 Hello, let us look at how we estimate the lift coefficient for a long range transport aircraft using Boeing 787 Dreamliner as an example. So here is a picture of this beautiful aircraft belonging to ANA Airways. Let us look at the color scheme in this presentation. The general instructions are going to be given in brown color. The specified values if there are any would be shown in the black color. The values which are assumed based on the existing literature or any online source which we follow as constant data or a given data will be in this light blue color. The values that have to be calculated would be shown in the red color with this symbol. So wherever you see this symbol in the screen it is a hint for you that you have to pause the video, do some calculations and then proceed further. The calculated values will be shown in this dark blue color and in the end if we do a comparison with any data which is reported then those comparison values will be shown in the green color. There is some useful data about Boeing B787-8 which we will use in this particular analysis. Notice that the bottom two entries in this table are the maximum lift coefficient at takeoff also called as CL max TO 1.91 and the maximum lift coefficient at landing CL max land 2.66. These two values are the ones that we are going to determine in this tutorial. So we will compare our calculated values with these two numbers in the last slide along with another parameter called as the wing Oswald efficiency E. So talking about Oswald efficiency there is a formula that can be used to estimate the value of Oswald efficiency for an aircraft while operating at any Mach number M. This formula taken from the textbook by Professor Dennis Howe, it is a long formula which has several aircraft related parameters. Let us look at these parameters one by one. The first parameter to be considered is the Mach number shown here by MA. The formula shows that the Oswald efficiency depends as the sixth power of Mach number and then we have lambda wing which is the wing taper ratio. We are going to define a parameter called as or a function called as F lambda wing which we will calculate. A wing is the wing aspect ratio, T by C wing is the wing thickness ratio. Now an aircraft has several thickness ratios there is one at the root, one at the tip, maybe there is one at the mid break. In this case we are going to use an average thickness ratio value. N E stands for number of engines, lambda 25 wing stands for the quarter chord wing sweep. So once we have the value of these parameters we can use this equation to calculate the value of E. Now the function F lambda wing is calculated in terms of the wing taper ratio lambda. So this is the formula and now let us calculate the values for Boeing 787-8. For that we need to recall some data. So it is given to us that the cruise Mach number is 0.85, the lambda wing is 0.1528 the taper ratio wing aspect ratio is 10.58, the wing T by C is 9.4 percent, number of engines is 2, the sweep at quarter chord is 32.2 degrees. So with this what we can do is we can divide this formula into 3 separate terms and these terms are color coded as the purple color the term representing A which is 1 plus 0.12 times M A to the power 6. The second term is the term B which is in the center the large term highlighted in the brown colored box and then there is a term C which is highlighted in the gray colored box. So what we need to do is we need to actually calculate the values of various parameters. So as you can see there is a pause button on the screen. So I would request you to pause the screen and calculate the value of F lambda wing where the formula for F lambda wing is shown in the bottom of the screen with the green colored box. So it is 0.005 times 1 plus 1.5 times lambda wing minus 0.6 the whole square. So I would request you to use your calculators and obtain this value for lambda wing of 0.1528. After that you can calculate the value of A, B and C and once you get the value of A, B and C then E will be nothing but A into bracket of 1 plus B plus 3 the whole item that you get 1 upon that. So let us get the values. So F lambda wing is 0.0065 the term A 1 plus 0.12 power 6 turns out to be 1.0453 the term B in the center turns out to be 0.2924 and the term C on the right gray colored box turns out to be 0.0821 putting all these values together the value of E turns out to be 0.6961. So we need to remember this value because we will compare this value with the quoted value. Let us first start with estimation of the lift curve slope for which we will use this formula where A is equal to dcl by d alpha. So the dcl by d alpha is estimated in terms of the parameters like wing aspect ratio A and then the parameter beta which is root of 1 minus m square and delta m which is the sweep of the maximum thickness line. So in our case beta will be root of 1 minus m crew square which will be equal to 1 minus 0.85 squares please pause your screens calculate the value and write it down beta turns out to be 0.5268. Next you calculate the lift curve slope A in which the values of the various parameters like A, beta and delta m have already been written down for you on the right hand side. So all you need to do is to just solve this expression and get the value of A the lift curve slope the answer is 6.327 which will come in per radians and if you want to convert it to degrees you will multiply by pi and divide by 180. So that will be 0.1104 per degree. Let us now try to estimate the cl max value for this aircraft during takeoff and during landing. We see on the screen 2 pictures of the Vietnam Airways Vietnam Airlines Boeing 787 aircraft just after takeoff in a very steep climb very clearly showing the working of the outboard flaps and the inboard flaps and here we have the same aircraft when it comes into land. Before we estimate the values let us get an idea about what to expect by looking at the past literature and searching the typical values of maximum lift coefficient. So if we first look at unflap wings the for a for a swept wing with the very high sweep of 60 degrees typically the values of cl max are nearly 0.75 if on the other hand you have a 30 degree sweep the value comes out to be nearly 1.25 and for 45 degree sweep the value turns out to be 1.0. So broadly speaking if you have unswept wings then the value is 1.5. So before we go ahead with calculating the values of the maximum lift coefficient let us get an idea about what are the typical values based on the literature that is available. If we look at unflap wings we see that wings with very large sweep of 60 degrees or so they have a very low value of cl max of nearly 0.75. If the sweep is reduced to 45 degrees it comes out to be approximately 1.00 if you reduce it further to around 30 degrees it comes to 1.25 and if you reduce it to 0 degrees or take an unswept wing then you get the value of cl max is 1.5. So what we see is that by sweeping of wing to about 60 degrees you are reducing by 50% the maximum lift coefficient of the base aircraft. As far as we are concerned our aircraft has a quarter chord sweep of around 32.2 degrees. So the value that we expect for cl max will be approximately 1.25. When you have flat wings then with a plane flap you get 1.75 if you have slots in the flap you get a substantial improvement and the cl max becomes nearly 2.25. For a fauler flap you have a further improvement because not only you create a slot but you also have larger area by moving the flap behind. If you put two slots in the flap then you get further improvement from single slot to 2.75 of course with additional complications because when you have multiple slots in the flaps then the mechanical gearing and the design of the system becomes a little bit more complicated and complex. On the other hand if you go for two slotted flaps double slotted flaps and if you use slats also you get a further improvement and cl max is around 3 similarly if you have a triple slotted flaps with slats you get get up to even 3.5. If you want to go beyond 3.5 then you need to resort to certain you know special approaches for example you can have a flap with upper surface blowing or USB where you can aim to get cl max values of nearly 5. In our case our flap is actually a slotted flap but with some improvements so the maximum value is expected to be in between the fowler flap but better than a double slotted flap. So, let us look at the estimation of the wing cl max for wings with low quarter quad sweep with aspect ratio more than 5 and with taper of nearly 0.5 and those who have very large flaps and that is true for aircraft like Boeing 787-8 which are shown in the screen. So for these aircraft the wing cl max can be almost 90% of the aerofoil cl max. So the 3D effects because of aspect ratio because of taper ratio and because of sweep are not going to substantially reduce the cl max value if the aerofoil gives you cl max of 0.9 that is a 2D cl max the 3D value will be almost 90% of that and most airliners that we see they fall in this category. But when you have partial span flaps not covering the entire span but partial span flaps then the cl max is calculated as approximately 90% of the cl max of the flap area into the ratio of s flap by s ref where s ref is the total wing reference area plus cl max for the unflapped area and the ratio of unflapped area upon the wing reference area. So in proportion to how much of the area is a flap area you can actually a portion the approximate value of the cl max to that particular portion and just add them together and take around 90% of that that would be a good estimate for cl max of the aircraft of the wing. Let us see the effect of sweep on cl max here is a graph taken from the seminal textbook by Daniel Raymer on aircraft design which showcases the effect of increase in the quarter chord sweep which is on the x axis on the maximum cl values first of all for an aerofoil with an aircraft with no wing with no flaps such as the bottom most line and then you can see how cl max is increasing when you have a plane flap and then you have a slotted flap okay then you have how the flap etc etc till you go to the double slotted follow flap and flat so if you do not have any sweep you can get the values as high as 3.5 3.4 and with the with the advent of sweep back these values are going to only version. Now in our case our aircraft has a sweep of 32.2 degrees and it has got a basically if you look at an unflap wing to get the base cl value you just take the sweep and we read off the value the value comes out to be nearly 1.25. So for us the wing will have the cl max of 1.25 without the effects of sweep and other parameters okay then we calculate the effect we calculate the flap area so for that we need to understand the definition of the flap area and the unflap areas. So whether you have a leading edge flap or a trailing edge flap or both any area that comes under the influence of flaps whether they are leading edge flaps or trailing edge flaps is called as a flap area and in case of a trailing edge flap the area upstream of the flap is also counted and in case of a leading edge flap the area downstream of the flap is also counted. This figure has been taken from the classic textbook by Brandt, Stiles, Bertin and Wittford. This is the top view of the wing of Boeing 787-8 and if you look closely at the wing we realize that we have both leading edge flaps and trailing edge flaps. So the leading edge flaps cover from this point to this point and trailing edge flaps cover pretty much from the almost from the root to this point. So the total area that is flapped area will be the area marked in the red color in this figure. So that would be the flapped area and the unflap area would be only the area inside the fuselage and that in the wing tips or to the outboard of the leading edge flap and the yeah because leading edge flaps are the ones which are having the largest distance along the span. So the area marked in red color turns out to be the flapped area. Now let us calculate the flapped area ratio of Boeing 787-8. So here is the same wing once again I show you the wing but only half the wing is shown and this particular wing can be replaced by a series of geometrical constructs. Incidentally the flapped area would consist of two trapezia. The first trapezium would be the one that goes from the root to the midspan break and the second trapezium would go from the midspan break to almost the wing tip portion. So once you calculate the area of these two trapezia this one and this one you can get the area of the flap. So S flap is going to be area of trapezium 1 and area of trapezium 2 the summation of these two quantities. Now let us calculate the area of the first trapezium for a trapezium we know that the formula for the area is half of the sum of the two parallel sides into the gap between the two parallel sides. So with that the value of a trapezium 1 turns out to be half of 119 plus 6.93 into 7.55. Now these numbers for the geometry of the Boeing 787-8 have come from the source as mentioned here. It is a very interesting source which talks about application of mathematics in aircraft design. I would urge all of you to check out this particular website which has a document containing this description and along with the document there is also a small video of about 7 and a half minutes which explains how the application of mathematics has been found in aircraft design. So we borrowed this figure from that calculation. So you calculate the value of the area of the first trapezium as 71.08 square meters. Similarly we need to calculate the area of the second trapezium which is half of this chord 6.93 plus 1.92 times the distance which is 17.27 centimeters. Please pause the video at this stage and have a look at the values that you obtained and match with the values that we have calculated. The number comes up to be 76.68. So if you add these 2 numbers together and then if you multiply by 2 because there are 2 wings in the aircraft you get the flat total flapped area is 2 times bracket 71.08 plus 76.68 into 2. So this area into 2. So the ratio S flat by S rep will be 2 times the calculated area divide by the wing reference area which is already specified to us as 359.9 for 35. So hence the value of S flat by S rep turns out to be 0.822. In other words 82.2 percent of the wing is under the influence of flaps whether leading edge flaps or trailing edge flaps. So let us calculate the max lift coefficient of the aircraft with partial span flaps. We first look at the formula that we will use. So CL max for the flapped region would be CL max no flaps that means the plane aircraft plus DCL by D alpha into the change in the angle of attack created by the deflection of the flaps change in the absolute angle because of the flap deflection. Now delta alpha is a parameter that can be calculated as delta alpha 2D that is for the 2D aerofoil times the flap area ratio times cos of the hinge line of the flap. So we have been given in this example that the takeoff condition you can assume delta alpha because of the flap deflection in the two dimensions to be 10 degrees and for landing 20 degrees. And for want of any other accurate estimate we will take the hinge line as the maximum of the inboard and the outboard. In any case the outboard flap is very heavily swept it has a 23 degree sweep. So therefore we will take we will ignore this 2 degrees sweep of the inboard portion and take it as 23 degrees. So we might err on the side of caution and the actual value may be slightly higher but that is fine. CL max as I already mentioned is 0.9 times the product of two terms the flapped CL max value as obtained by the formula shown in the slide into the ratio of flapped area by the reference area plus the formula CL max unflapped or the base aerofoil CL max base wing CL max unflapped wing CL max into the ratio of unflapped area by the reference area. So if this area is 0.822 this will be just 1 minus 0.822. So using these let us start calculating the max lift coefficient at takeoff condition. The formula is as shown on the screen we need to get the parameters SF by S and cos lambda HL. So we already know that this number is 10 this ratio we have already calculated as 0.82 and cos lambda HL also we took as a maximum of the both. So pause your video here and try to calculate delta alpha the value turns out to be 7.57. So the additional lift coefficient because of the flaps is as if the aircraft has an additional angle of attack of around 7.57 degrees. So therefore CL max F will be CL max with no flaps plus DCL by D alpha into delta alpha which is 1.25 is the CL max with no flap for the base wing unflapped wing and then you have a term 0.1104 CL alpha into 7.57. At this stage I would like you to pause the video and do this calculation the value turned out to be 2.35. So our estimate is that the CL max of the flap area is 2.35 and now you can get CL max for the whole wing with 0.9 times some of the 2 products. So for this it is very simple we just bring in the values of CL max flap which is 2.35 plus the ratio 0.822 plus CL max unflapped which is the basic wing with no sweep 1.25 and area of unflapped region upon the reference area which will be 1 minus 0.822. So please calculate this value it turns out to be 1.94. So our estimate is that the CL of the aircraft is 1.94 this is based on the basic calculations. During landing the same thing will be done except the difference is that we will use the value for landing. So delta alpha at landing is much more 20 degrees because flaps are deflected to a higher angle but SF by S remains the same cos lambda HL remains the same. So do calculate this value it comes to 15.13 degrees. Now CL max F is equal to CL max no flap plus CL alpha delta alpha. So CL max no flap was 1.25 delta alpha 15.13 as just calculated and CL alpha is 0.1104. So at this stage please pause the video and calculate this value it is 3.46 this is the CL max of the flap region of the wing and for the entire wing we will again go for 0.9 times the summation of 2 products. So in other words it is summation of 3.46 is the value obtained for CL max for flapped region SF by SRF is already known to you plus 1.25 is the value for the unflapped region and the SF S unflapped upon SF SR is already known to you it will be 1 minus 0.822. So do calculate this value and you will get the value of 2.76. So we estimate that this aircraft has got a CL max of 2.76 with the flaps deflected during landing and 1.94 with the flaps deflected during takeoff. Let us see how our estimates compare with the actual values that are quoted. So for comparison purposes I have used the values given by Dimitri Simos in his piano documentation available online. So first we look at the Oswald efficiency at cruise we have obtained the value as 0.6961 and Simos does not directly quote the value of E but if you look at the non-CDO components of the drag at cruise and if you actually just add all of them together and assume it to be the part of induced drag then you can back calculate the value of E at Mach number given he quotes the CL to be 0.508 and Mach number to be 0.85. So with that you can back calculate the value of E as 0.6682. So we see that we are only 4.2 percent off we have over predicted the value by approximately 4.2 percent. The next is the maximum lift coefficient at takeoff. If you recall we got the value as 1.91 whereas the value quoted in the literature by piano is 1.9 we got 1.94 and the quoted value is 1.91 there is only about 1.6 percent and finally max coefficient at landing we have got the value of 2.76 using our simplistic methods whereas the actual value quoted is 2.66. So again we are off by approximately 4 percent. So in conceptual design if you are off by about 4 to 5 percent it is not really a big deal. So we can say that we have been able to reasonably capture the values and from now on we are going to continue with these values 1.91 and 2.60 for all other practical calculations. Before I close I would like to acknowledge a few people. First of all the Timothy Simmonds whose piano sample data has been very useful in this particular work. I also want to thank the 4 authors Brandt, Stiles, Bertin and Wittford for their lovely book Aeronautics on Aeronautics introduction to Aeronautics by them on a design perspective. It is a very nice book and I borrowed one figure from that particular book and to Daniel Rehmer as always for his seminal book on aircraft design from where I have taken one chart regarding effect of various types of flaps and the sweep angle on to the CL max. Thank you so much for your attention.