 Let us have a look at how we can do the initial sizing of a civil transport aircraft and we will take Boeing 787-8 Dreamliner as our test case. In this presentation, we will follow a color scheme. Please understand this scheme so that you know what the various colors mean. General instructions like this will be in brown color. If there are any specified values from the database of Boeing 787 then they will be shown in black color. The blue color will be the assumed values. The calculations to be carried out will be marked with red color so that the moment you see a red color it means some calculations need to be done and this pause button will be a aid to your memory so that you can stop the video when you see this button, do the calculations and then resume and the values which are calculated will be shown in dark blue color. Towards the end, we will compare our data with some source and those comparison values will be shown in green color. Let us understand what is meant by initial sizing. At this point I hope you have already seen the video regarding initial sizing because this tutorial can only be done if you have already seen the video and if you understand what is initial sizing. So, if you have not done I would suggest you stop here, go back and watch those clips. If not you can go ahead. So, as we know initial sizing is the estimation of aircrafts design gross weight W0 before it starts the design mission. The mission profile is specified by the user but there are certain additional requirements which are given by the regulatory bodies. Let us look at the source of the data in comparison for this particular tutorial. We have used the output provided by Dimitri Simos of the Piano software. So, check out www.piano.arrow to where the code is described and we are going to use Boeing Dreamlander sample analysis carried out in 2005 as our baseline. Let us look at some useful data about Boeing 787-8 which we will need to do this tutorial and the source of this data is also from the output on piano by Simos. We are going to look at a baseline case with 224 passengers and the operational items which are going to be carried by the crew members to serve these passengers would be 963 pounds or 437 kg. So, we will add this in the crew weight. The maximum payload weight of the aircraft is 100,000 pounds or around 45.372 tons. It is assumed that the mass of the baggage and the passengers would be 210 pounds and for the crew members it would be 200 pounds each. Now, from the source given by Dimitri Simos, we know that the crews specific fuel consumption of this aircraft in loiter and cruise phases are specified as mentioned. We have mentioned here the units in the SI system. The data is actually provided in terms of pounds per hour divided by pound force but we have converted that number into milligrams per Newton seconds and of course the wing aspect ratio is 10.58. Let us look at the mission profile of Boeing 787. The mission profile that we will consider will be a simple one which will involve several segments. So, the first segment will be the warm-up, taxi out and take off. The next would be the climb segment. After that, we have a cruise segment and it is mentioned that the cruise altitude is 37,000 feet, mark number is 0.850 and the total distance to be covered in cruise is around 8,000 nautical miles and then we have the descent segment. We descend down to the loiter altitude and then there is a loiter for about 30 minutes after which you have a missed approach and then you divert the aircraft and travel a distance of 200 nautical miles at a mark number of 0.535 while flying at an altitude of 23,117 feet. These numbers have come from the source which I have already mentioned to you and finally the last segment is the approach and land segment. So, this table shows you the data to be used and because it is in black color, it is understood that this information is specified by the user as a mission profile. One more point you need to know which is the reserve fuel fraction, how much is the additional fuel to be carried beyond the mission fuel that is 5 percent. So, the first step initial sizing is the takeoff weight buildup. So, the takeoff weight can be considered to be a summation of 4 items, the crew weight which includes the items, the operational items, the payload, the fuel and the empty weight. The empty weight would be the weight of the structure, the engines, the landing gear, equipment, avionics, etc. Now, in this particular situation, W crew is available from operational requirements as well as some regulatory requirements and W payload is a specified number. So, therefore, if we can somehow estimate W fuel and W empty, then we can get a first estimate for W 0 which is the initial sizing. So, the problem now is to determine the values of W fuel and W empty. So, these are the two unknowns which are to be determined. The equations for initial sizing are very straightforward. As I mentioned, you can replace W 0 by a summation of 4 terms. But since 2 of these terms are specified by the requirements or the mission, we keep them in the numerator and we take the remaining 2 terms in the denominator and then we divide by W 0 on both sides. So, we get 2 ratios, the empty weight ratio and the fuel weight ratio. So, now since the numerator is known, therefore, to calculate W 0, we only need to calculate We bar and W F bar which are the remaining 2 unknowns. Let us see how the crew weight for Boeing 747-8 is estimated. There are 2 kinds of crew, there is a cockpit crew inside the cockpit and there is a cabin crew inside the cabin. As per regulatory requirements, there have to be minimum 2 pilots in the cockpit. But for durations more than 8 hours, we need at least 2 more. In our case, the flight is around 8000 nautical miles. So, surely it will travel for more than 8 hours. So, we will assume that occupy the entire space of 4 crew members. So, there will be 4 crew members in the cockpit. In the cabin, we require 1 crew member for every 35 passengers to ensure their safety during emergencies and also their comfort and to provide them services. In case of Boeing 787, there is a provision for 7 crew members. So, we will assume that all 7 are present, which means there are totally 11 crew members each of whom is 200 pounds for their weight and for their baggage. So, therefore, the crew weight turns out to be 1000 kilograms. But remember, we have to also carry operational items and this has been specified as 963 pounds. So, therefore, the total crew weight would be 1437 kilograms. Let us look at the W payload estimation. In this case, of course, it is already specified. But just to complete our information, the payload consists of passengers, weight of the passengers, weight of the baggage and the weight of cargo. In our case, we have been told that each passenger weighs 210 kilograms. So, when we calculate the payload weight in pounds, we will use this simple formula. Converting it into kilograms, we can use the formula as shown. For Boeing 787-8, we are told that the maximum payload is 100,000 pounds. So, since there are 224 passengers, therefore, the payload with all the passengers and their baggage will be 21347.2. And in our design mission, we assume that we do not carry any extra cargo. However, because we can carry a total of 100,000 pounds, therefore, if required, the cargo can be carried to the tune of 24 tons approximately. But in our case, we are going to assume for the design mission that you are only carrying the 224 passengers and their baggage. Now, let us see how we estimate the empty weight fraction. As I mentioned, two unknowns are remaining, the empty weight fraction and the fuel weight fraction. For the empty weight fraction, we normally resort to historical information. So, if you refer to the standard textbook by Daniel Rehmer, the formula is A, W0 power C into KVS, where A and C are constants. And their values are different for different aircraft types. And we obtain these values from statistical curve fits, as I will show you very shortly. The term KVS takes care of the additional weight due to variable sweep. So, in our case, there is no variable sweep. It is a fixed wing aircraft. So, this factor will be 1.0. However, if we design an aircraft with variable sweep, the additional weight because of that would be approximately 4% of the empty weight fraction. So, therefore, we have to add a factor of 1.04. So, if you look at the textbook by Daniel Rehmer, there is a table given where the coefficients A and C, both in metric units that is in kilograms and in pounds are specified. The coefficient C remains the same, whether you use the power coefficient C remains the same, whether you use the value in pounds or in kilograms. So, since we are going to work in SI system, we will not worry about the units in pounds. So, we will go into metric. And since our aircraft is a jet transport aircraft, we have to use the values as 0.97 for A when W0 is in kilograms and minus 0.06 for C. Let us have a look at how the empty weight fraction trends are there for various aircraft type. So, in this particular figure, the largest coefficients are generally for flying boats. You can see they can go as to as high as 0.7 followed by jet fighters, jet trainer, general aircraft, general aviation, twin engine aircraft, powered sail plane, twin turboprop, agricultural aircraft, home-built metal or wood aircraft, home-built composite aircraft, sail plane, unpowered, general aviation, single engine aircraft. This is the curve for jet transport with which we are interested. Notice that the empty weight fractions can go as high as nearly 0.53 or 0.54 depending on the aircraft weight. And the lightest fractions are for the military cargo, which are seldom beyond a value of 0.44 or 0.45. So, the highest empty weight fractions are seen for the flying boat and the lowest empty weight fractions are seen generally for the military cargo aircraft. So, this entire figure is shown in one shot in this particular graph, which is taken from the textbook by Raymer. So, we have seen, so what we need to do is we need to choose a line which is relevant to our type. In our case, it is jet transport. If we assume some value of W0, for example, if we assume the weight to be 100,000 pounds, 100,000 kgs, sorry, then we can estimate WE bar directly as a ratio. So, if the aircraft weight is 100,000 kgs, then the empty weight fraction is around 0.49. Now, what do we do for Boeing 787-8? Unfortunately, we do not know its gross takeoff weight, that is what we have to determine. So, therefore, we cannot use this method directly, we cannot get the empty weight fraction as a number, we have to use this expression because in our case, W takeoff is itself a variable which has to be determined. So, therefore, we will use the expression WE bar is equal to 0.97 WTO power minus 0.06. Let us look at how we estimate the next variable which is the mission fuel fraction. So, just a recap, the fuel in the aircraft is basically mission fuel and reserve fuel. The mission fuel depends on the type of the mission that you are flying, the aerodynamics of the aircraft and the engine SFCs. We need reserve fuel for three major contingencies, either for missed approach, diversion or hold or some errors in navigation because of and also root weather effects and plus a small amount of fuel will be trapped in the pipelines which will not be available to you. So, in our method that we follow for initial sizing, we have a very fundamental assumption that the mission fuel segment weight is proportional to the aircraft weight in that segment. So, a heavier aircraft is going to consume more fuel and head hence give a higher fraction and also we have to assume that the value of this view is independent of the general aircraft weight. It only depends upon the weight of the aircraft in that specific mission. It does not depend upon the weight of the aircraft in the other missions. So, going ahead, here is our basic simple mission profile. So, we number the segments from 0 to 7 with 0 being the start and 1 being the takeoff and 2 being etc. etc. So, for the ith segment, the mission segment weight will be wi upon wi minus 1 and the total fraction will be w7 by w0 in this case, which can be obtained by just a multiplication of the individual mission fuel weight segment fraction. So, because of the assumption that we have made this particular equation becomes very simple. So, all you need to do now is calculate using some method or by assumptions the weight of the aircraft at the end of the mission divided by the weight of the aircraft at the beginning of that particular segment. So, for every segment, we have to now calculate the ratio as weight at the end of the segment upon weight at the beginning of that segment. So, for takeoff, we are going to use some historical information. For climb also, we will use some historical information. For descent, we are going to ignore the weights. So, this number w4 by w3 will be equal to 1. And similarly, for the land for the approach and landing, we are going to assume from historical data. So, three of the mission segments are assumed from historical data. One of them is assumed to be a segment which does not consume any fuel. So, what is remaining now is three segments w3 by w2 that is the cruise segment, w5 by w4 which is the loiter segment and w6 by w5 which is the diversion segment. So, all we need to do now is to calculate the segment fuel fraction for these three segments. And then we can multiply all these terms together to get the total fuel fraction. So, as I mentioned for three of these segments, that is the warm-up takeoff and landing and also climb. We are going to assume it from this table given by Daniel Raymer. And we will also assume that the fuel consumed as well as the distance travelled during the descent segments is going to be ignored. So, what happens when you use historical data is that in this mission, this was the basic equation and three of these ratios, rather four of these ratios are now replaced by values available from historical data. So, if you multiply those three that is 0.97 into 0.985 into 0.995, you will get 0.9506. In other words, w7 by w0 is simply going to be a multiplication of the fuel fraction in three segments, the cruise segment, the descent and sorry the loiter segment and the diversion segment. So, before that we need to estimate the L by D max and again we look at historical data because there is no other way you can do it at this stage of the design when you do not know much about your aircraft. So, first of all let us look at what are the approximate values of L by D cruise L by D max. So, for a commercial jet transport it is generally specified that the values of L by D max would be nearly about 15 to 18 and the average would be 14.4. So, if you do not have any information or you do not have any data about the aircraft, you are safe to assume these values. However, please note that modern day aircraft have a very high L by D max values typically of the order of 20. So, here is a graph taken from Ramers textbook which talks about the wetted area ratio that is S wet by S ref for various aircraft types on the basis of their shapes. So, what we do is we eyeball our aircraft in this particular figure. We have a rough idea right now about how our aircraft will look like. So, you just locate where your aircraft will roughly fit and then just read the value of S wet by S ref. So, for Boeing 787-8 where do you think it will fit? So, I think the aircraft figure is very similar to Boeing 747 except that Boeing 787 has got a much more slender wing compared to Boeing 787. So, that is why it is positioned slightly below and slightly to the left of Boeing 747. So, it so turns out that the value of S wet by S ref at that particular point turns out to be 6. So, roughly it is a whole number. So, that means you calculate the reference area of this aircraft multiply by 6 you will get the wetted area. Now, once you know the wetted area you can calculate the wetted aspect ratio and then there is this figure which shows the typical L by D max values for various types of aircraft as a function of the wetted aspect ratio. So, first you calculate the wetted aspect ratio which is aspect ratio divided by the ratio that we got in the previous slide S wet by S ref. So, we locate the applicable line for us and then we estimate the L by D max. For Boeing 787 dash 8 the line would be the top most line which is for civil jets. We know that the wing aspect ratio is 10.58 and just now we have seen that the S wet by S ref ratio is going to be approximately 6. So, please pause the video here and calculate the value of the wetted aspect ratio it will be 10.58 divided by 6 the number comes out to be 1.76. So, then in this graph what we do is we start from 1.76 go to the appropriate line and then take a left turn and hit the vertical axis we see that the value of L by W is estimated to be approximately 21. So, this is how you can estimate the max L by D of the aircraft. Now, fuel weight fraction is estimated using mission profile and the historical data for engines. So, first let us look at there are 3 segments for which we need the fuel fraction in first is cruise the other is the loiter and third is the diversion which is also a cruise. So, for W3 by W2 ratio we assume that we can follow the Breguet range equation which is as shown in the screen. So, what you need is V cruise and SFC cruise L by D cruise and if you know the value of R then you can invert this expression and you can easily get the value of W i minus 1 by W i where R is the cruise range C cruise is the specific consumption per second. Remember the units are very important notice that the units of velocity are meters per second and the units of range are meters. So, therefore, to balance the units on both sides since L by D is dimensionless and log of weight ratio dimensionless the units of C cruise have to be per second then in that case it will balance the units on both sides. So, the value of C cruise given in the requirements or in specifications is not in the units of per second but we will convert it. And since we are looking at optimization of the fuel fraction for range of a transport aircraft we know that the L by D in cruise should be approximately 0.866 times L by D max. So, let us see how to estimate the W3 by W2. So, what are the specifications? It is specified that the R cruise is 8034 nautical miles which is 14887 kilometers. We also know the cruising altitude and you also know the cruising mark number. So, first thing we do is we calculate the value of sonic speed at that particular altitude. This you can get from the atmospheric tables or you can calculate based on the value of temperature. L by D max was just previously calculated as 21. SFC cruise is given as 14.9 to milligrams per Newton second. This is the value inputted from the piano data it has to be converted into per second by multiplied by 9.807 that is G and dividing by the 10 power 6 to convert the milligrams into kilograms. So, let us calculate the values of V by V L by D and W3 by W2. So, V is simply mark number into the sonic speed which is 250.81 and L by D cruise is 0.866 times 21 which is already known. So, by inverting the equation you can simply get W3 upon W2 is equal to e power minus R into C by V into L by D and we get the value of W3 by W2 as 0.6205. Moving ahead let us look at the fuel fraction and loiter segment W5 by W4. Again for this we use the Brighay endurance equation 1 by C L by D log of weight ratio. Here endurance E is in seconds and hence the SFC C lighter also has to be in per second. L by D and log weight ratio are dimensionless. Now, since we are looking at turbo jet engine aircraft the L by D in loader should be equal to L by D max. So, let us see how these numbers span out for Boeing 787-8. The endurance time is half an hour or 800 seconds and L by D is 21 SFC is already given in during the loiter segment. So, just calculate the value of W5 by W4 in the same fashion. So, first you calculate L by D in during the loiter which is equal to L by D max which is 21 and then you put in E power minus endurance into SFC divide by L by D you get the value of 0.9901. So, this is a very small loss very small loss in the fuel fraction during a half an hour cruise. Then you go to diversion segment. Diversion segment is like a cruise segment. So, therefore the same formula applies the same condition of 0.866 times L by D max also applies. Only thing is that the V cruise SFC cruise L by D cruise will change because it is at a different operating condition. So, let us see how it is calculated for diversion we are given that the distance to be travelled is 200 nautical miles at a height of 2 to 117 feet at a Mach number of 0.535. So, L by D max was 21 as obtained earlier SFC during diversion is given. So, you can convert it to per seconds and then you just calculate the values of V L by D and weight ratio. So, V is obtained by calculating T first as gamma 288.16 minus the lapse rate into distance. Then when you know the value of T, root of gamma RT is equal to A and since you know the Mach number you can get the value of V and L by D is also available to you as an input. So, with that you can get W6 by W5 is equal to 0.9824. Let us estimate the mission fuel fraction for Boeing 787. This is our equation which had the putting together of the various constant terms. So, at this point I think you should pause and have a look at the numbers and calculate the values. So, we get it as 0.5737 and to bring in the fact that we need to carry 5 percent more mission fuel, we are saying that W of I W0 is equal to W0 bar which you obtained earlier to be changed slightly and the index of point it has to be changed. The result fuel fraction is 5 percent. So, therefore WF bar is equal to 1 plus 0.05 into 1 minus 0.5737. Please take a pause and calculate the value. The value comes out to be 0.4476. So, this is our equation the master equation for the initial sizing in which we have replaced WE bar by AW0 power minus C WF bar remains and we have just reproduced here the values from the previous slides. So, W crew is the weight of the crew members plus the operating items which is 1437 kg. W pay is the design payload. This is the payload of these 224 passengers at the rate of 93.2 kgs each and the values of A and C from Ramers table are taken directly here for a jet transport aircraft and we have just calculated the value of fuel fraction, mission fuel fraction as 0.4476. So, what we do now is we now solve iteratively. We assume the first value of W0 as 4 times the payload plus crew. So, while doing that you can have LHS as that value then you calculate the value of 0.97 times W0 power minus 0.06 which is the WE bar and with that you can calculate the RHS. You can see there is a huge difference. So, in the next iteration what you do is you can take the average of the new RHS and the old LHS and repeat the calculations and you keep on doing this iteration the few times few more times and finally, finally, finally it is going to converge to a value of 247631 kgs. Now, let us see the comparison between what we got and what has been quoted by Piano. So, for the first segment that is a warm-up text editor we have estimated the weight to be 7431 kgs whereas Piano it is only 111 further is a gross overestimation. For the climb segment we have obtained 3604 whereas the value quoted is 4, 3, 2, 3. So, there is a under estimation bias. During the cruise the value obtained by us is overestimated by 22%. During the descent we have ignored the fuel consume whereas in Piano there is a fuel of 216 kgs. So, of course there is a very large mismatch. For the loiter segment there is a reasonably good match of around 11%. We have slightly underestimated but for the loiter segment we have overestimated by the same amount approximately 12%. So, this is also okay. And again if you go for the landing we have done a gross overestimation because we have used Reimers formula with that you get a number of 714 whereas the actual value given by Piano is only 181 kgs. Finally, the reserve fuel fraction will be overestimated because there is a gross mistake in the estimation of the fuel and hence there is also an overestimation in the mission fuel in the reserve fuel. So, the total fuel estimation is around 30% beyond. Now, we have to worry we also look at the ratios. So, empty weight ratio has been underestimated the actual value is 0.4959 but we have got 0.4604. Fuel weight ratio as I mentioned there is a 15% excess. Empty weight has also been underestimated and fuel weight has been overestimated. In short there is a 15% overestimate in the gross weight. Now, why are there so much errors in the mission fuel segments as compared to Piano? The reasons are very clear. First of all in the warm-up taxi out and takeoff Reimers assumes a constant number of 0.97 whereas actual values are much lesser. During climb in the Reimers approach we have again a constant fraction of 0.985 actual value is different because climb happens in many, many segments. And in cruise we have assumed a single cruise segment whereas if you look very closely at the data we find that the actual aircraft profile involves a step. So, there is a first cruise at 270 nautical miles at 37,000 feet then you climb up by around 4000 feet to 41,000 feet and then you cruise for the remaining portion. And in descent Reimers approaches the fuel in descent whereas that is actually consumed. So, we have to now look at how to modify our calculations and for that we will take the values for the, we will take the values from the table given by Piano and try to modify the values according to that. So, here is the modified mission profile as suggested by Simos in his output. So, the first one is the takeoff segment, the second one is the climb segment, the third one is a short cruise at 37,000 feet at a constant Mach number. Then there is a small gain in altitude of 4000 feet and then there is a bigger cruise of 5 to 45 nautical miles at 41,000 feet but at the same cruise Mach number and also the same speed and then there is a descent to the loiter altitude of 5000 feet. Then there is a loiter for half an hour after that there is a missed approach and then a diversion. Finally, we have landing. So, we are going to now calculate the mission, modified mission numbers for this particular profile. So, for better estimation of W1 and W0, the weights in the initial first segment and the last segment, first let us look at takeoff and initial climb. In this the first one is that the warm up and taxi out fuel is not included in the numbers, actually it is a separate value. Simos has provided the fuel in takeoff as 458 pounds and fuel in the initial climb of 352 pounds. So, with this we get some numbers and the takeoff weight is 476,000 pounds. Hence we can estimate that the weight in the initial segment ratio is only 0.9983 as compared to 0.97 given by Ramer. Similarly, in approach and landing, the landing fuel is neglected actually and the weight of the fuel consumed approach is 263 pounds and that in taxi in is 167 pounds. The landing weight at the end of the mission is given as 303524 pounds. Hence, the weight ratio during approach and landing is only 0.9986. Now, let us look at a better estimation for climb fuel fraction that is W2 by W1. Now, if you look at Ramer's textbook, there is a improved formula available where if the Mach number of the aircraft at the end of the climb is known that is called as MN climb, then you can calculate W2 by W1 by this particular expression. So, what you can do is for Boeing 787-8 what we will do is we know that the Mach number at the end of the climb is 0.85 and here is the graph from the textbook by Nikolai and Karikner where they have shown this particular fuel weight ratio for the accelerated climb or climb segment. So, in our case, the Mach number at the end of climb is 0.85. So, with that if you proceed on this line, you can get the weight fraction as 0.978 whereas Ramer has taken a weight fraction of 0.985. For better estimation of the cruise fuel fraction, there are two segments. In the first segment, we go from 2 to alpha where the range is 2722, the velocity is 488 knots to a true air speed, SFC is given and L by D is given. So, with that you can get the W alpha by W2 as 0.8682. Similarly, for the second cruise segment which is from beta to 3, we have been given the value of range as 5245, the cruise velocity is the same, SFC is slightly different and L by D also is slightly different because the aircraft is lighter. So, putting in the equation, we can get the weight fraction as 0.7580. So, if you want to get a better estimate of the weight ratio during the cruise, you just multiply these two numbers and whatever number you get would be a better approximation than the one that we got earlier. Similarly, if you look at the revised mission profile, now we have 1 by 0, 2 by 1, alpha by 2, beta by alpha, where beta by alpha stands for the small climb segment of 4000 feet, then 3 by beta, then 4 by 3, 5 by 4, 6 by 5, 7 by 6. Now, what we can do is looking at the number that we obtained earlier, we can just insert these numbers 1 by 1, W1 by W0, the fresh 1 was 0.9983, W2 by W1, the fresh value was 0.978, then W alpha by 2 was 0.8682, beta by alpha we ignore because it is only 4000 feet, there is of course some fuel consumed, but in this big picture we can ignore that and we have no other way of estimating it. The second cruise segment, we get a fuel fraction of 0.7580, descent segment is again 4 by 3, we again ignore the value, 5 by 4 is the same as before, 6 by 5 is the same as before, 7 by 6 is a modified 1.9986. So, with this, the new value comes to 0.6241, which I think you should calculate by multiplying all these numbers. So, therefore, we can get the Wf by W0 or W fuel bar, which would be 1 plus RFF times 1 minus weight ratio during the mission, the RFF remains same. So, pause and calculate the value of Wf bar, the number comes out to be 0.3947. Now, let us also look at how we can get a better estimate for the empty weight fraction. Empty weight fraction is very important because it is a very large fraction. Raymer gives a simple formula in terms of empty weight fraction, in terms of the max takeoff weight. Here is a graph taken from the textbook by Nikolay and Karikner, which plots the empty weight fraction, empty weight versus takeoff gross weight in pounds for various transport aircraft. These are all conventional metallic structure. Now, let us look at some aircraft from the Boeing family in this. So, these 4 dots correspond to 4 aircraft from the Boeing family, which were designed and used a little bit few years ago. And these 3 dots represent the diagram, the values for the aircraft which are little bit recent. So, we notice that the 3 aircraft that have been designed recently by Boeing, they follow a trend line which is slightly different from the general trend line. So, that number the general trend line is 0.911 times W0.947 that is what Nikolay and Karikner talks in general. But if we look at the data of these 3 contemporary aircraft and if you actually try to obtain the equation of this line, it turns out that the empty weight is equal to 0.775 weight times W0 power 0.964 and the empty weight fraction is as shown on the screen. So, if we use this equation, we will get a better estimate. So, the revised gross estimate can be calculated by using the same formula. This is the original number, originally we had 0.97 into W0 minus power minus 0.06 minus 0.4476. But now we know that the new fuel weight fraction is 0.3947 and the new coefficients for W0 are 0.7758 and minus 0.036. So, with this if you solve iteratively with the same initial assumption, finally, the value would converge to 213770 which is a much better estimate for the gross weight as compared to obtained earlier. Hope your calculations were matching with this. Thanks for your attention.