Richard Nakka’s Experimental Rocketry Web Site
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Introduction
to Rocket Design
8. Rocket Motor Sizing
Introduction
Every rocketry enthusiast is familiar with flight simulator
applications that are used to predict how
high their rocket will fly. RasAero, SOAR, EzAlt, OpenRocket and RockSim
are some of the more popular apps. The basic algorithm that these apps utilize
to predict peak altitude is essentially the same. Input your rocket details
such as dimensions, mass, drag characteristics, motor performance parameters,
etc. and the results are instantly and conveniently outputted.
What if, instead, we have an apogee
goal, and wish to know how big a
motor is needed to power our rocket in order to achieve that goal? Or
perhaps a burnout velocity goal, such as achieving Mach one. Of course, we can
still use the simulation apps, and apply different motor sizes to our rocket to
see what is needed…so called trial and error
method. It's a basic process whereby you attempt solutions, observe
the results, and adjust your approach based on feedback, repeating the cycle
until a satisfactory solution is reached. Also known as “brute force” method or
“guess and check”. Although this method may not be particularly elegant, it is
an acceptable engineering approach. However, a more rational approach is clearly
preferred whereby motor sizing is the result
of an appropriate analytical method.
The Total Impulse
that a rocket motor generates is the key parameter that determines how high a
rocket will fly. Total impulse is the integral of thrust over the burn time of
a rocket motor, essentially measuring the total change in momentum imparted to
the rocket. This approach to solving our problem has its challenges, as there
is not a unique solution. Specifically, an altitude goal can be achieved by an
infinite number of different rocket motor configurations, each delivering the same Total Impulse. This concept is
illustrated in Figure 1. Both idealized motors have a Total Impulse of 1000 N-sec.,
however, have significantly different burn profiles.
Figure 1: Two
idealized rocket motors delivering same Total Impulse
In the absence of
atmospheric drag, the apogee achieved by both of these motors would be very
similar (although not identical). For real rockets, which have to cope with the
effects of aerodynamic drag, there will be some difference in apogee. For most EX
rockets, the difference will be small.
Zero-drag Method
With the zero drag method of determining Total
Impulse to achieve an altitude or velocity goal, atmospheric drag force is assumed to be zero. The
article Simplified Method for Estimating the Flight Performance of a
Hobby Rocket describes the theory, which is based on the principle of conservation
of energy whereby work done by the rocket is equated to the kinetic energy
of the rocket, at burnout, or equated to the potential energy of the rocket at
apogee:
Work = Force ´ distance, or W = F d
Kinetic
Energy = ½ mass ´ velocity squared, or KE = ½ mV 2
Potential
Energy = mass ´ gravitational acceleration ´ height,
or PE = m g z
The work performed by the
rocket is the average thrust of the motor multiplied by the distance it travels
during motor burn.
Noting that Total Impulse (I ) of the rocket motor is given by:
I = F t
where F = average thrust (N. or lbf)
t = duration of thrust (sec.)
Utilizing this expression
for Total Impulse, the zero drag equations given in the referenced document may
be rearranged to solve for Total Impulse required to achieve a specified
burnout altitude, burnout velocity and apogee.
(1) Total impulse (IZD) to achieve a zero-drag burnout altitude (Z1):
(2) Total impulse to achieve a zero-drag
burnout velocity (V1):
(3) Total impulse to achieve a zero-drag
apogee (Z2):
where:
g = acceleration due to gravity (m/sec2 or ft/sec2)
m = rocket average mass, m = ½ mp + md
mp = propellant mass (kg. or slugs)
md = rocket dead (empty) mass (kg. or slugs)
The resulting units of
Impulse are Newton-seconds (N-sec.) or Pound-force seconds (lbf-sec.).
Drag Reduction Method
To account for losses in
flight performance due to aerodynamic drag (work is performed to overcome drag
force), drag reduction factors are applied to these ideal (zero drag) values,
as explained in the referenced document. The first step is to calculate a Drag Influence Number (N):
where
Cd = average drag coefficient
of rocket
D = rocket reference diameter associated with Cd (centimetres or inches)
K = units factor. For metric units, K = 1000, for U.S. units, K = 24353
Using N, the Drag Reduction Factors,
which reduce the burnout altitude, maximum velocity and apogee, (fzbo, fv & fz) are obtained from the
following chart which is presented in Figure 2:
Figure 2: Drag Reduction Factor chart
Apogee, taking atmospheric
drag into consideration, is given by:
Zap = fz Z2
where the zero-drag apogee
is obtained from (3) above:
Units of Z are metres or feet. The Total Impulse (I ) to achieve our apogee goal (Zap), taking atmospheric drag
into account, is given by:
The Drag Reduction Factor
for apogee can be conveniently curve-fitted using the following relation
involving the Drag Influence Number (N):
The coefficients a and b have the following numerical values:
a = 1.050
b = 0.00172
For convenience, the Drag
Influence Number can be represented by:
N = c V12
where:
The range of validity for N is nominally between 5 and 1000. Beyond this range, the accuracy of
the results is uncertain. However, example 5 suggests this method remains
accurate for values as high as N » 2000.
The zero-drag burnout velocity
(V1) is given by:
Using the above
expressions, the Total Impulse (I ) required to boost a
given rocket to an apogee goal (Zap), taking drag into
consideration, can be obtained:
(4) Total impulse (I ) to achieve a target apogee (Zap)
Note that, in order to
simplify the method, zero-drag velocity is employed. This is justified by the
velocity factor (fv) being reasonably close to
a value of one over the N range of interest, as is
seen in Figure 2, combined with our goal of obtaining an estimate of the Total Impulse required to achieve our goal. The examples
provided at the end of this webpage demonstrate that the method is surprisingly
accurate despite this and other simplifying assumptions.
To estimate the impulse
required to achieve a burnout altitude goal or a burnout velocity goal, use
equations (1) and (2). Although these are for the zero-drag condition, the
results will be reasonably accurate. This is the consequence of the rocket motor
thrust being much greater than opposing drag force (for most EX rockets) during
the boost phase.
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Propellant
Mass Fraction and Mass Ratio
The propellant mass fraction of a rocket, denoted by
the Greek symbol zeta 
and the mass ratio of a rocket, are two important
parameters of rocket design, especially with respect to space boosters and
high-performance sounding rockets. Propellant mass fraction is the ratio of the
mass of propellant to the total
mass of a rocket at launch.
This parameter is important
as it directly affects the rocket's ability to achieve its mission, including
the velocity needed to overcome gravitational forces and reach space. It
generally indicates a more efficient design with regard to mass allocation and
ability to achieve a velocity or altitude goal. The average mass of the rocket
is related to the initial mass and the propellant mass fraction as such:
(5)
The dead mass of a rocket
is related to the initial mass and the propellant mass fraction as such:
(6)
A related parameter is the
rocket mass ratio (MR). The definition of mass ratio is:
The highest theoretical
velocity that a rocket can achieve is given by the Tsiolkovsky rocket equation
which involves the mass
ratio:
or put in terms of mass ratio:
Two things to note. In this
equation, units of Isp must be N-sec./kg. or if
US units are employed, Isp (seconds) must be multiplied by g. Secondly, Sutton’s definition of mass ratio is the reciprocal, or MR = Md /Mo. We’ll use the above definition of mass ratio,
rather than that of Sutton, as it makes more intuitive sense whereby a larger
mass ratio is more desirable. Also note that Dv can be considered to be vmax
since the initial velocity
for single-stage rockets is zero. Figure 3 illustrates the nature of the Tsiolkovsky
equation as it relates to maximum velocity that a rocket can ideally achieve
based on mass ratio and specific impulse.
Figure 3: Mass ratio versus velocity plot
Figure 4 presents the ideal
velocity attainable for two specific impulse values of interest to the EX
rocketeer.
Figure 4: Mass ratio versus velocity for two Isp
values
Tsiolkovsky equation is of special
importance to EX rocketeers as it identifies the mass ratio needed to achieve a
certain velocity goal, for example, Mach 1
(supersonic) or Mach 5 (hypersonic). From Figure 4, it can be seen that to
achieve supersonic velocity (»340 m/s.), the mass ratio
would need to be (ideally) 1.32 for Isp=125 sec. and 1.17 for Isp=220 sec. To
achieve hypersonic velocity, the mass ratios would need to be 4.0 and 2.2,
respectively. A mass ratio in the order of 3 requires exceptionally lightweight
design of the rocket structure (including motor). As such, it is clearly not
feasible for a single-stage EX rocket powered by sugar
propellant (Isp = 125) to achieve hypersonic flight. Supersonic flight,
however, can be achieved with careful attention to minimizing structural mass.
Although important for a
velocity goal, with regard to apogee
goal, mass ratio is of lesser
significance, as mass ratio of a particular rocket is dependant upon the choice
of propellant and its specfic impulse.
The average mass of the
rocket is related to the initial mass and the mass ratio as such:
(7)
The dead mass of a rocket
is related to the initial mass and the mass ratio as such:
(8)
Table 1 presents the key
mass properties for various single-stage rockets, and the related flight
performance.
Table 1: Mass properties and related flight
performance for various rockets
Notes applicable to Table
1:
1) Apogees and velocities shown
are predictions from OpenRocket app for the model and
Mid/Hi Power rockets. Apogees for EX rockets are actual, measured by the
rocket’s flight computer. Apogees for the Sounding rockets are taken from
published data.
2) Vmax theo. is that calculated
using the Tsiolkovsky rocket equation. Isp values assumed for the calculations
are as follows:
BP 70 sec.
KNDX, KNSB 125 sec.
KNPSB 160 sec.
ANCP, APCP 200 sec.
Things of particular
interest to note from the data given in Table 1.
· Propellant mass fraction
values are similar for typical model rockets and Mid Power rockets, generally
in the range of 0.1 to 0.2. Hi Power rockets, such as the Blackhawk, can be
configured to achieve a respectfully large propellant mass fraction. EX rockets
tend to have lower propellant mass fractions, generally in the range of 0.05 to
0.15. High performance sounding rockets, which are designed to reach the upper
limits of the atmosphere, by necessity have high propellant mass fractions, in
the general range of 0.6 to 0.7. Such sounding rockets feature optimized lightweight
designs of both motor and airframe structure.
· EX rockets (and
interestingly the Alpha model rocket), which have a rather high ballistic
coefficient, tend to achieve maximum velocity in the 90-95% range of ideal
maximum velocity. Supersonic rockets suffer from relatively large drag losses
and do not achieve as high a velocity percentage.
To summarize, an estimate
of the Total Impulse required to boost a rocket to a target apogee, taking aerodynamic
drag into consideration, can be obtained from the expression given in (4). It
is necessary to first decide upon a best
estimate the following design
parameters:
1) The dead (or empty) mass of
the rocket, including empty motor
2) The propellant mass
fraction or mass ratio of the vehicle.
3) Average drag coefficient
and rocket diameter
4) Average motor thrust. As
mentioned earlier, the apogee that a rocket will achieve has a low sensitivity to which combination of thrust and burn time is
chosen. A rough estimate of thrust will therefore suffice. As a guideline, a
value of thrust to achieve a liftoff acceleration of 10 G’s is considered as a
minimum.
Isp Based Method
An alternative to specifying propellant mass fraction or mass ratio of
the rocket, it may be more convenient to specify the propellant Specific Impulse (Isp). In terms of specific impulse, the
average rocket mass (m) is given by:
(9)
Squaring both sides of equation
(4) and utilizing equation (9) for average mass allows for the Total Impulse to
be calculated using the following equation:
(10)
There is no closed-form
solution to equation (10) with m expressed in terms of Isp, as m is a function of I. The equation must be solved using an iteration method. Basically, plug in values
for I until the resulting expression on the left-hand side is (approximately)
equal to zero. This can be accomplished using an app such as MATLAB, MathCad,
python or Excel, or writing a simple code using a computer language such as
Basic or Fortran. Example 6 illustrates this method of calculating Total
Impulse using the Goal Seek function in Excel.
Examples
Example 1 – A target apogee of 2000 metres is chosen
for a 75mm diameter rocket with dead mass of 2.65 kg. Rocket mass ratio is aimed
at 1.25. Estimate what impulse/class of motor is required, assuming our motor
generates an average thrust of 425 N. Propellant is KNDX.
Example 2 – A target apogee of 2500 metres is chosen for a 150mm diameter rocket
with a dead mass of 10 kg. Rocket mass ratio is aimed at 1.2. Estimate what
impulse/class of motor is required, assuming our motor generates an average
thrust of 1100 N.
Example 3 – A target apogee of 500 metres is chosen for
a 70mm diameter rocket with a dead mass of 3.27 kg. Rocket mass ratio is
expected to be 1.07. Estimate what impulse/class of motor is required, assuming
our motor generates an average thrust of 400 N. Propellant is KNSB.
Example 4 – A target apogee of 6500 feet is chosen for
a 3 inch diameter rocket with a dead weight of 5.9 pounds and a mass ratio of
1.18. Estimate what impulse/class of motor is required, assuming our motor
generates an average thrust of 110 lbf.
Example 5 – A target apogee of 4100 metres is chosen
for a 155mm diameter rocket with a dead mass of 17.7 kg.
Mass ratio of 1.41 is targeted with careful weight control, as this rocket is
expected to achieve supersonic velocity. Estimate what impulse/class of
motor is required, assuming our motor generates an average thrust of 3000 N. Calculate
Drag Influence Number and maximum theoretical velocity.
Example 6 – A target apogee of 3000 metres is chosen
for a 89mm diameter rocket with a dead mass of 5 kg. The propellant is APCP
with a specific impulse of 205 seconds. Estimate what impulse/class of motor is
required, assuming our motor generates an average thrust of 600 N. Determine
propellant mass required, as well as propellant mass fraction and rocket mass
ratio.
Resources
Res.IM1 Derivation of equation 1.
Res.IM2 Derivation of equation 2.
Res.IM3 Derivation of equation 3.
Res.IM4 Derivation of equation 4.
Res.IM5 EzRocket software
Res.IM6 Impulse_(Isp)_Ex6.xlsx
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Last updated July 28,
2025
Originally posted July
9, 2025