Optimizing the Mass of Expansion Joints Used in …ode.engin.umich.edu/me555reports/2014/ME555-14-Goyal.pdfOptimizing the Mass of Expansion Joints Used in Thermal Power Plants Term

Optimizing the Mass of Expansion Joints Used in

Thermal Power Plants

Term Project Report

ME 555 - Design Optimization

Winter 2014

Abstract

Expansion Joints are widely used in different industries to absorb thermal expansion in operating

conditions. This project focusses on optimizing the design of 2000 NB expansion joints being used

in India’s first 2x800 MW supercritical power plant. The total weight of assembly of a single

expansion joint weighs from 5 kg to 2000 kg. Even 1% improvement can help in saving significant

amount of money considering total number of expansion joints used in a single power plant. The

model is developed using standards described in Expansion Joints Manufacturers Association,

manufacturer’s data and company’s data which is using this expansion joint. Many optimization

approaches are tried and the most efficient one, fmincon is reported. The results were verified by

running a separate loop for all possible combination of values of variables. Monotonicity analysis,

KKT conditions checking, sensitivity analysis, parametric studies, were carried out to get more

insight of the optimum point.

Section Instructor: Prof Yi(Max) Ren Abhishek Goyal

GSI: Alparslan Emrah Bayrak M.S.E in Mechanical Engineering

Department of Mechanical Engineering UM ID: 14157299

University of Michigan [email protected]

2

Term Project - ME 555 – Winter 14, University of Michigan

Contents Introduction ............................................................................................................................................. 3

Importance of Expansion Joint................................................................................................................. 3

Mathematical model ................................................................................................................................ 7

Model Analysis ..................................................................................................................................... 12

Design Optimization .............................................................................................................................. 15

Parametric Analysis ............................................................................................................................... 17

Discussions ........................................................................................................................................... 17

Acknowledgements: .............................................................................................................................. 18

Appendix A: Neural Network MATLAB Code ...................................................................................... 19

Appendix B: Main file ........................................................................................................................... 25

Appendix C: Constraints File ................................................................................................................. 26

Appendix D: Objective Function ........................................................................................................... 30

Appendix E: Experimental Values of Cd, Cp and Cf depending on design variables ................................ 31

List of Figures and Tables

Figure 1: Manufacturing of Expansion Joint ............................................................................................ 3

Figure 2: Importance of Expansion Joint .................................................................................................. 3

Figure 3: Alternative Solutions to Expansion Joints ................................................................................. 4

Figure 4: Working of Expansion Joint...................................................................................................... 5

Figure 5: Actual Installation of Expansion Joints ..................................................................................... 5

Figure 6: Detailed AutoCAD drawing of an expansion joint..................................................................... 6

Figure 7: Geometry of Expansion Joint for Mathematical Model Development ........................................ 7

Figure 8: Validation for Neural Network of Cd ....................................................................................... 13

Figure 9: Error Histogram for Neural Network of Cd .............................................................................. 13

Figure 10: Regression Plot for Neural Network of Cd ............................................................................. 14

Figure 11: Training Graph for Neural Network of Cd ............................................................................. 14

Figure 12: Parametric analysis of objective function with axial expansion .............................................. 17

Table 1: Manufacturing Constraints on Variables..................................................................................... 8

Table 2: Monotonicity Analysis ............................................................................................................. 12

Table 3: Design Parameters ................................................................................................................... 15

Table 4: Comparison of result with actual values ................................................................................... 15

Table 5: Testing of objective function with different starting points ....................................................... 16

3


Introduction

An expansion joint or bellow is an assembly designed to safely absorb the heat-induced expansion

and contraction of piping systems, to absorb vibration, to hold parts together, or to allow movement

due to pressure changes in the system. They are also known as compensators because they

‘compensate’ for thermal movement in the systems.

Depending on the application, they are made up of steel,

rubber, fabric or plastic. The metal expansion joints are the

most commonly used in power plant industry because of

their ability to withstand high temperature and pressure of

the processing equipment. This project focusses on metal

expansion joint because they contribute a significant

proportion in the total piping cost. They are used in

process industries, power plants, aerospace applications

etc. Below is described the importance and working of a

typical expansion joint.

Importance of Expansion Joint

Let us consider a simple system as shown below,

The equipment A is supplying superheated steam at 200oC and 2 bar pressure to equipment B.

They are placed 10 m apart. A pipe is connected to both the equipment’s nozzles. Since the pipe

is made up of steel (metal), it will try to expand linearly. The coefficient of linear expansion of

steel is 5

1.6 10 / om m C

. Based on the given data, we can calculate the change in length

as 510 1.6 10 200 0.032 32L L T m mm . The strain in the piping due to high

Figure 1: Manufacturing of Expansion Joint

Figure 2: Importance of Expansion Joint

http://en.wikipedia.org/wiki/Thermal_expansion

4


temperature will be 0.032

0.003210

LStrain

L

. From stress strain relationship, we can find

the stress generated in the piping. The Young’s Modulus of Elasticity for steel is 200 GPa.

2000.0032

StressE

Strain

StressGPa

640 310 ( )Stress MPa MPa Compressive Strength of Steel

Since the pipe’s natural tendency at such a high temperature is to expand and the equipment

connected to it is stopping it from expanding, therefore, compressive forces will be developed in

the pipe which, in this case, are exceeding its compressive strength. This will either result to the

cracks in the piping or damage to equipment’s nozzle. Therefore there is need to lower down the

maximum stress generated in the piping. It can be achieved in two ways:

1. Avoiding long straight runs and introduce loops.

From the above figure, it can be seen that as the temperature increases, the bends in the piping

absorb the axial expansion and thus reduce the maximum stress. But introducing loops has two

limitations.

a) Pressure loss in the fluid due to bend loss

b) When there are space constraints. For example, if there is a roof or overhead equipment

which hinders the installation of loop of necessary height.

Elevated Temperature Normal Temperature

Figure 3: Alternative Solutions to Expansion Joints

5


Figure 4: Working of Expansion Joint

2. Use expansion joints/bellows to absorb axial expansion: As the temperature increases, the

convolutions come closer and absorb expansion, just like as we increase the force on

spring, the coils come closer.

Therefore, we see that the expansion joints play an important role in the safety of piping. The size

of expansion joints varies from 50 NB to 2500 NB. The weight of the total assembly of expansion

joint varies from 5 kg to 2000 kg and the cost can vary anything from $500 to $200,000 per

assembly. This cost excludes cost of supporting structure for these expansion joints. Therefore

from cost perspective, it looks attractive to optimize. For this project, mass of 2000 NB will be

optimized. 2000 NB is chosen because of the permission of the manufacturer and company using

this expansion joint, to use the actual manufacturer’s data for this particular size. The optimization

approach developed can be extended to other diameter pipe sizes as well. The detailed CAD

drawing of a typical expansion joint which is currently being used in India’s first supercritical 800

MW power plant is shown on the following page.

Expansion

Joint

Equipment

Figure 5: Actual Installation of Expansion Joints

Abhishek

Rectangle

Abhishek

Rectangle

Abhishek

Rectangle

Abhishek

Textbox

Design Parameters

Abhishek

Textbox

Expansion Joint

Abhishek

Textbox

Focus of this project

7


Mathematical model

Figure 7: Geometry of Expansion Joint for Mathematical Model Development

The geometry of expansion consists of 5 independent variables. These are

1. w = height of a convolution

2. q = distance between each convolution

3. N = number of convolutions

4. n = number of plies

5. t = thickness of plies

For given operating conditions (expansion and pressure), the stresses in the bellow material,

fatigue life and column stability of the bellow to absorb the pressure vary according to chosen

values of these variables. The mathematical formula for the mass of an expansion joint can be

written as follows:

2 ( 2 ) 2 ( )m m m bm NntD w r r ntL D nt

Db = diameter of pipe, Dm = Mean diameter of bellow, rm= radius of convolution, L = length of

bellow, ρ = density of the material

Looking at the mass equation it can be easily said that as the values of these variable increases,

mass increases. Hence our mass function is monotonically increasing w.r.t our variables.

Since these expansion joints are subjected to cyclic loading, the fatigue life should be long enough

to match industry standards. The manufacturing of expansion joints is governed by the EJMA

(Expansion Joints Manufacturers’ Association) standards. The allowable values and mathematical

expressions for different stresses are taken from this standard.

8


The manufacturing of expansion joints depends on the usage environment which can be described

by axial expansion/compression in the pipe to be absorbed, temperature, pressure, pipe diameter,

pipe material. Therefore in our optimization study these will be taken as parameters.

Based on the manufacturing constraints for 2000 NB metal expansion joints, the bounds on

variables are

Table 1: Manufacturing Constraints on Variables

Variable Lower Bound Upper Bound

height of convolution, w 35 mm 40 mm

distance between each convolution, q 25 mm 30 mm

number of convolutions, N 6 11

number of plies, n 1 5

thickness of each ply, t 1 mm 3 mm

All variables can take only integer values, therefore all are discrete variables.

Engineering constraints are:

1. Circumferential membrane stress (S2) should be lower than allowable stress.

2

m rallowable

c

P D K q

A

P = Inside pressure

Dm = Outer diameter of expansion joint = Db + w + nt

Kr = circumferential stress factor = 1

Ac = Cross sectional area of convolution = 22 2

p

q qw t n

where tp is b

m

Dt

D

9


2. Sum of meridional membrane and bending stress due to pressure(S3 and S4) should be less

than the product of material strength factor and allowable stress

2

2 2p m allowable

p p

P w P wC C

n t n t

Cp, Cm = values to be extracted from the table given in the standard.

Dependent on q and w

3. Internal design pressure based on column instability should be greater than 4

2

0.344inC f

N q

where,

fin =

3

3

1.7 m b p

f

D E t n

w C

, Cθ = 1;

4. Internal design pressure based on in-plane instability and local plasticity at temperature

below creep range should be greater than 3.5

1.33.5

c y

r m

A S

K D q

where

Sy = Yield strength at design temperature after completion of bellow

forming

= 3044 kg/cm2

2 2 4

2

2

1 2 1 2 4

2

3

p

p

C w

n t

S

P

5. Fatigue life at room temperature should be greater than 10,000 cycles

10


10000c

a

c

t

N

cN

S b

3 4 5 6

2

5 3

3.4

54000

1.86 06

0.7( ) ( ) 14.2233

2

t

b p

f

a

b

c E

S S S S S

E t eS

w C

6 2

5

3

b p

d

E t eS

w C

xe

N

Cf, Cd = values to be extracted from the table given in the standard.

Dependent on q and w

6. Theoretical axial elastic spring rate per convolution should be such that the loads on the

connected equipment should be within allowable range. The maximum value is calculated

using pipe stress analysis software like CAESAR II.

3

3

1.7 m b p

allowable

f

D E t nk

w N C

Dimensional constraints will be:

7. Maximum convoluted bellow length should be less than or equal to twice of the diameter

of pipe

2 bD N q

8. The ratio of width and pitch should be greater than 0.6

0.6w

q

11


9. The ratio of width and pitch should be less than 1.6

1.6w

q

10. The convolution height should not be greater than 2 bDN

2 bDw

N

Therefore we have,

min weight = f (w,t,n,N,q) subject to

g1: 02

m rallowable

c

P D K q

A

g2:

2

02 2

p m allowable

p p

P w P wC C

n t n t

g3:

3

2 3

1.70.344 0

m b p

f

D E t nC

N q w C

g4: 1.3

3.5 0c y

r m

A S

K D q

g5: 10000 0cN

g6:

3

3

1.70

m b p

allowable

f

D E t nk

w N C

g7: 2 0bN q D

g8: 0.6 0w

q

g9: 1.6 0w

q

g10:2

0bDw

N

g11: 1 0n

g12: 5 0n

g13: 7 0N

g14: 11 0N

g15: 35 0w

g16: 40 0w

g17: 25 0q

g18: 30 0q

g19: 1 0t

g20: 3 0t

12


Model Analysis

The defined problem and constraints are highly non-linear and thus needs to be checked thoroughly

for any possible unboundedness. Although all our variables are bounded, but the monotonicity

analysis will help us to find if there is any concavity possibility in our design space. Below is

shown the monotonicity table.

Table 2: Monotonicity Analysis

w t n N q

f + + + + +

g1 + - -

g2 + - -

g3 + - - + +

g4 + - - +

g5 - + - - -

g6 - + + - -

g7 + +

g8 - +

g9 + -

g10 + +

g11 -

g12 +

g13 -

g14 +

g15 -

g16 +

g17 -

g18 +

g19 -

g20 +

13


From the monotonicity table, it can be seen that all our constraints are monotonic with involved

variables. We can also observe that based on the manufacturing constraint bounds, g7, g8, g9 and

g10 are redundant and will always be satisfied with given bounds. But in the present model, they

are not being removed because as we change the diameter of pipes, there will be different

manufacturing constraints are thus may play a role. Therefore, to make this optimization model

universal for every pipe size, they should be included. Based on our engineering intuition, g6 seems

to be active w.r.t w, which will be verified by checking Lagrange multipliers.

The design parameters in this particular model are taken from a super-critical power plant in India

with all requisite permissions. The generalized optimization model developed here, can be used

for other design parameters as well.

Let’s talk about Cf, Cp and Cd, used in our design constraints. These values depend on design

variables and material properties and obtained experimentally by the manufacturer. The

manufacturer’s data was available in tabular format as seen in Appendix E. To calculate the values

of Cf, Cp and Cd for different values of design variables, we have used Neural Networks because

there is no analytical formula which relates Cf, Cp and Cd with design variables. The MATLAB

code for these neural networks is attached in the Appendix A.

For Cd the graphs obtained after applying Neural Network are shown here.

Figure 9: Error Histogram for Neural Network of Cd Figure 8: Validation for Neural Network of Cd

14


Figure 10: Regression Plot for Neural Network of Cd

Figure 11: Training Graph for Neural Network of Cd

15


Design Optimization

Our design parameters are:

Table 3: Design Parameters

Parameter Value

Design Temp, T, °C 100

Design Pressure, P, (Kg/cmsq)) 2

Axial compression, x, (mm) 47.05

The material is SA240 Gr.304.

Although our design variables are discrete, we will first use fmincon and analyze the results. Since

the bounds on design variables are not too wide, the fmincon result can be easily verified by trying

all possible combinations of values for variables (~ 2700 combinations)

Table 4: Comparison of result with actual values

Variable Actual fmincon Combinations (in loop)

Height of convolution 39 37.73 38

Width of convolution 27 25 25

Number of plies 1 1 1

Thickness of each ply 1 1 1

Number of convolutions 11 7 7

Objective function 7326922 4725383 4749585

From our results we can see that the variables width of convolution, thickness of each ply, number

of convolutions and number of plies are hitting bounds. From our MATLAB results, it can be seen

that g6 is active w.r.t. w, height of convolution which was predicted after monotonicity analysis

also.

Activity of g6 can be explained in physical terms. g6 corresponds to the maximum force on

equipment nozzle. The equipment nozzle is designed separately from expansion joint, by the

equipment manufacturer. If we could ask equipment manufacturer to provide more strengthening

16


at the nozzle, it can potentially reduce the function value by 1625 for every 1kgf increment in force

absorbing capacity of nozzle.

It should be mentioned that as we got 4 variables on the lower bound, it does not mean that they

will not be always on lower bound. It will depend on the manufacturing constraints for

corresponding pipe diameter.

In fmincon result, local minimum was always found which satisfied all constraints. The values of

eigenvectors of the hessian of our objective function is greater than or equal to zero, hence we can

say that this is indeed a minimum and the Hessian is positive semi definite. The values of

eigenvectors of hessian is as follows:

[-3.03e-11 0.082522 528040.98 2762127.31 21932050.61]’

Checking gradient of constraints at optimal point, we also see that xopt is a regular point. µ6 =

1625 (>0), all other µ=0 and µigi = 0. Hence this point satisfies KKT condition. The first order

optimality point from fmincon is: 3.5048e-07 which is greater than 0 and very small. Hence it is

also proving KKT conditions are satisfied.

Below is the table of results when the fmincon is used with different starting points:

Table 5: Testing of objective function with different starting points

Initial Point xopt function value

[ 38 1 1 7 25 ] [37.7155 1 1 7 25] 4723112

[ 39 1 1 8 30 ] [37.7155 1 1 7 25] 4723112

[ 40 1 1 7 30 ] [37.7155 1 1 7 25] 4723112

[ 35 1 2 7 25 ] [37.7201 1 1 7 25] 4723537

[ 37 2 4 10 25 ] [37.7201 1 1 7 25] 4723537

The results were similar upto 4 decimal points when run with Interior Point, SQP and Active Set

algorithms. The results were also validated by running a loop for all combinations.

17


To further strengthen the validity of our results, we can do stress analysis and fatigue life analysis.

In case, the results do not match, further modifications in analytical formulation of different stress

calculations should be carried out.

Parametric Analysis

There are three parameters in our study. Temperature and pressure cannot be changed because they

are decided based on thermodynamic conditions of the fluid. There is one parameter, axial

expansion, which can be changed according to the different piping arrangements. For example, the

equipment are closer or farther, there is a loop in the piping etc. So we varied this parameter from

0 to 100. We can see that when the axial expansion is below 50 mm, objective function value is

not changing. This is because g6 is active constraint within this range. But as we increase the value

beyond 50 mm, g5 becomes active and objective function increases. It goes on increasing

monotonically with expansion till 100 mm. But at 100 mm axial expansion, there is no feasible

point. It means that there is

no possible set of values for

variables for which our

constraints will be satisfied.

Hence the piping designer

should incorporate another

piping routing or use two

bellows in series. Below is

the pattern observed in

objective function w.r.t

axial expansion.

Discussions

Based on the available data and design standards, we found that there is considerable amount of

difference between calculated optimal value of objective function and actual one. The assembly

design of expansion joints (like flange design, tie rods etc) is also affected by the convolutions and

hence the total weight will also reduce proportionately. But before jumping to any conclusions and

Figure 12: Parametric analysis of objective function with axial expansion

18


out rightly rejecting the actual design variable values, one should validate the results using

engineering analysis software like ANSYS and FLUENT. It might be possible, that the calculated

stresses using mathematical formulae may be not be similar to the stresses obtained through FEA.

Also there might be possibility of fluid flow interaction with convolutions which can create

disruptions, turbulence or low pressure zone inside convolutions. The thermal expansion of

convolutions was also not considered in mathematical models. It was assumed that the temperature

will not affect yield strength. Installation conditions and methods may also affect the design. There

may be some assembly manufacturing constraints due to which there might be some restrictions

in using optimum values of design variables.

Acknowledgements:

I would like to express my sincere gratitude to Prof. Yi (Max) Ren for his continuous

encouragement and guidance throughout the semester. He has been very helpful and patient in

teaching me important design optimization concepts to be used in this project. His untiring support

to students who were not very proficient in MATLAB must be appreciated. I am also thankful to

the GSI, Mr. Alparslan Emrah Bayrak who regularly helped me in resolving MATLAB coding

errors faced in homework assignments. I am very satisfied with the instructors and course content

and look forward to apply these newly learned concepts in practical world. I wish them good luck

in their respective academic careers.

19


Appendix A: Neural Network MATLAB Code

T1 = 0:0.05:1;

M = [0.2:0.2:1.6,2.0:0.5:4];

[t1,m] = meshgrid(T1,M);

t1 = reshape(t1,13*21,1);

m = reshape(m,13*21,1);

X = [t1,m];

y = [1.000 0.976 0.946 0.912 0.876 0.840

0.803 0.767 0.733 0.702 0.674 0.649

0.627 0.610 0.596 0.585 0.577 0.571

0.566 0.560 0.550

1.000 0.962 0.926 0.890 0.856 0.823

0.790 0.755 0.720 0.691 0.665 0.642

0.622 0.606 0.593 0.583 0.576 0.571

0.566 0.560 0.550

0.980 0.910 0.870 0.840 0.816 0.784

0.753 0.722 0.696 0.670 0.646 0.624

0.605 0.590 0.585 0.577 0.569 0.566

0.558 0.550 0.543

0.950 0.842 0.770 0.722 0.700 0.680

0.662 0.640 0.627 0.610 0.593 0.585

0.579 0.574 0.569 0.563 0.557 0.553

0.546 0.540 0.533

0.950 0.841 0.744 0.657 0.592 0.559

0.536 0.541 0.548 0.551 0.551 0.550

0.547 0.544 0.540 0.536 0.531 0.526

0.521 0.515 0.510

0.950 0.841 0.744 0.657 0.579 0.518

0.501 0.502 0.503 0.503 0.503 0.502

0.500 0.497 0.494 0.491 0.488 0.485

0.482 0.479 0.476

0.950 0.840 0.744 0.651 0.564 0.495

0.462 0.460 0.458 0.455 0.453 0.450

0.447 0.444 0.442 0.439 0.437 0.435

0.433 0.432 0.431

20


0.950 0.841 0.731 0.632 0.549 0.481

0.432 0.426 0.420 0.414 0.408 0.403

0.398 0.394 0.391 0.388 0.385 0.384

0.382 0.381 0.380

0.950 0.841 0.731 0.632 0.549 0.481

0.421 0.388 0.369 0.354 0.342 0.332

0.323 0.316 0.309 0.304 0.299 0.296

0.294 0.293 0.292

0.950 0.840 0.732 0.630 0.550 0.480

0.421 0.367 0.332 0.315 0.300 0.285

0.272 0.260 0.251 0.242 0.235 0.230

0.224 0.219 0.215

0.950 0.840 0.732 0.630 0.550 0.480

0.421 0.367 0.328 0.299 0.275 0.258

0.242 0.228 0.215 0.203 0.195 0.188

0.180 0.175 0.171

0.950 0.840 0.732 0.630 0.550 0.480

0.421 0.367 0.322 0.287 0.262 0.241

0.222 0.208 0.194 0.182 0.171 0.161

0.152 0.146 0.140

0.950 0.840 0.732 0.630 0.550 0.480

0.421 0.367 0.312 0.275 0.248 0.225

0.205 0.190 0.176 0.163 0.152 0.142

0.134 0.126 0.119

];

y = reshape(y,size(y,1)*size(y,2),1);

[n1,p] = size(X);

Cpnet = feedforwardnet(10);

Cpnet = train(Cpnet,X',y');

save Cpnet;

y1 = [1.000 1.116 1.211 1.297 1.376 1.451

1.524 1.597 1.669 1.740 1.812 1.882

21


1.952 2.020 2.087 2.153 2.217 2.282

2.349 2.421 2.501

1.000 1.094 1.174 1.248 1.319 1.386

1.452 1.517 1.582 1.646 1.710 1.775

1.841 1.908 1.975 2.045 2.116 2.189

2.265 2.345 2.430

1.000 1.092 1.163 1.225 1.281 1.336

1.392 1.449 1.508 1.568 1.630 1.692

1.753 1.813 1.871 1.929 1.987 2.049

2.119 2.201 2.305

1.000 1.066 1.122 1.171 1.217 1.260

1.300 1.340 1.380 1.422 1.465 1.511

1.560 1.611 1.665 1.721 1.779 1.838

1.896 1.951 2.002

1.000 1.026 1.052 1.077 1.100 1.124

1.147 1.171 1.195 1.220 1.246 1.271

1.298 1.325 1.353 1.382 1.415 1.451

1.492 1.541 1.600

1.000 1.002 1.000 0.995 0.989 0.983

0.979 0.975 0.975 0.976 0.980 0.987

0.996 1.008 1.022 1.038 1.056 1.076

1.099 1.125 1.154

1.000 0.983 0.962 0.938 0.915 0.892

0.870 0.851 0.834 0.820 0.809 0.799

0.792 0.787 0.783 0.780 0.779 0.780

0.781 0.785 0.792

1.000 0.972 0.937 0.899 0.860 0.821

0.784 0.750 0.719 0.691 0.667 0.646

0.627 0.611 0.598 0.586 0.576 0.569

0.563 0.560 0.561

1.000 0.948 0.892 0.836 0.782 0.730

0.681 0.636 0.595 0.557 0.523 0.492

0.464 0.439 0.416 0.394 0.373 0.354

0.336 0.319 0.303

1.000 0.930 0.867 0.800 0.730 0.665

0.610 0.560 0.510 0.470 0.430 0.392

0.360 0.330 0.300 0.275 0.253 0.230

0.206 0.188 0.170

22


1.000 0.920 0.850 0.780 0.705 0.640

0.580 0.525 0.470 0.425 0.380 0.342

0.300 0.271 0.242 0.212 0.188 0.167

0.146 0.130 0.115

1.000 0.900 0.830 0.750 0.680 0.610

0.550 0.495 0.445 0.395 0.350 0.303

0.270 0.233 0.200 0.174 0.150 0.130

0.112 0.092 0.081

1.000 0.900 0.820 0.735 0.655 0.590

0.525 0.470 0.420 0.370 0.325 0.285

0.252 0.213 0.182 0.152 0.130 0.109

0.090 0.074 0.061];

y1 = reshape(y1,size(y1,1)*size(y1,2),1);

[n1,p] = size(X);

Cfnet = feedforwardnet(10);

Cfnet = train(Cfnet,X',y1');

save Cfnet;

y2 = [1.000 1.061 1.128 1.198 1.269 1.340

1.411 1.480 1.547 1.614 1.679 1.743

1.807 1.872 1.937 2.003 2.070 2.138

2.206 2.274 2.341

1.000 1.066 1.137 1.209 1.282 1.354

1.426 1.496 1.565 1.633 1.700 1.766

1.832 1.897 1.963 2.029 2.096 2.164

2.234 2.305 2.378

1.000 1.105 1.195 1.277 1.352 1.424

1.492 1.559 1.626 1.691 1.757 1.822

1.886 1.950 2.014 2.077 2.141 2.206

2.273 2.344 2.422

1.000 1.079 1.171 1.271 1.374 1.476

1.575 1.667 1.753 1.832 1.905 1.973

23


2.037 2.099 2.160 2.221 2.283 2.345

2.407 2.467 2.521

1.000 1.057 1.128 1.208 1.294 1.384

1.476 1.571 1.667 1.766 1.866 1.969

2.075 2.182 2.291 2.399 2.505 2.603

2.690 2.758 2.800

1.000 1.037 1.080 1.130 1.185 1.246

1.311 1.381 1.457 1.539 1.628 1.725

1.830 1.943 2.066 2.197 2.336 2.483

2.634 2.789 2.943

1.000 1.016 1.039 1.067 1.099 1.135

1.175 1.220 1.269 1.324 1.385 1.452

1.529 1.614 1.710 1.819 1.941 2.080

2.236 2.412 2.611

1.000 1.006 1.015 1.025 1.037 1.052

1.070 1.091 1.116 1.145 1.181 1.223

1.273 1.333 1.402 1.484 1.578 1.688

1.813 1.957 2.121

1.000 0.992 0.984 0.974 0.966 0.958

0.952 0.947 0.945 0.946 0.950 0.958

0.970 0.988 1.011 1.042 1.081 1.130

1.191 1.267 1.359

1.000 0.980 0.960 0.935 0.915 0.895

0.875 0.840 0.833 0.825 0.815 0.800

0.790 0.785 0.780 0.780 0.785 0.795

0.815 0.845 0.890

1.000 0.970 0.945 0.910 0.885 0.855

0.825 0.800 0.775 0.750 0.730 0.710

0.688 0.670 0.657 0.642 0.635 0.628

0.625 0.630 0.640

1.000 0.965 0.930 0.890 0.860 0.825

0.790 0.760 0.730 0.700 0.670 0.645

0.620 0.597 0.575 0.555 0.538 0.522

0.510 0.502 0.500

1.000 0.955 0.910 0.870 0.830 0.790

0.755 0.720 0.685 0.655 0.625 0.595

0.567 0.538 0.510 0.489 0.470 0.452

0.438 0.428 0.420

24


];

y2 = reshape(y2,size(y2,1)*size(y2,2),1);

[n1,p] = size(X);

Cdnet = feedforwardnet(10);

Cdnet = train(Cdnet,X',y2');

save Cdnet;

25


Appendix B: Main file

clc

clear all

lb = [ 35 1 1 7 25 ]';

ub = [ 40 3 5 11 30]' ;

x0 = [ 39 1 1 11 27 ]';

options = optimset('FunValCheck','on');

[xopt,fa, exitflag, output, lambda, grad, hessian] =

fmincon(@func,x0,[],[],[],[],lb,ub,@nonlcon,options)

26


Appendix C: Constraints File

function [ c,ceq ] = nonlcon( x )

w = x(1);

t = x(2);

n = x(3);

N = x(4);

q = x(5);

P = 2;

Db = 2032;

d = 47.05;

sigma = 1144;

Dm = Db + w + t*n;

ex = d/N;

Kr = 1;

tp = t*((Db/Dm)^0.5);

Ac = (pi*q/2 + 2*(w-q/2))*tp*n;

s2 = (((P/100)*Dm*Kr*q)/(2*Ac))/0.01;

c(1) = s2 - sigma;

%%%%%%%%%%%%%%%%%%%%%%%%%%

j = q/2/w;

k = q/(2.2*((Dm*tp)^0.5));

load Cpnet;

27


load Cdnet;

load Cfnet;

Cp = sim(Cpnet,[j k]');

Cd = sim(Cdnet,[j k]');

Cf = sim(Cfnet,[j k]');

kf = 1;

ef =100*(((log(1+(2*w/Db)))^2+((log(1+(n*tp)/(2*q/4))^2)))^0.5);

ysm=1.5*(1+(9.94*10^-2*(kf*ef))-(7.59*10^-4*(kf*ef)^2)-(2.4*10^-

6*(kf*ef))+(2.21*10^-8*(kf*ef)^4));

Cm = ysm;

if Cm>3

Cm = 3;

end

s3 =(P/100*w)/(2*n*tp)/0.01;

s4 =0.01*P/(2*n)*((w/tp)^2)*Cp/0.01;

c(2) = s3 + s4 - Cm*sigma;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Ctheta = 1;

Ed = 1935199;

fin =(1.7*Dm*0.01*Ed*tp^3*n)/(w^3*Cf);

c(3) = 4-(0.34*pi*Ctheta*fin)/(N*N*q)/0.01;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

28


Sy = 3044;

delta = (Cp/(2*n)*((w/tp)^2))/(3*(s2/P));

alpha = 1 + 2*(delta^2) + (1-2*(delta^2)+4*(delta^4))^0.5;

c(4) = 3.5 - (1.3*Ac*0.01*Sy)/(Kr*Dm*q*(alpha^0.5))/0.01;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

a = 3.4;

b = 54000;

C = 1860000;

e = ex;

Er = 1989687;

s5 = (Er*0.01*(tp^2)*e)/(2*(w^3)*Cf)/0.01;

s6 = ((5*0.01*Er*tp*e)/(3*(w^2)*Cd))/0.01;

st = (0.7*(s3+s4) + (s5+s6))*14.22334;

if st>b

Nc = (C/(st-b))^a;

else

Nc = 100000000;

end

c(5) = 10000 - Nc;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

equip = 800;

c(6) = fin - equip;

29


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

c(7) = N*q-2*Db;

c(8) = 0.6-w/q;

c(9) = w/q-1.6;

c(10) = w-2*Db/N;

%%%%%% To make sure values for Cd, Cf, and Cp are within allowable

range

c(11) = -q/2/w;

c(12) = q/2/w - 1;

c(13) = -q/(2.2*((Dm*tp)^0.5));

c(14) = q/(2.2*((Dm*tp)^0.5))-1;

ceq = [];

end

30


Appendix D: Objective Function

function f = func(x)

w = x(1);

t = x(2);

n = x(3);

N = x(4);

q = x(5);

Dm = 2032 + w + t*n;

tp = t*((2032/Dm)^0.5);

f = (2*pi*N*n*t*Dm*(w-q/2+pi*q/4) + 2*3.14*n*t*50*(2032+n*t));

end

31


Appendix E: Experimental Values of Cd, Cp and Cf depending on design variables

32


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