# When is jet engine thrust maximum?

Assuming we have a convergent nozzle I've read that the maximum thrust is achieved just in the moment the nozzle exit -minimum area- is chocked, i.e., the nozzle is adapted in the sense the pressure in the exit area equals the ambient pressure. How can I demostrate this with formulas?

• With a subsonic aircraft and a choked convergent nozzle, the gas pressure in the exhaust will be higher than ambient pressure. What exactly do you want to catch in equations? Thrust, efficiency, exhaust pressure? Commented Jun 7, 2017 at 4:56

Let's look at two cases, choked exhaust and complete expansion:

1. Choked exhaust

With a convergent exhaust pipe, the jet engine thrust reaches a maximum at the speed of sound of the exhaust gas stream.

The velocity of the gas flow increases if it was subsonic at the entrance of the pipe. In a convergent nozzle the maximum gas exhaust velocity is M = 1, the speed of sound at the temperature of the hot exhaust gas. With a choked exhaust, at M = 1 in the exhaust outlet, the static pressure is higher than ambient pressure.

The exhaust area needs to be reduced until the gas exit velocity is M = 1, which at for instance 800 ºC is 657 m/s. The pressure $p_e$ at the exhaust outlet will then be:

$$p_e =\frac {\dot{m}\cdot R \cdot T_e}{V_e \cdot A_e }$$

which is greater than the ambient pressure $p_0$.

The net thrust $F$ of a pure jet engine is $$F = \dot{m} * (V_e - V_0) + A_e * (p_e - p_0)$$

$R$ is the gas constant. Parameters you need to know:

• outlet mass flow from the turbine $\dot{m}$ in kg/s

• gas outlet temperature $T_e$ in ºK

• speed of sound at $T_e$ in m/s, which for a choked exhaust is equal to $V_e$

• outlet area $A_e$ in $m^2$

• airspeed $V_0$ in $m/s$ and ambient pressure $p_0$ in $N/m^2$

If we use the following example of an aircraft flying M 0.85 at 30,000 ft, choked converging exhaust, exit area 0.1 $m^2$, mass flow 70 kg/s, exhaust temperature 1073 K, we get:

F = 70 * (657 - 258) + 0.1 * (328,106 - 30,100)

= 27,962 N from kinetic energy + 29,801 N from pressure difference, about the same amount.

2. Complete expansion

$p_e$ is now equal to $p_0$. This condition occurs if total pressure at the turbine exit $p_{Tt} \leq \epsilon_{kr} * p_0$, with $\epsilon_{kr}$ for a hot exhaust gas being around 1.95.

Analogous to the choked exhaust case: $$A_e =\frac {\dot{m}\cdot R \cdot T_e}{V_e \cdot p_e }$$

For the same conditions at 30,000 ft follows: $A_e$ = 1.09 $m^2$

and F = 70 * (657 - 258) = 27,962 N

The net thrust in this case is a lot lower because the turbine exit pressure is lower than in the choked case, and therefore the propulsive power of the jet engine is lower. Usually, in the case of a turbojet the turbine exhaust pressure $p_{Tt}$ is a lot higher than $\epsilon_{kr} * p_0$, which will lead to the choked exhaust case above.

Turbofans with a high bypass ratio have a low enough $p_{Tt}$ to allow complete expansion, most of the gas generator power being used for the bypass air compression.