I must be one of the few people who still uses the Fehskens-Malewicki equations. I have used them as refinements, in each interval, to digital calculations, and as A simple way to do drag coefficient and altitude backtracking in a spreadsheet. I discover that many individuals believe these equations to be much less accurate than they actually are for standard model rockets. Here are the recognized sources of inaccuracy, and the reasons why they don't matter for small rockets:
1) They do not depreciate air density with altitude. For rockets with altitudes less than about 1200 feet, the difference is trivial. Most home-brew (**Not**, for example, WRASP) digital routines out there have more inaccuracy from digital error than is in this exact integral, which doesn't depreciate air density.2) They use flat thrust curves. In fact, if boost times and altitudes are small, the difference in total altitude and velocity are trivial. Indeed, for very short burning times, the differences are absent. (G. Harry Stine made a big deal of thrust curve shape in his _Handbook_. His calculations, there, are wrong.) In practice, model rocket thrust curves that deviate most from flat lines are also the shortest in duration. Problem? What problem?
3) The boost phase equations use an average mass, equal to the dead mass plus half the propellant mass. This approximation, while technically incorrect, makes virtually no difference when propellant fractions are small, as they are in model rockets. In fact, the differences are trivial for mass ratios (LaunchMass/BurnoutMass) up to about 2. No standard model rocket has a mass ratio anywhere near that.On the plus side, the coast phase integral is exact, given the constant air density assumption. Elementary digital programs frequently make the same assumption and their computations are not exact.
I was happy with all of the above until I started playing with water rockets. (Go ahead and laugh, Jerry :-). That's when I ran afoul of item 3). Aside from this item, water rockets are splendid candidates for the F/M equations. They have very short thrust times and cutoff altitudes, and they don't go very high. They do, however, have extremely high propellant mass fractions.
Consider this hypothetical water rocket.
Bottle Size One-Liter Fluid Fraction 36.8% Gage Pressure 160 PSI Launch Mass 519 grams (Nose weighted) Diameter 8.25 CM Cd .65
Theory yields Impulse 15.86411 newton-seconds Water Mass 367.917 grams Operation Time .024468 seconds
Trajectory Computations (Assumptions: Flat thrust curve and constant air density)
Digital Old Fehskens- Malewicki YTotal 190.50 ft 169.61 ft YCutoff 2.56 ft 1.88 ft VCutoff 172.47 ft/sec 154.19 ft/sec
Yuch! I decided to get a more refined estimate of average mass From the rocket equation, which describes cutoff velocity In the absence of air or gravity.
CutoffVelocity = Ve*Ln(MassRatio)
Where Ln is the natural logarithm function. Equivalent mass come from
AverageAcceleration *BurningTime = Ve*Ln(MassRatio)
(Thrust / EquivalentMass)*BurningTime = Ve*Ln(MasRatio)
(Thrust / EquivalentMass)*BurningTime = Thrust/(dm/dt)*Ln(MasRatio)
We are using a constant thrust assumption, so a constant mass flow rate Is perfectly reasonable. Thus
dm/dt = PropellantMass/BurnTime
Substituting and rearranging:
*** EquivalentMass = PropellanMass/Ln(MassRatio) ***This seems like an unlikely quantity, but it works amazingly well. For Low mass ratios, the equivalent is almost exactly the same as the old average mass, as one would expect. When mass ratios get spacey (say, around 20) the equivalent mass levels off just above .4 times the average mass. Here are the results for the same water rocket using equivalent mass in the Fehskens/Malewicki equations.
Digital New FM Old FM YTot 190.50 ft 191.34 169.61 YCutoff 2.56 ft 2.12 1.88 VCutoff 172.47 ft/sec 173.26 154.19
Much better. Given that my digital routine was written on the spur of the moment, I'm not sure which is more accurate. For regular model rockets, the improvement is absolutely immaterial. For High mass ratios, the new equivalent mass is much more accurate.
Regards,
-Larry (Apologies, Len!) Curcio.