Air-headed Aviator
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Forward:
For a week I've been attempting to research whether there were physics based rules one could use to size the inlet and exhaust of EDF power systems. I've searched through a variety of theorems, principles and concepts and after many brain straining nights I think I've finally crafted a theory that could work. The purpose of thrust tubes on EDF is to assist the fan to give its thrust the highest velocity it can. Thrust is defined as the mass flow rate of air times its velocity. An thrust tube becomes too long when instead of helping thrust grow it creates pressure that restricts it. The purpose of inlets on EDFs is to deliver an even and laminar stream of air to the fan. The more even and smooth this air is the easier a time an EDF has at accelerating that air to create thrust. Inlets are too long when the air going through them starts to build to highly in pressure, once again creating a restriction that the fan has to work harder on to overcome. Additionally if either exhaust or inlet have extreme changes in cross section some pressure can actually be lost due to Bernoulli Principles, reducing thrust.
The current guidelines for thrust tube sizing most hobbyist will find online are on making the tube length four times as long as the fan diameter, and the exit area 90-80% of the fan area. Inlet sizing is only suggested as an entrance area of 100-110% of the fan diameter with no length guidelines. Taking advantage of my access to real world turbofan aircraft I adapted their sizing ratios to RC scale. Exhaust length became equal to the Fan circumference, with an exit area equal to the fan swept area (total fan area minus the area of the nose cone). The inlet sizing became equal in area still to the total fan area, now with a depth 70% of the fan diameter. While these guidelines have been effective, and corroborated with bench testing, they remained anecdotal and lacked the why behind those choices. How does one determine based off the capabilities of their fan the maximum size of inlet or exhaust that helps add to performance and avoids reducing it.
For my write up in this post all my calculations will be based on a FMS 64mm EDF, powered by a 28-36 brushless motor at 3300Kv, powered by a 4S battery at 35 amperes drawn. This EDF will lack inlet and exhaust elements in these calculations.
Power/Thrust Formula:
The first part of my approach to solving EDF physics began with this Power formula I learned from the book "Aerodynamics for Naval Aviators". This formula is meant to calculate how much horsepower is needed to achieve a certain velocity of flight with a certain amount of thrust. In this formula, P = Power (Hp), T = Thrust in pounds force, V = Velocity in mph. 325 is an important value that is the speed in miles per hour for one pound of force to overcome one pound of drag.
This formula is important because it can be used to calculate an EDFs efflux: the velocity of the thrust it produces. This works because of a handy equivalence in aerodynamics; controlling for drag, an aircrafts maximum possible speed can only at best be equal to the efflux of its powertrain. Thus if we have the Thrust and Power of our EDF, we can solve the equation for V and find its efflux.
For my EDF, its theoretical max mechanical power is 588 Watts, or 0.79 hp. Off of a test on the bench its thrust is 640 grams, or 1.41 lbs. Pass those numbers through the formula and the efflux we calculate is 182 mph, or 81.4 m/s.
Actuator Disk Theory:
Now with efflux determined I can use equations in the Actuator Disk Theory next to find the next ingredient. These theories attempt to describe how some form of thrust producing unit changes the pressure of the fluid it accelerates. It simplifies the fan or propeller as an infinitely thin disk that causes an instantaneous acceleration of the fluid. In my case the fluid is air, and my disk will be the EDF.
Here p_1 is the pressure ahead of the disk in Pascals, while p_2 is pressure behind the disk. Rho is the density of air in Kg/m^3. V_e is the exit velocity behind the disk in m/s, or the efflux, while V_0 is the initial velocity of the air. In the use of this equation we take our efflux as 81.4 and the initial velocity as 0 m/s for the case of determining static thrust ( which is the highest amount of thrust a fan can make). The density of air at sea level around room temperature is 1.204 Kg/m^3. Plug those values into the formula and you get 3,988.8 Pa, or 0.58 psi. This is how much more pressure the example 64mm adds to the air at full power. If p1 is taken as the ambient air pressure at sea level, the fan has changed 14.7 psi to 15.28.
For a week I've been attempting to research whether there were physics based rules one could use to size the inlet and exhaust of EDF power systems. I've searched through a variety of theorems, principles and concepts and after many brain straining nights I think I've finally crafted a theory that could work. The purpose of thrust tubes on EDF is to assist the fan to give its thrust the highest velocity it can. Thrust is defined as the mass flow rate of air times its velocity. An thrust tube becomes too long when instead of helping thrust grow it creates pressure that restricts it. The purpose of inlets on EDFs is to deliver an even and laminar stream of air to the fan. The more even and smooth this air is the easier a time an EDF has at accelerating that air to create thrust. Inlets are too long when the air going through them starts to build to highly in pressure, once again creating a restriction that the fan has to work harder on to overcome. Additionally if either exhaust or inlet have extreme changes in cross section some pressure can actually be lost due to Bernoulli Principles, reducing thrust.
The current guidelines for thrust tube sizing most hobbyist will find online are on making the tube length four times as long as the fan diameter, and the exit area 90-80% of the fan area. Inlet sizing is only suggested as an entrance area of 100-110% of the fan diameter with no length guidelines. Taking advantage of my access to real world turbofan aircraft I adapted their sizing ratios to RC scale. Exhaust length became equal to the Fan circumference, with an exit area equal to the fan swept area (total fan area minus the area of the nose cone). The inlet sizing became equal in area still to the total fan area, now with a depth 70% of the fan diameter. While these guidelines have been effective, and corroborated with bench testing, they remained anecdotal and lacked the why behind those choices. How does one determine based off the capabilities of their fan the maximum size of inlet or exhaust that helps add to performance and avoids reducing it.
For my write up in this post all my calculations will be based on a FMS 64mm EDF, powered by a 28-36 brushless motor at 3300Kv, powered by a 4S battery at 35 amperes drawn. This EDF will lack inlet and exhaust elements in these calculations.
Power/Thrust Formula:
The first part of my approach to solving EDF physics began with this Power formula I learned from the book "Aerodynamics for Naval Aviators". This formula is meant to calculate how much horsepower is needed to achieve a certain velocity of flight with a certain amount of thrust. In this formula, P = Power (Hp), T = Thrust in pounds force, V = Velocity in mph. 325 is an important value that is the speed in miles per hour for one pound of force to overcome one pound of drag.
Actuator Disk Theory:
Now with efflux determined I can use equations in the Actuator Disk Theory next to find the next ingredient. These theories attempt to describe how some form of thrust producing unit changes the pressure of the fluid it accelerates. It simplifies the fan or propeller as an infinitely thin disk that causes an instantaneous acceleration of the fluid. In my case the fluid is air, and my disk will be the EDF.
Specific Volume:
The last part of my theory depends on the concepts of Specific Volume. Mathematically equal to the inverse of density, this concept describes the minimum volume a certain mass of material can take up at constant density. Air itself has a specific volume of 820L/Kg - one kilogram of air at constant density, constant pressure takes up 820 liters of space. The reason I think this concept is important is because I think this is how I can identify the sizing for a thrust tube. When a thrust tube is installed on an EDF, the whole unit becomes "one" vessel. Thrust depends then on the exit pressure at the exit of the thrust tube. Specific Volume helps describe essentially what the minimum volume a chosen pressure can be maintained in.
When I take the initial pressure of 14.7 psi and divide that by the exit pressure calculated with Actuator Disk Theory (15.28 psi) I get 0.96. Since pressure is relatable to attempting to fit more air in smaller volumes, I can use this ratio to calculate Air's Specific Volume at 15.28 psi: 787.2L/Kg. I convert this volume to milliliters (our cubic millimeters) and then divide that by the area of the EDF. For 64mm fans that is 3125 square mm. This resulted in 251.87 mm in length, or 9.9 inches. This value in fact is very close to the experimental values calculated from the anecdotal guidelines ( 8 inches based off my circumference approach), giving me confidence in the potential accuracy of this method.
Through this approach the relationships become: The high in differential pressure the shorter the thrust tube can be all other variables constant. The greater the fan diameter the shorter the potential thrust tube can be (all other variables constant). Higher balances of power and thrust to the efflux generator will also see lower pressure differences and thus shorter thrust tubes. The value of these relationships give potential explanation to some of the many differences seen in aviation; like how very high bypass turbofans see outer fan cases that terminate sooner than the gas turbine core. Such large fans create a lower pressure ratio than other turbofans.
When I take the initial pressure of 14.7 psi and divide that by the exit pressure calculated with Actuator Disk Theory (15.28 psi) I get 0.96. Since pressure is relatable to attempting to fit more air in smaller volumes, I can use this ratio to calculate Air's Specific Volume at 15.28 psi: 787.2L/Kg. I convert this volume to milliliters (our cubic millimeters) and then divide that by the area of the EDF. For 64mm fans that is 3125 square mm. This resulted in 251.87 mm in length, or 9.9 inches. This value in fact is very close to the experimental values calculated from the anecdotal guidelines ( 8 inches based off my circumference approach), giving me confidence in the potential accuracy of this method.
Through this approach the relationships become: The high in differential pressure the shorter the thrust tube can be all other variables constant. The greater the fan diameter the shorter the potential thrust tube can be (all other variables constant). Higher balances of power and thrust to the efflux generator will also see lower pressure differences and thus shorter thrust tubes. The value of these relationships give potential explanation to some of the many differences seen in aviation; like how very high bypass turbofans see outer fan cases that terminate sooner than the gas turbine core. Such large fans create a lower pressure ratio than other turbofans.
Inlet Sizing:
The same equations used for the thrust tube don't appear to be usable for inlets. Missing information is how to determine the force of "sucking" caused by the front of a fan, at least without more specific data. However I've identified other solutions. Based off this paper from NASA the shape and depth of an inlet is desired to be as short as possible for drag, but effective enough to keep airflow attached at higher angles of attacks. This paper relates the design of fan inlets as similar to wings; having some level of design that helps keep the boundary layer attached (in addition to the use of a diffuser for engine performance). Based off these points, its suggested that I could use concepts of Reynold's Numbers to size an inlet.
Reynold's Numbers are the dimensional value that describes the viscosity of flowing fluids. The Reynold's of a surface like a wing can be calculated from multiplying the Chord of the wing (in inches) times the cruising velocity (mph) times the constant K, which depends on altitude. In this case to get the shortest inlet that fulfills the goals of attached even flow I have to match the inlet depth to the minimum Reynold's the aircraft operates at. The minimum Reynold's is based off the smallest chord on the aircraft, at the wing tips (typically). Taking an aircraft design I have as an example, the minimum Reynold's it operates at is 87,750. An important detail to keep in mind that the inlet construction will likely be significantly thinner than the wing chord of my design ( 1 foam bird thickness vs 2) so the Reynold's value it "stalls at" which be lower. Taking the 87,750 and dividing it in half gets 43,875. Divide that value by the minimum speed and the constant K at my altitude (about 780) I get 2.25 inches for this aircraft. This relationship suggests that the slower an aircraft flies the more shallow its inlet can be to be effective. This however conflicts with another relationship suggested with inlets, where the faster an aircraft flies the more shallow the inlet can be (if it could change). The relationships that battle is the sense that the slower an aircraft flies the easier for airflow to stay attached, but the quicker an aircraft flies the more inlet is needed to ensure oncoming air is organized before meeting the fan. One approach essentially creates the best conditions for low speed or static thrust, and one for retained efficiency at speed.
That said, another sense of corroboration comes from a different NASA paper that reveals, at least for the turbofan they use as a test, that the average depth of the inlet they used in their test was 1.1 times the fans diameter. This is very close to the values calculated through the Reynold's number approach, and lends confidence to my conclusions.
Reynold's Numbers are the dimensional value that describes the viscosity of flowing fluids. The Reynold's of a surface like a wing can be calculated from multiplying the Chord of the wing (in inches) times the cruising velocity (mph) times the constant K, which depends on altitude. In this case to get the shortest inlet that fulfills the goals of attached even flow I have to match the inlet depth to the minimum Reynold's the aircraft operates at. The minimum Reynold's is based off the smallest chord on the aircraft, at the wing tips (typically). Taking an aircraft design I have as an example, the minimum Reynold's it operates at is 87,750. An important detail to keep in mind that the inlet construction will likely be significantly thinner than the wing chord of my design ( 1 foam bird thickness vs 2) so the Reynold's value it "stalls at" which be lower. Taking the 87,750 and dividing it in half gets 43,875. Divide that value by the minimum speed and the constant K at my altitude (about 780) I get 2.25 inches for this aircraft. This relationship suggests that the slower an aircraft flies the more shallow its inlet can be to be effective. This however conflicts with another relationship suggested with inlets, where the faster an aircraft flies the more shallow the inlet can be (if it could change). The relationships that battle is the sense that the slower an aircraft flies the easier for airflow to stay attached, but the quicker an aircraft flies the more inlet is needed to ensure oncoming air is organized before meeting the fan. One approach essentially creates the best conditions for low speed or static thrust, and one for retained efficiency at speed.
That said, another sense of corroboration comes from a different NASA paper that reveals, at least for the turbofan they use as a test, that the average depth of the inlet they used in their test was 1.1 times the fans diameter. This is very close to the values calculated through the Reynold's number approach, and lends confidence to my conclusions.
Conclusions:
I pursued a lot of different principles in order to find a theory that could work. I looked at formulas on pressure loss through pipes, Rocket nozzle fundamentals, and attempted to find viscosity values. Many of these other approaches simply couldn't work because I lacked complex identities for various materials. Only in the methods above could I find solutions based off the information I had on hand to make an equivalence that holds mathematically. There are of course some shortcomings: For one, the only way to know for sure if these concepts are accurate is if I measured things. I lack things like pressure instruments to test these differences. The math suggests one thing, but reality is another. Additionally, these results depend on perfect geometries on paper. In reality the construction of these materials, even with like Additive printing tools can have imperfections. Thus a high chance of some energy and thrust being lost. Additionally, I lack the mathematics to predict how much higher thrust from the inclusions of the inlet and thrust tube will become. I have proven their inclusion increases thrust, these methods above perhaps reveal how to size them for their best effect, but their results allude me for now. The running theory on the "how" is that the thrust tube allows all the force from the fan on the air to go towards accelerating it vs losing energy from the ambient environment, swirl, and pressure loss. Calculating the potential retention of energy from a thrust tube likely depends on hard to find experimental data.
For now, I want to explore this process; Modify some of my current designs to see if any efficiencies are gained. In truth I need to purchase another spare EDF to test a variety of dimensions and see how accurate (or not) this theory is. If anyone else wants to explore this data as well, you have my full permission, if not also feel free to add your insights, ideas, and assistance to these ideas.
For now, I want to explore this process; Modify some of my current designs to see if any efficiencies are gained. In truth I need to purchase another spare EDF to test a variety of dimensions and see how accurate (or not) this theory is. If anyone else wants to explore this data as well, you have my full permission, if not also feel free to add your insights, ideas, and assistance to these ideas.
