A Really Deep Dive into propulsion

Overview

Propulsion is achieved through Newtons Second Law. Accelerating a mass in the opposite direction of where one wants to go. To make the mathematics easier, and certainly the instrumentation easier that is considered a mass rate and a velocity, (acceleration x time):

v = vi + a*t
a = dv / dt = (v₁ – v₀) / (t₁ – t₀)

Adding on to this, if we are looking at dv instead of a, then we get to look at m_dot, mass flow rate instead of mass.

We have an open tube with an area moving at a given speed through a medium with certain pressure, temperature and characteristics. Pushing a mass through an area in a certain time gives the total mass moved. As the velocity is implied by the Mach number, we get a fair indication of when this will be maximised, as Mach occurs twice in the equation, once as a square function.

When Mach is 1.0, the maximum flow rate is achieved. This is such a fundamental observation, that it has become a design limitation, well and truly baked into turbo-machinery design software; this is the point where mass flow rate choking will occur, when flow rate is at a maximum and achieved pressure declines. This then gives rise to a number of interesting turbo-machinery phenomena, and it defines the performance limit of compressors, turbines, fans etc. It’s considered to be inviolate.

Mass flow rate choking mathemetics

The mathematics on the relationship of mass flow rate, pressure and velocity are set in stone- mass flow rate:

where:

This expands to the components of density and velocity to be:

where:

From this equation, it’s apparent that the mass flow rate, M-dot, is related to A (area), total pressure, total temperature, gas constant, and specific heat ratio, and… Mach.

It follows that the maximum mass flow rate occurs when Mach, “M” = 1.0. Altering any condition affecting mass flow rate in a turbo-machinery simulator will result in the A component, the area, altering when local Mach number exceeds 1.0.

When M = 1.0, the equation collapses to the maximum mass flow rate:

As an observation, while this is the maximum mass flow rate, it is not exactly the maximum thrust outcome that can be achieved. higher Mach numbers > 1.0 will reduce the mass flow rate from its maximum, but as far as the thrust component goes, the V^2 bit gets interesting, and maximum thrust would occur at around Mach 1.6 when the free-stream velocity is considered as Newton’s 2nd law gives:

which expands to:

For a turbofan engine, this is the component from each part of the thrust sources. A propeller follows the same mathemetics, as exhaust from the power plant is additive to the thrust from the propeller disk itself.

replacing the “bypass” and “core” stations with the SAE ARP-755A standard values.

Station 13 is the bypass flow aft of the fan; station 2.1 is the intake to the core; wf is the fuel flow mass that is added to the core flow, which is related to the stochiometric ratio of the fuel control units setting, but is usually around 1:40, or 2.5% fuel to air, dependent on how hot the combustion chamber is able to be run without damage, and without excessive NOx production. Station 8 is the core nozzle, 18 is the bypass exit nozzle, and 0 is the environmental values.

The development of the jet engine came about as a means to avoid the constraints of the propeller as a propulsion system when TAS started to get to significant values. The propeller is similar to the fan conceptually, but evolved to have a variable pitch to maintain a semblance of efficiency as speed increased. The fan could be produced with a variable pitch fan blade, which would increase efficiency at high speeds, however, the material and loads problems of such a design are not for the faint hearted. A propeller takes a very high blade root load from acceleration, a propeller can have more than 12 tons of root load on the blade retention faces, but fan blade…

or more completely…

The load on a bearing becomes quite a serious issue for the fan, more than the propeller, as the fan has far more blades, and therefore lower area for a bearing to change pitch. A fan blade can have more than double the radial load that a propeller has… Of more concern is blade retention on any kinetic impact; propeller speeds are almost invariably subsonic, turbo fan blades at high power are well sonic, and so the kinetic energy from any impact is considerably greater, being KE=1/2mv^2… VP fan blades would be helpful, but are challenging.

But, for mass flow rate choking, the conventional constraints are not the only solution…

It is the obvious one, and the one that we have followed for the last 100 years of propulsion design, building on the fluid mechanics basics of the giants of the 19th century, who in turn stood on the shoulders of a long line of creative minds:

The consequences of mass flow rate choking are quite evident, as shown below:

and again… from an assessment of varying fan blade twists…

we get mass flow rate choking, still:

At this point, it is worthwhile to consider what is happening on any blade, whether propeller, rotor or fanblade. The total aerodynamic force that can be produced is not difficult to calculate using a blade element-momentum theory analysis, until compressibility occurs at which point all bets are off as the analysis becomes non -linear. Propeller-powered aircraft operators may suggest that their propellers are designed to not go sonic, and as far as that refers to tip velocity, that is quite true, going sonic on a propeller is a memorable occasion, and normally not done twice. Propeller tips are indeed generally subsonic; fan blades, not so, they are often sonic at higher RPMs with the shroud providing acoustic attenuation. Helicopter rotors definitely get untidy at sonic speeds. In all cases however, even a J-3 cub with a Continental 65 will have sonic flow on the suction face of the blade, normally around the first 25-30% of the chord of the blade, extending from near mid span to the tip. This flow is the consequence of acceleration due to the blade angle and the increase of the local rotational velocity due to this accelerated flow velocity. This flow acceleration is associated with a lower pressure field on the forward part of the blade.

The velocity near the trailing edge of the blade is low; characteristically, the flow is reversed at the trailing edge and usually separated, which increases local pressure towards ambient. Having a higher pressure at the trailing edge than near the leading edge occurs in all cases, and this results in an adverse pressure gradient along the chord that increases pressure as well, until the sonic flow decelerates to a local Mach number of 1.0. At Mach 1.0 local velocity, a compression shockwave occurs, known as a normal shockwave, being “normal” to the surface of the airfoil. At this shockwave, an immediate change in pressure and temperature occurs which is associated with a gain of entropy. Behind the normal shockwave, the boundary layer is severely impacted and a thickening of the turbulent boundary layer occurs behind the foot of the shockwave, described aptly as a Lambda foot. The whole impact is known as the “shockwave boundary layer interaction”, or SBLI, and usually this will exhibit substantial turbulent flow behind this point, and very early separation and reverse flow arising.

A characteristic of normal shockwaves is they are very sensitive to any change in angle of attack, so any variation in the inflow angle relative to the axis of rotation of the blade will result in highly unsteady shockwave locations on the blade, resulting in vibration from the harmonic periodic change in lift, drag and pitching moment on the blade.

Do high tech fan blades have normal shockwaves?

Absolutely, but only when above idle- usually. In simple terms, when developing higher thrust from a propeller, a rotor or a fan, substantial normal shockwave involvement will occur.

So, what?

Normal shocks are associated with mass flow rate choking, and limit performance while absorbing energy from your power plant, be that an ICE, turbine, electric motor or rubber band; your current propulsion system is constrained. Slow RPM propellers have additional issues in play, as true speed increases, the propeller blade angle, “beta” increases on variable pitch propellers, to absorb the torque applied from the power plant, and to convert that to thrust. Thrust is the resultant force acting in the direction of the axis of rotation of the propulsion disk, (normal to the plane of tip path rotation) and the torque demand is any force acting opposite the axis of rotation of the propulsion disk. As speed increases, aerodynamic forces which are acting relative to the induced inflow angle to the zero lift line of the blade (its a prop thing, they are almost all cambered sections) so torque required increases relative to any thrust, while thrust reduces as beta angle increases. The limit case is a feathered prop absorbing all the torque that can be mustered from your powerplant, and no thrust being achieved.

As limiting as this is, it is also a similar condition on the fan blade, except that the fan blade is generally fixed beta angle, and the only consolations are good nacelle design acting to lower intake velocity while increasing pressure, and the curious matter that blades “unwind” under aerodynamic loads.

Unwinding

Fan blades bend under aerodynamic loads, but are stiffened by radial acceleration. The blades also twist under aerodynamic loads, and due to the normal shockwave involvement, the lift is forward on the blade, forward of the axis of twist of the blade. That this is a true statement can be verified by looking at a CF6, TFE731, CFM56-3 or -5 blade, or any others that have mid span shroud “MSS” “snubbers” in their design to limit flutter effects, The MSS tabs under no load are a loose fit, the blades will rattle as they rotate slowly, as some MSS tap against another. The shape shows that they effectively lock as the blade twists to a higher beta angle in the area of the MSS. This is generally only a curiosity, until an analysis of the loads that are derived on a blade gets serious consideration.

OK, but why is this important?

Conventional propulsion design accepts that shockwave losses will occur, and are unavoidable. When trying to achieve propulsive thrust, invariably losses associated with normal shockwaves will occur.

“these are the facts, and they are incontestable…!”

Not so.

What we have been doing for the last 100 odd years is only one solution. deltaBurn applies a second solution, and does not have a normal shockwave structure on the same blade, and therefore, we get:

Lower Vibration. Always. Why? There’s no shockwave, and therefore the related inevitable flow asymmetry isn’t there.
More Thrust. A lot more. We have no normal shockwave losses, we don’t gain the entropy that is a natural part of every propeller, rotor and fan blade in use today. We will double-cross this bridge later, once the penny has dropped.
Less torque demand for a given thrust. To achieve the existing thrust from a propeller, or a fan, we can use a lower RPM for the fan blade, or lower torque for the propeller case.
More thrust is not always your friend. The thrust bearings on the turbofan N1 stage, the fan frame itself, etc., and the engine mounts are all designed for a (maximum) standard thrust. While far more thrust can be had from any turbofan engine with a bypass ration above 2.3, doing so has to be tempered to remaining within the structural limits of the original design.
This means that a lower N1 is used for takeoff to achieve the same overall (rated) thrust. As the engine core (the rubber band) is not altered at all, running the engine at a lower N1 means running the throttle at a lower point, with lower fuel flow, EGT, CO2, NOx particulates, water vapour, all of it, being produced.

There is nothing special generally about this outcome, running an engine at idle uses less energy and pollutes less, and that is about the sum of that, except- NOx; in the case of NOx, lower EGT comes from lower combustion temperatures, and the production of NOx is dependent on how much fuel is burned in a nitrogen-rich atmosphere, as well as the temperature of the combustion itself. Cooler means lower NOx production. For a one-third reduction in fuel burn to achieve the same thrust (which is about average for what we’ve been flying) the NOx change starts at that one-third less, and is further reduced by another one-third due to the ~120 C lower EGT, which means ~200 C lower CCT.

(Why not run an analysis of a GE90 or a Leap-1 in the GAS TURB 14 software? Use two-thirds of the regular fuel flow of the standard engine, then check the emissions).

Great Reading:

-o0o-

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