A previous analysis indicated that the POH fuel-tank recommendations reduced the wing root bending moment. This note attempts to quantify the lifespan impact of the estimated maximum 15% reduction in bending moment.
This video by Airframe Components provides an excellent visual understanding of the PA-28 and PA-32 spar structures. Please review this video before continuing.
The PA-28 aircraft use the NACA 65(2)-415 airfoil (c.f. Piper PA-28 Cherokee Wing Comparison: An Aerospace Engineering Perspective) with wing spar attach points as illustrated in Figure 1. With the spar at the maximum 15% thickness of a 63 inch chord, the spar height is about 9.5 inches. The moment of inertia is about 18.7 [in4] resulting in a stress of 0.253 [psi] per moment [in-lb]. However, the presence of holes significantly increases the stresses at the bolt holes by about 3x.
Figure 1: PA-28 Wing Spar at the Root
Aluminum has the unfortunate mechanical property that all loading -regardless of the stress magnitude- contribute to the metallic lifespan.
The Piper Cherokee PA-28-235 recommends the use of the tip tanks last. The purpose of this note is to estimate the reduction in wing root bending moments associated with this strategy. The PA28 model has a known AD history with wing root cracks; reducing the bending reduces the stress and increases the structural life. The POH for a C model says:
To familiarize yourself with the fuel available on the aircraft, the planform view of the aircraft from the POH with fuel tank overlays is illustrated in Figure 1. The aircraft has 4 tanks: two 25 gallon main tanks and two 17 gallon tip tanks for a total of 84 gallons of fuel. The main tanks are approximately 70 inches outboard from the centerline and the tip tanks are approximately 180 inches outboard.
Figure 1: Planform View
Aerodynamics: Using the process described in https://charles-oneill.com/blog/cherokee-tapered-wing-float/, the wing’s effective spanwise aerodynamic center is located at approximately 43% of the span. To remain conservative, we ignore the structural weight of the wing. The simplified configuration is given in Figure 2.
Figure 2: Simplified Summation of Moments
Summation of moments about the root (left side) gives the following equation. When substituting for the wing panel’s lift (L/2) and the weight of fuel, the overall moment at the wing root is
Plotting this wing root bending moment (per g) over the aircraft’s envelope provides a visual comparison of the differences in loading and operational techniques. All feasible loading and operating conditions lie within the gray shaded region. The worst case (red color) is a light 150 lbf -but perhaps not so bright- pilot with VFR minimum fuel and a maximum payload to achieve gross weight of 2900 lbs; notice that the Cherokee does NOT appear to have a zero fuel weight limitation (ZFW). On the other extreme, a light 150 lb pilot with no payload results in the conditions at lower left, where the POH loading suggestion (main 1st, then tips) is the lower black line. The green line shows the results when tips are fed first until dry and then the mains are fed.
Figure 3: Bending Moments and Loading Cases
Conclusion 1: Adding fuel reduces wing bending at the root. This immediately shows shows that the tip fuel is almost 5 times more effective at reducing the bending moment per gallon.
Conclusion 2: Each gallon of main fuel is structurally equal to a reduction in payload weight of 4 pounds.
Conclusion 3: Each gallon of tip fuel is structurally equal to a reduction in payload weight of 21 pounds. Adding fuel tip tanks (17 gallons) acts to reduce fatigue stresses similar to reducing the payload weight by 360 pounds. This is not a negligible amount.
Conclusion 4: The maximum benefit of feeding the mains before the tips is approximately a 15% reduction in bending moment.
Statement 1: This analysis does NOT include the effects of maneuvering speed. Please refer to the appropriate POH for guidance and remember that Va reduces at lower weights.
Statement 2: The 15% reduction in bending moment contributes FAR more than 15% to the aircraft lifespan. 15% is in fact a substantial number. Further analysis of this fact will be conducted later (See Cherokee 235 Tip Tanks and Fatigue).
Conclusion: The Piper POH’s suggestion to use the main tanks first does have a substantial structural fatigue benefit.
When should you start your turn inbound to exactly ensure the needle will be centered? This post explains and demonstrates a quick mental math solution. First, the answer:
Begin your turn no later than the CDI dot corresponding to one-half of the angle to intercept divided by 3 and divided by the time to pass from one CDI dot to the next.
This solution works for turning inbound & outbound for tracking a VOR radial, a LOC/ILS, or any situation where you can count intervals leading up to a desired track.
Example
You are on a 45 degree intercept. The needle is alive and you count 4 seconds between CDI dots.
The mental math is: 45 divided by 3 is 15….. 15 divided by 4 is about 4…. Half of 4 is 2. Turn no later than 2 dots.
Derivation:
The pilot needs a quicker and more intuitive solution. We can simplify this engineering-style intercept equation with a few steps.
Thus the pilot solution in words is: Begin your turn no later than the CDI dot corresponding to one-half of the angle to intercept divided by 3 and divided by the time to pass from one CDI dot to the next.
Experimentally Developed Solution:
Full disclosure: I did not initially develop this intercept solution based on the mathematics. Rather, I experimentally flew many intercepts and developed the following rule of thumb. By a complete stroke of luck, the experimental solution happens to be exactly identical to the mathematical solution above.
Disclaimer
This solution is only meant for intuition and the mathematical understanding of intercepts. You must follow ATC, the appropriate regulations and standard operating procedures when flying. Find and train with an instructor, preferably with a CFII. No exceptions!
The purpose of this page is to develop and distribute a simplified airspeed indicator calibration technique and computer program tool. Calibration from indicated (KIAS) to calibrated (KCAS) is required for certified and experimental aircraft (c.f FAR 23.1323 and FAR 25.1323). There are many techniques and flight test approaches available; however, the mathematics of generating a calibration chart or card can be daunting. This page provides a FREE self-contained airspeed calibration tool for Windows computers useful for subsonic aircraft with minimal calculation and with minimal equipment.
Requirements: You need an indicated airspeed, a GPS with track and groundspeed readouts, a thermometer, and your altitude. You will need to fly three independent headings (approximately 120 degrees apart) for each data point. You will need to download and enter your data into the windclover program.
Non-requirements: You do NOT need any ground references. You do NOT need to know your precise heading or magnetic variation. You do NOT need accurate timing or any clock. You do NOT need to calculate your true airspeed. You do NOT need an aerospace engineering background or on-board flight test engineers or hardware.
Cloverleaf Flight Profile: You will need to fly three lines that are approximately 120 degrees apart (e.g. 100, 220, 320). Maintain a constant heading, altitude, and indicated airspeed. Using your GPS, record your ground speed and track. Enter these data values into heading columns #1, #2, and #3. The program determines the wind direction/speed and the calibration from KIAS to KCAS.
This is an engineering structures parody of the X Files from 1999 found in my class notes. April Fools… once every 20 years is about right for this engineering joke.
So why is this related to structural engineering? Well, there is a shortcut method using so-called singularity functions to calculate the moment and shear in beams. Refer to any classical engineering textbook. As my initial undergraduate instructor in a structural analysis course, Dr. Wolf Yeigh, would say, it’s a good tool to have in your pocket. FYI, that was an intense course and professor, but one that I’m really lucky and glad to have taken. And yes, I made an A.
In this note, you will discover a trigonometric identity to assist maintaining a DME arc. The DME arc is a common maneuver for instrument approach procedures consisting of flying a specified distance from a DME site. The VOR/DME-A approach (Figure 1) uses an 18 nm arc from the CLL Vortac at (1) to provide two arcs (highlighted in yellow) from IAFs at OWANY at (2) and OWDIM at (4) towards JISPU at (3).
Figure 1: VOR/DME Arc approach plate
Using a Turn-Twist strategy, once on the arc, the heading to fly is tangent to the arc. This makes the no-wind control law: Turn to heading = Radial plus 90 deg when CW or Radial minus 90 deg when CCW. Unfortunately, this control strategy contains inherent divergence; in other words, the aircraft always tracks outside the desired arc (Figure 2, left). With a Turn 10/Twist 10 step, the cross track error is 1.5%. For example, a 20 nm arc with 10 degree radial steps, would give 0.3 nm error every step.
Is there a correction to exactly remain on the arc given a Turn-Twist step? Yes, and amazingly enough, the result is exact and a trigonometric identity. The right portion of Figure 2 derives a correction angle (gamma) such that the exact track is from point A to point B, both on the same arc. The result is that exactly half the Turn/Twist angle is applied inside the normal +-90 heading.
Figure 2: Derivation
For example, using a Turn 10/Twist 10 in a counter clockwise direction at the R-040 would require the heading be 305 degrees; this heading will precisely keep you on the exact DME arc at the R-030 radial.
Warning: The normal flying caveats apply: 1) This is only meant for insight and is not meant as instruction or as a change to your specific flight operations manual, 2) wind will require varying correction angles, and 3) aviate, navigate, communicate.
The study of classical PDEs is a useful and typical course for engineers and scientists to both appreciate and understand the behavior of physical systems.
As the (former) instructor of a course in PDEs, I reviewed classical solution techniques in a lecture titled A brief history of GES 554 PDE to prepare students for their final exam. This lecture makes an excellent refresher or rapid introduction.
If you want to review the entire 50 lecture course, visit here. Feel free to call it The Brief History of the World of PDEs in 50 Parts.
Topics covered are:
Motivation, classification & canonical forms
Diffusion, Elliptic, Hyperbolic, and Transport PDEs
Solution methods: Series, Separation of variables, Monte Carlo, finite difference, Ritz / Galerkin and Transforms
How much liquid water is generated in a fuel tank due to condensation? This is an engineering analysis focused on aviation, where water in the fuel tanks is a known cause of accidents. There is plenty of anecdotal evidence and opinions ranging from “Never happens” to “Critical in humid areas”. At the end of this note, you will be able to understand the physics and quantify the generation rate. Let’s get started.
Actual in-the-wild capture of Water in Avgas
Summary
Nearly empty light-aircraft tanks in extreme hot and humid environments with extreme temperature swings theoretically could condense approximately a couple of fluid ounces a week.
The generation rate linearly scales with empty tank volume and humidity, but exponentially with temperature.
Normally vented tanks substantially reduce the water influx rate, but do have a breathing mode that can pump moist air during temperature and pressure swings.
Condensation is more likely to be a long term storage threat; Large volumes of water are more likely to be ingress of liquid water.
Dry & Wet Air
Water is a key enabler of life and dramatically affects the behavior of air. We call “dry air” the mixture of mostly nitrogen (80%), oxygen (20%), and trace other constituents (Ar, carbon dioxide, etc). “Wet air” is what we normally encounter and is dry air + water. “Air” could also include particles + bugs + dirt. You can learn more at my course notes here for an introduction and here for non-standard atmospheres.
The important takeaways are:
Adding water decreases air density since water has a lower molecular mass than air. (Technical note: Water has 2 Hydrogen of mass 1 plus 1 Oxygen of mass 16 for a total of 18. Air on the other hand has 80% diatomic Nitrogen of mass 28 and 20% diatomic Oxygen at mass 32 for a total of 28.97.).
Increasing temperature substantially increases the absolute water carrying capacity of air. Water vapor at 100% relative humidity consists of 0.7% of the wet air mass at 50 F and 6% at 110 F, but a whopping 15% at 140 F. This is why pilots need to be much more concerned with high humidity at high temperatures and not so much at lower temperatures (This is in addition to the temperature effects on density altitude).
Figure 1: Water Carrying Capacity of Air & Figure 2: Saturation Pressure of Water Vapor
The saturation pressure (Ps) in Figure 2 is generated from the Arden-Buck approximation as an exponential function of temperature. This also indicates the pressure at which boiling occurs (e.g. 212 F at sea level pressures of 14.69 psi; and 200 F at 10000 ft with a pressure of approximately 10 psi).
Design a worst case scenario
Let’s pick a scenario where a nearly empty 25 gallon tank completely condenses the water vapor. Plus there is a complete air exchange/recharge of hot and humid (120 F and 100% relative humidity) air once per day. How much water is generated?
The answer is about 0.25 oz (7.5 ml) per day. Consider it one cupped-hand of water, or about a 2.5″ spot of water, or 0.75 seconds of fuel at 10 gal/hr. That’s enough to fully grab your attention.
The rate scales linearly with the tank’s air volume so keeping the tank 90% full reduces the generation rate by 90%. Being in 30% humidity air reduces the rate by 70%. Doubling the elapsed days doubles the generated volume. Notice that the rate exponentially scales with temperature (ps/T is exponential).
Critically, we can bound the generation rate. For example, at 120F and 100% humidity with a 20 gallon tank, the rate should be less than 0.20 oz per day.
If you are generating more than this, chances are extremely likely that there is another mechanism responsible. Go find it.
Maximum Condensing Case with an more Realistically Vented Tank
Or… How I learned to stop worrying and count the zeros
The dB decibel scale can often be very intimidating to others, so here’s a quick way to simplify (i.e. no logs or powers) your explanation to two steps. The fundamental point to make is that a Bell is how many zeros. A decibel is the number of zeros multiplied by 10.
Let’s convert a ratio to dB. Pick 100. This number has a number 1 followed by two zeros before the decimal point.
How many zeros? “2”. Multiply by 10. “20”
Say that number. “20 dB”
Let’s reverse the process and convert dB to a ratio. Pick 40 dB.
Divide by 10. “4”
Four zeros before the decimal place is: “10000… ten thousand”
How about a more complicated case. Convert 25 dB to a ratio.
Divide by 10. “2.5”
Two and a half zeros before the decimal place is? “more than 100 and less than 1000” Yes, and half a decimal place is about 3. “So 300?” You got it. “25 dB is about 300”
Now, convert 564 to dB.
How many zeros? “Almost the number 6 followed by two zeros. So 2” Yes, but we had a 6 in front of the zeros. 6 is worth about 75% of a decimal place. “So 2.75?” Exactly, now multiply by 10. “27.5”
Say that number. “27.5 dB”
This approach is much easier to explain than defining dB = 10 log(R) and the inverse operation using pow(10) and gives much better intuition. So, in field work, I tend to just use this approach. This may seem trivial to experts, but any trick to increasing understanding and explain-ability is worth your consideration.
This note determines the airspeed calibration card for a Piper Cherokee aircraft. The flight occurred on 6 December 2020 near College Station with a PA-28 140. Minimal onboard equipment was used: the airspeed indicator, the altimeter, and the outside air temperature. A personal uAvionix Sentry connected to Foreflight provided the GPS derived track and groundspeed.
Using a personal algorithm, the indicated to calibrated airspeed data points were reduced and plotted.
The trend is clear. At low speeds, the airspeed indicator reads too low (a common error). The errors at cruise are negative; the airspeed indicator reads too high. Only at around 70 MPH is the error near zero. Unfortunately, at the low end, there is more scatter than I hoped for. This scatter is likely resulting from the challenges of 1) precisely maintaining a specified indicated airspeed with an analog airspeed indicator, while 2) recording the average groundspeed and track. A future approach will use the raw GPS data points.
If we assume that the errors are solely resulting from errors in the static pressure (a reasonably good assumption), then we can determine the effective altitude errors associated with the static pressure error. These just barely meet the +-30 ft legal requirement at 100 kts.
For such a short flight, we were able to determine the overall character and approximate error curves of the airspeed indicator and altimeter. A more formal program would involve multiple people, multiple data points at the same test condition, and much more flight time.
