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Lecture – 41
Estimation of Empty Weight Fraction
This is the first step in the initial sizing of the aircraft and unfortunately, we do not have much Leave way here, we have to rely almost completely on history here. Because as I mentioned, the aircraft has not yet been designed, the aircraft is not in front of you, only what you know is the type of the aircraft and hence you have to go purely by historical information.
But we have some help available from historical data. So, according to the procedure suggested by Raymer the Empty weight fraction w´ e=AW0
C KVS KVS which stands for variable sweep.
(Refer Slide Time: 01:04)
A and C are constants which are a function of a particular aircraft type, their values are obtained by a statistical fit of existing data KVSis just a multiplication factor that tells us the effect of providing variable sweep. So, KVS= 1.04 basically tells us that if you provide variable sweep in a aircraft, which you would probably never provide in the transport aircraft, then you have to have a weight penalty of 4% of the total weight.
And if you do not provide variable 3 which is most commonly true for transport aircraft, then you only haveAW0C.
(Refer Slide Time: 01:54)
The values of A and C for various aircraft types have been specified by Raymer in his textbook, so, he categorizes the aircraft type under these various categories. And for each of these categories, there are suggested values of the coefficients A and C to be used. Please note that these values are to be used when the gross weight is in kilograms. There is a separate set of values which are specified in the textbook when the gross weight is in pounds.
Now let us observe a few interesting things here. What we notice here is that the exponent C is always going to be a negative number for all the aircraft types. So, what does this tell us.
This tells us that as W0 increases for a given aircraft type for all aircraft, as W0 increases, the empty weight fraction reduces. In other words, a smaller aircraft which weighs less will have a higher empty weight fraction, a larger aircraft which weighs a lot.
Will have a larger empty weight, but it will have a smaller empty weight fraction. The coefficient A is the one that tells you about the sensitivity of the gross weight to the increase in the size.
(Refer Slide Time: 03:37)
So, let us see this information in a graphical fashion. So, what we notice here is that the jet transport aircraft follow this particular line. Now, I want to put in a word of caution here. In this particular graph, you are seeing the figures where there are these symbols attached to each of the lines. And you may assume that these symbols correspond to actual points. But that is not true.
These symbols are just given to allow you to distinguish between the various types of aircraft because there are so many lines here. So, and we could not even use so many colors. So, we have just included a bunch of symbols, so please ignore the symbols, but use them only as an indicator. So, a jet transport aircraft typically has an empty weight fraction, which can be starting from approximately 0.55 or 55%.
And it can go as low as around 44%. This is only based on historical data. What do we see here, we see here that the flying boats have the highest empty weight fraction, never below around 0.64 and generally as high as 0.71, 072. And the military cargo aircraft, they have the least empty weight fraction, which can be as low as 35%, when they tend to be very large.
Another point to be observed is that this graph has a logarithmic scale on the x axis.
It is not a linear scale, there in mind, it is a logarithmic scale. And the y axis, of course, is just a linear scale. So, on a linear log plot, the trend lines appear to be linear. And we can use them to get an idea about what would be the expected empty weight. So how do we use this graph, let us say you are designing and a jet fighter aircraft and these jet fighter aircraft, gross
weight of this aircraft is expected to be some number, let us say 10000 kilograms.
Then it is empty weight fraction is likely to be around 0.65 by looking at this particular. So, what you do is you come up on the line and then go on the x axis and read the number this is how you use this particular graph. But in our case, unfortunately, we do not have an estimate of gross weight. On the other hand, we are actually going to estimate the gross weight. And
we need to know the empty weight fraction.
But empty weight fraction depends upon the gross weight. So therefore, we have to use this formula which was shown. This particular point shown in the graph this particular point corresponds to going 787 - 8, which has got an empty weight fraction of approximately 0.49. Remember I told you that many transport aircraft they have an empty weight fraction, roughly 50%, which is very much true for Boeing 787.
(Refer Slide Time: 07:05)
This graph taken from the book by Professor John fielding actually gives you better information about the spread. So, we notice here that the aircraft do not fall along a straight line. These lines are just such statistical inferences, you can see that there is a huge spread.
Even if you look at one particular type of aircraft, let us say for example, we are looking at you know, we are looking at large commercial aircraft.
The large commercial aircraft again, this is a log linear graph, they do not follow a straight line is not that they are all on this line. They are above and below this line different manufacturers they have solutions that you get are not exactly along the straight line. So therefore, one has to keep in mind that there are bands and what we use in our estimate is only a number which has come from history. Thanks for your attention. We will now move to the next section.

Lecture - 42
Estimation of Mission Segment Weights
(Refer Slide Time: 00:18)
Let us have a look at how we estimate the mission fuel weight fraction w´ f . Just as a recap, we started with the defining the gross weight of the aircraft into 4 components, the payload weight, the crew weight, the empty weight and the fuel weight, we then converted them into fractions. And we saw that empty weight can be obtained by or empty weight fraction can be
obtained using historical data.
Now, we look at how to get the mission fuel fraction. The fuel that you see in an aircraft or the fuel that an aircraft carries consists of 2 independent components. One is called as the mission fuel, which is the fuel that is planned to be used in the design mission and the other is the reserve fuel which is kept as reserved specified by the regulatory agencies. The mission
fuel depends on the type of mission that you are performing the aerodynamics of the aircraft and what kind of engine performance you have.
Reserve fuel is fuel which is kept as reserve you require it for missed approach, diversion and hold you require it for navigational errors or any weather related route changes which can happen also, there is some amount of fuel which is trapped in the pipelines and in the system.
And although it is there, it cannot be used because we need to have a continuity of fuel when we feed the engine roughly half to 1% of the total fuel is actually blocked in the pipelines and we call it as trapped fuel.
Now, we have to make an assumption here, when we want to estimate the mission fuel fraction and this is a very, very strong assumption. The assumption here is that the loss of weight of the aircraft as it carries out its mission is only because of the continuous fuel consumption and there is no sudden payload drop or there is no sudden change in the weight of the aircraft because of dropping of a payload or any other reason.
This assumption is valid for most transport aircraft, because we do not throw off passengers as we go nor do we lose the weight of the aircraft along the flight for any other reason. But in military aircraft, there are some situations for example, when you have an aircraft which goes on a combat mission and let us say drops bombs or even an aircraft that can have a drop tank and it drops it after the fuel is consumed.
Plus there could be a situation like an air to air refueling in which you acquire additional fuel while you are flying these kinds of missions in which we have a sudden addition or deletion of mass because of reasons like these, you may not be able to estimate their gross weight using this procedure for them there is a procedure that Reamer has suggested in his textbook, which can be taken up and we will discuss that later on when we go for the refined sizing.
So, we assume that the fuel used in each mission segment is proportional to the aircraft weight during that mission segment, because it is slowly being consumed. And hence w´ f is becomes independent of the aircraft weight it only depends.
(Refer Slide Time: 04:01)
So what we do is we number the segments of the mission and we denote a number like 0 at the mission start and then we just number all the segments and for any ith segment of the mission, the mission segment weight fraction will be the weight at the end of the segment upon the weight at the beginning of the segment. So, what we can do is for example, this is a simple cruise profile. So we have 0 to 1 takeoff, 1 to 2 cruise, etc, etc.
So there are totally 6 so what you can do is that total fuel weight fraction will be W6 W0 that means weight at the end of the mission divided by weight at the beginning of the mission and you can obtain this by a multiplication since we are not losing the weight of the aircraft due
to any reason other than consumption of fuel, you can come up with this particular equation where the mission segment fuel of each segment can be multiplied and with that you can get the total fuel weight fraction W6 W0
(Refer Slide Time: 05:29)
Taking this further now, we have to look at various segments can we get some estimate for these fractions. So, the warm up, the takeoff and landing weight tractions are estimated by historical trends and the fuel consumed and distance travel during these segments during the segments of descend is ignored, this is not really true because the aircraft can actually travel
substantial distance during descent, but this distance may be a very small number as compared to the total mission that aircraft carries.
So, for example, an aircraft that carry that travels for 7000 kilometers may cover maybe 150 to 200 kilometers in the descent segment. So, in front of 7000, 100 can actually be neglected, but if it is a very short range mission, if the mission itself is 250 to 300 kilometers, then we cannot assume that the descent segments are ignored. So, the ratios of the mission segment
for warm up and take off for the climb and for landing we for our typical transport aircraft, we can take it from past trends or historical data.
But here there is a word of caution these ratios are only true and meaningful when you look at a standard transport aircraft or an airliner we should not use these missions when we are designing let us say a UAV or even a small aircraft or regional or a commuter aircraft, we have to be very careful this example is basically meant for only airliners. So, therefore, these numbers should be used very carefully and with a lot of thought.
(Refer Slide Time: 07:28)
Now, in climb and acceleration, you can also use a slightly more sophisticated formulation.
This particular formulation is important because many aircraft they have an accelerated climb this is especially true for military aircraft. So, they may go from you know a very small Mach number of approximately 0.1 or 0.2 during the takeoff to a Mach number of let us say 0.9 or even 1 in the climb, and when they do such kind of missions, substantial amount of fuel is
consumed in climb.
In this graph, you have the X axis shows the Mach number at the end of the climb or at the end of the acceleration in a logarithmic scale and on the Y axis we have the fuel fraction of that segment in the linear scale. So, we notice that if you are accelerating up to around you know 0.2 Mach number or 0.15 to 0.2 Mach number, which is the typical value for a transport aircraft at the end of the climb, then the mission fuel fraction for that is indeed a very small number as shown in this particular graph.
But if your Mach number at the end of climb is something like 0.85 or so, then there is a much larger much larger fuel weight fraction.
(Refer Slide Time: 08:57)
So, in the mission profile that we have chosen a simple mission profile, let us see what is the effect of using historical data, we were we had to take we had to do the estimate for 6 weight fractions. But out of those 6 weight fractions, 3 of them are going to be now available to you as an input data or as an assumed value purely from historical trends. The fuel fraction, during the warm up taxi out the fuel fraction during the descent and the fuel fraction during the landing are going to be available for historical data.
So, the problem of determining the empty weight, the fuel weight fraction
W6 W0 for the whole mission is now converted into a problem of only finding out this value and this value. Now, we cannot assume fuel fractions for these values from historical data because if we do that, then we might as well take a fuel fraction from history and go ahead, that is not acceptable, because then we have not considered the requirements in our calculations.
The distance travelled in the cruise, the height at which you cruise and the Mach number at which you cruise these values strongly affect the ratio W3
W2 . And similarly, the amount of time that which you have to loiter the height at which you have to loiter and maybe if it is specified the speed of loiter that will determine the ratio W5 W4. So, therefore, these 2 ratios we
have to now calculate.
In other words, the problem of estimation of the mission fuel segments is now reduced to getting an estimate of these 2 only. And we need to do this because we need to look at the mission requirements to get to these numbers. So, if I just multiply these 3 numbers, I can get 1 number 0.95067. So, the total mission fuel fraction is equal to this number into the 2 ratios which we are going to now determine. Thanks for your attention we will now move to the next section.

Lecture - 43
Estimation of Fuel Weight Fractions
(Refer Slide Time: 00:15)
So, let us see how the fuel fraction is determined here what we do is we use mission profile information and also we used the historical data for the engines.
(Refer Slide Time: 00:28)
Now, to be able to determine the fuel fraction of the aircraft during either the cruise or during the endurance we can take recourse to the Breguet range and endurance equations. Let us have a look at the Breguet range equation basically, this equation helps us determine the relationship between the fuel consumed in a particular segment and certain important
attributes related to the aerodynamics of the aircraft and also to the propulsion and also to its structure.
So, if the fuel consumption basically by definition of tsfc now, tsfc has the thrust especially fuel consumption is defined actually as how much fuel is consumed per unit thrust per unit time and this negative sign indicates that there is a reduction in the fuel with time. So, from there you can get a quick idea that dW or the change in the weight of the aircraft which is assumed to be only because of fuel consumption is basically going to be dW=−tsfc∗T∗dt
And the distance travelled by the aircraft is actually going to be for a small amount of fuel dW the distance travelled will be ds which will be its speed assumed to be constant in that small segment into the time. So, therefore, if I just leave dt here and if I take these 2 parameters that side I get
d t= −dW tsfc∗T
So, if I have to multiply by V I have to put V here. So, you get this expression for the elemental distance covered in 1 particular small segment.
Now, during cruise we are going to assume that thrust is equal to drag and lift is equal to weight because we are looking at steady level cruise. And we also assume that as the aircraft fuel is consumed, while we proceed further we take care of that by assuming LD equal to constant. So, the lift is not going to be the same because the aircraft is going to lose fuel. So, if we assume LD is constant, then you can actually multiply by L and divide by D without making any great change.
So, therefore, you can get this expression that you know since L=W since, so, therefore, this L is equal to this W. So, in other words, you can get the expression ds=−(V∞ tsfc )(LD)dW W and now, if you actually integrate this expression to get the value of total distance assuming now, here the integration has to be done very carefully because depending on what you
assume as constant there are actually 3 different types of range and the Breguet range equation can actually be used.
So, you can assume for example, that LD is constant tsfc is constant and V is also constant.
So, if that is assumed constant then V ∞ tsfc LD are all going to come outside the integration sign.
(Refer Slide Time: 04:17)
And in that case you will get the value of range as R= a tsfc (M LD)lnWinitial
Wfin al
So, here what we have done is we have replaced Va=M So, otherwise it will be R=V c (LD )ln Winitial Wfinal but I have replaced the value of V a =M. Now, this is a very interesting equation. Here it captures a lot of important components of aircraft design.
So, in the denominator we have the tsfc which is a measure of the engine efficiency. We have our factor M LD , which is an indication of the aerodynamic efficiency and we have a ratio Winitial Wfinal
which is indication of the structural efficiency. So, the range is going to be a function of how small the value of tsfc is how large the value of M LD
is and also on the weight ratio.
So, the important point is that 1 equation is capturing the 3 important elements.
(Refer Slide Time: 05:45)
So, what we can do is we can say that the range of the aircraft will be available as a multiplication of the ratio of V cruise / c cruise where c is the SFC. Now, this is 1 place where a lot of students make a mistake in the calculation and I think it is very important for me to point out that particular mistake if you look at this equation, the units on the LHS R range which in SI system are going to be in meters. So, therefore, the units of the RHS also
have to be meters.
This is a ratio of 2 weights initial and final that is dimensionless this is the ratio of lift over drag which is also dimensionless. And here you have velocity over SFC and we want the velocities in meters per second and we want this to be in meters. Therefore, the units of SFC have to be in per second this is very important because, the value of SFC which is quoted by
the various agencies or the engine manufacturers is going to be in Newton’s per Newton's second or pounds per kg or pounds per pound hour etc.
We have to be very careful to use those units carefully and to correctly use now L/D in cruise should be the one that gives you the optimal condition of cruise. And it can be shown that if you have a propeller driven aircraft, turboprop piston prop, then you should use (LD)max when you go for the Breguet range equation and you should use 0.866*(LD)max when you use a jet engine aircraft.
This is a simple exercise in propulsion in aircraft performance, which can be actually done by the students themselves it is very easy to show that the optimum range of a propeller driven aircraft occurs when it is flying at a condition when (LD)max . And for a jet engine aircraft, actually the endurance is maximum and (LD)max condition is used and the range is maximum when L / D = 0.866*(LD)max value.
(Refer Slide Time: 08:21)
The next segment that we need is the loiter segment and again if you use a Breguet range equation, it can be easily shown that the value of endurance can be E= 1 cloiter (LD)loiter ln{Wi−1 Wi } this is nothing but dividing the Breguet range equation by the velocity. But again please notice that this is dimensionless, this is dimensionless. So, therefore, if the endurance has to
be in time unit of seconds, the loiter has to be in per second.
And once again this is a place where many students make mistake they will put the value blindly and they will get some very inappropriate answers. So, it is very important to caution them and here the L/D in loiter is exactly the opposite of that what you have in cruise for a propeller driven aircraft and endurance is maximized when you are flying the aircraft at an L/D corresponding to 0.866 ¿(LD)max.
And the endurance is maximized for jet engine aircraft if you fly at LD
=(LD)max So, in other words, we have to now find out the value of (LD
)max because that becomes a very useful parameter. This is something we do not know in fact, this is our target we are going to use this c loiter and (LD)max and (LD)loiter. So, the values of SFC in loiter and in range a cruise as well as LD in loiter and in cruise. We have to calculate the value (LD
)max . So we will take a short break here. And in the next section when we are back we are going to see how to estimate the value of (LD)max for an
aircraft, because that is needed in the calculations. Thanks for your attention we will now move to the next section.