Optimization of Flight Performance of PET Bottle Rockets by Integrated Analysis of Dynamics Simulation System and Parameter Design

PET bottle rocket is a rocket system that obtains thrust by injecting water with compressed air at high pressure, is usually manufactured using a pressure-resistant plastic bottle. Although it is compact and easy to manufacture, the technical elements and principles included in PET bottle rockets are many in common with the actual rocket system and it is an appropriate system as a teaching material for science education and research. In this research, we developed a simulator based on a numerical calculation program based on dynamic consideration about the flight performance of PET bottle rockets, and theoretically analyzed the optimum flight conditions for obtaining the flight distance and flight stability of the rocket. Based on the analysis results, factors that influence the flight performance of PET bottle rockets by applying the ”Parameter Design Method” were extracted, and the flight conditions to obtain the optimum maximum flight distance and flight stability were verified.


Introduction
PET bottle rocket is a rocket system that obtains thrust by injecting water with compressed air at high pressure (usually about 5atm), and is usually manufactured using a pressure-resistant plastic bottle (Fig. 1).Although it is compact and easy to manufacture, the technical elements and principles included in PET bottle rockets are many in common with the actual rocket system and it is an appropriate system as a teaching material for science education and research.However, as a PET bottle rocket is a flying object, attention must be paid to safety during launch.There is no problem if the water rocket flies on the specified trajectory and land in a safe area, but its flight path is often not stable, there is a possibility of damaging things and people on the ground.In the case of manufacturing and launching in a short time like the event, we will make a water rocket with a simple structure, so we need to think about a fuselage structure that fly stably even if made by anyone.For that purpose, numerical analysis of flight trajectory by simulation for evaluating flight characteristics and quantitative analysis on flight stability are important.However, there are many empirical factors such as basic aerodynamic data required for orbital analysis and guidelines for aerodynamic design / structural design, which are important in achieving stable flight.In addition, it is necessary to per- form many experiments by setting various parameter conditions such as adjusting the amount of water to be put in the bottle to obtain the flight distance.In reality, the optimum value is mostly data obtained as a result of repeating the launch, and its scientific basis has not been clarified at present [1] [2].Therefore, in this research, we developed a simulator based on a numerical calculation program based on dynamic consideration about the flight performance of PET bottle rockets, theory about the optimum flight condition for obtaining the flight distance and flight stability of PET bottle rockets are analyzed.Based on the analysis results, we approached optimization on flight performance by applying Parameter Design Method.Parameter design is a design method based on statistical theory that investigates the relationship between design parameters and the quality of an object and minimizes its effect.It can be said that it is an extremely effective methodology for flight performance evaluation of PET bottle rockets.
In this research, factors that influence the flight performance of PET bottle rockets were extracted, and flight conditions to obtain optimum maximum flight distance and flight stability were verified [3].

Dynamical consideration of PET bottle rockets 2.1 Rocket's equation of motion
The equation that determines the motion of the rocket thrust flight is expressed as follows [4].
Here, m: Mass of the rocket (including the mass of water) F: Thrust force R: Air resistance force β: Water discharge per unit time g: Gravitational acceleration θ: Orbital angle of the rocket v: Speed of the rocket x, y: Rocket coordinates

Thrust due to water jetting
The thrust of a PET bottle rocket includes one caused by water jetting and the other caused by the jetting out of air that occurs after the water runs out.This thrust is generated when the pressure P 0 in the bottle is larger than the atmospheric pressure P a , and water is accelerated by the pressure difference (Fig. 2) [4] [5].First, consider thrust obtained by squirting water.When water (density rho) is injected from the mouth of the cross-sectional area A to the rocket at the velocity u, the mass dm of water ejected during the minute time dt is, dm is injected backward from the rocket at the speed of u, so that the impulse received by the rocket F • dt is, Figure 2: Thrust model By substituting Eq. ( 4) into Eq.( 5) and rearranging it, we get the expression of thrust F of the rocket.
Here, if the cross sectional area A 0 in the tank, the flow velocity u 0 , and that at the injection port are A and u, the following expression is established from the Bernoulli equation and Continuous equation which is the basic equation of fluid dynamics.
Here, P a is the atmospheric pressure, and P 0 is the pressure in the pressure tank.g is the gravitational acceleration, and h is the height from the injection port to the water surface in the tank.From these two equations, the following equation is obtained.
If this is substituted into the expression of thrust F, thrust is given by the following equation.
The pressure P 0 in the pressure tank sharply decreases with the ejection of water, but this change can be calculated by considering it as adiabatic change.

Thrust due to air jetting
Compressed air still remains in the pressure tank at the time the water jetting has ended.By ejecting this air, the rocket will be further accelerated with thrust [6].Consider the case where the pressure and density of air flow out from the tanks with P 0 and ρ 0 to the outside air with pressure P a .Assuming that the pressure and density of the gas at the ejection port are P 1 and ρ 1 , respectively, the outflow velocity u 1 at the ejection outlet can be obtained from the equation of compressive fluid as follows.
γ is the specific heat ratio of air, and if γ = 1.4 is substituted and s rearranged, it is expressed by the following equation.
Here, the pressure P 1 at the jet port is not always equal to the outside air pressure P a .When the flow velocity u 1 of the injected gas does not exceed the sound velocity, it can be considered that the jet pressure P 1 is substantially equal to the outside air pressure P a .However, as the pressure difference between the inside and the outside becomes large, even if the flow velocity increases, it cannot exceed the sound speed, so the jet pressure P 1 does not decrease to the outside air pressure P a .
Such a state is called choking (flow blocking).The thrust F a as a reaction of the momentum given to the jet air, like the case of water jetting ρ 1 •A•u 2 1 and the following equation can be obtained by using the "Adiabatic expansion" equation.
In addition, when the pressure of the air injection port is larger than the atmospheric pressure, thrust F p (referred to as a pressure thrust) due to a pressure difference is added thereto.
In summary, thrust F can be obtained by the following equation.
Here, the following conditional expression can be substituted for P 1 .When 0.528 × P 0 ≧ P a , P 1 = 0.528 × P 0 , When 0.528 × P 0 < P a ， P 1 = P a

Flight trajectory simulation program
Based on the solution obtained from the above differential equation, we developed a simulator with a numerical calcu-lation program that applied the macro function of spreadsheet software (Excel).The developed simulator was applied to extract the optimal flight conditions for getting flight distance and stability of a PET bottle rocket [7].Table 1 shows the conditions and basic formulas in the numerical calculation program.It is possible to input flight conditions of a PET bottle rocket on the screen and simulate the output data of flight distance and altitude.

Flight characteristics analysis
The PET bottle rocket is a clear example in which the flight motion is established based on the law of action and reaction, and theoretical considerations derived from the law of impulse and momentum conservation [8].In general, experimental results have been reported that the amount of water is 1/3 of the bottle volume, the launch angle is 55 • , and the flight distance can be extended if the airframe is made as light as possible.However, the theoretical basis is not so clear.In order to clarify this, we examined how each condition affects the rocket's motion.By changing each initial condition, we estimated and analyzed the optimum condition to extend the flight distance.
4.1 Force acting on rocket and flight distance Fig. 3, Fig. 4 shows the relationship between launch angle water volume and flight distance of rocket.As can be seen from the graph, the trajectory gradually faces downward even if it is launched upward at a large angle.This phenomenon of crouching is called Gravity turn.
Fig. 5 shows various forces acting on rocket flight.The rocket's traveling direction is determined by the resultant force (F7), which is lower than the direction of thrust, and its orbit gradually changes downward.Also, from the graph of Fig. 4, the optimum water rate ratio for obtaining the flight distance for the bottle volume (1.5L and 1.0L) at the launch angle of 45 • and the initial pressure of 5atm is 1.5L: 35% ( About 530ml) and 1.0L: 40% (about 400ml) [9].Fig. 6 shows the calculation results of the time change of the internal pressure when the initial pressure is 5atm (gauge pressure), the launching angle is 45 • , and the amount of water added is changed from 200ml to 500ml using 1.5L bottle.From the internal pressure, we can calculate the volume of compressed air, and the injection velocity of water and air.Since the thrust received by the rocket is the product of the mass flow rate of the injected fluid and the injection speed, the time change of the thrust can also be determined.Fig. 7 shows the injection velocity, and Fig. 8 shows the thrust calculation results.It can be seen that the flight of the PET bottle rocket relies more heavily on water injection than air injection.It can be seen that in order to obtain a large momentum, it is better to increase the initial pressure to increase the injection speed, or to increase the amount of water introduced to extend the injection time [10].

Initial pressure of bottle internal air
If the internal pressure is expanded to the atmospheric pressure before all the water is injected, all water in the bottle will not be released and thrust cannot be obtained.Therefore, the initial pressure must be such that the internal pressure becomes greater than the atmospheric pressure when all the water is injected.The higher the initial pressure, the faster the injection velocity, the greater the thrust.Therefore, in order to fly away the PET bottle rocket, it is considered that the initial pressure should be made as high as possible [11].

Amount of water in the bottle
In the PET bottle rocket, water injection is important in flying.Actually, even if it is skipped with compressed air without putting in water, it is not able to obtain a large thrust, and it does not give much flying distance, so be sure to put water in the bottle.Therefore, it can be imagined that there is an optimum amount of water to fly far.The initial pressure changes the IIAE Journal, Vol.7, No.2, 2019 From this, it is thought that the optimum value changes with the initial pressure and the weight of the aircraft.We calculated the change in the flight distance when the water volume was changed from 7% (100 ml) to 55% (800 ml) of the bottle volume (Fig. 9).The initial pressure is fixed at 5atm and the weight of the aircraft is from 100g to 500g, calculation result is shown in Fig. 10.The vertical line in the figure shows the condition that the carry distance is the maximum.When the initial weight is increased with the aircraft weight constant, the flight distance has increased overall and the optimum amount of water is increasing.
In other words, it can be estimated that the ratio of 30% to 40% of the bottle volume is the optimum amount of water for flying away.

Aircraft weight and flight distance
The momentum that the rocket obtains depends on the thrust, but the flight speed after the end of injection varies with the weight of the aircraft.Fig. 11 shows the calculated flight distance when the water volume is fixed at 30% of the bottle volume and the initial pressure is 3 to 7atm.Fig. 12 shows the calculation results when the initial pressure is 7atm and the water volume is 20% to 50% of the bottle volume.From the calculation results, it was found that the optimum body weight is in the range of 100g to 130g.Therefore, in order to fly further, it can be estimated that the attachment should be as simple as possible and the aircraft should be lightweight.

Launch angle and flight distance
It has been reported that the PET bottle rocket has an increase in flight distance at a launch angle of about 55 • , but the scientific basis of the factor has not been clarified yet.Fig. 13 and 14 shows the variation of the flight distance against the launch angle when the ratio of water to bottle volume is set to 30% and 40%, respectively, and the initial pressure of compressed air is set to 3 to 7atm.
It can be estimated that the optimal launch angle for obtaining flight distance is about 45 • to 55 • .The rocket is considered to extend the flight distance by gliding during the descent.In order to obtain the optimum launch angle clearly, it is necessary to examine in detail the flight movement after the end of water injection.maximum flight distance and flight stability using a simulator.[12] [13].

Overview of Parameter Design
The concept of Robust design in quality engineering is shown in Fig. 15.Robust design is an idea that improves technology to bring it closer to what it should be, and Robust means stability in quality engineering.Parameter Design is one of the central methods of robust design, which is a method of evaluating the functionality and determining the parameter value of the system.
Parameters are design constants and components of the system, and are selected as control factors in parameter design experiments.Improve robustness by intentionally generating variations with noise factors among combinations of system parameters, and optimizing the level of strong con- In this research, the maximum flight distance and flight stability of a PET bottle rocket is the target function, and as the quality characteristic for extracting the optimum parameter condition for obtaining it, each characteristic value is set as the horizontal reaching distance Preferably large characteristic and Preferably target characteristic were applied [16] [17].For the quality characterization in this research, the following data analysis tool was adopted JUSE.StatWorks / V5 Quality Engineering Edition (Nikka Giken Co., Ltd.)The number of experiments this time is 9 × 2 = 18 times, and the upper row in the horizontal direction represents the type of control factor allocated.The numerical values and letters listed thereunder represent the level of each control factor.The merit of using an orthogonal table is the reduction in the number of experiments.When the orthogonal table L 9 is not used, the number of times of the experiment is 3 4 = 81 times.Four control factors (bottle diameter, launch angle, air pressure, water volume) were extracted.Table 5 shows the noise factors extracted in this study, it is taken as the temperature of water to be put in the bottle (normal temperature 10 • C, high temperature 90 • C).Fig. 16 shows the output data of the flight distance by simulation.From the output data, it can be seen that under certain conditions the difference in water temperature is a factor affecting the distance traveled.It can be inferred that the weight of the aircraft changes due to the change in the density of water depending on the temperature, which affects the flight distance.

Calculation of SN ratio
Table 6 shows the value of the SN ratio from the output data, and Table 7 shows the calculation process of the SN ratio calculated.
The factor effect of a harmful component represents a dispersion effect from an ideal function with a component whose output unintentionally changes.In the parameter design, the measure of the evaluation is obtained by the SN ratio, and ultimately the presence or absence of the effect is judged on the factor effect diagram to obtain the optimum condition.The calculation formula of the SN ratio in the preferably large characteristic applied to each characteristic value is shown below [20].SN ratio: an evaluation scale for minimizing variation, n: number of data, y: value of each data, SN ratio;

Factor effect and variance analysis
Fig. 17 shows the SN ratio for each control factor expressed in a factorial effect diagram.The factor effect table (Table 8) represents the influence of factors or factor combinations on characteristic values, and the diagram showing this is the factor effect diagram.The meaning of this diagram can be judged to be effective for the control factor whose SN ratio is widened in the vertical direction to be effective for extending the flight distance of the PET bottle rocket.In Fig. 17, the optimum level is the factor in the circled portion, BOTT.DIA:ϕ90mm, LAUN.ANG:45deg, AIR PRESS:7atm, WAT.VOL:500ml.
In addition, the obtained characteristic value varies from the average value for each level.This variation scale was evaluated in the variance analysis table (Table 9).From this table, it can be estimated that the influence degree to the flight distance is a high factor because the contribution ratio of the factors with large dispersion ratio (AIR PRESS.and WAT.VOL.) is also large.Furthermore, it was also possible to estimate that there exist factors with a high contribution rate other than the parameters extracted in this experiment.This means that it is necessary to add another parameter level to the optimization for obtaining the maximum flight distance.

Optimization of maximum distance
From the factor effect diagram in the previous section, Table 10 shows the combination of the optimum control factor level for obtaining the maximum flight distance in this experimental region with respect to the flight performance of PET bottle rockets.
Also, based on the comparison with the benchmark condition, since the optimum condition is a higher estimated value, it can be judged that the orthogonal table experiment applied in this experiment is reliable.

Optimization of flight stability by "Preferably target characteristic" 5.4.1 Determination of level table and various factors
Table 11 shows the experimental plan for this experiment.In this section, we evaluated the "Preferably target characteristic" with the target function of extracting the optimum flight condition concerning the flight stability of PET bottle rockets.The preferably target characteristic is a desirable characteristic evaluation as it is closer to the target value.Table 12 shows the noise factors extracted in this experiment, and it was set to 4 types of water temperature (normal temperature 10 • , high temperature 90 • ), duration (s), maximum velocity (km/h) to be put in the bottle.Fig. 18 shows the output data of the flight distance (normal temperature   , high temperature 90 • ), duration and maximum velocity by simulation.
It is understood from the output data that flight distance and Duration and maximum velocity are deeply involved in flight stability.

Calculation of SN ratio and Sensitivity
Table 13 shows the SN ratio and the sensitivity value from the output data, and Table 14 shows the calculation process of the calculated SN ratio and sensitivity.The preferably target characteristic is a characteristic that there is a finite target value and it cannot be smaller or larger than the target value.Analysis of the preferably target characteristic takes two steps: robustness (stability) improvement and output tuning work.The SN ratio is used for improving the robustness (stability), and sensitivity is used in the tuning work of the output.From the factor effect diagram, select the highest factor level of the SN ratio and evaluate the stability of the system.Factor of SN Ratio and Sensitivity use the effect diagram to extract factors that have a large influence on the SN ratio and have as little influence on the sensitivity as possible, and finally adjust the difference from the target value.The calculation formula of the SN ratio and sensi-tivity in the preferably target characteristic applied to each characteristic value is shown below [20].n: number of data, y: value of each data, Mean variation； Error dispersion； SN ratio； Sensitivity；

Factor effect and variance analysis
Fig. 19 shows the SN ratio and sensitivity for each control factor expressed in a factor effect diagram.The factor effect table (Table 15) shows the influence of factors or factor combinations on characteristic values, and the diagram showing this is the factor effect diagram.The significance of this diagram was determined to be a factor that has a large influence on the SN ratio and does not affect the sensitivity and that the level is highly significant for the flight stability performance of PET bottle rockets.
In Fig. 19, the optimum level is the factor BOTT. DIA : ϕ80mm, LAUN.ANG: 55deg, AIR PRESS: 7atm, WAT.VOL : 300ml .As for the diameter of the bottle, it can be inferred that a smaller diameter is more advantageous as a condition for obtaining flight stability.In addition, the obtained characteristic value varies from the average value for each level.This variation scale was evaluated in the vari-ance analysis table (Table 16).
From this table, it can be inferred that AIR PRESS.and WAT.VOL., which are large dispersion ratio factors, have high contribution ratios, so that the degree of influence on the flight stability performance is a high factor.
Furthermore, it was also possible to estimate that there exist factors with a high contribution rate other than the parameters extracted in this experiment.This means that it is necessary to add another parameter for optimization to obtain flight stability.

Optimization of flight stability
From the factor effect diagram in the previous section, Table 17 shows the optimum combinations of control factor levels for obtaining flight stability in the experimental range with respect to the flight performance of PET bottle rockets.In addition, from the comparison with the benchmark condition, the optimum condition is a higher estimated value, which is  a desirable condition regarding the stability of the flight trajectory [20].Therefore, it can be judged that the orthogonal table experiment applied in this experiment is reliable.

Conclusion
In this study, we developed a simulator based on a numerical calculation program based on dynamic consideration about flight performance of PET bottle rockets, and calculated time change of internal pressure, water and compressed air injection speed by calculation.Next, considering the initial conditions that can be set in the PET bottle rocket, we examine how the flight distance changes when changing them.Then, by calculating the equation of motion by simulation, the optimum condition for obtaining the maximum flying distance was estimated.
As a result, it has been found optimal that the initial pressure is 5 to 7atm, the water volume is 30 to 40% of the bottle volume (1.5 L), the aircraft weight is 100 to 130g, and the launch angle is 45 to 55degrees.
In addition, in the previous section, based on the analysis result, the extraction of factors that influence the flight performance of PET bottle rockets by applying the Parameter Design method, and the extraction of the factors affecting flight performance to obtain the optimum maximum flight distance and flight stability were verified.When comparing the analysis result of the flight characteristics with the evaluation result of the optimum condition based on the parameter design, almost the same results were obtained (Table 18).However, since various conditions can be set in the PET bottle rocket, these conditions are not independent one by one but are closely related to each other.Therefore, it is necessary to set so as achieve the best balance.Furthermore, it was possible to verify that it is necessary to add another parameter level to the optimization for obtaining the max- imum flight distance and flight stability from the analysis result of variance in the evaluation of flight conditions by parameter design.For example, as parameters not covered in this research, the size and shape of the tail fin and its attachment position, the length of the rocket body, etc. can be mentioned [20].If these are changed, it can be presumed that the lift, drag, center of gravity and aerodynamic center position change in addition to the weight of the aircraft, which will affect flight stability and flight after injection.Thus, from the scientific knowledge obtained on the flight characteristics of PET bottle rockets, it turned out to be an extremely complicated flight system.
As future research topics, we plan to add parameter conditions related to flight and further analyze the error between the simulation analysis result and the demonstration experiment result.We also acquire hydrodynamic demonstration experiment data on flight performance by analyzing the influence on flight stability due to vibration of the rocket body due to the behavior of water in the bottle which is the rocket propellant, and we are planning to further explore the chemistry research.

Figure 1 :
Figure 1: Main structural model of PET bottle rockets

Figure 5 :
Figure 5: Force acting on the rocket

Figure 9 :
Figure 9: Changes in water volume and flight distance

Figure 11 :
Figure 11: Changes in empty rocket mass and flight distance

Figure 13 :
Figure 13: Changes in launching angle and flight distance

Figure 14 :
Figure 14: Changes in launching angle and flight distance

Table 2 :
Results of analysis

Table 3 :
Control Factor Levels

Table 4 :
Preferably Large Characteristic [19] Journal, Vol.7, No.2, 2019Figure 16: Output Data Plot Figure 17: Effect Diagram and the experiment plan table.From this level table, Orthogonal table (L9) which is a statistical tool of optimization method was created to formulate an experiment plan.Orthogonal table is a table that defines the assignment such that combinations of levels of arbitrary factors appear the same number of times in combinations of experimental levels.The number of experiments is determined by the scale of the orthogonal table and is represented by the experiment number shown at the left end of the orthogonal table[18][19].

Table 11 :
Preferably Target Characteristic

Table 12 :
Combination of Noise Factor Levels

Table 15 :
Effect Table

Table 18 :
Summary of optimization analysis results