Configurations and pressure levels optimization of heat recovery steam generator using the genetic algorithm method based on the constructal design

Accepted Manuscript Research Paper Configurations and pressure levels optimization of heat recovery steam generator using the genetic algorithm method based on the constructal design Morteza Mehrgoo, Majid Amidpour PII: DOI: Reference:

S1359-4311(17)30028-5 http://dx.doi.org/10.1016/j.applthermaleng.2017.04.144 ATE 10293

To appear in:

Applied Thermal Engineering

Received Date: Revised Date: Accepted Date:

15 January 2017 30 March 2017 27 April 2017

Please cite this article as: M. Mehrgoo, M. Amidpour, Configurations and pressure levels optimization of heat recovery steam generator using the genetic algorithm method based on the constructal design, Applied Thermal Engineering (2017), doi: http://dx.doi.org/10.1016/j.applthermaleng.2017.04.144

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Configurations and pressure levels optimization of heat recovery steam generator using the genetic algorithm method based on the constructal design

Morteza Mehrgooa,1, Majid Amidpour a, 2 a

Department of Energy System Engineering, Faculty of Mechanical Engineering, K. N. Toosi

University of technology, Tehran, Iran

Abstract In last two decades, there was a great deal of attention on the optimum design and performance improvement of the heat recovery steam generator (HRSG) units. In the present work, considering different objective functions and utilizing the constructal design method, three configurations of HRSG are compared. The design method is based on the constructal theory and optimization technique is carried out by varying the geometric design parameters and steam pressure levels for different values of the exhaust gas temperatures. Optimum conditions of HRSG are obtained with the help of the genetic algorithm under the fixed total volume constraint. For each configuration of HRSG, optimal distribution of the heat surfaces (sizes) subject to the total volume constraint are derived such that the 1

[email protected] Corresponding author: Tel.:+98 21 8406 3222; fax: +98 21 8867 4748 E-mail address: [email protected] (M. Amidpour). 2

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objective function is optimum. It is shown that how the geometric and thermodynamic design variables of HRSG can be achieved, simultaneously. Features that resulted from the constructal design are the number of tubes, configurations and aspect ratios for the main sections, the tube diameters and rate of the steam production at each pressure level. The results revealed that variations in different objective functions are strongly affected by the hot gas inlet temperature. In addition, the use of several pressure levels in HRSGs causes a considerable increase in the power production, declines irreversibility in HRSGs and allows producing higher steam flow rate for all values of the inlet gas temperature. The constructal principle invoked in this paper represents that geometrical form of systems can be deduced from a single principle.

Keywords: Construtal Theory, Configuration, Efficiency, Heat recovery steam generator, Genetic algorithm, Power production

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1.

Introduction

Heat recovery steam generators (HRSG) are used to recover heat from high temperature exhaust gas leaving gas turbine and generate steam at different pressure levels. The efficiency of combined cycle power plant is affected by the design of all its components, which HRSG is one of the most important of them. So, any change in its design directly affects the cycle efficiency, its power generation, the global cost and many other variables in the cycle. Therefore, the design parameters of HRSG should be carefully selected in order to maximize the heat recovery and improve the overall performance of the combined cycle. A lot of efforts and work have been carried out regarding modeling and optimization of different thermal systems. Sanjay et al. [1] performed parametric energy and exergy analysis of reheat gas–steam combined cycle using closed-loop steam-cooling. It was shown that the reheat gas–steam combined cycle plant with closed-loop-steam-cooling enhanced thermal efficiency (around 62%) and plant specific work. Godoy et al. [2] optimized designs of a CCGT power plant characterized by maximum second law efficiency values for a wide range of power demands and different values of the available heat transfer area. Valdes et al. [3] performed a thermoeconomic optimization of combined cycle gas turbine power plants using a genetic algorithm. They proposed two different objective functions; aimed at minimizing the cost of production per unit electricity and maximizing the 3

annual cash flow. Mohagheghi et al. [4] developed a computer code to examine the competence for different types of HRSG by the thermodynamic optimization. They obtained a high rate of generating power in the steam cycle. Bracco et al. [5] developed a mathematical model to optimize one pressure level HRSG using first and second low approach. Different objective functions have been analyzed, some of which refer only to the exergy balance of the heat recovery steam generator while others involve the completely bottoming cycle. Tajik Mansouri et al. [6] investigated the effect of HRSG pressure levels on exergy efficiency of combined cycle power plants. Three types of combined cycles, with the same gas turbine as a topping cycle were evaluated. A double pressure, and two triple pressure HRSGs were modeled. They showed how an increase in the number of pressure levels of the HRSG affects the exergy losses due to heat transfer in the HRSG and the exhaust of flue gas to the stack. Naemi et al. [7] investigated the thermodynamic and thermoeconomic analyses to achieve the optimum operating parameters of a dual pressure heat recovery steam generator, coupled with a heavy duty gas turbine. An extensive sensitivity analysis is performed to compare optimum pinch point for different electricity and fuel prices. Massaldi et al. [8] proposed a mixed integer non-linear programming (MINLP) model to optimize the equipment arrangement and operating conditions of CCPPs. General Algebraic Modelling System (GAMS) was used to implement and solve 4

the mathematical model. Carapellucci et al. [9] undertook a thermoeconomic optimization of several types of HRSG adapted to three types of GT and investigated the effect of fuel price and capacity factor on the electricity cost. Several researchers have done further studies on the performance evaluation and optimization of the HRSG during the three last decades [10–15]. On the other hand, Constructal theory, introduced first by Bejan [16], deals mainly with shapes and generation of flow configurations. In last decade, the theory found a widespread usage in optimizing a large variety of engineering problems, to optimize shape and structure [17-18]. A growing body of work and literature illustrate the constructal theory applied to different topics (steam generator architecture [19-22], desalination [23-25], assembly of fins [26-27], Heterogeneous porous media [28], design and performance evaluation of different types of heat exchangers [29–33], design and modeling of heat and mass transfer in a solid–gas reactor [34-35], evolutionary design of condensers [36], electrokinetics systems and enclosures [37], solar pond fields [38] and Phase change heat storage [39]). As mentioned, there are several studies on HRSG unit most of which are restricted to a number of thermodynamic and thermoeconomic evaluation, mathematical modeling and performance improvement of the HRSG units. To the authors’ knowledge, there are limited studies on the geometric optimization of HRSG unit. Also, widespread applications of the constructal theory demonstrated that it can be 5

used to deliver the geometric features of engineering systems. Therefore, it can adequately be used for novel design and new design concepts of HRSG. The present research work is aimed at introducing a new way of conceptual design of HRSG units by applying the constructal law. The main motivation behind this work is the application of the constructal theory in several engineering systems. The originality of the paper is in the new looking at the design of the heat recovery steam generator systems. The other feature distinguishing this study is optimizing different objective functions (such as maximizing the power output) using the genetic algorithm method by varying the complex configurations of heat exchangers. The optimization work has been carried out under the constraint of finite size by using the combination of heat transfer, thermodynamics, and fluid dynamics. A mathematical model for HRSG system, which considered three configurations of HRSG with different numbers of the main heat exchangers, is developed. The related equations of the geometric parameters, thermodynamic variables and heat transfer equations are combined to identify and recognize all the feasible and competing configurations. Furthermore, the effects of various parameters such as the gas temperature on optimal configuration and power production of system are investigated.

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2. Process description and problem formulation The constructal law complements the existing principles (mass, momentum and energy conservation and the second law of thermodynamics) and provides new insights into finding the flow configuration. Therefore, constructal design of a HRSG unit based on the optimization of different objective functions under the condition of global constraint (the fixed total volume) is a robust design method which is represented in this section. The goal of this study is to simultaneously determine the optimal values of the pressure levels, geometric and thermodynamic design variables for several HRSG configurations according to different objective functions. Three configurations of HRSG with different numbers of pressure level are considered in this article that are represented in Figs. 1-4. As shown, each pressure level consists of three main sections including an economizer, an evaporator and a superheater. A reheater is added before high pressure superheater to reheat the LP steam leaving the HP turbine. All HRSGs are considered as water tube type with natural circulation evaporator. There are a large number of geometric variables for each section which significantly affect the operation and performance of the system. Geometric design parameters are different for each section and various dimensions and arrangements of the tubes (configurations) could be used. Obviously, each configuration has its 7

own heat transfer characteristics, pressure drop and entropy generation. Moreover, Power production and system efficiency are significantly changed by the tube configurations of units. So, obtaining the optimum values of the geometric and operating design parameters of HRSG, five objective functions are considered. For three configurations of HRSG, these objective functions are evaluated and optimized subject to the total volume constraint. Furthermore, optimum values of geometric variables (configurations) and thermodynamic parameters are obtained. As shown in Figs.1-4, the geometric design parameters for each section of HRSG (economizers, evaporators, reheater and superheaters) are different which consist of tube diameters, number of tube rows in the direction of flow, number of tubes per row, number of fins per unit length of tube. In order to have a HRSG with uniform section, it is considered that the length of the tubes (Ly) and the width of each section (Lz) to be the same for all sections.

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(a)

(b) Fig.1. Single pressure HRSG configuration: a) flow diagram, b) overall dimensions.

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For each section (heat exchanger) of the HRSG, there are three geometric design variables. So, a heat recovery steam generator which composed of three pressure levels (as shown in Fig. 3) has thirty-three geometric variables. These variables consist of: ten tubes diameters (dHS, dHR, dHB, dHE, dIS, dIB, dIE dLS, dLB, dLE), ten longitudinal pitch ratios (αLHS, αLHR, αLHB, αLHE, αLIS, αLIB, αLIE, αLLS, αLLB, αLLE), ten lengths for main sections of HRSG (LxHS, LxHR, LxHB, LxHE, LxIS, LxIB, LXIE, LxLS, LxLB, LxLE ), one transverse pitch ratio (αz), one width (Lz) and one height (Ly) which are shown in Figs.1-4. Thermodynamic variables are the water saturation pressure at each pressure level, the gas temperature at the outlet of the each heat exchanger, superheater temperature (Tsup) and water to steam flow rate ratio (ṁw/ṁg). It is necessary to consider these relatively large numbers of variables simultaneously to obtain the major operating parameters of the HRSG and optimize different objective functions. For all three configurations, the total volume of the HRSG is deemed to be fixed and has the same value. The geometry is free to vary according to this constant size obtaining the optimum value for different objective functions. Due to space limitation, only the modeling of triple pressure HRSG is presented.

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(a)

(b) Fig.2. dual pressure HRSG configuration: a) flow diagram, b) overall dimensions 11

(a)

(b) Fig.3. Triple pressure HRSG configuration: a) flow diagram, b) overall dimensions.

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Fig.4. Geometric configurations of tubes in the main sections of HRSG

(a) 13

(b)

(c) Fig.5. Temperature-enthalpy diagram: a) HRSG 1P, b) HRSG 2P, c) HRSG 3P. 14

Geometric equations of HRSG with three pressure levels and a reheat are as follows:

(13) (14)

Only the surfaces which heat transfer occurs are taken into account. Therefore, the volume of the drums and down comer tubes of the evaporators are not considered 15

in the total volume. Moreover, the vertical length of tubes (Ly) apart from the bowshaped tube for connecting is the same for main sections. Transverse length is the same for main sections as considered in Eq. (13). The transverse and longitudinal pitch ratios are αz and αL, respectively. The value of αz is considered to be equal for all exchangers but αL could be different for main sections. The energy balances are applied to the all components of HRSG, through ε-NTU method. Correlations of ε-NTU method can be obtained or derived from correlations given in the standard literature. Energy balance equations for the triple pressure HRSG in the above-mentioned heat exchangers are as follows [40]: Economizer:

Where

Evaporator:

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Reheater

Superheater:

The ε-NTU correlations are as follows [40]: For economizers, reheater and superheaters:

For evaporators: 17

The overall heat transfer coefficient for extended surface can be obtained from:

Where do, di and kt are tube outer and inner diameters, thermal conductivity of the tube wall, respectively. Fouling factors for inside and outside of the tubes are ffi and ffo. At, Ai and Aw are defined in Ref. [40] and fin efficiency (ηf) is defined in Ref. [41]. Forced convection model with turbulent flow is considered to evaluate the heat transfer coefficient inside the HRSG tubes (hi). Correlation for fully developed turbulent flow in the tube is expressed as [41]:

Where Re is Reynolds number that is expressed as

Where ṁw is mass flow rate of the water, di is inner diameter of the tube, μw is water viscosity, and Pr is Prandtl number that is defined as

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Where Cpw and kw are specific heat and thermal conductivity of the water, respectively. Heat transfer outside the HRSG tubes (ho) is considered as forced convection due to cross flow of the hot flue gas over fin tubes.

Where hf and Tf are the fin height and fin tip temperature. kg, µg and Cpg are thermal conductivity, viscosity and specific heat of flue gas, respectively. C1, C3 and C5 are defined in ref. [40]. Where G in Eq. (32) is called gas mass velocity and is defined as follows

For solid fins, the correlation between base tube wall and fin tip temperature is given by [40]:

The first and second orders of modified Bessel function are k1, I1, k0, I0, respectively. (mre and mr0 are defined in ref. [40]).

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The following expressions are used to evaluate the pressure drop for gas side (in-line arrangement) in each section of the HRSG: 39

Where NL and ρg are the number of rows deep and gas density. C2, C4, C6 and B are given in Ref. [40]. Nw, Ly and ST are the number of the transverse tubes, the length of the tube and transverse pitch, respectively. 3. Optimization approach The genetic algorithm (GA) is a population based optimization technique that searches the best solution of a given problem based on the concepts of natural selection, genetics and evolution. The search is made starting from an initial population of individuals, often randomly generated. An individual is considered a possible candidate solution for the optimization problem in hand. At each evolutionary step, individuals are evaluated using an objective function. Three types of operators do the evolution (i.e., the generation of a new population): breeding, mutation and selection while selection includes killing a given proportion of the population based on probabilistic “survival of the fittest”. 20

Killed individuals are superseded by children, which are created by breeding the remaining individuals in the population. For each child produced, breeding first requires probabilistic selection of two parent individuals, getting more chance to choose fitter individuals. Mutation allows new areas of the response surface to be explored by random alterations of optimization variables. GA iteratively improved the set of tentative solutions by applying the aforementioned stages to find a good solution. 3.1.

Objective functions

The following objective functions are considered and the above three proposed configurations of HRSG are optimized for each function. 3.1.1. Maximum power production (42)

3.1.2. Maximum heat recovery rate

3.1.3. Minimum entropy generation

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3.1.4. Maximum energetic efficiency

3.1.5. Maximum exergetic efficiency

3.2.

Decision variables

In thermal system design and optimization, it is convenient to identify two types of independent variables. These variables are decision variables and parameters. The decision variables may be varied in optimization process. However, the parameters remain fixed in a given application. All other variables are dependent variables. Their values are calculated from independent variables using thermodynamic relations. Genetic algorithm using Matlab software optimization toolbox is herein employed to optimize objective functions, decision variables and constraints. The tuning

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parameters of the optimization program are presented in Table 2 and the decision variables of the genetic algorithm, used in this study, are as follows: The length of the each heat exchanger

1.5(m) ≤ Lx,i ≤10(m), i=1:10

(48)

The length of the tubes

3 (m) ≤Ly≤12(m)

(49)

The width of each section

3 (m) ≤Lz≤12(m)

(50)

The longitudinal pitch ratios

1.5 ≤ αL,i ≤ 4.5, i=1:10

(51)

The transverse pitch ratio

1.5 ≤ αz ≤ 5, i=1:10

(52)

The tube diameters of the

do,i = 33.4, 42.2, 48.3,60.3, 73,

(53)

each heat exchangers

88.9 (mm), i=1:10

First pressure level (LP)

2 ≤PLP≤ 12 (bar)

(54)

Second pressure level (IP)

10 ≤PLP≤ 40 (bar)

(55)

Third pressure level (HP)

40 ≤PLP≤ 160 (bar)

(56)

To select the values of the tube diameters (discrete decisions), ten decision variable (30 ≤ do,i ≤ 90 (mm)) are defined and the following conditions are considered to select the conventional tube diameters for each section of the HRSG. The imposed conditions to select the discrete decision variables are: if 30 ≤ do,i ≤ 40 :

do,i= 33.4 (mm)

if 40 < do,i ≤ 45 :

do,i= 42.2 (mm)

if 45 < do,i ≤ 55 :

do,i= 48.3 (mm)

if 55 < do,i ≤ 70 :

do,i= 60.3 (mm)

if 70 < do,i ≤ 80 :

do,i= 73 (mm)

if 80 < do,i ≤ 90 :

do,i= 88.9 (mm) 23

3.3.

Assumptions

Input parameters, which are considered in modeling and optimizing of the system, are represented in Table.1. In addition, some of the main assumptions to develop the mathematical model are as follows:  System is at steady state.  The pressure drop in the water steam line is neglected.  There is no extraneous heat loss Table. 1 Input parameters which are used in the mathematical model Input Parameters T0

ṁg Pcond V Fin configurations in the economizers

Unit o C kg/s bar m3

Value 25 657.5 0.12 2500

cm

1.18/0.19/1.27

cm

1.57/0.19/1.27

cm

0.4/0.19/1.27

(No. of fins per (cm)/ fin thickness / fin height )

Fin configurations in the evaporator (No. of fins per (cm)/ fin thickness / fin height )

Fin configurations in the superheater (No. of fins per (cm)/ fin thickness / fin height )

3.4.

Constraints

The following limitations are considered for decision variables in the optimization study. These constraints are related to the operating restrictions HRSG and must be satisfied:

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Table. 2. Tuning parameters in the Genetic Algorithm Tuning parameters

Value

Population size

400

Maximum no. of generations

700

Minimum function tolerance

1e-5

Probability of crossover (%)

80

Probability of mutation (%)

1

Number of crossover point

2

Selection process

Tournament

Tournament size

2

4. Constructal Design purpose (or objective, function)

25

deduced

freedom

26

5. Results and discussion A computer program based on the procedure explained in the previous section was developed. The geometric parameters and values of the pressure levels are

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considered as decision variables. In addition, total volume of the HRSG is deemed fix and several constraints are considered to optimize the objective functions. Optimization is done using genetic algorithm method. In order to optimize the objective functions at each step, it is assumed that gas flow rate is fixed and optimization process is done for different values of the gas inlet temperature. As a way of illustration, ṁg and Tinw are considered as fixed parameters while Ting is changed from 350 (oC) to 650 (oC). For a given inlet gas temperature, each objective function is optimized using the genetic algorithm method subjected to the fixed volume constraint (Eq.1). Values of the decision variable and thermodynamic parameters are obtained by the above-mentioned procedure. It is noteworthy that thermodynamic and geometric variables are dependent. It means if all of the geometric variables are known, thermodynamic variables and gas pressure drop can be obtained. The main assumptions, optimization approach, constraints and the results are given below. 5.1.

Model Validation

To verify the validity of the suggested model, the model outputs are compared with different data from literature. The comparison between model results and Ref. [15] is represented in Table.3. It is shown that results are close to the data of ref. [15] and there is a slight difference between them, which is acceptable.

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Moreover, the results are validated by two other researches work (Refs. [4] and [13]) which are shown in Table.4 and Fig.6. The imposed constraints such as the limited total volume and pinch point criteria are reasons of the slight deviations presented in this section. In sum, there is an admissible conformity among the results and literature. Table. 3 Model validation by Ref. [15] Input Parameters

Unit

Ref. [15] Data

Model data

Exhaust gas mass flow rate

kg/s

657.5

657.5

C

644

644

bar

0.12

0.12

o

Exhaust gas temperature Condenser pressure

Output variables 1st level pressure

bar

4.2

4.319

2nd level pressure

bar

25.3

25.8

3rd level pressure

bar

142.5

152.83

1st level steam mass flow rate

kg/s

12.63

15.59

2nd level steam mass flow rate

kg/s

12.76

3rd level steam mass flow rate

kg/s

86.7

82.69

C

565

570.3

MW

145.5

151.81

Maximal steam cycle temperature Net output

o

21.33

Table. 4 Comparison of the power production ratio for different types of HRSGs 2 Pressure levels

Woptimum/Wmax,single pressue

3 Pressure levels

Ref. [4] Data

Model data

Ref. [4] Data

Model data

1.177

1.181

1.225

1.241

29

20

1P, Ref. [13] 1P, Constructal

18

2P, Ref. [13]

2P, Constructal

16

3P, Ref. [13]

ṁw/ṁg ,%

14

3P, Constructal

12 10 8 6 300

350

400

450

500

550

600

650

700

Ting ( °C) Fig.6. Comparison of the produced steam to gas flow rate ratio with Ref. [13].

5.2.

Optimal Designs

5.2.1. Maximum Power Production In order to avoid mechanical degradation and deterioration of the aerodynamic performances of the last stages of steam turbine, the steam fraction should be higher than 88%. To maintain a good operation of HRSG and turbine material, the steam pressure and temperature at HRSG exit should not exceed 160 bar and 580oC, respectively [13]. The optimal values of the steam pressure at inlet of the HP turbine for different values of the inlet gas temperature are depicted in Fig. 7. The limitation of the steam quality at outlet of the LP turbine restricts the maximum values of HP steam pressure. The superheated steam temperature has 30

lower values at lower amount of the inlet gas temperature that increases the possibility of reducing the steam quality at outlet of LP turbine. Therefore, the values of HP steam pressure are lower at lower Ting and increases by soaring the hot gas temperature. Moreover, for a given value of Ting, it is possible to reach higher steam pressure by adding a pressure level. Variations in the optimal power production as a function of the inlet hot gas temperature for the three different configurations of HRSG are represented in Fig.8. The power consumption of the pumps corresponding to the maximum power production are calculated and these values are subtracted from the total power production, for all three configurations. The isentropic efficiency of pumps is considered equal to 85%. As shown, power production strongly depends on the inlet gas temperature. Increasing the hot gas temperature, would enhance the driving force for heat transfer. Therefore, at higher temperatures, the steam production and output power increase. Furthermore, power production for the three HRSG configurations are compared. For all values of Ting, triple pressure HRSG leads to higher values of work than HRSG 2P and 1P, respectively.

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180 160 140

PHP (bar)

120 100 80 60 40 20 0 350

400

450

500

525

550

Ting ( °C)

575

600

3P

650

2P

1P

Fig.7. Optimum values of the high pressure (HP) versus Ting (OF: Maximum Power).

1P

180

2P

3P

160

Wmax (MW)

140 120

100 80 60 40 20 0 300

350

400

450

500

550

600

650

700

Ting ( °C) Fig.8. Optimal power production as a function of the inlet hot gas temperature. 32

Considering Fig.9, it is clear that gas inlet temperature has significant effect on the total steam fractions (the ratio of the total produced steam to gas mass flow rate) produced by the three types of HRSG. As shown, ṁw/ṁg increases by the growth of gas inlet temperature. For all the values of Ting, the triple pressure HRSG produces more steam as compared with dual and single pressure level HRSGs. 1P

2P

3P

20 18

ṁw,t/ṁg , %

16 14 12 10 8 6 300

350

400

450

500

550

600

650

700

Ting ( °C) Fig.9. Effects of the gas inlet temperature on the total steam fractions produced by the three types of HRSG (OF: Maximum Power).

For instance, the total produced steam to gas flow rate ratios for 1P, 2P and 3P at Ting= 600oC, are 13.53%, 15.86% and 16.88%, respectively. The total size of the HRSG (the total volume) is deemed to be fixed and has the same value for all three configurations. Therefore, the same size the HRSGs have, the same approximate cost would be.

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The steam production ratios for each pressure level versus the gas inlet temperature for 2P and 3P HRSGs are distinctively depicted in Fig. 10. The steam production at HP level is much higher than that of the IP and LP pressure levels. This difference increases by rising the gas inlet temperature. By way of illustration, steam production to gas flow rate ratios for LP, IP and HP pressure levels, at Ting= 550oC, are 2.53%, 3.53% and 9.14%, respectively. These values for LP and HP levels of the dual pressure HRSG are 5.8% and 8.8%, respectively. In the constructal design of the HRSG, the aim is to search for the best designation of the heat transfer surface among the heat exchangers. Optimal dimensions of the main heat exchangers for maximum power are shown in Table. 5. The results show that for the triple pressure HRSG, about 34.5% of the total size is allocated to the evaporators, 31.8% to the economizers, 24.7% to the superheaters and 9% to the reheater. For the dual and single pressure HRSGs, the highest value of the heat transfer area are dedicated to the evaporators, too. According to Table. 5, inlet temperature of the flue gas has negligible effect on the geometric parameters of the HRSG which the value of the heat transfer surface of the three configurations of HRSG remain approximately constant by changing Ting from 350 oC to 650 oC.

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0.2 0.18 0.16

ṁw,i/ṁg

0.14 0.12

0.1

ṁw1/ṁg

0.08

ṁw2/ṁg

ṁw/ṁg

0.06 0.04 0.02 0 350

400

450

500

550

600

650

Ting ( °C) (a) 0.2 0.18 0.16

ṁw,i/ṁg

0.14 0.12 ṁw1/ṁg

0.1

ṁw2/ṁg

0.08

ṁw3/ṁg ṁw/ṁg

0.06 0.04 0.02 0 350

400

450

500

550

600

650

Ting ( °C) (b) Fig.10. The steam production ratios for each pressure level versus the gas inlet temperature: a) 2P HRSG, b) 3P HRSG (OF: Maximum Power). 35

Table. 5. Optimal length for main heat exchangers of the HRSG a. Optimal dimensions for the three configuration of HRSG Ting (oC)

Lx (m)

1P Ly (m)

Lz (m)

Lx (m)

2P Ly (m)

Lz (m)

Lx (m)

3P Ly (m)

Lz (m)

350 400 450 500 550 600 650

30.91 31.68 31.63 32.06 32.01 31.80 33.08

9.23 9.08 9.11 9.11 7.81 8.61 9.01

8.77 8.69 8.67 8.56 10.00 9.13 8.39

31.70 31.64 31.62 31.83 31.89 31.65 31.77

9.94 9.98 9.98 9.91 9.87 9.94 9.95

7.93 7.91 7.92 7.93 7.94 7.94 7.91

43.28 43.46 43.30 43.84 43.29 43.01 43.68

8.62 8.02 8.46 8.22 9.10 8.51 9.37

6.70 7.18 6.83 6.94 6.35 6.83 6.11

b. Optimal aspect ratio for the single and dual pressure HRSG 1P*

2P*

Ting 350 400 450 500 550 600 650

16.81 16.81 40.90 25.48 15.98 15.98 43.82 24.22 16.13 16.13 43.29 24.45 15.74 15.74 44.66 23.86 16.07 16.07 43.51 24.35 16.01 16.01 43.70 24.27 15.13 15.13 46.79 22.94 * The unit of aspect ratio is ‘%’.

10.16 10.18 10.18 10.12 10.10 10.18 10.14

10.16 10.18 10.18 10.12 10.10 10.18 10.14

18.21 18.76 18.61 18.82 18.80 18.15 18.60

15.39 15.42 15.43 15.33 15.30 15.42 15.36

11.86 11.88 11.89 11.81 11.79 11.88 11.84

18.76 18.10 18.21 18.40 18.55 18.71 18.50

15.46 15.49 15.50 15.40 15.36 15.48 15.43

22.02 22.06 22.07 21.93 21.89 22.06 21.97

36.97 36.86 36.82 37.22 37.35 36.86 37.10

30.85 30.91 30.93 30.73 30.67 30.90 30.79

34.40 34.67 34.42 34.75 34.41 34.72 35.01

31.87 31.73 31.86 31.94 31.87 31.55 31.57

c. Optimal aspect ratio for the triple pressure HRSG 3P* Ting 350 400 450 500 550 600 650

7.44 8.96 12.36 10.00 7.41 8.93 12.42 9.95 7.44 8.96 12.96 9.99 7.35 8.85 12.15 10.36 7.44 8.96 12.24 10.00 7.29 9.02 11.92 10.01 7.37 8.88 8.69 9.90 * The unit of aspect ratio is ‘%’.

9.01 8.97 9.01 8.90 9.01 9.07 8.93

9.86 12.35 12.10 12.68 9.03 10.45 12.69

36

11.09 11.05 11.09 10.95 11.09 11.16 10.99

8.32 8.28 8.31 8.21 8.32 8.37 8.24

12.18 9.90 9.36 9.92 13.14 12.36 13.62

10.78 10.74 10.78 10.63 10.78 10.39 10.68

24.77 24.67 24.76 24.45 24.76 24.73 24.54

Fig. 11 shows the optimal values of the UA for maximum power production as a function of the inlet gas temperature for three layouts of the HRSG. The results show that for all configurations, most of the UA is dedicated to the evaporators and economizers. For instance, about 52% and 21% of the total UA for the dual pressure HRSG are earmarked to the evaporators and economizers. Moreover, it is obvious that Ting has a slight effect on allocation of UA to the heat exchangers. This results supports the one presented by Table. 5 which comes to the conclusion that the heat transfer areas of the main section of the HRSG have slender variation by the gas inlet temperature. Cart Title

0.07

0.08

0.09

0.07

0.08

0.09

0.07

0.08

0.09

0.07

0.07

0.09

0.34

0.21

0.23

0.34

0.21

0.23

0.34

0.21

0.23

0.34

0.21

0.24

0.34

0.21

0.06

0.07

0.23

0.34

0.21

0.23

0.35

0.08

0.06

0.06

0.08

0.06

0.21

0.23

0.8

0.07

1

0.05

1P 2P 3P 1P 2P 3P 1P 2P 3P 1P 2P 3P 1P 2P 3P 1P 2P 3P 1P 2P 3P

0.38

0.51

0.55

0.38

0.51

0.55

0.39

0.52

0.56

0.38

0.52

0.56

0.38

0.52

0.58

0.39

0.53

0.58

0.39

0.55

0.4

0.59

UAi/UAT

0.6

0.2 0.21

0.20

0.12

0.21

0.20

0.12

0.21

0.19

0.12

0.21

0.20

0.12

0.21

0.20

0.11

0.21

0.20

0.11

0.21

0.19

0.11 0

350 350 350 400 400 400 450 450 450 500 500 500 550 550 550 600 600 600 650 650 650

Ting ( °C)

UAEco/UAT

UAEva/UAT

UASup/UAT

UARH/UAT

Fig. 11. Optimal allocation of the UA for maximum power production. 37

5.2.2 Maximum Heat Recovery In this section, maximization of the heat transfer subject to the total volume constraint is considered as the objective function. The variations of the maximum heat recovery with the hot gas inlet temperature is depicted in Fig.12. This figure illuminates a high sensitivity of the heat recovery to Ting and also a moderate increase in its value by escalating the number of pressure level. As shown, heat recovery in the triple pressure HRSG has higher value in comparison with dual and single pressure HRSGs. For example, heat recovery in 3P HRSG when Ting=500oC is equal to 262.4 MW which is 6% and 35% higher than that of the 2P and 1P HRSGs. 400

1P

350

2P 3P

300

Qmax (MW)

250 200 150 100 50 0 300

350

400

450

500

550

600

650

700

Ting ( °C) Fig. 12. Variations of the maximum heat recovery with the hot gas inlet temperature. 38

5.2.3 Minimum Entropy Generation The entropy generation of HRSG can be evaluated by using Eq. (44). Considering Fig.13, it is clear that the inlet gas temperature has significant effect on the entropy generation number where Ns decreases by increasing the amount of Ting. Furthermore, the number of pressure levels is another parameter that effects on the entropy generation. Optimal values of Ns markedly decline by adding a pressure level to HRSG where entropy generation numbers are 0.25, 0.21 and 0.19 for 1P, 2P and 3P HRSGs, respectively. This is in accordance with the second law of thermodynamics that in heat exchangers with one cold and one hot stream, the irreversibility of the heat transfer process is less for small temperature differences between cold and hot streams, and the possibility of using exergy depending on the use of (heating or cooling uses) heat exchangers is more [4]. Therefore, distribution of heat exchangers in HRSG to different sections and using these heat exchangers in a specific temperature range decrease the irreversibility of the heat transfer process. The evolution of exergy destruction with the inlet gas temperature for three configurations of the HRSG are illustrated by Fig. 14. It can be seen that the HRSG with highest-pressure level has the lowest value of exergy destruction. Triple pressure HRSG destroys less exergy as compared with dual and single pressure

39

HRSG, for all Ting values. In addition, this figure represents that exergy destruction rate declines by increasing the gas inlet temperature. 0.40

1P 2P

0.35

3P

0.30

Ns

0.25 0.20 0.15 0.10 300

350

400

450

500

Ting ( °C)

550

600

650

700

Fig. 13. Optimized entropy generation number versus Ting. 1P 0.5

2P

3P 0.45 0.4

Ed

0.35 0.3 0.25 0.2 350

400

450

500

Ting ( °C)

550

600

650

Fig.14. Rate of exergy destruction as a function of inlet gas temperature for three configurations of the HRSG. 40

5.2.4 Maximum Thermal Efficiency Fig. 15 reveals variation of the optimal thermal performance as an objective function versus the inlet gas temperature. As it shows, for all three configurations, energetic efficiency increases by escalating Ting and the curves have upward trend. Moreover, increment of the number of pressure level improves the thermal efficiency. For single pressure HRSG, ??th at Ting=500oC is equal to 32.64%, whereas thermal efficiency of 2P and 3P HRSGs are 36.13% and 38.25% that shows about 3% efficiency improvement by adding each pressure level. This trend can be anticipated from Qmax and Wmax (Figs.8 and 12) where by soaring the inlet gas temperature and number of pressure level, heat recovery and power production have upward trend. So, energetic efficiency would have the same behavior. 0.45

0.4

0.35

ηth 0.3

0.25

0.2 300

350

400

450

500

550

600

Ting ( °C)

Fig.15. The optimal thermal performance of the systems. 41

650

1P

700

2P

3P

5.2.5 Maximum Exergy Efficiency The results of the second law efficiency optimization subject to the fixed total volume constraint is represented in Fig.16. According to this figure, single pressure HRSG has lowest value of the exergetic efficiency that its value remarkably increases by rising the inlet gas temperature and difference between exergy efficiency of the three configurations of the HRSG decreases by rising Ting. For example, at Ting=450 oC, the ??EX for 3P, 2P and 1P are 0.731, 0.671 and 0.61 while at Ting=650 oC, these values are 0.742, 0.765 and 0.787, respectively.

0.8

0.75

0.7

1P 2P

0.65

3P

𝜂EX 0.6

0.55

0.5

0.45 300

350

400

450

500

550

600

Ting ( °C)

Fig.16. Optimal exergetic efficiency versus Ting. 42

650

700

6.

Conclusion In summary, optimization of the three configurations of HRSG system on the

basis of the constructal law was performed using a comprehensive mathematical model. The model solves a set of linear and non-linear equations describing the performance of the components in HRSG units. Several objective functions were considered and optimized by using the genetic algorithm method subject to the total volume constraint. The results revealed that variations in different objective functions strongly affected by the hot gas inlet temperature. Also, the use of several pressure levels in HRSGs causes a considerable increase in the power production, declines irreversibility in HRSGs and allows producing higher steam flow rate for all values of the inlet gas temperature. Moreover, the following general conclusions can be used for all operation conditions of HRSG systems.  Steam production at HP level is much higher than that of the LP and IP levels.  Most of the heat transfer area is dedicated to evaporators.  For all configurations, most of the UA is allocated to the evaporators and economizers.  The heat recovery in the triple pressure HRSG has higher value in comparison with single and dual pressure HRSG.

43

 The HRSG with highest pressure level has the lowest value of exergy destruction.  Optimal values of entropy generation number markedly decline by adding a pressure level to HRSG.  The heat transfer areas of the main sections of HRSG have slender variation by the gas inlet temperature. This general constructal design proves to be very efficient in the case of HRSG design and can be used as a practical model in many engineering applications. Results show that the present design method can adequately satisfy the designer requirements and can be sufficiently used in problems with more complicated conditions. 7. References [1] A. Ongiro, V.I. Ugursal, A.M. Al Taweel, J.D. Walker, Modeling of Heat Recovery Steam Generator Performance, Applied Thermal Engineering 16 (1997) 427-444. [2] N. Subhramanyam, S. Rajaram and N. Kamalnathan, HRSGs for Combined Cycle Power plants, Heat Recovery System & CHP 15 (1997) 153-161.

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51

Nomenclature Ai

tube inner surface area (m2)

At

surface area of finned tube (m2)

Aw

average wall surface area (m2)

APP

Approach Point (K)

Cmin

minimum heat capacity rate (kJ kg-1 K-1)

Cp

specific heat (kJ kg-1 K-1)

C1, 3, 5

dimensionless factors, Eq. (32)

C2, 4, 6

dimensionless factors, Eq. (36)

CCGT

Combined Cycle Gas Turbine

CCPP

Combined Cycle Power Plant

GT

Gas Turbine

d

diameter (m)

ffi

fouling factors inside (W-1 m2 K)

ffo

fouling factors outside (W-1 m2 K1)

G

gas mass velocity (kg m-2 s-1)

h

heat transfer coefficient (W m-2 K-1)

hf

height of fin (m)

hfg

Latent heat (kJ kg-1)

I0

modified Bessel function of first kind (ν=0)

I1

modified Bessel function of first kind (ν=1)

K0

modified Bessel function of second kind (ν=0)

K1

modified Bessel function of second kind (ν=1) 52

Kt

thermal conductivity (W m-1 K-1)

L

length (m)

ṁ

flow rate (kg s-1)

NTU

number of transfer unit

NL

number of rows deep

Ns

entropy generation number

Nu

Nusselt Number

Nw

number of tubes through transverse direction

P

pressure (Pa)

 Q

rate of heat transfer (kJ)

Re

Reynolds number

Pr

Prandtl number

R

specific gas constant (kJ kg-1 K-1)

S

pitch (m)

T

temperature (K)

Ting

gas inlet temperature to high pressure superheater (K)

Tb

gas outlet temperature from high pressure superheater (K)

Tc

gas outlet temperature from reheater (K)

Td

gas outlet temperature from high pressure evaporator (K)

Te

gas outlet temperature from high pressure economizer (K)

Tf

gas outlet temperature from intermediate pressure superheater (K)

Tg

gas outlet temperature from intermediate pressure evaporator (K)

Th

gas outlet temperature from intermediate pressure economizer (K) 53

Ti

gas outlet temperature from low pressure superheater (K)

Tj

gas outlet temperature from low pressure evaporator (K)

Tk

gas outlet temperature from low pressure economizer (K)

Tsat,HP

high pressure saturation temperature (K)

Tsat,IP

Intermediate pressure saturation temperature (K)

Tsat,LP

low pressure saturation temperature (K)

Tsup,HP

high pressure steam temperature (K)

Tsup,IP

Intermediate pressure steam temperature (K)

Tsup,LP

low pressure steam temperature (K)

V

total volume (m3)

U

overall heat transfer coefficient (W m-2 K-1)

Greek symbols ρ

density (kg m-3)

µ

Viscosity (N s m-2)

ε

heat exchanger effectiveness

α

ratio of tube pitch to diameter

ΔP

gas side pressure drop (kPa)

ηf

fin effectiveness

ηth

Thermal efficiency

ηEX

Exergetic efficiency

Subscripts eco

economizer 54

eva

evaporator

f

fin

g

flue gas

HS

high pressure superheater

HB

high pressure evaporator

HE

high pressure economizer

i

inside

in

inlet (for stream)

IS

Intermediate pressure superheater

IB

Intermediate pressure evaporator

IE

Intermediate pressure economizer

L

longitudinal

LS

low pressure superheater

LB

low pressure evaporator

LE

low pressure economizer

o

outside

out

outlet (for stream)

RH

Reheater

sat

saturation

sup

superheater

T

total

w

water

x

length or longitudinal dimension

55

y

height

z

width or transverse dimension

0

environment

Abbreviation 1P

one pressure level HRSG

2P

two pressure level HRSG

3P

three pressure level HRSG

LB

total length of the all evaporator sections

LE

total length of the all economizer sections

LS

total length of the all superheater sections

OF

Objective Function

56

Figure captions Fig.1. Single pressure HRSG configuration: a) flow diagram, b) overall dimensions. Fig.2. dual pressure HRSG configuration: a) flow diagram, b) overall dimensions. Fig.3. Triple pressure HRSG configuration: a) flow diagram, b) overall dimensions. Fig.4. Geometric configurations of tubes in the main sections of HRSG Fig.5. Temperature-enthalpy diagram: a) HRSG 1P, b) HRSG 2P, c) HRSG 3P. Fig.6. Comparison of the produced steam to gas flow rate ratio with Ref. [13]. Fig.7. Optimum values of the high pressure (HP) versus Ting (OF: Maximum Power). Fig.8. Optimal power production as a function of the inlet hot gas temperature. Fig.9. Effects of the gas inlet temperature on the total steam fractions produced by the three types of HRSG (OF: Maximum Power). Fig.10. The steam production ratios for each pressure level versus the gas inlet temperature: a) 2P HRSG, b) 3P HRSG (OF: Maximum Power). Fig. 11. Optimal allocation of the UA for maximum power production. Fig. 12. Variations of the maximum heat recovery with the hot gas inlet temperature. Fig. 13. Optimized entropy generation number versus Ting. Fig.14. Rate of exergy destruction as a function of inlet gas temperature for three configurations of the HRSG. Fig.15. The optimal thermal performance of the systems. Fig.16. Optimal exergetic efficiency versus Ting.

57

Table captions Table.1. Input parameters which are used in the mathematical model. Table.2. Tuning parameters in the Genetic Algorithm Table.3. Model validation by Ref. [15]. Table.4. Comparison of the power production ratio for different types of HRSGs. Table.5. Optimal length for main heat exchangers of the HRSG: Optimal dimensions for the three configuration of HRSG, b) Optimal aspect ratio for the single and dual pressure HRSG, c) Optimal aspect ratio for the triple pressure HRSG.

Research Highlights

 Utilizing the constructal design method, three configurations of HRSG are compared.  Different objective functions are optimized using the genetic algorithm method.  Optimization is done by varying the geometric parameters and steam pressure levels.  The best configuration of each heat exchanger is derived without any extra step.  The effects of inlet gas temperature on the operating parameters is investigated.

58