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Tuesday, December 22, 2015
Friday, December 18, 2015
PitotStatic Leak
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So far, the pneumatic connections on our Pitotstatic 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 Pitotstatic 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 Tjoints. 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 shows a concentrated parameters leak model. Similarly to [1], the leak behavior is modeled as a flow between a vessel and the outside environment through a single hole. The air flow through a hole under a differential pressure of \(\Delta p\) will have a flow rate [2].
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 quasisteady 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)
$$q_{s}=q_{l}$$
Solving for \(p_m\)
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 counterexample, 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 nonzero angleofattack 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}
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_mp \\
&=\frac{r^2}{1+r^2}\Delta p\\
&=\frac{r^2}{1+r^2}(1/100\rho V^2)
\end{split}
\end{equation}
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_mp}{\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}
\begin{array}{cccc}
\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}
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.
Inspecting the equations 5 and 6 we observe that \(p_m < p_s \).
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 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 \(a_l=q/ \sqrt{p_mp}\). 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 \(q=\frac{M_{t=1}M_{t=0}}{10}=[kg/s]\).
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 Pitotstatic 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 nonpressurized 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 rewrite equation 3 using \(p_c\) instead of \(p\).
\begin{split}
\epsilon_{p_s}' & =p_mp \\
&=\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}
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 Pitotstatic 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 Pitotstatic system by taking pressure measurements \(p_i\). Refer to figure 6.
After connecting the Pitotstatic 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 0.25e3~\(m^3\) 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.
\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.
Inspecting Figure 8, we find out that with a fixed leak coefficient we get better results with bigger working volumes and thus record a lower pressure variation. Naturally, the test equipment contributes to the volume \(V\) and, to some point, to the total leakage. A contour graph of the same experiment can be found in Figure 7.
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]).
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
Manipulation of the flow equation (10) leads to \(a_s=2.45e6\). 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, McGraw Hill, 1996
[3] NASA, Technical Note D1724, 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 ISO5167 standard, Technical Note 12, Revision b.
So far, the pneumatic connections on our Pitotstatic 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 Pitotstatic 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 Tjoints. 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 quasisteady 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)
$$q_{s}=q_{l}$$
\( a_{s} \sqrt{p_sp_m}= a_{l} \sqrt{p_mp} \) (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 counterexample, 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 nonzero angleofattack 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_mp \\
&=\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_mp}{\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 Pitotstatic 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 PitotStatic 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 \(a_l=q/ \sqrt{p_mp}\). 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 \(q=\frac{M_{t=1}M_{t=0}}{10}=[kg/s]\).
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 Pitotstatic 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 nonpressurized 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 rewrite 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_mp \\
&=\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 Pitotstatic 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 Pitotstatic system by taking pressure measurements \(p_i\). Refer to figure 6.
Figure 6: Ground test pneumatic circuit
After connecting the Pitotstatic 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 0.25e3~\(m^3\) 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
 \(d=2e3~mm\), 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 \(a_s=2.45e6\). 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, McGraw Hill, 1996
[3] NASA, Technical Note D1724, 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 ISO5167 standard, Technical Note 12, Revision b.
Friday, November 13, 2015
Flow Rate in a Circular Pipe
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In this short article we will make an effort to calculate the flow rate of a volume of air via \(n\) multiple velocity measurements at a selected pipe section. Velocity profiles are useful when evaluating small DIY wind tunnels and nozzle designs. Although this article doesn't reflect any particular technology, the reader can imagine that velocity measurements themselves are carried out with a Pitot tube or with a Pitot rake placed at the section in question. There are different industrial standards that cover the topic, such as the ISO 3966: Measurement of fluid flow in closed conduits  Velocity area method using Pitot static tubes. In the following test we will not follow any particular standard. Still, those standards are interesting readings.
In this short article we will make an effort to calculate the flow rate of a volume of air via \(n\) multiple velocity measurements at a selected pipe section. Velocity profiles are useful when evaluating small DIY wind tunnels and nozzle designs. Although this article doesn't reflect any particular technology, the reader can imagine that velocity measurements themselves are carried out with a Pitot tube or with a Pitot rake placed at the section in question. There are different industrial standards that cover the topic, such as the ISO 3966: Measurement of fluid flow in closed conduits  Velocity area method using Pitot static tubes. In the following test we will not follow any particular standard. Still, those standards are interesting readings.
We are interested in calculating the volumetric flow rate, \(Q\). The volumetric flow rate is the volume \(\Delta V\) that flows across a predefined surface in an arbitrary interval of time \(\Delta t\), \(Q=\frac{\Delta V}{\Delta t}\).
Figure 1  Sectioned pipe and flowing fluid
Refer to Figure 1; A section perpendicular to the pipe axis defines a circular surface of area\(s\). On the assumption that the axial velocity at every point of the section is constant and equal to \(V_c\), then \(Q=V_c\cdot s\). An animation of the system at hand can be seen in the following video
Video 1  Volumetric Flow rate
However, in the real world the velocity is not constant across the section. The interaction of the fluid with the pipe surface generates a nonnegligible velocity gradient. In general, it stands that \(Q=\int v(x,y,)dt\). Bibliography comes handy and we will use a powerlaw velocity profile. With this model, we will build a reference pipe numerical model, which represents the pipe and the flow rate which we will measure. The realworld pipe is replaced by a numeric simulation. The commonly used model does not separate the viscous sublayer nor the transition layer; in fact it is discontinuous at the surface of the pipe and on the symmetry axis. Since we are not treating a particular realword case we're not concerned about the uncertainty introduced by the powerlaw model. Details about the viscous sublayer and the transition layer can be found in this link.
We end up using a fully developed turbulent axial flow model at high Reynolds number. The viscous layer is expected to be thin in typical cases (like a hair) and it is not calculated in this example. You can find a Scilab script in this file that will calculate the powerlaw velocity profile for you. The result is visible in the next figure.
Figure 2  Velocity profile, powerlaw approximation
The resulting velocity profile is dependent solely on the distance from the axis. You can see that the velocity gradient, as expected, is high near the pipe wall and low along the center line.
Now it is time to measure the flow rate. Refer to Figure 3. We assume having access to velocity measurements at \(n\) stations. Each station is at a different distance \(r(n)\) from the pipe center line. For the sake of simplicity, we divide the radius in \(n\) equal segments. For each segment we measure the velocity \(v(n)\) at its midpoint.
Figure 3  Velocity measurements position into the measurement section
We make the assumption that the velocity remains constant along each segment of the radius and, by extension due to axial symmetry, over the whole corresponding annular ring. The total volume that passes through the ring surface is the product of velocity with the annulus area. Iterating the calculation for all the segments we calculate the volumetric flow rate as \[Q=\sum_{i=1}^n v(n)\pi(r_{ext}^2(n)+r_{int}^2(n))\], where
 \(i\) is the annular ring index
 \(v(n)\) is the velocity of \(i\) ring
 \(r_{ext}\) is the outer ring radius of \(n\) ring
 \(r_{int}\) is the inner ring radius of \(n\) ring
Figure 4  Velocity profile reconstructed by 10 measurements
Figure 4 visualizes the velocity profile as reconstructed by our measurements. It is evident from the profile steps that we have a coarse approximation. There is a simple way to get better flow rate measurements: we can use the model to calculate the velocity profile within each annulus. In such a way, using our simulation data, we will get exactly the flow rate produced by our reference model. Driven by this result, we can even try to use only the data from one single measurement, for example at the center line. Doing so we will bring the very same result predicted by our reference model. Unfortunately the accuracy is constrained to the ability of the model to capture the real world with all accuracy. For that reason, in actual applications, in general, the accuracy should be expected to be higher when using multiple simultaneous measurements rather than model interpolations. But let's leave this problem to the professionals, for now.
In this article we have explored the basics of volumetric flow rate measurement with the velocityarea method. This topic is quite relevant to basic air data processing because it shows the impact of the flow pattern on simple calibration instruments like small size / DIY size wind tunnels.
Friday, October 9, 2015
Density and Total Air Temperature
Commonly the air density can be calculated with an equation of state or with an interpolated bestfit curve. In both cases, we have to deal with an equation of the form \(\rho=f(T,p,RH, others)\). Let's make things a bit simpler and start with a simple relationship between density and temperature:
$$\rho=f(T,Constant)$$
Temperature in the SI is measured in degrees Kelvin, but many equations of state may also be given with temperature expressed in \(Celsius=Kelvin – 273.15\). Quoted from BIMP: "The kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water.". The thermodynamic temperature is correlated to the state of a thermodynamic system and according to the third law of thermodynamics it defines an absolute temperature scale. For dry air, the ideal gas equation of state is
$$\rho=\frac{p}{R_{specific}T}$$where
\(\rho\) is the air density in \(kg/m^3\),
\(p\) is the absolute pressure in \(Pa\),
\(T\) is the absolute temperature in \(K\) and
\(R_{specific}=287.058\frac{J}{kgK}\) is the specific gas constant for air.
The main advantage of the equation of state is that it provides a closed form solution, which in turn can be incorporated in other computations. This is a very convenient property for analysis purposes. Let's proceed with a density calculation. Refer to figure 1; our idealized test setup is a room with two thermometers and a fan. The first thermometer measures the temperature of the air in the room at a point far away from the fan effects, while the second one is subject to the direct airflow from the fan.
Figure 1: Example layout
Prior to any calculation let's introduce the stagnation temperature, which indicates the temperature at a stagnation point within a flow field. At the stagnation point the local flow velocity is zero. We find a stagnation point on a Pitot tube tip or in a total air temperature probe.
In our example, the air from the fan which hits the thermometer is brought to rest on the surface of the sensing element. Consider an infinitesimal volume of the overall air stream traveling at a certain speed. This small volume has a corresponding kinetic and gravitational potential energy. If we assume that this volume does not exchange any form of energy with the surroundings then this energy must be converted when one of its forms is eliminated. In the process where the air slows down, kinetic energy is converted into heat, which causes a temperature rise. Assuming an adiabatic process we can describe the enthalpy at stagnation
$$h_s=h+\frac{V^2}{2}$$where the enthalpy term \(h\) is that of the nearest point along the streamline and
\(V\) is the speed of the air flow.
For an ideal gas it holds that
$$h=u+pv$$
where \(v\) is the specific volume and
\(u\) is the internal energy of the gas.
By manipulating the enthalpy expression and taking into account the constant pressure heat capacity definition \(Cp\) for an ideal gas \(h=C_pT\), we arrive at
$$T_s=T+\frac{V^2}{2C_p}$$
This equation defines how much the temperature rises because of the fact that is adiabatically brought to rest.
Returning to our example let's calculate the different values of stagnation temperature in correspondence of different airspeed values. You can find precalculated values in the table below.
T [K] Static  V [m/s]  V [km/h]  M  T0 [K] Stagnation  Deviation [K]  Deviation [%]  Parameters  

288.15

0

0

0

288.15

0

0

Cp

1005

J/Kg/K

288.15

2

7.2

0.01

288.15

0

0

c

340.3

m/s

288.15

3

10.8

0.01

288.15

0

0
 
288.15

4

14.4

0.01

288.16

0.01

0
 
288.15

5

18

0.01

288.16

0.01

0
 
288.15

6

21.6

0.02

288.17

0.02

0.01
 
288.15

7

25.2

0.02

288.17

0.02

0.01
 
288.15

8

28.8

0.02

288.18

0.03

0.01
 
288.15

9

32.4

0.03

288.19

0.04

0.01
 
288.15

10

36

0.03

288.2

0.05

0.02
 
288.15

11

39.6

0.03

288.21

0.06

0.02
 
288.15

12

43.2

0.04

288.22

0.07

0.02
 
288.15

13

46.8

0.04

288.23

0.08

0.03
 
288.15

14

50.4

0.04

288.25

0.1

0.03
 
288.15

15

54

0.04

288.26

0.11

0.04
 
288.15

16

57.6

0.05

288.28

0.13

0.04
 
288.15

17

61.2

0.05

288.29

0.14

0.05
 
288.15

18

64.8

0.05

288.31

0.16

0.06
 
288.15

19

68.4

0.06

288.33

0.18

0.06
 
288.15

20

72

0.06

288.35

0.2

0.07
 
288.15

21

75.6

0.06

288.37

0.22

0.08
 
288.15

22

79.2

0.06

288.39

0.24

0.08
 
288.15

23

82.8

0.07

288.41

0.26

0.09
 
288.15

24

86.4

0.07

288.44

0.29

0.1
 
288.15

25

90

0.07

288.46

0.31

0.11
 
288.15

26

93.6

0.08

288.49

0.34

0.12
 
288.15

27

97.2

0.08

288.51

0.36

0.13
 
288.15

28

100.8

0.08

288.54

0.39

0.14
 
288.15

29

104.4

0.09

288.57

0.42

0.15
 
288.15

30

108

0.09

288.6

0.45

0.16
 
288.15

31

111.6

0.09

288.63

0.48

0.17
 
288.15

32

115.2

0.09

288.66

0.51

0.18
 
288.15

33

118.8

0.1

288.69

0.54

0.19
 
288.15

34

122.4

0.1

288.73

0.58

0.2
 
288.15

35

126

0.1

288.76

0.61

0.21
 
288.15

36

129.6

0.11

288.79

0.64

0.22
 
288.15

37

133.2

0.11

288.83

0.68

0.24
 
288.15

38

136.8

0.11

288.87

0.72

0.25
 
288.15

39

140.4

0.11

288.91

0.76

0.26
 
288.15

40

144

0.12

288.95

0.8

0.28
 
288.15

41

147.6

0.12

288.99

0.84

0.29
 
288.15

42

151.2

0.12

289.03

0.88

0.3
 
288.15

43

154.8

0.13

289.07

0.92

0.32
 
288.15

44

158.4

0.13

289.11

0.96

0.33
 
288.15

45

162

0.13

289.16

1.01

0.35
 
288.15

46

165.6

0.14

289.2

1.05

0.37
 
288.15

47

169.2

0.14

289.25

1.1

0.38
 
288.15

48

172.8

0.14

289.3

1.15

0.4
 
288.15

49

176.4

0.14

289.34

1.19

0.41
 
288.15

50

180

0.15

289.39

1.24

0.43
 
288.15

51

183.6

0.15

289.44

1.29

0.45
 
288.15

52

187.2

0.15

289.5

1.35

0.47
 
288.15

53

190.8

0.16

289.55

1.4

0.48
 
288.15

54

194.4

0.16

289.6

1.45

0.5
 
288.15

55

198

0.16

289.65

1.5

0.52

Table 1. Stagnation temperature vs static temperature at different V values. Download spreadsheet.
By inspection of the table, the deviation between the two temperature measurements increases with the air speed.
The air density in the room should be calculated using the static temperature, in our case, for e.g. \(p=101325 Pa\) the corresponding density is \(\rho=1.225\ kg/m^3\). The same calculation using the total temperature with a fan pushing the air at 55 m/s gives \(\rho=1.218\ kg/m^3\), which is about a 0.5% of error on density.
Inspect the relationship between static and total temperature for a total air temperature probe; the first formula in this link.
$$T_{total}/T_s=1+\frac{\gamma1}{2}M^2$$
where the “total” subscript indicates the adiabatic temperature and “s” subscript is for the static temperature. This formula is in accordance with our example formula. To reach this formula from our derived temperature equation, substitute in this formula the definition of the specific heat ratio \(\gamma=C_p/C_v\), the Mayer's relation \(C_p – C_v = R\) and the Mach number expression \(M=V/\sqrt{\gamma R_{specific}T}\).
The total air temperature computed up to now here is somewhat ideal. All the kinetic energy is converted and contributes to a temperature rise but it should be accounted with an appropriate recovery factor (Commercial example here, equation 3). The relationship of the recovery factor with airspeed and other environmental factors is not trivial and for best accuracy it should be experimentally found. Nonetheless, the tight relationship between airspeed and temperature can be exploited to have a true air speed indication as per equation 9 in the previous link. Consequently, a total air temperature probe working together with a Pitotstatic tube offers a useful redundant airspeed indication.
Even at low airspeeds, to achieve top performance, the temperature measurement rise due to airspeed is existent. However, the impact of this effect on temperature measurements can be neglected in many applications with moderate to no loss of accuracy. Unfortunately, this is not the only source of uncertainty in a total air temperature probe. There are other aspects that hinder much more the performance of the total air temperature probes. Thermal aspects will be considered in a following article.
Further reading
General description
Full developed calculation example
Saturday, September 19, 2015
Boundary Layer Part Two
This is the second part of boundary layer miniseries.
Further reading:
Frederick H.Abernathy, Fundamentals of Boundary Layers, Hardward
Video for the above document
The Reynolds number can be used to predict a flow pattern. The onset of very fascinating flow patterns, for example vortex streets (Tutorial here) can be detected with the use of Reynolds numbers.
The Reynolds number is the ratio between inertial and viscous forces. A low Reynolds number indicates that inertial forces are overwhelmed by viscous forces. Conversely a high Reynolds number indicates that inertial forces are driving the flow. We can distinguish three principal flow patterns: laminar, transitional and turbulent.
We have a laminar flow when the fluid flows in parallel layers, each layer having little interaction with the adjacent layer. In a laminar flow it is possible to distinguish parallel flowing streamlines. An extreme example of laminar flow is shown on the following video.
Video 1  Laminar flow twisting and untwisting
This flow has a very low Reynolds number, well below unity. As the flow is strongly laminar the operator can rewind the system to a state visually identical to the original state
Turbulent flow is characterized by an apparent randomness, circulation, eddies and high Reynolds number.
A transitional flow is somewhat in between the two previous cases: some parts of the fluid can have a laminar behavior and some others a turbulentlike behavior. Determining the Reynolds number for each pattern in the transtional flow is not trivial, but fortunately for many notable cases there is an approximate math model. In the next video a complete transition between laminar and turbulent flow into a pipe is shown.
Video 2 – Transition between patterns
In the relevant literature, one of the most wellstudied boundary layer patterns is the one formed by a flow over a flat plate. Under some assumptions, the boundary layer expression can be analytically found, for example there is the Blasius solution. Have a look at Figure 1. One important thing to observe is that the boundary layer is developing along the plate \(x\) coordinate, on the horizontal direction of the image. The Reynolds number expression is \(Re=\rho Ux/\mu\) and \(x=0\) corresponds to the leading edge of the plate or the beginning of the boundary layer itself. Initially, the flow is laminar and as \(x\) increases the Reynolds number increases as well. At some point, the boundary layer transforms into a transitional pattern and finally ends with a turbulent configuration. If one is able to increase \(x\) at will, the flow will become turbulent sooner or later. Depending on many factors, like fluid type and plate roughness the transition to turbulent flow will happen at a different Reynolds number. A plausible interval of values is from 1e5 to 3e6.
Figure 1 – Boundary layer over a flat plate
The larger the xcoordinate of the section we examine is, the wider the corresponding boundary layer will be. Given a section at a coordinate \(x\), we observe that the thickness of the corresponding boundary layer decreases as the free stream speed increases. We can observe the relationship between freestream speed and boundary layer thickness on the following video.
Video 3  Boundary layer experiment water tube
In the video, the transition to turbulent pattern is triggered by an aerodynamic obstacle. Pressure gradients can cause transitions of boundary layers. Pressure gradients are likely to occur on external air temperature probes inside the diffuser section. You can see the geometry of a Kiel probe here.
In the general case, boundary layer shape is time and position dependent and thus no overall closed form solution is available.
In the next article we will investigate in more detail the simplest ways that pressure gradients affect boundary layers and we will examine a numerical example.
Frederick H.Abernathy, Fundamentals of Boundary Layers, Hardward
Video for the above document
Monday, September 14, 2015
Newsletter
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Saturday, August 22, 2015
Boundary Layer
When a body moves through the air, its interaction with the surrounding air mass generates a region where the air conditions are different from the freestream conditions. This region is named boundary layer. In basic air data applications there is a need to make accurate freestream conditions measurements, so we're really concerned about boundary layers. Boundary layers have been studied since the very beginning of aerodynamics. In 1904 a twentynineyearold Ludwig Prandtl laid down the groundwork for modern aerodynamics with a short boundary layer presentation in Heidelberg [1].
Figure 1 – Lovely Heidelberg. I've been there. Birthplace of the boundary layer
Let's visualize the boundary layer over a flat plate, refer to Figure 2. We have depicted a laminar air flow, with vectors highlighting the velocity profile. The air flowing near the wall is slower than the air far from the wall, so the local velocity \(u(y)\) is a function of the distance from wall \(y\). The boundary layer is defined as the region where the local velocity values \(u(y)\) are less or equal than the 99% value(or other reference value) of the freestream value \(u_{infty}\). The speed at the wall is zero.
Figure 2. Laminar flow boundary layer velocity profile
Air is viscous so to move one air layer with respect to other is necessary to apply a force. Let's consider the flow (Couette flow) between two parallel flat plates of area \(A\) at a distance \(y\), one fixed to the ground and the other free to move. The force \(F\) needed to maintain the speed \(u\) of the moving plate is \(F=\mu Au/y\). The newly introduced parameter \(\mu\) is the air dynamic viscosity, with SI unit of measurement the \(Pa \cdot s\). It is evident that energy is required to maintain a boundary layer of given characteristics between the two plates.
Video 1 – Turbulent and laminar flow
IIn general, boundary layers are affected by the relative speed of the fluid, the fluid properties and body geometry; see the following video that visualizes an example of laminar and turbulent flow.
Figure 3  Boundary layer velocity profiles in CFD pipe
The simplest way to visualize a boundary layer is to run a CFD simulation of a tube. You can find a tutorial here. From the figure 3 it is evident that the boundary layer is present and changes its shape along the pipe, the boundary changes along the length of the pipe, so the flow is said to be developing along the longitudinal axis. Let's suppose we want to calculate the flowrate across the pipe measuring the speed of the fluid at a single point inside the pipe. Flowrate is the integral over time of the velocity across a fixed section multiplied by the fluid density (Eq 12.2). To reconstruct the velocity values on the whole integration section based on a single measurement we need to know the analytical expression of the velocity profile. In many cases, that information can be gathered from precedent research work. It is evident that if we use the only measured velocity value as mean speed, neglecting the boundary layer, then we get a wrong flowrate measurement value.
Let's change the scenario and suppose that we want to know the true airspeed of a racing car, refer to figure 4. The issue here is that airspeed measured within the boundary layer is lower than the freestream airspeed. How far should our Pitot probe protrude from the car body to work properly? Visit this page for an analytical introductory approach.
Figure 4 – Pitot rake on a race car
At a glance to deal with a boundary layer we need to know
 The boundary layer thickness
 The boundary layer velocity distribution and flow status: turbulent, laminar or transitional?
 Flow development patterns: where will the flow change from laminar to turbulent?
In our typical application we're dealing with a flying platform, where unfortunately the boundary layer characteristics change with the aircraft attitude and airspeed. For example the wing downwash affects all the posterior parts of the fuselage. The magnitude and pattern of the airflow are dependent on the angle of attack.
In the real world it is not trivial to find a position that is not affected by the aircraft attitude. To push the measurement performance further, CFD analysis, wind tunnel studies and inflight calibration are needed.
Until now we focused our attention to the air velocity field, but the same definition of boundary layer can be used to define a temperature boundary layer. both boundary layers impact on basic air data measurements.
In the next post we will go into detail with a numerical example.
Further readings
[1] John D. Anderson Jr(2005), Ludwig Prandtl's Boundary Layer, December Physics Today 2005
Friday, July 10, 2015
Power System Reliability Considerations Part 2
F16 Aerobatic maneuver, Thunderbirds Aerobatic Team
This article is a followup on the previously presented results. We will examine the power system of a model airplane from the aspect of reliability. Common equipment configurations will be used as examples.
In a vehicle where the primary power source are electric batteries, the most crucial power consumers are the motor, the avionics and the servos. From now on, for the sake of simplicity, we will refer to the powerplant of the aircraft (which is tasked with producing thrust) as the “thrust” system, whereas the rest of the electric power consumers on the aircraft will be called the “control” system. We will carry out a high level analysis, since too much detail would eventually distract the reader from the most relevant factors.
Refer to Figure 1: our first layout consists of a single battery pack connected to the ESC. Control and thrust are powered by the ESC. Our working hypothesis is that the different failures modes can be considered independent.
Figure 1. Single battery system
Let's select the battery pack reliability value at \(R_p=0.96\) and the reliability value of the ESC at \(R_{ESC}=0.98\). The battery and the ESC should be working at the same time, so for the first layout, the reliability is \(R_{l1}=R_p \cdot R_{ESC}=0.940\). We notice that the overall reliability is lower than the reliability values of each single component. Moreover, our system is not redundant, since the failure of each component causes the failure of the overall system.
Our second layout consists of two independent batteries, one for thrust and one for control. One battery is physically connected to the motor's ESC and the other is connected to a BEC circuit that supplies the control module. Usually, the power related to the thrust system has more capacity and nominal voltage than the control battery.
Figure 2. One battery for thrust, one battery for control
Let's set the reliability value of the battery packs to \(R_p=R_c=0.96\), the reliability value of the ESC to \(R_{ESC}=0.98\) and the reliability of the BEC to \(R_{BEC}=0.98\).
Under nominal system operation, the two batteries, the ESC and the BEC should be operational at the same time. From this aspect, the reliability of the second layout is \(R_{l2}=R_p\cdot R_{ESC}\cdot R_c\cdot R_{BEC}=0.885\). At first, this result seems wrong: Despite having used more equipment, the overall reliability is now lower than the initial 0.940 value from Layout 1. Until now we haven't considered in detail what happens when a part of our system fails and that led us to noncomparable results. In fact, under careful examination, the second layout has extended capabilities. In layout 1 we have a probability \(1R_{l1}=0.06=6\%\) of a total power loss, and if this unfortunate event happens then we will lose control of the vehicle as well as any ability to safely (crash) land the unit. Revisiting layout 2, we calculate the probability to lose completely the vehicle control. The cases that lead to a catastrophic failure are those that include a simultaneous failure of both the batteries or both the ESC and BEC. The following table presents all such failure cases.
References
[1]
For example
SmartFly  PowerSystem Eq6 Turbo Plus Battery input protected
[2]
Mc Dowall (2005), Lies Damned Lies and Statistics: The Statistical treatment of Battery Failures , Retrieved 09/07/2015
Case #  Battery P  Battery C  ESC  BEC 

1  fail  fail  fail  fail 
2  fail  fail  ok  fail 
3  fail  fail  fail  ok 
4  fail  fail  ok  ok 
5  ok  fail  fail  fail 
6  fail  ok  fail  fail 
7  ok  ok  fail  fail 
Table 1. Failure modes that lead to total power loss equivalent to Layout 1
Combining the probabilities of the cases indicated in the table, we get the following expression for reliability \(R_{l2flat}=1((1R_p)\cdot(1R_c)+(1R_{ESC})\cdot(1R_{BEC}))=1(0.0016+0.0004)=0.998=99.8\%\). Now the odds changed to being favorable to Layout 2. However from a user's point of view, it is more interesting to know the value of the probability that the vehicle is still controllable (at least to some degree) after a failure. The necessary condition for controllability is that the BEC and its battery are still working properly.
Refer to the next table. In cases 8 to 11 the pilot will have a chance to land the aircraft safely.
Case #  Battery P  Battery C  ESC  BEC 

8

fail

ok

fail

ok

9

ok

ok

fail

ok

10

fail

ok

ok

ok

11

ok

ok

ok

ok

Table 2. Failure modes that lead to a controllable (crash) land.
The reliability related to this minimum guaranteed performance is \(R_{l2user}=R_c\cdot R_{BEC}=0.940\). This value is the same as the value of the first layout. With this method of analysis, the advantage of layout 2 over layout 1 not so clear anymore. However, the situation can be radically different if there is a relationship between the reliability of batteries/ESC/BEC and the corresponding capacity/maxcurrent/etc or there is a dependency among the reliabilities of single items.
All things said, however, by inspection of layout 2, it is evident that it does not offer any physical redundancy, so statistics apart, it's wise to not expect any sudden reliability increase.
Figure 3. One battery for thrust, dual batteries for control
In this layout number 3, we have a battery that goes straight to the ESC and a redundant voltage regulator, powered by two separated batteries, which feed the control system. The working hypothesis is that the redundant voltage regulator will continue to work even if one battery fails. The most tricky failure to handle for the voltage regulator is a battery cell short. Fortunately even regulators at RC grade can handle this condition [1].
Back to the math, this layout is more reliable as the system composed by voltage regulator with \(R_v=0.98\) and two batteries with \(R_c=0.96\).
\(R_{l3}=1((1R_c\cdot R_v)(1R_c\cdot R_v))=10.0035=0.996=99,6\%\) [2]
Using the same battery pack type, layout 3 offers augmented reliability, and that result was reached by means of physical components redundancy.
Typically, the weakest link in the chain affects the system reliability the most, so prior to purchasing or building an expensive or complex reliable thrust system, an analysis of the reliability of the whole aircraft system should be performed. It will be useless to have an amazing thrust system with undersized servos.
References
[1]
For example
SmartFly  PowerSystem Eq6 Turbo Plus Battery input protected
[2]
Mc Dowall (2005), Lies Damned Lies and Statistics: The Statistical treatment of Battery Failures , Retrieved 09/07/2015
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