Hi, everyone, and welcome to this new class in performance and specifically Brigette Equation's,
so we will do a bit of an overview of these Brigette equations.
OK, and my idea is actually to introduce some of the mathematical concepts which are useful to the
optimization of a specific aircraft.
So this class I haven't you this is a bit more mathematically oriented.
But however, it's kind of an overview of some different topics that we have covered already in order
for you to see that they can they can be applied directly to the performance and to the optimization
of performance of an aircraft.
Now, let's start.
Breguet equations are mathematical models which will relate a range of an aircraft.
So the distance an aircraft can travel to other parameters, such as geometry of the wings on flight
conditions, for instance, it is usually used to visualize which optimizations can be done to the performance
of the aircraft and how to design the aircraft to fulfill a given range.
So it's kind of a way to relate the geometry of an aircraft in general terms and conditions to the actual
range or capability of the aircraft itself.
This mathematical derivation of this equations is actually quite straightforward and it can be used
as a first approximation of the optimization of the whole aircraft.
Now, if you want to derive mathematically the different equations that we all see, you can just look
for them or you can ask me and I'll try to find and the exact derivation in some in some paper or another,
it usually is not difficult to follow.
OK, so you'll probably get it.
All right.
And in this case, I just wanted to you to visualize the equations themselves.
OK, so well, in fact, study basically the range equation, their range Breguet equation, which is
just defined as a distance, you know that the ranges of distance.
So a distance is equal to velocity.
In this case, it's the velocity of the aircraft multiplied by the time which will be flying.
Right.
So the range is equivalent to the velocity of the aircraft times, the time of flight.
So the idea is in order to optimize the range, we have to optimize these parameters.
Right.
And optimizing this is actually the same as optimizing this, because the flight time can be defined
after some mathematical derivation as the ratio between the lift and drag.
The specific impulse and dysfunction, which is the Napoleone logarithm of the relation between the
initial mass initial wait and final weight of the aircraft.
Now, it's interesting to note that in order to maximize in order to optimize the range, what we will
have to do is actually optimize this product and in this case, we'll just focus on optimizing the flight
time.
So this would be optimizing these elements over here now, optimizing the flight time implies by optimizing
I mean increasing as much as possible.
So if we want a bigger range, what do we have to do is actually increase the flight time.
Right.
So basically, we need to increase as much as possible or have the biggest number as an outcome of these
multiplications.
And by this, what I mean is we need to maximize the lift to drag ratio.
We need to maximize the specific impulse and we need to maximize the Napoleon logarithm of the ratio
between the initial weight and final weight.
Now, let's start by maximizing the lift to drag ratio.
This radio will actually depend on the wind design.
As we saw, airfoils is all the way to relate in the lift and drag.
And in this case, we will.
What we will do is actually discuss the polakov, which is a curve which relates the coefficient of
lift and coefficient of drag for different angles of attack.
OK, so the idea is just to evaluate the value of a coefficient of lift, a certain angle of attack
and evaluate the conversion of drag at the same angle of attack.
And by evaluating the different coefficient of lift and drag as following the angle of attack, we actually
get a curve of all the ratios between the left and right.
Now, what's interesting in this is that you can actually get a minimum drag coefficient of drag, so
you can actually get a small the smallest, the smallest value of drag, which would be something like
this.
You can see that because we have a curve which is defined by coalition of left and coefficient of drag.
This is the minimum since the coefficient of drag.
Is this axis over here.
Right.
So it's like the first value that we get that the smallest value that we get with a coefficient of drag.
Now, what I wanted to say is that if we want to maximize lift versus drag, it is quite intuitive that
we need the smallest drag and the biggest lift.
So we could actually focus on minimum drag as the maximum, the maximum lift to drag ratio.
But this is not the case.
OK, it's quite interesting to note that this is the value for six degrees of angle of attack is actually
the value of maximum lift to drag ratio.
Even if the government of drag is not the smallest.
This actually gives the most the biggest value, the maximize the value of the cooperation of divided
by coefficient of drag.
You can actually see it in this curve over here is actually the same.
This curve is actually the same, but you actually get directly the lift to drag ratio as a function
of the angle of attack.
Right.
So here we have the angle of attack, which is implicit.
We just calculated and we have this curve relating coefficient of lifting, coefficient of drag.
And in this case, we directly get the lift to drag ratio, which is equal to the coefficient divided
by the coalition of drag to angle of attack.
Right.
So we just get the curve itself.
And as you can see, for minimum drag, we get a value.
But it's not optimal.
It's not the best one.
In terms of lift the drag radio, this would be the most the biggest ratio between lift and drag for
six degrees of angle of attack.
Now, the idea then is that we have to design the wing.
Of course, the wing will actually provide this curve.
OK, but what we have to do is ensure that our aircraft will actually be flying at this specific point,
because at this point we will have maximum efficiency in terms of lift to drag ratio and therefore we
will increase.
We will have optimized our flight time.
So it's important to take into account that this parameter actually depends on the wing design, but
also depends on the flight conditions as a function of the angle of attack.
OK, well, that's what we meant to do also is maximize the specific impulse of our engine.
Now, as I mentioned, this depends on the engine design.
This case depends on the wing design and the specific impulse depends on the engine design.
No specific impulse is a definition, OK?
It's just a mathematical definition almost in order to measure the engine efficiency.
And it's just related to the effective velocity of ejection, the increase of the velocity due to the
air going through the engine.
OK, so basically we have just the aircraft in total aircraft which is travelling in a certain speed.
So we have a relative speed between the wind and the engine.
OK, which would be the speed over here and.
Because we have the engine and we actually get more power, we get thrust from the engine, then the
exhaust velocity, which means the velocity at which the air or the wind is actually getting out of
the engine is actually bigger than the relative velocity of the wind with respect to the engine.
OK, so we just get more velocity due to the air going to the through the engine.
And so specific impulse is just a measure.
It is multiplied by a constant, but it's just a measure of the relative velocity increase.
OK, so in this case, you can see that all of this velocity would be the velocity at the end and this
velocity is the initial velocity.
You just make the calculation of the difference between exhaust velocity and initial velocity and you
get kind of the specific impulse.
So you can see that the specific impulse depends on the engine design as a way to define at which velocity
the exhaust velocity is actually going it.
OK, so it's just a measure of how much the engine is able to accelerate the fluid through the engine.
Right.
So this is the specific impulse.
It's just kind of an efficiency for the engine itself.
What else we wanted to do also is maximize this function over here, which is the natural logarithm
of the way, the initial word and the final weight.
Now, this will depend on structural design, basically.
And what this means is that basically, if we want to maximize this function over here, it is equal
to maximizing, OK, all of this here.
Now, let's let's see this.
This is the initial weight and this is the final weight.
So initial weight will be actually the same as the final weight plus the fuel.
Right.
Because throughout the flight we actually consume some of this fuel.
So initially we will have the same weight as finally but with also the fuel.
Right.
So, yeah, the idea is that the difference between the initial weight and the final weight is just
the amount of fuel that we had before and which we have consumed or used in order to propel our engine.
Right.
So in a way, you can actually mathematically you can see that this term cancels with this one.
Right.
But we still have the weight of the fuel divided by the final weight of the aircraft.
And so what this means is that we actually get the Napoléon logarithm of this function over here.
Right.
So imagine that we had no fuel at all if we have no fuel.
OK, this would imply that this is zero because we have no weight of the fuel.
And if this is zero, then this is the logarithm, the napoleone logarithm of one.
And if we go to this, which is the function of the logarithm that Napoléon logarithm.
OK, this is the you know, the geometry of this function.
We can see that for a given X, X would be one.
Right, because this is this was had a zero.
If we consider that there's no fuel, we would actually get the result of zero.
So in this case, we this equation in the case that we have no fuel actually is stating that this is
zero.
So all of this is zero and therefore the flight time is zero.
Does this make sense?
Well, it kind of does or doesn't it?
Because it means that if we have no fuel, then the flight time is zero.
Well, that's kind of you know, it's an interesting analysis of this mathematical form.
Now, let's see the generate the most generic case, which is to have some fuel.
OK, and the relation between the fuel and the final weight is actually important in terms of how much
we can actually maximize the flight time.
Now, in order to maximize this function, which is what we want to do, we have to maximize this function
over here.
And as you can see, this function, which is the algorithm, is actually increases with the amount
of fuel that we put at, because if the fuel is, say, one tenth of the final weight, then this coefficient
would be zero one.
So we would have one plus zero at one and this would be one point one.
So the logarithm then logarithm of one point one we can go to this graph would actually be this value
over here.
However, if we have almost, say, 50 percent of the total weight of our aircraft is actually fuel,
which is improbable, but, you know, just a theoretical model, then this means that this over here
would be fine because the amount of weight of the fuel is actually 50 percent of the total final weight.
So this relation would be five.
If we add one plus not five, this is equal to one point five and one point five would be over here
and you can see that the function is very much higher.
So in the end, what we have to optimize is the weight fuel divided by the final weight, which in other
terms means optimizing the amount of fuel that we have with respect to the total mass of the aircraft.
So another way of seeing this is that we need to decrease as much as possible the final weight in order
for the fuel to be an important factor with respect to the final weight.
So the you know, the solution of this is actually to decrease as much as possible the final weight.
And this is why this will depend on the structural design.
So we need a structural design, which is as light as possible, and it kind of makes sense to it.
Right.
So the flight time will be higher.
If our aircraft is actually has actually less mass, then that kind of makes sense, because if the
mass is very high for the aircraft, you can imagine intuitively that the flight time will be smaller.
So this is the idea.
OK, optimize this radio over here, which means decreasing as much as possible, the final weight and
so decreasing the structural mass of the aircraft.
This is what you need to know.
OK.
So, you know, as a summary of what we saw in order to optimize the range, what do we have to do is
optimize the flight time, which means maximizing the lift to drag ratio, maximize specific impulse
and maximize this factor over here.
OK, so basically we have to design a high lift to drag ratio wing, OK, so it is important to study
which airfoil and 3D configuration will give us the best radio.
Also, we have we need to have a high velocity of ejection for the engines, which we discussed for
the specific impulse.
So high compression ratios is actually very important in the design of the engine and will affect how
much velocity we can actually get out of the engine.
So as I mentioned, so high compression ratios in the compressor and high-temperature fluid due to the
combustion chamber.
That's what we need, high-temperature fluid and high compression radio.
So combustion chamber has to be effective.
And also the compressor has to pressurize the the wet the air in a big way.
OK, we have we need a big compression also, as we mentioned, maximizing the ratio between fuel and
mass and aircraft mass.
So basically studying how to reduce the mass of the structure of the aircraft.
Right.
So I think this is all for the duration of range.
As always, if you have any doubts, if you want to have more insight into the equation or any other
topics, just let me know.