# Bypass Turbofan Engine: Formulation

Typically, in this book, a long-duct nacelle is preferred to obtain better thrust and fuel economy and to offset the weight gain as compared to short-duct nacelles (see Figure 10.21). The pressure increase across the fan (i. e., secondary cold flow) is substantially lower than the pressure increase of the primary airflow. The secondary airflow does not have the addition of heat as in the primary flow. The cooler and lower exit pressure of the fan exit – when mixed with the primary hot flow within the long duct – reduces the final pressure to lower than the critical pressure, favoring a perfectly expanded exit nozzle (pe = p»). Through mixing, there is a reduction in the jet velocity, which provides a vital benefit in noise reduction (see Chapter 15) to meet airworthiness requirements. The long-duct nacelle exit plane can be sized to expand perfectly.

Primary flow has a subscript designation of p and secondary flow has a subscript designation of s. Therefore, Fp and Vep denote primary flow thrust and exit velocity, respectively, and Fs and Vsp denote bypass flow thrust and its exit velocity, respectively. Thrust (F) equations of perfectly expanded turbofans are computed separately for primary and secondary flows and then added to obtain the net thrust, F, of the engine (i. e., a perfectly expanded nozzle):

F = Fp + Fs = [(mp + mf) X Vep mp X V00] + [ms x (Ves 1»)]

Specific thrust in terms of primary flow becomes (f = fuel-to-air ratio), or:

F/mp = [(1 + f) x Vep + BPR x Ves – V» x (1 + BPR)] (10.12)

If the fuel flow is ignored, then:

F/mp = [Vep – V»] + BPR x (Ves – V») (10.13)

For kinetic energy (KE):

ke = mp^Vp – V2)] + ms[1h{VSp2 – V»2)] or (10.14)

KE/thp = [i/2(Vep2 – V2)] + BPR x [i/2(Vsp2 – V»2)]

At a given design point (i. e., flight speed V»), BPR, fuel consumption, and mp are held constant. Then, the best specific thrust and KE are found by varying the fan exit velocity for a given Vep, setting the differentiation relative to Ves equal to zero. (This may be considered as trend analysis for ideal turbofan engines; real engine analysis is more complex.)

Then, by differentiating Equation 10.13:

d( F/m p)/d(VeS) = 0 = d(Vep)/d(VeS)+BPR (10.15)

Equation 10.14 becomes:

d(KE/thp)/d(VeS) = 0 = Vepd(Vep)/d(VeS) + BPR x VSp (10.16)

Combining Equations 10.15 and 10.16:

-BPR x Vep + BPR x Vsp = 0

Because BPR = 0, the optimum is when:

Vep = Vsp (10.17)

That is, the best specific thrust is when the primary (i. e., hot core) exit-flow velocity equals the secondary (i. e., cold fan) exit-flow velocity.

Equation 10.4 gives the turbojet propulsive efficiency, np = v2VV, for a simple turbojet engine; however, for the turbofan, there are two exit-plane velocities – for the hot-core primary flow (Vep) and for the cold-fan secondary flow (Ves). Therefore, an equivalent mixed turbofan exit velocity (Veq) can substitute for Ve in the previous equation. Fuel-flow rates are minor and can be ignored. The equivalent turbofan exit velocity (Veq) is obtained by equating the total thrust (i. e., a perfectly expanded nozzle) as if it were a turbojet engine with total airmass flow (mp + ms).

Entropy, e

Figure 10.13. Afterburning turbojet and T-s diagram (real cycle)

Thus:

(mp + ms) X (Veq Vc») — mp X (Vep V(Xi’) + ms X (Ves Vc»)

or (1 + BPR) X (Veq – VTO) — (Vep – VTO) + BPR X (Ves – Vx)

or (1 + BPR) X Veq — (Vep – VTO) + BPR X (Ves – VTO) + VTO X (1 + BPR)

— Vep + BPR X Ves

or Veq — [Vep + BPR X Ves]/(1 + BPR) (10.18)

Then, turbofan propulsive efficiency:

—- *- m |

2VTO

npf V V

eq + to

Large engines could benefit from weight savings by installing short-duct turbofans; some smaller aircraft also use short-duct nacelles.