Genetic algorithm to optimize the design of main combustor and gas generator in liquid rocket engines

Journal of Thermal Science Vol.23, No.3 (2014) 259268

Received: September 2013 Jaye Koo: Professor This work was supported by the National Research Foundation of Korea grant funded by the Korean Government (MSIP) NRF- 2012M1A3A3A02033146 and NRF-2013M1A3A3A02042434

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DOI: 10.1007/s11630-014-0704-8 Article ID: 1003-2169(2014)03-0259-10

Genetic Algorithm to Optimize the Design of Main Combustor and Gas Gen-erator in Liquid Rocket Engines

Min Son1, Sangho Ko2, Jaye Koo2

1. Graduate Student, Korea Aerospace University, Goyang, Republic of Korea

2. School of Aerospace and Mechanical Engineering, Korea Aerospace University, Goyang, Republic of Korea

© Science Press and Institute of Engineering Thermophysics, CAS and Springer-Verlag Berlin Heidelberg 2014

A genetic algorithm was used to develop optimal design methods for the regenerative cooled combustor and

fuel-rich gas generator of a liquid rocket engine. For the combustor design, a chemical equilibrium analysis was

applied, and the profile was calculated using Rao’s method. One-dimensional heat transfer was assumed along the

profile, and cooling channels were designed. For the gas-generator design, non-equilibrium properties were de-

rived from a counterflow analysis, and a vaporization model for the fuel droplet was adopted to calculate resi-

dence time. Finally, a genetic algorithm was adopted to optimize the designs. The combustor and gas generator

were optimally designed for 30-tonf, 75-tonf, and 150-tonf engines. The optimized combustors demonstrated su-

perior design characteristics when compared with previous non-optimized results. Wall temperatures at the nozzle

throat were optimized to satisfy the requirement of 800 K, and specific impulses were maximized. In addition, the

target turbine power and a burned-gas temperature of 1000 K were obtained from the optimized gas-generator design.

Keywords: Liquid Rocket Engine, Main Combustor, Gas Generator, Optimization, Genetic Algorithm

Introduction

In a turbopump-fed liquid rocket engine, the main combustor and gas generator are hot components, and both should be given preferential focus during design of the engine. Because the combustor and gas generator are operated in a closed loop with other components, a pre-liminary gas-generator design should be tested before the overall design is completed and the actual manufacturing process is carried out. Such tests help avoid failures, but they require an optimal design method that provides fast feedback with a simple profile at the initial design level. In most previous studies on individual components, analyses were only performed at the system level, and aside from shape design, few performance variables were

considered. Typical examples of this approach can be seen in SCORES [1,2] and REDTOP [3]. Moreover, these studies did not consider the conditions of chemical nonequilibrium condition. Nevertheless, to satisfy tem-perature restrictions on the turbine, a gas generator is operated in either fuel-rich or oxidizer-rich conditions. Son et al. have suggested simple design methods for the main combustor and gas generator with respect to the profile design and a corresponding performance analysis [4,5]. However, many design variables are dependent and interact with other components. For example, some tur-bine parameters, which are directly related, have to be applied to the gas-generator design. In addition, because there are many variables, an optimization algorithm should be applied to easily optimize these variables. In

260 J. Therm. Sci., Vol.23, No.3, 2014

Nomenclature tvap

Vaporization time of droplet (s)

AR Aspect ratio of cooling channel 1 2 3, ,w w w Weighting factors

cp

Specific heat at constant pressure (J/kg-K) Greek symbols

c* Characteristic velocity (m/s) Specific heat ratio

G Mass flux (kg/s-m2) Efficiency

h Heat transfer coefficient, W/m2 Density, (kg/m3)

Isp

Specific impulse (s) Subscripts

L Power, (W) aw Adiabatic wall

Le Emission length (m) c Coolant side

L* Characteristic length of combustor (m) co Coolant

mr Ratio of mass flow rate cond Conductive heat transfer

m Mass flow rate (kg/s) conv Convective heat transfer

N Number of cooling channels CO2 Carbon dioxide

OF Oxygen/fuel ratio g Burned-gas side

P Pressure (Pa) gg Gas generator

Pr Pressure ratio at turbine H2O Water

q Heat flux (W/m2) o Overall

T Temperature (K) rad Radiative heat transfer

tchem Chemical reaction time (s) turb Turbine

tres Residence time in combustor (s) w Wall

this study, a simple genetic algorithm was adopted to achieve optimization and results were compared with non-optimized designs of the combustor and gas generator.

Main combustor design method

Performance analysis and profile design

We adopted a main design concept similar to the re-sults of Cho [4]. Chemical equilibrium was assumed for combustion in the main combustor. A chemical equilib-rium code called chemical equilibrium application (CEA), originally developed by the National Aeronautics and Space Administration (NASA), was used [6]. Thermo-dynamic properties of the equilibrium mixture can be calculated from free energy minimization, and a simple analysis of rocket performance is available. To handle pro-pellant properties in supercritical conditions, SUPERTRAPP was adopted; this code was developed by the National Institute of Standards and Technology (NIST) and is based on an extended corresponding states model that uses propane as the standard fluid [7]. The combustor profile design was created using the parabolic approxi-mation proposed by Rao [8].

Heat transfer analysis and cooling channel design for regenerative cooling

A regenerative cooling system was adopted for com-bustor cooling. In general, cooling channels have com-plicated structures, including bifurcated branches and

variable cross sections, but in this study the channel shape was simplified. The channels have rectangular cross sections with constant heights and widths. In addi-tion, it was assumed that coolant would flow from the end of the nozzle to the injector plate. The heat transfer from hot gases in the combustor to the coolant was sim-ply modeled one-dimensionally, as in Fig. 1.

At first, heat generated from the burning gas is trans-ferred to the wall through a boundary layer and a depos-ited carbon layer. Convective heat transfer is defined as

, ,g conv g g w gq h T T (1)

Fig. 1 Schematic of heat transfer and temperature distribution near combustor inner wall

Min Son et al. Genetic Algorithm to Optimize the Design of Main Combustor and Gas Generator in Liquid Rocket Engines 261

We used the convective transfer coefficient proposed by Bartz [9,10]. When kerosene is used as fuel, carbon is deposited and forms a layer that offers thermal resistance to heat transfer; this thermal resistance was estimated by Cook [10].

Radiative heat was also included in this study and was assumed to be emitted from heteropolar bonding gases, such as water and carbon dioxide. The emitted radiative heat from steam and carbon dioxide was estimated by the following equations [11].

2 2, , ,g rad rad CO rad H Oq q q (2)

2 2

3.53.5,

3, 3.5

100 100w gaw

rad CO CO e

TTq P L

(3)

2 2

3.53.5,0.8 0.6

, 3.5100 100

w gawrad H O H O e

TTq P L

(4)

Finally, the heat flux from the hot gas to the wall was determined by

, , ,g o g conv g radq q q (5)

Through the wall, heat is transferred by conduction. Because oxygen-free high-conductivity (OFHC) copper has high thermal conductivity, OFHC copper was used in this study [12,13]. The heat transfer coefficient used here was that proposed by Sieder and Tate [14].

Coolant running through channels is heated by heat transferred from the burning gas in the straight channel [15]. In the cooling channels, the pressure drop from fric-tion losses was calculated using the friction coefficient from Chen [16].

Fuel-rich gas-generator design method

The basic concept model was adopted from Son [5]. Propellants injected into the combustor are vaporized, mixed, and then burned. This combustion process, which is shown in Fig. 2, can be simplified to two steps without the mixing effect. The whole process should be com-pleted in the chamber before the gas exits to the turbine. Thus, the length of the gas generator has to be deter-mined to allow the propellants to undergo both steps. The total time required defines the residence time,

res chem vapt t t (6)

From this definition of residence time, a characteristic length of the combustion chamber is defined by

**gg res

gg gg

P tL

c

(7)

Curves fitted from the chemical non-equilibrium analysis were used to obtain burned-gas properties and chemical reaction time. A vaporization model for a kero-sene droplet was used to calculate the vaporization time. Finally, the residence time was used to estimate the

characteristic length, and a conceptual profile was deter-mined.

Fig. 2 Schematic of combustion process in gas generator

Droplet vaporization model

Kerosene, which is a liquid fuel, needs a longer time than liquid oxygen to be vaporized, so the vaporization time is strongly dependent on the kerosene time [17]. For this reason, only the kerosene vaporization time was con-sidered in the Spalding model [18,19].

The actual process of combustion progresses in a su-percritical condition, and in this condition, the enthalpy of vaporization converges to zero. Thus, the original transfer number in the Spalding model is not suitable, and a new transfer number with the critical temperature for the supercritical condition was used [20,21].

Non-equilibrium analysis

To satisfy temperature restrictions on the turbine, a gas generator is operated in either a fuel-rich or an oxi-dizer-rich condition. Thus, the properties of the gas being burned in the gas generator should be estimated when designing the gas generator. Studies on the gas property have been conducted by a number of researchers. Foel-sche [22] used the perfectly stirred reactor (PSR) model, which is a chemical nonequilibrium analysis for a pre-mixed kerosene-rich condition. A droplet vaporization model presented by Spalding was adopted, and the re-sults were similar to experiment results. Yu [23] used Foelsche’s method with a reaction mechanism from Da-gaut [24] and obtained good results compared with the experimental results of Lawver [17] for temperatures and species. However, the studies of Foelsche and Yu were conducted at low-pressure conditions and were not suffi-cient to simulate the actual operating conditions of a gas generator. For the analysis of combustion in this study, a counterflow flame analysis proposed by Lutz [25] was performed in a premixed condition. In previous studies, the maximum temperature along the axis was assumed as the reaction termination point, and properties at this point were used as the burned-gas properties at the gas- generator exit [5]. The counterflow flame analysis has the advantage of creating a stationary flame; this allows the properties to be easily determined. A schematic of the premixed counterflow flame is shown in Fig. 3.

262 J. Therm. Sci., Vol.23, No.3, 2014

Fig. 3 Schematic of premixed counterflow flame Properties of kerosene fuel, which is liquid in its stan-

dard state, can be calculated from an equation of state (EoS) that applies to supercritical fluids. In the super-critical condition, the fluid cannot be assumed to be a perfect gas, so an EoS for real fluids should be applied. Thus, the Soave modification of the Redlich–Kwong EoS (SRK EoS) [26,27] was adopted.

Design optimization

Genetic algorithm

Optimization algorithms are generally classified into two approaches: the first is derivative-based methods that include the gradient method, Newton’s method, and the conjugate gradient method, and the second is non- derivative-based methods that include the simplex method, random search, and genetic algorithms [28]. A genetic algorithm was used in this study. It was originally designed to mimic evolutionary selection and proposed by Holland in 1975 [29]. The genetic algorithm simulates a chromosome that undergoes reproduction, crossover, and mutation similar to biological evolution. In addition, a fitness estimation, which plays the role of the natural environment, is performed to select a well-fitted object and to deselect an ill-fitted object.

Unlike the conventional gradient method or Newton method, both of which use a single equation in the search space, the genetic algorithm uses a number of solution populations and improves the solution through virtual evolution. The advantage of a genetic algorithm is that there is no concern for the search to become trapped in a local solution, provided enough solutions are used. In addition, a genetic algorithm can be used in regions that cannot be searched by gradient-based algorithms because feasibility is independent of the continuity or differenti-ability of the search space. However, due to the use of a number of solution populations, a genetic algorithm can have dispersed solution populations close to an optimized value rather than at the exact value [30].

The flowchart for the algorithm used in this study is shown in Fig. 4. First, initial variable populations are

generated and initial solution populations are calculated from the design method. The initial solution group is evaluated using the fitness estimation, and convergence is checked. In this step, a scaling window method is adopted for the fitness estimation and the scaling factor is fixed to a unit, as in Grefenstette’s research [31]. If the solutions do not converge, the variables are reproduced by an operator, which is performed by a gradient-like selector. This selector was proposed by Pham and Jin [32]. Basically, a roulette-wheel selector reduces the ge-netic diversity in early generations due to replication of the excellence object, which happens several times. However, the disadvantages can be overcome when the gradient-like selector drags weaker objects to the optimal point and keeps stronger objects in place [32]. The re-produced variables undergo a crossover process using the modified simple crossover method [28]. Next the vari-ables mutate using the dynamic mutation method devel-oped by Janikow [33] and Michalewicz [34] to achieve detailed adjustments. Using the design method, the solu-tion is newly calculated, and an elite strategy is per-formed from the second iteration. If an optimal object stored from the previous generation is destroyed in the current generation, the stored object will be exchanged with the weakest object. Finally, the loop is iterated with the fitness estimation. To develop off-line performance, De Jong suggested that population size, crossover prob-ability, and mutation probability should be selected as 50, 0.6, and 0.001, respectively [35].

Fig. 4 Genetic algorithm flowchart

Objective function

An objective function is a criterion used to determine


optimum values. The objective function should be for-mulated to find maximum or minimum values of per-formance variables. In combustor design, there are three objectives. First, the maximum specific impulse is re-quired. Second, the proper wall temperature, which is below the material limitation, is sought. Third, the pres-sure drop should be minimized in the cooling channels. The objective function and constraints for combustor optimization are shown in Table 1. Table 1 Objective function and constraints for main combus-tor design

Objective function (minimization):

1 2 3, , , 500 1 csp w req

co

PF OF AR N G w I w T T w

P

Design variable and constraint

2.2 2.8OF

1.0 4.0AR

50 200N

0.01 0.03G

In this study, 0.1, 1.0, and 0.1 were used as the weighting factors. A mixture ratio (O/F ratio), aspect ra-tio of the cooling channels, number of cooling channels, and coolant mass flux were used as the design variables. The constraints of each design variable were determined within a suitable range on the basis of suggestions in [11,36]. Some design parameters, such as chamber pres-sure, expansion ratio of the nozzle, and pressure drop of the injectors, were fixed to constants. Those constant values were taken from Cho’s study [4] because they should be chosen on the basis of the design circum-stances.

The design objectives can be summarized as the gas temperature for the material limitation and the generating power of the turbine. The design variables were the mix-ture ratio (O/F ratio) and the relative ratio of the mass flow rate to the main combustor. The chamber pressure of the gas generator was fixed to the same value as that in the main combustor because the performance was less affected by the chamber pressure [5]. However, a higher chamber pressure allows for a shorter gas generator, and the chamber pressure can be limited by system con-straints such as pump performance. The objective func-tion for the gas-generator design and constraints are given in Table 2, with the weighting factors 1.0 and 0.5.

Table 2 Objective function and constraints for optimal design of gas generator

Objective function (minimization):

1 2, ,

, 1 1g turbr

g req turb req

T LF OF m w w

T L

Design variable and constraint 0.25 0.5AR

0.01 0.1rm

For the gas-generator design, a turbine design should be included. Thus, (8) was applied to calculate turbine power. Moreover, the required power for the turbine was obtained from existing engine data [37] for the criterion of the objective function, as shown in Fig. 5, using an approximate expression related to engine thrust.

1

,1

1

gg

gg

turb gg p gg turb ggr

L m c TP

(8)

Fig. 5 Actual engine data for turbine power with respect to engine thrust [37]

Results and discussion

Design optimization of main combustor

The design parameters are specified in Table 3 for de-sign optimization of the gas generator as per the required thrust. The parameters were the same as those of a pre-vious study that was conducted to compare optimized results with nonoptimized results.

The values of the objective function for the entire population converged before 15 iterations, as shown in Fig. 6. All specific impulses converged to optimal values, Table 3 Design parameters for optimal design of regenera-tive-cooled combustor

Engine class (tonf) 30 75 150

Assumed design parameters

Combustor pressure (bar) 60

Inner wall thickness (mm) 1.0

Nozzle exit pressure (bar) 0.45

Injector pressure drop (bar) 12

Requirements

Specific impulse (s) Maximum

Maximum gas-side wall temperature (K) 800

Pressure drop in cooling channel (bar) Minimum

264 J. Therm. Sci., Vol.23, No.3, 2014

as shown in Fig. 7, but the optimal values did not con-verge perfectly to maximum values because of low sensi-tivity near the maxima. In Fig. 8, the population moved to the required wall temperature and the minimum pres-sure drop. Optimized results for the main combustor are summarized in Table 4. For the nonoptimized design, input parameters were fixed to constant conditions re-gardless of the engine class.

Fig. 6 Values of objective function for main combustor design at iterations of optimization

Fig. 7 Specific impulses (dots) at different O/F ratios and optimal values (stars) for 30-tonf, 75-tonf, and 150- tonf engines

In the optimized results, values for input parameters

were determined according to the optimal values. The O/F ratios and the aspect ratios increased in the opti-mized results, and the number of cooling channels was adjusted to be proportional to the engine class. In all cases, the specific impulses increased to 312 s, and the maximum wall temperatures at the nozzle throat de-creased to about 800 K, which is the design requirement. Pressure decrease in the cooling channels dramatically

reduced to about 28% of the nonoptimized values for the 30-tonf engine. Finally, the designed profiles are shown in Fig. 9.

Fig. 8 Pressure drops (dots) in cooling channels with respect to wall temperature at nozzle throat and optimal values (stars) for 30-tonf, 75-tonf, and 150-tonf engines

Design optimization of fuel-rich gas generator

The gas generator was designed using a genetic algo-rithm with the basic parameters in Table 5. Requirements for the gas generator are also presented in Table 5. The turbine inlet temperature, which is the same as the burned-gas temperature of the gas generator, was as-sumed to be 1000 K for the material limitation, and the required power was calculated from Fig. 5 according to the engine thrust.

As shown in Fig. 10, the objective functions con-verged before about 10 iterations for all cases. The O/F ratios were determined to satisfy the requirement on the burned-gas temperatures in Fig. 11. Figure 12 shows the turbine powers that were calculated from (8). The turbine powers were also selected to satisfy the required values, which were actual engine data, in Fig. 5. The optimized results are summarized in Table 6. The optimal O/F ratios in each class were almost the same. The mass flow rates of the gas generator were about 3.5% of the mass flow rate of the main combustor; the higher class engine re-quired a lower mass flow ratio. The designed profiles of the gas generator are presented in Fig. 13.

Conclusions

In this study, optimization was achieved by using a genetic algorithm for the regenerative cooled combustor and fuel-rich gas generator in a liquid rocket engine. The main combustor was designed using a modified chemical equilibrium analysis, and the cooling system was simul-taneously analyzed using one-dimensional modeling. The


Table 4 Comparison of optimized and nonoptimized design results for the main combustor


Non-optimized results

O/F ratio of propellant 2.55

Aspect ratio of cooling channel 2.0

Number of cooling channels 100

Input parameters

Mass flux of coolant per unit cooling channel (kg/s-m2) 0.03

Mass flow rate of propellant (kg/s) 96.84 242.21 484.64

Specific impulse in vacuum (s) 309.5 309.3 309.1

Maximum gas-side wall temperature (K) 813.37 841.63 864.95

Pressure drop in cooling channels (bar) 43.96 36.16 31.56

Combustor length (m) 1.304 1.879 2.503

Nozzle throat diameter (m) 0.189 0.298 0.422

Expansion ratio 17.09 16.97 16.88

Width of cooling channel (mm) 2.13 3.37 4.77

Design parameters

Height of cooling channel (mm) 4.27 6.75 9.55

Optimized results

O/F ratio of propellant 2.61 2.58 2.73

Aspect ratio of cooling channel 3.44 3.09 3.10

Number of cooling channels 79.07 122.17 158.85

Input parameters

Mass flux of coolant per unit cooling channel (kg/s-m2) 0.0182 0.0191 0.0205

Mass flow rate of propellant (kg/s) 96.08 240.44 480.53

Specific impulse in vacuum (s) 311.78 311.47 311.77

Maximum gas-side wall temperature (K) 800.05 800.00 800.00

Pressure drop in cooling channels (bar) 12.47 14.85 18.67

Combustor length (m) 1.384 1.960 2.631


Expansion ratio 16.815 16.544 17.165

Width of cooling channel (mm) 2.33 3.05 3.58

Design parameters

Height of cooling channel (mm) 8.00 9.45 11.12

gas generator was designed using residence time and nonequilibrium properties. Optimal design methods were developed using a genetic algorithm for feedback and redesign, and the combustor and gas generator were op-timally designed for three classes of engines: 30 tonf, 75 tonf, and 150 tonf of engine thrust.

For the combustor, design wall temperatures satisfied the 800-K requirement, and specific impulses converged to maximum values. Thus, the combustor profiles and the dimensions of the cooling channel were well determined compared to previous nonoptimized results. For the gas-generator design, empirical data for the required tur-bine power and the 1000-K material limitation were as-sumed as optimal points, and the results agreed with these requirements. It is expected that this design ap-proach can be used to design an entire engine system that

Fig. 9 Combustor profiles for 30-tonf, 75-tonf, and 150-tonf engines

266 J. Therm. Sci., Vol.23, No.3, 2014

Table 5 Design parameters for optimal design of fuel-rich gas generator


Assumed design parameter

Gas-generator chamber pressure (bar) 60 60 60

Mass flow rate of main combustor (kg/s) 96.08 240.44 480.53

Pressure ratio of turbine 16

Contraction ratio 10 10 10

Initial diameter of fuel droplet ( m ) 50 50 50

Initial temperature of fuel droplet (K) 300 300 300

Initial velocity of fuel droplet (m/s) 50 50 50

Requirements

Turbine inlet temperature (K) 1000 1000 1000

Required turbine power (kW) 1122.60 2464.31 4700.49

Fig. 10 Values of objective function for gas-generator design at iterations of optimization

Fig. 11 Burned-gas temperatures (dots) at different O/F ratios and optimal values (stars) for 30-tonf, 75-tonf, and 150-tonf engines

Fig. 12 Turbine powers (dots) with respect to mass flow rates

of propellants and optimal values (stars) for 30-tonf, 75-tonf, and 150-tonf engines

Fig. 13 Gas-generator profiles for 30-tonf, 75-tonf, and 150-tonf engines.


Table 6 Optimized design results for fuel-rich gas generator


O/F ratio 0.273 0.273 0.273 Input parameters

Mass flow ratio of gas generator to main combustor 0.0375 0.0329 0.0314

Turbine inlet temperature (K) 1000.21 1000.00 1000.00

Turbine power (kW) 1122.56 2464.30 4700.36

Length (m) 0.599 0.604 0.607

Chamber diameter (m) 0.075 0.111 0.153

Design parameters


combines design modules for other components, includ-ing turbopump, turbine, and feeding systems.

Acknowledgement

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean Government (MSIP) (NRF-2012M1A3A3A02033146) and (NRF-2013M1A3A3A02042434).

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Documents

Genetic algorithm to optimize the design of main combustor and gas generator in liquid rocket engines