gordonaut
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Centrifugal Impellor in place of Axial Ducted Fan
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WonderousMountain
Well-Known Member
It's an order of operations thing, it you devide pi/AR first your drag will increase with larger AR number. If you work left to right, you method works fine. Highspeed used brackets (equation) to denote the order of operations so nothing becomes confused. You may find the correction trivial, but mathmatics is really only an international language when spoken properly.
My work is really imprecise, when you can get me to show it.
Mountain
My work is really imprecise, when you can get me to show it.
Mountain
gordonaut
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WonderousMountain
Well-Known Member
All that's well and good, but do you realize a 300hp aircraft piston engine will weight in excess of 300lbs (most likely). Auto conversions are likely to weight even more. That means our take off weight is at least doubled if not trippled over a single passenger, and our thrust/weight ratio is only some better than a light 100hp engine. If the recuperator weighs 100 pounds it's self and is no more compact than the piston engine we've gained little, from a planemakers perspective.
300mph is a respectable speed, if not blow your mind fast, it should satisfy most sport pilots.
300mph is a respectable speed, if not blow your mind fast, it should satisfy most sport pilots.
gordonaut
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gordonaut
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WonderousMountain
Well-Known Member
That's good, didn't know recuperation could be done so light.
Write it your way for all I care, it's just an equation-easier done in the head to all relavent digits anyway.
Does that mean you can make 100 hp with 25lb's with a five pound recuperator? I'm thinking twin, I always think twin.
Just for the record, I'm thinking more sport than utilitarian. Gaining comparative efficiency to a piston engine is beyond expectation.
Write it your way for all I care, it's just an equation-easier done in the head to all relavent digits anyway.
Does that mean you can make 100 hp with 25lb's with a five pound recuperator? I'm thinking twin, I always think twin.
Just for the record, I'm thinking more sport than utilitarian. Gaining comparative efficiency to a piston engine is beyond expectation.
gordonaut
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highspeed
Well-Known Member
The way I read your equation is CDi equals Lift coefficient squared divided by pi divided by aspect ratio. Which seems to indicate you are dividing pi by the aspect ratio, which would be incorrect. I just wanted to clear it up.
An excellent example of a nonplanar wing on a transport category aircraft.
autoreply
Well-Known Member
Mathematically, both equations are identical. You can check it by filling in some random numbers for the variables and calculate it left to right.
A/B/C =A/(B*C)
Nonplaner right? This plane needs it's dihedral written down in a matrix:
http://soaringcafe.wpengine.netdna-cdn.com/wp-content/uploads/images/2012/01/Quintus-M-10.jpg
:gig:
highspeed
Well-Known Member
I stand corrected. Wish I would have figured that out earlier, it would make my spreadsheets significantly less complicated. Thanks guys.
bob989
Member
Hi Gordon,
Your comments and criticisms are always appreciated. I will attempt to respond as rigorously as possible. I'm really trying to understand why the industry does not opt for small engines, and what you say is relevant.
The issue about nullifying lift-induced drag is a matter of semantics. When I say "nullify", I mean that the amount of propulsion allocated to each small portion of the span would be sufficient to overcome the drag generated by that portion. Propulsion would be distributed over the wing in the same way that drag is distributed. Since wings tend to generate more lift near the root, and therefore more drag, I would place more propulsion near the root. Toward the tip, the wing generates less lift and less drag, and at some point I think it would make sense to forego propulsion altogether.
I'm not saying that a small-engine system eliminates drag. I'm saying that it would locally compensate for drag. Therefore, viewed from larger scales, drag and propulsion are locally balanced, so that what you see at the scale of the entire wing is lift produced by fuel consumption. That is the mathematical integration of the problem. We would no longer talk about drag separately and thrust separately when we're talking about the whole system in steady cruising flight. So instead of (L/D)/(g SFC), we would have L/(g mdot_fuel), where mdot_fuel is the mass flow rate of the fuel. Drag and thrust would be everywhere equal, and therefore the only thing left to optimize is the lift per unit fuel consumption rate.
When you look at the fanwing, for example, you don't see a separate wing, with its L/D, and a propulsion system, with its SFC. What you see is an integrated system. If you try to apply textbook formulas for wings and propulsion systems, you find that they don't really make sense. It is, of course, fair to ask how good a wing is a fanwing when the engine is turned off, and that answer can be expressed as an L/D or a glide angle. It's not surprising that the fanwing is a rather poor wing when evaluated purely as a wing. But of course it was never designed to be purely a wing.
The optimization of the integrated lift / propulsion system is a problem that the industry has never, to my mind, really attacked. I believe there is some optimum arrangement of the airfoil surfaces and propulsion intakes and ducts that maximizes endurance and range. But the industry has always viewed the airframe/wing/airfoil separately from the propulsion system. The industry says, an airframe has a lift-to-drag ratio. And the industry says, a propulsion system has a thrust-specific fuel consumption rate. There are whole textbooks on airframes that don't say anything about engines, and there are whole textbooks on engines that don't say anything about airframes. It's as if they were completely separate subjects, completely separate industries. There are airframe manufacturers, and there are engine manufacturers.
When we start mixing propulsion and lift intimately, we have to call into question their mathematical framework. Locally, there will be lift and drag and fuel consumption and thrust, but if we balance thrust and drag locally, then globally, what's left is just lift and fuel consumption.
The motivation for locally balancing thrust and drag is that it leaves the wake as clean as possible locally. If we don't do that, there's no way to correct for the messed-up wake by doing something elsewhere. When the air leaves the trailing edge, it's too late to fix it, and nothing that happens elsewhere on the span is going to make up for the local inefficiency.
I agree with you that we have to do the math. But we can't apply textbook formulas out of context and then say that they prove something. We have to actually do the math for the problem at hand. We can't borrow somebody else's math for somebody else's problem.
You suggest that thickening the wing to accommodate the propulsion system would be aerodynamic suicide. But it's going to be the same total intake area, for the cruise problem, as for the large-engine paradigm. You don't say that the large intake area on existing turbofans is aerodynamic suicide. Why not? When I look at a Boeing 777, I see two great big turbofan engines sticking right out there, and I know, as does the whole industry, that those big engines create drag. (I'm looking at one of those beautiful images where the marketing people have airbrushed out the flock of birds that the airplane is about to collide with, and the fire and rescue boats down on the river. In marketing land, every day has sunshine and clear blue skies, and the fact that the airplane has only two engines is never a problem.)
You also suggest that the structural problems associated with small engines would be nightmarish. But you don't seem to have nightmares when you look at a great big turbofan engine hanging from a skinny pylon attached to the wing at one position. Do you think that 100,000 pounds of thrust wobbling around in turbulent air doesn't lead to structural problems? Concentration of stress and fatigue, perhaps?
The aerodynamic and structural nightmares associated with great big turbofan engines don't seem like nightmares to Boeing engineers because they are nightmares that the industry has already confronted. The industry has solutions for those nightmares.
But there are aerodynamic and structural solutions available for the small-engine paradigm. In the small-engine paradigm, the wing structure would have a linear array of engine receptacles. If I offer you two structures problems, one in which you have a large weight attached to a beam at one position, and the other in which you have a large number of small weights attached to the beam at equally-spaced positions, which problem would you prefer?
Start with a biplane wing. This has two surfaces and an empty space between. The structure is a set of struts and cords, and it's more efficient than an I-beam. That is, it has more bending and torsional strength and stiffness for the weight.
Take any large structure. Take the Golden Gate Bridge. Why is that not an I-beam? Because it's more structurally efficient to have towers and cables. Look at all the structures around you that are not solid metal plates. Then ask yourself why the structure inside a wing can't be more like those structures.
Look at the structure inside the great big turbofan nacelle. It's composed of a few struts that hold the engine and fan in place without seriously interfering with the airflow.
Regarding your calculation of the number of engines, you are using takeoff thrust requirements. If we look first at cruise, a 700,000 lb airplane with an L/D of 19 would only need 36,800 lb of cruise thrust. But in order to determine the number of cruise-class engines, we would need a rating for the cruise thrust of a Williams FJ44 at the cruise conditions of a Boeing 777. Once we have calculated the cruise requirement, we can then use the rated takeoff thrust of those engines to see how well we are doing with the takeoff requirement. I think we'll find that some extra takeoff thrust will be needed.
I would be reluctant to use turbofans for the takeoff problem, because the low fan pressure results in a large area. I envision a higher pressure ratio for the fans, with the objective of reducing intake area. I see this as an application for single-stage centrifugal fans. Steal as much air as possible from the leading edge near the root, and don't worry so much about fuel efficiency. Just make sure it is small enough to fit inside the wing.
But there remains the problem that your general approach is overly constrained. You're trying to take a wing which was never designed to be integrated with a propulsion system, and a propulsion system which was never designed to be integrated with a wing, and put them together without any design modifications. You don't consider changing the dimensions of the wing to accommodate the engines because you believe that that would make it "worse". Considered purely as a wing, you're probably right. But so what? That would only be a problem if all, or a substantial portion, of the engines were to fail simultaneously, which is exponentially improbable, and even then, the combined system would still function as a wing, just not a very good one.
The formula you exhibit for lift-induced drag is a textbook formula for a wing. Just a wing, pure and simple. It's typical of the math that the industry uses. I can't ask: how does the propulsion system enter into your formula? Where is the fuel consumption rate? Where is the thrust? The obvious textbook answer is: a wing doesn't have a propulsion system. It has an aspect ratio, a coefficient of lift, and a coefficient of induced drag.
With an integrated wing / propulsion system, we might be able to develop simple, useful equations if we use concepts like "propulsion-generated lift". How much of the lift is generated by fan air acting against the interior surfaces of the ducts? And how much of the exterior surface of the airfoil can be eliminated because of the aerodynamic contribution of the interior surfaces?
So I still think that the integration of small engines into the wing could offer substantial cost and safety advantages, and it doesn't seem to pose any insurmountable aerodynamic or structural issues.
-bob989
Your comments and criticisms are always appreciated. I will attempt to respond as rigorously as possible. I'm really trying to understand why the industry does not opt for small engines, and what you say is relevant.
The issue about nullifying lift-induced drag is a matter of semantics. When I say "nullify", I mean that the amount of propulsion allocated to each small portion of the span would be sufficient to overcome the drag generated by that portion. Propulsion would be distributed over the wing in the same way that drag is distributed. Since wings tend to generate more lift near the root, and therefore more drag, I would place more propulsion near the root. Toward the tip, the wing generates less lift and less drag, and at some point I think it would make sense to forego propulsion altogether.
I'm not saying that a small-engine system eliminates drag. I'm saying that it would locally compensate for drag. Therefore, viewed from larger scales, drag and propulsion are locally balanced, so that what you see at the scale of the entire wing is lift produced by fuel consumption. That is the mathematical integration of the problem. We would no longer talk about drag separately and thrust separately when we're talking about the whole system in steady cruising flight. So instead of (L/D)/(g SFC), we would have L/(g mdot_fuel), where mdot_fuel is the mass flow rate of the fuel. Drag and thrust would be everywhere equal, and therefore the only thing left to optimize is the lift per unit fuel consumption rate.
When you look at the fanwing, for example, you don't see a separate wing, with its L/D, and a propulsion system, with its SFC. What you see is an integrated system. If you try to apply textbook formulas for wings and propulsion systems, you find that they don't really make sense. It is, of course, fair to ask how good a wing is a fanwing when the engine is turned off, and that answer can be expressed as an L/D or a glide angle. It's not surprising that the fanwing is a rather poor wing when evaluated purely as a wing. But of course it was never designed to be purely a wing.
The optimization of the integrated lift / propulsion system is a problem that the industry has never, to my mind, really attacked. I believe there is some optimum arrangement of the airfoil surfaces and propulsion intakes and ducts that maximizes endurance and range. But the industry has always viewed the airframe/wing/airfoil separately from the propulsion system. The industry says, an airframe has a lift-to-drag ratio. And the industry says, a propulsion system has a thrust-specific fuel consumption rate. There are whole textbooks on airframes that don't say anything about engines, and there are whole textbooks on engines that don't say anything about airframes. It's as if they were completely separate subjects, completely separate industries. There are airframe manufacturers, and there are engine manufacturers.
When we start mixing propulsion and lift intimately, we have to call into question their mathematical framework. Locally, there will be lift and drag and fuel consumption and thrust, but if we balance thrust and drag locally, then globally, what's left is just lift and fuel consumption.
The motivation for locally balancing thrust and drag is that it leaves the wake as clean as possible locally. If we don't do that, there's no way to correct for the messed-up wake by doing something elsewhere. When the air leaves the trailing edge, it's too late to fix it, and nothing that happens elsewhere on the span is going to make up for the local inefficiency.
I agree with you that we have to do the math. But we can't apply textbook formulas out of context and then say that they prove something. We have to actually do the math for the problem at hand. We can't borrow somebody else's math for somebody else's problem.
You suggest that thickening the wing to accommodate the propulsion system would be aerodynamic suicide. But it's going to be the same total intake area, for the cruise problem, as for the large-engine paradigm. You don't say that the large intake area on existing turbofans is aerodynamic suicide. Why not? When I look at a Boeing 777, I see two great big turbofan engines sticking right out there, and I know, as does the whole industry, that those big engines create drag. (I'm looking at one of those beautiful images where the marketing people have airbrushed out the flock of birds that the airplane is about to collide with, and the fire and rescue boats down on the river. In marketing land, every day has sunshine and clear blue skies, and the fact that the airplane has only two engines is never a problem.)
You also suggest that the structural problems associated with small engines would be nightmarish. But you don't seem to have nightmares when you look at a great big turbofan engine hanging from a skinny pylon attached to the wing at one position. Do you think that 100,000 pounds of thrust wobbling around in turbulent air doesn't lead to structural problems? Concentration of stress and fatigue, perhaps?
The aerodynamic and structural nightmares associated with great big turbofan engines don't seem like nightmares to Boeing engineers because they are nightmares that the industry has already confronted. The industry has solutions for those nightmares.
But there are aerodynamic and structural solutions available for the small-engine paradigm. In the small-engine paradigm, the wing structure would have a linear array of engine receptacles. If I offer you two structures problems, one in which you have a large weight attached to a beam at one position, and the other in which you have a large number of small weights attached to the beam at equally-spaced positions, which problem would you prefer?
Start with a biplane wing. This has two surfaces and an empty space between. The structure is a set of struts and cords, and it's more efficient than an I-beam. That is, it has more bending and torsional strength and stiffness for the weight.
Take any large structure. Take the Golden Gate Bridge. Why is that not an I-beam? Because it's more structurally efficient to have towers and cables. Look at all the structures around you that are not solid metal plates. Then ask yourself why the structure inside a wing can't be more like those structures.
Look at the structure inside the great big turbofan nacelle. It's composed of a few struts that hold the engine and fan in place without seriously interfering with the airflow.
Regarding your calculation of the number of engines, you are using takeoff thrust requirements. If we look first at cruise, a 700,000 lb airplane with an L/D of 19 would only need 36,800 lb of cruise thrust. But in order to determine the number of cruise-class engines, we would need a rating for the cruise thrust of a Williams FJ44 at the cruise conditions of a Boeing 777. Once we have calculated the cruise requirement, we can then use the rated takeoff thrust of those engines to see how well we are doing with the takeoff requirement. I think we'll find that some extra takeoff thrust will be needed.
I would be reluctant to use turbofans for the takeoff problem, because the low fan pressure results in a large area. I envision a higher pressure ratio for the fans, with the objective of reducing intake area. I see this as an application for single-stage centrifugal fans. Steal as much air as possible from the leading edge near the root, and don't worry so much about fuel efficiency. Just make sure it is small enough to fit inside the wing.
But there remains the problem that your general approach is overly constrained. You're trying to take a wing which was never designed to be integrated with a propulsion system, and a propulsion system which was never designed to be integrated with a wing, and put them together without any design modifications. You don't consider changing the dimensions of the wing to accommodate the engines because you believe that that would make it "worse". Considered purely as a wing, you're probably right. But so what? That would only be a problem if all, or a substantial portion, of the engines were to fail simultaneously, which is exponentially improbable, and even then, the combined system would still function as a wing, just not a very good one.
The formula you exhibit for lift-induced drag is a textbook formula for a wing. Just a wing, pure and simple. It's typical of the math that the industry uses. I can't ask: how does the propulsion system enter into your formula? Where is the fuel consumption rate? Where is the thrust? The obvious textbook answer is: a wing doesn't have a propulsion system. It has an aspect ratio, a coefficient of lift, and a coefficient of induced drag.
With an integrated wing / propulsion system, we might be able to develop simple, useful equations if we use concepts like "propulsion-generated lift". How much of the lift is generated by fan air acting against the interior surfaces of the ducts? And how much of the exterior surface of the airfoil can be eliminated because of the aerodynamic contribution of the interior surfaces?
So I still think that the integration of small engines into the wing could offer substantial cost and safety advantages, and it doesn't seem to pose any insurmountable aerodynamic or structural issues.
-bob989
gordonaut
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gordonaut
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gordonaut
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gordonaut
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gordonaut
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gordonaut
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gordonaut
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bob989
Member
Hi Gordon,
I think it's worth examining some of your objections to small engines. I don't see that you have a clear, well-reasoned argument based on fundamental physics or some other very basic objection like the properties of air or the strength of materials. I don't claim that small engines would necessarily be more efficient than or even as efficient as large engines, but you seem to have concluded that they are so much less efficient as to be unreasonable on that basis alone. I don't agree.
To my mind, it's clear that (a) we could design a flat, wide duct, (b) we could build it into the wing, (c) we could take in air at the leading edge, (d) we could use small fans, and (e) each fan could be powered by its own small engine. It's clear that it can be done.
You are arguing that it cannot be done reasonably efficiently.
To start with, you make an argument that in the transonic regime the air coming at the slot in the leading edge would decrease the lift, and you express this idea in terms of a dynamic pressure and a Reynolds number. Without defining those quantities more precisely, the argument can't be made. If the dynamic pressure is defined in terms of airspeed and ambient density, then it doesn't change just because we cut slots into the wing. Same for the Reynolds number. You seem to be confusing the dynamic pressure, which is an abstract single scalar property of the overall airspeed and ambient density, with the real pressure distribution around and inside the wing. The pressure distribution will generally be everywhere proportional to the dynamic pressure, but when you change the airfoil, you don't change the dynamic pressure. You change the pressure distribution.
It's true that above about Mach 0.7 there is "too much air" and the turbofan engine has to slow the air down before it hits the fan. You say that a slot in the leading edge wouldn't work because it would result in less lift. I don't buy that. By opening slots in the leading and trailing edges, effectively we would be dividing the wing into two wings. That doesn't necessarily reduce lift. It changes the flow field and the pressure field, and therefore the distribution of pressure force over the surfaces, but there is still the question of how big the slots are and what the air is doing. The slot width could be calculated for cruise conditions, based on say an L/D of 19 for the unmodified wing and a propulsion velocity speed-up of say 15%. That would mean stealing about 1/3 of the air ((1/19)/15%). The flow approaching the leading-edge slot would be slowing down and diverging vertically, in analogy to the flowtube in front of a normal turbofan, but the slot width would be adjusted to take that into account.
This is another variant on your objection that "stealing" air from the wing lowers lift. But the stolen air doesn't disappear. It interacts with the interior surfaces of the duct, generating lift and drag as it does so. The main difference between it and exterior air is that it is pressurized by the fan. Otherwise, it's just good old, plain old, everyday, garden-variety air.
Your observation about the tail cone is interesting, something I too had considered some time ago. It's a simple, clean way to do boundary-layer ingestion. One big engine wrapped around the circular fuselage, optimized for cruise. Strictly from an efficiency point of view, I never saw any problem with that. And there's no safety issue if you have takeoff-class engines to keep you in the air. You mentioned one small issue, the clearance during rotation and flaring, but that's an easy problem. I also worried that a single large fan would be difficult to hold rigid enough to have acceptably close tolerances relative to the non-rotating fuselage and nacelle. If you do it as a ring, the bearings wouldn't be tight enough, and if you do it as some sort of spoked wheel, then you're slicing up the fuselage structurally and possibly introducing flutter. One reason I like small rotors is that I see less grief in keeping them structurally rigid. But I don't think these issues with the tail cone engine are insurmountable.
I see my small-engine approach as taking the tail cone approach and cutting it open, unrolling it, and spreading it out over the wings. The inefficiencies resulting from smaller engines and fans do not seem all that extreme.
A possible compromise would be to take the small-engine approach and work it into the tail cone idea. A big nacelle wrapped around the tail cone, but filled with small engines and fans. With boundary-layer ingestion, what happens to the choice of axial versus centrifugal fans?
Regarding cruise versus takeoff, you of course have noticed that the parameters driving the cruise problem are airspeed and altitude, whereas the parameters driving the takeoff problem are runway length and altitude. That's why I tend to look first at cruise, because the runway parameters are not under the airplane designer's control. If you design a good cruising airplane for some specified altitude and airspeed, the runway parameters then force you to adjust it in some way. If general, you're forced to add flaps and takeoff thrust.
It's true that the extra takeoff thrust results in dead weight, but that is always the case, even for large engines. A large engine sized for takeoff is bigger and heavier than a large engine sized for cruise. The cruise engines, by contrast, are never dead, because they contribute to total takeoff thrust. So we are really only talking about the dead weight of the takeoff class during cruise. I think of that as a cost of doing business, sort of like the dead weight of the landing gear and flaps, which are also useless for cruising.
I should mention, while we are on the subject of turbofans, that the fan disk does not necessarily have to be perpendicular to the flow. If the flow interacts with the fan at a shallow angle, then we can avoid shock waves on the tips. If you follow this idea to its logical extreme, you would have a spiral flow coming into a centrifugal fan, which could be laid horizontal, and then the flow would spiral out on the other side. The air would never hit any solid surface at a relative Mach number greater than 1. The flat, horizontal fan would fit more neatly into the wing. This idea seems implausible at first glance, because the airflow would follow one complete turn of a helix while inside the duct, and intuitively, that seems more problematic than the standard technique of simply whacking the fan at a 90 degree angle and accepting the consequences. But it might be more efficient.
Your calculation for the Concorde is enlightening. You have to wonder why they never took a hard, hard look at engine efficiency when their fuel was more than half the takeoff weight. Come on, people, weight is weight! How many fuel-efficiency tricks could they have justified purely on the basis of the weight of the fuel they would have saved? If you're going to cruise at Mach 2, you might as well do it efficiently.
But on the other hand, if you've already decided that you're going to leave a sonic boom all the way across the North Atlantic, then you're probably not really interested in fuel economy anyway, right? Obviously the energy in that wave has to come from someplace...
More generally, arguments about energy efficiency in airplanes tend to run aground, because an airplane doesn't acquire energy during flight. When I'm parked on the tarmac at the destination, my kinetic energy is zero and my gravitational potential energy is the same as it was at the origin (assuming the same elevation). Therefore 100% of the energy that was in the fuel is gone. It's up there in the atmosphere someplace. From that perspective, overall efficiency is always 0%. When you talk about the efficiency of the Concorde, you're talking about how efficiently it produces that sonic boom (among other things). But the passengers weren't paying for the sonic boom. They were paying for transportation.
To put it another way, in steady, level cruising, Thrust = Drag. So when you define efficiency as (Thrust * Velocity) / (Fuel Flow per second * Fuel Energy content), it sounds as if (Thrust * Velocity) is some kind of "benefit" that you get in exchange for burning the fuel. But of course, if you substitute (Drag * Velocity), which is mathematically equal, you see immediately that the supposed "benefit" you're getting isn't really a benefit.
That's why I tend to use the pre-factor in the Breguet equation to evaluate a transport airplane, and the pre-factor in the endurance equation for a hovering or loitering airplane. Those pre-factors tell me something useful. The Breguet pre-factor explains why long-range transports fly at Mach 0.8 or so and use turbofan engines. You maximize that pre-factor by increasing the airspeed and altitude along a contour of constant dynamic pressure until you start flirting with Mach 1. Then you use tricks (Whitcomb airfoil, turbofan engines) to get closer to Mach 1 without suffering too much drag.
For loitering, there's no value in high airspeed, and I find it curious that the StratoLaunch concept would use turbofan engines. That doesn't seem like the best way to maximize the endurance or the operating ceiling. I think they just decided to use off-the-shelf engines, without giving it too much further thought. That means they are proposing to design and build a huge special-purpose airplane, which is quite a pricey proposition in and of itself, without carefully considering the choice of propulsion system. A clumsy approach, IMHO.
But as to HOW to optimize an airplane, I look first at the immediate wake. On a y-z plane (x being the flight direction), in the stationary reference frame not moving with the airplane, and just after the airplane has gone through, what does the energy distribution look like? Where's the kinetic energy on that y-z plane, where's the enthalpy, where's the sound energy including shock waves? Once I have a good idea what the energy profile of the wake looks like, I then work forward to the airplane and try to modify it to improve the wake. That approach is not the standard textbook way of thinking about airplane efficiency, but it has the advantage of simplicity. The wake contains only a few kinds of energy. An airplane, by contrast, is generally a complicated beast with lots of little details that have to be thought about. Like the rule that sharp corners, e.g. a sharp corner where the wing joins the fuselage, should be avoided because they generate drag. That's a detail, but to understand it you have to jump through hoops and read literature. It's easier just to look at the wake.
-bob989
I think it's worth examining some of your objections to small engines. I don't see that you have a clear, well-reasoned argument based on fundamental physics or some other very basic objection like the properties of air or the strength of materials. I don't claim that small engines would necessarily be more efficient than or even as efficient as large engines, but you seem to have concluded that they are so much less efficient as to be unreasonable on that basis alone. I don't agree.
To my mind, it's clear that (a) we could design a flat, wide duct, (b) we could build it into the wing, (c) we could take in air at the leading edge, (d) we could use small fans, and (e) each fan could be powered by its own small engine. It's clear that it can be done.
You are arguing that it cannot be done reasonably efficiently.
To start with, you make an argument that in the transonic regime the air coming at the slot in the leading edge would decrease the lift, and you express this idea in terms of a dynamic pressure and a Reynolds number. Without defining those quantities more precisely, the argument can't be made. If the dynamic pressure is defined in terms of airspeed and ambient density, then it doesn't change just because we cut slots into the wing. Same for the Reynolds number. You seem to be confusing the dynamic pressure, which is an abstract single scalar property of the overall airspeed and ambient density, with the real pressure distribution around and inside the wing. The pressure distribution will generally be everywhere proportional to the dynamic pressure, but when you change the airfoil, you don't change the dynamic pressure. You change the pressure distribution.
It's true that above about Mach 0.7 there is "too much air" and the turbofan engine has to slow the air down before it hits the fan. You say that a slot in the leading edge wouldn't work because it would result in less lift. I don't buy that. By opening slots in the leading and trailing edges, effectively we would be dividing the wing into two wings. That doesn't necessarily reduce lift. It changes the flow field and the pressure field, and therefore the distribution of pressure force over the surfaces, but there is still the question of how big the slots are and what the air is doing. The slot width could be calculated for cruise conditions, based on say an L/D of 19 for the unmodified wing and a propulsion velocity speed-up of say 15%. That would mean stealing about 1/3 of the air ((1/19)/15%). The flow approaching the leading-edge slot would be slowing down and diverging vertically, in analogy to the flowtube in front of a normal turbofan, but the slot width would be adjusted to take that into account.
This is another variant on your objection that "stealing" air from the wing lowers lift. But the stolen air doesn't disappear. It interacts with the interior surfaces of the duct, generating lift and drag as it does so. The main difference between it and exterior air is that it is pressurized by the fan. Otherwise, it's just good old, plain old, everyday, garden-variety air.
Your observation about the tail cone is interesting, something I too had considered some time ago. It's a simple, clean way to do boundary-layer ingestion. One big engine wrapped around the circular fuselage, optimized for cruise. Strictly from an efficiency point of view, I never saw any problem with that. And there's no safety issue if you have takeoff-class engines to keep you in the air. You mentioned one small issue, the clearance during rotation and flaring, but that's an easy problem. I also worried that a single large fan would be difficult to hold rigid enough to have acceptably close tolerances relative to the non-rotating fuselage and nacelle. If you do it as a ring, the bearings wouldn't be tight enough, and if you do it as some sort of spoked wheel, then you're slicing up the fuselage structurally and possibly introducing flutter. One reason I like small rotors is that I see less grief in keeping them structurally rigid. But I don't think these issues with the tail cone engine are insurmountable.
I see my small-engine approach as taking the tail cone approach and cutting it open, unrolling it, and spreading it out over the wings. The inefficiencies resulting from smaller engines and fans do not seem all that extreme.
A possible compromise would be to take the small-engine approach and work it into the tail cone idea. A big nacelle wrapped around the tail cone, but filled with small engines and fans. With boundary-layer ingestion, what happens to the choice of axial versus centrifugal fans?
Regarding cruise versus takeoff, you of course have noticed that the parameters driving the cruise problem are airspeed and altitude, whereas the parameters driving the takeoff problem are runway length and altitude. That's why I tend to look first at cruise, because the runway parameters are not under the airplane designer's control. If you design a good cruising airplane for some specified altitude and airspeed, the runway parameters then force you to adjust it in some way. If general, you're forced to add flaps and takeoff thrust.
It's true that the extra takeoff thrust results in dead weight, but that is always the case, even for large engines. A large engine sized for takeoff is bigger and heavier than a large engine sized for cruise. The cruise engines, by contrast, are never dead, because they contribute to total takeoff thrust. So we are really only talking about the dead weight of the takeoff class during cruise. I think of that as a cost of doing business, sort of like the dead weight of the landing gear and flaps, which are also useless for cruising.
I should mention, while we are on the subject of turbofans, that the fan disk does not necessarily have to be perpendicular to the flow. If the flow interacts with the fan at a shallow angle, then we can avoid shock waves on the tips. If you follow this idea to its logical extreme, you would have a spiral flow coming into a centrifugal fan, which could be laid horizontal, and then the flow would spiral out on the other side. The air would never hit any solid surface at a relative Mach number greater than 1. The flat, horizontal fan would fit more neatly into the wing. This idea seems implausible at first glance, because the airflow would follow one complete turn of a helix while inside the duct, and intuitively, that seems more problematic than the standard technique of simply whacking the fan at a 90 degree angle and accepting the consequences. But it might be more efficient.
Your calculation for the Concorde is enlightening. You have to wonder why they never took a hard, hard look at engine efficiency when their fuel was more than half the takeoff weight. Come on, people, weight is weight! How many fuel-efficiency tricks could they have justified purely on the basis of the weight of the fuel they would have saved? If you're going to cruise at Mach 2, you might as well do it efficiently.
But on the other hand, if you've already decided that you're going to leave a sonic boom all the way across the North Atlantic, then you're probably not really interested in fuel economy anyway, right? Obviously the energy in that wave has to come from someplace...
More generally, arguments about energy efficiency in airplanes tend to run aground, because an airplane doesn't acquire energy during flight. When I'm parked on the tarmac at the destination, my kinetic energy is zero and my gravitational potential energy is the same as it was at the origin (assuming the same elevation). Therefore 100% of the energy that was in the fuel is gone. It's up there in the atmosphere someplace. From that perspective, overall efficiency is always 0%. When you talk about the efficiency of the Concorde, you're talking about how efficiently it produces that sonic boom (among other things). But the passengers weren't paying for the sonic boom. They were paying for transportation.
To put it another way, in steady, level cruising, Thrust = Drag. So when you define efficiency as (Thrust * Velocity) / (Fuel Flow per second * Fuel Energy content), it sounds as if (Thrust * Velocity) is some kind of "benefit" that you get in exchange for burning the fuel. But of course, if you substitute (Drag * Velocity), which is mathematically equal, you see immediately that the supposed "benefit" you're getting isn't really a benefit.
That's why I tend to use the pre-factor in the Breguet equation to evaluate a transport airplane, and the pre-factor in the endurance equation for a hovering or loitering airplane. Those pre-factors tell me something useful. The Breguet pre-factor explains why long-range transports fly at Mach 0.8 or so and use turbofan engines. You maximize that pre-factor by increasing the airspeed and altitude along a contour of constant dynamic pressure until you start flirting with Mach 1. Then you use tricks (Whitcomb airfoil, turbofan engines) to get closer to Mach 1 without suffering too much drag.
For loitering, there's no value in high airspeed, and I find it curious that the StratoLaunch concept would use turbofan engines. That doesn't seem like the best way to maximize the endurance or the operating ceiling. I think they just decided to use off-the-shelf engines, without giving it too much further thought. That means they are proposing to design and build a huge special-purpose airplane, which is quite a pricey proposition in and of itself, without carefully considering the choice of propulsion system. A clumsy approach, IMHO.
But as to HOW to optimize an airplane, I look first at the immediate wake. On a y-z plane (x being the flight direction), in the stationary reference frame not moving with the airplane, and just after the airplane has gone through, what does the energy distribution look like? Where's the kinetic energy on that y-z plane, where's the enthalpy, where's the sound energy including shock waves? Once I have a good idea what the energy profile of the wake looks like, I then work forward to the airplane and try to modify it to improve the wake. That approach is not the standard textbook way of thinking about airplane efficiency, but it has the advantage of simplicity. The wake contains only a few kinds of energy. An airplane, by contrast, is generally a complicated beast with lots of little details that have to be thought about. Like the rule that sharp corners, e.g. a sharp corner where the wing joins the fuselage, should be avoided because they generate drag. That's a detail, but to understand it you have to jump through hoops and read literature. It's easier just to look at the wake.
-bob989
