Friday, February 7, 2014

Pitot test and calibration part 3

In two precedent articles a DIY pitot test rig has been introduced. Now some relevant aspects of test rig design are examined and a closed form solution for piston sizing is provided. All the files for simulation are free downloadable.
Refer to the following figure for the rig typical layout

P.28.1 Test rig layout
The precedent pneumatic circuit can be realized using every kind of standard valves, tubing and fittings available on the market. No matter what is the material of this components as long as it is undeformable. The test rig on the next picture have been made of stainless steel, in the past the author used plastic drip irrigation components.
P.28.2 Example test rig
 
Every component, except the pressure source , is available at the hardware store . Initial design guidelines excluded the possibility to use an electronic pressure controller. This devices can be directly piloted by a micro-controller to provide a closed loop regulated pressure,as per this pdf page 2, they are the best choice for full automated operation. As requirements for the calibration rig included low cost and manual calibration then this device have been replaced by a simple hand operated piston.
Figure P.28.3 Example piston drawing , link to download
Figure P.28.4 One piston prototype with the “reservoir” valve
Video V.28.4 Exploded view of piston

As the test rig will be suitable for Pitot and Altimeters calibration we should choice the desired maximum altitude and airspeed. Let's pretend that we want to calibrate altimeter up to 3000 meters, so using the barometric altitude formula  we can esteem that our cylinder should be capable to generate 70 kPa starting from air at 101,325 kPa otherwise stated a variation of pressure of -31,325 kPa. Set the maximum calibrated airspeed \(V\) to 50 m/s, using the basic pitot formula we attain a variation of pressure of \(1531  Pa=\frac{1}{2} \rho_{base}V^2\).
Now you should take the higher differential pressure value, typically the altitude differential value will be higher than airspeed differential.
Now that you have the differential value let's increase this value by a 1,5 security factor, in our example case we get 46988 Pa.
At this point we need to esteem the volume of the pneumatic circuit, you need only considerate the volume of the tubing between the piston and the pressure tap. We don't need  any precise calculation of valves internal volume or any other contribute, we are happy with rough value. Let's pretend we're using a tube with an internal diameter radius \(r\) of 3e-3 m and with a length lt of 0,4 m. We can calculate the approximate pressure line volume \(PL=0,0000113044  m^3=  \pi r^2L\)
Note that the piston assembly at figure P28.3 is provided with a knob that can accurately move the plunger forth and back, is possible to increase or decrease the pressure in a very smooth manner. To relief the o-ring from friction is a good idea to lubricate the piston internal surface, if you use a nitrile o-ring a good choice for lubricant is silicone grease.
Let's start assume an internal piston diameter D of 21 mm.
So what is the piston stroke L needed to reach the required pressure?
When the plunger is moved the volume at the disposal of the air into the pressure line is changed, consequently the pressure changes.  Let's assume that this process is adiabatic, no heat is exchanged with the environment.
We define \(V_1\) as the initial air volume and \(V_2\) as the final air volume, the air have a molar mass of 28,96 \(kg/kmol\) and heat capacity ratio http://en.wikipedia.org/wiki/Heat_capacity_ratio  \(\gamma_air\)  of 1,4. \(P_1\) is the pressure of air with volume \(V_1\) and  \(P_2\) is the pressure of air with volume \(V_2\).  To determine piston stroke dimension we calculate the maximum differential pressure that
Under the adiabatic compression assumption and using the gas law we get
$$P_2=P_1(V_2/V_1)^{-\gamma_{air}}$$

Launching adiabatic.sce Scilab file, and imposing an initial guess value of 25 mm for full stroke lenght value, we get \((P_2-P_1)=123256.8 Pa\)

P1 Initial pressure 101325 Pa
P2 terminal pressure 224582 Pa
Maximum differential pressure 123257 Pa
Differential pressure after one knob turn 4371 Pa
P2/P1 2.2

Table T.28.1 adiabatic.sce output

\((P_2-P_1) >> 46988 Pa\) so the selected stroke is suitable for our requirements.
Such a simple model does not account for piston leaking, this problem will not affect calibration as long as the “reservoir” valve is shut off during the pressure measurements. For a comfortable use of the test rig oversize L to compensate for leaks, there will no need to refill the piston with atmospheric air during a basic calibration progress.

Another parameter to set is the plunger screw pitch.  The parameter we should set to size the pitch is the desired pressure variation per knob turn, a too high value can lead to a piston really uncomfortable to be used. A good initial value seems to be 2000 Pa, that will lead to \(5,5 Pa/deg=2000/360\) that lead to choice a piston with an internal diameter around 15mm. That should be sound clear that the air pressure will not increase exactly 5,5 Pa each degree of knob rotation. A piston with 21 mm of internal diameter has been used with no loss of operator comfort. For best comfort use a big knob diameter, the bigger the better.

Figure P.28.5 Top, cylinder diameter vs screwpitch deltap turn constant
Bottom, sizing chart piston stroke vs diameter, deltap constant

Reduction of the internal diameter lead to the need to increase the piston stroke to achieve the previous desired pressure range. A graphical representation of sizing parameters relationship is reported in figure P28.4.
Analytic solution of sizing is implemented in the last part of adiabatic.sce.
A sizing example, using the graphs at figure 28.4 will follow in the next post.
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