# What is the optimal jet engine size for maximum range? [closed]

I under stand the flying wing has been calculated to have an optimal length. Is their an optimal blade length within a turbine engine for planes as well?

• Wouldn't it depend on the size of the aircraft? Commented Jan 3, 2023 at 19:54
• It's hard to imagine that engine that will produce the max range for a single seat jet carrying a few thousand pounds of fuel will look very much like the engine that will produce the max range for a jumbo jet, carrying a few hundreds of thousands of pounds of fuel. And a quick visit to an aviation museum, or their website, will indeed confirm the intuition that the engines for the two aircraft do, indeed, look vastly different... with the fan blades of the one as long as or greater than the entire diameter of the other. Good reasons they don't use F-16 engines in the A-380!
– Ralph J
Commented Jan 3, 2023 at 20:24

Starting with your title question, check out this paper.

The optimal engine size for maximum range is based on the thrust hook of an engine (Fig 2c in the paper). The thrust hook is the relationship between TSFC and thrust as you adjust the throttle.

Engines have poor TSFC (relative fuel consumption) at flight idle and lower throttle settings. They have better TSFC at nearly full power (near the design point for the engine).

Many engines pay a slight fuel economy penalty to get to 100% thrust -- such that their optimal fuel consumption is at about 85% throttle.

A jet's specific range (instantaneous fuel economy) is (Eq 2):

$$SR=\frac{V}{TSFC D}$$

Typically discussed in nmi/lb of fuel -- think like mpg for a car.

In cruise, $$L=W$$ must be true, so we'll often think of range in terms of the range parameter (Eq 3):

$$RP=\frac{V}{TSFC}\frac{L}{D}$$

We've multiplied both sides by $$W$$ and substituted $$L=W$$ on the RHS. The range parameter shows us that for best range of a jet aircraft, we want to maximize $$V L/D$$ and minimize $$TSFC$$.

Fun games can be played by multiplying by the speed of sound divided by the speed of sound ($$a/a=1$$) -- allowing us to write the velocity in terms of Mach number -- replacing $$V L/D$$ with $$M L/D$$ which you may have seen around. In this case, the speed of sound gets absorbed into the assumed altitude variation of TSFC, where $$TSFC=TSFC_0 \sqrt{\theta}$$ and the speed of sound variation between sea level and altitude provides the $$\sqrt{\theta}$$.

To maximize range, you want to simultaneously maximize $$M L/D$$ and minimize $$TSFC$$. This requires matching the aircraft (drag polar and wing loading) to the operating condition (speed and altitude) while sizing the engine such that the cruise throttle achieves best TSFC while $$T=D$$.

If the engine is too big, it will cruise near flight idle and will pay a substantial TSFC penalty. If the engine is too small, it will cruise at maximum throttle and a small TSFC penalty. If the engine is smaller than that, it won't be able to reach your desired flight condition (altitude/speed). This tradeoff is illustrated in Fig. 4 of the paper.

The other answer (that says the engine does not enter into the equation) is based on a handful of simplifications. Specifically, TSFC being constant with throttle, speed, and altitude (none of which are true) and the drag polar ($$C_{D,0}$$ and $$K$$) being constant with speed and altitude (also not true).

First, the most simple thrust equation for a jet engine is $$F_n=\dot{m} \Delta V$$. Where $$\dot{m}$$ is the mass flow through the engine and $$\Delta V$$ is the change in velocity of the flow through the engine.

The mass flow $$\dot{m}=\rho A V$$ is the product of the density, the area, and the velocity. If you are flying at a given flight condition, the freestream density and velocity ($$\rho$$ and V) are basically determined. As a designer, our best tool to change $$\dot{m}$$ is the engine size -- which we relate as the capture area $$A$$.

We can also effect thrust by increasing $$\Delta V$$, but know that a high $$\Delta V$$ engine will be much less efficient than a low $$\Delta V$$ engine. So, for fighters and other high performance aircraft, we design for high $$\Delta V$$ -- whereas for transport aircraft, we design for low $$\Delta V$$.

For thrust at best efficiency, you want to accelerate a lot of air (high $$\dot{m}$$) a little (low $$\Delta V$$).

So when we think about engine size for a given cycle (i.e. given $$\Delta V$$), we think about changing the capture area -- and engine thrust is proportional to area.

Once you get into the design of the blades, things get a lot more complex. You likely need to keep the blade tips from going too fast (high transonic or low supersonic). In the compressor or fan, you need to get a certain pressure rise out of each rotor/stator pair (a stage). To get a larger pressure rise, you'll want more turning (determined by blade angles and camber) and more stage solidity (blade area per annulus area). But you can't work a stage too hard without causing stall. Because of this, engines are designed with multi-stage compressors that work together to compress the flow as much as is needed.

As you have probably guessed, the length of the blades is mostly set by the area at each engine flow station. The rest of the blade design gets messy.

For a jetliner the range is maximised when it flies at a specific speed called velocity for best range and which has the following value:

$$V_{range}=\sqrt{\frac{2W}{\rho S}\sqrt{\frac{3K}{C_{d_0}}}}$$

where $$W$$ is the airplane's weight, $$\rho$$ is air's density, $$S$$ wing surface, and $$C_{d_0}$$ and $$K$$ the two usual terms expressing the drag coefficient $$C_d=C_{d_0}+KC²_l$$.

The relevant drag for best range can also be calculated and it is:

$$C_{d_{range}}=\frac{4}{3}C_{d_0}$$

What is the optimal jet engine size for maximum range?

As it can be seen, the engine does not enter in this equation in any way, so you simply would choose the smallest jet engine able to push the jetliner at that speed and supply enough thrust to win that drag. Obviously an engine thus selected might not guarantee to meet other requirements like the need to take off or to manoeuvre the aircraft.