# Range and Endurance revisited

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Guys,Following are the calculations for estimating range and endurance for my Flight Performance gauge. As far as the actual velocities for Vbe and Vbr - I think they're correct, but the problem is estimating and displaying the actual range in nm and maximum flight time at those calculated velocities.Since CURRENT range and flight time can at least be superficially estimated based on the current fuel flow and fuel remaining, I am not concerned about that. But the numbers I'm calculating for range and flight time at Vbr and Vbe are way off when those speeds are matched and compared with current conditions.So my question comes down to this: How can one effectively estimate what the fuel flow, and in turn, the range and flight time is at a velocity and altitude other than your current flight conditions?I assume that the only way to do that is to estimate the DRAG which will lead to the thrust and then the fuel flow, which is what I'm doing, but it's wrong:(Code follows, and I apologize for the RPN syntax. I'll be happy to convert it to a more traditional format if that helps. I'm just looking for any insight as to how this might be wrong.5004.040.0580447150.9951734010.6990053820.5470943661 pi @kAR * @ke * /(A:Turb eng1 fuel flow pph, pounds per hour) (A:Turb eng2 fuel flow pph, pounds per hour) +(A:Turb eng1 jet thrust, pounds) (A:Turb eng2 jet thrust, pounds) +(A:Ambient density, slug/ft3)(A:Ambient density, slug/ft3) (A:Airspeed True, feet per second) (A:Airspeed True, feet per second) * *(A:Airspeed True, feet per second)pi 2 * (A:Incidence Alpha, radians) * @K @Cl @Cl * *@kCd0 @Cdi +@Cl @kS * @rho 0.5 * @V @V * *@Cd @kS * @rho 0.5 * @V @V * *@Cl @Cd /@FF @T /(A:Fuel Total Quantity Weight,pounds) @TSFC @T * /@V @Tmax *(A:Fuel Total Quantity Weight,pounds) @TSFC @DVbr * /(A:Total Weight, pounds) @rho 0.5 * @kS * @kClme * / sqrt@Vbe 1.316 *@Vbr @TVbr *@Cd @kS * @rho 0.5 * @Vbr @Vbr * * *--Hope you enjoy a puzzle:)--Jon

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"Guys,"Following are the calculations for estimating range andendurance for my Flight Performance gauge. As far as theactual velocities for Vbe and Vbr - I think they're correct,but the problem is estimating and displaying the actual rangein nm and maximum flight time at those calculated velocities."Since CURRENT range and flight time can at least besuperficially estimated based on the current fuel flow andfuel remaining, I am not concerned about that. But thenumbers I'm calculating for range and flight time at Vbr andVbe are way off when those speeds are matched and comparedwith current conditions."So my question comes down to this: How can one effectivelyestimate what the fuel flow, and in turn, the range andflight time is at a velocity and altitude other than yourcurrent flight conditions?" I wish I knew how to use XML macros. I calculate Drag for both Jet and Prop XML test gauges. However, I had to reconstruct the 'Mach Drag' table for the Mach numbers involved and add Cdm to Cdo. Which I set as constants when initializing my gauge(s). I calculated Cdi as IDK * AoAe * (Adj_Factor * m)^2. Where 'm' is the Lift Slope of TBL 404 and Adj_Factor is near 1.0. 0.985 was appropriate for a 707 AIR file to get Cdi very close. While 'IDK' is 1/2Pi*AR*e. 'e' is the Oswald efficiency set in aircraft.cfg; FS calculates 'IDK' this way. Finally, 'AoAe' is AoA - AoA_CL=0. Compared to Thrust, Drag came within 0.2% once speed stabilized. However, it's not easy to calculate Best Endurance or Best Range for jets. Compressablity effects complicate matters. Now if one flies a jet transport at the appropriate Mach number and changes FL as required, it's easier to calculate overall range. However, it's a graphical technique that requires Drag curves. Unless one knows LRC Mach (FM tables generally give this). Of course, the FS model must also be close to real performance. I think "ln (Wto/Wland)" is involved in these range equations. But, that only gives the relative range, and assumes the proper flight profile is used. And, doesn't account for climb and descent (thought, they tend to balance out). An approximation to total range would be to set fuel to 1/2 TO fuel and altitude to mid flight altitude. Check 'NAM' - nm/1000lb and divide by TO fuel to estimate total range. Prop AC are simpler if one can assume constant prop efficiency and power. Equations were derived around 1930. Best Range is at same IAS as Best Endurance, though one would probably fly a bit faster and reduce Range only slightly. In this case Cdo = Cdi; of course, the AIR file has to be set correctly for it to have the appropriate Cdi variation running in FS. When set 'correctly' I see absolutely no difference between FS9 and FS8. Ron

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Ron,Thanks much for the info. After evaluating your calculations for IDK, I re-wrote mine to look like this: 1 / pi * AR * e.Drag is now calculated as:(Cd0 * dynamic pressure * planform area) + ((IDK * Weight^2) / (dynamic pressure * planform area)).This seems to be producing fairly good results for thrust in non-accelerated flight, although it takes awhile for the results to stabilize.BTW, macros are deceptively simple:Anything within the tags is interpreted literally with the exception of certain (A:Velocity world X, m/s) (A:Velocity world Z, m/s) atg2 rddgNow anytime I want to use that expression, I just use @Sideslip instead.Macros can be passed arguments as well:s0 100 * near 10000 % d 10 % r 10 / int d 10 % r 10 / int d 10 % r 10 / int 16 * + 16 * + 16 * +In this case s0 was passed from another place in my code as a decimal, and the result is BCN. (although I should point out I did not even need to use s0 in this case. The typical way to pass them an argument is to use @1 @2 @3 within the macro and it will look at the stack entries preceeding the call and insert the appropriate values.(L:Scratchpad, enum) @ToBCN (>K:COM_RADIO_SET)Macros can be used as constants just by inserting a value.Macros can be nested within macros.Macros can be used within strings.Macro names are case sensitive.The important thing to remember, and why they are so simple, is that everything within the tags is essentially inserted into the place where you called the @macro verbatim. They're beautiful for breaking down complex RPN gibberish into something halfway readable:)--Jon

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"Thanks much for the info. After evaluating your calculationsfor IDK, I re-wrote mine to look like this: 1 / pi * AR * e."Drag is now calculated as:(Cd0 * dynamic pressure * planform area) + ((IDK * Weight^2)/ (dynamic pressure * planform area))." I just sent you an email through AVSIM. I used 'effective' AoA, though I wrote from memory. That is how FS calcualtes Cdi, and it accounts for TBL 401, important at higher Mach Numbers. I may have written 'Lg' when I meant 'Ln' for Natural Log. One can use any base, it just means a different constant is required. This was relative to 'actual range'. I've noted the MS RPN is very similar to the basic HP Calc RPN. Only one has 50 registers to store/load from. And, an indefinite stack depth. I found I had to 'c' (clear) the stack at times to keep things working right. So, I add a 'c' at the top of each block I calculate parameters in. I'll save the message so I can refer to the info on Macros. Looks very useful, and gets ride of the long L:Variable terms once they are defined. But, won't work in FS2K2. I've managed to get 'Drag' within 0.2% of Thrust in level flight. Close enough. There are all kinds of useful things one can calculate and display that aid in seeing of one's flight model is realistic. Often they are simply quotients of things already calculated. Thats why I had to learn something about "Delta's and Theta's", used a lot in Jet Performance and Turbines. In fact, they are easier to calculate with the Parameters provided in XML than in C gauges. Ron

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Jon,That looks quite simple, but..Can you place the macro's anywhere or must they be in the same gauge just like in the GPS?Jan"Procul Negotiis"

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My thoughts and observations. Please correct me if I am wrong.Endurance as in staying in the air the longest possible time. Such as the Predator on orbit above.... waiting to kick off a couple of Hellfire missiles. Wouldn't this endurance solution be obtains by reducing fuel rate until it is at an absolute min and still be able to fly "stable"? Got to be careful on some aircraft that you don't get caught in the coffin corner. High AoA - high drag - low speed and thus requiring high fuel rate. (high power that may be more then available)The lowest possible fuel rate has to yield the most endurance?And then Max Range.I plotted fuel rate (lph) vs. velocity (knts) and had Excel fit a 3rd degree polynominal to the data. It yielded a very good fit with rsquared=.9992, better then you expect in real world data collection. If any one runs that good of data in the rw, it is a dry lab job, cooked data!Wanting to know the point of inflection for the data curve, I should take the first derivative, set it equal to zero and solved for fuel rate. Not wanting to solve a quad' by hand, we used Excel's "Solver" setting the desired result equal to a min value. If you use zero, as you should, it will not converge using an iteration method AND the parameters I selected. But, it got close enough, 65 LPH, at Vdot of 1.6 knts/(lth squared).Now back to the simpler math, dividing fuel remaining by the inflection point value for fuel rate, will yield the time in hours of fuel remaining. I will leaving the rest, for the rest of us.The solution is only correct for a given gross weight/cg location.But does it change with density altitude? Increased AoA, thus higher drag, required for same fuel rate as altitude is increased? That can be a novel problem for me. ;-) Regards,BobSSeems the rage to talk about the "size and speed" of each others computer. Beat this if you can for solving novel/unique problems.. ..Have K&E and know how to use it!

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>My thoughts and observations. Please correct me if I am>wrong.>>Endurance as in staying in the air the longest possible time.>Such as the Predator on orbit above.... waiting to kick off a>couple of Hellfire missiles. I have a pretty good Predator FM. I found the curves for the Rotex engine on the WEB and also enough other info to set other things. I get best enduranace at about the right speed (seems about 65 kts) and can climb to 25,000 ft also. I think I found I could stay aloft 40 hrs with the big fuel tanks. They don't say what the max endurance is but basic aerodynamics should give a good value. ;) >Wouldn't this endurance solution be obtains by reducing fuel>rate until it is at an absolute min and still be able to fly>"stable"? Got to be careful on some aircraft that you don't>get caught in the coffin corner. High AoA - high drag - low>speed and thus requiring high fuel rate. (high power that may>be more then available) Normally Max Endurance occurs when L/D is minimum. Note Thrust = W*[L/D] and it is minimum when L/D is minimum. Either the total L and total Drag (in lbs) or the Coefficent forms can be used, since Lift(lbs)= q*S*CD, while Lift (lbs) = q*S*CL. 'q*S' cancels in the ratio. However, without ALT hold, I find I can't trim for stable flight at best L/D. If speed drops slightly drag increases, speed drops more. eventually the AC pitches down and speed picks up. Possibly a diverging Phugoid. I expect a real AC would be similar. Though the pilot can keep adjusting pitch to fly more slowly than L/D min. He has to to land. ;) ALT hold lets me set speed a bit above L/Dmin. A FP prop AC might slow and stall if speed is slightly below best L/D with ALT hold. A CS prop AC would more likely still be stable, since RPM and power stay more constant and thrust increases more if speed drops. >The lowest possible fuel rate has to yield the most>endurance? This assumes prop or turbine efficiency is constant when q changes some at best L/D. It also assumes the simple, classical form for drags due to Cdo and Cdi. Note S is wing area, q varies with CAS^2 (until compressability comes in). While dropping speed below best L/D reduces q, L/D changes faster, so it turns out L/Dmin is still the point for minimum thrust, assumed for minimum fuel burn. I get confused by this, since with a CS prop thrust might increase some even though SHP could be dropped as one slows below L/D min. I think Drag still increases faster than one can reduce HP. Regardless, if one can't get more Thrust than Drag, he is 'behind the curve' and will have to descend before he picks up enough speed to climb. More noticable in low powered AC. However, props tend to get more efficient as speed increases, I've noted my prop AC can climb as fast, or faster at a higher speed than where L/D is maximum. That uses more fuel, but one gets to altitude faster also. >And then Max Range. That makes similar assumptions. In a prop AC, Thrust drops with TAS for constant Power. It turns out that Best Range still occurs at 'Best Endurance'. However, that's generally rather slow, and one loses little in range by flying 20% faster. Which also means your fuel runs out faster. The above don't require one to know weight, temperature, altitude, density, etc. L/D will vary with various things, but at the speed it is minimum at still gives best endurance (and range in prop AC).>I plotted fuel rate (lph) vs. velocity (knts) and had Excel>fit a 3rd degree polynominal to the data. It yielded a very>good fit with rsquared=.9992, better then you expect in real>world data collection. If any one runs that good of data in>the rw, it is a dry lab job, cooked data!>Wanting to know the point of inflection for the data curve, I>should take the first derivative, set it equal to zero and>solved for fuel rate. Not wanting to solve a quad' by hand,>we used Excel's "Solver" setting the desired result equal to a>min value. If you use zero, as you should, it will not>converge using an iteration method AND the parameters I>selected. But, it got close enough, 65 LPH, at Vdot of 1.6>knts/(lth squared). There is a Sport's Aviation SS that is good for calculating performance of small prop AC. One has to iterate "flat plate area" some to get the appropriate IAS for L/D min. AFSD data can be further analysed and new variables calculated in Excel. I'm not very good at that. Note there are plenty of texts that analyse the basic drag, etc. matters. But, they typically assume constant prop efficiency, no compressability effects, etc. I compare my FS FM's with the basic theory and note where they vary. Note FS props have variable efficiency and I try to set the prop efficiency table so climb and cruies efficencies are appropriate. Also, my jets (in fact, all FS jets) reduce Cdi as Mach number increases; it's due to TBL 401. While the default TBL 401 used in Jets appears reasonable at low Mach numbers, it goes too high before dropping rapidly around Mach 0.8. I found for jets, Theoretical Best Range occures at Cdo = 3Cdi, which results in 2^1/3 = 1.26 for CAS for Best Range, relative to Best Endurance. However, that's only true when Mach is so low TBL 401 is insignificant. I've often carefully checked speed for Best Endurance (also, watch L/D in AFSD) below 10,000 ft, then find where 'Specific Range' peaks. It is usually around 1.22 the IAS for best endurance. I figure it's different from the theoretical 1.26 due to TBL 401 (Mach Drag, tbl 430, is still zero). But, I haven't tried setting TBL 401 flat to see if it increased to the 'simple theory' value. This takes some time, and fuel is being burned, so W is dropping. One would do better to set the fuel_flow_scalar low so W stays nearly constant, then account for the better SFC independently. Most of the formulas involve 1/x, x^2, etc. so it's not surprising a polynominal or other fit works out well in many cases. More creative series might fit better, for example one of the form a + b/x + c/x^2 + ... A series with negative powers of x. Excel doesn't find such a series directly. Some of us have tried to fit turbine tables to a simple series. This is especially appropriate when simple therory implies an x^2, x^3, etc. variation. Or, a combination, where the exponent is '2' at slow values and goes to '3' at higher values. >Now back to the simpler math, dividing fuel remaining by the>inflection point value for fuel rate, will yield the time in>hours of fuel remaining. I will leaving the rest, for the rest>of us.>>The solution is only correct for a given gross weight/cg>location. Yes, simple endurance or range is based on current weight, speed, and altitude. But, one climbs in a real AC as weight decreases. PoH tables show range may not increase much at high altitudes if one tries to fly at the same TAS. Perhaps because friction HP varies with RPM, and becomes a greater part of SHP as one goes to higher altitudes. However, FS doesn't model pph correctly for that, though it does reduce SHP appropriately as one climbs.>But does it change with density altitude? Increased AoA, thus>higher drag, required for same fuel rate as altitude is>increased? That can be a novel problem for me. ;-) Fortunately, L/Dmin is still the point for lowest drag (though not exactly so at high FL's for jets). I mentioned altitude, etc. don't change the L/D min for best endurance. Howver, L/D will vary with those factors, so if L/Dmin moves to a higher TAS one has to fly faster to get the same relative range. I emailed my XML Jet Test files to several guys, it displays L/D and many other parameters, including Rs and Es. I experimented in flight tests to see how close the simple theory fit my FM's. When things vary it's likely due to the fact that the 'best' of anything generally occurs when all the postive derivatives and all the negative derivatives sum to zero. That is a relative min or max. Since TBL 401 will drop rapidly at one point, which Cdm (TBL 430) increases rapidly near Mach 1.0, the speed where all the 'rates' cancel isn't so simple. To stay at the 'best range' speed one may have to change Mach signficantly in a fighter as he changes altitude. Also, climb rate depends a lot on TAS since thrust increases with speed. This is a further factor beyond just the airframe drag. This is were we are lucky to have flight simulators, SS's, etc. to work out more complex matters. >Regards,>BobS I also have an XML Prop Test gauge. However, one has to set some constants for the specific AC it is used with to display everything right. AFSD reads the AIR and aircraft.cfg files to get them directly. I've mentioned I suspect "Ln(W/Wend)" is likely a factor that would result in True Range or Endurance to the fuel level that gives Wend. However, one would have to climb as W decreased and maintain the proper 'speed' relative to L/D min. That might account for ISA atmosphere variations, of course, if temperature, etc. changes away from IAS during a flight one loses accuracy. Headwinds are the worse. ;) Airlines get professional Flight Plans with predicted winds and temperatures along the route. They assume LRC or some FM FL and Speed is used but account for the leg times with the predicted winds, etc. Flight time varies, one reason fuel reserves are carried. Ron

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