PLAIN AND CIVIL: EXAMPLE 4.1.1. ORIFICE WITH PIEZOMETER

Tuesday, 19 March 2019

The 50mm-diameter orifice is with $\mathbf{C_{c}=0.63}$ and $\mathbf{C_{v}=0.97}$ . The diameter of the pipe is 150mm. The liquid is water and stands at a height of 3m in the piezometer. Compute the following:

Velocity head at the orifice
Discharge at the jet
Diameter of the jet at vena contracta

Using Bernoulli's Energy Equation:

$\frac{v_{1}^{2}}{2g}+\frac{p_{1}}{\gamma }+z_{1}=\frac{v_{2}^{2}}{2g}+\frac{p_{2}}{\gamma }+z_{2}$

Both elevation heads are zero because the pipe placed horizontally. The pressure head at station 2 is also taken as zero because the jet sprays out the atmosphere. These conditions leave the equation as:

$\frac{v_{1}^{2}}{2g}+\frac{p_{1}}{\gamma }=\frac{v_{2}^{2}}{2g}$

With the piezometer level rising to 3m, station 1's pressure head is 3m.

$\frac{v_{1}^{2}}{2g}+3m=\frac{v_{2}^{2}}{2g}$

$\frac{v_{2}^{2}}{2g}-\frac{v_{1}^{2}}{2g}=3m$

With the constant Q equation, we know that:

$A_{1}v_{1}=A_{2}v_{2}$

$\frac{\pi }{4}\left ( 0.15 \right )^{2}v_{1}=\frac{\pi }{4}\left ( 0.05 \right )^{2}v_{2}$

$v_{1}=\frac{1}{9}v_{2}$

Substituting to the previous equation:

$\frac{v_{2}^{2}}{2g}-\frac{\left ( \frac{1}{9}v_{2} \right )^{2}}{2g}=3m$

$\frac{80}{81}\frac{v_{2}^{2}}{2g}=3m$

${\color{Red} \mathbf{\frac{v_{2}^{2}}{2g}=3.0375m}}$

The discharge at the jet is taken as the actual Q (both coefficients are given)

$Q=C_{c}C_{v}Av$

where:

$v_{2}=\sqrt{3.0375\left ( 2 \right )\left ( 9.81 \right )}=7.72\frac{m}{s}$

$Q=0.63\left ( 0.97 \right )\left [ \frac{\pi }{4} \left ( 0.05 \right )^{2}\right ]\left ( 7.72 \right )$

${\color{Red} \mathbf{Q=0.00926\frac{m^{3}}{s}}}$

The diameter of the jet at vena contracta will be computed using the coefficient of contraction:

$C_{c}=\frac{A_{a}}{A_{t}}$

$A_{a}=C_{c}{A_{t}}$

$\frac{\pi }{4}d^{2}=0.63\left [ \frac{\pi }{4}\left ( 0.05^{2} \right ) \right ]$

${\color{Red} \mathbf{d=0.0397m}}$

PLAIN AND CIVIL

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Tuesday, 19 March 2019

EXAMPLE 4.1.1. ORIFICE WITH PIEZOMETER

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