Perspective Chapter: Technical and Economic Analysis of Reverse Osmosis Desalination System

Abstract

Economic thermodynamic analysis is a branch of engineering science that is derived from economic laws. The goal of economic thermodynamic analysis of systems is the lowest price. Price calculation in a system includes the following steps: Determining the actual price of products. Provide a reasonable way to price products. Providing information on which calculations are made. The overall investment cost of a project includes fixed investment costs, including the costs related to the purchase of land, the construction of the necessary facilities and equipment, and the purchase and installation of machinery, as well as the initial costs related to the investment, including a series of other side costs. It is possible that their relationships and the percentage of their allocated costs in the project are explained separately and finally the estimation equations of each part of the power plant cycle as well as the economic modeling of the RO system and the effective input parameters such as the input salt concentration, discharge Feeding and input water, ambient pressure, number and type of membrane, etc. are stated along with their relationships. Finally, a RO system design flowchart and how to solve its algorithm are explained in detail.

1. Introduction

The need for water all over the world has increased due to the growth of the population and also due to the growth of the industry, and the water resources are rapidly being depleted. Since 1990, more than 80 countries are facing the problem of water shortage, while more than 70% of the earth’s surface is covered with water; But only one percent of these resources are suitable for use, and 97.5% of them are oceans. The only solution; The use of salt water desalination techniques can solve the problems of water shortage. The two main types of desalination techniques that are widely used are evaporation methods and membrane methods. Evaporation methods such as MSF and MED are common in regions such as the Middle East that have huge energy resources. In 2007, all over the world, 66% of sweetening was done with the MSF process, and only 22% was using reverse osmosis (RO) [1]. In 2011, this statistic changed to 60% for reverse osmosis and only 27% for MSF, which shows the increasing importance of the reverse osmosis process [2]. Reverse osmosis is a separation process whose driving force is pressure, in which salt water is purified by pressure by passing through a semi-permeable membrane. This process depends on the resistance of the membrane and the concentration of water impurities. Reverse osmosis is a process that consumes a lot of energy. The costs of a reverse osmosis system, which includes investment and operating costs, are classified in the diagram “Figure 1”. Due to the operation of reverse osmosis membranes at high pressure, the electric energy of feed pumps is an important part of the operating cost of these systems. However, the pressure drop in reverse osmosis systems is low and the concentrated water flow leaves the last membrane with a pressure equal to 80 to 90% of the supply pressure. If the concentrated water of the system is directed to the surface water, this excess pressure must be dissipated before discharge. The pressure that is lost in the flow of concentrated water through the control valve is wasted energy, because it does not do any useful work in the purification system. Due to the high level of pressure and flow rate of condensed water, the amount of wasted energy is significant.

Figure 1.

Classification of seawater reverse osmosis costs for fresh water production (produced water flow rate 125 m³/h, system recovery 40%, one pass and with a life of 10 years) [2].

2. Estimation of total capital investment (TCI)

In order to estimate the total investment cost of a project, the costs related to the purchase of land, the construction of necessary facilities and equipment, and the purchase and installation of special machinery and equipment that are used to make the system work should be calculated. These costs are Fixed Capital investment (FCI); But the initial costs related to the investment of a project, in addition to these fixed costs, also include a series of other side costs, the sum of these costs and fixed costs is called total investment costs (TCI).

2.1 Cost estimate of purchased equipment (PEC)

Estimating the cost of purchasing equipment is the first and most important step in estimating the costs of a project. It is clear that the accuracy of estimating these costs depends on the amount of information available to engineers. Of course, in addition to the amount of available information, the time and the budget given to the relevant engineers for price estimation will also affect the accuracy of cost estimation. The best way to estimate the cost of purchasing equipment is to refer to the sellers of these items and use their information in this regard. It should be noted that the costs related to the transportation and installation of the equipment should be added separately to the cost of purchasing the equipment. Another way to estimate the cost of purchasing equipment is to refer to previous purchases and use the information in them. In such a case, by referring to professional and experienced people whose job is to estimate the price, or by using the information that engineering companies often make available to users, the cost of purchasing equipment can be estimated. In addition to these methods, software packages designed for this purpose can also be used, although it should be noted that the costs estimated by these softwares may be higher than which are calculated through different charts, do not have more accuracy and advantage.

2.1.1 Use of price estimation charts

In cases where referring to equipment sellers is not very useful or the time and budget required for price estimation is insignificant, in such cases, by referring to various brochures that are mostly presented in the form of price estimation charts. Can be estimated the cost of purchasing equipment. Such charts are obtained experimentally and are provided to users by different manufacturers.

2.1.2 The effect of the size of parts on the price of equipment

In all price estimation charts, the purchase cost of equipment is shown in a logarithmic chart according to their size changes. The lines drawn in such diagrams have a slope of α. The value of α plays a very important role in estimating the cost of purchasing equipment; Therefore, you should be careful in choosing it. The relationship that relates the cost of purchasing equipment to α is [3];

$${\mathrm{PEC}}_{\mathrm{y}}={\mathrm{PEC}}_{\mathrm{w}}{\left(\frac{{\mathrm{X}}_{\mathrm{y}}}{{\mathrm{X}}_{\mathrm{w}}}\right)}^{\mathrm{\alpha }}$$E1

By using this relationship, the cost of purchasing equipment (${\mathrm{PEC}}_{\mathrm{y}}$) for a desired capacity or size (${\mathrm{X}}_{\mathrm{y}}$) can be calculated by having the cost of purchasing equipment (${\mathrm{PEC}}_{\mathrm{w}}$) for a known capacity or size (${\mathrm{X}}_{\mathrm{w}}$) achieved.

For processes that deal with heat, α is usually smaller than 1; This means that the percentage increase (or decrease) in the purchase cost of equipment is less than the percentage increase (or decrease) in their size or capacity. If there is no information about the desired design, α = 0.6 is used. This work is known as the sixteenth law.

2.1.3 Cost indices

All the prices that are examined in the economic analysis must be stated relative to the same year in which those prices were estimated; That is, if we want to use the data related to a specific year for the present, we must also consider the price index and the inflation rate. For this purpose, the following relationship can be used [3];

$$\begin{array}{l}\text{Current time in price equipment}=\text{The price of the equipment in the desired year}\\ \times \text{(Price index for the present)}/\text{(Price index for the desired year)}\end{array}$$E2

2.2 Purchased equipment installation

The cost of installing the equipment actually includes the cost of transporting the goods from the factory, workers’ wages, the cost of emptying the cargo at the place of installation of the equipment, the insurance of the workers and related goods, the foundation of the intended place for the installation of the equipment, and in general, it includes all the costs that have been purchased for the installation of the equipment. They find communication [3].

2.2.1 Piping cost

The cost related to piping includes the cost of the pipe used and also the wages of the workers during the period when the piping of the system is completed [3].

2.2.2 The cost of instrumentation and control

The multiplier value that is considered for these costs depends on the degree of automation of the devices. The more advanced the regulating and controlling devices are, the higher the cost of using them will certainly be. Of course, in cases where the use of advanced computers and complex control systems is more common, this coefficient will have a higher value [3].

2.2.3 The cost of electrical equipment and materials

These costs include the cost of parts used in power distribution lines, current replacement levers, control centers, emergency power stations, etc. [3].

2.2.4 The cost of purchasing or renting land

The cost of purchasing or renting land is significantly dependent on the geographical location of the place in question, and unlike other costs that have been studied before, this cost is likely to increase over time. But it never decreases [3].

2.2.5 The cost of civil, structural and architectural work

This category of costs includes the general costs of construction as well as other services such as the construction of streets, sidewalks and fences in the desired location and the development of green spaces. As seen in Table 1, the costs related to this part are variable depending on whether the construction is related to the construction of a new system inside the site, or a new unit inside the site, or the development of a site.

Type of Activity	Building a new system inside a new site	Building a new unit on a site that has already been built	Development of an already established site
A process that deals only with solid materials	83%	40%	30%
A process that deals with both solid and liquid materials	62%	44%	22%
A process that only deals with fluids materials	60%	20–33%	21%

Table 1.

Costs related to civil, construction and architectural affairs as a percentage of the cost of purchasing equipment [3].

2.3 Costs related to auxiliary equipment

The costs related to auxiliary equipment include all the costs that must be spent so that the main equipment can perform optimally. These costs are often spent on fuel, water, steam, electricity, cooling and sewage management. Eliminating garbage, controlling environmental pollution, providing firefighting equipment, first aid, and building dining halls are among the other uses of these expenses [3].

2.4 Costs related to engineering and supervision and supervision

This category of costs includes costs such as development and construction, preparation of appropriate maps and plans of the desired location, and other costs that are related to engineering matters; Such as the cost of purchasing engineering equipment, the cost of supervision and supervision, the implementation of construction plans, and the wages of consulting engineers [3].

For a better understanding of the issue, the costs and the ratio of each percentage to the cost of purchasing equipment are shown separately in Tables 1 and 2.

Type of Activity	% of the purchase of equipment (PEC)	Different conditions
cost of installing parts	Normally, 20–90%	If there is not much information about the type and amount of equipment, the cost of installing parts can be considered equal to 45% of the cost of purchasing equipment
Piping Cost	10–70%	1. where more solid materials are used and there is no need for piping (10–20%) of the cost of purchasing equipment; 2. for systems that mostly work with fluids is 50–70% of the cost of purchasing equipment; 3. power plants that are provided with coal fuel, the cost of piping is equal to 16% of the cost of purchasing equipment; 4. for systems that work with both solids and fluids, the cost of piping is equal to 31% of the cost of purchasing equipment; 5. and for systems that only work with Fluids work; Like Heat Recovery Steam Generators (HRSGs), the cost of piping is estimated to be equal to 66% of the equipment purchase cost
The cost of Instrumentation and control	often allocate 2–30%	1. In general and taking into account expensive and advanced control systems, the common range for the cost of setting and controlling the system is equivalent to 6–40% of the cost of purchasing parts; 2. For old steam power plants that use traditional control systems, the amount of this coefficient is 6–10% of the equipment purchase cost; 3. In the absence of appropriate information to select the price coefficient of system controls and adjustments, the average and common value of 20% of the purchase cost of parts can be used
The cost of Electrical equipment and materials	approximately 10–15%	1. The average and common amount used for these costs is equivalent to 11% of the equipment purchase cost; 2. in some cases where electronic equipment plays a major role in the design; Like power plants, these costs reach up to 48% of the equipment purchase cost
The cost of purchasing or renting land	10%	—
Costs related to auxiliary equipment	30–100%	The average amount that is often considered for these costs is 65% of the equipment purchase cost
Costs related to engineering and supervision and supervision	25–75%	The common amount that is usually considered for these costs is 30% of the equipment purchase cost

Table 2.

Costs related to a percentage of the cost of purchasing equipment [3].

2.5 The cost of constructing a building including the contractor’s wages

These costs include the cost of all temporary equipment and facilities. Among the examples of these equipments, we can mention the living place of the workers, which is temporarily built inside the site, the insurance fee and the wages of the construction workers. It should be noted that these costs are in addition to the construction costs that were mentioned in the previous sections. It should be noted that in the costs related to this part, the contractor’s profit and wages are also calculated. The experimental estimate of these costs is equivalent to 15% of direct cost (DC) [3].

2.6 The cost of possible accidents

Sometimes, unpredictable events such as weather changes, sudden stoppage of work, sudden changes in market prices and problems caused by the transportation of goods may have some effect on the estimated costs. For this purpose, a cost is usually included as a cost caused by possible incidents [3].

2.7 Startup costs

The Startup Costs (SUC) of system includes workers’ wages, the cost of materials and equipment, and other additional costs that must be spent during the setting up of the system. Of course, to these costs, the costs resulting from the decrease in income due to the system being shut down or its operation under partial load should also be added [3].

2.8 Working capital cost

The Working capital (WC) cost of the system depends on the average period of time required to produce the product and reach the customer; The meaning of products reaching the customer is when money is received from the customer for the sale of the products [3].

For a better understanding of the issue, the costs and the ratio of each percentage to the fixed investment cost (FCI) and the total investment cost (TCI) are shown separately in Table 3.

Type of Activity	% of Fixed Capital Investment (FCI)	Different conditions
The cost of possible accidents	5–20%	—
Startup Costs	1–5%	If there is not enough information about the Startup Costs, the conventional value of 10% of the Fixed Capital investment (FCI) is used for the Startup Costs
Working capital cost	% of the Total Investment Cost (TCI).	Usually, a conventional value is used to calculate the Working capital cost of the system. This amount is equivalent to 15% of the total investment cost
	10–20%

Table 3.

Costs related to a percentage of the cost of fixed capital investment (FCI) and the Total investment cost (TCI) [3].

2.9 The cost of obtaining a licensing, research and development department

If there is a desire to use franchise to obtain a work permit, in this case, the cost of obtaining a licensing, research and development (LRD) department are directly dependent on the process that is carried out inside the system. In fact, for any type of industrial activity and according to the extent of the activity, these costs have a specific range, so we should add these amounts to the total investment cost. Therefore, there is no standard or conventional amount for these costs.

2.9.1 The cost due to the lack of budget estimated during the construction

The time it takes for a project from the initial design stage to be put into operation and its equipment is launched is between 1 and 5 years. During this period, some of the investment costs should be spent on providing the salaries of the system design engineers and civil engineers, as well as the purchase and installation of system equipment and things like this. In the same way, a large amount of initial capital is spent without obtaining any income. This money may be withdrawn from the company’s fund, or taken as a loan from a bank or institution, or a combination of these two cases. In any case, some of the money intended for investment will be spent on things that will not bring any profit or income to the company. In fact, the costs of this section are applied due to the change in the value of money during the construction period; That is, the longer the project takes, the higher the costs of this part will be.

3. Simplified relationships related to the initial investment of the project

The purpose of stating the contents mentioned so far is to provide simple methods for estimating the initial investment cost of a new plan or expansion of old plans. After examining the various factors that are effective in estimating the cost of a reverse osmosis system and a multi-stage distillation system, in this part, by creating a mathematical relationship between these factors, we will arrive at simple relationships to estimate the cost of these systems.

As seen, total capital investment (TCI), Fixed Capital investment (FCI), Startup Costs (SUC), Working capital cost (WC), licensing, research and development department cost (LRD) and the estimated cost of underfunding during construction (AFUDC) also direct costs (DC) of the project are calculated, which includes onsite cost (ONSC) and offsite cost (OFSC) [4].

$$\mathrm{TCI}=\mathrm{FCI}+\mathrm{SUC}+\mathrm{WC}+\mathrm{LRD}+\text{AFUDC}$$E3

$$\mathrm{DC}=\text{ONSC}+\text{OFSC}$$E4

$$\text{OFSC}=\left\{\begin{array}{c}1.2\text{ONSC}\mathrm{new}\text{system}\\ 0.45\text{ONSC}\text{expansion}\end{array}\right.$$E5

$$\mathrm{WC}=0.15\mathrm{TCI}$$E6

$$\mathrm{SUC}=0.1\mathrm{FCI}$$E7

If it is assumed:

$$\mathrm{LRD}=\text{AFUDC}+0.15\mathrm{FCI}$$E8

In this case, by combining relations (3),(8) and (6), (7) the following relation is obtained:

$$\mathrm{TCI}=1.47\mathrm{FCI}$$E9

It follows from relations (4), (5), (9):

$$\mathrm{TCI}=1.84\mathrm{DC}=1.84\left(\text{ONSC}+\text{OFSC}\right)$$E10

By combining relations (6) and (10), the following relation is obtained:

$$\mathrm{TCI}=\left\{\begin{array}{c}4.05\text{ONSC}\mathrm{new}\text{system}\\ 2.67\text{ONSC}\text{expansion}\end{array}\right.$$E11

Experience has shown that the cost of fixed investment in a new system is between 2.8 and 5.5 times the cost of purchasing equipment [4]. Therefore:

$$\mathrm{FCI}=\left\{\begin{array}{c}2.8\sim 5.5\mathrm{PEC}\mathrm{new}\text{system}\\ 2.83\mathrm{PEC}\text{expansion}\end{array}\right.$$E12

By combining the above relations and relations (12), the following general relation is obtained:

$$\mathrm{TCI}=\left\{\begin{array}{c}4.12\sim 8.09\mathrm{PEC}\mathrm{new}\text{system}\\ 4.16\mathrm{PEC}\text{expansion}\end{array}\right.$$E13

As it is clear from relations (12) and (13), by having equipment purchase cost (PEC) and internal site costs (ONSC), the total investment cost (TCI) can be estimated. Therefore, you should be as accurate as possible in estimating the cost of purchasing equipment and the internal costs of the site; Because the more accurately the costs are estimated, the more accurate the overall investment cost will be.

Several methods to express the cost of purchasing equipment in terms of design parameters are stated in the equation, but here by using the cost functions for multi-stage distillation and reverse osmosis and other components extracted from references [3, 4, 5] respectively has been used.

Z_k is the purchase cost for the k-th component, N is the number of operating hours per year, φ is the maintenance factor, if there is no comprehensive information, the value of 1.06 can be used, and the investment return factor (CRF) which depends on the r_n percentage of inflation and the estimated life It is the equipment that is determined from the following relationship:

$${\mathrm{Z}}_{\mathrm{K}}=\frac{\mathrm{TCI}\times \mathrm{CRF}\times \mathrm{\phi }}{\mathrm{N}\times 3600}$$E14

$$\mathrm{CRF}=\frac{r_n{\left(1+r_n\right)}^{\text{year}}}{{\left(1+r_n\right)}^{\text{year}}-1}$$E15

where year is the useful life of the power plant. With the purchase cost of equipment (Z_k) and the internal costs of the site, the total investment cost can be estimated. The operating cost of multi-stage distillation and reverse osmosis system will be explained in the form of the relationships presented below.

4. Economic modeling of RO and MED system

With the equipment purchase cost (CC) and internal site costs (ONSC), the total investment cost (TCI) can be estimated. Equations for estimating the price of each component of the reverse osmosis and MED system are shown in Tables 3 and 4, respectively, and the cost of other available equipment such as steam turbine and condenser are described below. It is worth mentioning that the fixed parameters of the economic analysis are shown in Tables 5 and 6.

parameter

Capacity factor MED + RO (fc) [7]

The price of each membrane (Cm[$])

Electricity price (Ce [$/(kW h) − 1])

Inflation percentage [7]

System operation time [7]

Price of pressure vessel($) [7]

Table 4.

To estimate the cost of purchasing equipment, the functions related to the price estimation of the mentioned system can be extracted from Table 5.

Equation	Component
CCSWIP = 996 ${\left({\mathrm{Q}}_{\mathrm{f}}\right)}^{0.8}$	capital cost of the seawater intake and pre-treatment
CChpp = 52 $\left({\mathrm{Q}}_{\text{fhpp}}∆{\mathrm{P}}_{\mathrm{f}}\right)$	capital cost of high pressure pump and pre-treatment
${\mathrm{Z}}_{\text{Pump}}={\mathrm{a}}_1{\left(\text{mVΔP}\right)}^{{\mathrm{a}}_2}{\mathrm{f}}_{\mathrm{m}}.{\mathrm{\varphi }}_{\mathrm{\eta }}$; ${\mathrm{\varphi }}_{\mathrm{\eta }}=1+{\left(\frac{1-{\overline{\mathrm{\eta }}}_1}{1-{\mathrm{\eta }}_1}\right)}^{{\mathrm{a}}_3}$; ${\mathrm{f}}_{\mathrm{m}}=\left\{\begin{array}{c}\text{Cast iron}=1\\ \text{Steel}=1.41\end{array}\right.$; That: ${\mathrm{f}}_{\mathrm{m}}$: correction factor of pipe material, in this case: ${\mathrm{f}}_{\mathrm{m}}$ =1.41; ${\mathrm{\varphi }}_{\mathrm{\eta }}$: first law efficiency correctionfactor; ${\mathrm{a}}_1=549.13\frac{\text{\$}}{\mathrm{K}{\mathrm{w}}^{0.71}}$ ، ${\mathrm{a}}_2=0.71$ ، ${\mathrm{a}}_3=3$ ، ${\overline{\mathrm{\eta }}}_1=0.8$	Pump
CCm = ${\sum }_{{\mathrm{N}}_{\mathrm{RO}}}{\mathrm{C}}_{\mathrm{k}}{\mathrm{n}}_{\mathrm{m},\mathrm{j}}{\mathrm{n}}_{\mathrm{PV},\mathrm{j}}+{\sum }_{{\mathrm{N}}_{\mathrm{RO}}}{\mathrm{C}}_{\mathrm{PV}}{\mathrm{n}}_{\mathrm{PV},\mathrm{j}}$	Total membrane module cost
CCpx = 313.47 ${{\mathrm{Q}}_{\mathrm{px}\mathrm{hin}}}^{0.58}$	Pressure exchanger

Table 5.

Price of RO cycle components [6, 7].

Equations	Descriptions
CA = 140 × AE&C	Area cost ($)
Cequipment = 4 × CA	Instrument cost (evaporator, condenser…) ($)
Csite = 0.2 × Ceq	Site cost ($)
Ctr = 0.05 × (CA+ Ceq + Cs)	Transportation costs ($)
Cb = 0.15× Ceq	Building costs ($)
Cb = 0.1× Ceq	Engineers and salary costs ($)
Cc = 0.1 × (CA+ Ceq + Cs)	Contingency costs ($)
CCMED = CA+ Cequipment + Csite + Ctr + Cb + Cb + Cc	Capital costs ($)

Table 6.

MED price [5].

which in these relations A_E&C is the total area of condenser and effects.

According to the model stated in the previous part, the maintenance cost can also be determined by calculating the cost of each component.

The operating cost of the reverse osmosis system is calculated as follows:

$$\mathrm{OCm}=0.2\times \mathrm{CCm}$$E16

$$\text{OCinsurance}=0.005\times \mathrm{TCI}$$E17

$$\text{OClaor}=\mathrm{Qp}\times 24\times 365\times \mathrm{fc}\times 0.01$$E18

$$\text{OCmain}=\mathrm{Qp}\times 24\times 365\times \mathrm{fc}\times 0.01$$E19

$$\text{OCch}=\mathrm{Qp}\times 24\times 365\times \mathrm{fc}\times 0.0225$$E20

$$\mathrm{OCO}\&\mathrm{M},\mathrm{RO}=\text{OCinsurance}+\text{OClaor}+\text{OCmain}+\text{OCch}$$E21

$$\text{AOCRO}=\mathrm{OCm}+\mathrm{OCO}\&\mathrm{M},\mathrm{RO}$$E22

which in these equations OC_m is the replacement cost. OC_{O& M} is the total cost of operation, which includes OC_laor, OC_main, OC_ch, OC_insurance, which are the annual cost of the laboratory, the annual cost of repairs, the annual cost of chemicals and the insurance cost, respectively.

The operating cost of the MED system is calculated as follows [4];

$${\mathrm{C}}_{\mathrm{el}}={\mathrm{c}}_{\mathrm{el}}\times \mathrm{P}\times {\mathrm{f}}_{\mathrm{c}}\times {\mathrm{Q}}_{\mathrm{p}}\times 365$$E23

$${\mathrm{C}}_1=0.1\times {\mathrm{f}}_{\mathrm{c}}\times {\mathrm{Q}}_{\mathrm{p}}\times 365$$E24

$${\mathrm{C}}_{\mathrm{ch}}=0.04\times {\mathrm{f}}_{\mathrm{c}}\times {\mathrm{Q}}_{\mathrm{p}}\times 365$$E25

$${\mathrm{C}}_{\text{in}}=0.005\times {\mathrm{C}}_{\mathrm{A}}$$E26

$${\mathrm{AOC}}_{\mathrm{MED}}={\mathrm{C}}_{\mathrm{th}}+{\mathrm{C}}_{\mathrm{el}}+{\mathrm{C}}_{\mathrm{l}}+{\mathrm{C}}_{\mathrm{ch}}+{\mathrm{C}}_{\text{in}}$$E27

which C_el is the cost of electricity, C_l is laboratory costs, C_ch is chemical costs, C_in is insurance costs, and finally AOC_MED is the annual operating cost. In relations (23) to (25), Q_p is equal to the flow rate of permeate water and P represents the power of the pumps. The operating cost of other components have also been calculated according to reference [3, 4]. Finally, the annual total of exploitation is calculated as follows:

$${\mathrm{AOC}}_{\text{Total}}={\mathrm{AOC}}_{\text{Other}}+{\mathrm{AOC}}_{\mathrm{RO}\&\mathrm{MED}}$$E28

The normalized total cost is also determined from the eq. (29):

$$\mathrm{TAC}=\left(\mathrm{TCI}/\mathrm{CRF}\right)+{\mathrm{AOC}}_{\text{Total}}$$E29

Finally, the unit cost of fresh water production is calculated as follows.

$$\mathrm{UPC}=\frac{\mathrm{TAC}}{\mathrm{Qp}\times 24\times 365}$$E30

where in:

UPC: production cost of one m³ of produced water ($/m³).

TCC: Investment Cost ($).

AOC: annual operating cost ($/year).

OC: Operating Cost ($).

$∆{\mathrm{P}}_{\mathrm{hpp}}$: High pressure pump pressure drop (MPa).

$∆{\mathrm{P}}_{\mathrm{Tb}}$: Turbine pressure drop (Francis, Pelton) (MPa).

E_w energy consumption (KW).

Also, regarding the costs of other components of the cycle, we can refer to the suggested formulas of the reference which is stated below.

4.1 Gas turbine cycle cost

The gas turbine is made up of various components, the relationships related to the cost estimation of compressor (AC), combustion chamber (CC) and gas turbine (GT) can be expressed in the Table 7.

GT	${\mathrm{Z}}_{\mathrm{GT}}=\left(\frac{{\mathrm{a}}_{31}{\mathrm{m}}_{\mathrm{g}}^{.}}{{\mathrm{a}}_{32}-{\mathrm{\eta }}_{\mathrm{GT}}}\right)ln\left(\frac{{\mathrm{P}}_{\mathrm{C}}}{{\mathrm{P}}_{\mathrm{D}}}\right)\left[1+exp\left({\mathrm{a}}_{33}{\mathrm{T}}_{\mathrm{C}}-{\mathrm{a}}_{34}\right)\right]$; ${\mathrm{a}}_{31}=479.34\frac{\text{\$}.\mathrm{s}}{k\mathrm{g}}$; ${\mathrm{a}}_{32}=0.92$; ${\mathrm{a}}_{33}=0.036{\mathrm{K}}^{-1}$; ${\mathrm{a}}_{34}=54.4$
CC	${\mathrm{Z}}_{\mathrm{CC}}=\left(\frac{{\mathrm{a}}_{21}{\mathrm{m}}_{\mathrm{a}}^{.}}{{\mathrm{a}}_{22}-\frac{{\mathrm{P}}_{\mathrm{C}}}{{\mathrm{P}}_{\mathrm{B}}}}\right)\left[1+exp\left({\mathrm{a}}_{23}{\mathrm{T}}_{\mathrm{C}}-{\mathrm{a}}_{24}\right)\right]$; ${\mathrm{a}}_{21}=46.08\frac{\text{\$}.\mathrm{s}}{k\mathrm{g}}$; ${\mathrm{a}}_{22}=0.995$; ${\mathrm{a}}_{23}=0.018{\mathrm{K}}^{-1}$; ${\mathrm{a}}_{24}=26.4$
AC	${\mathrm{Z}}_{\mathrm{AC}}=\left(\frac{{\mathrm{a}}_{11}{\mathrm{m}}_{\mathrm{a}}^{.}}{{\mathrm{a}}_{12}-{\mathrm{\eta }}_{\mathrm{AC}}}\right)\left(\frac{{\mathrm{P}}_{\mathrm{B}}}{{\mathrm{P}}_{\mathrm{A}}}\right)ln\left(\frac{{\mathrm{P}}_{\mathrm{B}}}{{\mathrm{P}}_{\mathrm{A}}}\right)$; ${\mathrm{a}}_{11}=71.10\frac{\text{\$}.\mathrm{s}}{\mathrm{kg}}$; ${\mathrm{a}}_{12}=0.9$

Table 7.

Gas turbine cycle costs.

4.2 Steam turbine cycle cost

The Steam turbine is made up of various components, the relationships related to the cost estimation of Steam turbine (ST), Pump (P) and Condenser (CON) can be expressed in the Table 8.

ST	${\mathrm{Z}}_{\mathrm{ST}}={\mathrm{a}}_1{\mathrm{w}}^{{\mathrm{a}}_2}{\mathrm{\varphi }}_{\mathrm{\eta }}.{\mathrm{\varphi }}_{\mathrm{T}}$; ${\mathrm{\varphi }}_{\mathrm{\eta }}=1+{\left(\frac{1-{\overline{\mathrm{\eta }}}_1}{1-{\mathrm{\eta }}_1}\right)}^{{\mathrm{a}}_3}$; ${\mathrm{\varphi }}_{\mathrm{T}}=1+{\mathrm{a}}_4.exp\left(\frac{{\mathrm{T}}_1-\overline{{\mathrm{T}}_1}}{{\mathrm{a}}_5}\right)$; That: ${\mathrm{\varphi }}_{\mathrm{\eta }}$ first law efficiency correction factor; ${\mathrm{\varphi }}_{\mathrm{T}}$correction factor for inlet steam temperature; ${\mathrm{a}}_1=3880.5\frac{\text{\$}}{\mathrm{K}{\mathrm{w}}^{0.7}}$; ${\mathrm{a}}_2=0.7$; ${\mathrm{a}}_3=3$; ${\mathrm{a}}_4=5$; ${\mathrm{a}}_5=10.42$; ${\overline{\mathrm{\eta }}}_1=0.95$; $\overline{{\mathrm{T}}_1}=866$; ${\overline{\mathrm{\eta }}}_1=0.9$
P	${\mathrm{Z}}_{\text{Pump}}={\mathrm{a}}_1{\left(\text{mVΔP}\right)}^{{\mathrm{a}}_2}{\mathrm{f}}_{\mathrm{m}}.{\mathrm{\varphi }}_{\mathrm{\eta }}$; ${\mathrm{\varphi }}_{\mathrm{\eta }}=1+{\left(\frac{1-{\overline{\mathrm{\eta }}}_1}{1-{\mathrm{\eta }}_1}\right)}^{{\mathrm{a}}_3}$; ${\mathrm{f}}_{\mathrm{m}}=\left\{\begin{array}{c}\text{Cast iron}=1\\ \text{Steel}=1.41\end{array}\right.$; $f_m=\left\{\begin{array}{c}\text{Cast iron}=1\\ \text{Steel}=1.41\end{array}\right.$; That: Fm ${\mathrm{f}}_{\mathrm{m}}$: correction factor of pipe material, in this case: ${\mathrm{f}}_{\mathrm{m}}$ =1.41; ${\mathrm{\varphi }}_{\mathrm{\eta }}$: first law efficiency correction factor; $a_1=549.13\frac{\mathit{\text{\$}}}{K{w}^{0.71}}$; $a_2=0.71$; $a_3=3$; ${\overline{\eta }}_1=0.8$
CON	${\mathrm{Z}}_{\mathrm{Con}}=\frac{{\mathrm{a}}_1{{\mathrm{Q}}^{.}}_{\mathrm{Con}}}{\mathrm{k}.{\mathrm{\Delta T}}_{\text{in}}}+{\mathrm{a}}_2{\mathrm{m}}_{\mathrm{m}}^{.}+70.5{{\mathrm{Q}}^{.}}_{\mathrm{con}}\times \left(-0.6936ln\left({\overline{\mathrm{T}}}_{\mathrm{cw}}-{\mathrm{T}}_{\mathrm{b}}\right)+2.1897\right)$; That: ${\mathrm{a}}_1=280.74\text{\$}.{\mathrm{m}}^{-2}$; ${\mathrm{a}}_2=746\text{\$}.{\left(\mathrm{Kg}.\mathrm{s}\right)}^{-1}$; $\mathrm{k}=2200\mathrm{W}.{\mathrm{m}}^{-2}.{\mathrm{K}}^{-1}$

Table 8.

Steam turbine cycle costs.

5. Input parameters for RO system modeling

In a practical process, several stages are used for the RO system, each stage includes several parallel Pressure Vessel (PV) that work with the same conditions. Each PV consists of several membrane elements connected in series. The feed water enters the first element and after purification, the condensed water (water coming out of the membrane) enters the second element and continues in the same way until the last element. The output of the products of all the elements are connected to each other and finally the water output of the final product is collected. The number of elements of the numerical series is between 2 and 8.

RO system modeling is done according to the entries in Table 9.

Parameter	Unit
Inlet salt concentration	(mg/lit)
Feed water flow rate	(m^3/h)
Inlet water temperature	(°C)
Ambient pressure	(bar)
Number of PVs	(−)
Membrane type	(−)
Pure water permeability constant	(−)
Salt permeability constant	(kg/m^2.s.Pa)
Outer radius of the fiber bundle	(kg/m^2.s)
Inner radius of the fiber bundle	(m)
length of fiber bundle	(m)
Feed space	(m)
Pure water permeability constant	(m)
RO system stage number	(−)
The amount of total recovery of fresh water considered	(%)

Table 9.

Input parameters for RO system design.

The assumptions used in this modeling are expressed as follows:

1. Constant consideration of temperature during the process.

2. Constant considering the permeability coefficients for water and different salts for each membrane [6, 7, 8].

5.1 Relevant equations for modeling an RO system.

At first, we should determine the average current intensity of the membrane (f) according to the type of water entering the membrane, then the total number of required elements of the N_E system is calculated by eq. (31) and the number of PVs is calculated from eq. (32) is determined. Using the tables in the DOW catalogs and the number of elements, we can determine the number of steps needed to achieve the desired recovery. By determining the number of stages using the stage ratio eq. (33), the number of pressure pipes in each stage is obtained eq. (34, 35) [4].

$${\mathrm{N}}_{\mathrm{E}}=\frac{\left({\mathrm{Q}}_{\mathrm{p}}\right)}{\mathrm{f}\times {\mathrm{S}}_{\mathrm{m}}}$$E31

In this equation, Q_p is the permeate flow rate (m³/h) and S_m is the membrane area (m²).

$${\mathrm{N}}_{\mathrm{V}}=\frac{{\mathrm{N}}_{\mathrm{E}}}{{\mathrm{N}}_{\mathrm{E}\mathrm{pv}}}$$E32

N_v is the total number of pressure pipes and N_Epv is the number of series elements in each pressure pipe.

$$\mathrm{RR}={\left[\frac{1}{1-R}\right]}^{\frac{1}{n}}$$E33

RR is the step ratio, R is the system recovery and n is the number of steps.

$${\mathrm{N}}_{\mathrm{V}\left(1\right)}=\frac{{\mathrm{N}}_{\mathrm{v}}}{1+{\mathit{RR}}^{-1}}$$E34

$${\mathrm{N}}_{\mathrm{V}\left(2\right)}=\frac{{\mathrm{N}}_{\mathrm{v}}\left(1\right)}{1+\mathit{RR}}$$E35

N_v(i) is the number of pressure tubes in the i-th stage.

According to the mass transfer relations, it can be seen that the flow rate of water and salt through the membrane will be in the form of eqs. (36) and (37) and also the average velocity in each element of the membrane in relation (38) will be determined [6, 7, 8]. The amount of salt concentration in produced water is determined from the eq. (39) [6]. Also, according to the concentration polarization phenomenon of the mass transfer process, the salt concentration near the wall is calculated based on the film theory in the form of eq. (40) [6]. According to the continuity equation, eqs. (41) and (42) can be used for water, and eq. (43) can be used for salt.

$${\mathrm{J}}_{\mathrm{w}}=\text{A}\times \text{TCF}\left[\left(P_f-P_p-\frac{∆P_f}{2}\right)-\left({\pi }_w-{\pi }_p\right)\right]\times {10}^6$$E36

$${\mathrm{J}}_{\mathrm{s}}=\mathrm{B}\left({\mathrm{C}}_{\mathrm{w}}–{\mathrm{C}}_{\mathrm{p}}\right)$$E37

$${\mathrm{V}}_{\mathrm{w}}=\frac{\mathrm{Jw}+\mathrm{Js}}{{\rho }_p}$$E38

$${\mathrm{C}}_{\mathrm{p}}=\frac{\mathrm{Js}}{\mathrm{Vw}}\times 1000$$E39

$${\mathrm{C}}_{\mathrm{w}}={\mathrm{C}}_{\mathrm{p}}\times \left[\left(P_f-P_p-\frac{∆P_f}{2}\right)-\left({\pi }_w-{\pi }_p\right)\right]\times {10}^6$$E40

$${\mathrm{Q}}_{\mathrm{p}}={\mathrm{V}}_{\mathrm{w}}{\mathrm{S}}_{\mathrm{m}}$$E41

$${\mathrm{Q}}_{\mathrm{B}}={\mathrm{Q}}_{\mathrm{F}}-{\mathrm{Q}}_{\mathrm{P}}$$E42

$${\mathrm{C}}_B=\frac{{\mathrm{Q}}_{\mathrm{F}}{\mathrm{C}}_{\mathrm{F}}-{\mathrm{Q}}_{\mathrm{P}}{\mathrm{C}}_{\mathrm{P}}}{{\mathrm{Q}}_{\mathrm{B}}}$$E43

In the above equations, A and B and S_m is the permeability coefficient of water and salt in the membrane and its area, respectively. Permeation coefficients for different membranes are fixed and considered based on the Dow catalog. The relationships that can be used to reduce the number of adjectives are as follows [7, 8, 9];

$$\mathrm{k}=0.04\times {Re}^{0.75}\times \mathrm{S}{\mathrm{c}}^{0.33}\times \frac{{\mathrm{D}}_{\mathrm{s}}}{\mathrm{d}}$$E44

$$∆{\mathrm{P}}_{\mathrm{f}}=\frac{0.0033{\mathrm{Q}}_{\mathrm{a}}{\mathrm{L}}_{\mathrm{PV}}\mathrm{\mu }}{\mathrm{W}{\mathrm{d}}^3}$$E45

$${\mathrm{Q}}_{\mathrm{a}}=\frac{{\mathrm{Q}}_{\mathrm{b}}+{\mathrm{Q}}_{\mathrm{f}}}{2}$$E46

$$\mathrm{\pi }=\frac{0.2641\times \mathrm{C}\left(\mathrm{T}+273\right)}{1.0\times {10}^6-\mathrm{C}}$$E47

and in these relationships Re is Reynolds Number, D_s is the salt permeability (m²/S), d is the distance of the feeds from each other, and Q_a is the average flow rate, which is calculated from the eq. (45). L_PV=m. L_m where L_m is the length of the membrane $∆P_f\le 0.35\mathit{MPa}$. π is Osmotic pressure and C in this relationship is salt concentration.

In order to estimate the average pressure drop, the Hagen-Poisey equation is used. Sherrod’s number is calculated according to the eq. (44) and (45); by which concentration polarization can be calculated.

For a spiral membrane element, each of the feed and product water flows can be considered as a flow between two parallel planes with length L, width W and distance d; and based on that, he calculated the pressure drop on the supply side. For the spiral element, the width of the membrane W can be calculated with the following relationship based on the area of the membrane and the number of sheets (N_l) ${\mathrm{S}}_{\mathrm{m}}=\mathrm{W}\times \mathrm{l}\times {\mathrm{N}}_{\mathrm{l}}$ [6]. According to the above equations from (36) to (43), the number of adjectives for each equation can be determined in the following Table 10. In some equations, the adjectives are repeated, we have tried to mention the adjectives that are expressed for the first time in the table in order to determine the number of variables completely.

Adjectives	equation
${\mathrm{J}}_{\mathrm{w}},{\mathrm{P}}_{\mathrm{f}},{\mathrm{C}}_{\mathrm{w}},{\mathrm{C}}_{\mathrm{p}}$	(36)
${\mathrm{J}}_{\mathrm{s}}$	(37)
${\mathrm{V}}_{\mathrm{w}}$	(38)
—	(39)
—	(40)
${\mathrm{Q}}_{\mathrm{p}}$	(41)
${\mathrm{Q}}_{\mathrm{B}}$	(42)
${\mathrm{C}}_B$	(43)

Table 10.

Adjectives in RO system modeling equations.

As can be seen, there are 8 equations with 9 variables, which can solve 8 equations and 8 nonlinear adjectives by assuming the input pressure and correcting it.

5.2 RO system design flowchart

In some references, to solve these equations, the rate of salt rejection (for example, in Ref. [6]) or in some other references, the recovery of any RO system is assumed (reference [10]) and the equations are solved based on that. Nader et al. [8] presented another solution that solves equations with the same number of adjectives, which is based on the trial and error method. Here, due to the fact that an accurate and comprehensive modeling is used, the minimum assumptions of the method presented by Nader et al. [8], have been used, whose values can be seen in Table 8. But in the following, according to the determination of the inlet water pressure and the requirement to solve the equations by repetition method, the duration increases, and as a result, the equations are solved according to the solution of nonlinear equations by Newton’s method. Finally, according to the repetition method, the amount of feed water inlet pressure has been determined by trial and error; According to the explanations given, the problem solving flowchart is presented in “Figure 2”.

Figure 2.

Algorithm for solving equations in RO desalination system design.

In the solution process, to increase the accuracy, each element is divided into smaller components, and the output from one component will be the input to another component. “Figure 3” shows the method of dividing an element into smaller components to increase accuracy, which is chosen based on the method proposed by Nader et al. [8].

Figure 3.

Schematic of each element for RO system modeling [4].

6. Conclusions

In this chapter, the technical and economic analysis of the reverse osmosis desalination system was discussed. Relationships and parameters that are important in discussing the cost of produced water in reverse osmosis desalination systems as well as different parts of the power plant were explained in detail. Also, relationships, parameters, and mathematical models used to simulate reverse osmosis water desalination were also explained.