Cost Effectiveness Optimization of Component ...

Proceedings of the ASME 2015 International Mechanical Engineering Congress and Exposition IMECE2015 November 13-19, 2015, Houston, Texas

IMECE2015-52567

COST EFFECTIVENESS OPTIMIZATION OF COMPONENT IMPROVEMENTS OF A TURBOFAN ENGINE USING GENETIC ALGORITHM

Jason Yobby Southern Illinois University at Edwardsville Edwardsville, Illinois, USA [email protected]

Daniel S. Raja Southern Illinois University at Edwardsville Edwardsville, Illinois, USA [email protected]

Terry X. Yan Department of Mechanical Engineering Southern Illinois University at Edwardsville Edwardsville, Illinois, USA [email protected] thrust and specific fuel consumption) and eight degrees of freedom (i.e. diffuser pressure ratio, fan polytropic efficiency, compressor polytropic efficiency, burner efficiency, burner pressure drop ratio, turbine polytropic efficiency, fan duct exit nozzle pressure ratio, core duct exit nozzle pressure ratio) is employed. As a result, we have found the lowest overall cost associated with the improvements for each of the components.

ABSTRACT In this paper, we examined an imaginary underperforming prototype of a real high bypass-ratio turbofan gas turbine engine that has been assembled to specifications. The prototype was designed and assembled to generate a predetermined value of specific thrust while consuming fuel at a predefined specific fuel consumption value. To meet the required performance targets, improvements needed to be made to one or more of the engine components. In most real world scenarios, the improvement of any or all engine parameters pertaining to its performance is tied to a cost per percentages of improvements of individual component in the engine. There is always a narrow room for improvement in each or all of the components. However, the improvements come with high costs, since the engine has been designed in an efficient way to begin with. The cost of improvement of each component is indexed by a dollar cost per percent value of the component performance characteristics. It is of technical and economic importance to find a combination of performance improvements of each of the components that yields the lowest overall rework cost, thereby the total design cost, given the specified engine performance criteria. To achieve this goal, simulations for a real gas turbine turbofan cycle are performed in conjunction with the genetic algorithm (GA). A single objective (i.e. total improvement cost) GA with two constraints (i.e. desired values of specific

NOMENCLATURE ao 𝐶𝑝𝑐 𝐶𝑝𝑡 ec e′c et f FA g0 h M0 ṁc

1

Speed of sound at required altitude and temperature Specific heat upstream of compressor Specific heat capacity of fluid downstream of the turbine Polytropic efficiency of compressor Polytropic efficiency of fan Polytropic efficiency of turbine Fuel – air ratio Thrust generated by the engine Acceleration due to gravity Fuel heating value Flight Mach number Mass flow rate through the core section of the engine

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ṁF P0 P9 P9′ Pt9 Pt9′ Rc S T0 T9 T9′ 𝑇𝑡4 u0 u9 u′9

Mass flow rate through the fan section of the engine Ambient Pressure upstream of engine Static pressure of fluid leaving the core nozzle Static pressure of fluid leaving the fan nozzle Stagnation pressure of fluid leaving the core nozzle Stagnation pressure of fluid leaving the fan nozzle Gas constant Thrust specific fuel consumption Ambient temperature Static temperature of the fluid leaving the core nozzle Static temperature of the fluid leaving the fan nozzle Turbine inlet temperature Velocity of the air entering the inlet diffuser Velocity of the air leaving the core nozzle Velocity of the air leaving the fan nozzle

engineering analysis points of view, is to complete a thorough survey of each component fabrication process and identify the possible room for improvement in each component and the cost associated with it. Since modern gas turbine engines are already better designed and manufactured using the state of art technologies, the margin of improvement is consequently small and costly. The key to resolve this issue is to find the improvement in parameters that could yield the least rework cost. The sooner these values are found, the sooner the engine could go into production, thereby preventing a delay in the development of the engine. This combination of improved parameters will improve the performance of the underperforming engine to the desired value of specific thrust and specific fuel consumption. Optimization through improvement of multiple components, through redesign and rework becomes increasingly important. Genetic algorithm, particle swarm optimization, ant colony optimization and simulated annealing are all viable methodologies to accomplish this task. The GA was selected for this optimization because of its ability to deal with a large number of variables, simultaneously search a wide sampling of the cost area, provide a list of optimum variables, not just a single solution (if needed), and not require derivative information thus making it a good fit to use for this optimization purpose. The GA is a robust search technique that mimics nature’s evolution according to Darwin’s laws for natural selection. It uses a series of steps that allow an optimal solution to be found. The first step is to define the cost function (the actual function to be optimized), cost (the value used to rank the solution sets), and variables and to select the constraints for the GA. A random initial population of a set (e.g. 100) size is then created. The solution sets would then be ranked and, based on eliteness, the sets with the highest fitness values will be kept while the remaining solutions are discarded. The crossover fraction determines the percentage of mating and mutation used to refill the population back to the original size. Mating uses a crossing of two of the best kept answers to refill the population while mutations copy one of the kept answers and slightly changes one of the variables in the solution set. After mating and mutation, a convergence check is conducted. If the optimization solution is within the tolerance level or the maximum number of generations is exceeded, the GA is complete, if not it will repeat the process. A plan of improvement combination of all the components, with a given component improvement cost index, was found as a result that yielded the least cost. The results obtained from this computation of a zero dimensional thermodynamic model successfully demonstrated the viability of GA method applied to practical problem with multiple degree of freedom and with multiple constraints. With genetic algorithms gaining widespread use for optimization, there is a large amount of literature of GA in engineering application. One of the closely related literature included a system optimization of a turbofan engine via GA by Homaifar et al. [1]. In their paper, the GA was implemented on a high bypass turbofan engine. The system was optimized for four cases, thrust per unit mass flow rate (TMA) with subsonic Mach number, overall efficiency with subsonic Mach number, TMA and overall efficiency combined with subsonic Mach

GREEK SYMBOLS 𝛼 γc γ′c γt ηb 𝜂𝑚 πc π′c πr πb πd πn π′n 𝜏𝜆 τr τc τ′c

Bypass ratio Adiabatic index upstream of the compressor Adiabatic index of the air upstream of the fan Adiabatic index downstream of the turbine Burner efficiency Mechanical efficiency Stagnation pressure ratio just after and just before the compressor Stagnation pressure ratio just after and just before the fan Reference stagnation pressure ratio Stagnation pressure ratio just after and just before the burner Stagnation pressure ratio just after and just before the diffuser (inlet) Stagnation pressure ratio just after and just before the core nozzle Stagnation pressure ratio just after and just before the fan nozzle Design limitation parameter (Ratio of Turbine Inlet Stagnation enthalpy to ambient stagnation enthalpy) Reference temperature ratio Compressor temperature ratio Fan temperature ratio

INTRODUCTION Modern aviation employs high bypass ratio turbofan gas turbine engines that generate a large amount of thrust at an optimized value of fuel consumption. Engine manufacturers invest large sums of money in designing and fabricating these engines. Should a prototype engine of a new design underperform in terms of specific thrust and specific fuel consumption when tested, an improvement program is called to augment the performance through component improvements. The individual cost in the form of rework of each component depends on many factors such as reorganization, redesign and retooling, etc. A systematic approach, from both business and

2



Figure 1 shows the various stages of a high bypass turbofan engine as per AIAA designations. This figure was taken from the Gordon C. Oates’ book [4]. The region between 0 and 1, is the upstream section of the fan and core sections (engine). The region between 1 and 2 is the inlet (diffuser), the region between 2 and 3’ is the fan, and that between 2 and 3 is the compressor. The region between 3 and 4 is the burner. The region between 4 and 5 is the turbine. The region between 6 and 7 is the afterburner. The region between 7 and 8 is the core nozzle. The region between 6’ and 7’ is the afterburner in the fan section of the engine. The region between 7’ and 9’ is the nozzle of the fan section. Note that in this paper, the analysis is performed on a non-afterburning high bypass real turbofan engine. Therefore it can be assumed that regions between 6 and 7 and 6’ and 7’ are thermodynamically and mechanically identical. The performance evaluation equations for a real high bypass turbofan gas turbine engines differ from those of an ideal engine due to the following factors. 1. Pressure ratio in the inlet is less than one, due to the presence of wall friction and flow separations 2. Compressors, fans and turbines efficiencies are always less than 100%, since the process is not isentropic 3. Combustion in the burner does not occur at constant pressure, and the efficiency of the combustion process is less than 100% due to heat losses 4. Pressure ratio in the nozzles is less than one, due to over- or under expansion and boat- trail losses 5. The efficiency of all mechanical components is less than 100%

number, and TMA and overall efficiency combined with supersonic Mach number. The optimization of gas turbine engines is discussed by Guha [2]. Here he explains the difficulty of designing a turbine engine for certain parameter criteria when optimizing that criteria affects the performance of other critical parameters. He clearly breaks each component optimization down and then explains the effects of that optimization on other critical parameters to design the system. He then shows a methodology flowchart on how to optimize the system for the lowest specific fuel consumption while retaining other critical flight parameters, including optimal fan pressure, specific thrust, cost, weight, and noise. Casalino and Pastrone [3] optimized a high bypass turbofan using three different optimization methods in parallel. A hybrid technique that involved the use of genetic algorithm, differential evolution, particle swarm optimization and cooperative algorithm was used to optimize their system. They reported that the synergy of the three processes reduced the computation time significantly. In this paper, we are trying to find a combination of engine parameter improvements to restore the performance of an imaginary underperforming high bypass turbofan engine to its designed and desired values. The parameters considered for improvement are those associated with a “real” engine, in other words the effects of temperature loss, pressure loss, friction, mechanical loss are taken into account. It is worth mentioning that the parameters considered for the optimization are directly associated with the component of the engine itself and not the operating conditions of the engine. Therefore, the engine can still be used at the altitude at which it was initially designed for, and at the speed it was designed for, while carrying the designed payload. Also for obvious reasons, these improvements must be favored economically. To do this, a single objective GA with dual constrains was employed. Therefore, the aim of this paper is to find a set of improvement values of engine parameters that yields the lowest cost using GAs and show the potential of using GA in the aviation industries.

EQUATIONS AND PERFORMANCE AND OPERATION PARAMETERS The ratios of stagnation pressure 𝝅 Eqn. 1 and stagnation temperature 𝝉 Eqn. 2 were useful to solve the problems related to gas turbines. This concept was taken from the Oates book [4]. 𝛑=

TURBOFAN ENGINE DESIGN A turbofan engine is effectively a turbojet engine with a larger fan area that divides the incoming airflow into two streams. The “hot” stream flows into the core and is burned with fuel. The “cold” stream bypasses the engine and is compressed in the bypass stage. A modern high bypass turbofan gas turbine engine comprises of 5 key subcomponents. These include a diffuser (inlet), a low pressure compressor (fan or bypass compressor), a high pressure compressor (compressor), a burner (combustor or flame holder), a turbine, and nozzles (fan and core). In this analysis, we assume the engine exhausts to be subsonic and that it has separate exhausts for the fan and core streams. These two streams result in two different exit pressure ratios and exit velocities.

𝐬𝐭𝐚𝐠𝐧𝐚𝐭𝐢𝐨𝐧 𝐩𝐫𝐞𝐬𝐬𝐮𝐫𝐞 𝐥𝐞𝐚𝐯𝐢𝐧𝐠 𝐜𝐨𝐦𝐩𝐨𝐧𝐞𝐧𝐭 𝐬𝐭𝐚𝐠𝐧𝐚𝐭𝐢𝐨𝐧 𝐩𝐫𝐞𝐬𝐬𝐮𝐫𝐞 𝐞𝐧𝐭𝐞𝐫𝐢𝐧𝐠 𝐜𝐨𝐦𝐩𝐨𝐧𝐞𝐧𝐭

𝛕 𝐬𝐭𝐚𝐠𝐧𝐚𝐭𝐢𝐨𝐧 𝐭𝐞𝐦𝐩𝐞𝐫𝐚𝐭𝐮𝐫𝐞 𝐥𝐞𝐚𝐯𝐢𝐧𝐠 𝐜𝐨𝐦𝐩𝐨𝐧𝐞𝐧𝐭 = 𝐬𝐭𝐚𝐠𝐧𝐚𝐭𝐢𝐨𝐧 𝐭𝐞𝐦𝐩𝐞𝐫𝐚𝐭𝐮𝐫𝐞 𝐞𝐧𝐭𝐞𝐫𝐢𝐧𝐠 𝐜𝐨𝐦𝐩𝐨𝐧𝐞𝐧𝐭

(1 ) (2 )

The following equations constitute the operating and design parameters of a real high bypass non-afterburning turbofan engine with separate exhaust. They were taken from the book authored by Gordon C. Oates [4] and were used in this computation of a high bypass turbofan engine with separate exhaust. γc − 1 Rc = ( ) Cpc γc

(3)

a o = √γc R c T0

(4)

Eqn. 3 and 4 help to find the speed of sound at the cruising altitude of the aircraft. R c is the universal gas constant of the air at cruising altitude, and a o is the corresponding speed of sound.

FIGURE 1: STAGES OF A HIGH BYPASS TURBOFAN ENGINE [4]

3



γt

The subscript ‘C’ is to denote that the property of air that enters the compressor of the engine. In should be noted that the physical properties of the air that enters the compressor is assumed to be equal to those upstream of the engine. T0 is the ambient temperature of the air at cruising altitude.

Since in most subsonic transport application of separate stream turbofan engines, the pressure ratio across both the primary and secondary nozzles is not very large. Often, only convergent nozzles are employed. These terms account for the choking phenomenon in the secondary nozzle and the primary nozzle as expressed in Eqn. 13 and 14 respectively.

𝑇𝑡 γc − 1 (5) τr = 1 + ( ) ∗ Mo2 = 0 2 𝑇0 γc (6) 𝑃𝑡 γ −1 πr = τ r c = 0 𝑃0 τr and πr expressed in Eqns. 5 and 6 are exceptions to the above defined τ and π in Eqns. 1 and 2. The subscript ‘R’ denotes the free stream reference ratios of temperature and pressure respectively. It can be considered as the representation of the effects of the flight Mach number M0 γc −1 γ ∗ec

(7)

γc −1 γc ∗e′c π′c

(8)

τ c = πc c τ′c =

𝐶𝑝𝑡 𝑇𝑡4 𝐶𝑃𝑐 𝑇0

πt =

Cpc ′ T9′ ( Cpt ) τr τc = γ′c −1 T0 Pt9′ γ′c ( ′) P9

(17)

Cpc T9 ( Cpt ) τλ τr = γt −1 T0 Pt9 γt ( ) P9

(18)

(10) M0

γt −1 Pt9 γt

u9 2 (τ τ ) (1 − ( ) =( u0 γc − 1 λ r P9

M0

(11)

u′9 = u0

))

(

−(

𝜏𝑡 and πt are the stagnation temperature ratio and stagnation pressure of the turbine respectively, as expressed in Equations 11 and 12. 𝜂𝑚 is the mechanical efficiency of the shaft that connects the turbine to the compressor and fan. ‘𝛼’ is the bypass ratio of the engine. 𝑒𝑡 is the polytropic efficiency of the turbine. γc

(19)

2 (τ τ′ ) (1 γc − 1 r c

(12)

Pt9′ γc + 1 γc −1 =( ) P9′ 2

1 2

(20)

+ 𝛼(𝜏𝑐′ − 1)] γt (γt −1)et τt

(16)

Eqn. 17 and 18 express the static temperature ratio of the air exiting the fan duct to the air entering the engine and the air exiting the core duct to the air entering the engine respectively.

‘f’ is the fuel – air ratio and is expressed in Eqn. 10. ‘h’ is fuel heating value or calorific value of the fuel and ηb is the efficiency of the burner. 1 𝜏𝑟 𝜏𝑡 = 1 − ( ) ( ) [(𝜏𝑐 − 1) 𝜂𝑚 (1 + 𝑓) 𝜏𝜆

γt

Eqn. 15 and 16 express the static pressure ratio of the air entering the engine to the air leaving the engine in the secondary duct and the primary duct respectively. These pressure ratio also account for the choking of the flows in their respective ducts.

(9)

τλ − τr τc hηb − τλ Cpc T0

(15)

γ + 1 γt−1 ( t ) P0 2 = P9 πr πd πc πb πt πn

Another exception is 𝜏𝜆 expressed in Eqn. 9. However, it is appropriate to include a design limitation. In this paper, that limitation is the maximum allowable turbine inlet stagnation enthalpy 𝐶𝑝𝑡 𝑇𝑡4 which is manifested in 𝜏𝜆 . f=

γc

γc + 1 γc −1 ) P0 2 = P9′ πr πd π′c π′n (

τc and τ′c expressed in Eqns. 7 and 8 and are the stagnation temperature ratios of the compressor and fan respectively. Likewise 𝜋𝐶 and 𝜋𝐶′ are the compressor and fan stagnation pressure ratios. In this paper, these parameters are considered to be design parameters. 𝑒𝐶 and 𝑒𝑐′ are the polytropic efficiency of the compressor and fan respectively. 𝜏𝜆 =

(14)

Pt9 γt + 1 γt −1 =( ) P9 2

Pt9′ ) P9′

γ′c−1 γ′c

1 2

) )

Eqn. 19 and 20 account for the specific thrust lost by virtue of irreversibility’s in the engine and inertia of the components that engine make up the engine. Eqn. 19 accounts for the loss in the core duct and Eqn. 20 accounts for the fan duct.

(13)

4



operation, i.e., close to those values at which the underperforming engine operates.

FA 1 u9 = 𝑎0 ( ) (1 + f) (M0 ) (ṁc + ṁF ) 1+α u0

Mach No. Altitude Ambient Air Temperature Ambient Air Pressure Ambient Air Density Fuel Heating Value Upstream Specific Heat Capacity Downstream Specific Heat Capacity ϒc ϒt

{

− M0 + (1 ( + f) (

−

1

T9 u9 ) T0 (1 γC [M0 ] u0

P0 ) P9 )

Engine Performance / Output

′

u9 − M0 u0

Specific Fuel Consumption ((mg /s)/N) Specific Thrust (N/(kg/s))

[

( −

T9′ (1 u′ T γC [M0 9 ] 0 u0 1

+

Desired or Designed Value

Tested Value or Obtained Value

197.1

191.3

TABLE 2: ENGINE OUTPUT VALUES

P0 ) P9′ )]}

Eqn. 21 calculates the specific thrust of a non-ideal (real) turbofan engine without afterburning in the core as well as the fan duct. f (106 ) S= FA (1 + α) [ ] (ṁc + ṁF )

1.4 1.3

TABLE 1: OPERATING FLIGHT CONDITIONS

(21)

+ α M0

0.87 11.6 km 216.7 K 20.7 kPa 0.335 kg/m3 43031 kJ/kg 1.005 kJ/(kg K) 1.245 kJ/(kg K)

(22)

Eqn. 22 calculates the thrust specific fuel consumption of the engine, which is represented as ‘S’. The flight condition of the airplane fitted with this engine i.e., the operating conditions of the engine is shown in Table 1. The speed and altitude was arbitrarily chosen. The properties of air mentioned in the table correspond to the properties of air at the altitude that the airplane is flying at. These values were taken from Appendix A on page 435 through to 437 of the book authored by Gordon C. Oates [4]. The values of the engine output that are below par when compared to the desired values are shown in the second column of Table 2, along with the desired or required engine output value as shown in the first column. The parameters and properties of the underachieving engine are shown in Table 3. A list of possible improvement percentages of components and the associated cost per percent is shown in Table 4. As stated earlier in the paper, the room for improvement is very narrow, thereby giving us a very small “wiggle” room. These values are arbitrarily set, but care was taken to remain as close as possible to the original value of

5

Turbine inlet temperature

1645 K

Compressor Pressure Ratio

26

Fan pressure Ratio

2

Bypass ratio

7

Diffuser Pressure Ratio

0.97

Nozzle Pressure Ratio

0.98

Burner Pressure Ratio

0.96

Mechanical Efficiency

0.99

Burner Efficiency

0.98

Fan Efficiency

0.9

Compressor Efficiency

0.9

Turbine Efficiency

0.87

Specific Thrust

191.3 (N/(kg/s))

Specific Fuel Consumption

21.87 ((mg/s)/N)



TABLE 3: UNDERACHIEVING ENGINE CHARACTERISTICS

Component Diffuser Pressure Ratio (πd) Fan Efficiency (ηc') Compressor Efficiency (ηc) Burner Efficiency (ηb ) Burner Pressure Ratio (π b ) Turbine Efficiency (ηt ) Core Nozzle Pressure Ratio (π n ) Fan Nozzle Pressure Ratio (π n ')

% Possible Cost ($M) per Improvement % Improvement 1

18

1.5

12

1.5

14

1

21

1

21

2

20

1

15

1

15

TABLE 4: IMPROVEMENT PERCENTAGE AND COST

GA MODEL OF PARAMETERS Genetic algorithms are based on Darwin’s laws for natural selection. Haupt, et al, state there are four main concepts that make up these laws. They are a) offspring inheriting multiple traits of the parents, b) variations of these traits being passed down through the generations, c) a small percentage of offspring surviving into adulthood, and d) offspring survival depending on the inherited traits (survival of the fittest) [5]. Understanding how a GA works requires some knowledge of terminology and genetics. First is the gene, which is the unit of heredity, and then the chromosome, which is a pair of genes in the form of DNA. Each parent will contribute one gene to be combined with the other parent’s gene to create the offspring’s chromosome. The genetic code is the sequence of the DNA that make up the offspring’s chromosomes. The crossover point is where each parent’s genes will contribute to the offspring’s DNA. Mutations are small random changes in the characteristics of a gene. Finally, the population is the group of interbreeding individuals. The GA uses a specific set of steps according to Darwin’s laws. GA’s are broken down into three types, binary, continuous, and a combination of both. Binary GA’s are used when the variables for the algorithm are whole numbers and whole number answers are required (e.g. number of fins, number of engines, number of connections, etc.), while continuous GA’s are used for finding values for fractional or percentage variables. A combination type of GA combines both binary and continuous if a whole value and a percentage is to be used, (i.e. number of fins plus a bore to wall thickness ratio is to be found). These are also called Double Vector (continuous, Bit String (binary), and Hybrid (combination) populations.

FIGURE 2: GA FLOWCHART

The process of the genetic algorithm with constraints can be seen in Figure 2. The first step in the GA is to define the cost function, the cost, and variables. The GA parameters and constraints are then selected. Next, the initial population is randomly selected followed by finding the cost of each chromosome and ranking them in order. From here, the mating selection is performed and then the mating and mutation process occurs. A constraint check is performed during the mating and mutation and after the population is refilled a convergence check is performed. If the function converges within the specified tolerances or the maximum number of generations is exceeded, the GA is complete, otherwise the GA will return to the cost ranking of the chromosomes and repeat the process.

6

COST FUNCTION The cost function is the program written to determine the cost of the variables we are trying to optimize. The cost is the actual minimization function. In this case, the cost function was that of actual dollar cost of improvement for the turbofan engine in millions. The design parameters used were cost per percentage of improvement for eight component variables. Those percentages, percent of possible improvement, and cost per improvement percentage are shown in Table 4. The maximum cost, when all components were maximized, would be a total of Copyright © 2015 by ASME


$169M resulting in a specific thrust of 20.16 lbf/lbm/s and a fuel consumption of 0.7434 lbm fuel/h/lbf thrust. Table 2 shown the tested values and the desired constraints that are needed for improvement. The deficiencies can be seen in the specific fuel consumption and specific thrust causing the need for component improvement.

Crossover Function Child

0

1

0

1

1

0

0

1

A 2 C

4

5 F G

8

TABLE 5: SCATTERED CROSSOVER SAMPLE

POPULATION SIZE AND INITIAL POPULATION For the GA, the population size was set to 100. This will be the maximum size of the number of sets of variables the GA will use to determine fitness. The initial population was also 100 so that the GA started out with a full population. A double vector population type was used because a continuous GA is better suited than a binary GA in this case.

CONSTRAINT CHECK During the mating and mutation processes, a constraint check is performed on the new offspring. If the values are within the constrained boundaries, they will be kept as offspring for the new generation. If the values are not within the constraints, the values will be thrown out and mating or mutation will occur again. The two constraints in the current analysis are the thrust per mass flow and the specific fuel consumption. This will happen until the population is full from the mutations or mating.

CHROMOSOME COST AND SCALING The chromosome cost, or fitness, is the value given to each chromosome in order to rank them. The setting of Rank type was used for the fitness scaling function. Scaling converts the raw scores into values the GA is able to handle. Rank scaling removes the spread effect of the raw scores.

STOPPING CRITERIA The stopping criteria check for the GA to quit running when the cost function is met. There were different factors set to stop the GA. These factors were if the solution converged to within a specified tolerance (function tolerance), constraint tolerance, number of generations and stall generations. The values for these criteria are set at, respectively, 1E-50, 1E-50, 100, and 50. The value that always stopped this GA function was the function tolerance, or the converging of the solution to a specified tolerance of 1E-50.

MATE SELECTION The method used to select individuals for mating was set to stochastic uniform. This is the choosing of parent genes to refill the population with. An elite count of 2 was used, which keeps the top two answers and discards the rest of the population to make room for repopulation. CROSSOVER FRACTION AND MUTATION The crossover fraction was chosen as 0.4. This fraction determines what percentage of the repopulation group will be from mating or from mutations. The 0.4 represents that 40% of the repopulation will be from mating and 60% will be from mutations. Mutations are a small change in a copy of one of the parent’s genes. The mutations were constraint dependent which chooses if the mutations are Gaussian or adaptive feasible. This particular problem called for adaptive feasible because of the constraints placed on the specific thrust and specific fuel consumption.

SELECTION OF GA CRITERIA For the GA used in the turbofan improvement cost analysis, a population of 100 was used. Our GA eliteness was kept at 2 resulting in keeping only 2% of the population. The crossover fraction was set to 0.4 in this study. The settings for both the tolerance function and constraint function of 1E-50 were selected. RESULTS The GA was set in a loop to run 900 times and write the values to a tab delimited .txt file for the values of the component percentage of improvements, the cost in $M for the improvement, the number of generations, and the computation time for each run. The average and standard deviation for these values are given in Table 6. From here the cost is $130.58M for improvements of each component, represented by x (i), (i = 1 to 8). The standard deviation for the cost was $186000 for the project cost. The GA algorithm is performed with Matlab’s global optimization. The algorithm ran between 10 and 11 generations in Matlab R2009A on average with an average computation time of between 37 and 38 seconds on an Acer Aspire laptop with an AMD A8-4500M 1.90 GHz quad core APU, 8.00 GB of RAM, running Windows 8.1, 64-bit OS.

CROSSOVER (MATING) Crossover is how the new genes will be produced from combining the parents’ genes to refill the population. The GA uses the scattered type of mating. In this type of mating, the GA creates a random binary string. It assigns one of the two parents to the 0’s and the other parent to the 1’s. The resultant gene will be a combination of both parents’ genes based on the genetic value dictated by the binary string. In this case, the binary string will be 8 bits long, one bit for each variable. This is demonstrated in Table 5. Haupt, et al, shows that parent 1 is assigned to 0’s and parent 2 is assigned to 1’s. The resultant offspring would then have a mix of variables from both parents. [5]

Parent 1

A B C D E F G H

Parent 2

1

2

3

4

5

6

7

8 7



Optimizing Parameter

Average

Std. Deviation

Diffuser pressure ratio

x(1)

0.9797

2.25E-06

Burner pressure ratio

x(2)

0.96002

2.07E-05

Burner efficiency

x(3)

0.9898

1.84E-06

0.98925

1.03E-03

0.9898

2.88E-06

0.90285

1.14E-03

0.91349

1.03E-05

0.8874

4.57E-06

130.58 10.3 37.63

0.186 1.44 5.846

Core nozzle pressure x(4) ratio Fan nozzle pressure x(5) ratio Compressor polytropic x(6) efficiency Fan polytropic x(7) efficiency Turbine polytropic x(8) efficiency Cost ($Million) No. of Generations Computation Time (s)

FIGURE 4: AVERAGE DISTANCE BETWEEN INDIVIDUALS

TABLE 6: GA RESULTS

The best and mean values of the fitness function can be seen in Figure 3. The best score is the fitness score of the best individual in each generation while the mean score is the average of each generation’s scores for the fitness function. Figure 4 is the average distance between individuals of each generation’s population. The best, worst and mean scores are represented in Figure 5. FIGURE 5: BEST, WORST AND MEAN SCORES

The values of the specific thrust generated and the specific fuel consumption of the engine using the values of the optimized parameters obtained by the GA (from Tab.6) are in concurrence with those obtained by the Engine Performance Software Oates provided along with the book authored by Gordon C. Oates [4] for the same set of input values. The numerical values differ slightly due to the limitations of software bits during the development of the software. The Oates software will input only 6 digits including the decimal points. Ergo, there is a small discrepancy in the input values. The figures shown below Figure 6 and Figure 7 are the input window and output window of the Oates software respectively.

FIGURE 3: BEST & MEAN FITNESS SCORES

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Flight Mach number Ambient temperature (K) Turbine inlet temperature (K) Fuel Heating Value (kJ/kg) Specific heat of fluid upstream of compressor (kJ/kg-K) Specific heat capacity of fluid downstream of the turbine (kJ/kg-K) Adiabatic index upstream of the compressor Adiabatic index downstream of the turbine Stagnation pressure ratio just after and just before the diffuser Stagnation pressure ratio just after and just before the burner Stagnation pressure ratio just after and just before the core nozzle Stagnation pressure ratio just after and just before the fan nozzle Polytropic efficiency of compressor Polytropic efficiency of turbine Polytropic efficiency of fan Burner (Combustion) Efficiency ηb Shaft (Mechanical) Efficiency η m P 0/P 9 (0=Convergent Nozzle) P 0/P 9`(0=Convergent Nozzle) Stagnation pressure ratio just after and just before the compressor Stagnation pressure ratio just after and just before the fan Engine Bypass Ratio

opposed to the brute force method which took several hours to converge at the solution in our initial analyses. The benefit is obviously seen if using GA to tackle problems like such with multiple variables (degrees of freedom).

0.87 216.7 1645 43031 1.005 1.245 1.4 1.3 0.9797 0.96 0.9893 0.9898 0.9029 0.8874 0.9135 0.9898 0.99 0 0 26 2 7

CONCLUSIONS In this study, a high bypass real turbofan engine with performance values sub-par to the desired performance values was investigated to find an economically favorable combination of component improvement values quickly, to meet the specified specific thrust and specific fuel consumption requirements (constraints). The objective of this paper was to implement a GA in a read world scenario and provide some insight of the relevance of a powerful optimization tool in everyday problems. The components that were optimized were diffuser, burner, core nozzle and fan nozzle pressure drop ratios and the burner, compressor, fan and turbine polytropic efficiencies. The total cost of improvement was dependent on the percentage of component improvement. Using genetic algorithms, the components were successfully optimized to meet the constraints with the lowest cost.

FIGURE 6: INPUT VARIABLE USED TO VERIFY ON OATES SOFTWARE Tt0 /T0

1.1514

Tt4 /T0

9.4039

Tt3 /Tt2

2.804

Polytropic efficiency of compressor

85.19%

Tt3 `/Tt2

1.2421

Polytropic efficiency of fan

90.47%

Tt5 /Tt4

0.5812

Polytropic efficiency of turbine

91.55%

U9 `/a0

1.0917

U9 /a0

1.888

M9

1

a0

295.15 m/s

Thrust generated by the engine

197.61 N/(kg/s)

Thrust specific fuel consumption

20.9826 (mg/s)/N

Pt0 /P0

1.6378

Fuel – air ratio

0.033172

Pt3 /Pt2

26

Pt3 `/Pt2

2

Pt5 /Pt4

0.0707

Pt9 `/P9 `

1.8929

P0 /P9 `

0.5959

Pt9 /P9

1.8324

P0 /P9

0.6545

M9`

1

Propeller Efficiency

76.72%

Thermal Efficiency

18.22%

Overall Efficiency

13.98%

In conclusion, we executed a continuous genetic algorithm with a single objective and multiple constraints to a real world situation. Again, the GA was chosen for this task because of its ability to quickly find solutions for large variable problems without requiring derivative information. For the given set of possible component improvements, the GA successfully performed an optimization, with the constraints of set specific thrust and specific fuel consumption, on the total cost of the performance improvement, which corresponds to the global minimum of the cost function in the GA while consuming less computation time. The number of generations that the GA takes to converge is sensitive to the selection of the elite number and mating (cross over) ratio. A small elite section and large cross over ratio led to a fast convergence. This work adds to the collective knowledge of turbofan component optimization techniques for cost analysis under nonlinear constraints, both in numerical data and methodology. REFERENCES [1] A. Homaifar, H. Y. Lai and E. McCormick, "System Optimization of Turbofan Engines Using Genetic Algorithms," Applied Mathmatical Modelling, vol. 18, pp. 72-83, February 1994. [2] A. Guha, "Optimisation of aero gas turbine engines," The Aeronautical Journal of the Royal Aeronautical Society, pp. 345-358, July 2001. [3] L. Casalino and D. Pastrone, "Optimization of Civil Turbofan with Evolutionary Algorithms," in 48th AIAA/ASME/ASEE Joint Propulsion Conference and Exhibit, Atlanta, 2012. [4] G. C. Oates, Aerothermodynamics of Gas Turbine and Rocket Propulsion, Washington D.C.: American Institute of Aeronautics and Astronautics, Inc., 1988.

FIGURE 7: OUTPUT VARIABLES VERIFIED BY OATES SOFTWARE

The computation time suggests that the GA converged at the solution in less than three quarters of a minute. This is as

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[5] R. L. Haupt and S. E. Haupt, Practical Genetic Algorithms, 2nd ed., Hoboken, New Jersey: John Wiley & Sons, Inc., 2004.

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