Optimizing the Mass of Expansion Joints Used in. Thermal Power Plants

Transcription

1 Optimizing the Mass of Expansion Joints Used in Thermal Power Plants Term Project Report ME 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 GSI: Alparslan Emrah Bayrak Abhishek Goyal M.S.E in Mechanical Engineering Department of Mechanical Engineering UM ID: University of Michigan abgoyal@umich.edu

2 Contents Introduction... 3 Importance of Expansion Joint... 3 Mathematical model... 7 Model Analysis Design Optimization Parametric Analysis Discussions Acknowledgements: Appendix A: Neural Network MATLAB Code Appendix B: Main file Appendix C: Constraints File Appendix D: Objective Function Appendix E: Experimental Values of C d, C p and C f depending on design variables 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 C d Figure 9: Error Histogram for Neural Network of C d Figure 10: Regression Plot for Neural Network of C d Figure 11: Training Graph for Neural Network of C d Figure 12: Parametric analysis of objective function with axial expansion Table 1: Manufacturing Constraints on Variables... 8 Table 2: Monotonicity Analysis Table 3: Design Parameters Table 4: Comparison of result with actual values Table 5: Testing of objective function with different starting points

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, Figure 1: Manufacturing of Expansion Joint The equipment A is supplying superheated steam at 200 o C 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 5 o steel is m / m C. Based on the given data, we can calculate the change in length as Figure 2: Importance of Expansion Joint 5 L L T m 32mm. The strain in the piping due to high 3

4 temperature will be L Strain From stress strain relationship, we can find L 10 the stress generated in the piping. The Young s Modulus of Elasticity for steel is 200 GPa. Stress E Strain Stress 200GPa Stress 640MPa 310 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. Normal Temperature Elevated Temperature Figure 3: Alternative Solutions to Expansion Joints 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. 4

5 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. Figure 4: Working of Expansion Joint Expansion Joint Equipment Figure 5: Actual Installation of Expansion Joints 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 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. 5

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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: m 2 NntD ( w 2 r r ) 2 ntl( D nt) m m m b 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. 7

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. P Dm Kr q 2 A c allowable P = Inside pressure Dm = Outer diameter of expansion joint = Db + w + nt Kr = circumferential stress factor = 1 q q Ac = Cross sectional area of convolution = 2 w t p n 2 2 where tp is t D D b m 8

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 P w P w C C 2 n t p 2 n t p 2 p m allowable 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 C f in 2 N q 4 where, fin = m b p D E t n, 3 Cθ = 1; w C f 4. Internal design pressure based on in-plane instability and local plasticity at temperature below creep range should be greater than Ac Sy 3.5 where K D q r m Sy = Yield strength at design temperature after completion of bellow forming = 3044 kg/cm C p w 2n t p S2 3 P 2 5. Fatigue life at room temperature should be greater than 10,000 cycles 9

10 N N c c c St b a 3.4 b c 1.86E 06 2 b p 5 3 a S 0.7( S S ) ( S S ) S S t E t e 2 w C 5 E t p e 3 w C b f d x e 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 m b p 3 w N C f D E t n k allowable Dimensional constraints will be: 7. Maximum convoluted bellow length should be less than or equal to twice of the diameter of pipe 2 Db N q 8. The ratio of width and pitch should be greater than 0.6 w 0.6 q 10

11 9. The ratio of width and pitch should be less than 1.6 w 1.6 q 10. The convolution height should not be greater than w 2 Db N 2 Db N Therefore we have, min weight = f (w,t,n,n,q) subject to P Dm Kr q g1: allowable 0 2 A c w g9: q g2: g3: g4: 2 P w P w Cp Cm allowable 0 2 n t p 2 n t p D E t n 4 0 N q w C 3 C m b p 2 3 f 1.3 Ac Sy K D q r m 2 D g10: w b 0 N g11: 1 n 0 g12: n 5 0 g13: 7 N 0 g14: N 11 0 g5: N c Dm Eb t p n g6: 0 3 k allowable w N C g7: N q 2 D b 0 g8: 0.6 w 0 q f g15: 35 w 0 g16: w 40 0 g17: 25 q 0 g18: q 30 0 g19: 1 t 0 g20: t

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 g g g g g g g7 + + g8 - + g9 + - g g11 - g12 + g13 - g14 + g15 - g16 + g17 - g18 + g19 - g

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 C d Figure 8: Validation for Neural Network of C d 13

14 Figure 10: Regression Plot for Neural Network of C d Figure 11: Training Graph for Neural Network of C d 14

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) 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 Width of convolution Number of plies Thickness of each ply Number of convolutions Objective function 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 15

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 ] 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: e-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 [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 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. 16

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

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. 18

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 = [

20 ]; 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 = [

21

22 ]; 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 = [

23

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; 24

25 Appendix B: Main file clc clear all lb = [ ]'; ub = [ ]' ; x0 = [ ]'; options = optimset('funvalcheck','on'); [xopt,fa, exitflag, output, lambda, grad, hessian] = fmincon(@func,x0,[],[],[],[],lb,ub,@nonlcon,options) 25

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; 26

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 = ; 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; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 27

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) = (1.3*Ac*0.01*Sy)/(Kr*Dm*q*(alpha^0.5))/0.01; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% a = 3.4; b = 54000; C = ; e = ex; Er = ; 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))* ; if st>b Nc = (C/(st-b))^a; else Nc = ; end c(5) = Nc; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% equip = 800; c(6) = fin - equip; 28

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 29

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 = 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 30

31 Appendix E: Experimental Values of C d, C p and C f depending on design variables 31

32 32