How to determine the maximum forward airspeed of a plane in an unaccelerated flight?

Question

What is the equation needed to determine the maximum forward airspeed of a propeller driven plane in an unaccelerated level flight? I need an example with units of measurements. The engine is rated at 55-75 hp.

Answer 1

EDIT: I added a calculation at the end of the answer

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Thrust $T$ produced by a propeller is:

$T=\eta \frac{P}{V}$

where $\eta$ is the propeller's efficiency, $P$ the power supplied to the propeller and $V$ the flying speed.

This thrust wins drag, which can be as usual expressed as:

$D=½\rho V²SC_d$

where $\rho$ is air's density, $S$ wing surface and $C_d=C_{d_0}+\frac{C²_l}{\pi Ae}$.

$C_l$ can be derived from the usual equation for lift $L$ that equals weight $W$:

$L=½\rho V²SC_l=W \Rightarrow C_l=\frac{W}{½ \rho V²S} \Rightarrow C²_l=\frac{W²}{(½ \rho V²S)²}$

Substituting and equating $T=D$ we get:

$\eta \frac{P}{V}=½\rho V²S(C_{d_0}+\frac{W²}{(½\rho V²S)²(\pi Ae)})$

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So far so good.

This equation can be obviously solved mathematically but a graphical representation is easier; we just have to plot that equation after having rewritten it as:

$0=-\eta \frac{P}{V}+½\rho V²S(C_{d_0}+\frac{W²}{(½\rho V²S)²(\pi Ae)})$

This equation is 0 where it intersect the x-axis and that intersection is the $V$ we are looking for.

Let's use then the values given in the question:

• $\rho=0.96kg/m³$; I really couldn't convert the given $q$ in a standard number so I use the ISA density at 1'500m (5'000ft).
• $S=3.5m²$.
• $C_{d_0}=0.021$
• $\eta= 0.8$; this value is ok if we consider a variable-pitch propeller or a propeller working at its design point; otherwise it can be as low as 0.4 or even less.
• $P=56'000W$ (75hp).
• $W=3'700N$.
• $A=7.65$.
• $e=0.821$.

Substituting we get:

$0.0355V²-44'800/V+110.81/V²$

whose graphical solution is some 108m/s 210kts (general shape in the first picture and zoom of the solution in the second picture):

Just out of curiosity, with an $\eta=0.4$ the speed reduces to some 85.8m/s 167kts.

Answer 2

Your "givens" include a table of Cl/Cd versus airspeed.

Therefore it is simple to prepare a graph of Cd/Cl versus airspeed.

From this, you can prepare a graph of (Weight * (Cd/Cl)) versus airspeed.

Thrust Required = Weight * (Cd/Cl). So this is your graph of Thrust Required versus airspeed.

You'll also need to prepare a graph of Available Thrust versus airspeed. For this, you'll need to know more information than you've given us. For example, you might have a curve of engine horsepower versus RPM, along with some indication of prop efficiency. Or you might even have some way of measuring thrust delivered by the motor to the prop in actual flight. Without knowing more about what your "knowns" are, it's hard to be specific about how you should prepare the graph of Available Thrust versus airspeed. (It's going to make a tremendous difference whether the propeller is fixed-pitch or variable-pitch (constant speed), and if the former, for what airspeed it is optimized -- i.e. "climb prop" versus "cruise prop".)

An absolute upper bound for your Available Thrust at any given airspeed would simply be Power / airspeed, assuming that the full nominal value of (75 hp? It's unclear what "55-75 hp" means) is available regardless of airspeed. You'll never have this much thrust available in actual practice, but it's a starting point.

At any rate, the airspeed (well to the right on your graphs, i.e. well above the minimum-thrust-required airspeed) where the Thrust Required curve crosses the Available Thrust curve, is the maximum sustainable airspeed. If you've used the theoretical "upper bound" expression for Available Thrust given above, this airspeed figure will be overly optimistic. (But if this was a homework problem, perhaps this is the approach you were expected to take.)