Mate,
Go back and read the original email to gather some *context* and please
dispense with the semantics FFS. Ed was basically asking why the condition for
Pe>Pa doesn't produce more thrust than Pe=Pa given the equation in every text
is F = q × Ve + (Pe - Pa) × Ae
It's a fair enough question and I gave it my best shot at answering it. Okay,
I might not have worded everything perfectly, but given the context of the
original post, I assume most people would have known what I was getting at.
Troy.
-----Original Message-----
From: arocket-bounce@xxxxxxxxxxxxx [mailto:arocket-bounce@xxxxxxxxxxxxx] On ;
Behalf Of Alexander Ponomarenko
Sent: Friday, 28 October 2016 4:59 PM
To: arocket@xxxxxxxxxxxxx
Subject: {Spam?} [AR] Re: Rocket thrust change with altitude, RPA
As Pe increases higher than Pa (under expansion), Ve drops and to alesser
extent Ae drops too thereby reducing F
why does Pe=Pa produce more thrust than Pe>Pa for the same mass flowand throat area
As Pe increases higher than Pa, Ve drops and to a lesser extent Aedrops too
As Pe reduces lower than Pa (over expansion) Ve increases,
Just to clear this up a bit more:
F = q × Ve + (Pe - Pa) × Ae
where F = Thrust
q = Propellant mass flow rate
Ve = Velocity of exhaust gases
Pe = Pressure at nozzle exit
Pa = Ambient pressure
Ae = Area of nozzle exit
So, why does Pe=Pa produce more thrust than Pe>Pa for the same mass flow and
throat area:
(1) Ve is inversely proportional to Pe Due to Ve = SQRT((2 * k / (k -
1)) * (R' * Tc / M) * (1 - (Pe / Pc)(k-1)/k)) And
(2) Ae must also drop as Pe increases
Ae =
At/(((k+1)/2)^(1/(k-1))*(Pe/Pc)^(1/k)*SQRT((((k+1)/(k-1)))*((1-((Pe/Pc
)^((k-1)/k))))))
So, for F = q × Ve + (Pe - Pa) × Ae
As Pe increases higher than Pa (under expansion), Ve drops and to a
lesser extent Ae drops too thereby reducing F
As Pe reduces lower than Pa (over expansion) Ve increases, but pressure
thrust reduces more as it goes negative and coz that negative number is
multiplied with Ae which itself increases, the resultant negative is more
than the increase in Ve alone.
Troy
Ed (Kelleher),
The confusion here is that you're specifically looking at the pressure
thrust side of the equation what relates specifically to pressure imbalances
and yes, it is true that for an under expanded nozzle, the pressure thrust
will be higher than for an optimally expanded nozzle. However, the momentum
thrust will be reduced even more because the velocity component of that also
accounts for the pressure balance between Pc,Pe - in particular this part of
the momentum thrust equation : "(1 - (Pe / Pc)^(k-1)/k)" As you can see, as
you increase Pe, the contribution from this part is lower as it's being
subtracted from 1. In the end, this influence overrides the pressure thrust
contribution.
Here is probably the best explanation you're ever likely to see:
http://www.braeunig.us/space/sup1.htm
Troy.
