The One Hour Flight Test Program to Determine Aircraft Climb Performance

Bob Waldmiller

Originally published May 1993

When we left our story last time, Rocky and Bullwinkle were flying precariously close to powerlines and rising terrain in a Piper Cherokee 140 loaded to max gross weight with Aircraft Spruce Catalogs purchased at the Copperstate Fly-in. Having overestimated their aircraft's performance based on data presented in the operating manual means almost certain doom for our heroes. Little did they know that the manual had been written by the evilest of villains, Boris Badenov...aka..."I want to sell more airplanes so I'll fudge my performance numbers a bit--Piper."

"Look Boris, moose and squirrel are headed right at that mountain!"

"Yeah, isn't it wonderful Natasha, moose and squirrel will smack into mountain and we get Aircraft Spruce Catalogs for free."

As the picture looks more and more bleak, Rocky speaks the obvious. "I think the performance manual for this airplane was a little optimistic, Bullwinkle."

"I'll say! And in a few minutes we won't have much use for it either!"

In a fit of panic, Bullwinkle tossed the operating manual out the window. Miraculously, the airplane started to climb which allowed our heroes to return safely home. However, in their next flight in the same airplane, Rocky and Bullwinkle found themselves in a similar situation 7500' MSL over the Mojave Desert. This time 120 lbs lighter, the airplane barely achieved a 200 ft/min rate of climb. Additionally, the engine was turning about 100 to 150 RPM less than required for the 75% power setting they desired.

"Boy, Rocky, this performance is depressing. Maybe if I work a little magic with the engine controls...Nothing up my sleeve...Presto!"

"What'd you get, Bullwinkle?"

"Oh, not much. Just this magazine...the Pacific Flyer--Swimsuit Edition."

"That's not what I mean, Bullwinkle. How's the engine?"

"Don't bother me now, Rocky, I'm reading..."

Does this mean their airplane will always have marginal high altitude performance or is there an unknown engine problem?

Given the above information, it appears that the engine is only producing 65% power rather than the desired 75% at wide open throttle. The first thing we'll do is give the engine a thorough inspection. Let's see, new spark plugs, new ignition harness, mag timing checked, cylinder differential pressure is in the mid 70s over 80 on all four cylinders, carb heat box rebuilt to replace a worn out seal and bushings, throttle plate moves to the fully open position, carb venturi looks ok, fuel and air filters all clean, and the mixture control operates normally. During run-ups the engine leans very smoothly like it always has, i.e. there is no engine shaking or backfiring as the engine is over-leaned which indicates to me that all four cylinders are getting pretty much the same mixture. Without the benefit of an EGT or a CHT it's difficult to be precise in my analysis, but I believe there are no defects in the engine itself. A tachometer check indicates that my tach is reading 50-60 RPM low throughout the entire operating range so I'm turning close to the required static RPMs although it's still 25 RPM low. Given the fact that we're 2300' above sea level while doing the run-up might have something to do with that however.

Now, what about the lack of climb performance? Time for the one hour flight test program! My objective is to learn as much as I can about the airplane's performance by doing a simple rate of climb check. Since I don't have calibration charts to correct for pitot-static system errors or instrument errors, I'll make the BIG assumption that the errors are reasonably small. Also, I presume Piper did a reasonable job of locating the pitot-static system on the airplane since certifying an airplane to Part 23 standards requires the airspeed system error to be within 3 percent or 5 kts--whichever is greater. Still, I'm making a BIG assumption here. To generate a little credibility however, I'll fly the airplane through a low speed ground course to verify my assumptions. 'Nuff said about instrument errors.

The procedure is simple. After setting the altimeter to 29.92, I'll climb at the best rate of climb speed (85 mph) from 500' AGL to 8000' pressure altitude. At 500 foot increments, I'll note the total elapsed time on the stopwatch (starting from a reference altitude) and the outside air temperature. During the climb, I'll keep adjusting the mixture to achieve best power on the engine and all the while, the throttle will be wide open. Simple, huh?

Still with me? Great! Before I started my climb, I performed an airspeed calibration test by flying both ways through a 4 mile ground course. The first two runs were done at 80 mph indicated in both directions and the second two runs were done at 85 mph indicated. A few minutes of electronic whiz-wheel time to calculate the average ground speed through the course along with my true airspeed revealed that my total airspeed system errors are less than 1 mph at both 80 and 85 mph! That's fantastic! How'd Piper do that?

Ok, back to the rate of climb data. I took the liberty and already did some number crunching in order to correct my data to standard atmosphere conditions. In addition, I calculated the actual rate of climb (ROC_act) for each altitude based on the corrected altitudes and the stopwatch timing values. Since there is some scatter in the data due to vertical air mass movement during the test flight and altimeter errors, I applied a linear regression fit to the ROC_actdata to smooth it out (ROC_lr). Finally, I determined the horsepower required to climb at those rates using the following equation:

Where ROC is the rate of climb in feet/minute and Wt is the aircraft weight in pounds. Table 1 is a summary of the raw flight test data and the calculated data.

Since the purpose of this exercise is to determine what the aircraft performance will be at 2150 pounds (gross weight) and compare it with the data that Piper published, I need to determine how much horsepower is consumed by aircraft drag in the climb configuration. Unfortunately, I can't measure drag directly nor do I know how much thrust the propeller is generating to measure it indirectly. Therefore, I'm forced to make a few more assumptions...oh, oh, there's that word again! Conservatively, those assumptions are:

1. The engine produces 150 HP at sea level at 2700 RPM and 75% power at 7500' at 2600 RPM (per the operating manual).
2. The engine power output varies linearly with RPM from 2300 RPM to 2700 RPM.
3. The propeller efficiency is 75% and is constant with altitude, RPM, and true airspeed. This is not entirely true. However, changes should be relatively small and in fact will probably fall to something less than 75 percent at the higher true airspeeds. Therefore this assumption is very conservative.
4. The airframe parasitic drag is constant at 85 mph indicated in the climb configuration and only the induced drag will vary with aircraft weight. This is a good assumption until reaching the higher Mach numbers. (Come on now...high Mach numbers in a Cherokee???)

First, I'll need to determine how much available horsepower I have at 75% power at 7500' standard altitude. From Table 1 we see that the engine is turning 2390 RPM at approximately 7554 feet standard altitude. From the above assumptions we defined the propeller efficiency as 75 percent and the desired RPM as 2600. So using the following equation,

we find that HP_avail = 77.6 HP. From that we can subtract HP_climb at 7554 feet standard altitude (20.4 HP) to get the horsepower required just to overcome the aircraft drag (HP_drag) which is 57.2 HP. Now, according to the fourth assumption, HP_drag is not dependent on altitude which allows us to do a quick check at sea level values. Adding HP_climb and HP_drag together gives 108.8 for HP_avail at sea level. Since I can only get about 2450 RPM at sea level at 85 mph, the HP_avail should only be 102.1 HP. Between the errors in the curve fit and my assumptions, I deem this 6.7 HP error acceptable.

The next step is to determine how much more horsepower I must give up at the higher gross weight to overcome the additional induced drag. To determine that, I must first calculate the lift coefficient at both weights. Using the well known formula,

out pops the two numbers of C_{L 1670} = .603 for the 1670 pound airplane and C_{L 2150} = .776 for the 2150 pound airplane. (Oh, by the way, S = 150 ft², V = 124.7 ft/sec, and rho = .0023769 slug/ft³.) The change in the induced drag coefficient can be expressed as follows:

Where e = Oswald's Efficiency factor of 0.8 for a rectangular wing, and AR is the wing's aspect ratio of 6. The result is a change in the induced drag coefficient of 0.0158 which doesn't sound like very much but when expressed as a change in horsepower,

it works out to 9.9 horsepower, which is very significant! By adding this 9.9 HP to each HP_drag term, which in turn means there's 9.9 fewer HP for each HP_climb term, we can calculate a rate of climb value for each altitude for the higher gross weight. Table 2 is a summary of that data.

After all this work, we can finally show all the relevant data in a single graph. Figure 1 allows us to see the original flight test data at 1670 lbs, the linear regression curve fit applied to this data, the predicted data at 2150 lbs gross weight, and Piper Aircraft's data for a 2150 lb airplane as published in the operating manual.

Figure 1

Check out the Sequel "Frostbite Falls Flyer Foils Foul Fossil Fueled Fallacy"

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