# For any given modern turbofan engine, does bypass ratio stay approximately constant during a typical flight?

Bypass ratio is the quotient of mass flow rate through the bypass duct over mass flow rate through the engine core. Does it stay constant during the typical flight of an airliner?

If it does not stay virtually constant, what are the driving factors?

There are two factors that determine the bypass ratio (or rather, how it may change):

1. The first is the relative diameter of the fan, compared to the core. Obviously, this is fixed in the design process, and doesn't change during flight
2. The second is the relative difference in RPM speed between the two (or even 3) shafts that exist in the engine. This does change during a flight, from the takeoff condition, to a lower thrust setting at cruise. So let's look at this in detail.

The compressor acts like a pump, increasing the pressure of the air, whether it's sucking at the intake like in takeoff, or RAM feed, such as in cruise flight. Also like a pump, the faster a spool (compressor) spins, the larger the mass flow it pumps (creates). So, if we consider a CFM56-7B engine for example, its shaft speeds at 100% are 5,175 rpm for N1 and 14,417 rpm for N2. That's a ratio of 2.78. At takeoff, the engine will be close to this condition. But at cruise thrust at 35,000ft, the engine will be throttled back. What tends to happen in a gas turbine, is the N1 spool slows down relatively more than N2, as the throttle is reduced. So at cruise, N1 is maybe 3,880 rpm (75%), while N2 is maybe 11,534 rpm (80%). The ratio between them is now 3.41. The pumping capacity of the core has only reduced 20% from the takeoff condition, while the pumping capacity of the fan has reduced 25%. Hence, the bypass ratio will be lower at cruise than at takeoff.

Now, my values of 80% and 75% at cruise are just suggestions. But it is to illustrate the point, that if the ratio between the rpm of the two spools change, the bypass ratio will change.

In essence, the core flow is being determined by the rpm of both spools, while the fan duct flow is just determined by the rpm of the fan. So changes in the rpm ratio will impact the bypass ratio.

{Why is the N2 operating range less than the N1 range?: The CFM56 engine has a single stage fan, and depending on the model, either a 3 or 4 stage booster (or LPC), with a 9 stage HPC. These compressors are making the air flow in the direction of increasing pressure. The air doesn't really want to do that. So, if the design tries to increase the pressure ratio per stage too much, the blade is likely to stall. Getting aerodynamic conditions which avoid stall for the first stage, and the last stage, for the same operating condition is difficult. It's even harder when there are 9 stages on the shaft, than 3 or 4 (plus a fan). So the range of conditions that a 9 stage compressor will operate over is generally less than the range of conditions that a 3 or 4 stage compressor will operate over.}

• Would a good substitute for "ram-feed" be "inertia feed", since the air kind of stumbles into the first rotor at cruise? There might also be a slight drop in static pressure at the inlet of the first rotor, which could help with some suction even during cruise. But I don't know if that actually is the case. Do you, by any chance, have a reference where I could read up on what you've said. It appears that there's some reasoning with equations behind it, and I would like to read up on this. – user7241 Dec 16 '17 at 10:40
• Hi @jjack. I would not say "inertia feed" rather than ram feed, as the axial velocity of the air in the compressor is about Mach 0.4, while the aircraft in cruise is doing about Mach 0.8. So the air isn't really stumbling into the first rotor stage. It is decelerating in the intake, and converting dynamic pressure into static pressure, so the pressure is increasing even before entering the compressor. Sorry, I do not have a reference, but if I find one I will add it. I will add a paragraph explaining why the N2 operating range is less than the N1 range. – Penguin Dec 16 '17 at 11:53
• Hi, thanks a bunch. – user7241 Dec 16 '17 at 11:58

Mass flow rate is obtained as

$$\dot{m} = A \cdot \rho \cdot v$$

where $A$ is the area, $\rho$ is the density, and $v$ is the flux speed.

The bypass ratio is then

$$\frac{\dot{m}_{bypass}}{\dot{m}_{core}} = \frac{A_{bypass} \cdot \rho_{bypass} \cdot v_{bypass}}{A_{core} \cdot \rho_{core} \cdot v_{core}}$$

but $v_{bypass} = v_{core}$, and $\rho_{bypass} = \rho_{core}$ (since new mass is not created in the core, we can look at the values at the beginning of the core section).

This leaves us with

$$\frac{\dot{m}_{bypass}}{\dot{m}_{core}} = \frac{A_{bypass}}{A_{core}}$$

but $\frac{A_{bypass}}{A_{core}}$ is constant, because the geometry of the engine is generally not variable in an airliner.

This leads to $\frac{\dot{m}_{bypass}}{\dot{m}_{core}}$ being (fairly) constant with these conditions.

Assumptions that I deem reasonable:

• fuel flow mass is negligible w.r.t. air flow mass
• engine has fixed geometry
• air flow speed and density (page 130 of this book (*) ) are fairly homogeneous across the fan outlet

This last assumption means that I assume that the engine is working at or near its design point, as mentioned in the comments, changing the fan speed will affect the velocity distribution after the fan, and thus the BPR. This mostly happens only during take-off or landing, while the vast majority of the flight is usually cruise, where the assumption is generally satisfied.

(*) Note how the book, in the following chapter, explains that the BPR is a fundamental design parameter. And a few pages earlier (page 125) it shows how an engine would be designed starting from some requirements, and these include a specific (not variable) BPR. This is because engines are primarily designed around the conditions in which they work for the longest time, i.e. cruise.

• I remember an illustration where the separating streamline between core flow and bypass flow moved above the horizontal and below when looking at it from the side. However if I remember right then that effect was referred to as minor. – user7241 Dec 15 '17 at 8:43
• Do you think the additional "sucking" effect of the compressor would have much effect? – user7241 Dec 15 '17 at 9:09
• @jjack the fan acts on all the flow, bypass and not. the compressor does not "suck", it pushes back. – Federico Dec 15 '17 at 9:10
• The reason why I'm asking is that I have plots of bypass ratio over reduced fan speed for a given design. The BPR there also depends on altitude and flight Mach number. Only close to a reduced fan speed of one is the bypass ratio constant for all Mach numbers and altitudes. The different BPR lines converge from low reduced fan speeds to a point at a reduced speed of one. Below one, bypass ratio varies significantly. I don't see how fan speed would play a role unless it significantly changes the axial velocity distribution across the blade. – user7241 Dec 15 '17 at 9:23
• So basically, the answer, is, the mass flow ratio does change, but is fairly constant when the engine is operated at its design point (which really should be no surprise, as most things stay constant at a fixed operation point)? – Sanchises Dec 15 '17 at 12:57

The bypass ratio is optimised for the cruise, which is the design point, and it changes with changing eternal conditions. Off-design, the bypass ratio changes:

• During take-off;
• During climb;
• During approach.

The book Gas Turbine Theory by Saravanamuttoo-Rogers-Cohen contains a section on determining off design procedures of the turbofan. From section 9.4:

The approach described in section 9.2 is also applicable to turbofans, but in this case we must take into account the division of flow between the bypass duct and the gas generator, which will vary with off-design operating conditions.

Bypass characteristic is contained in the following figure, also from the book.

Input parameters for the procedure are pressure and temperature at the entry of the LP compressor, and the bypass ratio therefore also changes during climb in cruise. These two parameters change as a function of:

• True air speed
• Ambient temperature and pressure
• N1 and N2 of the engine.

This answers the second part of OP question If it does not stay virtually constant, what are the driving factors?

Note that the book also contains a worked out example on a twin spool turbofan - from example 32 on page 123 and further:

In both cases the area can be calculated from continuity, i.e. $m = \rho A C$. The density is obtained from $\rho = p/RT$, where $p$ and $T$ are the static values in the plane of the nozzle;

Density $\rho$ is a function of pressure and temperature, and statements on density and velocity anywhere in the engine cannot be made without also considering $p$ and $T$, which the book does of course.