Citation: Pietrasanta, A.M.; Mussati,
S.F.; Aguirre, P.A.; Morosuk, T.;
Mussati, M.C. Optimization of
Cogeneration Power-Desalination
Plants. Energies 2022,15, 8374.
https://doi.org/10.3390/en15228374
Academic Editor: Xiaolin Wang
Received: 19 September 2022
Accepted: 25 October 2022
Published: 9 November 2022
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energies
Article
Optimization of Cogeneration Power-Desalination Plants
Ariana M. Pietrasanta 1, Sergio F. Mussati 1, Pio A. Aguirre 1, Tatiana Morosuk 2,* and Miguel C. Mussati 1
1INGAR Instituto de Desarrollo y Diseño (CONICET-UTN), Avellaneda 3657, Santa Fe 3000, Argentina
2Institute for Energy Engineering, Technishe Universität Berlin, Marchstrs. 18, 10623 Berlin, Germany
*Correspondence: tetyana.mor[email protected]
Abstract:
The design of new dual-purpose thermal desalination plants is a combinatory problem
because the optimal process configuration strongly depends on the desired targets of electricity
and freshwater. This paper proposes a mathematical model for selecting the optimal structure, the
operating conditions, and sizes of all system components of dual-purpose thermal desalination plants.
Electricity is supposed to be generated by a combined-cycle heat and power plant (CCHPP) with
the following candidate structures: (a) one or two gas turbines; (b) one or two additional burners
in the heat recovery steam generator; (c) the presence or missing a medium-pressure steam turbine;
(d) steam generation and reheating at low pressure. Freshwater is supposed to be obtained from
two candidate thermal processes: and (e) a multi-effect distillation (MED) or a multi-stage flash (MSF)
system. The number of effects in MED and stages in MSF are also discrete decisions. Different case
studies are presented to show the applicability of the model for same cost data. The proposed model
is a powerful tool in optimizing new plants (or plants under modernization) and/or improving
existing plants for desired electricity generation and freshwater production. No articles addressing
the optimization involving the discrete decisions mentioned above are found in the literature.
Keywords:
combined-cycle heat and power plant; multi-effect distillation; desalination; muti-stage
flash desalination; MINLP; optimization
1. Introduction
Seawater desalination represents a pivotal technology to meet the freshwater supply
required for rapid population growth. More than twenty thousand desalination plants are
currently under operation in 150 countries. The majority of the large-scale seawater desali-
nation plants are dual-purpose ones. For power generation, either steam or combined-cycle
heat and power plants (CCHPP) are used. The steam is extracted at a temperature that
is required by a thermal seawater desalination plant. Dual-purpose power desalination
plants (DPPDP) offer several benefits over stand-alone desalination plants: significant re-
ductions in costs and increases in overall energy efficiencies. The electricity and freshwater
demands are the major design specifications, which can be met with several process struc-
tures and designs, leading to a combinatory problem. The power-to-water cogeneration
plant (El-Nashar [
1
]). Then, optimizing the process schemes (configurations) and operating
conditions play an important role in proposing cost-effective designs of DPPDPs.
The study on DPPDPs was carried out considering:
•
different types of power generation plants and desalination systems—Shahzad et al. [
2
],
Eveloy et al. [
3
], Mokhtari et al. [
4
], Al-Zahrani et al. [
5
], Ansari et al. [
6
], Wu [
7
],
Tian et al. [8], Eltamaly et al., 2021 [9], Ali et al., 2021 [10], and
•
several computational tools—ASPEN (Luo et al. [
11
]), EES (Tamburini et al. [
12
]), MAT-
LAB and Termoflex (Modabber and Manesh [
13
]), GAMS (Manassaldi et al. [
14
], and
Mussati et al. [15]).
Shahzad et al. [
2
] studied a CCHPP and multi-effect distillation (MED) desalination
system by applying an exergy-based analysis to develop an improved fuel cost estima-
tion method. The authors found that the exergy destruction of the desalination unit is
Energies 2022,15, 8374. https://doi.org/10.3390/en15228374 https://www.mdpi.com/journal/energies
Energies 2022,15, 8374 2 of 22
about 2–7% of the total exergy destruction. Eveloy et al. [
3
] investigated the integration
of a pressurized solid oxide fuel cell–gas turbine (SOFC-GT) hybrid system and a reverse
osmosis (RO) plant. With the help of a genetic algorithm (GA), they conducted the multi-
objective optimization using exergetic efficiency and total cost as objective functions. They
coupled the ASPEN process simulation with a non-dominated sorting multi-objective GA
supported by MATLAB. The different working fluids for the organic Rankine cycle (ORC)
were considered. Mokhtari et al. [
4
] and Al-Zahrani et al. [
5
] studied integrated systems
consisting of a GT system and MED and RO desalination processes. Al-Zahrani et al. [
5
]
implemented a mathematical model of the entire process using Engineering Equations
Solver (EES) to evaluate the values of the exergy destruction in the process components.
The GT combustion chamber showed the highest irreversibility, followed by the HRSG.
The MED process with thermal vapor compression (TVC) contributed 18% to the total
exergy destruction. Several authors investigated the integration of the pressurized water
reactor (PWR) in nuclear power plants and desalination systems (Ansari et al. [
6
], Wu [
7
],
and Tian et al. [
8
]). Ansari et al. [
6
] conducted the optimization using GA. Three opti-
mization problems were considered: a single-objective thermodynamic, a single-objective
thermoeconomic, and a multi-objective. In the multi-objective optimization (MOO), the
minimization of the product costs (electricity and freshwater) and the maximization of the
overall exergetic efficiency were solved using the Pareto frontier. In the thermoeconomic
optimization, the cost of generated power and freshwater production was reduced by
13.4% and 27.5%, respectively, with respect to a selected base case. Eltamaly et al. [
9
]
and Ali et al. [
10
] investigated hybrid renewable energy systems combining solar and
wind energies with reverse osmosis desalination units. Eltamaly et al. [
9
] applied different
optimization approaches, such as particle swarm optimization (PSO), bat algorithm (BA),
and others, based in social mimic technique. Optimization results show the preference of
usage BA algorithms compared to the other ones.
By using ASPEN Plus, Luo et al. [
11
] investigated a DPPDP consisting of a chemically
recuperated gas turbine (CRGT) and a MED-TVC system. They proposed to replace
the superheater of the HRSG by a steam methane reformer (SMR) to produce syngas.
The property estimation packages supported in ASPEN ‘RK-SOAVE’, ‘STEAM-TA’, and
‘ELECNRTL’ were used to calculate the properties of the working gas fluid, water, and
seawater, respectively. The authors found that the proposed CRGT system is economically
attractive only if a low-cost source of water is available. Using EES, Tamburini et al. [
12
]
studied the retrofitting of existing CHP systems considering a MED process with TVC
technology. They developed an analytical model to simulate plant operation under different
operating conditions.
In addition, some publications addressing different optimization methods should be
mentioned: metaheuristic approaches (Wu et al. [
16
], Shakib et al. [
17
], Hosseini et al. [
18
],
Modabber and Manesh [
19
]) and deterministic approaches (Zak [
20
], Manassaldi et al. [
14
],
Mussati et al. [
15
]. Wu et al. [
16
] proposed a mixed-coded GA to solve a mixed-integer
nonlinear programming (MINLP) model to optimize the process configuration and op-
eration conditions to satisfy specified electricity and freshwater demands at a minimum
total annual cost (TAC). They proposed a boiler and two candidate steam turbines for the
power plant—a back-pressure turbine and an extraction–condensation turbine—which are
modeled as discrete decisions. They proposed modifications to the classic GA to consider
these discrete decisions. For seawater desalination, a hybrid MSF/RO system is considered.
The resulting model and solution strategy were applied to several case studies considering
different freshwater demand levels. Shakib et al. [
17
] investigated a DPPDP consisting of
a GT with and without an air preheater (APH), HRSG, and MED-TVC. After simulating
the process and performing a thermoeconomic analysis, a multi-objective genetic algo-
rithm (MOGA) is applied to achieve the optimal design at the minimum cost of products
and maximal exergy efficiency. Zak [
20
] highlighted the need for numerical optimization
and detailed modeling to obtain cost-effective DPPDPs. One of the main advantages of
metaheuristic-based optimization approaches is that there is no need for the analytical
Energies 2022,15, 8374 3 of 22
knowledge of the equations system (i.e., gradient information of the design variables, in-
cluding the objective function), and there is a requirement for low computational resources
in providing solutions. However, they are derivative-free approaches at the same time,
which is a disadvantage from a rigorous optimization point of view, because the optimality
of the solutions cannot be guaranteed. Then, deterministic and gradient-based optimization
approaches are preferred over metaheuristic approaches. Manassaldi et al. [
14
] recently
developed a deterministic MINLP technique to address the optimal revamping of an exist-
ing DPPDP. Several optimization scenarios were investigated by using the simple branch
and bound (SBB) [
21
] as the derivative-based MINLP solver. One of them consisted in
optimizing the HRSG of the integrated CCHP/MSF desalter system by keeping the same
size of the GT and the same configuration of the MSF desalter as in the existing plant. The
influence of three-pressure (3P), two-pressure (2P), and one-pressure (1P) heat recovery
steam generators (3P-, 2P-, and 1P-HRSG, respectively) on the overall energy efficiency
was investigated.
The novelty of this work is to develop a deterministic mathematical model of combined
power and desalination systems that allows a systematic optimization of the configuration,
the sizes of the process components, and operating conditions to meet desired electricity and
freshwater demand at a minimum total annual cost. In this work, the number of candidate
configurations is much higher than those considered in [
14
,
15
,
20
]. For instance, besides
considering several candidate configurations involved in the CCHP, two thermal desalting
processes (MED and MSF) are the candidates to produce freshwater, significantly increasing
the combinatory nature of the problem and the degrees of freedom—the associated trade-
offs between the variables—for optimization. In addition, this work differs from [
20
] in the
application of a deterministic optimization approach instead of a metaheuristic algorithm.
2. Process Description
Figure 1shows a general configuration of a DPPDP consisting of a CCHP and a MED
desalination process.
The CCHP consists of a compressor (COMP), a combustion chamber (CC), and a
gas turbine (GT). The exhaust gases are used for the 2P-HRSG to produce steam. A part
of the steam leaving the steam turbine ST1 (S) is forwarded to the thermal desalination
(multi-effect distillation or multi-stage flash desalination units) to be used as the heating
source. The main design specification of any DPPDP is the ratio of the required electricity
to freshwater production (PWR), which strongly influences the optimal structure and
operating conditions. Typical PWR values expressed in MW power generated per million
gallons per day of water produced range from 3 to 20.
The simplest configuration of a combined-cycle power and desalination plant involves
a back-pressure steam turbine with a one-pressure heat recovery steam generator (1P-
HRSG). The HRSG can be designed for one, two, or three pressure levels influencing the
steam turbine network. For high PWR values, the power plant design is more critical
than the design of the desalination plant. For instance, for PWR values higher than 8,
the structure of the HRSG could involve two or three pressure levels with reheating of
steam at medium (or low) pressure level and/or auxiliary boilers. For low PWR values, the
design of the desalination process is more critical than the design of the power plant. For
instance, for PWR values lower than 3, the freshwater demand could be satisfied with a
large MED unit or a medium-small MSF unit involving a simple HRSG, i.e., one pressure
or two pressure levels. For values of PWR between 3 and 8, the designs of the power and
desalination plants have the same importance.
In Figure 1, the CCHPP was coupled with a MED desalination system, which can be
replaced with a MSF system, obtaining other configurations of DPPDPs.
Energies 2022,15, 8374 4 of 22
Energies 2022, 15, x FOR PEER REVIEW 4 of 22
Figure 1. Dual-purpose power and desalination plant (DPPDP).
2.1. Multiple Stage Flash (MSF) Desalination System
Figure 2a illustrates a simplified schematic of a MSF desalination system, and Figure
2b shows the representative system for the mathematical modeling.
The MSF system involves several stages. Each stage includes a preheater (HEX), pri-
mary flashing chambers (PFC, brine flashing), secondary flashing chambers (SFC, distil-
late flashing), and the main brine heater (MBH). In the HEXs, the incoming seawater
stream F is heated from TSW to T1F to reach the maximum allowable temperature (Tmax) in
the MBH by using steam extracted from the CCHP cycle. The heated seawater F enters the
flashing chamber of the first stage PFC1, where a flash boiling of a stream is carried out.
The vapor formed in the first stage PFC1 condenses in the associated pre-heater HEX1,
pre-heating the incoming seawater F. The distillate leaving the HEX1 (freshwater) is col-
lected in the corresponding distillate plate of stage SFC1 and is passed to the next stage
flowing in parallel with the brine stream B1. The brine leaving the first stage (B1) enters
the second stage PFC2 and the vapors formed are mixed with the vapor formed by the
distillate stream SFC2, and the resulting vapor stream is used as a heating source in the
HEX2. The boiling/condensation process of the brine and distillate streams is repeated un-
til the last stage. The concentration profile increases from the first to the last stage. To
reduce the incoming seawater SW and the associated pretreatment cost, a part of the brine
leaving the last stage is often recycled by mixing it with the incoming seawater. A desali-
nation plant operating in this mode is often referred to as a “brine recycle” plant.
Figure 1. Dual-purpose power and desalination plant (DPPDP).
2.1. Multiple Stage Flash (MSF) Desalination System
Figure 2a illustrates a simplified schematic of a MSF desalination system, and Figure 2b
shows the representative system for the mathematical modeling.
The MSF system involves several stages. Each stage includes a preheater (HEX), pri-
mary flashing chambers (PFC, brine flashing), secondary flashing chambers (SFC, distillate
flashing), and the main brine heater (MBH). In the HEXs, the incoming seawater stream
F is heated from T
SW
to T
1F
to reach the maximum allowable temperature (T
max
) in the
MBH by using steam extracted from the CCHP cycle. The heated seawater F enters the
flashing chamber of the first stage PFC
1
, where a flash boiling of a stream is carried out.
The vapor formed in the first stage PFC
1
condenses in the associated pre-heater HEX1,
pre-heating the incoming seawater F. The distillate leaving the HEX
1
(freshwater) is col-
lected in the corresponding distillate plate of stage SFC1 and is passed to the next stage
flowing in parallel with the brine stream B1. The brine leaving the first stage (B1) enters the
second stage PFC
2
and the vapors formed are mixed with the vapor formed by the distillate
stream SFC
2
, and the resulting vapor stream is used as a heating source in the HEX
2
. The
boiling/condensation process of the brine and distillate streams is repeated until the last
stage. The concentration profile increases from the first to the last stage. To reduce the
incoming seawater SW and the associated pretreatment cost, a part of the brine leaving the
last stage is often recycled by mixing it with the incoming seawater. A desalination plant
operating in this mode is often referred to as a “brine recycle” plant.
Energies 2022,15, 8374 5 of 22
Energies 2022, 15, x FOR PEER REVIEW 5 of 22
(a)
(b)
Figure 2. (a) Schematic of a MSF desalination system; (b) representative system for mathematical
modeling.
2.2. Multi-Effect Distillation (MED) Desalination System
Figure 3a illustrates a simplified schematic of a MED desalination system, and Figure
3b shows the representative system for mathematical modeling. Despite the working prin-
ciple of the MED system involving evaporation of brine and condensation of vapor as in
the MSF system, the evaporation/condensation processes and the heat transfer mechanism
are different. In the MED units, the evaporation process is carried out from a seawater
film in contact with a heat transfer area, while in the MSF units, the evaporation is carried
out from a flow of brine flashing due to the pressure drop applied to each stage without
using a heat exchanger. Thus, the brine B is sprayed as a thin film on the tube’s external
surface, and the steam formed in the previous effect V flows inside the tube providing the
energy required by the evaporation process.
Additionally, compared to the MSF process, the MED process operates at lower tem-
peratures (70–90 °C), which is beneficial for reducing tube corrosion and scale formation
on the tube surfaces. In addition, the MED technology might be preferred over the MSF
technology for lower freshwater production rates because it could involve lower total
costs. For higher production rates, the MSF technology could be preferred over the MED
technology because of its lower risk and consolidation in the market. Thus, the selection
of the desalination system depends on the design specifications (freshwater production in
a single desalination plant and freshwater production and electricity generation in a dual-
purpose desalination plant).
Figure 2.
(
a
) Schematic of a MSF desalination system; (
b
) representative system for mathemati-
cal modeling.
2.2. Multi-Effect Distillation (MED) Desalination System
Figure 3a illustrates a simplified schematic of a MED desalination system, and Figure 3b
shows the representative system for mathematical modeling. Despite the working principle
of the MED system involving evaporation of brine and condensation of vapor as in the
MSF system, the evaporation/condensation processes and the heat transfer mechanism are
different. In the MED units, the evaporation process is carried out from a seawater film
in contact with a heat transfer area, while in the MSF units, the evaporation is carried out
from a flow of brine flashing due to the pressure drop applied to each stage without using
a heat exchanger. Thus, the brine B is sprayed as a thin film on the tube’s external surface,
and the steam formed in the previous effect V flows inside the tube providing the energy
required by the evaporation process.
Additionally, compared to the MSF process, the MED process operates at lower
temperatures (70–90
◦
C), which is beneficial for reducing tube corrosion and scale formation
on the tube surfaces. In addition, the MED technology might be preferred over the MSF
technology for lower freshwater production rates because it could involve lower total
costs. For higher production rates, the MSF technology could be preferred over the MED
technology because of its lower risk and consolidation in the market. Thus, the selection
of the desalination system depends on the design specifications (freshwater production
in a single desalination plant and freshwater production and electricity generation in a
dual-purpose desalination plant).
Energies 2022,15, 8374 6 of 22
Energies 2022, 15, x FOR PEER REVIEW 6 of 22
(a)
(b)
Figure 3. MED desalination system: (a) schematic; (b) representation for mathematical modeling.
3. Problem Statement
Figure 4 shows the superstructure DPPDPs that are used for structure optimization.
In the proposed superstructure, several candidate configurations are simultaneously em-
bedded for optimization. For instance, regarding the gas turbine cycle, the superstructure
in Figure 4 includes two candidate gas turbines (GT1–39.1 MW and GT2–64.3/67.5 MW),
but only one gas turbine must be selected. As it will be presented later, the selection of the
gas turbine involves a discrete decision, precisely, a binary variable that is associated to
the gas turbine in the node N1. These gas turbines differ in the power capacity, pressure
ratio, fuel consumption, and conditions of the exhaust gases (pressure, temperature, and
flow rate). Regarding the HRSG, the following candidate options are considered: (a)
burner BURN1 and/or BURN2, (b) steam reheating at the medium-pressure level through
the splitter SP1 (indicated in blue color in Figure 4), and (c) steam generation and reheat-
ing at low-pressure (indicated in green color). Finally, regarding seawater desalination,
fresh water can be produced by a MED or MSF system. The selection of the desalination
unit is carried out in the splitter SP_DES through a binary variable, as it will be described
in the following section. Combining all the mentioned options leads to a total number of
feasible process combinations higher than 50. The higher the number of combinations, the
higher the chances of finding cost-effective designs are. In the Section 4, the constraints
used to model each one of the discrete decisions embedded in Figure 4 are presented.
Figure 3. MED desalination system: (a) schematic; (b) representation for mathematical modeling.
3. Problem Statement
Figure 4shows the superstructure DPPDPs that are used for structure optimization.
In the proposed superstructure, several candidate configurations are simultaneously em-
bedded for optimization. For instance, regarding the gas turbine cycle, the superstructure
in Figure 4includes two candidate gas turbines (GT1–39.1 MW and GT2–64.3/67.5 MW),
but only one gas turbine must be selected. As it will be presented later, the selection of the
gas turbine involves a discrete decision, precisely, a binary variable that is associated to the
gas turbine in the node N1. These gas turbines differ in the power capacity, pressure ratio,
fuel consumption, and conditions of the exhaust gases (pressure, temperature, and flow
rate). Regarding the HRSG, the following candidate options are considered: (a) burner
BURN1 and/or BURN2, (b) steam reheating at the medium-pressure level through the
splitter SP1 (indicated in blue color in Figure 4), and (c) steam generation and reheating
at low-pressure (indicated in green color). Finally, regarding seawater desalination, fresh
water can be produced by a MED or MSF system. The selection of the desalination unit is
carried out in the splitter SP_DES through a binary variable, as it will be described in the
following section. Combining all the mentioned options leads to a total number of feasible
process combinations higher than 50. The higher the number of combinations, the higher
the chances of finding cost-effective designs are. In the Section 4, the constraints used to
model each one of the discrete decisions embedded in Figure 4are presented.
Energies 2022,15, 8374 7 of 22
Energies 2022, 15, x FOR PEER REVIEW 7 of 22
Figure 4. The superstructure of the dual-purpose power and desalination plant (DPPDP).
The optimization problem is stated as follows:
Min TAC (annCAPEX + OPEX)
Subject to:
• mass balances;
• energy balances;
• design equations;
• cost model;
• process conditions (seawater temperature and salinity);
• design specifications (desired levels of electricity and freshwater production).
By solving the proposed model, the following results are simultaneously obtained:
• the minimum total annual cost (TAC);
• optimal distribution among annCAPEX and OPEX;
• optimal selection of the configuration of the entire process (electricity generation
plant + desalination process);
• optimal sizes of all process components selected;
• optimal operating conditions of all process streams.
4. Modeling Assumptions and Mathematical Model
The main assumptions considered as a first approximation for modeling the MSF and
MED desalination systems and the CCHPP are presented below.
4.1. Thermal Desalination Systems
• The following assumptions were used for the simulation of the thermal desalination
systems:
• The number of distillation effects in the MED and number of stages in the MSF are
treated as continuous variables.
• Average salinity and temperature values of the brine at operating conditions are con-
sidered for estimating the boiling point elevation.
Figure 4. The superstructure of the dual-purpose power and desalination plant (DPPDP).
The optimization problem is stated as follows:
Min TAC (annCAPEX + OPEX)
Subject to:
•mass balances;
•energy balances;
•design equations;
•cost model;
•process conditions (seawater temperature and salinity);
•design specifications (desired levels of electricity and freshwater production).
By solving the proposed model, the following results are simultaneously obtained:
•the minimum total annual cost (TAC);
•optimal distribution among annCAPEX and OPEX;
•
optimal selection of the configuration of the entire process (electricity generation
plant + desalination process);
•optimal sizes of all process components selected;
•optimal operating conditions of all process streams.
4. Modeling Assumptions and Mathematical Model
The main assumptions considered as a first approximation for modeling the MSF and
MED desalination systems and the CCHPP are presented below.
Energies 2022,15, 8374 8 of 22
4.1. Thermal Desalination Systems
The following assumptions were used for the simulation of the thermal desalina-
tion systems:
•
The number of distillation effects in the MED and number of stages in the MSF are
treated as continuous variables.
•
Average salinity and temperature values of the brine at operating conditions are
considered for estimating the boiling point elevation.
•
The heat load and heat transfer area of the pre-heaters in the MSF process are consid-
ered as optimization variables (Al-Mutaz and Wazeer [22]).
•
The same optimization variable is considered for the heat loads and heat transfer areas
along the pre-heaters in the MSF process are assumed (Al-Mutaz and Wazeer [22]).
•
The heat load and heat transfer area of the evaporation effects in the MED process are
considered as optimization variables (Al-Mutaz and Wazeer [22]).
•
The same optimization variable is considered for the heat loads and heat transfer
areas along the evaporation effects in the MED process are assumed (Al-Mutaz
and Wazeer [22]).
•
Vapor streams leaving the MED effects and MSF stages are salt-free (El-Dessouky and
Ettouney [23], Al-Mutaz and Wazeer [22]).
•
An effective driving force for the heat transfer in the evaporation effect/stage repre-
sents an optimization variable (Al-Mutaz and Wazeer [22]).
•
The same optimization variable is associated with the effective driving forces for the
heat transfers along all effects/stages (Al-Mutaz and Wazeer [22]).
•
The evaporation effects/stages are optimized under adiabatic conditions (El-Dessouky
and Ettouney [23]; Al-Mutaz and Wazeer [22]).
4.2. Combined Cycle Heat and Power Plant
The following assumptions were used for the simulation of the combined cycle heat
and power plants:
•Steady-state condition is considered.
•A fixed and known value of pressure drop in the HRSG is assumed.
•
Pinch-point temperature differences in all heat exchangers (economizers, evaporators, su-
perheaters, and condensers) are optimization variables with imposed lower bounds [
14
].
•
Complete combustion with excess air is assumed. CO
2
, H
2
O, O
2
, and N
2
are present
in the combustion gas.
•Fixed overall heat transfer coefficients are assumed [14].
•
Heat transfer areas are estimated using the approximation from [
24
] to overcome
numerical difficulties arising from the logarithm mean temperature difference (LMTD)
computation.
•
Dependence of the ideal gas thermodynamic properties of the combustion gases with
temperature is considered [14].
The DPPDP mathematical model was developed taking into consideration the as-
sumptions listed above and the nomenclature included in Figures 2b, 3b and 4. A set of
equations describing the MSF and MED processes and the CCHPP are included in the
Appendix A. Here, the main constraints used to model the discrete decisions associated
with the candidate structures embedded in Figure 4is presented.
4.3. Selecting the Optimal Gas Turbine (GT1 or GT2)
As mentioned, a gas turbine must be selected from two options: GT1–39.1 MW or
GT2–64.3/67.5 MW. Then, an optimization binary variable yGT1 associated to GT1 in the
node N1 is defined and used in Equations (1)–(5) to calculate the values of P
2
,
˙
m
Air
,
˙
m
Fuel
,
ηAC, and ηGT in terms of parameter values characterizing GT1 and GT2:
P2=[RPGT1·yGT1+RPGT2·(1−yGT1)]·P1(1)
Energies 2022,15, 8374 9 of 22
mAir =mAir,GT1·yGT1+mAir,GT2·(1−yGT1)(2)
mFuel =mFuel,GT1·yGT1+mFuel,GT2·(1−yGT1)(3)
ηAC =ηAC,GT1·yGT1+ηAC,GT2·(1−yGT1)(4)
ηEXP =ηEXP,GT1·yGT1+ηEXP,GT2·(1−yGT1). (5)
If y
GT1
= 1, then GT1 is selected and, according to Equations (1)–(5), the values of
P
2
,m
Air
,m
Fuel
,
ηAC
, and
ηGT
are calculated with the parameter values corresponding to
GT1 (P
2
= RP
GT1·
P
1
,
˙
m
Air
=
˙
m
Air,GT1
,
˙
m
Fuel
=
˙
m
Fuel,GT1
,
ηAC
=
ηAC,GT1
, and
ηGT
=
ηGT1
);
otherwise, with the parameter values corresponding to GT2. Then, with these values, the
corresponding electrical power required by the air compressor and the power generated by
the expander are calculated.
It is important to note that, as the gas turbine can be selected from two options, only
one binary variable is needed. If more than two gas turbine types are candidates, then
Equations (1)–(5) are no longer valid, and the definition of a binary variable for each gas
turbine type y
GT
is needed. For this case, the Equations (1)–(5) should be replaced by
Equations (1a)–(5a):
P2=
n
∑
GT=GT1
RPGT ·P1·yGT (1a)
·
mAir =
n
∑
GT=GT1
·
mAir,GT ·yGT (2a)
·
mFuel =
n
∑
GT=GT1
·
mFuel,GT ·yGT (3a)
ηAC =
n
∑
GT=GT1
ηGT ·yGT (4a)
ηEXP =
n
∑
GT=GT1
ηEXP ·yGT (5a)
It should be mentioned that the complete set of equations describing the gas turbine is
included in the model in order to have the possibility to optimize the size and operating
conditions for new designs, i.e., without using data taken from manufacturer catalogues.
In this work, only two candidate gas turbines are proposed in order to see how well the
entire model works from the convergence point of view. In future works, the model will
be extended to include more gas turbine candidates by considering Siemens F-Class and
H-Class types taken from the literature [25].
4.4. Selection/Removal of Additional Burners and Steam Generation and Reheating at
Low-Pressure Level
The selection/removal of the burners BURN1 and BURN2 and the steam generation
and reheating at the low-pressure level, indicated in green color in Figure 4, does not
require the use of binary variables because they can be selected directly from the mass and
energy balances of the process units associated to them. For instance, consider the mass and
energy balances around the burner BURN1, which are expressed in Equations (6) and (7):
·
m7=·
m5+·
m6(6)
·
m7·h7=·
m5·h5+·
m6·h7. (7)
If the optimal value for the fuel mass flow rate
·
m6
is zero, then the burner BURN1 is
removed, and according to the mass and energy balances in Equations (6) and (7):
·
m5
=
·
m7
and h5= h7. Otherwise, it is selected by the optimization algorithm.
Energies 2022,15, 8374 10 of 22
Equations (8) and (9) are proposed for the selection of the burner BURN2, similarly to
that proposed for BURN1:
·
m12 +·
m7=·
m13 (8)
·
m12 ·h12 +·
m7·h8=·
m13 ·h13. (9)
Then, if
·
m12
= 0, then the burner BURN1 is removed, and according to the mass and
energy balances in Equations (8) and (9):
·
m7
=
·
m13
and h
8
= h
13
. Otherwise, BURN2 is
selected by the optimization algorithm.
On the other hand, Equations (10)–(12) are proposed for the selection of the steam
generation at the low-pressure level:
·
m10 =·
m11 +·
m11a(10)
·
m11 ≤MUP ·yREC (11)
·
m11 ≥MLO ·yREC. (12)
If the optimal value is y
REC
= 0, then, according to constraints Equation (11) and (12)
·
m11
= 0 (
·
m10
=
·
m11a
), indicating that no steam reheating is selected; otherwise, the reheating
is included and the optimization variable m11 is bounded between MLO and MUP.
4.5. Selection of the Optimal Desalination System: MED System or MSF System
In a similar way, a binary variable y
MSF
is defined and associated with the steam
required by the MSF desalination unit. Then, the following constraints are derived from
the splitter SP_DES (Figure 4):
·
m42,MSF ≤MUP ·yMSF (13)
·
m42,MSF ≥MLO ·yMSF (14)
·
m42,MED ≤MUP ·(1−yMSF)(15)
·
m42,MED ≥MLO ·(1−yMSF). (16)
If the optimal value is y
MSF
= 0, then, according to constraints Equations (13) and (14),
˙
m
42_MSF
= 0, indicating that no steam is supplied to the MSF unit and, therefore, it is removed
from the optimal solution. At the same time, according to constraints Equations (15) and (16),
˙
m42_MED > 0, thus assuring that steam is supplied to the MED unit.
5. Model Implementation Aspects
The resulting MINLP model for the superstructure-based representation was imple-
mented in general algebraic modeling system (GAMS), which is a high-level modeling
system for mathematical programming and optimization. It deals with algebraic equations
that are solved simultaneously. Discrete and continuous optimizer (DICOPT) code was
used as the MINLP solver. By employing an iterative process, it solves a series of nonlinear
programming (NLP) and mixed-integer linear (MIP) sub-problems. The optimization algo-
rithm stops when the difference in the solutions obtained from these two problems is less
than a pre-defined tolerance.
6. Results
Once the model was implemented and successfully verified, it was used to solve the
optimization problem stated in Section 3by using the parameter values listed in Tables 1and 2.
Energies 2022,15, 8374 11 of 22
Table 1. Main parameter values.
Specification Design
Net electrical power generation (MW) 80.0
Freshwater production rate (m3/h) 700.0
Process data
Seawater temperature (K) 298.15
Seawater salinity (ppm) 42,000
Cooling water temperature (K) 298.15
Table 2. Parameter values used in the mass and energy balances.
MED and MSF Desalination Systems
Specific heat capacity of seawater (kJ/(kg·K)) 4.2
Boiling point elevation (K) 1.5
Latent heat of vaporization (kJ/kg) 2333
Overall heat transfer coefficient in the effects
(kW/(m2·K)) 3.0
Overall heat transfer coefficient in the
condenser (kW/(m2·K)) 2.0
CCHP plant
Steam turbine isentropic efficiency (%) 85
Overall heat transfer coefficients (W/(m2K))
Superheater 50.0
Evaporator 43.7
Economizer 42.6
Pinch temperature (K) 5
Fuel cost (USD/MJ) 0.00386
Figure 5shows the optimal solution obtained by minimizing the TAC. A minimum
TAC value of 45.944 MM USD/y (5743.2 USD/h) was obtained, where the CCHPP rep-
resents around 85%. As shown in Figure 5, the optimization algorithm selected from the
proposed superstructure is the gas turbine GT1–39.1 MW, the first burner BURN1, steam
reheating at the medium-pressure level RH1, and the MSF desalination process. The gas
turbine GT2–64.3/67.5 MW, the second burner BURN2, steam reheating at the low-pressure
level EVP2/SH2, and the MED desalination unit were removed. Regarding the electrical
power generation, the steam turbine STs generate 41.91 MW, of which 6.92 MW are gener-
ated in HPST, 25.34 MW in MPST, and 9.65 MW in LPST, while the remaining electrical
power is generated in the selected gas turbine GT1 (39.1 MW). The optimal pressure value
for the HP level is 131.0 bar and 60.0 bar for the MP level. In the MPST, the steam expands
from 60.0 bar to 3.29 bar.
The total heat load recovered by the HRSG is 393.6 MW, with the following distribution
among its components: 43.9 MW in two superheaters, 41.9 MW in the evaporator, and
40.1 MW in two economizers. This total heat load requires 26,787 m2of heat transfer area.
Regarding the desalination process, the selected MSF system requires 42 MW as a
heating source, which is extracted from the CCHPP before passing through LPST, at a flow
rate of 18.55 kg/s. The total heat transfer area required by the MSF is 64,093 m
2
, which is
distributed in 24 flashing stages.
Energies 2022,15, 8374 12 of 22
Energies 2022, 15, x FOR PEER REVIEW 12 of 22
Figure 5. Optimal configuration of the integrated power/desalination process obtained by minimiz-
ing the total annual cost to generate 80.0 MW of net electrical power and 700 m3/h of freshwater.
Regarding the desalination process, the selected MSF system requires 42 MW as a
heating source, which is extracted from the CCHPP before passing through LPST, at a
flow rate of 18.55 kg/s. The total heat transfer area required by the MSF is 64,093 m2, which
is distributed in 24 flashing stages.
Several optimization problems were solved for the same design specifications, pro-
cess data, and cost model with the aim of obtaining suboptimal solutions for investigating
how much better the optimal configuration presented above is with respect to other con-
figurations. To achieve it, different process configurations were fixed by properly setting
the values of the discrete decisions. The main results are compared in Table 3. Figure 6
illustrates a sub-optimal solution obtained by considering GT2 instead of GT1 and keep-
ing the MSF unit but including BURN1 and BURN2.
Compared to the optimal solution values, the differences in TAC range between 1.2%
and 21.6%, observing that the higher differences are obtained when the MED unit is con-
sidered. The TAC value increases from 45.944 MM USD/y to 49.184 MM USD/y (5743.2 to
6148.1 USD/h) when the GT1 is replaced with the GT2, keeping fixed the remaining con-
figuration. The TAC value significantly increases when the MSF unit is replaced with the
MED unit. By keeping the same optimal configuration in the CCHPP but replacing the
MSF unit with the MED unit, the TAC value increases by around 13%. If both the GT1 and
MSF unit are replaced with the GT2 and MED unit, respectively, and the second burner is
included, the TAC increases by around 21%.
Table 3. Comparison of the costs obtained by the optimal and suboptimal configurations.
Config. GT1
# GT2
## BURN1 BURN2 RH1 SH2/EV2 MSF MED
TAC
(MM
USD/y./USD/h)
Difference
(%)
Optimal X - X - X - X - 45.944/5743 -
#1 - X - - X - X - 46.488/5811 1.2
#2 - X X - X - X - 49.184/6148 7.1
Figure 5.
Optimal configuration of the integrated power/desalination process obtained by minimiz-
ing the total annual cost to generate 80.0 MW of net electrical power and 700 m3/h of freshwater.
Several optimization problems were solved for the same design specifications, process
data, and cost model with the aim of obtaining suboptimal solutions for investigating
how much better the optimal configuration presented above is with respect to other con-
figurations. To achieve it, different process configurations were fixed by properly setting
the values of the discrete decisions. The main results are compared in Table 3. Figure 6
illustrates a sub-optimal solution obtained by considering GT2 instead of GT1 and keeping
the MSF unit but including BURN1 and BURN2.
Table 3. Comparison of the costs obtained by the optimal and suboptimal configurations.
Config. GT1 #GT2 ## BURN1 BURN2 RH1 SH2/EV2 MSF MED TAC
(MM USD/y./USD/h) Difference (%)
Optimal X - X - X - X - 45.944/5743 -
#1 - X - - X - X - 46.488/5811 1.2
#2 - X X - X - X - 49.184/6148 7.1
#3 - X X X X - X - 49.784/6223 8.35
#4 X - X - X - - X 52.184/6523 13.6
#5 - X - - X - - X 54.448/6806 18.5
#6 - X X X X - - X 55.888/6986 21.6
#GT1–39.1 MW ## GT2–64.3/67.5 MW.
Energies 2022,15, 8374 13 of 22
Energies 2022, 15, x FOR PEER REVIEW 13 of 22
#3 - X X X X - X - 49.784/6223 8.35
#4 X - X - X - - X 52.184/6523 13.6
#5 - X - - X - - X 54.448/6806 18.5
#6 - X X X X - - X 55.888/6986 21.6
# GT1–39.1 MW ## GT2–64.3/67.5 MW.
The comparison of the solutions presented in Figures 5 and 6 shows that the total
electricity generation by the steam turbines in Figure 6 (sub-optimal solution) is 27.39 MW
(16.52 MW vs. 43.91MW) lower than that generated in Figure 5 (optimal solution) because
the net electricity generation of GT2 is 28.40 MW (67.5 MW vs. 39.10 MW) higher than
GT1 (Figure 5). The total heat transfer area in Figure 6 is 16,342 m2 lower than in Figure 5
because less energy is neede to be recovered (304 MW). The selection of the gas turbine
affects not only the design and operating conditions of the HRSG and steam turbines, but
also the MSF unit. The sub-optimal solution in Figure 6 requires 30.0 kg/s to run the HP
steam turbine while the optimal solution in Figure 5 requires 37.71 kg/s. Despite the heat-
ing utility required, the MSF unit in Figure 6 is 10.07 MW higher than that required in
Figure 5 (52.37 MW vs. 42.30 MW) the total area required by the MSF unit is 18,191 m2
lower (45,902 m2 vs. 64,093 m2).
Figure 6. Sub-optimal configuration of the integrated power/desalination process obtained by min-
imizing the total annual cost to generate 80.0 MW of net electrical power and 700 m3/h of freshwater
(Config. #3).
The proposed mathematical model is robust enough from the convergence point of
view. For desired design specifications (electricity and freshwater demand), users can ap-
ply the proposed model to find the optimal solution (configuration, dimensions, and op-
erating conditions of all process units) by considering several candidate configurations.
The model is based on the first law of thermodynamics (conservation of energy principle),
and a conventional method was used to calculate the total annual cost of the entire system.
However, it should be mentioned that there are recent advanced methods that take into
account exergy destruction as the value basis. For instance, the advanced energetic and
Figure 6.
Sub-optimal configuration of the integrated power/desalination process obtained by
minimizing the total annual cost to generate 80.0 MW of net electrical power and 700 m
3
/h of
freshwater (Config. #3).
Compared to the optimal solution values, the differences in TAC range between
1.2% and 21.6%, observing that the higher differences are obtained when the MED unit is
considered. The TAC value increases from 45.944 MM USD/y to 49.184 MM USD/y (5743.2
to 6148.1 USD/h) when the GT1 is replaced with the GT2, keeping fixed the remaining
configuration. The TAC value significantly increases when the MSF unit is replaced with
the MED unit. By keeping the same optimal configuration in the CCHPP but replacing the
MSF unit with the MED unit, the TAC value increases by around 13%. If both the GT1 and
MSF unit are replaced with the GT2 and MED unit, respectively, and the second burner is
included, the TAC increases by around 21%.
The comparison of the solutions presented in Figures 5and 6shows that the total
electricity generation by the steam turbines in Figure 6(sub-optimal solution) is 27.39 MW
(16.52 MW vs. 43.91MW) lower than that generated in Figure 5(optimal solution) because
the net electricity generation of GT2 is 28.40 MW (67.5 MW vs. 39.10 MW) higher than
GT1 (Figure 5). The total heat transfer area in Figure 6is 16,342 m
2
lower than in Figure 5
because less energy is neede to be recovered (304 MW). The selection of the gas turbine
affects not only the design and operating conditions of the HRSG and steam turbines,
but also the MSF unit. The sub-optimal solution in Figure 6requires 30.0 kg/s to run the
HP steam turbine while the optimal solution in Figure 5requires 37.71 kg/s. Despite the
heating utility required, the MSF unit in Figure 6is 10.07 MW higher than that required
Energies 2022,15, 8374 14 of 22
in Figure 5(52.37 MW vs. 42.30 MW) the total area required by the MSF unit is 18,191 m
2
lower (45,902 m2vs. 64,093 m2).
The proposed mathematical model is robust enough from the convergence point of
view. For desired design specifications (electricity and freshwater demand), users can apply
the proposed model to find the optimal solution (configuration, dimensions, and operating
conditions of all process units) by considering several candidate configurations. The model
is based on the first law of thermodynamics (conservation of energy principle), and a con-
ventional method was used to calculate the total annual cost of the entire system. However,
it should be mentioned that there are recent advanced methods that take into account
exergy destruction as the value basis. For instance, the advanced energetic and exergoeco-
nomic methods reported in [
26
] provide additional information useful for improving the
design and operation of the entire process by considering splitting the exergy destruction
into unavoidable and avoidable parts. In this context, the model presented in the current
work represents the initial step, and it will be extended to include all the equations required
to apply the advanced exergy-based method developed by [
26
]. Thus, the current model
and results will allow for finding a feasible initial solution at a low computational cost
(fewer iterations and CPU time) for the advanced exergy-based method.
7. Conclusions
This paper addressed the optimization of dual-purpose power and desalination plants
from the perspective of process systems engineering. Several candidate configurations
result from the combination of different combined cycle configurations with two alternative
thermal desalination processes. The integrated plants were optimized to find the optimal
structure and operation conditions simultaneously. To this end, a mixed-integer nonlinear
mathematical programming model was developed, which included the possibility of
selecting one of two different types of gas turbines, several alternative arrangements of the
heat recovery steam generator, and two alternatives for the thermal desalination processes
for freshwater production. In order to show the strengths of the developed model, a case
study considering a freshwater production of 700 m
3
/h and electricity generation of 80 MW
was presented. It was found that a minimum TAC value of 5743 USD/h and an optimal
configuration consisting of the gas turbine GT1 and the MSF process as the main subsystems.
Then, the optimal solution was compared with suboptimal solutions obtained for other
configurations different from the optimal one. The TAC value increased 405 USD/h when
the GT1 was replaced with the GT2, keeping fixed the remaining configuration. However,
the TAC value significantly increased when the MSF unit was replaced with the MED unit.
By keeping the same CCHPP configuration as in the optimal configuration but replacing
the MSF unit with the MED unit, the TAC value increased 1174 USD/h.
The presented model will be extended in order to include more candidate processes.
For example, a reverse osmosis unit for freshwater production will be included in the
superstructure-based representation, resulting in a higher number of alternative flowsheets
for finding optimal integrated power/desalination facilities. Additionally, models of
CO
2
capture plants and absorption refrigeration systems already implemented will be
included in the current model to address the study of polygeneration systems with zero
greenhouse emissions.
Author Contributions:
A.M.P.: software; investigation. S.F.M.: methodology; investigation; writing
the original draft. P.A.A.: reviewing and editing; T.M.: reviewing and editing. M.C.M.: methodology;
supervision. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.
Conflicts of Interest: The authors declare no conflict of interest.
Energies 2022,15, 8374 15 of 22
Nomenclature
Ae Heat transfer area of an effect, m2.
annCAPEX Annualized capital expenditure, USD/y.
·
BFlowrate of the discharge brine stream, kg/s.
BPE Boiling point elevation, K.
Ccivil Civil work cost, USD.
Ceq Total cost of the equipment associated to the MSF and MED desalination plants, USD.
CpSW Averaged heat capacity of the inlet seawater stream, kJ/(kg K).
CpDAveraged heat capacity of the distillate (freshwater) stream, kJ/(kg K).
CpBAveraged heat capacity of the discharge brine, kJ/(kg K).
CRF Capital recovery factor, yr−1.
·
DFlowrate of the distillate (freshwater) stream, kg/s.
hSpecific enthalpy, kJ/kg.
˙
mAir Mass flowrate of the inlet air stream, kg/s.
˙
mAir,GT1 Mass flowrate of the air stream in the gas turbine GT1, kg/s.
˙
mAir,GT2 Mass flowrate of the air stream in the gas turbine GT2, kg/s.
˙
mFuel Molar flowrate of the fuel stream, kmol/s.
˙
mFuel,GT1 Molar flowrate of the fuel stream in GT1, kmol/s.
˙
mFuel,GT2 Molar flowrate of the fuel stream in GT2, kmol/s.
LMTDCOND Logarithmic mean temperature difference of condenser, K.
MPfMolecular weight, kg/kmol.
MLO Lower value used in the constraints involving binary variables
MUP Upper value used in the constraints involving binary variables
NNumber of evaporation stages in MSF or effects in MED
·
nFFlowrate of the fuel stream, kmol/s.
OPEX Operating expenditure, USD/yr.
OPEXmant Maintenance cost, USD/yr.
OPEXtreat Pretreatment cost of the seawater stream, USD/yr.
P2Outlet pressure at the air compressor, bar.
RPGT1 Pressure ratio at the gas turbine GT1, dimensionless.
RPGT2 Pressure ratio at the gas turbine GT2, dimensionless.
·
SW Flowrate of the inlet seawater stream, kg/s.
TAC Total annual cost, USD/y.
TBTemperature of the discharge brine, K.
THTAMSF Total heat transfer area of the MSF desalination unit, m2.
THTAMED Total heat transfer area of the MED desalination unit, m2.
TS Temperature of the steam, K.
XFMass composition of the feed seawater, ppm.
XBMass composition of the discharge brine, ppm.
ZCOM Investment cost of the combustion chamber, USD.
ZHE Investment cost of heat exchangers, USD.
ZST Investment cost of steam turbines, USD.
ZDRUM Investment cost of the drum, USD.
ZPUMP Investment cost of pumps, USD.
ZGT Investment cost of gas turbine, USD.
yGT1 Binary variable to select or remove the gas turbine type 1, dimensionless.
yGT2 Binary variable to select or remove the gas turbine type 2, dimensionless.
yMSF Binary variable to select or remove the MSF desalination unit
∆tTotal temperature difference of the stage (MSF), K.
∆TTemperature difference between the heating utility temperature in the first effect TS
and the discharge brine temperature TB, K.
∆tc Driving force for the heat transfer, K.
∆te f f Effective driving force for heat transfer in the evaporation effects, K.
∆tf Driving force for the flashing process, K.
ηAC Efficiency of the air compressor, dimensionless.
ηGT Efficiency of the gas turbine expander, dimensionless.
Energies 2022,15, 8374 16 of 22
Abbreviations
AC Air compressor
BURN Burner
CC Combustion chamber
CCHPP Combined cycle heat and power plant
COMP Compressor
COND Condenser
DPPDP Dual-purpose power desalination plants
EC Economizer
EVP Evaporator
EVP2 Evaporator at the low-pressure level
GT1 Gas turbine Type I
GT2 Gas turbine Type II
HEX Pre-heater
HRSG Heat recovery steam generator
MED Multi-effect distillation desalination
MINLP Mixed integer nonlinear
MSF Multi-stage flash
ORC Organic Rankine cycle
RH1 Re-heater of the steam at high-pressure level
SH2 Superheater at the low-pressure level
Appendix A
Figure A1 illustrates the multi-stage flash (MSF) desalination process and includes the
nomenclature. The main model constraints used in this work were taken from [14,15].
Energies 2022, 15, x FOR PEER REVIEW 16 of 22
DPPDP Dual-purpose power desalination plants
EC Economizer
EVP Evaporator
EVP2 Evaporator at the low-pressure level
GT1 Gas turbine Type I
GT2 Gas turbine Type II
HEX Pre-heater
HRSG Heat recovery steam generator
M
ED Multi-effect distillation desalination
M
INLP Mixed integer nonlinear
M
SF Multi-stage flash
ORC Organic Rankine cycle
RH1 Re-heater of the steam at high-pressure level
SH2 Superheater at the low-pressure level
Appendix A
Figure A1 illustrates the multi-stage flash (MSF) desalination process and includes
the nomenclature. The main model constraints used in this work were taken from [14,15].
Figure A1. Schematic of a multi-stage flash MSF seawater desalination process.
Appendix A.1. Overall Mass Balances
SW D B=+
(A1)
⋅=⋅
SW B
SW x B x (A2)
where ,SW D
, and B
refer, respectively, to the flowrates of the inlet seawater, distillate
(freshwater) and discharge brine streams. The associated concentrations are xsw and xB.
Appendix A.2. Overall Energy Balance
00
42, 42,
0
() ()
()
SW SW D D
MSF MSF
BB
mSWCpTTDCpTT
BCp T T
⋅+⋅ −=⋅⋅−
+⋅ ⋅ −
λ
(A3)
where 42,
M
SF
m
represents the steam flowrate extracted from the HRSG and λ42_MSF the la-
tent heat of condensation. The parameters CpSW, CpD, and CpB refer to the averaged heat
capacities of the seawater, distillate, and discharge brine, while T0 represents the reference
temperature.
Figure A1. Schematic of a multi-stage flash MSF seawater desalination process.
Appendix A.1. Overall Mass Balances
·
SW =
·
D+
·
B(A1)
·
SW ·xSW =
·
B·xB(A2)
where
·
SW,
·
D
, and
·
B
refer, respectively, to the flowrates of the inlet seawater, distillate
(freshwater) and discharge brine streams. The associated concentrations are xsw and xB.
Appendix A.2. Overall Energy Balance
·
m42,MSF ·λ42,MSF +
·
SW ·CpSW(TSW −T0) =
·
D·CpD·(TD−T0)
+
·
B·CpB·(TB−T0)
(A3)
Energies 2022,15, 8374 17 of 22
where
·
m42,MSF
represents the steam flowrate extracted from the HRSG and
λ42_MSF
the
latent heat of condensation. The parameters Cp
SW
, Cp
D
, and Cp
B
refer to the averaged
heat capacities of the seawater, distillate, and discharge brine, while T
0
represents the
reference temperature.
Appendix A.3. Overall Balances in the Main Brine Heater
·
m42,MSF ·λ42,MSF =
·
F·CpF·∆t(A4)
where
∆
trefers to the total temperature difference of the stage that is calculated as the
difference between the outlet temperature in the HEX and the inlet temperature at the PFC
as expressed in Equation (A5).
∆t=TB
1−TF
1(A5)
The variable
∆
tis divided into two temperature differences [
14
]:
∆
tf associated with
the driving force for the flashing process, and
∆
tc associated with the heat transfer’s driving
force, as expressed in Equations (A6)–(A8):
∆t=TB
1−TF
1(A6)
∆t f =TB
1−TB
2(A7)
∆tc =TB
2−TF
1(A8)
Appendix A.4. Total Heat Transfer Area
By assuming constant temperature differences
∆
tand
∆
te in all the stages, the total
heat transfer area (THTAMSF) can be expressed as follows:
THTAMSF =
·
F·CpF
U
·Nln∆t−BPE
∆te N
(A9)
where Nrefers to the number of stages which is related to the
∆
tf and
∆
tby Equation
(A10). BPE refers to the boiling point elevation and
∆
te represents the effective temperature
difference for the heat transfer, which is calculated by Equation (A11):
N·∆t f =TB
1−(TF−∆t)(A10)
∆te =∆tc −BPE (A11)
Appendix A.5. Fresh Water Production
Then, the total fresh water production (
·
D) can be calculated by Equation (A12):
·
D=
·
F·1−1−CpB·∆t f
λ (A12)
where λrepresents the heat of condensation of water.
Appendix A.6. Multi-Effect Distillation (MED) Desalination Process
Figure A2 illustrates the multi-effect distillation (MED) desalination process and
includes the nomenclature. The main model constraints used in this work were taken
from [27].
Energies 2022,15, 8374 18 of 22
Energies 2022, 15, x FOR PEER REVIEW 18 of 22
Figure A2. Schematic of a multi-effect distillation (MED) desalination process.
Appendix A.7. Overall and Component Mass Balances
=+
CW
SW M B D (A13)
⋅= ⋅+⋅
FCWF B
SW x M x B x (A14)
where
SW ,
CW
M,B
, and
D
refer to the mass flowrate of the feed seawater, cooling water,
discharge brine, and distillate streams, respectively, and XF and XB refer to the mass com-
position of the feed seawater and discharge brine, respectively.
Appendix A.8. Heating Steam in the First Effect E1
The steam required as a heating utility 42,
M
ED
m
in E1 is calculated as follows [22]:
42,
1
0.8
⋅=⋅
MED
Nm D
(A15)
where N refers to the number of evaporation effects.
Appendix A.9. Effective Driving Force for Heat Transfer in the Evaporation Effects
The effective driving forces for heat transfer in the evaporation effects ∆𝑡𝑒𝑓𝑓 is cal-
culated by Equation (A16) in terms of the number of evaporation effects N, the tempera-
ture difference ∆𝑇 between the heating utility temperature in the first effect TS and the
discharge brine temperature TB (Equation (A17)), and the boiling point elevation BPE:
(1)teff N T N BPEΔ⋅=Δ−−⋅ (A16)
B
TTSTΔ= − (A17)
Appendix A.10. Heat Exchange in Evaporation Effects
The heat transfer area of an effect Ae is calculated from Equation (A18) taken into
account N, ∆𝑡𝑒𝑓𝑓, D, the evaporation heat λ, and the overall heat transfer coefficient U.
e
DNUAte
ff
⋅= ⋅⋅ ⋅Δ
λ
(A18)
Then, the total heat transfer area associated to the evaporation effects A is expressed
as follows:
e
AAN=⋅
(A19)
Figure A2. Schematic of a multi-effect distillation (MED) desalination process.
Appendix A.7. Overall and Component Mass Balances
·
SW =
·
MCW
·
B+
·
D(A13)
·
SW ·xF=
·
MCW ·xF+
·
B·xB(A14)
where
·
SW
,
·
MCW
,
·
B
and
·
D
refer to the mass flowrate of the feed seawater, cooling water,
discharge brine, and distillate streams, respectively, and X
F
and X
B
refer to the mass
composition of the feed seawater and discharge brine, respectively.
Appendix A.8. Heating Steam in the First Effect E1
The steam required as a heating utility ·
m42,MED in E1 is calculated as follows [22]:
N··
m42,MED =1
0.8 ·
·
D(A15)
where Nrefers to the number of evaporation effects.
Appendix A.9. Effective Driving Force for Heat Transfer in the Evaporation Effects
The effective driving forces for heat transfer in the evaporation effects
∆te f f
is calcu-
lated by Equation (A16) in terms of the number of evaporation effects N, the temperature
difference
∆T
between the heating utility temperature in the first effect TS and the discharge
brine temperature TB(Equation (A17)), and the boiling point elevation BPE:
∆te f f ·N=∆T−(N−1)·BPE (A16)
∆T=TS −TB(A17)
Appendix A.10. Heat Exchange in Evaporation Effects
The heat transfer area of an effect Ae is calculated from Equation (A18) taken into
account N,∆te f f ,D, the evaporation heat λ, and the overall heat transfer coefficient U.
·
D·λ=N·U·Ae·∆te f f (A18)
Then, the total heat transfer area associated to the evaporation effects Ais expressed
as follows:
A=Ae·N(A19)
Energies 2022,15, 8374 19 of 22
Appendix A.11. Energy Balance and Heat Transfer Area of the Condenser
The energy balance in the condenser is expressed by Equation (A20) and its heat
transfer area is calculated by Equation (A21):
(
·
F+
·
MCW )·CpF·(TF,N−TF) =
·
D
N·λ(A20)
(
·
F+
·
MCW )·CpF·(TF,N−TF) = UCOND ·ACOND ·LMTDCOND (A21)
where the logarithmic mean temperature difference LMTD
COND
is calculated by Equation (A22):
LMTDCOND =(TB−BPE −TF)−(TB−BPE −TF,N)
ln (TB−BPE−TF)
(TB−BPE−TF,N)
(A22)
Appendix A.12. Combined Cycle Heat and Power Plant
The mathematical model of the combined cycle and power plant consists of the
equations needed to describe the mass and energy balances and calculate the sizes of the
gas turbine (air compressor, combustion chamber, and expander), heat recovery steam
generator (economizers, evaporators, and superheaters), and steam turbines. These model
equations can be found elsewhere [28].
Appendix A.13. Cost model for the Entire Integrated Process
The main equations considered for the cost model were taken from Ulrich and Vasude-
van [29].
Appendix A.14. Combustion Chamber and Burners
The capital investments of the combustion chamber and burners are calculated as follows:
ZCOMB ($)=418 ·46.08 ··
nF·MPF·(1+exp0.018 ·T−26.4(A23)
where
·
nF
and MP
f
represent the flowrate and molecular weight of the fuel expressed in
kmol/s and kg/kmol, respectively. Additionally, T refers to the temperature of the flue gas
expressed in K.
Appendix A.15. Heat Exchangers
The capital investments of the economizers, evaporators, and superheaters of the heat
recovery steam generator HRSG are expressed in Equation (A24)
ZHE ($) = 3.6 ·83.43 ·CHE ·(0.004 ·P+0.9)
1.34 ·373.1
385.9 (A24)
where C
HE
is expressed in USD/m
2
and varies with the type of heat exchanger (35 USD/m
2
for economizers, 41.71 USD/m
2
for evaporators, and 83.43 USD/m
2
for superheaters).
HTA refers to the heat transfer area (m2) and Pis the operating pressure (bar).
Appendix A.16. Steam Turbines
The capital investments of the steam turbines are calculated in terms of the electrical
power generation WST expressed in kW.
ZST ($) = 0.1358 ·WST
1000 4+3.085 ·WST
1000 3−
−3666.08 ·WST
1000 2+351064.2 ·WST
1000 +226726.56
8·ηST
(A25)
Energies 2022,15, 8374 20 of 22
- Drums
ZDRUM ($) = CPDRUM ·(1.62 +1.47 ·FPDRUM)·437.4
427.4 ·0.8696 (A26)
where CPDRUM and FPDRUM are calculated from Equations (A27)–(A29).
CPDRUM =1542.9 +946.5 ·LDRUM (A27)
FPDRUM =1.31145 +0.03755 ·(PDRUM ·10)(A28)
2··
nDRUM =LDRUM ·2.7 ·3.1415 ·DDRUM (1/(PDRUM ·10)0.7
DDRUM ·(HDRUM ·0.42993)·1.235 =(PDRUM ·10)·14.503 ·16.0184 (A29)
where
·
nDRUM
and P
DRUM
refer to the flowrate (kmol/s) and operating pressure at the
drum (MPa).
Appendix A.17. Pumps
The capital investments of pumps are calculated as follows:
ZPUMP ($) = CPPUMP ·(1.8 +1.5 ·FPPUMP)·663.7
615.9 ·0.8696 (A30)
where CP
PUMP
and FP
PUMP
are calculated taken into account the electrical power consump-
tion WPUMP (MW) and high pressure PHigh (MPa).
CPPUMP =EXP(3.593 +0.3208 ·log10(WPUMP) + 0.0285 ·(log10(WPUMP))2(A31)
FPPUMP =0.1682 +0.3477 ·log10(PHigh ·10)+0.4841 ·(log10PHigh ·10))2(A32)
Appendix A.18. Gas Turbines
The capital investment of the gas turbine is expressed in Equation (A33).
ZGT ($)=ZGT1·yGT1+ZGT2·yGT2(A33)
where Z
GT1
and Z
GT2
represent the cost of the turbines (17.4
×
10
6
USD for GT1 and
11.3 ×106USD for GT2) and ZGT1 and ZGT2 refer to the corresponding binary variables.
Appendix A.19. Desalination Processes
The total annual costs of the two desalination processes are calculated from Equa-
tions (A34)–(A48) using a base model presented in [30].
TACj=annCAPEXj+OPEXjj=MSF,MED (A34)
where annCAPEX and OPEX represent the total capital expenditure and total operating
costs and are calculated by Equations (A35) and (A43):
annCAPEXj=CRF ·CTCj(A35)
where CRF refers to the capital recovery factor given by Equation (A36). CTC represents
the total capital cost and is expressed in Equation (A37) considering the direct CAPEX
(Cdirect) and indirect CAPEX (Cindirect) as expressed as follows:
CRF =i·(1+i)n
(1+i)n−1(A36)
CTCj=Cdirecj+Cindirecj(A37)
Energies 2022,15, 8374 21 of 22
Cdirectj=Ceqj+Ccivilj(A38)
where Ceq and Ccivil refer, respectively, to the total costs of the desalination plant and civil
work, expressed in Equations (A39) and (A40)
Ceqj=Cmatj·Kmatj·THTA 0.54
j+50 ·24 ·3600 ·
·
Fj/ρSW,j(A39)
where F
j
and
ρSW,j
represent the flowrate of the incoming seawater (kg/s) and den-
sity (kg/m3).
Ccivilj=0.15 ·Ceqj(A40)
The indirect CAPEX is given by Equation (A41)
Cindirecj=0.25 ·Cdirectj. (A41)
The total operating OPEX is expressed in Equation (A42)
OPEXj=OPEXmantj+OPEXtreatj(A42)
where OPEXmant and OPEXtreat represent the costs associated to the maintenance and
seawater treatment.
OPEXtreatj=Ctreatj·THY ·
·
Fj/ρb,j·3600 (A43)
where THY,
˙
Fand
ρb,j
represent, respectively, the total hours per year (8000 h/y), the in-
coming seawater flowrate (kg/s) and seawater density (kg/m
3
). A value of 0.024 USD/m
3
is assumed for Ctreat.
The maintenance cost is expressed in Equation (A44):
OPEXmantj=0.001 ·CTCj(A44)
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