Description of formulas for my simple but working (I hope so) numerical model of SSTO

Even if with the current values , my SSTO don't go to LEO, I hope it will be an good help to every one interested in SSTO .
1) Why using a model on spreadsheet and not something done in Mathematica or C++?
Because Mathematica or C++ required highly trained people and Mathematica is a costly commercial product that no one use at home.
Instead everybody who use Internet can read Excel or Works files (or even download compatible spreadsheets).
Plus : Spreadsheet have proven along the years an capability to make quickly models of various kind and help to test instantly various hypothesis, increasing dramatically productivity of mere mortals!
Spreadsheet success story exist because there is no loop in spreadsheet, all loops are developed and you are not required to be Einstein or Hawkings to understand complex phenomenon because they are exposed under your eyes.
In some ways Visicalc inventors changed more our world than Nasa scientists! (and it cost far least :-)
2) If you don't use Mathematica, C++, Fortran (giggle) or spreadsheets you must do mathematics analysis using calculus.
Even if this is something every educated young adult studied in classroom, the same adult nearly forgot all this stuff at the age of 30, because no one use calculus in real life, even engineers use numerical analysis because of the high computational power of today computers (old timers remember that Pentium series is more powerful than the 80' Cray I or II).
Numerical analysis must follow the same rules than calculus : Work on small quantities and be cautious about the use of formulas and units (ever see endless and frightful quarrels about ISP or remember Mars Polar Lander?). You must not use short cuts like delta V, this is not correct but lazy thinking, there is no excuses, after all it's not you who really work hard but the computer!
I test every formulae against another to hunt bugs, but I am sure there is a lot of bugs there even after all my efforts!


ALL references apply to rocket1.htm

1) Just for fun there is some errors, good luck!
2) To avoid circular references, formulae in some cells refer not to the same row, but to the previous row.
3) Precision can be attained only with small steps.

Exhaust speed = 3500 : The value in m/sec for Kerosene and Liquid oxygen (RP1 and LOX)
Length of combustion chamber = volume of combustion chamber / Aerodynamic surface of the rocket (m^2)
volume of combustion chamber =35*(3,14*(Throat diameter/2)^2)
Length of tanks (LOX+RP-1) = ((Mass of propellant /1080)*9,8) / Aerodynamic surface of the rocket

The various mass are in kilograms. Don't confuse weight and mass!
For the sake of simplicity the total mass is added from the payload mass and the propellants mass.

Ejected Mass per second = This is the important stuff!
It's the mass of gas that is ejected backward by the rocket, by the law of the conservation of impulse and knowledge of Exhaust speed of the gas you can compute the speed of the rocket (provided that you compute the new mass of the rocket because of the loss of the gas mass).

Theoretical speed with rocket equation = The rocket equation help compute the speed of a rocket from an initial state to a final state with three variables : Exhaust speed, initial mass and final mass.
This is the theoretical speed in vacuum and far away from any gravity well.

Real speed (w. air drag and weight) = an attempt to compute a more realistic value for speed with respect to gravity and air drag.

Force de propulsion (dm*Ex_speed) = A weak point in my calcs! I never retrieve where I found this formulae but it work better than another derivation of Newton's law so...

The last six formulas are an attempt to verify the efficiency of rockets as an engine motor. Carnot must be happy even steam engines make better use of energy!
Someone really smart must start thinking on something less dispendious of this world fossils energy.
In my dreams I sometime think that a conventional thermal motor using mechanic or electric mean to accelerate exhaust gas of the thermic motor to 6000m/sec might be something as highly disgracious but as useful as a railway engine!

Air drag = =(Real speed^2)*(Aerodynamic surface)*(Atmospheric Density)*0,4
This formulae is the classical one for a rocket speed slower than the sound speed. There is other formulas one for larger than sound speed, and one another for air drag in semi vacuum.
I use this one because of simplicity and there was so much simplifications elsewhere and so much things not incorporated in the model (wind by example) that it was inconsistent to use here a complicated formulae.

Atmospheric Density = (last step atmospheric density)*EXP(-(Altitude above ground) / 25000)
This is not the usual formulae, this one is simpler and seem as accurate.










System :m/kg/sec

 

Fact. rég. l'accélération

 

0,0045

Vertical depuis le sol

 

Ratio dry_M/M_plein

 

0,090909091

Surface aérodyn. (m^2)

3

Vmax (m/sec)

 

0

Exhaust speed

3500

Gmax

 

1,0

Length of combustion chamber

volume of combustion chamber

 

#VALEUR!

Length of tanks (LOX+RP-1)

7,56

 

Throat diameter

#VALEUR!

 

 

 

 

 

Sens du vol

V

V

V

V

Time in sec after launch

0

2

5

10

Total Weight

27005,00

SI(C11*(C29-(C29*(B14/11000)))> 0;C11*(C29-(C29*(B14/11000)));0)

SI(D11*(D29-(D29*(C14/11000)))> 0;D11*(D29-(D29*(C14/11000)));0)

SI(E11*(E29-(E29*(D14/11000)))> 0;E11*(E29-(E29*(D14/11000)));0)

Total mass

2750

C12+$L1

D12+$L1

E12+$L1

Mass of propellant

2500

SI(B12-(C13*(C9-B9))> 0;B12-(C13*(C9-B9));0)

SI(C12-(D13*(D9-C9))> 0;C12-(D13*(D9-C9));0)

SI(D12-(E13*(E9-D9))> 0;D12-(E13*(E9-D9));0)

Ejected Mass per second

8,00

SI(B12>0;(B11*$E$1);0)

SI(C12>0;(C11*$E$1);0)

SI(D12>0;(D11*$E$1);0)

Real speed (w. air drag and weight)

0

((C18*(C9-B9))/C11)+B14

((D18*(D9-C9))/D11)+C14

((E18*(E9-D9))/E11)+D14

Theory with rocket equ.

 

$B4*LN($B11/C11)

$B4*LN($B11/D11)

$B4*LN($B11/E11)

Apparent number of G

1,00

((C18/C11)+C29)/9,82

((D18/D11)+D29)/9,82

((E18/E11)+E29)/9,82

Force de propulsion (dm*Ex_speed)

0,00

(C13*$B4)

(D13*$B4)

(E13*$B4)

Force de propulsion nette

0,00

C17-C25-C10

D17-D25-D10

E17-E25-E10

Kinetic energy gained (millions of Joules)

0

(0,5*C11*(C14*C14))/1000000

(0,5*D11*(D14*D14))/1000000

(0,5*E11*(E14*E14))/1000000

Equiv motor in KW gained

 

(C19/(3,6))/((C9-$B9)/3600)

(D19/(3,6))/((D9-$B9)/3600)

(E19/(3,6))/((E9-$B9)/3600)

Caloric energy used

0

42*($B$12-C12)*9,82

42*($B$12-D12)*9,82

42*($B$12-E12)*9,82

Equiv motor in KW used

 

(C21/(3,6))/((C9-$B9)/3600)

(D21/(3,6))/((D9-$B9)/3600)

(E21/(3,6))/((E9-$B9)/3600)

Kinetic energy in propellant

 

((0,5*C13*($B$4^2))/1000000)+B23

((0,5*D13*($B$4^2))/1000000)+C23

((0,5*E13*($B$4^2))/1000000)+D23

Ratio gained/used

 

C19/(Somme($B21:C21))

D19/(Somme($B21:D21))

E19/(Somme($B21:E21))

Air drag

0

(B14^2)*($B$3)*(C28)*0,4

(C14^2)*($B$3)*(D28)*0,4

(D14^2)*($B$3)*(E28)*0,4

Altitude above ground

0

SI(C8 "V";(C14*(C9-B9))+B26;SI((C14*(C9-B9))> (B14*(B9-A9));(C14*(C9-B9)*pente)+B26;B26-(C14*(C9-B9)*pente)))

SI(D8 "V";(D14*(D9-C9))+C26;SI((D14*(D9-C9))> (C14*(C9-B9));(D14*(D9-C9)*pente)+C26;C26-(D14*(D9-C9)*pente)))

SI(E8 "V";(E14*(E9-D9))+D26;SI((E14*(E9-D9))> (D14*(D9-C9));(E14*(E9-D9)*pente)+D26;D26-(E14*(E9-D9)*pente)))

Max alt possible ballisticly

 

SI(C13>0;(C14^2)/(2*C29)+C26;B27)

SI(D13>0;(D14^2)/(2*D29)+D26;C27)

SI(E13>0;(E14^2)/(2*E29)+E26;D27)

Atmosphéric Density

1

B28*EXP(-B26/25000)

C28*EXP(-C26/25000)

D28*EXP(-D26/25000)

G at altitude

 

9,82*((($L$4*1000)^2)/((B26+($L$4*1000))^2))

9,82*((($L$4*1000)^2)/((C26+($L$4*1000))^2))

9,82*((($L$4*1000)^2)/((D26+($L$4*1000))^2))

Altitude from Earth center

6500

C26+$L$4

D26+$L$4

E26+$L$4

Horizontal distance from start point

 

SI(C8 "H";pente*(C26-B26);0)+B31

SI(D8 "H";pente*(D26-C26);0)+C31

SI(E8 "H";pente*(E26-D26);0)+D31

Horizontal distance in hour

 

#VALEUR!

#VALEUR!

#VALEUR!



Dernière mise à jour le 30/12/99
Par Le Rouzic Jean Pierre
Email: jean-pierre.lerouzic@wanadoo.fr