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So far, the pneumatic connections on our Pitot-static system were considered perfectly airtight. For a change, in this article we examine typical leakage scenarios and their impact on airspeed and altitude measurements. We will focus on leakage solely in the static system; study of the leakage in static pressure piping is important, since at high airspeeds, deviation in static pressure will lead to airspeed measurement errors. Perhaps in the future we will be able to dedicate some time on leakage on the total pressure port as well. In this article, our approach will be to use a lumped parameters leakage model to get preliminary results on the behaviour of our system under leak conditions. The acquired results shouldn't be held as an absolute truth and this analysis isn't exhaustive. Bulletproof results for a particular setup would require an experimental approach [1].

In a basic setup, the Pitot-static pneumatic system is composed by the static piping and the total pressure piping. Both run from the probe to the pressure sensor. Static piping carries the static pressure \(p_s\) measured by the static pressure ports on the probe circumference and the total pressure piping carries the total pressure \(p_t\) measured at the probe tip, the stagnation point. At this level of analysis we do not need to define to a great detail the specifics of the pneumatic circuit, such as the T-joints. Our main variables will be the pneumatic lines volumes, \(V_s\) for the static pressure pneumatic circuit volume and \(V_t\) for the total pressure pneumatic circuit volume. These volumes are assumed to be constant, in other words, all the piping should withstand operating conditions without geometric deformations in all frequencies. See figure 3 for a typical layout.

Figure 1: Leak Model

Figure 2: Pneumatic circuit of a static pressure measurement unit with a leak

\(q=a\sqrt{\Delta p}\) (1)

The flow coefficient \(a\) depends on fluid properties, state and orifice geometry. A high \(a\) value means that the airflow can move relatively freely through the hole. Low \(a\) values mean that the air is restricted while moving through the hole, for example, when the hole has a very small section. Keep in mind that \(a\) is used to simplify the expression, where in fact, in a standard flow equation [2] \(a\) is the product of several terms. In this model, when \(\Delta p\) is halved, the flow rate is divided by four.

Let's examine how the static pressure measurement is affected by a leak in the pipe which routes pressure to a pressure sensor port. Refer to Figure 2. We indicate as \(p_m\) the pressure at the sensor port. The atmospheric pressure around the measurement equipment is \(p\). The leak has been modeled as a single orifice and thus it's an ideally concentrated pressure loss. Similarly, the static pressure port inlet itself is represented as a concentrated pressure loss as well. Distributed pressure losses in the pipe are neglected. Note that we are performing calculations on an equilibrium point: we have modeled a quasi-steady system. This is evident from the fact that volumes are not used. Hence, the air flow \(q\) escaping through the leak orifice is the same as the one which enters the probe static port, using equation (1)

$$q_{s}=q_{l}$$

\( a_{s} \sqrt{p_s-p_m}= a_{l} \sqrt{p_m-p} \) (2)

Solving for \(p_m\)

\(p_m=\frac{1}{1+(\frac{a_{s}}{a_{l}})^2}\Big( p+\Big(\frac{a_{s}}{a_{l}}\Big) ^2 p_s \Big)\) (3)

This formula will yield correct results, as long as we can get the values of \(a_s\) and \(a_l\), the equivalent flow coefficients for the static and leak ports. Equation (3) highlights how the ratio \((a_{s}/a_{l})\) affects the pressure measurement \(p_m\).

In the case where \(a_{s}/a_{l} \gg 1\), the air meets much greater resistance going through the leak hole than going through the static port hole. This is the situation we should aim for.

As a counter-example, let's consider the case where \(a_s=a_l\). \(p_m\) will then be the arithmetic mean of \(p\) and \(p_s\). In physical terms, there is a significant leak flow escaping through a hole comparable to the static pressure orifice, also called a "huge leak". At the sensor port, we measure a pressure value which is far from the desired \(p_s\) value.

Now, let's introduce some uncertainty in the static pressure value. During typical flight conditions, the static pressure present at the static ports \(p_s\) can deviate from the actual atmospheric pressure \(p\), for example in non-zero angle-of-attack conditions. In short, \(p_s = p+\Delta p\), where \(\Delta p\) is the static pressure error. A conservative, but not exaggerated error sizing could be \(\Delta p=1/2\rho V^2 \frac{2}{100}=2\%\bar{q}\) [3]. We will use \(\Delta p=2\%\bar{q}\) as a demonstration value in our calculations, but its value should be adapted to each installation.

In turn, in basic air data applications, the pressure altitude above sea level \(h_p\) is calculated using the static pressure value, so the \(\Delta p \) error will introduce an altitude error \(\Delta h_p\) [3].

Realistically, with \(a_s \gg a_l\) we have

\begin{equation}\label{4}

\begin{split}

\lim_{(r^2) \to \infty} p_m =& \frac{p}{1+r^2}+\frac{r^2}{1+r^2}p_s\\

=& \frac{p}{1+r^2}+\frac{r^2}{1+r^2}p+\frac{r^2}{1+r^2}\Delta p \\

=& p+\Delta p \\

=& p_s

\end{split}

\end{equation}

Equation 4

where, \(r=a_s/a_l\)Combining the static pressure error expression with equation (4) unveils the relationship between measurement uncertainty and current vehicle airspeed \(V\).

\begin{equation}

\begin{split}

\epsilon_{p_m}& =p_m-p \\

&=\frac{r^2}{1+r^2}\Delta p\\

&=\frac{r^2}{1+r^2}(1/100\rho V^2)

\end{split}

\end{equation}

Equation 5

As we can see, the static pressure measurement deviation depends upon the airspeed and the ratio of the leak coefficients. The deviation expression of the measurement relative to dynamic pressure value is

\begin{equation}

\begin{split}

\frac{\epsilon_{p_t}}{\bar{q}}& =\frac{p_m-p}{\bar{q}} \\

&=\frac{r^2}{1+r^2}\frac{\Delta p}{\bar{q}} \\

&=\frac{r^2}{1+r^2}\frac{2}{100} \\

\end{split}

\end{equation}

Equation 6

Figure 3: Basic Pitot-static pneumatic connection

\hline Scenario & Leak Point & From & To \\\hline

1 & 1 & p_s & p \\\hline

2 & 2 & p_s & p_c \\\hline

3 & 3 & p_t & p_c \\\hline

4 & 4 & p_t & p \\\hline

5 & 5 & p_t & p \\\hline

\end{array}

Table 1: Basic Pitot-Static pneumatic connection

The asymptotic value of \(\epsilon_{p_t}/{q}\) in terms of \(r\) is 2/100: if no leak is present then the measurment error is the static pressure error itself.

Figures 4 and 5 are plots of the deviation as a function of \(r\) and airspeed \(V\). The related Matlab script file can be found here.

Figure 4: Deviation Contours

Figure 5: Deviation Surface

Since the static pressure is used for the calculation of barometric altitude, we can project the static pressure error to the altitude calculations. In case of leakage at the static line we know that:

- The measured pressure altitude is higher than the real pressure altitude.
- The deviation of the measurement is constant for a given altitude.
- The deviation increases with airspeed. However, in [1, p.23] it is stated that the airspeed dependence is low, which agrees with our formulas. At a low angle of attack \(\epsilon_{p_s}\) will have a very small value. Consequently, at low speeds and at low angles of attack this dependence become insignificant.
- The magnitude of the deviation of \(p_s\) decreases for increasing altitude, since at higher altitudes \(\rho\) decreases.
- The magnitude of the deviation of \(p_s\) decreases for increasing temperatures, since at higher temperatures \(\rho\) decreases.

The flow coefficients ratio \(r\) must be estimated experimentally. Equation (3) can be solved for \(r\) directly but to determine the value of \(a_l\) it is necessary to measure the flow rate and then use equation (1) to calculate it as \(a_l=q/ \sqrt{p_m-p}\). A method to measure such leak flows using related apparatus is described in [1]. Summarily, the mass flow rate can be found by calculating the total mass of air present in a known volume at two different time instances and then approximating the time derivative of their difference. Using a time interval of one second the calculation becomes \(q=\frac{M_{t=1}-M_{t=0}}{1-0}=[kg/s]\).

The mass of air present in a container can be calculated with the use of an equation of state for the air. To get the air density from an equation of state we need to know the pressure and the temperature of the fluid during the measurements.

Refer to Figure 3. The possible leakage points have been numbered from 1 to 5. Three leak points, labeled respectively 1,4 and 5, are located outside the body of the aircraft, which for convenience we call the cabin. The remaining leak points labeled 2 and 3 are located inside the cabin. In this article we deal only with the static pressure part of the circuit, in particular leak points 1 and 2. The most complex is the scenario number 5, where there is a leak between the static and total lines. This deserves a dedicated article.

The Pitot-static electronic sensor (also known as a differential pressure sensor) is usually placed in the cabin. Let's pretend that our aircraft cabin is pressurized at a pressure value of \(p_c\), which is not typical in RC applications. In the case of a non-pressurized aircraft we assume that \(p_c=Kp_s\), where \(K\) is an unknown coefficient but expected to be near unity. Also, typically \(p_s\approx p\). Having defined all pressure variable, we can fill Table 1 which reports all the different leakage scenarios.

If the cabin is pressurized we have to consider at what altitude we are flying. Most often than not, pressure altitude is higher than cabin pressure altitude and a leakage in the cabin will cause a measurement with a lower than true altitude value.

To handle this leakage scenario, we re-write equation 3 using \(p_c\) instead of \(p\).

\(p_m'=\frac{1}{1+(\frac{a_{s}}{a_{l}})^2}\Big( p_c+\Big(\frac{a_{s}}{a_{l}}\Big) ^2 p_s \Big)\) (7)

\begin{split}

\epsilon_{p_s}' & =p_m-p \\

&=\frac{1}{1+(\frac{a_{s}}{a_{l}})^2}\Big( p_c+\Big(\frac{a_{s}}{a_{l}}\Big) ^2 p_s \Big)-p\\

\end{split}

Equation 8

In this case, the pressure deviation magnitude depends on cabin pressure. If cabin pressure is higher than the external pressure then we will read a pressure altitude that is lower than the actual pressure altitude. The relationship with airspeed is weak like in the first scenario.

It is good practice to verify the proper operation of the Pitot-static system when the hosting vehicle is on the ground. If we want our airplane to comply to FAR 23.1325 regulations, the following test should be passed. This excerpt is from this web site.

Unpressurized airplanes. Evacuate the static pressure system to a pressure differential of approximately 1 inch of mercury or to a reading on the altimeter, 1,000 feet above the aircraft elevation at the time of the test. Without additional pumping for a period of 1 minute, the loss of indicated altitude must not exceed 100 feet on the altimeter.

In an effort to comply to this specification, we can test the leakage of our Pitot-static system by taking pressure measurements \(p_i\). Refer to figure 6.

Figure 6: Ground test pneumatic circuit

After connecting the Pitot-static circuit with our leak testing equipment, we have an overall, cumulative air vessel with total volume \(V\), which is the sum of the volume of the static circuit \(V_s\) and the connection tubes and other internal volumes \(V_i\). In RC setups, \(V\) usually is under 0.25e-3~\(m^3\) and essentially a very small mass of air \(M\) is contained inside of the test circuit. Using the ideal gas law we can relate the variation in pressure to the variation of the entrapped air mass.

Figure 7: Contour plot of pressure loss percentage

\begin{equation}

\begin{aligned}

p_i&=\rho R^*T \\

&=M/V R^*T \Rightarrow\\

\Delta p_i&=\Delta M/V R^*T

\end{aligned}

\end{equation}

where \(R^*=R/MW_{air}\).

We start our test with a maximum pressure differential of 450 Pa, equivalent to about 100 feet of altitude loss. We normalize to the initial differential pressure measured and acquire \(P_l\%\), the percentage pressure loss. Refer to the plot in Figure 8 which simulates the experiment for various leak coefficients and volumes. Pressure loss exceeding 100\% represents a test failure.

Figure 8: 3D Plot of pressure loss percentage

The last piece of the puzzle is the estimation of the coefficient \(a_s\) of equation 2. There are standard formulations of (2) the case of a single orifice [2]. The advantage of using a standard formula for the mass flowrate \(q_m\) is that bibliography is relatively abundant ([5],[2]).

\(q_m=\frac{C}{\sqrt{1-\beta^4}}\epsilon\frac{\pi}{4}d^2\sqrt{2\Delta p \rho_{air}}\) (10)

You can see that equation (1) has the same form as (10). We leave to reference [2] the details about the theory and variables definition but we should mention the \(C\) coefficient, which expresses the ratio between the actual flowrate and the theoretical flowrate. Let's assume that its values are in the range \(C\in(0.5, 1)\). Typical values in standard conditions for the rest of the parameters are

- \(d=2e-3~mm\), the diameter of the static port hole.
- \(\epsilon=0.9999999\)
- \(\beta=1\), the pipe hole to the atmosphere is equal to the internal static pipe diameter.

Manipulation of the flow equation (10) leads to \(a_s=2.45e-6\). Combining that value of \(a_s\) with acceptable values of \(a_l\) from Figure 7 we observe that \(r\in(10e+2,10e+3)\). As a comparison, the constraint on leakage from FAR code is more strict than the bounds resulting from equation (5).

Due to the low volumes involved in RC miniature test equipment, the leak tests require a high degree of airtightness. A large static pressure port diameter seems to mitigate the leak impact. Not all values of \(r, a_l, a_s\) are physically acceptable; \(r\) should be greater than unity. Valid \(a_s\) and \(a_l\) values will not produce any negative root arguments into the pressure loss percentage calculation.

*Bibliography and Further Reading*

[1] AFFTC, Investigationon Pitot and Static System Leak Effects, Edwards Air Force Base, California, 1976

[2] R.W. Miller, Flow Measurement Engineering Handbook, Mc-Graw Hill, 1996

[3] NASA, Technical Note D-1724, NASA, 1963

[4] Laksana Guntur Harus, Maolin Cai, Kenji Kawashima and Toshiharu Kagawa , Systems Modeling and Simulation, Chapter "Dynamical Model of Leak Measurement by Pneumatic Pressure Change", Springer, 2007

[5] Arian Control and Instrumentation, Theory overview of flow measurement using differential pressure devicesbased on ISO-5167 standard, Technical Note 12, Revision b.