The Pitot Tube

Nowadays, a Pitot tube would be of use as a speedometer on aeroplanes. It is also found useful in industries where velocity measurements are required, where an anemometer may not be the most efficient instrument to use. There are three types of pitot tubes: Pitot tubes, Static tubes, and Pitot-Static tubes.

The simple pitot tube essentially consists of a tube bent at – usually – 90°, with an open end pointing directly towards the fluid flow. As the fluid flows in the tube, it becomes stagnant since there is no direct opening at the other end for it to exit from. As the inert fluid rises, it creates a pressure of its own. This pressure is equivalent to the dynamic pressure, which can be seen as the kinetic energy of the fluid per unit weight.

F.X. Pitot originally used this device to analyze the pressure created by the stagnating fluid at the other end of the tube. He did this by calculating the sum of the dynamic pressure and static pressure. The pressure created by the stagnant fluid (stagnation pressure) is found where the velocity component is zero. Using these principles, it is possible to determine a fluids velocity.

This has manifested itself in the form of the Bernoulli equation. It describes this phenomenon as it states that an inviscid fluid’s increase in velocity is accompanied with a concurrent decrease in pressure or in the fluid’s potential energy. However, this equation works on two assumptions – the first, that the fluid is incompressible; the second, that friction caused by viscous forces are to be considered negligible. The Bernoulli equation can nevertheless be used for compressible flows, but only at low Mach numbers.

In the experiment, we measured the radial velocity profile at a cross-section of a pipe using the Pitot tube. As the Pitot probe is shifted along the pipe, we can record the stagnation pressure and static pressure at that cross-section of the pipe. The velocity of the inviscid fluid can now
be found using both pressures.

The smaller the diameter of the Pitot tube is, the more accurate the flow velocity. If the Pitot tube has too large a diameter, results for the fluid’s flow could be inaccurate due to the difference in velocity between the fluid against the tube walls and the fluid in the middle of the tube.

Background and Theory

One of the most fundamental equations in the field of fluid mechanics is the Bernoulli Equation:

As stated in the introduction, this equation is based on the assumption that the density and velocity of the fluid are constant.

If we then multiply both sides of the Bernoulli equation (eq.1) by the denominator density (), the result is:

We can assume, for the Venturi tube, that , as it is placed on a horizontal surface and therefore does not change height. This means that height stays constant. Subsequently, eq. 2 can be rephrased into:

At this point, a number of things can be done to the Bernoulli equation. It is possible to make use of the relationship found in the Continuity equation, which states that the volumetric flow rate – the volume of fluid passing a point in the system, per unit time – is equal to the product of the cross-sectional area (A) and the average flow of velocity (v):

The substitution is performed as such, and the new equation can be rearranged with respect to :

Here, we can benefit from the basic equation relating volumes (V), mass (m) and density (:

By dividing by time (t) both sides of equation (eq. 6) by end up with:

and therefore:

() = Mass Flow Rate

If we now substitute eq. 6b into eq. 5, the outcome becomes:

Pitot tubes are used to calculate pressure at a certain point. To do this, two points are selected inside the airflow – the first, situated at the entrance of the tube, which is also referred to as the “stagnation point”; the second is situated further away from the tube.

We can apply the Bernoulli equation (eq. 1) to both these points. As they are both at the same level height (z) is constant. Therefore:

The stagnation point is called so because it is the point where there is no air flow, where v = 0. With this new information, we can once more re-arrange the formula to be:

The cross section of the pipe used in the experiment should now be looked at.
Its is represented below:

(Fig. 1)

When r = 0 you will find that velocity reaches a maximum, with the opposite effect when R=0. Taking from the continuity equation (eq. 4) again, and by substituting it into (eq. 6b) we get:

Rewriting this with respect to the flow moving through the circular gives us:

r = radius
= density of air
= increment of mass flow rate
= velocity (at distance r from the centre line)

The formula for the area of a cross sectional pipe is known as:

Substitute (eq. 12) into (eq. 11):

Here, if we could define over , then integrating the above equation would give us . However this is not the case. We must sum up the cylindrical elements where each measurement was taken. This can be written as:

is the measurement at
is the measurement at

Apparatus

Experimental Procedure

1. Record the room temperature from the barometer.

2. Set up the apparatus as shown above in the diagram.

3. Calibrate the electronic manometer used making sure the power supply is turned off.

4. Once the manometer has been calibrated turn on the power supply.

5. In order to take the measurement of the pressure difference across the Venturi tube, the pipe connecting the Pitot tube to the manometer must be sealed.

6. Once the pipe has been sealed, turn on the power supply to obtain a low speed and take the reading.

7. Then maintaining the same flow, seal off the tube connecting the venture with the manometer and open the tube connecting the Pitot tube.

8. Making sure that the Pitot tube is at the lowest height, take the reading.

9. After that increase the height of the pitot tube by 2mm until

10. Once the pitot tube readings has been obtained, close the pitot tube and open the venture tube and change the speed to a medium flow and repeat step 7-9.

11. Once step 10 has been completed, change the flow to a higher speed and repeat the step 7-9 to obtain the final set of data.

Results

Table 1 – High air flow

|Height / z |Radius / r |Pressure / p |Pressure difference |v(ri ) |v(ri)r | |(x10-2 m) |(x 10 -3 m) |(mmH2O) |(∆p ) |(ms-1) |(m2s-1) | |26.1 |13.05 |60.0 |588.60 |31.32 |0.41 | |26.3 |13.15 |66.2 |649.42 |32.90 |0.43 | |26.5 |13.25 |69.4 |680.81 |33.69 |0.45 | |26.7 |13.35 |73.6 |722.02 |34.69 |0.46 | |26.9 |13.45 |83.3 |817.17 |36.90 |0.50 | |27.1 |13.55 |86.7 |850.53 |37.65 |0.51 | |27.3 |13.65 |87.5 |858.38 |37.82 |0.52 | |27.5 |13.75 |87.9 |862.30 |37.91 |0.52 | |27.7 |13.85 |88.5 |868.19 |38.04 |0.53 |

|Height / z |Radius / r |Pressure / p |Pressure difference |v(ri ) |v(ri)r | |(x10-2 m)
|(x 10 -3 m) |(mmH2O) |(∆p ) |(ms-1) |(m2s-1) | |26.1 |13.05 |42.00 |412.02 |26.20 |0.34 | |26.3 |13.15 |46.60 |457.15 |27.60 |0.36 | |26.5 |13.25 |53.90 |528.76 |29.69 |0.39 | |26.7 |13.35 |56.80 |557.21 |30.47 |0.41 | |26.9 |13.45 |59.10 |579.77 |31.09 |0.42 | |27.1 |13.55 |62.30 |611.16 |31.92 |0.43 | |27.3 |13.65 |64.50 |632.75 |32.47 |0.44 | |27.5 |13.75 |65.70 |644.52 |32.77 |0.45 | |27.7 |13.85 |66.30 |650.40 |32.92 |0.46 |

Table 2 – Medium air flow

|Height / z |Radius / r |Pressure / p |Pressure difference |v(ri ) |v(ri)r | |(x10-2 m) |(x 10 -3 m) |(mmH2O) |(∆p ) |(ms-1) |(m2s-1) | |26.1 |13.05 |0.0296 |290.38 |22.00 |0.29 | |26.3 |13.15 |0.033 |323.73 |23.23 |0.31 | |26.5 |13.25 |0.0371 |363.95 |24.63 |0.33 | |26.7 |13.35 |0.0402 |394.36 |25.64 |0.34 | |26.9 |13.45 |0.0405 |397.31 |25.73 |0.35 | |27.1 |13.55 |0.0423 |414.96 |26.30 |0.36 | |27.3 |13.65 |0.0435 |426.74 |26.67 |0.36 | |27.5 |13.75 |0.0441 |432.62 |26.85 |0.37 | |27.7 |13.85 |0.0448 |439.49 |27.06 |0.37 |

Table 3 – Lower air flow

For Table 1,2 and 3:

Refer to appendix 1 for raw set of data.

Refer to appendix 2 to see how radius (r) in the above tables were obtained.

Refer to appendix 3 to see how pressure difference was calculated using the raw data

Refer to appendix 4 to see how the velocity (v(ri)) was calculated with the use of equation 8 from the above derivation.

Venturi Tube

Table 4:

|Flow |VENTURI READING OF PRESSURE (mmH2O) |Pressure difference ()/Pa |Mass flow rate ()/kgs-1 | |Setting | | | | |Low |43 |191.31 |0.015 | |Medium |60 |47.11 | | |High |85 |4.91 |2.37 |

Refer to appendix 5 to see how the mass flow rate was calculated for the venturi tube with the help of equation (6) from the derivation part.

Graphs

References

http://www.grc.nasa.gov/WWW/K-12/airplane/bern.html accessed on 28/12/10 at 12:20

http://www.centennialofflight.gov/essay/Theories_of_Flight/Ideal_Fluid_Flow/TH7G3.htm accessed on 13/11/10 at 17:08

http://gallica.bnf.fr/ark:/12148/bpt6k408489d.image.f354 accessed on 28/12/10 at 13:31

Munson et al, 2009, Fundamentals of Fluid Mechanics, 6th edition, Wiley: Hoboken

http://www.grc.nasa.gov/WWW/K-12/airplane/pitot.html# accessed on 28/12/10 at 13.35