Rocket Regression Rate and Sizing PDR
In order for our four stage rocket to reach lower earth orbit, calculations needed to be done regarding how much solid propellant was needed, and the velocity needed to reach the proper altitude. The propellant diameter, thickness, and length was found using the equations shown in Appendix A. To begin these calculations an exit velocity had to be found, along with a total velocity using Tsiolkovsky’s basic rocket equation:
where mo is the initial total mass of the rocket
m1 is the final total mass of the rocket
ve is the rocket’s exhaust velocity
∆v is the rocket’s change in velocity
This essentially tells us the maximum rocket velocity change that can be achieved by expelling a known amount of mass (mo-m1) at a known velocity.
The mass of the propellant was the first dimension of the propellant we found. By using the specific heat and gravity we found the exit velocity of stage four of our rocket to be 2,300 m/s. This velocity times each of our four stages gives a total velocity that needs to be achieved of 7,600 m/s, which is adjusted for gravitational effects and other forces that will pull our rocket out of it trajectory, resulting in a final total burnout velocity of 9,200 m/s. This number used in conjunction with our four stages resulted in the rocket’s change in velocity to be 2,300 m/s.
After finding the total velocity, the masses were calculated using three different parts of the rocket to make up the total mass. These three parts included the payload mass of 1 kg satellite, a structural mass of 0.75 kg, and an electrical component mass of 0.3 kg. When theses masses were added up they gave a total mass of 2.05 kg. By using the above rocket equation, plugging in the exit velocity, velocity change, and the total mass, an initial mass of the rocket was found to be 5.572 kg. By subtracting the initial mass of the rocket from the total mass, a propellant mass was found to be 3.522 kg.
The next step was to use regression rate analysis that will later be used to find the diameter of the solid propellant. The regression rate equation (as shown below) that was found from extensive research in a paper by Chiaverini, was used to find an estimate of the regression rate for the last stage of our four stage rocket:
r is the regression rate
L = length
Go is the oxidizer mass flux (kg/m2-s)
x is the axial location (m)
k is the gas absorption coefficient (m-MPa)-1
p is the pressure (MPa)
h is the port height between fuel slabs (m)
n,m,k,C1,C2 are all constants whose value is given above
This equation was put in a spreadsheet and analyzed, resulting in an output that was not used due to the errors found in using this equation. Further research was done, and a paper by the same man, Chiaverini, showed another regression rate equation:
Ea is activation energy and is given as 20.557 kJ/mol
A is the Arrhenius pre-exponential constant, given as 11.04 mm/s
Ts is the surface temperature of the fuel grain
is the universal gas constant which is equal to 8314.3 J/(mol-K)
r is the solid propellant regression rate [mm/s]
This equation was also put into a spreadsheet and resulting in a regression rate of 0.9315 mm/s. This number was a closer match to the regression rates that had been found through research. The exact regression rate is not able to be calculated until actual testing of the solid propellants has been done. Due to this non-exact regression rate analysis, for ease of calculation purposes, the regression rate we decided to use is 1 mm/s. For the purpose of our rocket this adjustment should put us in a range of having slightly extra fuel should our theoretical regression rate be slightly off due to human calculation error.
After the regression rate was found, the next step was finding the mass flow rate. First the assumption of L/D = 10 and a 0.03 m inner diameter was made, which is accurate with other theoretic calculations we ran across. An outer diameter was calculated using the regression rate, a burn time of one second, and the inner diameter. The outer diameter after one second was found to be 0.032 m, which was used in conjunction with the inner diameter and length to find the volume of HTPB. The volume of HTPB was then multiplied by the density of the HTPB to give the mass flow rate after one second for our solid propellant of 0.0272 kg/s.
Using an oxidizer-to-fuel ratio of 8:1, the mass for HTPB was calculated. By taking 1/9 (due to the ratio) of the fuel mass that was calculated earlier, the HTPB mass came out to be 0.3913 kg. By placing the mass of HTPB over the mass flow rate after one second of HTPB, the burn time was calculated to be 14.4 seconds.
Finally, a back calculation was done to find an outer diameter for 14.4 seconds. This was done by adding to the inner diameter the burn time times the regression rate times twice that for each side of the solid propellant. The resulting outer diameter came out to be 0.0635 m. We increased this outer diameter so that we could have extra fuel to insulate the combustion chamber walls due to the heat that the combustion chamber will produce. Also extra fuel was used to allow standard pipe sizes for the combustion chamber to be used in our design.
An oxidizer flow rate was calculated as shown in the Appendix using the remaining 8/9 ratio of fuel over the burn time, resulting in 0.2174 kg/s of oxidizer.
Results of solid propellant calculations which will be used in our test design next quarter:
Inner Diameter = 0.03 m
Outer Diameter = 0.0635 m
Length = 0.3 m
Weight = 3.522 kg