Numerical Simulation of Chemical Propulsion Systems: Survey and Fundamental Mathematical Modeling Approach
Abstract
:1. Introduction
 Shortening the development period and reducing development cost.
 Protect potential problems and predict propulsion performance in various conditions.
 Generate a database for the possible failure modes from FMEA and develop a reliable health monitoring system.
 Reduce maintenance cost for RLVs.
2. Overview of Chemical Propulsion Systems
3. Simulation Modeling Trend of Chemical Propulsion Systems
3.1. Literature Review
3.1.1. LiquidPropellant Rocket Engines
3.1.2. SolidPropellant Rocket Motor
3.1.3. HybridPropellant Rocket Motor
3.1.4. State of a Chemical Propulsion System
3.2. Analysis of the Trend
3.2.1. Perspective of the Modeling Approach
 Nonlinear modeling: In general, nonlinear modeling of CPS models has been used for simulation and analysis. The physics of propulsion system components are generally described using thermofluiddynamic and mechanical conservation equations. Each component is usually not developed to its full complexity since the most accurate model is not the target. The main input data to the model are propellant tank outlet pressure and temperature, geometric and thermal properties, and valve settings. CPS parameters are initially set using design points determined in preliminary design and critical design. After that, they are upgraded and tuned through estimation through generalized residual sum of squares using data obtained from actual tests.
 Linearized modeling: Linearized modeling is the most common modeling by linearizing a nonlinear thermodynamic model (mostly for design control). The approach is linearized around the design points or previously computed equilibrium points. Generally, after each component of a CPS is linearized based on the equilibrium points, all linearized models are combined. However, sometimes parts of all components, which are nonlinear equations that are hard to describe due to complexity or data lack problems, are linearized and combined with other nonlinear forms to make them simple.
 Linear identification: Some researchers have determined a mathematical model using the data obtained from systemlevel simulations or actual tests rather than developing a model based on thermodynamic equations. The approach mainly considers each valve opening angle as an input and pressure, temperature, and turbopump speed as outputs. The approach requires preliminary information about the nonlinearity and bandwidth of the system. The responses point to valve nonlinearity, which can be isolated and removed to identify the main system. Through this work, the equations by linear identification have the transferfunction structure between the jth input and the ith output as:$${\mathbf{H}}_{ij}\left(s\right)={\mathbf{C}}_{i}{\left(s\mathbf{I}\mathbf{A}\right)}^{1}{\mathbf{B}}_{j}$$
 Nonlinear identification: There are several nonlinear identification approaches, including the Volterra series model, blockstructured model, neural network model, NARMAX model, and statespace model [125]. However, in the five models, mostly a CPS is modeled using an artificial neural network (ANN) approach, which is adequate for realtime monitoring, diagnosis, and control. Using the ANN approach to represent a CPS requires training with a database from actual tests to provide the correct output determined by the user.
Modeling Approach  Types  Refs. (Selected) 

Nonlinear modeling  LPRE  [16,34,39,40,53,57,60,61,67] 
SPRM  [77,78,79,84,87]  
HPRM  [96,98,99,100]  
Linearized modeling  LPRE  [55,126,127] 
SPRM  [128]  
HPRM  [97,129,130]  
Linear identification  LPRE  [131,132,133] 
SPRM  [134]  
HPRM  [105,111]  
Neural network approach  LPRE  [135,136] 
SPRM  [137,138]  
HPRM  [139,140] 
3.2.2. Perspective of Pipe Modeling Method
Method  Refs. (Selected) 

Lumped model method  [34,40,60,67] 
Method of characteristics  [37,141] 
Volumejunction method (Lumped parameter method)  [29,39,47,142] 
3.2.3. Others
Country  Institute  Simulation Toolbox  Refs. (Selected) 

U.S.  NASA  ROCETS  [28] 
Europe  ESA  EcosimPro + ESPSS  [31,32] 
China  NUDT  LRETMMSS  [47] 
China  HUST  Selfdeveloped Toolbox (Modelica)  [48] 
China  BUAA  Selfdeveloped Toolbox (MATLAB)  [52] 
Iran  KNTU  Selfdeveloped Toolbox (MATLAB)  [52] 
Korea  KAU  Selfdeveloped Toolbox (MATLAB)  [61,62] 
Main Category  Subcategory  Subsubcategory 

Fluid Properties    Ideal gas, Simplified liquids, Real fluids 
Fluid Flow 1D  AbstracJunction  Jun_TMD, DeadEnd, Filter 
Time dependent Boundaries  VolPT_TMD, VolPx_TMD, VolTx_TMD  
Cavities  Chamber, Volume1, Volume2, Volume5  
AbstacJunctionLoss  Juntion, ValveCheck, Valve, ValvePressRegDown, VallvePressRegUp, ValveCheck_Dynamic, VolPsTsVs_TMD  
Sensor  SensorJun, SensorPipe, SensorVol  
Channel  Pipe, Tube, Pipe_res, Pipe_Rect, Tube_Rect  
Etc.  WorkingFluid, VTee, WorkingFluuid, HeatExchanger, Nozzle, ColdThruster  
Tanks  Propellent Tank  Tank_single, Tank_Sphere, Cylinder_ins, Sphere_ins, Tank_CylDomes, Dome_ins, Tank_Bladder, Tank_CylDomesSph 
Combustion Chambers  Combustor  ABSCombustor_eq, ABS−Combustor_rate 
Preburner  PreBurnerCoat_eq, PreBurner_eq, PreBurner_rate, reBurnerCoat_rate  
Nozzle  Nozzle, Nozzle_Ex, Nozzle_Ex2  
CobustChamber_Nozzle  CombustChamberNozzle_eq, CombustChambbeerNozzleCoat_eq, CombustChamberNozzle_rate, CombustChambberNozzleCoat_rate  
Cooling Jacket  CoolingJacket, CoolingJacket_simple, CoolingJacket_tore  
Injector    
Turbo Machinery  Compressor  Compressor, Compressor_gen 
Pump  Pump, Pump_gen, Pump_vaccum  
Turbine  Turbine, Turbine_gen 
4. Fundamental Mathematical Modeling Approach
4.1. Governing Equations
4.1.1. Rotational Dynamics
4.1.2. Mass Flow Rate
4.1.3. Pressure Dynamics
4.1.4. Density Equation
4.1.5. Energy Balance in Heat Exchangers
4.1.6. Heat Transfer Equations
4.1.7. Time Delay Equation
4.2. Algebraic Equations
4.2.1. The Combustion Gas Mass Flow Rate
4.2.2. Injector Pressure Decremental Equation
4.2.3. The Power or Torque of the Pump
4.2.4. The Power or Torque of the Turbine and Motor
4.3. Characteristics Equations
4.3.1. Pump Pressure Incremental Equation
4.3.2. Propellant State Equation
4.4. Mathematical Modeling of Chemical Propulsion Systems
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
A  Crossarea of pipe 
${A}_{b}$  Burning surface area of solid propellant 
${A}_{n}$  Nozzle throat area 
${A}_{pp}$, ${B}_{pp}$, ${C}_{pp}$  Coefficients of pump 
${C}_{d}$  Discharge coefficient 
${C}_{mt}$  Torque coefficient of motor 
D  Diameter of pipe 
${I}_{p}$  Moment of inertia of pump rotor 
K  Resistance coefficient of pipe 
${K}_{i}$  Injector resistance coefficient 
${K}_{m}$  Resistance coefficient 
${K}_{t}$  Coefficient of state change 
L  Length of pipe 
${\mathrm{OF}}_{c}$  Oxidizer to fuel mixture ratio 
P  Pressure 
${P}_{c}$  Combustion chamber pressure 
${P}_{in}$  Inlet pressure 
${P}_{out}$  Outlet pressure 
${P}_{te}$  Turbine outlet pressure 
${P}_{ti}$  Turbine inlet pressure 
$\Delta {P}_{i}$  Injector pressure decrement 
$\Delta {P}_{p}$  Pump pressure increment 
Q  Flow rate 
${\dot{Q}}_{w}$  Heat flow rate between the walls and the coolant fluid 
${\dot{Q}}_{c}$  Heat flow rate between the hot fluid and the wall 
R  Gas constant 
${R}_{c}$  Gas constant of combustion gas 
T  Temperature 
${T}_{c}$  Temperature of combustion gas 
${T}_{w}$  Hot wall temperature 
V  Volume 
${V}_{c}$  Volume of combustion chamber 
${W}_{p}$  Power of pump 
${W}_{t}$  Power of turbine 
$\overline{V}$  Pipe volume ratio of total and filled 
a  Speed of sound 
${a}_{p}$  Empirical constant 
${a}_{v}$  Acceleration of vehicle 
${c}_{v}$  Specific heat of the wall material 
f  Friction factor of pipeline 
h  Enthalpy of fluid 
${i}_{m}$  Current of motor 
${k}_{c}$  Heat specific ratio of combustion gas 
m  Mass of the wall 
$\dot{m}$  Mass flow rate in pipe 
${\dot{m}}_{c}$  Combustion gas mass flow rate 
${\dot{m}}_{i}$  Injector mass flow rate 
${\dot{m}}_{in}$  Inlet mass flow rate 
${\dot{m}}_{out}$  Outlet mass flow rate 
${\dot{m}}_{p}$  Mass flow rate in pump 
${\dot{m}}_{t}$  Mass flow rate in turbine 
n  Burning rate exponent 
${r}_{b}$  Burning rate 
${t}_{ignite}$  Ignition time 
u  Internal energy of fluid 
$\u03f5$  Amount of the time delay 
${\eta}_{p}$  Pump efficiency 
${\eta}_{t}$  Turbine efficiency 
$\kappa $  Fluid compressibility 
$\rho $  Density of fluid 
${\tau}_{in}$  Torque produced by turbine or motor 
${\tau}_{m}$  Torque of motor 
${\tau}_{out}$  Torque absorbed by pump 
${\tau}_{p}$  Torque of pump 
${\tau}_{t}$  Torque of turbine 
$\omega $  Angular velocity of pump 
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Types  Specific Impulse  Thrust/Weight Ratio  Thrust Duration 

Chemical Rocket  170–465  1–10  Minutes 
Electrothermal  300–1500  <${10}^{3}$  Months (steady) Years (intermittent) 
Electromagnetic  1000–10,000  <${10}^{4}$  Months (steady) Years (intermittent) 
Electrostatic  2000–100,000  <${10}^{4}$–${10}^{6}$  Months/years (steady) 
Nuclear (thermal)  750–1500  1–5  Hours 
Model of CPS  Characteristics (Perspective of ODEs) 

SPRM 

HPRM 

LPRE 

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Cha, J. Numerical Simulation of Chemical Propulsion Systems: Survey and Fundamental Mathematical Modeling Approach. Aerospace 2023, 10, 839. https://doi.org/10.3390/aerospace10100839
Cha J. Numerical Simulation of Chemical Propulsion Systems: Survey and Fundamental Mathematical Modeling Approach. Aerospace. 2023; 10(10):839. https://doi.org/10.3390/aerospace10100839
Chicago/Turabian StyleCha, Jihyoung. 2023. "Numerical Simulation of Chemical Propulsion Systems: Survey and Fundamental Mathematical Modeling Approach" Aerospace 10, no. 10: 839. https://doi.org/10.3390/aerospace10100839