Rocket Motor Sizing – Example 5

A HiPer class EX rocket is being designed to achieve an apogee of 4100 metres. Mass ratio of 1.41 is targeted with careful weight control, as this rocket is expected to achieve supersonic velocity. The applicable design parameters are:

· Body diameter 155mm

· Dead mass of rocket 17.7 kg

· Mass ratio target 1.41

· Average drag coefficient 0.45

· Average motor thrust 3000 N.

· Propellant Isp = 210 sec.

Figure 1 – Example 5 rocket

Problem Statement:

1. Determine total impulse of motor to achieve target apogee

2. Determine propellant mass required

3. Calculate Drag Influence Number (N)

4. Calculate maximum theoretical velocity

5. Verify apogee prediction using RASAero sim tool

Solution:

(1) The total impulse (I ) needed to achieve an apogee goal (Z_ap) is given by:

Z_ap = 4100 m.

g = 9.808 m/sec²

F = 3000 N.

a = 1.050

b = 0.00172

m_o = 1.41 ´ 17.7 = 25 kg.

The average mass is calculated from eqn.7:

giving:

This represents a 36% N-Class motor.

(2) The propellant specific impulse is 210 seconds. The mass of propellant, therefore, required to deliver a total impulse of 13921 N-sec. is:
m_p = 13921/(210´ 9.808) = 6.76 kg.

(3) The Drag Influence Number is given by:

The burnout velocity is given by:

and the burnout altitude is given by:

Thrust time (t) is given by the total impulse divided by the average thrust:

t = 13921/3000 = 4.64 sec.

Plugging in the numbers:

= 605 m/sec.

This is well beyond the nominal range of validity (5-1000).

(4) The maximum theoretical velocity of this rocket is given by the Tsiolkovsky equation:

(5) RASAero app was used to simulate the flight of the rocket. A rocket motor file was created to produce a total impulse of 13924 N-sec. (1100 N. thrust over duration of 4.64 seconds). The simulation tool was run and the following results were obtained:

Max. altitude = 4168 metres (13,675 ft.) which is within 1% of the apogee goal. This is despite the Drag Inflence Number being well beyond the nominal range of validity.
Maximum velocity (at burnout) is Mach 1.32

References:

1. Rocket Motor Sizing webpage

2. Ex5.CDX1

3. rasp_ex.eng