# Why does thrust-to-weight ratio depend from max Mach number?

Why, for jet airplane design, does maximum Mach (or maximum speed) affect thrust-to-weight ratio so strictly? (as you can see for example in Daniel P. Raymer "Aircraft Design: A conceptual Approach" Table 5.3 page 80, where you can find $\frac{T}{W_o}=a\cdot M_{max}^C$, where “$a$” and “$C$” are empirical factors, for jet transport $a=0.267$ and $C=0.363$)

Just now I did notice to detail that Raymer say : "The equations were developed by the author using data from Ref 1" Ref1: i.e. the famous annual book : "Jane's All the world Aircrafts" . But anyway "maximum speed" (or maximum Mach number) in this case the maximum speed is ...the speed that the aircraft is not able to exceed or.... the speed that is forbidden for the pilot to try to exceed because it can be dangerous? Because, if it's the first case it's more understandable; in fact, if the other parameters are fixed, with the same fixed distance, if T/W is low the acceleration is lower and so before the end of the distance, you can arrive to low speed. At the opposite if T/W is high the acceleration is higher and so before the end of the distance, you can arrive to high speed. What do you think? So, in Janes book, what is the definition of Maximum Mach number?

• Sounds backwards... T:W would drive the max achievable speed. Even that assumes, though, that your Max achievable speed isn't limited by an MMO or VMO. An F-15 probably doesn't care; many airliners can reach MMO in level flight even at max weight.
– Ralph J
Mar 8, 2016 at 13:46
• @RalphJ it just depends on which way you look at it. When designing an aircraft, max Mach would be a typical requirement, and the given equation will allow you to estimate the T:W ratio needed to achieve it. Mar 8, 2016 at 15:24
• but, why does each affect other? Mar 9, 2016 at 9:49
• How can we demonstrate the dependence each other (theoretically, with formula not only with empirical factors)? Apr 21, 2016 at 15:59

Disclaimer: I have no copy of Raymer's book, so I have to guess what this table looks like.

There should be no direct dependency, except that all aircraft need to operate in the real world. If you look at the wing loading of regular jet transports, you will notice that they bunch together around 500 - 600 kg/m². This is the maximum wing loading which still allows them to operate from normal airports - go to higher values an you need a dry lakebed for take-offs and especially landings.

To push the resulting wing to higher flight speeds needs more thrust - hence the dependency. If you start with a higher wing loading (say, 800 kg/m²), the thrust/weight ratio for reaching the same Mach number is lower. The approach and landing speeds grow with wing loading as well, which in turn affects the resulting landing distance. I guess there is a maximum landing distance requirement for all aircraft in the table which is not explicitly mentioned.

• Peter you say: " If you start with a higher wing loading (say, 800 kg/m²), the thrust/weight ratio for reaching the same Mach number is lower. " Is not the opposite? May 9, 2016 at 14:29
• @d.pensopositivo: No, the smaller wing creates less drag and needs less thrust. Once the aircraft flies fast enough, a high wing loading will make it more efficient. May 9, 2016 at 20:59
• @d.pensopositivo Peter Kämpf is correct. At high speeds parasitic drag dominates, and this is proportional to the wing area. If thrust and weight are held constant, higher wing loading -> less wing area -> less drag -> higher speed. See the formula and derivation in my answer for exact details. May 12, 2016 at 22:24

In a short answer: In practice Mmax and Thrust to weight ratio are not correlated because other factors are more important, but in theory there is some dependency. The answer depends if you're trying to compare planes, in which case you're looking at correlations among dozens of performance parameters and trying to fit estimates into simple equations. Or you could be designing a plane, in which case you'd run equations to make sure the plane has enough thrust to meet the target Mmo. The case presented by Raymer seems to be a mix of the two. He's using industry data to predict necessary T/W early in the design process.

## Design Theory

The most confusing part here is that normally we think of thrust working against drag to provide acceleration, yet here we're talking about steady-state speed, not acceleration. The catch here is that drag is not constant as speed increases. As you accelerate you reach a point where drag becomes strong enough to counteract the maximum engine thrust, and you can't go any faster in level flight (see the first graph [source])). Assuming thrust is constant with respect to velocity, then from the formulas for drag as a function of speed we can derive the maximum velocity. This maximum velocity (in a thrust-limited level flight) can be written as a function of thrust-to-weight ratio and wing loading: (see derivation here (pdf), compare with here (pdf)) $$V_{max}=\sqrt{\frac{\frac{T_{A,max}}{W}\frac{W}{S}+\frac{W}{S}\sqrt{(\frac{T_{A,max}}{W})^2-4C_{D_{0}}K}}{\rho _{\infty }C_{D_{0}}}}$$ Where $T_a$ is available thrust, $W$ is the weight of the aircraft, $\frac{W}{S}$ is wing loading, $K$ is the correction factor for induced drag, and $C_{D_0}$ is the drag coefficient of aircraft when lift equals zero.

However, you'll notice that there are other parameters in this formula besides thrust to weight ratio like wing loading and drag coefficient. In practice these are not trivial considerations, especially when comparing different planes. Raymer himself notes "Later in the design process, aerodynamics calculation of drag at the design maximum speeds will be used, with other criteria, to establish the required T/W." There are also factors this formula doesn't consider like the variation in thrust as speed changes. These are the reasons thrust to weight ratio and maximum speed are not correlated when comparing different planes.

## Comparing Planes

Let's see if this relationship actually holds out when comparing different planes. What real-world perfomance parameter should we reference for Vmax? Mmo, high speed cruise TAS, or something else? Technically speaking, Mmo is the certified max operational speed and is a safety limit, not just a performance limit. During descents you can pick up energy very quickly and exceed the maximum level flight speed and Mmo. However, in practice (for business jets at least) the Mmo, high speed cruise TAS, and maximum level flight speeds are correlated. After all, you don't design many planes for their diving performance and many jets are designed to cruise close to their Mmo.

The truth is that none of these speeds is closely correlated with thrust/weight when comparing jets. In my data set of 31 business jets from "Business Jets 2012", the correlation between thrust to weight and Mmo has a pitiful R-squared value of 0.20 (see the second graph). I'm not sure what data Raymer was using and if anyone has better data for showing a trend let me know. I should note that once we account for the effects of wing-loading, there is a weak relationship between thrust to weight ratio and maximum speed. So the effect of thrust/weight on Mmax in the above equation is very weak but present (see the third graph).

There are a lot of parameters that loosely correlate with Mmo besides thrust, including aircraft type, cruise altitude, wing loading, and range. These parameters need to be engineered in conjunction to make sure the airplane meets all the performance expectations. Mmo may be reduced or increased so that one of these parameters reaches its target and vice versa.

In theory, there is a correlation, but I have to say my stats don't agree with Raymer's on the variables being "closely related" when comparing different jets. Other variables like drag and wing loading are too important to be neglected.

• Talking about "maximum Mach number" I didn't understand if Raymer means Mach number coming from V(Emax) (for levelling unaccelerate flight) or someone among limit speeds. I suppose, because it recalls Referes from Jane's, that we can find in Jane's, but till now.... May 15, 2016 at 15:19
• @d.pensopositivo I believe Raymer is referring to the maximum speed you want the engines to be capable of achieving in level flight during the design process. I'm not expert on how to choose that number and what it is in relation to Mmo, but I do know that for existing planes these top speeds and high speed cruises tend to be pretty closely related. May 16, 2016 at 7:05