Supercritical carbon dioxide as a new working medium for pneumatic launch:A theoretical study

Hong-xin Yao ,Xue-zhong Wei ,Hong Ye

Department of Thermal Science and Energy Engineering,University of Science and Technology of China,Hefei,230027,People’s Republic of China

Keywords: Pneumatic launch Supercritical carbon dioxide Restrictive relation of the launch readiness state Optimal launch readiness state High working capacity

ABSTRACT Compared with the conventionally gaseous or liquid working media,the speci fic internal energy of supercritical carbon dioxide(SCD)is higher at the same temperature and pressure,and the critical temperature of carbon dioxide is close to room temperature,making SCD a potential new working medium for pneumatic launch.To analyze the feasibility of this conception,an analytical model of a pneumatic catapult is established on basis of the conservations of mass and energy.The model consists of a high-pressure chamber and a low-pressure chamber connected by multiple valves,and there is a movable piston in the low-pressure chamber that can push an aircraft to accelerate.The effects of the launch readiness state of SCD in the high-pressure chamber,the initial volume of the low-pressure chamber and the valve control on the movement of the aircraft are analyzed.It is found that there is a restrictive relation between the temperature and pressure of the launch readiness state of SCD,i.e.,there is a maximum allowable launch readiness pressure when the launch readiness temperature is fixed.If this restrictive relation is not satis fied,the working medium in the low-pressure chamber will drop to its triple point within a few milliseconds,leading to a launch failure.Owing to this restrictive relation,there is an optimal launch readiness state of SCD with the highest working capacity for any allowable launch readiness temperature.The pressure of the low-pressure chamber will decrease signi ficantly as the initial volume increases,leading to a decreased acceleration of the aircraft.The acceleration can be controlled below a critical value by a designed sequential blasting technique of multiple valves.The calculated results show that a 500 kg aircraft can be accelerated from 0 to 58 m/s in 0.9 s with 36 kg of carbon dioxide.This research provides a new technique for the controllable cold launch of an aircraft.

1.Introduction

Aircraft are widely used in civil and military applications[1-3].To improve their endurance and economy,aircraft trajectory optimization has become a research hotspot in the aviation and aerospace industry[4,5].The aircraft launch is not only an essential part of the aircraft trajectory optimization but also the premise of the mission execution.Taking unmanned aerial vehicle(UAV)for example,its launching methods include conventional runway launch,rocket-assisted launch [6],bungee launch [7]and pneumatic launch[8].The use of a pneumatic catapult provides the user with several distinct advantages,i.e.,no explosives or dedicated runways are required to operate the catapult.This means the catapult is safe and easy to transport and can be used in a wide range of locations and terrains[9].

Pneumatic launch relies solely on a compressed gas to provide the force needed to accelerate an aircraft to its flying velocity.Fahlstrom[10]compared various launch methods of an aircraft and concluded that the pneumatic launch with air as the working medium was only applicable to an aircraft below 500 pounds,and the amount of air needed to launch a heavier aircraft could be signi ficantly increased.In addition,there is a peak acceleration when the existing pneumatic catapult is used,which is not conducive to accelerate an aircraft smoothly[11].Increasing the working capacity of the pneumatic catapult and controlling the maximum acceleration of the aircraft are the cruxes of optimizing pneumatic launch.The working medium of a pneumatic catapult is the most important factor affecting its working capacity.Considering the low working capacity of air,a new working medium is needed.The speci fic internal energies of air,oxygen,nitrogen,and carbon dioxide are compared in the temperature range of 300-400 K and the pressure range of 10-30 MPa,and the results are given in Table 1.The data of carbon dioxide,nitrogen,and oxygen are taken from Ref.[12],and the air data are calculated from the nitrogen and oxygen data based on Dalton’s law of partial pressure.The comparison shows that the speci fic internal energy of the carbon dioxide in the supercritical state is the highest.In addition,the critical temperature of carbon dioxide is close to room temperature,and the temperature of carbon dioxide will gradually decrease in working,hence the cold launch of an aircraft can be realized by controlling the launch readiness temperature of the working mediumwhen supercritical carbon dioxide(SCD)is used as the working medium for pneumatic launch.The acceleration of the aircraft can be affected by the structure of the pneumatic catapult and the launch readiness state of the working medium,hence we can control the maximum acceleration of an aircraft by designing the above parameters.

Table 1 Speci fic internal energy of common gases(kJ/kg).

In this work,a pneumatic catapult using SCD as the working medium is conceived.The catapult consists of a high-pressure chamber and a low-pressure chamber connected by multiple valves,and there is a movable piston in the low-pressure chamber that can push an aircraft to accelerate.To analyze the feasibility of this conception,an analytical model of the pneumatic catapult is established on basis of the conservations of mass and energy.The effects of the launch readiness state of SCD in the high-pressure chamber,the initial volume of the low-pressure chamber and the valve control on the movement of the aircraft are analyzed.

2.Launch model

The conceived pneumatic catapult with SCD as the working medium is shown in Fig.1.The SCD in the launch readiness state is stored in the high-pressure chamber.The initial state of the lowpressure chamber is the same as the ambient.When the valve between the high-pressure chamber and the low-pressure chamber is opened,the high-pressure carbon dioxide in the highpressure chamber is ejected into the low-pressure chamber through the valve and expands rapidly.At this time,the phase state of the carbon dioxide in the low-pressure chamber is very complex and may be composed of vapor,liquid or even solid.The rapidly expanding carbon dioxide in the low-pressure chamber can push the aircraft to accelerate.

Fig.1.Pneumatic catapult with SCD as the working medium.

The uneven distributions of the CO2in the high-pressure chamber and the low-pressure chamber are not conducive for us to obtain the theoretical results of the aircraft movement.Drawing on Ren’s[11]and Hammer’s[13]methods to solve this problem,we construct a zero-dimensional model of the pneumatic catapult.It is assumed that the carbon dioxide distributes evenly in the highpressure chamber or the low-pressure chamber.Assuming that the pneumatic catapult is airtight,the mass conservation of the CO2can be described as:

wheretis time,m1andm2are the masses of the CO2in the highpressure and low-pressure chambers,is CO2flow rate through the valve.

The kinetic energy and potential energy of the CO2are ignored,and it is assumed that the pneumatic catapult is adiabatic because of the short duration of the launch,hence the energy conservation of the pneumatic catapult can be described as:

whereU1andU2are the total internal energies of the CO2in the high-pressure and low-pressure chambers,Ekis the mechanical energy of the aircraft,the ambient pressurePamb=101 kPa,Aloadis the cross-sectional area of the aircraft,which is the same as that of the piston,υloadis the velocity of the aircraft.The term on the left side of Eq.(2)represents the rate of the total energy reduction of the CO2in the high-pressure chamber.The first,second,and third terms on the right-hand side of Eq.(2)represent the rate of the total energy increase of the CO2in the low-pressure chamber,the rate of the mechanical energy increase of the aircraft and the power of the CO2in the low-pressure chamber overcoming the ambient pressure to push the aircraft,respectively.

The acceleration of the aircraft can be calculated by Newton’s second law:

where the friction between the piston and the low-pressure chamber is ignored and the piston mass is not considered,υloadis the velocity of the aircraft,P2is the CO2pressure in the lowpressure chamber,mloadis the mass of the aircraft,θis the angle between the pneumatic catapult and the horizontal plane,the gravitational accelerationg=9.8 m/s2.The flow rate through the valve with the same cross-sectional areas at the inlet and outlet can be calculated using a simpli fied valve equation[14]:

whereAvalis the cross-sectional area of the valve,P1is the pressure of the working medium in the high-pressure chamber,ρ1is the density of the working medium in the high-pressure chamber,and the resistance coef ficientξis related to valve type and diameter[15].

The maximum value of mass flux is the choked flow rate[16].It can be obtained by[16]:

where the accuracy and complexity.The pressure of Feng’s equations can be described as:

whereciandbiare given in Table 2.

Table 2 Parameters of Feng’s equations.

The speci fic enthalpy of Feng’s equations can be described as:

whereγis the isentropic coef ficient.According to the theory of molecular dynamics,the typical isentropic coef ficient of diatomic molecules such as oxygen and nitrogen is 1.4,and that of carbon dioxide is 1.3,and that of air is 1.3-1.4[16,17].The isentropic coef ficients of the above common gases are nearly identical,but the density of carbon dioxide is much higher than that of air,oxygen and nitrogen at the same temperature and pressure.According to Eq.(5),when carbon dioxide is used as the working medium,the choked flow of the valve will increase,which can increase the speed of working medium transfer between the high-pressure chamber and the low-pressure chamber,and thus shorten the duration of the launch.

The critical temperatureTcand critical pressurePcof carbon dioxide are 304.13 K and 7.38 MPa[18].Considering that a real gas equation is needed to describe carbon dioxide during high pressure injection[19],we compare various cubic equations[20-22]and semi-empirical equations[23,24]to find a suitable real gas equation,and Feng’s equations[23]is selected based on the balance of whereis the enthalpy of ideal gaseous carbon dioxide,and the speci fic heat capacitycpcan be described as[16]:

whereC0=0.45,C1=1.67,C2=-1.27,C3=0.39,η=

The speci fic internal energyucan be obtained by

wherePis pressure,ρis density.

The entropy of Feng’s equations can be described as:

The Gibbs free energyGcan be obtained by

The comparison between the calculated results of pressure,enthalpy and entropy by Feng’s equations and NIST data[12]is shown in Fig.2.As can be observed,the results agree well.

Fig.2.Comparison of the results calculated by Feng’s equations with NIST data.

As shown in Eqs.(1)-(4),the crux of solving the movement of the aircraft is obtaining the state parameters of the CO2in the highpressure and low-pressure chambers.The state parameters of CO2in a single-phase state can be determined by Feng’s equations.However,when the CO2is in the state of vapor-liquid equilibrium,the densities of the vapor and liquid CO2are different.In this case,Feng’s equations cannot be used to obtain the state parameters of CO2.Hence it is necessary to establish the CO2phase change governing equations.When the CO2is in the state of vapor-liquid equilibrium,the sum of the mass fractions of the vapor and liquid CO2is 1:

wherexvandxfare the mass fractions of the vapor and liquid CO2.The pressures of the vapor and liquid CO2are the same:

where ρvandρfare the densities of the vapor and liquid CO2.The Gibbs free energies of the vapor and liquid CO2are the same[13]:

Considering the vapor and liquid CO2occupy the chamber(High-pressure chamber or Low-pressure chamber),the average density of the CO2can be obtained by

The average speci fi c internal energy of the CO2can be obtained by

Because of the lack of literature on the pneumatic launch with SCD as the working medium,we cannot directly verify the reliability of our model.Considering the crux of solving the movement of the aircraft is to obtain the state parameters of the CO2in the high-pressure and low-pressure chambers,we can indirectly verify the reliability of the model by verifying its accuracy in solving the above parameters.The case of the free jet of carbon dioxide in the literature[13]is calculated.In this case,a cylindrical tank with a diameter of 0.2 m and a height of 1.0 m is considered,and the initial states of the CO2in this tank areP01=10 MPa andT01=300 K,and the ambient conditions arePamb=0.1 MPa andTamb=293.15 K.The CO2fl ow rate is calculated by

whereKv=5×10-7m2is the fl ow coef fi cient of the valve.Because of the small opening of the tank,the duration of the jet is long,hence an equation that incorporates heat transfer should be considered.The heat-transfer rate between the tank and the ambient is calculated by

whereκis heat-transfer coef fi cient andSis the surface area of the tank,andκSis 1.0 W/K.The comparison between the calculated results of our model and the calculated data in the literature[13]is shown in Fig.3.As can be observed,the results agree well,showing that our model can accurately solve the state parameters of carbon dioxide.

Fig.3.Comparison of the calculated results with the calculated data from literature[13].

3.Design of the key parameters of the launch model

A 500 kg cylindrical aircraft with a diameter of 0.3 m is considered.Considering the structural strength of the aircraft,the maximum acceleration of the aircraft is limited below 80 m/s2.To increase the launch velocity of the aircraft and controlling the maximum acceleration of the aircraft,the key parameters of the launch model need to be optimized.Because SCD is the working medium of the pneumatic catapult,its launch readiness state is the most important factor affecting the fi nal velocity of the aircraft.The valve control and the initial volume of the low-pressure chamber can affect the acceleration of the aircraft.Hence they are also key parameters.Here,we take a globe valve with a diameter of 100 mm as an example,although it’s not the most suitable,to explore the optimization method and process of the above key parameters.ξis 4.1 when the diameter of a globe valve is 0.1 m[15].After the valve is selected,we should fi rst analyze the effect of the initial volume of the low-pressure chamber on the movement of the aircraft,and determine an initial volume of the low-pressure chamber according to the launch requirements of the aircraft,because the initial volume of the low-pressure chamber belongs to the geometric characteristic of the launch model and can affect the design of the launch readiness state.The valve control can be designed at the last step,because it has little effect on the design of other key parameters.In addition,the pressure in all cases is absolute.

3.1.Analysis of the initial volume of the low-pressure chamber

The initial volume of the low-pressure chamberis an important factor affecting the acceleration of the aircraft.To analyze the effect of the initial volume of the low-pressure chamber on the movement of the aircraft,three cases with initial volumes of 0.1,0.5 and 1.0 m3are calculated.In the three cases,the launch readiness state is(320 K,8 MPa)and the volume of the highpressure chamber is 0.1 m3.The diameter of the valve is 100 mm.The initial state of the low-pressure chamber is set as(300 K,101 kPa).

Fig.4 shows the time variations of the velocity and acceleration of the aircraft.For each of the initial volumes,the velocity of the aircraft increases gradually with time and the acceleration increases first and then decreases.The maximum velocity,maximum acceleration and maximum displacement of the aircraft are given in Table 3.As the initial volume of the low-pressure chamber increases,the maximum velocity,maximum acceleration,and maximum displacement of the aircraft decrease.Fig.5 shows the time variations of the pressure,temperature and flow rate at different initial volumes of the low-pressure chamber.For each of the initial volumes,the pressure and temperature of the highpressure chamber and the actual flow rate of the valve decrease gradually with time,and the actual flow rate of the valve remains below the choked flow,and the pressure of the low-pressure chamber increases first and then decreases.As the initial volume of the low-pressure chamber increases,the temperature of the lowpressure chamber changes more gently,and the overall trend is downward.

Table 4 The maximum velocity,maximum acceleration and maximum displacement of the aircraft and total mass of CO2 or air at the optimal launch readiness state.

Fig.4.Time variations of the velocity and acceleration of the aircraft at different initial volumes of the low-pressure chamber.

Table 3 The maximum velocity,maximum acceleration and maximum displacement of the aircraft at different initial volumes of the low-pressure chamber.

Fig.6 shows the trajectories of the CO2phase in the phase diagram.The CO2in the high-pressure chamber changes from the supercritical state to the vapor state and then to the state of vaporliquid equilibrium,hereafter,follows the vapor-liquid saturation line.The CO2in the low-pressure chamber changes from the vapor state to the state of vapor-liquid equilibrium,hereafter,follows the vapor-liquid saturation line until the CO2temperature drops to the three-phase point temperature.As the initial volume of the lowpressure chamber increases,the temperature of the CO2in the low-pressure chamber reaching the state of vapor-liquid equilibrium decreases,but that of the CO2in the high-pressure chamber reaching the state of vapor-liquid equilibrium is less affected.

Fig.7 shows the distributions of energy when the launches are completed.In the three cases with the initial volumes of the lowpressure chamber of 0.1,0.5 and 1.0 m3,the energy conversion efficiencies of the CO2internal energy into the mechanical energy of the aircraft are 13.11%,7.41%and 3.61%,respectively.The reason for the low energy conversion ef ficiency is that the low-pressure chamber must maintain a certain temperature and pressure to accelerate the aircraft,hence most of the CO2internal energy is left in the low-pressure chamber.

In conclusion,as the initial volume of the low-pressure chamber increases,the maximum velocity,maximum acceleration and final displacement of the aircraft and the energy conversion ef ficiency decrease.Considering the variety of aircrafts,their launch requirements can be different.The maximum launch velocity is more important for some,but the maximum launch acceleration is more important for others.Therefore,differentV02can be chosen for different aircraft.We takeV02=0.5 m3as an example to design the launch readiness state and the valve control to satisfy the launch requirements of our aircraft.

Fig.6.Trajectories of the CO2 phase in the phase diagram at different initial volumes of the low-pressure chamber.

3.2.Design of the launch readiness state

As shown by the trend of the temperature of the case withV02=0.1 m3in Fig.5,we found that the temperature of the CO2in the low-pressure chamber drops sharply and then rises slowly at the early stage of the launch,forming a in flection point of the temperature.Several cases with obvious characteristics of the inflection point of temperature are selected to discuss the phenomenon of the in flection point of the temperature.Fig.8 shows the time variations of the CO2temperatures in the high-pressure and low-pressure chambers when the launch readiness temperature is fixed at 320 K,but the launch readiness pressure(P01)is changed from 8.00 to 9.32 MPa.We can see the in flection point of the temperature(as indicated by the green box)drops sharply and is close to the triple point temperature with the increase of the launch readiness pressure.When the in flection point reaches the triple point temperature,the launch will fail.We de fine this phenomenon as the restrictive relation of the launch readiness state(RRLRS).

The reason why the in flection point of the temperature is formed is that the speci fic internal energy of the CO2in the highpressure chamber is lower than that in the low-pressure chamber at the beginning of the launch.When the valve is opened,a large amount of the CO2with low speci fic internal energy is injected into the low-pressure chamber,leading to a sharp drop of the temperature of the CO2in the low-pressure chamber.When it drops to a certain level,the speci fic internal energy of the CO2in the highpressure chamber is the same as that in the low-pressure chamber.Then,there’s a flow work of the CO2into the low-pressure chamber when the carbon dioxide is injected from the highpressure chamber into the low-pressure chamber,leading to a temperature rise of the CO2in the low-pressure chamber.The flow work of the CO2always exists in the launch,but it has a small effect on the temperature change of the CO2in the low-pressure chamber when the valve has just been opened.At this time,the difference of the speci fic internal energy between the high-pressure chamber and the low-pressure chamber plays a major role in the temperature change of the CO2in the low-pressure chamber.In addition,we can conclude that for any initial volume of the low-pressure chamber,the in flection point of the temperature can be formed as long as the launch readiness pressure reaches a ceiling when the launch readiness temperature is fixed.

Fig.7.Distributions of energy when the launches are completed at different initial volumes of the low-pressure chamber.

To increase the launch velocity of the aircraft,the CO2in the high-pressure chamber should have as much total internal energy as possible.For the cold launch,there is an allowable launch readiness temperature.Considering that the volume of the highpressure chamber is fixed,when a launch readiness state satis fies RRLRS and the launch readiness temperature requirement and the internal energy per unit volume of the CO2in this launch readiness state is the highest,it can be defined as the optimal readiness launch state.

Fig.8.Time variations of the CO2 temperature at different launch readiness pressures.

For example,when the allowable launch readiness temperature is 323 K,the contours of the internal energy of carbon dioxide per unit volume are drawn in the supercritical region to find the optimal launch readiness state,as shown in Fig.9.According to the de finition of the optimal launch readiness state,it is determined to be(323 K,9.76 MPa),as shown by the red circle sign.We calculate a case to study the working capacity of this state.In this case,the launch readiness state is(323 K,9.76 MPa),and the volume of the high-pressure chamber is 0.1 m3.The diameter of the valve is 100 mm.The initial state of the low-pressure chamber is(300 K,101 kPa),and the initial volume is 0.5 m3.Besides,to prove the superiority of CO2as a working medium,a comparative case using air as the working medium was calculated with the same operating condition of the CO2case in this section.The solution of this case will be stopped when the acceleration of the aircraft ends.The modi fied Virial equation[26]is chosen to describe air because its accuracy has been veri fied in the literature[11].To verify the reliability of the results of the air case,the case of the pneumatic launch using air as the working medium in the literature[11]is calculated.The comparison between our calculated results and the calculated data in the literature[11]is shown in Fig.10.As can be observed,the results agree well.

Fig.9.Contours of the internal energy of carbon dioxide per unit volume in the supercritical region.

Fig.10.Comparison of the calculated results with the calculated data from literature[11].

Fig.11.Time variations of the velocity and acceleration of the aircraft for different working media.

Fig.11 shows the time variations of the velocity and acceleration of the aircraft for different working media.For each of the working media,the velocity of the aircraft increases gradually with time and the acceleration increases first and then decreases.Compared with the air case,the aircraft of the CO2case has a shorter acceleration duration,with a higher acceleration and a higher maximum velocity.The maximum velocity,maximum acceleration and maximum displacement of the aircraft and the total mass of the working medium are given in Table 4.Compared with the air case,the maximum velocity of the aircraft of the CO2case is much higher and the acceleration distance of the aircraft is much shorter.Therefore,compared with the air,the CO2can increase the working capacity and improve the maneuverability of the pneumatic catapult.Fig.12 shows the time variations of the pressure,temperature and flow rate for different working media.For each of the working media,the pressure and temperature of the high-pressure chamber and the actual flow rate of the valve decrease gradually with time,and the actual flow rate of the valve remains below the choked flow,and the pressure of the low-pressure chamber increases first and then decreases.There is an in flection point of the temperature of the low-pressure chamber in the CO2case,but not in the air case.

Fig.12.Time variations of the pressure,temperature and flow rate for different working media.

Fig.13 shows the energy distribution when the launch is completed.For the CO2case,the energy conversion ef ficiency is 9.22%.For the air case,the energy conversion ef ficiency is 36.66%.Compared with the CO2case,the energy conversion ef ficiency of the air case is higher,but it’s working capacity is lower.As shown in Fig.11,for the CO2case,when the solution is stopped,the aircraft still has a high acceleration,and the reason why the solution is stopped at this time is the temperature of the CO2in the lowpressure chamber reaches the triple point temperature,as shown in Fig.12.Hence the triple point temperature of CO2limits the working capacity of the CO2working medium.Compared with the CO2case,the air case has a larger temperature range to improve the conversion ef ficiency of the internal energy of the working medium into the mechanical energy of the aircraft.However,compared with the air,the working capacity of the CO2is higher and the launch duration is shorter,which can improve the maneuverability of the pneumatic catapult.

Fig.13.Distributions of energy when the launch is completed for different working media.

3.3.Design of the valve control

In section 3.2,the maximum acceleration of the aircraft of the CO2case is much higher than the designed acceleration(80 m/s2).To maintain the acceleration of the aircraft at the designed acceleration,a sequential blasting technique of multiple valves is designed,i.e.,multiple valves are gradually opened in a speci fied sequence.Assuming that the values of the resistance coef ficients of the valves are fixed at 4.1,the step functions of the diameters of these valves are given in Table 5,which are obtained by calculating a large number of cases.In Table 5,the valve with diameterdis converted into several valves with smaller diametersd1,d2……dn,which is defined as the diameter conversion.The flow rate is the same before and after the diameter conversion under the same differential pressure and CO2density.

Fig.14 shows the distribution of the valves at the top of the highpressure chamber,where the diameters of the valves 1-5 are 37.30,30.21,40.80,67.04,and 166.24 mm,respectively.Fig.15 shows the time variations of the velocity and acceleration of the aircraft before and after the valve control for the CO2case in section 3.2.We can see the problem of excessive acceleration of the aircraft has been solved.The maximum velocity,maximum acceleration and maximum displacement of the aircraft before and after the valve control are given in Table 6.After the valve control,the maximum velocity of the aircraft decreases slightly,and the maximum acceleration decreases signi ficantly and is maintained at the designed acceleration(80 m/s2).Fig.16 shows the time variations of the pressure,temperature and flow rate after the valve control.Because of the in fluence of the valve control,the flow rate has taken a step at several times.Fig.17 shows the distribution of the energy when the launch is completed after the valve control.After the valve control,the energy conversion ef ficiency is 6.66%,which is lower than that before the valve control.The reason is that the valve control limits the acceleration of the aircraft and more the CO2internal energy is trapped in the low-pressure chamber than before the valve control.

Table 5 Step functions of the diameter of the valves.

Fig.14.Distribution of the valves at the top of the high-pressure chamber.

Fig.15.Time variations of the velocity and acceleration of the aircraft before and after the valve control.

Table 6 The maximum velocity,maximum acceleration and maximum displacement of the aircraft before and after the valve control.

Fig.16.Time variations of the pressure,temperature and flow rate after the valve control.

Fig.17.Distribution of energy when the launch is completed after the valve control.

4.Conclusions

SCD is used as a new working medium for pneumatic launch,and an analytical model of a pneumatic catapult is established.RRLRS is found.Owing to RRLRS,there is an optimal launch readiness state of the CO2with the highest working capacity for any allowable launch readiness temperature.The velocity and acceleration of the aircraft can decrease signi ficantly with the increasing initial volume of the low-pressure chamber.The acceleration can be controlled below a critical value by a designed sequential blasting technique of multiple valves.For our launch model,the optimal launch readiness state is determined to be(323 K,9.76 MPa),and the valve control strategy is designed to control the maximum acceleration of the aircraft.The calculated results show that a 500 kg aircraft can be accelerated from 0 to 58 m/s in 0.9 s with 36 kg of carbon dioxide.During launch,the maximum acceleration of the aircraft does not exceed 80 m/s2and the CO2temperature does not exceed 323 K.Compared with the traditional working medium(air),the advantage of CO2as a new working medium is that it has a much higher working capacity with the same launch readiness state and the same volume of the high-pressure chamber.With the same working capacity,the CO2working medium requires a smaller volume of the high-pressure chamber,which can improve the compactness of the pneumatic catapult.

Declaration of competing interest

We declare that there is no con flict of interest.

Acknowledgment

This work was funded by the National Natural Science Foundation of China(No.51576188).

Nomenclature

Mathematical symbols

a:acceleration(m/s2)

cp:speci fic heat capacity(kJ/(kg?K))

g:gravitational acceleration(9.8 m/s2)

h:speci fic enthalpy(J/mol)

m:mass(kg)

mass flow rate(kg/s)

s:entropy(J/(mol?K))

t:time(s)

u:speci fic internal energy(J/mol)

v:velocity(m/s)

x:mass fraction

A:cross-sectional area(m2)

Ek:mechanical energy(J)

G:Gibbs free energy(J/mol)

Kv:valve flow coef ficient(m2)

P:pressure(Pa)

heat-transfer rate(W)

Rg:gas constant(J/(g·K))

S:surface area(m2)

T:temperature(K)

U:total internal energy(J)

V:volume(m3)

Greek symbols

γ:isentropic coef ficient

κ:heat-transfer coef ficient(W/(m2·K))

θ:angle between the pneumatic catapult and the horizontal plane

ρ:density(kg/m3)

ξ:valve resistance coef ficient

Subscripts

0:ideal fluid

1:high-pressure chamber

2:low-pressure chamber

c:critical state

amb:ambient

f:liquid

v:vapor

val:valve

load:aircraft