FSX Aircraft climb performance

[bina]

Ok, first things first... I'm doing my very first steps in learning aircraft flight dynamics, so please be patient and bear with me!

For the last two months, I'm developing a VBA-based Excel sheet (basically a navigation log) trying to ease a workload of a regular FS pilot like me with time-consuming calculations required during pre-flight planning such as determining true and magnetic tracks, fuel consumption, airspeeds for various phases of flight, etc. I'm not an "FDE guy" by all means but upon my quest for a reasonable explanation of aircraft climb performance and particularly a relationship between power settings and IAS and R/C change rate with altitudes, I've stumbled upon Yves's Flight Model Documentation and an excellent FDE Workbook by Jerry Beckwith. For sure, they both are based on the results of scrupulous analysis and testing but are of little value for me at current stage...

Anyway, going back to questions, I'll try to be laconic but am kindly asking you to be as descriptive as possible considering my zero experience with aircraft flight dynamics:

(1) The attached image taken from POH of C182Q Skylane II shows that an aircraft climbing from sea-level to higher altitudes loses IAS (approx. 1-2 knots per 2000') and rate of climb (linearly, 110 fpm per 2000'). Trying to find an answer (basically, a formula behind rate of change) to these changes, I've noticed that:

But this sounds both reasonable and confusing at the same time... Assuming nil winds @ standard ISA conditions (t = 15°C; ρ = 1.225 kg/m3), the power required to climb an aircraft from sea-level at climb gradient of 12.78% (1010 fpm according to POH) will be:

Engine's rated power: 230 BHP (~172 kW)
Aircraft's weight: 2,950 lbs (1,338.1 kg)
Vy (best-rate-of-climb speed): 78 knots (40.13 m/s)
RC @ Vy: 1010 fpm (5.13 m/s)
Climb gradient: RC / Vy = 12.78%
sin(θ): sin(12.79) = 0.221267

P = mgV sin(θ) => 1,338.1 x 9.80665 x 40.13 x 0.221267 = 116,519 J/s = ~117 kW

If I'm not mistaken, this implies that an aircraft engine is working @ 117/172 = ~68% of it's rated power (230 BHP). But unless the speed and climb gradient are higher, the power required to climb the aircraft remains... constant, doesn't it?!? At least, the formula doesn't show a sign of decreasing power...

So, the questions are:
* How can I calculate or derive both power available and power required (excess power) using the above figures/values?
* Can you show/explain a logic how and, most importantly, at which rate an airspeed and rate of climb diminishes with diminishing excess power? Any formula(s)?!?

(2) Yves Guillaume's FMD states that FS works with 3 fixed aircraft reference points in order to calculate flight dynamics: VMO, Reference Datum Position, and WAC where VMO is the starting point for all offsets specified by the designer of the visual
model. And the location of the visual model origin is coded in the .mdl-file and cannot be changed without having the source files for the visual model.

If so, can I still change volume of fuel tanks, for example?!? Let's say the author of model wrongly defined in "aircraft.cfg" that both fuel tanks are capable of 48 US gallons total. Can I increase this number to real-world book value of 88 gallons with a noticeable change in aircraft performance?

Thank you very much for your patience reading my long opus!

 

[jx_]

If I'm not mistaken, this implies that an aircraft engine is working @ 117/172 = ~68% of it's rated power (230 BHP).

No, this implies the power being used for climbing is 22% of the current power setting. The cosine, 97.5%, is being used to maintain forward constant airspeed.

Now before continuing, there are two factors you must understand.

1) At any time you are climbing while maintaining or increasing true airspeed, you are in fact accelerating in space.
2) Thrust is what you climb on, not lift... without thrust, power, or momentum, there can be no climb. This is due to Lift = Weight * load factor. There is no excess lift available to climb nor can lift accelerate (as in #1).

How are you accelerating if speed is constant? First, because distance is measured at the surface. When you are at altitude, the vertical lines from two measuring points on the ground are further apart due to the surface curvature and your flight path is arced; thus for your true airspeed to remain constant, you must accelerate as altitude increases. Secondly, as the air thins, true airspeed must increase to maintain lift for a given IAS. So, even if your IAS airspeed is decreasing in a climb, if your TAS is not decreasing, you are accelerating. Excluding atmospheric changes, TAS increases with altitude, and that acceleration only occurs during climbs.

This is usually negligible in level flight, but will cause a discrepancy in climbs/descents which is corrected through what's called "acceleration factor".

Acceleration factor of zero in a climb is possible if your true airspeed is constant, but this is normally inefficient or impossible due to aircraft limitations. This is when thrust = drag + rearward weight component and excess thrust = 0. Without excess thrust, the aircraft's IAS will decelerate until it levels off, descends, or stalls. This is what was explained in the paragraph you posted about the absolute ceiling.

But unless the speed and climb gradient are higher, the power required to climb the aircraft remains... constant, doesn't it?!?

No, power available is fairly constant, but power required for a constant gradient & speed is based on drag, weight, and acceleration factor. For a heavy jet, this leads to the need for step climbs (weight), for a Cessna, propeller efficiency could play a big role (drag), while all airplanes must overcome acceleration. Wing design could lead to different angles of attack at various altitudes which also changes drag. Compressibility comes into play for high speed aircraft.

If indicated airspeed remains constant in a climb, mach drag will increase. If indicated airspeed falls during a climb, induced drag increases. If indicated airspeed increases, of course parasite drag increases... so no matter what, power required is constantly changing.

At least, the formula doesn't show a sign of decreasing power...

Actually it does. mgV is dependent on thrust. Thrust is based on the amount of air moved. Because air density is constantly decreasing during the climb, so is the power being produced.

P = mgV sin(angle) simply means; the amount of weight that thrust must negate to allow a climb to happen is = P.

So, the rear component of Weight * Velocity = P

or...

Power = Thrust * Velocity.

and since...

Power required = Thrust required * Velocity. Thrust required = Drag. Drag = CD * q * S.

Then, Power required equal CD * q * S * V.

At constant Velocity, q is dependent on air density.

Can you show/explain a logic how and, most importantly, at which rate an airspeed and rate of climb diminishes with diminishing excess power? Any formula(s)?!?

Yes, it decreases due to lower density which leads to lower thrust, and faster acceleration in the thinner air, which also leads to lower thrust.

Below are the equations that may help you.

Power P = Thrust * Velocity

Power required = Thrust required * Velocity

Thrust required in climb = Drag + [Weight * load factor * sin(flight path angle)]

Forward Thrust Component TX = T * cos(flight path angle)

Excess Power PS = (TX - D / W) * V = PAvail - PReq'd / W

acceleration factor = 1 + V/g * dV / dh


PS * acceleration factor = Rate of Climb


Flight Path Angle y = sin^-1 [ (T - D / W) / acceleration factor ]

Rate of Climb = Vtrue * sin( y )

Gradient = 100 * tan( y )

If so, can I still change volume of fuel tanks, for example?!? Let's say the author of model wrongly defined in "aircraft.cfg" that both fuel tanks are capable of 48 US gallons total. Can I increase this number to real-world book value of 88 gallons with a noticeable change in aircraft performance?

In the aircraft cfg under [fuel]:

[fuel]
LeftMain=-8.900, -16.000, -1.600, 1686.000, 3.000
RightMain=-8.900, 16.000, -1.600, 1686.000, 3.000

Tank name = forward/rear distance from reference datum, left/right distance, up/down distance, capacity in gallons, unusable in gallons.


Aircraft performance change will be noticeable. The added weight and placement of that weight is modeled by the sim.

 

[mgh]

@bina

The Rate of Climb (ROC) depends on the Power Required (Pr) the Power Available (Pa) and the aircraft's weight (W).

ROC = (Pa - Pr) / W

Both Pa and Pr vary independently with altitude and speed because they are effectively Thrust * Speed and Drag * Speed.

You correctly calculated the power required to climb as 117kW which is Pa - Pr. The information you have is insufficient to calculate Pa and Pd separately.

I attach a figure taken from U.S. NAVAL TEST PILOT SCHOOL FLIGHT TEST MANUAL USNTPS-FTM-NO. 108 (PRELIMINARY) FIXED WING PERFORMANCE. It illustrates the variation of Pa and Pr with true airspeed at a given altitude. There would be similar but different curves for other altitudes.

You would need thrust and drag values for the specific aircraft covering the range of speeds and heights in order to calculate the powers and rate of climb.

 

[jx_]

To find HPA for any condition is very simple...

HPA = 230 BHP * density ratio * Prop Efficiency Losses %

This gives Pa under any condition.



PR can be found easily using the chart you've provided.

Rate of Climb FPM * Weight LB / 33000 = HPA + HPR. Determine HPA as stated above to extract HPR. Again, applies to any condition.


At MSL@78kts & 1010FPM, Power Available is 214HP and Power Required is 140HP. (this is ~117kW * efficiency losses)

At 10,000 feet@72kts & 460FPM, Power Available is 158HP and Power Required is 116HP.

ROC = (Pa - Pr) / W

This is only true in unaccelerated climbs, but if you want to use something more generic, use ROC = 33,000(HPA – HPR)/W.

There would be similar but different curves for other altitudes.

Power Available for a recip only changes by density ratio * Prop Efficiency Losses versus all Sea Level velocities (what is shown on that kind of chart). No need for additional charts.

 

[mgh]

@Bina

Here's a link to U.S. NAVAL TEST PILOT SCHOOL FLIGHT TEST MANUAL USNTPS-FTM-NO. 108 (PRELIMINARY) FIXED WING PERFORMANCE I mentioned. Chapter 7 is particularly relevant. You'll find it offers reliable advice.

 

[jx_]

Bina,

Be sure to read Chapter 5 first. You need to understand the concepts behind it before you get the wrong idea.

Also, I was under the impression you wanted a walk through as opposed to a text book. Perhaps I misunderstood? If you need some training material I can see if I can find anything I can share in the company binder. The material can be confusing when taken out of context and without reference to real scenarios.

 

[bina]

Well, I was thinking to release the first version of program before Christmas but with so much information to "digest" thanks to your efforts (!) and involvement of His Majesty POWER in equations and probably tons of other factors not clearly visible at current stage, I better postpone a bit... So, please, don't consider this as my actual response to your explanations - I'll definitely return to this discussion later, hopefully by the end of this week.

Thanks a lot!!!

 

[bina]

I didn't have much time to explore but here is the results of my first expedition into the World of Flight Dynamics:

1- Not that I'm weak in maths and physics but I definitely need a few good resources on aerodynamics targeted at beginners. My hairs turn grey the more I try to derive basic variables (read: variables "well-known" for a layman like me - true airspeed, weight, etc.) from multi-layered aerodynamic formulas. At first glance this PPT presentation looks informative but straight on slide no.10 it shows some pre-calculated figures for equation without much details.

I started with this article by Michael K. Zyskowski of Microsoft and relevant chapters of "Fixed Wing Performance". Still reading them now...

2- I also need some reading on relationship between .AIR file and aircraft.cfg. Which parameters can be changed through each of these files?

3- The initial goal was to understand a relationship between the power, rate of climb and IAS changing with altitude. But now I'm totally lost...

In the light of your explanations above, I'm trying to "decode" Jerry's FDWB and the sections for drag and propeller efficiency calculations in particular. But I'm basically coming down to this simple understanding that drag (D) and drag coefficient (Cd) {which is always calculated in wind tunnels} are interrelated. Not sure that this is what I'm looking for in FDWB but Cd_0 (zero-lift drag) looks like the one I need. But Cd_0 requires knowing of propeller efficiency. Hopping from one sheet/cell to another, I've finally arrived at sheet "PROP!B75:B85" which doesn't depend on particular formulas - rather look like a bunch of (pre-calculated?) constants... Therefore, I cannot get to the root of calculation. Again, lost in nowhere!

Any help is highly appreciated!!!

Thanks!

 

[jx_]

What is it you are trying to accomplish? A spreadsheet? A gauge?

 

[bina]

It's a spreadsheet similar to the attached one but heavily dependent on VBA coding. As I mentioned above, it all started with changing IAS and R/C with altitude - other than that the program would be complete with an accent on simple fuel consumption and time calculations. But my desire for perfection...

 

[mgh]

@bina

some clarifications which my be helpful. In what follows I’m assuming incompressible flow.

The drag coefficient is given by
sCd =Drag / (0.5 * rho * S * V^2) or, when calculating drag

where rho is ambient density, S is an area, (conventionally the wing area for an aircraft), and V is speed.

or

D = Cd * (0.5 * rho * S * V^2)

when calculating drag.


As you say, Cd0 is the zero-lift drag. This doesn’t depend on propeller efficiency. The total drag depends on Cd0 and Cdi, which is the additional drag, caused by lift.

(Incidentally, Jerry Beckwith no longer supports or updates his Flight Dynamics Work Book because it “may be difficult to use.” He wrote a note on how FS models propellers here.)

FS got this wrong because propeller efficiency falls to zero at zero forward speed – zero Advance Ratio. This is shown in .air file Table 511.

FS’s workaround is to set a low_speed_theory_limit in the .cfg file. This is the speed in ft/sec at which “low-speed propeller theory gets blended into the high speed propeller theory”. This limit is 80 ft/sec – 47 kt for many default aircraft. Unfortunately, there’s no explanation of what this low-speed theory is. FS should have provided tables of Thrust Coefficient and Torque Coefficient instead of Efficiency (Table 511) and Power Coefficient (Table 512)

 

[bina]

mgh,

Thanks for your help!

As I tried to explain above, the spreadsheet, at its current stage, works with POH (book) values for Vy, RPM, fuel flow, weights, etc. Considering how rapidly a flight environment can change and affect aircraft performance starting from takeoff and upward, the real challenge for me, however, is to have a "unified" approach to calculation of these parameters "on the fly" regardless of their book values. Not sure how well the term "unified" fits in here but I definitely do NOT mean "simple". Instead, I would prefer, if a bunch of relatively explicit parameters such as BHP, propeller diameter, wing span and area, etc. could help to derive the implicit ones such as drag, thrust, Vy and ROC at certain altitudes and power regimes, to list a few.

Well, you can say that a difference of a few knots in speed or a few fps in climb rate is not that significant for pre-flight calculations when a pilot is essentially interested to know his fuel requirements and observe the weight balance but my aim here is to ensure two principal advantages:
# allow the end-user to extend capabilities of spreadsheet on his own by inserting additional aircraft to the database without getting into details of aerodynamic laws - using only explicit parameters. If it's not possible due to individual approach to different aircraft types (props vs jets) or propeller models (fixed-pitch vs constant-speed), then I will be forced to extend the database myself per user requests.
# fine-tune pre-flight performance calculations, e.g. considering the effects of density, altitude and power changes on indicated airspeed and rate of climb. Currently I want to derive (not just to know how they're calculated since the formulae are pretty much available over the net) Pr and Pa to be able to get min/max IAS for Vy and ROC. I still peruse a possibility of using Jerry's calculations put in FDWB but as you fairly mentioned it "may be difficult to use" at my current level of knowledge of aerodynamics in general and FS' FDE model in particular. Just hopping from one cell to another doesn't yield too much, unfortunately...

Now, what I got until now is tons of multi-layered formulae where most of them share the same implicit parameters. E.g., a calculation of drag and thrust (required to know "power required" and "power available") depends on coefficients of zero-lift, and induced drag, or propeller efficiency (for thrust), respectively. On another occasion I read two very good written articles on Bootstrap Approach to aircraft performance here and here. But for simplicity sake the author aimed at real-world pilots with hands on flying experience.

The matters get even more complicated when you have to differentiate between power and thrust, or deal with separate approaches to fixed-pitch and constant-speed propellers, turboprops and jets! And you soon realize that your hairs turn grey... slowly but steadily.

And before you reply, I kindly ask you to consider the following points:

# Keyword: derivation of implicit aerodynamic parameters from explicit ones!
# I'm interested in a generic/unified approach to consistent calculations in FS-world vs. real-world. IOW, a method or algorithm which would allow to derive parameters from what is known already in a same way the FS flight model does with the help of .AIR and .CFG files.

E.g., when the same "Bootstrap Approach" referenced above requires a pilot to take a measuring tape and write down a width of each blade station to arrive at Total Activity Factor (TAF) which in turn will lead him to efficiency value... Come on, guys, this won't work in FS world, will it?!?

And the same applies to drag, thrust or lift coefficients when a website or book simply states that these coefs are measured in wind tunnels... Well, I do believe these statements are true but I also believe that it's possible to obtain them in a simulated world through two FS files (obviously, not only these two!): AIR-file and aircraft.cfg. Plus Jerry's FDWB, or Hervé's AFSD, or Klotz's AAM, to list a few.
# As a rookie in such a vast topic as aerodynamics, I am pretty much motivated to learn on a step-by-step or reference (i.e., RTFM) basis rather than wasting your time and trolling in this thread.
# English is NOT my native language but I try to be as descriptive and consistent as possible.

Thank you very much!!! And Merry Christmas!!!

 

[jx_]

Everything you are discussing is fairly straight forward, but your questions are too broad. You would need a specific list of the things you need based on what you're trying to accomplish.

As for those sites and books that state coefficients are measured in wind tunnels, they are simplified versions of reality. Assuming "wind tunnels" includes computer simulations, coefficients are used as a simple way to compare different dimensional elements in a non-dimensional way.

In actuality, coefficients for a real, fully assembled airplane, are measured in flight...and charted the old fashion way, by hand. Once charted, you have what you need to make this kind of spread sheet tool. We use various performance programs such as InFlight, Report, APM, etc. that do exactly what you are discussing.

I'll start with drag as quick example.

First, start with MGH's post:

Drag = Cd * (0.5 * rho * S * V^2)

Note that 0.5 * rho * V^2 = dynamic pressure "q" which is also equal to:

1481.4 * mach^2 * pressure ratio = q

density ratio * KTAS^2 / 295.369 = q


This greatly simplifies our work because we can now avoid the conversions of Velocity to feet per sec and finding ambient rho at any altitude.


So,

Drag = CD * q * S
q = density ratio * KTAS^2 / 295.369

Assuming our airplane has an area of 500 and is traveling at 150 KTAS at 5000 feet MSL where CD = 0.023....


Drag = 0.023 * q * 500

q = density ratio * 150^2 / 295.369

density ratio at 5000 feet is 0.8617

65.64 = 0.8617 * 150^2 / 295.369




Drag = 0.023 * 65.64 * 500

Drag = 754 pounds of Force

This is thrust required for level flight.

Remember, Pr = Tr * V

so Pr = 754 * (150 * 1.6878)

(((1.6878 converts knots TAS to feet per second)))

Pr = 190,890 foot pounds / sec * 550 = 347 horsepower


As you can see, the only thing we would needed was the wing area and the coefficient of drag. These would be found in a chart. All you need for this example is a drag polar, which is a charted graph of drag coefficient by airspeed.

 

[jx_]

This [drag] doesn’t depend on propeller efficiency.

I'm guessing he was confused, but it reminded me that I missed a step. To find power required for the engine (what you command with throttle) you would need prop efficiency to up convert power required at the prop.

So in my last post I should have added a step, or at least clarification.

The 347 horsepower solved for is at the propeller, not the engine. To find the engine power setting, divide by prop efficiency and then shaft (if gear driven) efficiency.

 

[mgh]

In general there isn’t a single figure for propeller efficiency. It varies with propeller rotational speed and aircraft forward speed. The key parameter is the Advance Ratio, J, which is defined as;

J =V / (n * D)

where (in Imperial units)


V = aircraft speed (ft/sec)
n = propeller speed (rev/sec)
D = propeller diameter (ft)

Large J implies high aircraft speed/low propeller speed:. Small J implies low aircraft speed /high propeller speed.

Air file Table 511 shows Efficiency (Eta) against Advance Ratio. Microsoft cut a few corners here. Although the FSX default C172 has a fixed pitch propeller (beta = 20 deg), its Table 511 shows Beta = 15, 20 and 25 deg.

Air file Table 512 shows the propeller Power Coefficient (Kp) against Advance Ratio for Beta = 15, 20, 25, 30, 35, 40 deg! Power coefficient, Kp is defined as:

Kp = Power / (rho *n^3 * D^5)

where

rho = ambient density (slugs / ft^3)

Kp becomes negative at larger J (high aircraft speed and low propeller speed) because the propeller is windmilling, absorbing power and creating drag not thrust.

The figure shows how Efficiency (Eta) and Power Coefficient (Kp) vary with J for the C172. Propeller Efficiency and Power could be calculated from these for specific flight conditions. However, the propeller power would have to be matched to the available engine power.

 

[jx_]

try these,

nP = (Tpounds * VKTAS) / (HP * 326)

nP = T * V / PSHAFT


these may also be helpful,


Thrust = (HP * nP *326) / KTAS

HP = (Thrustpounds * VKTAS) / (326 * nP)


nP = prop efficiency



Of Velocity, Thrust, HP, and nP --- if you posses any two you can easily find any third.


I haven't look at the prop tables in a while so I don't remember if they are any good. I'll leave that to you guys.

P.S.: Kp = Power / (rho *n^3 * D^5) <<< Maybe I'm wrong but I believe it is RPM to the third power

 

[mgh]

P.S.: Kp = Power / (rho *n^3 * D^5) <<< Maybe I'm wrong but I believe it is RPM to the third power

You are right. I have corrected my post.

 

[mgh]

I modelled the FSX C172 propeller performance in a spreadsheet.

The attached figure show the variation of the power to turn the propeller for a range of engine rpm and aircraft speeds based in .air file Table 512. I’ve also added the engine power based on the rated power (180 HP) and rated speed (2700 rpm), assuming the variation in linear with rpm. That’s a reasonable assumption for the range of rpm considered. Anyway, I don’t know of any other way to get a power curve from the .air and .cfg files.

The steady state rpm occurs when the propeller power required equals the engine power available. This explains why on takeoff. rpm increases from about 2250 at 40 kts to about 2650 at 120 kts. It also shows that the rated power isn’t achieved in general At a typical climb speed (80 kts) the power is only about 160 HP at 2400 rpm.

The actual propulsive power is the engine/propeller power multiplied by the propeller efficiency and is significantly less.

 

[bina]

I've started this topic almost 2 years ago but it's still on my agenda. Anyhow, I've digged out this article which helps to determine KTAS and fuel consumption for a typical C172N using POH figures only. So, basically, with this approach in hand (assuming that POH figures for cruise performance are the result of numerous tests and substantially accurate), it becomes much easier to determine speed and fuel numbers instead of tedious aerodynamic calculations for coefficients of drag/lift and the power required to achieve the same purpose. And this suits my little program fine as well. Quite a good read actually...

However... I tried to clarify a few questions with original author (Kenneth) but it seems he left the blog more than a year ago, I thought maybe you guys can help.

Near to the end of his article the author says:

Based on what I know about aerodynamics, and using some curve fits, I reversed engineered this curve.

And this is followed by equation (11) which basically helps to derive coefficient of lift (CL) from the function f(CL).

It seems the numbers used in this particular equation are "reversed engineered" for C172N only! So, the question is: how can I generalize this equation, in other words - get the coefficient of lift (CL) to use for other types of fixed prop aircraft (e.g. Piper Cherokee)?

Thank you!