Venturi & Orifice Meter

1. INTRODUCTION

As we learned in the Impact Jet lab, the speed of a moving jet of water is proportional to the water’s velocity.  This is evident in a turbine flow meter, where the impact force of the water on the blades sets the rotor in motion.  Once there is a steady rotational speed, we can say the speed is proportional to the velocity of the fluid.  Measuring the flow rate of a system would be also be important for doctors who want to understand a patients’ blood flow in their circulatory system. For the purposes of this lab, we will study the flow rate within a pipe where the volume is the amount of fluid each second that passes through a cross-sectional area of that pipe.  It will be necessary to assume uniform flow of the fluid within the pipe, the flow rate, which is known as Q.  The flow rate is proportional to the velocity of the moving fluid, V and the formula is:

Q=V₁A₁=V₂A₂                                                        (1)

Where 1 and 2 are varying locations within the system. 

This lab will investigate the venturi meter and orifice meter, both of which indirectly measure flow rate.  They measure the flow rate by means of a pressure change.  The venturi meter has a converging section of pipe that causes the fluid’s velocity to increase, which creates a drop in the pressure.  The orifice meter utilizes an orifice plate that contains a small hole, which also increases the fluids velocity.  We will also take readings from a rotameter, which directly measures flow rate using a floatation device and that has a scale for pressure printed on it.

Since we will be only recording the pressure for the three devices, we will need to relate the pressure to velocity.  We will employ the Bernoulli equation, which is for steady, incompressible flow, without viscous effects, in the control volume.

$\frac{P_1}{\rho }+ \frac{V_1^2}{2}+g{z}_1=\frac{P_2}{\rho }+ \frac{V_2^2}{2}+g{z}_2$

   (2)

Where g equals the force of gravity, z equals the height, P equals the pressure, and

$\mathit{\rho }$

equals the density of water.

We are interested in the actual flow rate and ideal flow rate. We will need the discharge coefficient to account for the viscous losses that we assumed were negligible. The discharge coefficient is the ratio of actual to theoretical. The discharge coefficient for a venturi meter is around 0.9 and the orifice meter is around 0.6.  The venturi meter has a larger discharge coefficient because it contains a gradual change in pipe diameter whereas the orifice plate contains a sudden and drastic change in diameter where the fluid is flowing.

2. THEORY

The Venturi effect illustrates a reduction in fluid pressure as the fluid flows through the convergence of the pipe.  As the fluid flow through the pipe, the fluid’s velocity will increase to satisfy the equation of continuity.  The pressure will simultaneously decrease to satisfy the conservation of energy equation.  The equation for the pressure drop caused by the Venturi effect is Bernoulli’s principle and the equation of continuity.  The orifice meter has a similar pressure drop.  We will measure how long it takes the bucket system to fill with 30 lbs of water.  The time it takes to accumulate the known mass is known as the mass flow rate that is calculated as follows:

$\dot{m}=\frac{∆m}{∆t}$

                             (3)

We are also interested in the volumetric flow rate so that we can compare this value to the theoretical flow rate to verify our results. The volumetric flow rate is calculated by dividing the mass flow rate by the density of water.   The actual flow rate is shown:

$Q_{\mathit{actual}}=\frac{∆m}{\rho ∆t}$

                           (4)

Since we will be recording the height difference from the manometers, we can relate the height difference to the pressure drop as follows:

$P_1–P_2\mathrm{ }=\mathrm{ }{\rho }_{w}g\mathrm{ }(h_1–h_2)$

  (5)

Where h is the height of the manometer at the varying sections of the pipe, P1 and P2 are the pressure difference, g is the acceleration of gravity, and

$\rho$

is the density of water. The ideal flow rate is found using Bernoulli’s equation (Eq 2) and by using the fact that the pipe is horizontal.  Therefore, we can neglect the forces of gravity and the height:

$\frac{V_1^2}{2}+\frac{P_1}{\rho }=\frac{V_2^2}{2}+\frac{P_2}{\rho }$

 (6)

The ideal flow rate considers the ratio of the different sections of the pipe and we can substitute this value into equation 6:

$Q_{\mathit{ideal}}=V_2{A}_2=A_2\sqrt{\frac{2(P_1–P_2)}{\rho (1–\frac{d^4}{D^4})}}$

    (7)

The discharge coefficient is the ratio of the actual flow rate to the ideal flow rate:

$C_d=\frac{Q_{\mathit{actual}}}{Q_{\mathit{ideal}}}$

 (8)

The discharge coefficient is a dimensionless number that we use to identify the pressure loss behavior of different nozzles or orifices in the types of fluid systems under investigation.  As mentioned earlier, the discharge coefficient accounts for the frictional losses that we assumed were negligible.  We will need the discharge coefficient, Cd, to compare it to the calculated Reynolds number:

$\mathit{Re}=\frac{\mathit{\rho VD}}{\mu }$

 (9)

Where ρ is the density of the fluid, V is the velocity, D is the diameter, and µ is the viscosity.  We can expect a Re> 4000 because the flow is turbulent.  The velocity of the fluid for both the orifice and venturi meters and can be seen in the Appendix section of this report.

3. EXPERIMENTAL APPARATUS AND PROCEDURE

(3)