Turbofans and bypass ratio

The turbofan fixes the turbojet's weakness by adding a second, slower stream. A large fan at the front, driven by a low-pressure turbine behind the core, compresses a big mass of air that goes around the core instead of through it, the bypass stream. The core still runs a Brayton cycle, but most of its energy now goes into the fan instead of the core jet. The result is a much larger total mass flow with a lower average jet speed: higher propulsive efficiency and lower fuel burn. Every airliner and business jet flying today uses a turbofan. The Swift 900 study has two candidate engines, one with a bypass ratio of about 4 and one of about 6, and Dr. Kenji Watanabe, who leads the high-speed group, wants the propulsion group's view on which suits the aircraft better.

Bypass ratio and the two streams

The bypass ratio \(\alpha\) is the mass flow of the bypass stream divided by the mass flow through the core. Business jet engines run from about 3 to 6; large airliner engines from 8 to 12 or more; military fighter engines below 1. Each stream has its own nozzle in a separate-exhaust engine, the usual arrangement on larger engines, or the two are mixed before a common nozzle, common on business jets because mixing slightly improves efficiency and reduces noise.

Most turbofans have two shafts, or spools. The high-pressure spool joins the high-pressure compressor and turbine, turning fast; the low-pressure spool joins the fan, any booster stages and the low-pressure turbine, turning slower because the fan's large diameter limits its tip speed. Some engines use three spools to let each compressor turn at its best speed, and geared turbofans put a reduction gearbox between the low-pressure turbine and the fan so that both can turn at their own best speeds. Gearing lets the fan grow larger and slower without making the low-pressure turbine slow and heavy.

The ideal turbofan

The ideal turbofan analysis extends the turbojet equations. The turbine must now drive both the compressor and the fan, so the turbine temperature ratio becomes

where \(\tau_f = \pi_f^{(\gamma-1)/\gamma}\) is the fan temperature ratio. The core jet velocity follows from the turbojet formula with this \(\tau_t\), and the bypass jet from the fan pressure ratio alone. The specific thrust per kilogram of total air is

The fuel-air ratio of the core is the same as in the turbojet. For each bypass ratio there is a best fan pressure ratio. Raising the fan pressure ratio puts more energy into the bypass stream and takes it from the core jet; the minimum fuel consumption comes, for an ideal engine, when the two jets have the same velocity. A fan pressure ratio that is too high leaves the core jet slow and the bypass jet fast; too low, the reverse.

# Ideal turbofan with separate exhausts (cycle analysis in the style of Mattingly):
# how bypass ratio changes specific thrust, fuel consumption and propulsive efficiency
# at the Swift 900 cruise point. Fan pressure ratio is chosen near its optimum for each bypass ratio.
import math
g, R, cp, hPR = 1.4, 287.05, 1005.0, 43.2e6
M0, T0 = 0.80, 216.65
a0 = math.sqrt(g * R * T0); V0 = M0 * a0
def turbofan(alpha, pi_c, pi_f, Tt4=1500):
    tr = 1 + (g - 1) / 2 * M0**2; tl = Tt4 / T0
    tc = pi_c ** ((g - 1) / g); tf = pi_f ** ((g - 1) / g)
    tt = 1 - tr / tl * (tc - 1 + alpha * (tf - 1))           # turbine drives compressor and fan
    if tt <= 0 or tr * tc * tt <= 1: return None
    v9 = math.sqrt(2 / (g - 1) * tl / (tr * tc) * (tr * tc * tt - 1))   # core exit, in units of a0
    v19 = math.sqrt(2 / (g - 1) * (tr * tf - 1))                        # bypass exit
    Fs = a0 / (1 + alpha) * (v9 - M0 + alpha * (v19 - M0))              # N per kg/s of total air
    f = cp * T0 / hPR * (tl - tr * tc)
    S = f / ((1 + alpha) * Fs)
    eta_p = 2 * M0 * ((v9 - M0) + alpha * (v19 - M0)) / ((v9**2 - M0**2) + alpha * (v19**2 - M0**2))
    return Fs, S * 1e6, eta_p, v9 * a0, v19 * a0
def best_fan(alpha, pi_c):
    best = None
    for k in range(101, 600):
        r = turbofan(alpha, pi_c, k / 100)
        if r and (best is None or r[1] < best[1][1]): best = (k / 100, r)
    return best
print("bypass  fan PR  F/mdot N s/kg  TSFC mg/(N s)  eta_p   core Ve m/s  bypass Ve m/s")
for alpha in (3, 4, 6, 8, 10, 12):
    pf, r = best_fan(alpha, 24)
    print(f"{alpha:5.0f}   {pf:5.2f}     {r[0]:7.1f}       {r[1]:6.2f}      {r[2]:.3f}     {r[3]:6.0f}       {r[4]:6.0f}")
mdot_needed = 3600 / best_fan(4, 24)[1][0]
print(f"\nan ideal engine of bypass ratio 4 needs {mdot_needed:.0f} kg/s of air for 3.60 kN at cruise;")
print("component losses roughly halve the real specific thrust, so the real engine swallows about twice that")

bypass  fan PR  F/mdot N s/kg  TSFC mg/(N s)  eta_p   core Ve m/s  bypass Ve m/s
    3    5.04       350.7        14.83      0.574        587          587
    4    3.85       299.3        13.90      0.612        536          535
    6    2.75       233.6        12.72      0.669        472          469
    8    2.25       192.9        11.98      0.710        431          429
   10    1.97       164.8        11.48      0.741        400          401
   12    1.79       144.1        11.10      0.766        379          380

an ideal engine of bypass ratio 4 needs 12 kg/s of air for 3.60 kN at cruise;
component losses roughly halve the real specific thrust, so the real engine swallows about twice that

Arjun's program finds the best fan pressure ratio for each bypass ratio by searching for the lowest TSFC, and the result confirms the equal-velocity rule: at every bypass ratio the two jets leave within a few metres per second of each other. Going from a bypass ratio of 4 to 8 lowers the ideal TSFC from 13.9 to 12.0 mg/(N s), a 14 percent saving. The best fan pressure ratio falls from 3.85 to 2.25, and the specific thrust falls from 299 to 193 N s/kg, so the bypass-8 engine needs 55 percent more air for the same thrust. Propulsive efficiency rises from 0.61 to 0.71. Real engines are less efficient than these ideal numbers, but the trends are the same.

The last lines size the airflow. An ideal bypass-4 engine needs about 12 kg/s for the Swift 900's 3.6 kN cruise thrust; the real engine, with losses, needs about twice that, the 26 kg/s used in Lesson 7. Dr. Watanabe's question is therefore one of balance. The bypass-6 engine saves fuel, but its fan is about 20 percent larger in diameter, which means a heavier engine, more nacelle drag and harder installation under the Swift's low wing or on its rear fuselage. Arjun's recommendation, written up for the programme review, is that for a business jet flying about 400 hours a year the fuel saving of the larger engine does not repay its weight and drag on the typical 1500 km mission, and that bypass ratio 4 to 5 with a mixed exhaust is the better match.

Limits on bypass ratio

If higher bypass is always more efficient, why stop? Several things push back. The fan and nacelle get larger and heavier, and nacelle drag grows with the wetted area; at some bypass ratio the extra drag and weight cancel the propulsive gain. The low-pressure turbine must drive a larger fan at lower speed, so it needs more stages unless a gearbox is added. Ground clearance under a wing limits fan diameter, which is one reason recent airliners have raised landing gear or engines mounted forward and higher. And as the fan pressure ratio falls, the fan becomes harder to operate stably at low power, because the bypass nozzle is no longer choked and the fan's operating point moves toward stall; some designs consider variable-area nozzles or variable-pitch fans. Open rotors, unducted fans with effective bypass ratios of 30 or more, remove the nacelle altogether and are again being studied for future aircraft, with noise and installation as the main obstacles.

Mixed exhausts and noise

In a mixed-flow turbofan the hot core stream and cold bypass stream meet in a mixer, often a lobed ring that interleaves the two, before leaving through a common nozzle. Mixing a hot, fast stream with a cool, slower one at the same total pressure produces a little more thrust than expanding them separately, typically 1 to 3 percent, because it reduces the kinetic energy wasted in the faster jet. The mixer and the longer nacelle add weight, so mixing is most worthwhile on lower bypass engines like those of business jets, where the two streams differ most.

Bypass ratio also governs noise. Jet noise rises steeply with jet velocity, roughly with its eighth power, so the slower jets of high-bypass engines are much quieter. As jets have become quieter, fan noise, from the fan blades and their wakes striking the outlet guide vanes, has become the dominant source, controlled with acoustic liners in the nacelle, careful choice of blade and vane numbers and swept fan blades. Aircraft noise is certified against ICAO standards adopted by Transport Canada, and quieter engines also matter for community acceptance at airports like Winnipeg's.