Sequential Quadratic Programming Algorithm Based Optimization of Shell and Tube Type Heat Exchangers

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Sequential Quadratic Programming Algorithm Based Optimization of Shell and Tube Type Heat Exchangers

Alok Kumar Maurya1

Department of mechanical engineering MIET, Greater Noida, UP, India

Vikas Kumar Wankar2

Department of mechanical engineering MIET, Greater Noida, UP, India

Prashant Kumar Sharma3 Department of mechanical engineering MIET, Greater Noida, UP, India

Sanjay Singh Bhadoria4 Department of mechanical engineering MIET, Greater Noida, UP, India

Abstract-Shell and Tube type heat exchangers are having special importance in boilers, oil coolers, condensers and pre-heaters. These are also widely used in process applications as well as the refrigeration and air conditioning industry. The robustness and medium weighted shape of Shell and Tube type heat exchangers make them well suited for high pressure operations. The basic configuration, the thermal analysis and design of such exchangers form an included part of the mechanical, thermal and chemical engineering scholars for their curriculum and research activity.

Traditional design approaches using graph sheets are time consuming, these may not considered all the variables and constraints simultaneously. On the other hand some new evolutionary algorithms viz. Genetic Algorithm (GA), Particle swarm optimization(PSO),Imperialist competitive algorithm (ICA) are not simple to understand by every designer and are not easy to be implemented. Therefore, in present work, a new shell and tube heat exchanger optimization design approach is discussed based on sequential quadratic programming (SQP). The SQP algorithm has some good features in reaching to the global minimum in comparison to other evolutionary algorithms. In present study, SQP technique has been applied to minimize the total cost which includes capital investment and total discounted operating cost. The design variables considered in the present work are tube outer diameter, shell diameter and baffle spacing. A matlab code is developed based on SQP for optimal design of shell and tube heat exchangers. The different test cases are solved using code to demonstrate the effectiveness and accuracy of the proposed algorithm. The results using developed

code are compared to those obtained from previous literatures. It is found that the SQP algorithm is simple and it can be successfully applied for optimal design of shell and tube heat exchangers with higher accuracy.

Key Words: Shell and tube type heat exchangers, Optimal Design, Sequential Quadratic Programming.

  1. INTRODUCTION

    A shell and tube heat exchanger is a class of heat exchanger designs. It is the most common type of heat exchanger in oil refineries and other large chemical processes, and is suited for higher-pressure applications. As its name implies, this type of heat exchanger consists of a shell (a large pressure vessel) with a bundle of tubes inside it. One fluid runs through the tubes, and another fluid flows over the tubes (through the shell) to transfer heat between the two fluids. The set of tubes is called a tube bundle, and may be composed of several types of tubes: plain, longitudinally finned, etc.

    Shell and tube type heat exchanger is probably the most used and widespread type of the heat exchangers classification. It is used most widely in various fields such as oil refineries, thermal power plants, chemical industries and many more. This high degree of acceptance is due to the comparatively large ratio of heat transfer area to volume and weight, easycleaning methods, easily replaceable parts etc. Shell and tube type heat exchanger consists of a number of tubes through which one fluid flows. Another fluid flows through the shell which encloses the tubes and other supporting items like baffles, tube header sheets, gaskets etc. The heat exchange between the two fluids takes through the wall of the tubes. A schematic diagram of shell and tube type heat exchanger is given below: [1,2]

    Fig 1.1: A Shell and Tube Type Heat Exchanger [1] NOMENCLATURE

    a1,a2,a3 numerical constant Ntnumber of tubes as cross sectional area normal to flow direction (m2)Ppumping power (W)

    1. baffles spacing (m) Pr Prandtl number

      Cl clearance (m) Pttube pitch (m)

      Cp specific heat(kJ/kg K) Rffouling resistance (m 2 K/W) Ci capital investment (Rs) Qheat duty (W)

      Ce energy cost (Rs /kW hr) Re Reynolds number

      Co annual operating cost (Rs/yr) S heat transfer surface area (m ) 2

      Cod total discounted operating cost (Rs) Ttemperature (ºC)

    2. tottotal annual cost (Rs)U overall heat transfer coefficient (W/m2K)

    Ds

    shell diameter (m)

    Greek symbols

    f

    friction factor

    P pressure drop (Pa)

    F

    correction factor

    T logarithmic mean temperature difference (ºC)

    h

    heat transfer coefficient (W/m2 K)

    dynamic viscosity (Pa s)

    H

    annual operating time (hr/yr)

    kinematic viscosity (m2/s)

    i

    annual discount rate (%)

    density (kg/m3)

    k

    thermal conductivity (W/m K)

    overall pumping efficiency

    K1

    numerical constant

    Subscripts

    L

    tubes length (m)

    i inlet

    m

    mass flow rate (kg/s)

    o outlet

    n

    number of tubes passages

    s belonging to shell

    ny

    equipment life (year)

    t belonging to tube

    n1

    numerical constant

    Ds

    shell diameter (m)

    Greek symbols

    f

    friction factor

    P pressure drop (Pa)

    F

    correction factor

    T logarithmic mean temperature difference (ºC)

    h

    heat transfer coefficient (W/m2 K)

    dynamic viscosity (Pa s)

    H

    annual operating time (hr/yr)

    kinematic viscosity (m2/s)

    i

    annual discount rate (%)

    density (kg/m3)

    k

    thermal conductivity (W/m K)

    overall pumping efficiency

    K1

    numerical constant

    Subscripts

    L

    tubes length (m)

    i inlet

    m

    mass flow rate (kg/s)

    o outlet

    n

    number of tubes passages

    s belonging to shell

    ny

    equipment life (year)

    t belonging to tube

    n1

    numerical constant

    do tube diameter (m) v fluid velocity (m/s)

  2. LITERARTURE REVIEW

The basic configuration of shell and tube heat exchangers, the thermal analysis and design of such exchangers form an included part of the mechanical, thermal,chemical engineering scholars for their curriculum and research activity.In recent past year, the improvements in computing cost have increased the interest of engineers and researchers to simulate heir problems with computational and numerical methods. A lot of computational tools and methods have been developed in the last decades to analyse fluid dynamics, combustion, and different modes of heat transfer.

Srivastava A.K., Dubey V.V.P., Verma R.R., Verma P.S. have presented an overview of shell and tube

type heat exchanger, constructional details, design methods and the reasons for the wide acceptance of shell and tube type heatexchangers [1]. ShahR.K. and SekulibD.R. have given the classification of shell and tube type heat exchangers based on heat transfer process, constructional features and flow arrangements [2].Sinnot R.K. has presented various chemical processing equipment theory and design, (e.g. heat exchanger,) [3]. Kern D.Q. has discussed various types of heat transfer processes and design of engineering equipment explained [4].

CaputoA.C., PelagaggeM.P., SalineP.,have presented a procedure for optimal design for shell and tube heat exchangers which utilized a genetic algorithm to minimize the total discounted cost of the equipment

including the capital investment and pumping related annual energy expenditures [5].Taal M., Bulatov I., Klemes J., Stehlik P. have given the most common methods used for cost estimation of heat exchange equipment in the process industry and the sources of energy price projections andconsideredten methods for heat exchanger costing procedure [6].Peters MS, Timmerhaus

K.D. have presented methods of plant design and economics. Further these methods are used for calculation of heat exchangers total annual cost and pressure drop at shell side [7].PhilipG.E., LaurentJ.O., Michael L.W., LindaP.R., Sharmad V., have proposed a

sequential quadratic programming (SQP) method for the optimal control of large-scale dynamical systems and various steps of sequential quadratic programming method algorithm is also discussed [8]. Philip G.E., Wong E., have proposed the sequential quadratic programming (SQP) method for the solution of constrained nonlinear optimization problems and also compared with other optimization methods [9].Patel V.K., Rao R.V., have discussed a non-traditional optimization technique; called particle swarm optimization (PSO), for design optimization of shell and tube heat exchangers from economic view point and minimization of total annual cost is considered as an objective function [10].Hadidi A., Hadidi A., Nazari A., have presented a new design approach for shell and tube heat exchangers using imperialist competitive algorithm (ICA) from economic point of view.ICA technique has been applied to minimize the total cost of the equipment including capital investment and the sum of discounted annual energy expenditures related to pumping of shell and tube heat exchanger. Finally the results are compared to those obtained by other literature approaches [11].

3.1 SEQUENTIAL QUADRATIC PROGRAMMING ALGORITHM

Sequential Quadratic Programming (SQP) is one of the most successful methods for the numerical solution of constrained nonlinear optimization problems. It relies on a strong theoretical foundation and provides powerful algorithmic tools for the solution of large-scale technologically relevant problems. The problem which is considered to solve is to minimize some objective or cost function, f(x), subject to constraints ai(x) = 0 for i

= 1,2,…,p and cj(x) > 0 for

j =1,2,…,q. The f(x) can be a linear or nonlinear objective function.ai(x) and cj(x) are constraints which are functions of x and can be nonlinear. f(x),ai(x) and cj(x) are assumed to be continuous and have continuous second partial derivatives, and the feasible region of this problem is assumed to nonempty. A solution of the such type of problem generally requires an iterative procedure to establish a direction of search at each major

iteration. This is usually achieved by the solution of an LP, a QP, or an unconstrained subproblem.The quadratic programming subproblem is created using initial objective function as quadratic and linearizing constraints about a starting point. This method uses constraints steepest descent (CSD) method for search direction. The solution of quadratic programming problem is used as starting point for next iteration; therefore this method is called Sequential Quadratic Programming method.

It is important to note that method work equally well when initiated from feasible or infeasible points. It can also treat equality and inequality constraints.

    1. NUMERICAL DATA AND RESULTS

      The effectiveness and validity of the suggested approach in this work is assessed by analyzing some relevant case studies taken from the literature, in order to have reliable reference sizing data for the sake of comparison. The following two different test cases, representative of a wide range of possible applications, are considered. The first case study of this work is a heat exchanger for methanol and brackish water, taken from [3]. The heat load is 4.34 MW. This heat exchanger has two tube side passages with triangle pitch pattern and one shell side passage. The second case study is taken from [4] is a heat exchanger which transfers a heat load of 0.46 MW between distilled waterraw water heat exchanger and has two tube side passages with triangle pitch pattern and one shell side passage. The same configuration of above cases is retained in the present approach. For each case the original design specifications, shown in Table [2,4], were taken as input to the optimization algorithm and the resulting optimal exchangers design parameters given by the SQP method were compared with the original design suggested in literatures [3] and [4].

      The following upper and lower bounds for the optimization variables were imposed: Tubes outside diameter do ranging from 0.01 m to 0.051 m; Shell internal diameter Ds ranging between 0.1 m and 1.5 m; Baffles spacing B ranging from 0.05 m to 0.5 m.

      All values of discounted operating costs were computed with ny=10 years, annual discount rate=10%, Energy cost Ce = 0.12 Rs/kW hr, And annual amount of work hours

      =7000 hr/yr similar to other researches. [3, 7, 10, 11]

    2. Case study 1:This case study is taken from Sinnot

[3] and the process inputs and physical properties are given in table 4.1:

Case-1

m (kg/s)

Ti (c)

To (c)

(kg/m3)

Cp(kJ/kg)

µ (Pa-s)

k (W/mK)

Rf (m2K/W)

Shell side: methanol

27.8

95

40

750

2.84

0.00034

0.19

0.00033

Tube side: sea water

68.9

25

40

995

4.2

0.00080

0.59

0.00020

Table 4.2: Parameters of the optimal shell and tube heat exchangers for case study 1 using different optimization methods

Parameters

Literature [3]

GA [5]

PSO [10]

ICA [11]

SQP [present work]

do (m)

0.02

0.016

0.015

0.015

0.015

Ds (m)

0.894

0.83

0.81

0.879

0.786

B (m)

0.356

0.5

0.424

0.5

0.5

L (m)

4.83

3.379

3.115

3.107

3.2115

Pt (m)

0.025

0.02

0.0187

0.01875

0.0188

Cl (m)

0.005

0.004

0.0037

0.00375

0.0037

De (m)

0.014

0.011

0.0107

0.011

0.0107

Nt

918

1567

1658

1752

1550

vt (m/s)

0.75

0.69

0.67

0.699

0.7885

Ret

14925

10936

10503

10429

11769

Prt

5.7

5.7

5.7

5.7

5.7

2

ht (W/m K)

3812

3762

3721

3864

4814.5

ft

0.028

0.031

0.0311

0.031

0.030

Pt (Pa)

6251

4298

4171

5122

7449.8

2

as (m )

0.032

0.0831

0.0687

0.0879

0.0786

vs (m/s)

0.58

0.44

0.53

0.42

0.4718

Res

18381

11075

12678

9917

10928

Prs

5.1

5.1

5.1

5.1

5.1

2

hs (W/m K)

1573

1740

1950.8

1740

1957

fs

0.33

0.357

0.349

0.362

0.3569

Ps (Pa)

35789

13267

20551

12367

14318

U (W/m2K)

615

660

713.9

677

740.4033

S (m2)

278.6

262.8

243.2

256.6

234.4616

Ci (Rs)

3863025

3694425

3483975

3627750

3389100

Co (Rs/year)

158325

71025

77902.5

73125

82897.5

Cod (Rs)

972975

436350

508365

449625

509362.5

Volum

e 8, IssCutoet (1R0 s)

4836000

41P3u0b7l7is5hed

by, w39w9w2.3ij3e2rt..5org

4077450

3898462.5

221

Fig 4.1: Cost comparison for case study 1

Table 4.3: The process input and physical properties for case study 2

Case-2

m (kg/s)

Ti (c)

To (c)

(kg/m3)

Cp(kJ/kg)

µ (Pa-s)

k (W/mK)

Rf (m2K/W)

Shell side: distilled water

22.07

33.9

29.4

995

4.18

0.00080

0.62

0.00017

Tube side: raw water

35.31

23.9

26.7

999

4.18

0.00092

0.62

0.00017

Parameters

Literature[4]

GA[5]

PSO[10]

ICA[11]

SQP[present work]

do (m)

0.019

0.016

0.0145

0.015

0.015

Ds (m)

0.387

0.62

0.59

0.66

0.576

B (m)

0.305

0.440

0.423

0.5

0.5

L (m)

4.880

1.548

1.45

1.467

1.717

Pt (m)

0.023

0.020

0.0181

0.01875

0.0187

Cl (m)

0.004

0.004

0.0036

0.00375

0.0037

De (m)

0.013

0.015

0.0103

0.011

0.0107

Nt

160

803

894

897

781

vt (m/s)

1.76

0.68

0.74

0.745

0.8001

Ret

36409

9487

9424

10390

10425

Prt

6.2

6.2

6.2

6.2

6.2

ht (W/m2 K)

6558

6043

5618

5412

4489.8

ft

0.023

0.031

0.0314

0.031

0.0311

Pt (Pa)

62812

3673

4474

3497

4442

as (m2 )

0.0236

0.0541

0.059

0.0657

0.0576

vs (m/s)

0.94

0.41

0.375

0.36

0.3851

Res

16200

8039

4814

5130

10059

Prs

5.4

5.4

5.4

5.4

5.4

hs (W/m2 K)

5735

3476

4088.3

5239

6337.9

fs

0.337

0.374

0.403

0.3998

0.3614

Ps (Pa)

67684

4365

4721

4696

5022.9

U (W/m2K)

1471

1121

1177

1243

1221.5

S (m2)

46.6

62.5

59.15

62.05

63.163

Ci (Rs)

1241175

1437225

1396050

1431975

1378230

Co (Rs/year)

334950

20400

20700

20475

21121.515

Cod (Rs)

2058000

125325

127200

125925

129780

Volu

Ctot (Rs)

e 8, Issue 10

3299175

1562550

Published

1523250

by, www.ijert.org

1557900

1508010

222

Parameters

Literature[4]

GA[5]

PSO[10]

ICA[11]

SQP[present work]

do (m)

0.01

0.016

0.0145

0.015

0.015

Ds (m)

0.387

0.62

0.59

0.66

0.576

B (m)

0.305

0.440

0.423

0.5

0.5

L (m)

4.880

1.548

1.45

1.467

1.717

Pt (m)

0.023

0.020

0.0181

0.01875

0.0187

Cl (m)

0.004

0.004

0.0036

0.00375

0.0037

De (m)

0.013

0.015

0.0103

0.011

0.0107

Nt

160

803

894

897

781

vt (m/s)

1.76

0.68

0.74

0.745

0.8001

Ret

36409

9487

9424

10390

10425

Prt

6.2

6.2

6.2

6.2

6.2

ht (W/m2 K)

6558

6043

5618

5412

4489.8

ft

0.023

0.031

0.0314

0.031

0.0311

Pt (Pa)

62812

3673

4474

3497

4442

as (m2 )

0.0236

0.0541

0.059

0.0657

0.0576

vs (m/s)

0.94

0.41

0.375

0.36

0.3851

Res

16200

8039

4814

5130

10059

Prs

5.4

5.4

5.4

5.4

5.4

hs (W/m2 K)

5735

3476

4088.3

5239

6337.9

fs

0.337

0.374

0.403

0.3998

0.3614

Ps (Pa)

67684

4365

4721

4696

5022.9

U (W/m2K)

1471

1121

1177

1243

1221.5

S (m2)

46.6

62.5

59.15

62.05

63.163

Ci (Rs)

1241175

1437225

1396050

1431975

1378230

Co (Rs/year)

334950

20400

20700

20475

21121.515

Cod (Rs)

2058000

125325

127200

125925

129780

Volu

Ctot (Rs)

e 8, Issue 10

3299175

1562550

Published

1523250

by, www.ijert.org

1557900

1508010

222

Table 4.4: Parameters of the optimal shell and tube heat exchangers for case study 2 using different optimization methods

m

Fig 4.2: Cost comparison for case study 2

The figure shows results and graphs of case study 2 by SQP used in optimization tool of Matlab. It may be observed that the objective function converges within 9 iterations for case

2.The first graph shows the optimization variables

values, second graphs shows the optimization function value with respect to the iterations and third graph shows maximum constraint violation with respect to the iterations.

5.1 CONCLUSIONS AND FUTURE SCOPES

Identifying the best and cheapest heat exchanger for a specific heat duty is a tough decision making task. The present work focuses upon total cost minimization of shell and tube type heat exchanger. The total cost includes capital investment cost and discounted operating cost. The design variables tube diameter, shell diameter and baffle spacing along with bounds and a nonlinear constraint have been considered. The resulting optimization problem has been solved using sequential quadratic programming (SQP)

algorithm, which is simple and easy for implementation.

A code has been implemented in Matlab for optimization purposes. Two cases are considered from previous literatures. The code developed in present work using SQP converges to optimum value of the objective function within quite few iterations. This feature signifies the importance of SQP. In each case study, starting point may either be feasible or infeasible, optimization problem converges to same optima.

CASE STUDY1: lower bound = [0.015 0.1 0.05] and upper bound = [0.051 1.5 0.5]

STARTING POINT

FINAL POINT

IMPROVEMENT IN TOTAL COST WITH RESPECT TO

FEASIBLE POINT [0.02 0.2

0.1]

[0.015 0.786

0.5]

ORIGINAL DESIGN[3]

GA[5]

PSO[10]

ICA[11]

INFEASIBLE POINT [0.01 0.2

0.1]

[0.015 0.786

0.5]

19.38%

5.62%

2.35%

4.389%

CASE STUDY 2: lower bound = [0.015 0.1 0.05] and upper bound = [0.051 1.5 0.5]

STARTING POINT

FINAL POINT

IMPROVEMENT IN TOTAL COST WITH RESPECT TO

FEASIBLE POINT [0.02 0.2 0.1]

[0.015 0.576

0.5]

ORIGINAL DESIGN[4]

GA[5]

PSO[10]

ICA[11]

INFEASIBLE POINT [0.01 0.2 0.1]

[0.015 0.576

0.5]

54.291%

3.49%

1.00%

3.202%

It may be concluded that total cost of a shell and tube type heat exchanger is decreased by using SQP in each case study and the obtained results show improvement as compare to those presented in previous literatures.In present work, the total cost of shell and tube type heat exchanger is optimized considering three design variables, two tube side passages and one shell side passage using SQP. In future, number of design variables such as

REFERENCES

  1. Srivastava AK, Dubey VVP, Verma RR, Verma PS, Shell & Tube Type Heat Exchangers: An Overview IJRAME ISSN (ONLINE): 2321-3051.

  2. Shah RK and Sekulib DR, University Of Kentucky, Heat exchanger classification.

  3. Sinnot RK, Coulson and Richardsons chemical engineering, chemical engineering design. vol. 6. Butterworth-Heinemann; 2005.

  4. Kern DQ, Process heat transfer. New York: McGraw-Hill; 1950.

  5. Caputo AC, Pelagagge PM, Salini P, Heat exchanger design based on economic optimization. Appl Therm Eng 2008;28:11519.

  6. Taal M, Bulatov I, Klemes J, Stehlik P, Cost estimation and energy price forecast for economic evaluation of retrofit projects. Appl Therm Eng 2003;23: 181935.

  7. Peters MS, Timmerhaus KD, Plant design and economics for chemical engineers. New York: McGraw-Hill; 1991.

  8. Gill E. Philip, Jayb O. Laurent, Leonardc W.

    Michael, Petzoldd R. Linda , Sharmad V, An SQP methodfor the optimal control of large-scale dynamical systems, Journal of Computational and Applied Mathematics 120 (2000) 197213.

  9. Gill E. Philip, WONG E, Sequential Quadratic Programming Methods, UCSD Department of Mathematics Technical Report NA-10-03

    August 2010.

  10. Patel VK, Rao RV, Design optimization of shell-and-tube heat exchanger using particle swarm optimization technique. Appl Therm Eng 2010;30:141725.

  11. Hadidi A.,Hadidi M. and Nazari A, A new approach for shell and tube heat exchangers using imperialist competitive algorithm, energy conversion and management 67 (2013)

    66-74.

  12. length, pitch, tube side passages and shell side passages may be increased, for the optimization of total cost. Further, the algorithm used in present work may also be utilised in some other applications like in maximizing the total revenue by a hydroelectric power plant.

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