Comparative Analysis of Airflow Measurement Techniques: Venturi Meter and Pitot Tube

Introduction

The aim of this experiment is to measure the flow of air in a pipe using a venturi and to compare this value with the flow rate calculated from a velocity traverse of the pipe using a Pitot tube. There may be differences in the values obtained as different methods are used for either piece of apparatus. This experiment is based around the concepts of Bernoulli’s principle and continuity.

Theoretical Background

Bernoulli’s principle applies in this experiment as it states that a given area of fast flowing fluids applies more pressure on its surroundings than that of slow flowing fluids.

This concept can be seen in the venturi meter, whereby there is an evident deviation in the pressure of a fluid as it enters from the cylindrical inlet into the throat and back out through the diverging section. The increase in fluid velocity within the converging section occurs in order to satisfy the concept of continuity. This takes place so that the total flow rate remains the same, which is essentially what continuity is; the flow of fluid into a system must be equal to the flow of fluid that exits the system at a given time.

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Since the cross-sectional area of the converging section is smaller than that of the cylindrical inlet, the rate of flow within this region must increase, allowing for the required number of molecules to get through in the specified time.

In order to calculate the flowrate in the Venturi meter, several assumptions must be made.

The flow is assumed to be horizontal (therefore it can be stated that z1= z2 as there is no change in elevation), steady, inviscid and incompressible. In doing so, the Bernoulli equation becomes: (p127)

p1+1/2ρv12 =p2+1/2ρv22 (1)

Where: p=Pressure v=Velocity ρ=Density

Assuming that the profiles of velocity are uniform at either section of the Venturi meter, the continuity equation can be expressed as:

Q = A1v1 = A2v2 (2)

Where: Q=Volumetric flow rate, A=Cross-sectional area, v=Velocity

Equation (2) could then be rearranged so that v2 becomes the subject:

v2= v1A1/A2 (3)

The following step in the calculation of the flow rate in the Venturi meter would be to substitute equation (3) into equation (1), such that the equation is expressed in the following manner:

p1+1/2ρv12 =p2+1/2ρ(v1A1/A2)2

The final step involves rearranging equation (4) so that v1 is made to be the subject of the equation and this is then placed back into the Q= A1v1 form, allowing for the volumetric flow rate to be calculated:

√(2( p1-p2)/(ρ[1-(A2/A1 )² ))

The Venturi meter measures the average flow rate as it can be assumed that everything is constant along the cross-section. In contrast, the Pitot tube measures the velocity at just a certain point whilst the fluid is flowing. Therefore, in order to accurately compare the results from the Venturi meter and Pitot tube, several measurements must be taken at different radii so that an average flow rate can be calculated.

Flow rate can be determined using a Pitot tube as it converts the kinetic energy in a moving fluid into potential energy. Therefore, both stagnation and static pressures must be calculated as the difference between them is used in order to obtain a value for the velocity of the fluid. Similar to the Venturi meter method the initial equation used is the Bernoulli equation [equation (1)], however v2=0 in this case. This is because after the initial momentary motion of the fluid has stopped, the fluid essentially becomes trapped in the Pitot tube and so it remains stationary, therefore velocity is zero. Substituting v2=0 into equation (1) provides the equation:

p2= p1+1/2ρv12 (5)

Where: p1=Static pressure p2=Stagnation pressure

The equation can then be rearranged such that velocity, which is the variable required, becomes the subject:

v1=√((2(p2-p1))/ρ)

Determining the mass flow rate from the velocity traverse is the final calculation needed before the values for the Venturi meter and Pitot tube can be compared. The flow distribution does not depend on the value of the circumference of the pipe cross-section; therefore, the velocity will be the same for any fixed distance ‘r’ from the centre of the pipe where the values of ‘r’ range from 0 to the very edge of the pipe. An equation can be formed for the volumetric flow rate:

δQ = v(r)δA, (7)

Where: v(r)= Velocity at a distance r from the pipe centre and δA= An element of area

The area of the element can be equated to be:

δA = 2πrδr (8)

Where: δr=Thickness at distance r from the pipe centre

Equation (8) is then substituted into equation (7) to formulate a new equation for δQ:

δQ = v(r)2πrδr (9)

As the data values available are discrete and limited in number, the integral must be approximated such that the cylindrical elements from the pipe centre to the edge of the pipe are added together. In the limit there can be infinitely many measurement points, as a result the following integral will be obtained:

lim┬(n→∞)⁡(Q=) 2π ∫_0^R(v(r)rdr)

Discussion

The differences in flow rate measurements can be attributed to several factors, including the assumption of uniform velocity profiles, measurement errors, and the inherent characteristics of each measuring device. The Pitot tube's measurements, being localized, may not fully capture the flow dynamics within the pipe, contrasting with the Venturi meter’s integrated approach.

Conclusion

This experiment underscores the practical application of Bernoulli’s principle and the continuity equation in measuring fluid flow within a pipe. Through comparative analysis of the Venturi meter and Pitot tube measurements, we gain insights into the complexities of fluid dynamics and the considerations necessary for accurate flow rate measurement.