Russ Erb
Originally published October 1993
Fuel Flow and Power
Performance Charts
Maximum Endurance
Maximum Range
Optimum Cruising Speed (Carson's Speed)
How Do I Find These Speeds Without A Drag Polar For My Airplane?
Final Thoughts
References
This article was inspired by the excellent CAFE Foundation flight test article on the RV-6A in the September 1993 Sport Aviation. On page 37, the author (there are so many, I don't know who wrote this part) talks about an optimum cruise speed he call's "Carson's Speed." Well, in 13 years of studying and working in flight mechanics (performance, stability and control), I have never heard of "Carson's Speed." For that matter, neither had any of the flight mechanics gurus in the USAF Academy Department of Aeronautics. Needless to say, this new concept caught our attention.
Fortunately, the author cited the AIAA paper where Carson developed this concept (AIAA-80-1847). So I cruised on over to the USAF Academy library and picked up a copy. After studying the paper, this seemed like a good time to review the basic performance speeds.
Of course, Norm would prefer that I go through a detailed, rigorous mathematical derivation of these concepts, including stuff such as triple integrals as found in Green's Theorem:
(Whoa! I'm starting to have flashbacks to my partial differential equations course!) Norm seems to think this would improve our standing for next year's McKillop award. Unfortunately, the only thing I want to do less than read something like that is to write something like that. Therefore, we'll take the "Math-lite" or qualitative approach for this explanation. If you want the heavy duty mathematics, get a copy of Carson's paper.
Before launching into the various performance speeds, we need to first establish a relationship between power and fuel flow. As it turns out, fuel flow is directly proportional to power output. The constant of proportionality is the specific fuel consumption (SFC), defined as such:
The SFC is normally a constant, and will be considered such for these analyses. A typical value of SFC for reciprocating engines is 0.5 lbs of fuel/BHP-hour. The important concept for these analyses is that fuel flow is proportional to power output, that is:
Note also that propeller efficiency will be considered constant.
To analyze our cruise performance, we start out with the Drag Polar. This is shown in the bottom half of Figure 1. The drag polar is simply a plot of aircraft drag vs. aircraft velocity. The drag polar shown is the same as the RV-6A drag polar presented on page 37 of the September 1993 Sport Aviation. Note that the velocity is expressed in calibrated airspeed. For you aeronautical engineering purists out there, this is technically the equivalent airspeed. However, for the airspeeds we are talking about, the calibrated and equivalent airspeed are essentially equal. To get the drag back to coefficient form, you would need to use the density for sea level.
Figure 1
Since thrust (T) is equal to drag (D) in level flight at constant speed, the lower chart also tells us the thrust required to overcome drag to maintain a given velocity. Thus, it is also called the thrust required chart.
The upper chart of Figure 1 is the power required chart. Power is simply thrust multiplied by velocity (TV), so we get the power required chart by taking each point on the thrust required chart and multiplying by the corresponding velocity. This chart tells us the power required to maintain a given velocity.
To maximize the endurance, we want to maximize the amount of time that we can stay in the air. In order to do this, we must minimize the fuel flow. Since the fuel flow is proportional to the power required, the fuel flow will be minimized at the point where the power required is a minimum. The speed corresponding to the bottom of the power required curve is the speed for maximum endurance (Figure 1).
To maximize the range, we want to get the maximum distance for each pound of fuel burned. Starting with our basic relation
or, splitting out the units
Now if we divide both sides by velocity, we get
Dividing out the hours, we get
Note that the left side is what we wanted, pounds of fuel per nautical mile. To minimize the pounds of fuel per nautical mile, we can minimize the ratio of power over velocity. Looking at the power required chart, a line from the origin to any point on the curve has the slope of power over velocity (P/V). As you trace a line from the origin to each point on the curve, the slope will be a minimum when the line is tangent to the power required curve. Therefore, the maximum range airspeed occurs where a line from the origin is tangent to the power required curve. This also corresponds to the minimum point on the thrust required curve (drag polar).
Unfortunately the maximum range airspeed is generally a lot slower than most people wish to fly. After all, you built an airplane to get places fast. Since we are also interested in getting places fast, we must consider speed. So consider a parameter of fuel flow per knot (Fuel Flow/knot). This would tell us how much fuel per hour we are burning for each knot of velocity. The optimum speed would then be the speed where this parameter is a minimum. Mathematically, the derivative with respect to velocity would equal zero, or
Qualitatively, at this point, if we increase the velocity, the parameter (Fuel Flow/knot) would increase. Likewise, if we decrease the velocity, the parameter (Fuel Flow/knot) would increase. Thus, this would seem to be the optimum cruise speed.
To find this speed, consider that since fuel flow is proportional to power, we would have
Well, power divided by velocity is just thrust, so we can use the thrust required curve (drag polar). A line from the origin to any point on the thrust required curve has a slope of T/V. If we divide both sides of the above equation by velocity, we get
The left side is what we want to minimize, and the right side is the slope of the line from the origin to a point on the thrust required curve. The minimum slope of this line occurs when it is tangent to the thrust required curve. This velocity, then, is the optimum cruising speed (Carson's Speed).
You can find the speed for the maximum lift to drag ratio (L/Dmax) right in your Pilot's Operating Handbook (if you have one). Look under emergency procedures for engine failure. Find the recommended glide speed for maximum range. This glide speed is the speed for L/Dmax (I'll save you the derivation). This speed also corresponds to the bottom of the drag polar. After all, if lift is a constant for level flight, L/Dmax will occur at the minimum drag. According to Carson, these three speeds are related by factors of 31/4, or about 1.316. So this means:
Vmax range = Best Glide Speed
Voptimum cruise = (Best Glide Speed) * 1.316
So now you can find these speeds for your airplane.
Aircraft are not designed solely for cruising; they must also takeoff and climb. As a result, your airplane is carrying around a lot more power than is required for cruising. Many production aircraft ended up cruising around the optimum cruise speed by empirical power sizing. For instance, the optimum cruise speed for a Piper Tomahawk would be (70)*1.316 = 92 knots, which corresponds to about 75% power. Of course, many of today's homebuilts, like the RV-6A, have a much higher power to weight ratio, and are capable of speeds well in excess of the optimum cruise speed. Check out the reported cruise speeds for the RV-6A in Sport Aviation compared to the reported optimum cruise speed. Just remember the time honored axiom of all forms of transportation - Speed cost money...how fast do you want to go?
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