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Aerospace

Exit speed with a nozzle efficiency

Exhaust speed from a nozzle with efficiency ηn, the ratio of actual to ideal isentropic kinetic energy for the same inlet total state and exit static pressure pe.

Ref: AER-46
Rev: v1
Field: Aerospace
Result: V_e [m/s]

V_{e} = \sqrt{2 η_{n} c_{p} T_{t} [1 - {(\frac{p_{e}}{p_{t}})}^{(γ - 1) / γ}]}

The ideal exhaust speed from isentropic expansion to pe is √{2cpTt[1 − (pe/pt)^((γ−1)/γ)]}. A nozzle efficiency ηn is the ratio of the actual to the ideal exit kinetic energy for the same inlet state and exit static pressure, so Ve = √(ηn) times the ideal speed. Real nozzles have ηn of 0.92 to 0.99. The exit pressure must be one that the nozzle can really reach.

Exhaust speed from a nozzle with efficiency ηn, the ratio of actual to ideal isentropic kinetic energy for the same inlet total state and exit static pressure pe. It needs 0 < pe ≤ pt, 0 < ηn ≤ 1 and constant cp.

Calculator

Solve for

V_eExit speedComputed

m/s

η_nNozzle efficiency

0.81

c_pSpecific heat at constant pressure

J/(kg·K)

9001,300

T_tInlet total temperature

2503,000

γHeat-capacity ratio

1.11.67

p_eExit static pressure

20,000200,000

p_tInlet total pressure

200,0001,000,000

Ideal exit speed (ηn = 1)465.4 m/s
Exit kinetic energy per kg, ½Ve²102,900 J/kg
Pressure ratio pe/pt0.5

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Exit speed (m/s) current value 453.6 m/s

p_eExit static pressureKey driver

0743

T_tInlet total temperature

2931,010

p_tInlet total pressure

743

γHeat-capacity ratio

264527

c_pSpecific heat at constant pressure

429516

η_nNozzle efficiency

416465

The gas leaves at 453.59 m/s, against an ideal speed of 465.37 m/s.
The expansion from 200 kPa to 100 kPa converts 17.07 % of the total enthalpy cpTt into kinetic energy.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing (m/s)	Local step	Change in result
p_e Exit static pressure	20,000 to 200,000 Pa	0	743	743	+10%	−6.51 %
T_t Inlet total temperature	250 to 3,000 K	292.8	1,014	721.5	+10%	+4.88 %
p_t Inlet total pressure	200,000 to 1,000,000 Pa	453.6	743	289.4	+10%	+5.96 %
γ Heat-capacity ratio	1.1 to 1.67	264.4	527.3	262.8	+10%	+9.59 %
c_p Specific heat at constant pressure	900 to 1,300 J/(kg·K)	429.3	516	86.66	+10%	+4.88 %
η_n Nozzle efficiency	0.8 to 1	416.2	465.4	49.13	+10%	+4.88 %

Sweep of Exit static pressure

Exit static pressure (Pa)	Exit speed (m/s)
20,000	743
42,500	639.9
65,000	560.8
87,500	490.8
110,000	424
132,500	356.5
155,000	283.6
177,500	195.9
200,000	0

Sweep one variable

Vary

Exit speed (m/s)Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

V_e

Exit speed

Result · m/s

What the equation gives you. Everything else feeds it.

η_n

Nozzle efficiency

· typical 0.8 to 1

The nozzle efficiency, actual over ideal exit kinetic energy. Raises the exit speed with its square root. Raising it by 10 % (to 1.045) moves the result by +4.88 %, from 453.6 to 475.7 m/s.

c_p

Specific heat at constant pressure

J/(kg·K) · typical 900 to 1,300

The specific heat at constant pressure, in J/(kg·K). Raises the exit speed with its square root. Raising it by 10 % (to 1,105) moves the result by +4.88 %, from 453.6 to 475.7 m/s.

T_t

Inlet total temperature

K · typical 250 to 3,000

The inlet total temperature, in K. Raises the exit speed with its square root. Raising it by 10 % (to 660) moves the result by +4.88 %, from 453.6 to 475.7 m/s.

Heat-capacity ratio

· typical 1.1 to 1.67

The heat-capacity ratio cp/cv of the gas, about 1.4 for air. A lower γ (hotter, more complex gas) changes the compressible-flow exponents and the sizes of the shock and expansion effects. Raising it by 10 % (to 1.54) moves the result by +9.59 %, from 453.6 to 497.1 m/s.

p_e

Exit static pressure

Pa · typical 20,000 to 200,000

The exit static pressure, in Pa. A lower exit pressure (more expansion) raises the exit speed, with diminishing returns. Raising it by 10 % (to 110,000) moves the result by −6.51 %, from 453.6 to 424 m/s.

p_t

Inlet total pressure

Pa · typical 200,000 to 1,000,000

The inlet total pressure, in Pa. A higher total pressure for the same exit pressure raises the exit speed. Raising it by 10 % (to 220,000) moves the result by +5.96 %, from 453.6 to 480.6 m/s.

Worked example with the values above

Pressure term: 1 − (pe/pt)^((γ−1)/γ) = 1 − 0.5^0.28571 = 0.179665.
Energy: 2ηn cp Tt × term = 2 × 0.95 × 1,005 × 600 × 0.179665 = 205,739.4 m²/s².
Ve = 453.585 m/s.

Assumptions and limits

Steady, adiabatic flow with constant γ and cp, and a nozzle that is choked where the entry says so.
Uniform properties across the throat and exit plane, unless a survey is used.

Common mistakes

Using static instead of total pressure upstream.
Using the choked formula when the pressure ratio is below the critical value.
Forgetting that a convergent-divergent nozzle has different regimes from a convergent one.

Statistics and uncertainty

Ideal nozzle formulas overestimate real performance: discharge coefficients of 0.95 to 0.99 lower mass flow by 1 % to 5 %, and nozzle efficiency of 0.9 to 0.98 lowers exit speed by 1 % to 5 %. Total conditions are the key measurement, and a 1 % error in total pressure gives 1 % in mass flow.

Where it is used

Sizing throats and predicting thrust of jet and rocket nozzles.
Estimating mass flow through orifices, valves and venturis.

Maths you need

Critical pressure ratio: (2/(γ+1))^(γ/(γ−1)), about 0.528 for air
Momentum flux: ρu²A, the momentum carried by a stream
Nozzle efficiency: actual divided by ideal kinetic energy

Related formulas

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