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CHAPTER 5

RESULTS & DISCUSSION

The axial flow compressor inlet stage optimization problem is formulated

as multi objective optimization problem. The stage performance

parameters i.e.; stage efficiency, inlet stage specific area and stall margin

coefficient are considered as objective functions with centrifugal stress as

constraint. The problem is solved using Nelder-Mead Simplex, Classical

Genetic Algorithm, NSGA-I and NSGA-II techniques. The results obtained

by optimizing the multi objective optimization problem using these

techniques are presented in this chapter. The results are compared with

the experimental results available in literature [A Massardo and A Satta

(24)].

5.1 Performance evaluation of various optimization techniques

The objective function values without constraint i.e. Stage

efficiency, inlet stage specific area and stall margin coefficient,

obtained from classical Genetic algorithm and Nelder Mead simplex

techniques, are tabulated from table 5.1 to 5.3, along with

corresponding design variables. From the tables, it is found that

the Genetic Algorithm technique obtained higher objective function

values compared to Nelder Mead simplex technique. The

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percentage increase in case of stage efficiency is 5.8, inlet stage

specific area is 9.7 and stall margin coefficient is 9.18. The

variations in the objective function values are due to the

generation of different design variables or solution vectors in

Genetic Algorithm and Nelder-Mead techniques. The variations in

the design variables are due to different convergence criteria and

the strategies adopted by these techniques to generate successive

design variables from previous generations.

The transformed objective function values i.e. combined Stage

efficiency, inlet stage specific area and stall margin coefficient for

set-1 and set-2 weighing coefficients with and without constraint

obtained from Nelder-Mead simplex and Genetic Algorithm

techniques are presented from table 5.4 to 5.7, along with the

design variables. From the tables, it is noticed that the

transformed objective function value is minimum in case of

Genetic algorithm compared to Nelder Mead simplex technique.

The percentage decrease in the un constrained transformed

objective function values in case of Genetic Algorithm compared to

Nelder Mead technique is 10.8, for set-1 weighing coefficients. The

percentage decrease in the unconstrained transformed objective

function value in case of Genetic algorithm compared to Nelder-

Mead simplex technique is 6.41, for set-2 weighing coefficients.

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The percentage decrease in the constrained transformed objective

function value obtained from Genetic Algorithm compared to

Nelder-Mead technique is 13.2, for set-1 weighing coefficients.

Similarly, the percentage decrease in the constrained transformed

objective function value obtained from Genetic Algorithm compared

to Nelder-Mead technique is 12.5, for set-2 weighing coefficients.

The variations in the objective function values obtained from

Genetic algorithm and Nelder-Mead simplex techniques are due to

the differences in the design variables generated from these

techniques. The variations in the design variables are due to the

differences in the convergence criteria implemented by these

techniques and the strategies adopted by these techniques to

generate successive design variables from previous generations.

It is also observed that set-1 weighing coefficients obtained

minimum value to the transformed objective function in

comparison to set-2. The percentage decrease in the unconstrained

transformed objective function value obtained from Genetic

Algorithm for set-1 weighing coefficients in comparison to set-2 is

71. Similarly, the percentage decrease in the constrained

transformed objective function value obtained from Genetic

algorithm for set-1 weighing coefficients in comparison to set-2 is

69.8. In case of Nelder-Mead approach, the percentage decrease in

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the unconstrained transformed objective function value for set-1

weighing coefficients compared to set-2 is 70.3. Similarly, the

percentage decrease in the constrained transformed objective

function value for set-1 weighing coefficients compared to set-2 is

70. This drastic variation in the values of objective functions are

due to the arbitrary choice of the scalar weighing coefficients. The

correct determination of the values of scalar weighing coefficients

can only be known by estimating the relative importance of each of

the objective functions from the designer’s stand point.

From the table 5.4 to 5.7, it is observed that the transformed

objective function value for the constrained case is minimum

compared to the unconstrained case, considering both the sets of

weighing coefficients. In case of Genetic Algorithm, the percentage

decrease in the constrained transformed objective function value

compared to the unconstrained case is 4.71, for set-1 weighing

coefficients. Similarly, the percentage decrease in the constrained

transformed objective function value compared to the

unconstrained is 11.4, for set-2 weighing coefficients. In case of

Nelder-Mead technique, the percentage decrease in the constrained

transformed objective function value obtained compared to the

unconstrained is 7.12, for set-1 weighing coefficients. Similarly,

the percentage decrease in the constrained transformed objective

function value compared to the unconstrained is 3.75, for set-2

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weighing coefficients. This variation in the values of transformed

objective functions is due to the influence of interior penalty

parameter. The constraint is incorporated in the transformed

objective function using interior penalty parameter. The value

allocated to the penalty parameter serves as another weighing

coefficient which influences the variations in the transformed

objective function values.

The non-dominated objective function values of stage efficiency,

inlet stage specific area and stall margin coefficient obtained from

NSGA-I and NSGA-II techniques for constrained and

unconstrained cases are tabulated from table 5.8 to 5.13 along

with the design variables. From the tables, it is noticed that the

NSGA-II technique produced higher non-dominated objective

function values compared to NSGA-I technique. The percentage

increase in the performance parameters i.e. stage efficiency, stall

margin coefficient and inlet stage specific area are 1.14, 5.1 and

1.2 respectively. The differences in the objective function values

obtained from NSGA-I and NSGA-II techniques are due to

variations in design variables obtained from these techniques. The

variations in the design variables are due to the differences in the

size of the binary population used in NSGA-I and NSGA-II

techniques. The variation may also be due to the explicit elite

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solution preserving strategy adopted by NSGA-II technique and the

influence of sharing parameter in NSGA-I technique.

From table 5.8 to 5.13, it is also found that the non-dominated un

constrained objective function values of stage efficiency, inlet stage

specific area and stall margin coefficient are higher compared to

the values obtained for the constrained objective functions. The

percentage increase in stage efficiency, stall margin coefficient and

inlet stage specific area are 4.59, 14.15 and 20 respectively. The

disparities in the objective function values are due to the influence

of adaptive penalty parameter. In NSGA-I and NSGA-II techniques,

the constraint is incorporated in each of the objective functions

using adaptive penalty parameter method. The value of the

parameter allocated to each of the objective functions depends on

the influence of the constraint on that particular objective

function. The inlet stage specific area and stall margin coefficient

are largely affected by the constraint centrifugal stress. Therefore,

a large value of penalty parameter is allocated to these objective

functions to incorporate the constraint. Hence, there is a steep

variation between the constrained and unconstrained values of

inlet stage specific area and stall margin coefficient.

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The objective function values obtained by Massardo [24] are

compared with the objective function values obtained from the

present techniques in table 5.1 to 5.3 and 5.8 to 5.13. The

objective function values obtained from the present techniques and

Massardo [24] are well collaborated.

The design variables shown in table 5.1 to 5.13 are the values

obtained at the point of convergence of various algorithms

implemented i.e.; Nelder-Mead simplex, classical Genetic

Algorithm, NSGA-I and NSGA-II respectively. The variations in the

design variables obtained from these techniques is due to

differences in approach methodologies and also due to differences

in convergence criteria adopted by these techniques. How ever, the

initial input to various techniques is chosen from the unique

variable range specified in 3.1.2.

Table 5.1 Design variables and Stage efficiency withoutconstraint for different techniques

Technique

Design Variables

StageEfficiency

D

(m)

Ø

(ratio)

α1

(rad)

N

(rps)

GA 0.313 0.575 0.310 480.55 0.859

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Nelder-Mead 0.3 0.421 0.260 422.29 0.798

Massardo[24] 0.301 0.44 0.29 456.8 0.845

Table 5.2 Design variables and inlet stage specific area withoutconstraint for different techniques

Technique

Design Variables Inlet stagespecific Area

(sqm)D

(m)

(rr/rt)

(ratio)

GA 0.382 0.433 206.35

Nelder-Mead 0.365 0.401 185.15

Massardo[24] 0.357 0.413 186.33

Table 5.3 Design variables and Stall margin coefficient withoutconstraint for different techniques

Technique

Design Variables

Stall MarginCoefficientD

(m)

N

(rps)

α1

(rad)

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GA 0.331 481.12 0.314 0.979

Nelder-Mead 0.326 422.08 0.297 0.894

Massardo[24] 0.321 418.3 0.289 0.834

Table 5.4 Design variables and transformed objective function values for set-1weighing coefficients without constraint for different techniques

Technique

Design Variables

TransformedObjectivefunction

D

(m)

Ø

(ratio)

α1

(rad)

N

(rps)

(rr/rt )

(ratio)

Nelder-Mead 0.379 0.543 0.3256 482.22 0.425 123.44

GA 0.362 0.519 0.3008 462.11 0.409 111.98

Table 5.5 Design variables and transformed objective function values for set-1weighing coefficients with constraint for different techniques

Technique Design Variables TransformedObjectivefunction

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D

(m)

Ø

(ratio)

α1

(rad)

N

(rps)

(rr/rt)

(ratio)

Nelder-Mead 0.358 0.516 0.3179 469.67 0.411 119.89

GA 0.354 0.482 0.3012 449.97 0.402 106.38

Table 5.6 Design variables and transformed objective function values for set-2weighing coefficients without constraint for different techniques.

Technique

Design VariablesTransformed

ObjectivefunctionD

(m)

Ø

(ratio)

α1

(rad)

N

(rps)

(rr/rt )

(ratio)

Nelder-Mead 0.367 0.529 0.3211 476.52 0.419 415.98

GA 0.334 0.481 0.2898 400.29 0.406 391.56

Table 5.7 Design variables and transformed objective function values for set-2weighing coefficients with constraint for different techniques.

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Technique

Design VariablesTransformed

ObjectivefunctionD

(m)

Ø

(ratio)

α1

(rad)

N

(rps)

(rr/rt)

(ratio)

Nelder-Mead 0.370 0.537 0.3307 487.92 0.430 401.15

GA 0.322 0.451 0.3001 412.11 0.401 351.23

Table 5.8 Design variables and non-dominated objective function values withoutconstraint for different techniques.

TechniqueDesign Variables Objective Function values

D Ø 1 N StageEfficiency

Stall MarginCoefficient

NSGA-I 0.309 0.58 0.31 487.4 0.866 1.06

NSGA-II 0.304 0.6 0.33 492 0.873 1.13

Massardo[24] 0.301 0.44 0.29 456.8 0.845 0.945

Table5.9 Design variables and non-dominated objective function values withoutconstraint for different techniques.

Technique Design Variables Objective Function valuesD Ø 1 N (rr/rt) Stage Inlet stage

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Efficiency Specific Area

NSGA-I 0.391 0.57 0.29 486.7 0.469 0.866 224.75

NSGA-II 0.388 0.59 0.34 492 0.441 0.873 227.87

Massardo [24] 0.357 0.51 0.24 432.8 0.413 0.845 186.33

Table 5.10 Design variables and non-dominated objective function valueswithout constraint for different techniques.

TechniqueDesign Variables Objective Function values

D 1 N (rr/rt)Stall MarginCo-efficient

Inlet stagespecific Area

NSGA-I 0.368 0.299 458.5 0.443 1.1 211.05

NSGA-II 0.374 0.327 478.4 0.425 1.24 219.55

Massardo[24] 0.321 0.289 418.3 0.410 0.834 190.67

Table 5.11 Design variables and non-dominated objective function values withconstraint for different techniques.

TechniqueDesign Variables Objective Function values

D Ø 1 N StageEfficiency

Stall Margincoefficient

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NSGA-I 0.301 0.424 0.224 441.7 0.815 0.799

NSGA-II 0.303 0.438 0.265 450.6 0.832 0.907

Massardo[24] 0.301 0.44 0.29 456.8 0.845 0.945

Table 5.12 Design variables and non-dominated objective function values withconstraint for different techniques.

TechniqueDesign Variables Objective Function values

D Ø 1 N (rr/rt)Stage

EfficiencyInlet Stage

Specific Area

NSGA-I 0.310 0.46 0.21 386.7 0.403 0.814 178.35

NSGA-II 0.342 0.49 0.24 411.2 0.410 0.838 181.88

Massardo [24] 0.357 0.51 0.24 432.8 0.413 0.841 186.33

Table 5.13 Design variables and non-dominated objective function valueswith constraint for different techniques.

Technique Design Variables Objective Function valuesD 1 N (rr/rt) Stall Margin Inlet Stage

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Coefficient SpecificArea

NSGA-I 0.302 0.231 350.5 0.401 0.813 179.56

NSGA-II 0.313 0.273 371.6 0.409 0.827 189.35

Massardo [24] 0.321 0.289 418.3 0.410 0.834 190.67

5.2 The effect of design variables generated over a number ofgenerations on the objective function values

The variations in the unconstrained objective function values viz

stage efficiency, inlet stage specific area and stall margin

coefficient, obtained from classical Genetic Algorithm, NSGA-I and

NSGA-II techniques against a number of generations are shown

from Fig.5.1 to 5.3. From the figures, it is observed that the

objective function values varied drastically from 0 to 50th

generation. The objective function values after 50th generation

started stabilizing and became constant after 1200th generation.

The initial drastic variation in the objective function values is due

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to the poor solution fitness of the randomly generated binary

design vectors or variables. The fitness of these design variables in

successive generations improved due to operations like selection,

cross over and mutation in Genetic algorithm and also with the

strategies like fitness sharing and solution diversity preservation in

NSGA-I and NSGA-II techniques. The changes in objective function

values are found to be negligible after 1200th generation, due to

minor variation in the fitness of design variables. These minor

variations in fitness of design variables are because of different

probabilities of cross over and mutation. Owing to the probabilistic

selection of cross over and mutation, some portion of the binary

population only undergoes crossover and mutation.

The variations in the unconstrained and constrained transformed

objective function values for set-1 and set-2 weighing coefficients,

obtained from Genetic Algorithm and Nelder Mead techniques

against a number of generations are shown from Fig. 5.4 to 5.7.

From the figures, it is observed that the objective function values

obtained from Nelder-Mead simplex converged at 40th generation,

while the values obtained from classical Genetic algorithm at 700th

generation. It is because of the difference in the convergence

criteria adopted by these techniques. The Nelder-Mead simplex

converges when the standard deviation value obtained from the

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newly generated design variables and the centroidal point is less

than the given resolution. On the other hand, the genetic algorithm

converges when differences in the fitness of the newly generated

design variables from the previous generation is negligible.

5.3 Non-dominated solution fronts obtained for the objectivefunctions from NSGA-I and NSGA-II techniques

The non dominated solution fronts for stage efficiency, stall

margin coefficient and inlet stage specific area with and without

constraint, obtained using NSGA-I and NSGA-II techniques are

shown from Fig.5.8 to 5.13. From the figures, it is observed that

the NSGA-II technique generated better trade-off solution fronts to

the objective functions in comparison to NSGA-I technique. This

is because of the explicit elite solution preserving strategy adopted

by NSGA-I technique.

From Fig.5.11 and 5.13, it is observed that the non-dominated

solution fronts changed drastically between stage efficiency, inlet

stage specific area and between stall margin, inlet stage specific

area. It is because of the predominant influence of the constraint

centrifugal stress on inlet stage specific area.

5.4 The effect of change in design variables on the sensitivity of

objective function values

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The influence of mean diameter of compressor stage on the

sensitivity of stage efficiency is shown in Fig.5.14. From the figure,

it is observed that the stage efficiency decreased from 0.946 to

0.837 with the increase in mean diameter from 0.3 to 0.4 meters.

Owing to increase of mean diameter, the work input absorption

capacity of the compressor stage increased, which resulted in

reduction of stage efficiency. The sensitivity of stall margin

coefficient with variation in mean diameter of stage is also shown

in Fig.5.14. From the figure, it is observed that the stall margin

coefficient increased from 0.74 to 1.24, with increase of mean

diameter of stage from 0.3 to 0.4 meters. As mean diameter of

compressor stage increases, large volumes of working fluid enters

the compressor inlet and it decreases the probability of stall

inception or in other words increases the stall margin coefficient.

The effect of variation in mean diameter of the stage on the

sensitivity of inlet stage specific area is shown in Fig.5.15. From

the figure, it is found that the inlet stage specific area increased

from 145 to 225 m2 with the variation in the mean diameter of

stage from 0.3 to 0.4 meters. As the mean diameter increases, the

height of blades increases. In order to maintain constant

clearance between the blade tip and casing, the frontal area or

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inlet stage specific area is increased. From the figure, it is

observed that the percentage increase in transformed objective

function value is 30, with the variation in the mean diameter from

0.3 to 0.4 meters. It is because of the most predominant influence

of mean diameter on inlet stage specific area which forms a

significant part of the transformed objective function.

The sensitivity of stage efficiency owing to changes in flow

coefficient is shown in Fig.5.16. From the figure, it is noticed that

the stage efficiency increased from 0.801 to 0.947 due to variation

in flow coefficient from 0.2 to 0.6. At very low values of flow

coefficient, the angle of incidence of air increases, which increases

the stall inception probability. Therefore, the stage efficiency gets

reduced. By gradually increasing the value of flow coefficient with

in the design limits, the stage efficiency is improved.

Fig 5.17 shows the sensitivity of inlet stage specific area with the

variations in hub-tip radius ratio of the blade. From the figure, it

is found that the inlet stage specific area reduced from 303 to 211

m2 with increase in the hub-tip radius ratio of the blade from 0.4

to 0.9. The increase in hub tip radius ratio of the blade increases

the weight of the stage. The shaft speed for the fixed input power

gets reduced on account of increased stage weight which reduces

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the mass flow rate entering the frontal area. From the figure, it is

observed that the percentage decrease in the transformed

objective function value is 19, due to variation in hub-tip radius

ratio from 0.4 to 0.9. It is because of the most predominant

influence of hub-tip radius ratio on centrifugal stress, which is

incorporated as a constraint in the transformed objective function.

Fig 5.18 shows the sensitivity of stage efficiency with the

variations in shaft speed. From the figure, it is found that the

value of stage efficiency decreased from 0.923 to 0.836 due to

increase in shaft speed from 350 to 500 rps. As the shaft speed

increases, the centrifugal and bending stresses developed in the

blades increases which reduces the stage efficiency. From the

figure it is also noticed that the stall margin coefficient increased

from 1.02 to 1.71 with the increase in shaft speed from 350 to 500

rps. At high shaft speeds, the mass flow rate of working fluid

entering the compressor with high velocities increases, which

results in delayed stall propagation.

The effect of change in air inlet angle on the sensitivity of stall

margin coefficient is shown in Fig.5.19. From the figure, it is

observed that the stall margin coefficient increased from 0.95 to

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1.26 with the increase in the air inlet angle to rotor from 0 to9

radians. The increase in air inlet angle reduces the stagnation

pressure loss and increases the stall margin coefficient.

Fig. 5.1 Variation of Stage Efficiency against No of Generations

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Fig. 5.2 Variation of Inlet Specific Area against No of Generations

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Fig.5.3 Variation of Stall margin coefficient against No of Generations

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Fig. 5.4 Variation of un constrained transformed objective functionagainst number of generations for set-1 weighing coefficients

Fig. 5.5 Variation of constrained transformed objective functionagainst number of generations for set-1 weighing coefficients

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Fig. 5.6 Variation of unconstrained transformed objective functionagainst number of generations for set-2 weighing coefficients

Fig. 5.7 Variation of constrained transformed objective functionagainst a number of generations for set-2 weighing coefficients

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Fig. 5.8 Non dominated Solution front between Stage efficiencyand Inlet stage specific Area without constraint.

Fig. 5.9 Non dominated solution fronts between Stage Efficiencyand Stall Margin Coefficient without constraint.

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Fig. 5.10 Non dominated solution front between Stall Marginand Inlet Stage Specific area without constraint.

Fig. 5.11 Non dominated Solution Front between Stage Efficiencyand Inlet stage specific area with constraint.

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Fig. 5.12 Non dominated solution front between stage efficiencyand stall margin coefficient with constraint.

Fig. 5.13 Non dominated solution front between Stall margin andinlet Stage Specific area with constraint.

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Fig 5.14 Sensitivity of objective functions Stage efficiency and Stall margincoefficient to variations in mean diameter of the stage.

Fig. 5.15 Sensitivity of transformed objective function and the objective functioninlet stage specific area to variations in mean diameter of the stage.

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Fig. 5.16 Sensitivity of transformed objective function and the objective functionstage efficiency to variations in flow coefficient.

Fig. 5.17 Sensitivity of transformed objective function and the objective functioninlet stage Specific area to variations in hub-tip radius ratio of blade.

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Fig. 5.18 Sensitivity of objective functions stage efficiency, stall margin coefficientand the transformed objective function to variations in shaft speed.

Fig. 5.19 Sensitivity of objective functions Stage efficiency, Stall margin coefficientand the transformed objective function to variations in air inlet angle.