 Open Access
 Authors : Alok Kumar Maurya, Vikas Kumar Wankar, Prashant Kumar Sharma, Sanjay Singh Bhadoria
 Paper ID : IJERTCONV8IS10057
 Volume & Issue : ENCADEMS – 2020 (Volume 8 – Issue 10)
 Published (First Online): 18072020
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
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
AbstractShell and Tube type heat exchangers are having special importance in boilers, oil coolers, condensers and preheaters. 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.

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 higherpressure 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)

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 (ÂºC)

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 (ÂºC)
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 (ÂºC)
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)


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 largescale 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 nontraditional 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 largescale 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.

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]

Case study 1:This case study is taken from Sinnot
Case1 
m (kg/s) 
Ti (c) 
To (c) 
(kg/m3) 
Cp(kJ/kg) 
Âµ (Pas) 
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
Case2 
m (kg/s) 
Ti (c) 
To (c) 
(kg/m3) 
Cp(kJ/kg) 
Âµ (Pas) 
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
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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.