Genetic Algorithm to Optimize the Design of Main Combustor and Gas ...

Journal of Thermal Science Vol.23, No.3 (2014) 259268

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 Generator 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 derived from a counterflow analysis, and a vaporization model for the fuel droplet was adopted to calculate residence 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 superior 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 preliminary 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 temperature 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 turbine 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

Received: September 2013 Jaye Koo: Professor This work was supported by the National Research Foundation of Korea grant funded by the Korean Government (MSIP) NRF2012M1A3A3A02033146 and NRF-2013M1A3A3A02042434 www.springerlink.com

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Nomenclature AR Aspect ratio of cooling channel cp Specific heat at constant pressure (J/kg-K) Characteristic velocity (m/s) c* G Mass flux (kg/s-m2) h Heat transfer coefficient, W/m2 Specific impulse (s) Isp L Power, (W) Emission length (m) Le L* Characteristic length of combustor (m) Ratio of mass flow rate mr  m Mass flow rate (kg/s) N Number of cooling channels OF Oxygen/fuel ratio P Pressure (Pa) Pressure ratio at turbine Pr q Heat flux (W/m2) T Temperature (K) tchem Chemical reaction time (s) Residence time in combustor (s) tres this study, a simple genetic algorithm was adopted to achieve optimization and results were compared with nonoptimized 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 results of Cho [4]. Chemical equilibrium was assumed for combustion in the main combustor. A chemical equilibrium code called chemical equilibrium application (CEA), originally developed by the National Aeronautics and Space Administration (NASA), was used [6]. Thermodynamic properties of the equilibrium mixture can be calculated from free energy minimization, and a simple analysis of rocket performance is available. To handle propellant 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 approximation proposed by Rao [8]. Heat transfer analysis and cooling channel design for regenerative cooling A regenerative cooling system was adopted for combustor cooling. In general, cooling channels have complicated structures, including bifurcated branches and

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

tvap w1 , w2 , w3

Vaporization time of droplet (s) Weighting factors Greek symbols Specific heat ratio  Efficiency  Density, (kg/m3)  Subscripts aw Adiabatic wall c Coolant side co Coolant cond Conductive heat transfer conv Convective heat transfer Carbon dioxide CO2 g Burned-gas side gg Gas generator H2O Water o Overall rad Radiative heat transfer turb Turbine w Wall variable cross sections, but in this study the channel shape was simplified. The channels have rectangular cross sections with constant heights and widths. In addition, 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 simply modeled one-dimensionally, as in Fig. 1. At first, heat generated from the burning gas is transferred to the wall through a boundary layer and a deposited carbon layer. Convective heat transfer is defined as



q g ,conv  hg Tg  Tw, g

Fig. 1



(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

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]. q g ,rad  qrad ,CO2  qrad , H 2O (2)  T 3.5  T 3.5  w, g qrad ,CO2  3.5 3 PCO2 Le  aw      100  100      3.5 3.5   Tw, g   0.8 0.6  Taw  qrad , H 2O  3.5PH 2O Le     100    100     

(3)

(4)

Finally, the heat flux from the hot gas to the wall was determined by (5) q g , o  q g , conv  q g , rad 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 friction 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 completed in the chamber before the gas exits to the turbine. Thus, the length of the gas generator has to be determined to allow the propellants to undergo both steps. The total time required defines the residence time, (6) tres  tchem  tvap From this definition of residence time, a characteristic length of the combustion chamber is defined by  Pgg  tres  L*   (7)    c *gg   gg     Curves fitted from the chemical non-equilibrium analysis were used to obtain burned-gas properties and chemical reaction time. A vaporization model for a kerosene droplet was used to calculate the vaporization time. Finally, the residence time was used to estimate the

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characteristic length, and a conceptual profile was determined.

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 considered in the Spalding model [18,19]. The actual process of combustion progresses in a supercritical 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 oxidizer-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. Foelsche [22] used the perfectly stirred reactor (PSR) model, which is a chemical nonequilibrium analysis for a premixed kerosene-rich condition. A droplet vaporization model presented by Spalding was adopted, and the results were similar to experiment results. Yu [23] used Foelsche’s method with a reaction mechanism from Dagaut [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 sufficient 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 gasgenerator 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.

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Fig. 3

Schematic of premixed counterflow flame

Properties of kerosene fuel, which is liquid in its standard state, can be calculated from an equation of state (EoS) that applies to supercritical fluids. In the supercritical 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 nonderivative-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 differentiability 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 genetic 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 reproduced variables undergo a crossover process using the modified simple crossover method [28]. Next the variables mutate using the dynamic mutation method developed by Janikow [33] and Michalewicz [34] to achieve detailed adjustments. Using the design method, the solution is newly calculated, and an elite strategy is performed 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 probability, 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

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

optimum values. The objective function should be formulated to find maximum or minimum values of performance variables. In combustor design, there are three objectives. First, the maximum specific impulse is required. Second, the proper wall temperature, which is below the material limitation, is sought. Third, the pressure 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 combustor design

263

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.  gg 1     1   gg  Lturb  m gg c p , ggturbTgg  1    (8)    Pr    

Objective function (minimization):

F  OF , AR, N , G   w1 I sp  500  w2 Tw  Treq  w3 1 

Design variable and constraint

Pc Pco

2.2  OF  2.8 1.0  AR  4.0 50  N  200 0.01  G  0.03

In this study, 0.1, 1.0, and 0.1 were used as the weighting factors. A mixture ratio (O/F ratio), aspect ratio 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 pressure, 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 circumstances. 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 mixture 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 constraints such as pump performance. The objective function 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 Tg Tg ,req

 w2 1 

Design variable and constraint

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 design optimization of the gas generator as per the required thrust. The parameters were the same as those of a previous 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 regenerative-cooled combustor Engine class (tonf)

Lturb Lturb ,req

0.25  AR  0.5 0.01  mr  0.1

30

75

Assumed design parameters Combustor pressure (bar)

60

Inner wall thickness (mm)

1.0

Nozzle exit pressure (bar)

0.45

Injector pressure drop (bar)

Objective function (minimization): F  OF , mr   w1 1 

Fig. 5

12

Requirements Specific impulse (s) Maximum gas-side wall temperature (K) Pressure drop in cooling channel (bar)

Maximum 800 Minimum

150

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

as shown in Fig. 7, but the optimal values did not converge perfectly to maximum values because of low sensitivity near the maxima. In Fig. 8, the population moved to the required wall temperature and the minimum pressure drop. Optimized results for the main combustor are summarized in Table 4. For the nonoptimized design, input parameters were fixed to constant conditions regardless of the engine class.

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

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 150tonf 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 optimized 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 decreased to about 800 K, which is the design requirement. Pressure decrease in the cooling channels dramatically

Design optimization of fuel-rich gas generator The gas generator was designed using a genetic algorithm 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 assumed 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 converged 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 required 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 simultaneously analyzed using one-dimensional modeling. The

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

265

Table 4 Comparison of optimized and nonoptimized design results for the main combustor Engine class (tonf)

30

75

150

Non-optimized results Input parameters

O/F ratio of propellant

2.55

Aspect ratio of cooling channel

2.0

Number of cooling channels

100 2

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

Design parameters

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

Height of cooling channel (mm)

4.27

6.75

9.55

O/F ratio of propellant

2.61

2.58

2.73

Aspect ratio of cooling channel

3.44

3.09

3.10

79.07

122.17

158.85

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

0.0182

0.0191

0.0205

Mass flow rate of propellant (kg/s)

96.08

240.44

480.53

Optimized results Input parameters

Number of cooling channels 2

Design parameters

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

Nozzle throat diameter (m)

0.189

0.299

0.423

Expansion ratio

16.815

16.544

17.165

Width of cooling channel (mm)

2.33

3.05

3.58

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 optimally 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 turbine power and the 1000-K material limitation were assumed as optimal points, and the results agreed with these requirements. It is expected that this design approach can be used to design an entire engine system that

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

266 Table 5

J. Therm. Sci., Vol.23, No.3, 2014 Design parameters for optimal design of fuel-rich gas generator Engine class (tonf)

30

75

150

60

60

60

96.08

240.44

480.53

Assumed design parameter Gas-generator chamber pressure (bar) Mass flow rate of main combustor (kg/s) 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

Turbine inlet temperature (K)

1000

1000

1000

Required turbine power (kW)

1122.60

2464.31

4700.49

Requirements

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.

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

267

Table 6 Optimized design results for fuel-rich gas generator Engine class (tonf) Input parameters

Design parameters

30

75

150

O/F ratio

0.273

0.273

0.273

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

Nozzle throat diameter (m)

0.024

0.035

0.048

combines design modules for other components, including 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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