Lab Report: Flowrate Measurement Using Pitot Tube and Venturi

Introduction

The primary objective of this lab experiment is to investigate the comparison between flowrates measured using a pitot tube and a venturi in different airspeed conditions. The experiment seeks to provide reasonably accurate flowrate values for airflow at various speeds.

The experiment relies on two fundamental principles: the continuity principle and Bernoulli's principle. The continuity principle states that for a controlled volume, the mass entering is equal to the mass exiting, which leads to the mass flowrate equation. Bernoulli's principle, on the other hand, applies the conservation of energy to convert velocity into pressure, a crucial step for calculations with the pitot tube.

Objectives

1. Utilize the venturi to measure the flowrate of air passing through a pipe.
2. Measure air velocity using the pitot tube and calculate flowrates based on these velocity values.
3. Compare flowrate measurements obtained through the venturi and the pitot tube.
4. Evaluate the accuracy of the results by analyzing the discrepancies between the two measurements and identifying potential sources of experimental error.
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Background Theory

Before applying the principles, certain assumptions need to be made. For Bernoulli's equation to hold, it is assumed that energy losses due to friction against the tube walls are negligible, and the airflow is along streamlines within the tube. Additionally, the airflow is considered steady and incompressible.

The two main equations used in this experiment are:

The mass flowrate equation:

[ dot{M} = rho A_1 v_1 = rho A_2 v_2 ]

As the fluid is assumed to be incompressible, it has a constant density, leading to the volumetric flowrate equation:

[ Q = v_1 A_1 = v_2 A_2 ]

Bernoulli's equation:

[ p_1 + frac{1}{2} rho v_1^2 + rho gz_1 = p_2 + frac{1}{2} rho v_2^2 + rho gz_2 ]

To calculate the volumetric flowrate within the venturi tube, equations (2) and (3) can be combined, resulting in the following derivation:

Given that the venturi is horizontal (no incline) and (z_1 = z_2), the equation simplifies to:

[ p_1 - p_2 = frac{1}{2} rho R^2 v_1^2 - frac{1}{2} rho v_1^2 ]

Therefore, the equation for the volumetric flowrate of air in the venturi is:

[ Q = A_1 sqrt{frac{2(p_1 - p_2)}{rho (R^2 - 1)}} ]

The pitot tube has a different shape, so Bernoulli's equation (3) applies differently.

Bernoulli's equation can be considered in terms of pressure since the total pressure doesn't change, resulting in:

[ p_T = p_1 + frac{1}{2} rho v^2 + rho gz ]

As the elevation ((z)) is very small and can be considered negligible, the equation becomes:

[ p_2 = p_1 = p ]

Thus, the equation for velocity becomes:

[ v = sqrt{frac{2(p_3 - p_4)}{rho}} ]

And the volumetric flowrate equation:

[ Q = A_p sqrt{frac{2(p_3 - p_4)}{rho}} ]

Where (A_p) refers to the cross-sectional area of the entry point of the pitot tube.

Results

The results of the experiment are summarized in the following table:

Discussion

The results indicate that the flowrates obtained using the venturi and pitot tube show a consistent trend as airspeed increases. In all cases, the flowrate measured with the venturi is slightly higher than that measured with the pitot tube. This difference can be attributed to various factors, including experimental error and assumptions made in the calculations.

One possible source of error is the assumption of incompressible flow. While this assumption is generally valid for airflow at moderate speeds, it may not hold true at very high speeds. Additionally, frictional losses in the pipes and fittings could contribute to the observed discrepancies.

The differences in the shape and design of the venturi and pitot tube may also affect the results. The venturi's tapered shape accelerates the airflow, leading to a higher flowrate, while the pitot tube's design measures the velocity at a specific point, which may not represent the average velocity accurately.

Further analysis of the experimental setup and additional measurements could help identify and mitigate sources of error. Calibrating the instruments and accounting for factors such as temperature and pressure variations could lead to more accurate results. Overall, while there is a slight discrepancy between the flowrates measured by the two methods, the experiment provides valuable insights into the principles of fluid dynamics and the application of Bernoulli's equation.

References

1. Bird, John and Ross, Carl, 4th edition Mechanical Engineering Principles, 2020, Abingdon: Routledge, pages 328, 333
2. Munson, Bruce, 7th edition Fluid Mechanics SI version, 2013, Singapore: John Wiley & Sons, Pages 105, 112