefficient, $C_d = 0.714$
The velocity coefficient is close to $1$ and falls within the expected range of $0.95$ to $0.99$. It indicates a minimal energy loss meaning that the actual stream velocity is nearly equal to theoretical velocity. The contraction coefficient suggests that the stream cross-section at vena contracta is $88\%$ of orifice area implying significant contraction.
The discharge coefficient should be the product of velocity coefficient and contraction coefficient:
$$
C_{d\_expected} = 0.985\times 0.880 = 0.867
$$
However, the determined value of discharge coefficient is significantly lower at $0.714$. This could be due to several potential errors that are listed below.
### Sources of errors
The potential sources of errors include:
1. **Measurement errors:** Inaccurate measurement of stream diameter can produce a very different value of contraction coefficient. It is possible that we measured the diameter somewhere below the vena contracta where diameter was higher.
2. **Head fluctuations:** While static pressure head was fairly stable; pitot head was fluctuating and its heading can play a role for incorrect values.
3. **Pitot tube positioning:** Incorrect positioning of the pitot tube may have affected velocity measurements.
4. **Energy Losses:** Friction losses not accounted for in theoretical calculations.
5. **Human Error:** Delays in starting/stopping stopwatch or reading of manometer.