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Use of Python to automate FlexPDE scripts to simulate heat flow & trajectory of a solid booster rocket.
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ENGPHYS 2CM4: Computational Multiphysics
McMaster University Faculty of Engineering
Instructors: Dr. Minnick
TA: Sam Bovey
Submitted on: April 14 th^ , 2021
Dabeer Abdul-Azeez | abdulazd / 400261347
Your NASA engineering team unearthed an early Space Shuttle Solid Rocket Booster prototype (SRBP), and they tested it out for a small payload delivery. The design was able to generate immense thrust, but since the team back then was still determining the appropriate amount of insulation necessary to protect the rocket, the heat generated from the ignition of the fuel managed to make its way towards the metal outer casing (or “outer wall”) and compromised its structural integrity, causing a horrible crash. Your team is planning to rebuild the rocket once more. You unfortunately do not have any extra insulation, but you are
able to modify the fuel flow rate (𝑞𝑞 �
kg s �) of the rocket. Your job is to optimize the fuel flow rate to maximize the distance the rocket travels while ensuring the outer metal casing is not compromised.
Figure 1: Simplified diagram of a solid-fuel rocket (reprinted from [1]). 1. Solid propellant packed into the rocket, with a cylindrical hole in the middle. Insulation material (usually a type of rubber) would be found between this layer and the outer metal casing. 2. An igniter which combusts the surface of the propellant. 3. The cylindrical hole in the propellant which acts as a combustion chamber. 4. The hot exhaust is choked at the throat, dictating the thrust produced. 5. Exhaust gases.
Figure 2: Space Shuttle Solid Rocket Boosters (SRBs, adapted from [2]).
The two major categories of rockets include liquid-fuel rockets and solid-fuel rockets. Solid-fuel rockets (such as the SRBs in Figure 2 or the simplified version in Figure 1) generate thrust by igniting solid propellant and exhausting the combustion products down a column spanning the length of the rocket. The solid rocket fuel is used up from the inside to the outside until the fuel supply is depleted. Solid-fuel rockets are known for being capable of generating large amounts of thrust at a low cost. Just two of them provided 80% of the thrust required for the Space Shuttle to achieve liftoff [3]. However, they are generally less efficient and controllable than liquid fuel rockets, which is why the Space Shuttle uses a liquid fuel rocket for its main first stage engine, with two solid rocket boosters to accompany it [4]. Combustion temperatures of solid booster rockets can reach temperatures of up to 3300°C [3], which is well above the melting temperatures of many common alloys used to build the outer casings of the rockets. As such, a layer of insulation is often found between the propellant and the outer casing to prevent a rocket meltdown [5]. This layer is usually made up of some rubber compound [5]. Extra insulation adds unnecessary weight, so minimizing its usage in a solid rocket motor design is key [5].
Simple projectile motion involves an object of a fixed mass 𝑚𝑚 (kg) travelling through the air under only the force of gravity 𝐅𝐅𝐤𝐤 = −𝑚𝑚𝑚𝑚𝒚𝒚� where 𝑚𝑚 = 9.81 ms 2 is the gravitational acceleration near the surface of the
Earth and 𝒚𝒚� is the unit vector pointing directly upwards from the ground. According to Newton’s second law, the net acceleration of an object (𝐚𝐚 ) being acted upon only by gravity is very simple to calculate (see equation set (1)).
𝐅𝐅𝐤𝐤 = 𝐅𝐅𝐧𝐧𝐧𝐧𝐧𝐧 (−𝑚𝑚𝑚𝑚𝒚𝒚�) = 𝑚𝑚𝐚𝐚
𝐚𝐚 =
Where 𝐚𝐚 �
m s 2 �^ is the acceleration vector of an object,^ 𝑚𝑚^ =^9.^81
m s 2 is the gravitational acceleration near the surface of the Earth, and 𝒚𝒚� is the unit vector pointing directly upwards from the ground.
The position of the object can be calculated via the differential equations of motion seen in equation set (2).
𝑑𝑑𝐚𝐚 𝑑𝑑𝑑𝑑
Where 𝐚𝐚 �sm 2 � is the acceleration vector of an object, 𝐯𝐯 �ms � is its velocity vector, 𝐫𝐫 (m) is its position vector, and 𝑑𝑑 (s) is the time elapsed.
Calculations quickly become complicated when considering a rocket’s trajectory because a rocket generates its own thrust force (𝐅𝐅𝐧𝐧) in the direction opposite fuel ejection and its mass varies over time in accordance with the amount of fuel it is ejecting (see equation sets (3) and (4)).
𝑡𝑡
t 0 = 𝑚𝑚𝑤𝑤 − 𝑞𝑞(𝑑𝑑 − 𝑑𝑑 0 ) (constant fuel flow rate)
Where 𝑚𝑚 (kg) is the mass of the rocket over time, 𝑚𝑚𝑤𝑤 (kg) is the initial mass of the rocket, 𝑞𝑞 �kg s � is the fuel flow rate, 𝑑𝑑 (s) is the time passed since launch, and 𝑑𝑑 0 is the initial launch time.
Where 𝐅𝐅𝐧𝐧 (N) is the thrust force generated by the rocket at a point in time, 𝑞𝑞 �kgs � is the instantaneous fuel flow rate and 𝐯𝐯𝐟𝐟𝐟𝐟𝐧𝐧𝐟𝐟 represents the velocity vector of the ejected fuel
Another key component to consider with rocket trajectories is the drag force placed on the rocket by the surrounding air (see equation sets (5) and (6)).
Where 𝐯𝐯 �m s � is the velocity vector of the object travelling through the fluid, 𝐯𝐯𝐟𝐟𝐟𝐟𝐟𝐟𝐟𝐟𝐟𝐟 �ms � is the
velocity vector of the fluid, and 𝐯𝐯𝐫𝐫𝐧𝐧𝐟𝐟 �ms � is the relative velocity vector of the object travelling through the fluid.
Where 𝐅𝐅𝐟𝐟 is the drag force acting on the object, 𝜌𝜌 � (^) mkg 3 � is the density of the surrounding fluid, 𝐶𝐶𝑑𝑑 (dimensionless) is the drag coefficient of the object travelling through the fluid, 𝐴𝐴𝑓𝑓 is the object’s frontal area (i.e. the area of its projection seen from the direction of fluid flow), and 𝐯𝐯𝐫𝐫𝐧𝐧𝐟𝐟 is the relative velocity vector of the object travelling through the fluid.
Summing the effects of all these forces together, Newton’s second law can be rewritten to determine an equation for the acceleration of the rocket over time (see Equation set (7)).
𝐅𝐅𝑤𝑤𝑓𝑓𝑡𝑡 = 𝑚𝑚𝐚𝐚 �𝐅𝐅𝐤𝐤 + 𝐅𝐅𝐧𝐧 + 𝐅𝐅𝐟𝐟� = 𝑚𝑚𝐚𝐚
(−𝑚𝑚𝑚𝑚𝒚𝒚�) + (−𝑞𝑞𝐯𝐯𝐟𝐟𝐟𝐟𝐧𝐧𝐟𝐟) + (−
In mathematics and physics, the [time dependent] heat equation is a partial differential equation that models how heat flows through a material (see equation set (8)).
Where 𝑇𝑇 (°C or K) is the temperature of the material at a certain location, 𝑑𝑑 (s) is the time elapsed, 𝜌𝜌 � (^) mkg 3 � is the density of the material, 𝑐𝑐𝑝𝑝 � (^) kg⋅KJ � is the specific heat capacity, 𝑞𝑞̇𝑉𝑉 � (^) mW 3 � is
the volumetric heat generation, and 𝑘𝑘 � (^) mW⋅K� is the thermal conductivity.
In short, the heat equation says that the change in temperature over time depends on the change in temperature over space (∇𝑇𝑇), any heat generation within the material (𝑞𝑞̇𝑉𝑉), and some material parameters (𝜌𝜌, 𝑐𝑐𝑝𝑝, 𝑘𝑘). There exists a steady state heat equation as well (see equation set (9)) which models the heat
distribution in a material after an infinitely long duration.
from equation set (2). The initial position and velocity of the rocket were implemented as initial conditions on the primary dependent variables 𝒓𝒓 and 𝒗𝒗.
While realistically any rocket would have an initial velocity of zero, the FEM model does not account for guided control systems on the rocket which would help prevent it from tilting over and hitting the ground. Also, when an exactly zero initial velocity is entered into the model, no calculations are possible because the 𝐯𝐯� vector cannot be calculated (since it depends on the inverse of the magnitude of velocity, 𝑣𝑣). Inputting a small initial velocity surpasses the zero-division error, but it eventually leads to the rocket tilting over and crashing if the initial launch angle is not perfectly 90° from the ground. Thus a reasonably large initial velocity is required to acquire any meaningful results from the simulation.
To elaborate further, when a small initial velocity is used, the 𝑦𝑦-component of the rocket’s thrust is mitigated by gravity, but the 𝑥𝑥-component is not. As such, if the rocket is set off at, for example, a 45° angle, then even if its 𝑦𝑦-component of thrust is enough to overcome gravity, the 𝑥𝑥-component of thrust will continue to tilt the rocket without restriction until too much thrust has been diverted away from the ground to counteract gravity. At this point, the rocket starts to tilt further towards the ground, and the rocket’s thrust begins pointing upwards and pushing the rocket downwards, increasing its acceleration towards an unfortunate crash landing. By contrast, a launch angle perpendicular to the ground means that the rocket will not have any 𝑥𝑥-component of thrust available to tilt it over, so assuming the thrust force is enough to overcome gravity, the rocket will continue to accelerate upwards until its fuel runs out. This reasoning was verified via a comparison of the rocket trajectories for an initial velocity of 1 ms with both a 45° launch
angle and a 90° launch angle (see Figure 4 and Figure 5). The 45° launch angle resulted in a quick crash after about 9 seconds, while the 90° launch angle continued accelerating upwards well past 100 seconds.
Launch angle: 45° Launch angle: 90°
Figure 4: Rocket y displacement over time for an initial velocity of 1 𝑚𝑚/𝑠𝑠 at a 45° launch angle from a starting point of ( 1000 , 1000 ) m
Figure 5: Rocket y displacement over time for an initial velocity of 1 𝑚𝑚/𝑠𝑠 at a 90° launch angle from a starting point of ( 1000 , 1000 ) m.
Since the rocket will not be modelled with an automatic guiding system but it still needs to launch at a 45° angle, I included a much higher initial velocity to simulate how the rocket would be travelling through the air at some time point after liftoff. This avoids the complications of low initial velocities for non-vertical launch angles, as discussed before. For my specific simulation, the rocket was given an initial velocity of 600 ms. Literature is scare regarding the velocities that the SRBs would have travelled at alone considering
they were attached to the main Space Shuttle during launch, so I assumed the SRBP’s velocity near the beginning of their trajectory would be faster than a Space Shuttle at the same point in time after launch. Table data for the ascent of the STS-121 rocket recorded a velocity of 433 ms after 60 seconds at an altitude
of 11.6 km [21]. Based on this, the SRBP was given an initial velocity of 500 ms at a starting location of
(𝑥𝑥, 𝑦𝑦) = (11.6,2.5) km. The horizontal distance from the origin was assumed not to be as large as the vertical distance to mimic how rockets travel primarily in the vertical direction at launch and then slowly tilt horizontally over time in order to travel large distances.
FlexPDE requires a mesh to be created for any simulation since it is an FEM solver, but since this kinematics simulation was not modelled as spatially dependent (and thus is more of an ordinary differential equation simulation), a small, low resolution mesh was created to help decrease the simulation time. The simulation was set to halt when the y-component of the rocket’s position dropped below zero (i.e. indicating a landing).
The FlexPDE model was verified by conducting some tests for known cases. Firstly, with no fuel flow rate, the fuel in the rocket cannot produce its own thrust, and with a drag coefficient of 𝐶𝐶𝑑𝑑 = 0, the rocket will not experience any drag force. In this case, the SRBP must follow a parabolic trajectory in accordance with the concepts of simple projectile motion. This was verified as seen in Figure 6.
Figure 6: Rocket trajectory for a fuel flow rate of 0 𝑘𝑘𝑘𝑘𝑠𝑠 and a drag coefficient of 𝐶𝐶𝑑𝑑 = 0_._
The rocket followed a parabolic trajectory, as expected, and the launch angle of 45° seems to be accurately modelled, taking into consideration the scales of the axes. The rocket landed almost 210 km from its starting location and travels almost 50 km above the ground. This seems reasonable on the scale of rocket transportation, considering the rocket is so massive and had a very high initial speed.
Next, by adding some drag force, the rocket should travel a shorter horizontal distance because the drag force will oppose its relative motion through the fluid (i.e. the air). This was verified by reinstituting a non- zero drag coefficient and plotting the trajectory (see Figure 7).
Figure 9: Rocket acceleration ( 𝑐𝑐 𝑠𝑠 2 , x component in green, y component in yellow) and mass ( 𝑘𝑘𝑚𝑚 , blue) over time, with thrust included in the model at a fuel flow rate of 𝑞𝑞 = 3500 𝑘𝑘𝑘𝑘𝑠𝑠. Acceleration is scaled by a factor of 104 to view it appropriately on the graph.
With the help of the thrust generated by the fuel, the rocket travelled much further, both horizontally and vertically. The trajectory again deviated from a parabolic trajectory since now both drag and thrust forces modified the acceleration of the rocket over time. The mass of the rocket decreased linearly from the total mass 𝑚𝑚𝑑𝑑𝑑𝑑𝑑𝑑 + 𝑚𝑚𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 = 590 700 kg towards the dry mass of the rocket 𝑚𝑚𝑑𝑑𝑑𝑑𝑑𝑑 = 90 700 kg by around 150 s.
This is in line with the assumption of a constant fuel rate. Past this, the rocket’s mass remained constant at the dry mass, indicating that there was no more fuel left to burn.
The acceleration of the rocket is a bit more complex to analyze. The acceleration in the x direction (𝑎𝑎𝑥𝑥) started off positive, increased until around 75 s, and then started dropping. This indicates the rocket’s thrust accelerating it in the 𝑥𝑥 direction and the rocket tilting forwards to the point where drag started pushing it back. At around 150 s (when the fuel ran out), 𝑎𝑎𝑥𝑥 immediately became negative as there was no longer a thrust force to oppose drag. The drag forces horizontally died out as the rocket began tilting and accelerating towards the ground. The acceleration in the y direction (𝑎𝑎𝑑𝑑) started off negative, meaning that the rocket’s thrust was not strong enough in the vertical direction to overcome gravity. It was the initial velocity instead that maintained its ascent. Once the fuel ran out, there was no thrust to oppose gravity, so gravity took over and caused a negative spike in 𝑎𝑎𝑑𝑑. This spike then died out as the rocket accelerated towards the ground and reached terminal velocity as a result of drag forces opposing its descent.
As demonstrated through this thorough analysis, gravitational force, drag force, and thrust force were all appropriately implemented into the kinematics simulation.
The heat flow FEM model, like most FEM models, required four main components: the finite element mesh, the boundary conditions, the initial conditions, and the defining differential equation.
Figure 10: Representative 2D view of the domain over which the heat flow simulation was calculated, adapted from FlexPDE. This represents a section of the main fuselage from the inner combustion chamber to the outer atmosphere ( 𝑥𝑥 represents the thickness direction from the center of the rocket outwards). This diagram is not to scale and is for visual purposes only.
Figure 10 shows a 2D representation of the different layers from the inside to the outside of the SRBP. An analysis of the heat transfer from the center of a solid-fuel rocket towards the outer casing, assuming no material parameters or initial heat distributions vary in the width or depth direction, is a 1-dimensional analysis, so only a 1-dimensional mesh was used for the simulation. This mesh operated on the domain of 𝑥𝑥, which represents the thickness direction from the center of the rocket outwards.
The boundary conditions on this simulation are summarized in Figure 11. Firstly, the outer casing was modelled to lose heat due to convection with the ambient air (see equation (11)). This is highly relevant for this simulation considering the rocket is flying through the air at a high velocity. For this model, I did not incorporate velocity data to vary the heat loss over time and instead modelled the heat transfer at the wall
extremity with a constant convective heat transfer coefficient of 500
W m^2 ⋅K, assuming that the ambient temperature was held constant at −57°C. This ambient temperature was derived from NASA’s stratospheric temperature formula [22] at 25 km above the Earth’s surface (see equation (10)). Temperature variance with the height of the rocket travel over time was not included in the model. Secondly, at the boundary of the insulation and the outer wall, no boundary condition was necessary, and so none was applied in FlexPDE. Lastly, the combustion chamber air-fuel and fuel-insulation boundaries were maintained at the combustion temperature (𝑇𝑇𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐) according to the provided equation 𝑇𝑇𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 2500 + 2 𝑞𝑞. 3 (°C). The air-fuel boundary condition describes how the air is always at the
combustion temperature when in contact with the fuel. The fuel-insulation boundary condition was implemented in accordance with how I had to modify the initial conditions of my simulation (discussed later).
Table 1), so the FEM model thus represents a worst-case scenario. Due to this compromise, the fuel- insulation boundary was set at the combustion temperature as well, whereas with the initial plan to model the fuel burning up, there would not have been any extra boundary condition implemented here.
Finally, the defining differential equation in the simulation was set as the time dependent heat equation (see equation set (8)).
To verify that heat was not being generated arbitrarily within the model, the initial temperature distribution was set to a random fixed temperature (here, 22°C) everywhere, and all temperature boundary conditions were removed. This resulted in an average temperature (calculated using equation (12)) which remained constant at the fixed temperature throughout the simulation (see Figure 12), verifying that no heat was arbitrarily being generated within the model.
Where 𝑇𝑇 is the temperature of the material at a location 𝑥𝑥 along a material and 𝑇𝑇𝑎𝑎𝑐𝑐𝑘𝑘 is the average temperature of the material along 𝑥𝑥.
Figure 12: Average temperature history plot with the entire mesh set to 22°𝐶𝐶 initially, with all temperature boundary conditions removed.
Then the model was checked to see if the material parameters were implemented properly. If all materials are given a very high thermal conductivity, and the heat loss on the outer rocket wall is removed, then steady state equilibrium should be reached quickly, and the entire mesh should transition to the steady state temperature. To test this, the air-fuel and fuel-insulation boundary conditions were implemented, and all
the materials were given a thermal conductivity of 𝑘𝑘 = 10^6 m⋅KW. The results of the time dependent
simulation (see Figure 13) were compared with a steady state simulation using equation set (9), and both confirm that the steady state temperature (i.e. the combustion temperature 𝑇𝑇𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 2500 + 2 𝑞𝑞. 3 = 4456°C)
was reached.
Figure 13: Temperature plot over 𝑥𝑥 from a time dependent simulation (after 226.32 seconds) with 𝑘𝑘 = 1 × 10^6 𝑐𝑐𝑊𝑊 (^2) ⋅𝐾𝐾 for all materials and fuel flow rate 𝑞𝑞 = 4500 𝑘𝑘𝑘𝑘𝑠𝑠. Convective heat flow at the outer wall has been removed, while the air-fuel and fuel- insulation boundaries have been set to a combustion temperature of 𝑇𝑇𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 2500 + 2 𝑞𝑞. 3 = 4456°𝐶𝐶
Keeping the same high thermal conductivity for all materials and adding back in the convective heat flow at the outer wall should cause the temperature to drop near the outer wall because the ambient temperature is much colder than the combustion temperature. Convection will only be able to remove heat from the insulation and outer wall since they are the only materials not fixed in temperature via any boundary conditions. The fuel and air should remain at the combustion temperature due to the temperature value boundary conditions. This was verified via a simulation (see Figure 14), which showed a combustion temperature, 𝑇𝑇𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 2500 + 2 𝑞𝑞. 3 = 4456°C that is maintained from the combustion
chamber towards the insulation, but which drops once passing the insulation and nearing the outer wall. The temperature drops by about 0.1°C at the outer wall extremity. These results are identical to a steady state simulation that was run. This drop in temperature near the outer wall is very low because the outer atmosphere is not able to cool the rocket exterior down quickly enough with a convective heat transfer
coefficient of only ℎ = 500 (^) mW (^2) ⋅K. A convective heat transfer of 5 × 10^7 mW (^2) ⋅K demonstrates a much greater
capability to cool down the exterior wall of the SRBP (see Figure 15), showing that convection was included properly.
Figure 16: Flowchart for the Python script used to collect and analyze data from the FlexPDE simulations.
A Python script was designed according to the flowchart seen in Figure 16. The full Python code can be seen in Appendix: Python Script to Automate the FlexPDE Scripts. The main purpose of the script was to combine the kinematics and heat flow FlexPDE simulations together to generate a holistic analysis of the maximum safe fuel flow rate that the rocket can use. Not only did the script have to run the kinematics simulation to determine the horizontal range travelled, but it also had to run the heat flow simulation with the same fuel flow rate and determine if that fuel flow rate would cause the rocket to overheat.
The FlexPDE scripts were run using the Python subprocess module, which permits terminal commands to be run from a Python script. FlexPDE offers terminal commands to run its script files, and these were used within the Python script. The Python os module was also used to iterate through the directory of the Python script to look for template FlexPDE files. Matplotlib was used to generate the plots for the trajectory data, and the Labellines module was used to add labels to each trajectory curve for clarity.
The Python script operates using “template” FlexPDE script files. These template files are FlexPDE files containing string formatting codes which Python can recognize and replace with whatever fuel flow rate is being tested in the current iteration of the Python loop. The Python script reads the code from these template files and generates a new FlexPDE script file (the “testing” file) after substituting in the appropriate fuel flow rate into the simulation. For the kinematics simulation, only the fuel flow rate was substituted into the template code, while for the heat simulation the landing time from the kinematics simulation was also inputted so that the rocket only heated up for as long as the rocket was in the air (past this, the rocket would have crashed, so the heat flow simulation would not have been as insightful). This template file method was used so that the FlexPDE code could be stored within a FlexPDE file instead of within the Python script itself. The testing files also allow for quick testing and modifications without having to worry about modifying the files used by the Python script.
Extraction of data from the FlexPDE files was accomplished by including export statements within the FlexPDE template code. These allowed for the trajectory data from the kinematics simulations and the maximum temperature over time from the heat simulations to be outputted to separate text files, which could be read by the Python script. These text files were reused for each simulation to extract new data from each run.
The maximum temperature was set at 250°C as per the design requirements. The range of fuel flow rates to check was determined by first looking at the fuel flow rate for the SRB, which was calculated to be about
3500
kg s considering the^ 127 s^ burn time of the rockets which hold^ 550 000 kg^ of fuel [2]. With^3500
kg s as a baseline fuel flow rate, values around this were checked for the simulation, giving a final range of fuel
flow rates from 1000
kg s to^4500
kg s.
When running the script, all FlexPDE plots which were not outputting data to a text file were commented out within the template code, while those that were outputting data to a text file had a print-only keyword used so that FlexPDE would not have to expend resources generating graphical results which would not be read by the Python script. Performing these steps sped up the Python script significantly.
Once the script ran the kinematics and heat flow simulations for all fuel flow rates, it was able to return the fuel flow rate which resulted in the maximum horizontal range without surpassing the maximum temperature. Once this was done with the initial fuel flow rate range, subsequent, narrower ranges of fuel flow rates could be tested to increase the precision of the calculation for the optimal fuel flow rate. Determining these increasingly narrow ranges of fuel flow rates automatically is not trivial in this example as the rocket’s horizontal distance does not necessarily increase with increasing fuel flow rate (as is seen in Figure 17). As such, the plan was for these narrowing ranges of fuel flow rates to be determined manually after analyzing the results from the initial range of fuel flow rates.
Figure 17: Compilation of rocket trajectories generated using the Python script to automate the FlexPDE simulations. Fuel flow rates of 1000 𝑘𝑘𝑘𝑘𝑠𝑠 < 𝑞𝑞 < 4500 𝑘𝑘𝑘𝑘𝑠𝑠 were tested in increments of 500 𝑘𝑘𝑘𝑘𝑠𝑠. Red lines indicate overheating in the heat flow simulation, while blue lines indicate no overheating.
The optimal fuel flow rate for the SRBP was determined to be 2550
kg s , which is on the same order of magnitude as the fuel flow rates for the actual SRBs. The current model predicts overheating of the SRBP
at fuel flow rates of 4000
kg s and above, though this is not significantly relevant to the design of the rocket boosters (at least, as standalone rockets) as the optimal fuel flow rate to maximize the distance is much lower. At higher fuel flow rates, the effects of drag seem to limit the rocket’s horizontal range, thus making them less effective at travelling greater distances.
The distances that the rocket travelled in the kinematics simulations seem reasonable for a rocket of this magnitude, though literature on the SRBs being used as standalone rockets is scarce, so finding exact numbers to compare with is difficult. The rocket may have been capable of entering a low earth orbit around 160 km [23], but this cannot be validated with this model as the gravitational field was modelled at a constant 9.8 ms 2 , and the curvature of the Earth was not included.
One important distinction with the SRBP and the SRBs is the amount of insulation. The amount of insulation in this model is only 7 mm as opposed to the SRBs’ insulation which is around 50 mm thick. This major difference allows the temperatures of the SRBP to reach past 200°C which, though still under the maximum temperature as laid out by my hypothetical team of rocket scientists, may still not be optimal as the titanium alloy could become compromised. It should be noted, though, that the heat simulation here was assuming a worst-case scenario where the fuel material was already at combustion temperature at the start of the launch. A more realistic model which encapsulated the fuel burning away over time may have resulted in lower outer wall temperatures. Regardless, high outer wall temperatures are not ideal, as the SRBs are meant to be recovered after every run [4], so care must be taken to ensure the outer wall is not subject to fatigue after only a few launches. With the proper amount of insulation, the SRBPs could withstand much greater fuel flow rates, though as discussed that would not help them increase their maximum horizontal range in this model.
There were other limitations of the FEM models which hindered the accuracy of the simulations.
Ultimately, although both the kinematics and heat flow models have many limitations when compared to the actual SRBs, they were able to demonstrate how rocket kinematics differ greatly from simple projectile motion and how insulation is paramount to ensuring the heat of the combustion gases in the centers of solid- fuel rockets are funnelled out at the bottom and not the sides of the rocket. The optimal fuel flow rate for
the SRBP was determined at 2550 kgs , where the rocket exterior reached a maximum temperature of
205.99°C and travelled a total of 400.061 km across the ground. These show promising results for the SRBP to operate as a standalone rocket, though its thin layer of insulation would likely limit its capability for reuse in realistic scenarios. Rigorous computer modelling, brilliant engineering design, and meticulous testing are cornerstones that mark human exploration into the depths of space. Though there are many factors to consider when launching a vehicle destined for outer space, with thorough analysis and careful planning, even trips to other galaxies can become a reality.
