# A Stochastic Optimization Approach to the Design of Shale Gas/Oil Wastewater Treatment Systems with Multiple Energy Sources under Uncertainty

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## Abstract

**:**

## 1. Introduction

^{3}in 2007 to 24 trillion ft

^{3}in 2018, and the recent estimation shows that the cumulative production would be more than 400 trillion ft

^{3}over the next two decades [1,2]. This growth has spurred the development of various shale-gas monetization industries for the production of chemicals and fuels such as methanol, olefins, aromatics, and liquid transportation fuels [3,4,5]. Shale gas production is associated with utilizing substantial amounts of freshwater (estimated to be between 7000 and 18,000 m

^{3}per well [6,7]) for construction, drilling, hydraulic fracturing, and well closure operations. Major portions of the injected water leave the well as flow-back and produced water (FPW), which contains high concentration of dissolved solids and contaminated materials [8,9]. The direct injection of FPW into underground disposal wells was the most common option for wastewater management during the past decades. However, the disposal option is facing significant obstacles due to a high transportation cost and environmental risks. Therefore, there is a growing interest in developing treatment and reuse strategies [10,11,12].

## 2. Problem Statement

- A gas-and-oil production facility;
- A candidate solar-energy collection system to be integrated with the fossil energy;
- An external power grid;
- Candidate wastewater treatment systems (multi-effect distillation “MED” and reverse osmosis “RO”);
- A candidate cogeneration system;
- A candidate thermal-energy storage system;
- An adjacent industrial process with known size and data on hot and cold streams.

- The flowrate and characteristics of produced shale gas, flared shale gas, flow-back and produced wastewater, and freshwater demand during stimulating a few shale-gas wells by hydraulic fracturing operations.
- A set of ${\mathrm{N}}_{\mathrm{C}}$ industrial process cold streams (to be heated) and a set of ${\mathrm{N}}_{\mathrm{H}}$ industrial process hot streams (to be cooled). Given also are the heat capacity (flowrate × specific heat) of each process cold stream, ${\mathrm{fc}}_{\mathrm{p},\mathrm{v}}$, and of each process hot stream, ${\mathrm{FC}}_{\mathrm{p},\mathrm{u}}$; the inlet (supply) temperature of a cold stream, ${\mathrm{t}}_{\mathrm{v}}^{\mathrm{s}}$; the inlet (supply) temperature of a hot stream, ${\mathrm{T}}_{\mathrm{u}}^{\mathrm{s}}$; the outlet (target) temperature of a cold stream, ${\mathrm{t}}_{\mathrm{v}}^{\mathrm{t}}$; the outlet (target) temperature of a hot stream, ${\mathrm{T}}_{\mathrm{u}}^{\mathrm{t}}$, where $\mathrm{v}$ = 1, 2, …, ${\mathrm{N}}_{\mathrm{C}}$, and $\mathrm{u}$ = 1, 2, …, ${\text{}\mathrm{N}}_{\mathrm{H}}$.
- An external power grid demand.
- The solar data for a system site such as hourly dry bulb temperature, hourly wet bulb temperature, hourly direct normal solar irradiance, and hourly solar incidence angle.
- The forecast price of natural gas over the considered horizon.
- The direct capital cost of parabolic trough collector items (based on LS-3 collector type).
- The characteristics of a thermal storage system media.
- The techno-economic data for RO and MED
- The unit costs of freshwater acquisition, primary and secondary treatments of wastewater, disposal of wastewater, and transportation of wastewater.
- A percentage contribution of each water treatment plants in the total desalinated water production.

- Solar energy is utilized as a source of heat. The useful thermal power of solar collectors fluctuates dynamically during the year. The size (design area) and cost of the concentrated solar energy system are unknown and are to be specified through optimization formulation.
- A set ${\mathrm{N}}_{\mathrm{HU}}$ of heating utilities;${\text{}\mathrm{H}}_{\mathrm{UTILITY}}$ = {$\mathrm{h}|\mathrm{h}=1,2,\dots ,{\text{}\mathrm{N}}_{\mathrm{HU}}\}$; the temperature ${\mathrm{T}}_{\mathrm{h}}^{\mathrm{H}}$ and the cost ${\mathrm{C}}_{\mathrm{h}}^{\mathrm{H}}$ are known for each heating utility, and a set ${\mathrm{N}}_{\mathrm{CU}}$ of cooling utilities; ${\mathrm{C}}_{\mathrm{UTILITY}}$ = $\left\{\mathrm{c}|\mathrm{c}=1,2,\dots ,{\text{}\mathrm{N}}_{\mathrm{CU}}\right\}$; the target temperature ${\mathrm{t}}_{\mathrm{c}}^{\mathrm{t}}$ and the supply temperature ${\mathrm{t}}_{\mathrm{c}}^{\mathrm{s}}$ are known for each cooling utility, while heating and cooling utilities flowrates are unknown.
- The cogeneration process exploits a steam turbine to generate power and the surplus steam that leaves the turbine as a heat source for several heating purposes. The optimal values of generated power and produced steam are to be determined.

- The optimal mix of solar energy, thermal storage energy, and fossil fuel for the entire system that meets the system requirements of electric and thermal power;
- The minimum total annual cost of the entire system;
- The maximum annual profit of the entire system;
- The economic feasibility of the system;
- The optimal design and operation of the system;
- The impact of the system on environmental aspects.

## 3. Approach

#### 3.1. Generating Scenario Tree for Uncertain Parameters

^{n}scenarios, where n is the number of random variables in the model. This technique that can match the first (2N−1) statistical moments statistical features (mean, variance, max, kurtosis, and skewness) of the continuous distribution, where N is the number of discrete points [43,44,45,46,47,48]. The first task for modeling the operating system mathematically under uncertainty is to represent uncertain parameters using probabilistic scenarios, which is defined as scenario generation.

#### 3.2. Two-Stage Stochastic Optimization Model

#### 3.3. Modeling Formulation

#### 3.3.1. Solar Collection Process

#### 3.3.2. Thermal Energy Storage

#### 3.3.3. Cogeneration Process

#### 3.3.4. Desalination Process

#### 3.3.5. Economic Assessment

#### 3.4. Optimization Formulation

#### 3.4.1. Solar Collection Process

#### 3.4.2. Thermal Energy Storage

#### 3.4.3. Cogeneration Process

#### 3.4.4. Desalination Process

#### 3.4.5. Industrial Process

#### 3.4.6. Objective Function

^{s}= annual value of treated water + annual value of avoided cost of discharging wastewater + annual value of electricity supplied for an external grid + annual value of an industrial process (midstream) productions;

^{s}= annualized operating cost of the cogeneration process + annualized operating cost of the parabolic trough collectors + annualized operating cost of the thermal energy storage + annualized operating cost of MED plant + annualized operating cost of RO plant + annualized operating cost of an industrial process;

^{®}[76]. Additionally, it is noteworthy that the selection of optimal design and operational patterns for the system can be also evaluated for sustainability and safety metrics by altering the economic objective function (economic metric) using the sustainability and safety weighted return on investment metrics [77,78].

## 4. Case Study

_{water}[80]. A large number of wells in a shale play and the heavy regulations of storing wastewater in containers can contribute to obtaining approximately a constant flow of flow-back and produced water (FPW) because there is always a compensation for declination in the amount production of FPW in a well from other wells and the capability of providing constant flow of wastewater to desalination plants from containers directly. Thus, to estimate the amount of FPW returned from a shale play to the surface that can be considered as an input to desalination plants calculations in addition to avoid the uncertainty, the average of an FPW flow for 10 plays in the Eagle Ford Basin between the early 2000s to 2015 is estimated from the total FPW quantity (151.22 × 10

^{6}m

^{3}) during this period [7]. Table A9 provides techno-economic data for two desalination technologies [81,82,83] including RO and MED plants, which are utilized to ensure removal salt and non-salt impurities and typical exploitation for energy sources.

_{2}emissions that can be reduced by exploiting flared gas as a fuel for the cogeneration process. Particularly, in Eagle Ford basin, around 13% of the gas in the formation was flared, which is equivalent to 4.4 billion cubic feet of natural gas [87].

## 5. Results and Discussion

_{o}), as shown in Equation (128), which is considered as a stochastic parameter because it is a function of a period of year, seasons, climatic conditions, and geographic site [88].

^{®}for the feed stream condition and composition. The key results of the simulation such as the stream data, heat duty, and stream temperature were summarized in Table 2 and Table 3.

^{®}[76] and MS-Excel 2016 on Intel Core i7-6700 CPU with 16 GB RAM.

_{2}, which can be reduced during the operational mode for the system, as described in Figure 12.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

${\mathrm{AFC}}^{\mathrm{cogen}}$ | Annualized fixed capital cost of the cogeneration system |

${\mathrm{AFC}}_{\mathrm{EQ}}$ | Annualized fixed capital cost of equipment |

${\mathrm{AFC}}^{\mathrm{MED}}$ | Annualized fixed capital cost of the multi-effect desalination |

${\mathrm{AFC}}_{\mathrm{PR}}$ | Annualized fixed capital cost of an industrial process |

${\mathrm{AFC}}^{\mathrm{RO}}$ | Annualized fixed capital cost of the reverse osmosis |

${\mathrm{AFC}}_{\mathrm{S}}$ | Annualized fixed capital cost of supplements |

${\mathrm{AFC}}^{\mathrm{SC}}$ | Annualized fixed capital cost of the solar collector |

${\mathrm{A}}_{\mathrm{httfe}}$ | Heat transfer area for tubes of HTFFE for nth effect |

${\mathrm{ANI}}_{\mathrm{Cogen}}$ | Annualized income of the cogeneration process |

${\mathrm{ANI}}_{\mathrm{PR}}$ | Annualized income of processing facilities (midstream) production |

${\mathrm{ANI}}_{\mathrm{TW}}$ | Annualized income of the treated water |

${\mathrm{ANI}}_{\mathrm{WW}}$ | Annualized value of avoided cost of discharging wastewater |

${\mathrm{AOC}}_{\mathrm{PR}}$ | Annualized operating cost of an industrial process |

${\mathrm{A}}_{\mathrm{S},\mathrm{m}}$ | Membrane area per module |

${\mathrm{A}}^{\mathrm{SC}}$ | Effective surface area of the solar collector |

${\mathrm{A}}_{\mathrm{SF}}$ | Solar field aperture area |

A | Permeability |

a, b, c | Coefficients for the LS-3 collector |

$\mathrm{AFC}$ | Total annual fixed cost |

$\mathrm{AOC}$ | Total annual operating cost |

$\mathrm{Bbl}$ | Barrel |

${\mathrm{C}}_{\mathrm{CO}}$ | Cost of a column |

${\mathrm{C}}_{\mathrm{D}}$ | Salt fraction in distillate flowrate |

${\mathrm{C}}_{\mathrm{F}}$ | Salt fraction in feed flowrate |

${\mathrm{C}}_{\mathrm{FB}}$ | Salt fraction in brine flowrate |

${\mathrm{C}}_{\mathrm{HE}}$ | Cost of a heat exchanger |

${\mathrm{Cp}}_{\mathrm{ms}}$ | Specific heat of the molten salt |

${\mathrm{Cp}}_{\mathrm{oil}}$ | Specific heat of oil |

${\mathrm{C}}_{\mathrm{S}}$ | Average solute concentration in shell side |

${\mathrm{C}}_{\mathrm{TR}}$ | Cost of a tray |

Cy | Fixed charge cost |

${C}_{\mathrm{h}}^{\mathrm{H}}$ | Cost for each heating utility |

$\frac{{\mathrm{D}}_{2\mathrm{M}}}{\mathrm{K}\mathsf{\delta}}$ | Salt flux constant |

DNI | Direct normal irradiance |

d_{o} | Outer diameter of the receiver pipe |

${\mathrm{e}}_{\mathrm{MED}}$ | Electric energy requirements of MED |

${\mathrm{e}}_{\mathrm{RO}}$ | Electric energy requirements of RO |

${\mathrm{E}}_{\mathrm{T}}$ | Turbine shaft power output |

${\mathrm{E}}_{\mathsf{\theta}}$ | Expectancy operator |

$\mathrm{F}$ | Focal length of the collectors |

FC_{p,u} | Heat capacity of each process hot stream |

fc_{p,v} | Heat capacity of each process cold stream |

$\mathrm{FPW}$ | Flowback and produced water |

${\mathrm{F}}_{\mathrm{B}}$ | Volumetric flow rate of reject |

${\mathrm{F}}_{\mathrm{D}}$ | Volumetric flow rate of permeate |

${\mathrm{F}}_{\mathrm{F}}$ | Volumetric flow rate of feed |

${\mathrm{F}}_{\mathrm{f}}$ | Soiling factor (mirror cleanliness) |

${\mathrm{f}}^{\mathrm{o}}$ | Capital cost function of the design |

${\mathrm{ft}}^{3}$ | Cubic feet |

GOR | Gained output ratio |

${\mathrm{h}}_{\mathrm{act}}^{\mathrm{out}}$ | Actual outlet enthalpy of the turbine |

${\mathrm{h}}^{\mathrm{in}}$ | Inlet enthalpy of the steam |

${\mathrm{h}}_{\mathrm{is}}^{\mathrm{out}}$ | Outlet isentropic enthalpy |

${\mathrm{J}}_{\mathrm{water}}$ | Water flux |

${\mathrm{J}}_{\mathrm{solute}}$ | Solute (salt) flux |

$\mathrm{K}\left(\mathsf{\u019f}\right)$ | Incidence angle modifier |

${\mathrm{k}}_{\mathrm{f}}$ | Annualized factor for investment |

${\mathrm{k}}_{\mathsf{\gamma}}$ | Annual operation time |

${\mathrm{L}}_{\mathrm{SCA}}$ | Length of a single collector assembly |

${\mathrm{L}}_{\mathrm{spacing}}$ | Length of spacing between troughs |

MED | Multi-effect distillation plant |

MILNP | Mixed integer nonlinear program |

$\mathrm{MM}$ | Million |

$\dot{\mathrm{m}}$ | Inlet turbine steam flowrate |

${\mathrm{m}}_{\mathrm{B}}$ | Mass flowrate of brine |

${\mathrm{m}}_{\mathrm{D}}$ | Mass flowrate of distillate |

${\mathrm{m}}_{\mathrm{F}}$ | Total mass flowrate |

${\mathrm{m}}^{\mathrm{max}}$ | Maximum mass flowrate of the turbine |

${\mathrm{m}}_{\mathrm{ms}}$ | Mass flow rate of molten salt |

${\mathrm{m}}_{\mathrm{oil}}$ | Mass flowrate of oil |

${\mathrm{N}}_{\mathrm{CU}}$ | A set of cooling utilities |

${\mathrm{N}}_{\mathrm{EQ}}$ | Number of major equipment |

${\mathrm{N}}_{\mathrm{HU}}$ | A set of heating utilities |

${\mathrm{N}}_{\mathrm{P}}$ | Factor to account for the operation pressure of the boiler |

${\mathrm{N}}_{\mathrm{PS}}$ | Number of processing steps |

${\mathrm{N}}_{\mathrm{T}}$ | Factor accounting for the superheat temperature of the boiler |

N | Service life of the property in years |

N | Number of MED effects |

N | Number of discrete points |

N_{C} | An industrial process cold stream |

NGLs | Amount of natural gas liquids |

N_{H} | An industrial process hot stream |

N_{s} | Finite number of probabilistic scenarios |

$\mathrm{NSRDB}$ | National Solar Radiation Data Base |

${\mathrm{O}}_{\mathrm{EL}}$ | Optical end loss |

${\mathrm{P}}_{\mathrm{B}}$ | Pressure of reject |

${\mathrm{P}}_{\mathrm{D}}$ | Pressure of permeate |

${\mathrm{P}}_{\mathrm{F}}$ | Pressure of feed |

${\mathrm{P}}_{\mathrm{g}}$ | Gauge pressure of the boiler |

${\mathrm{P}}_{\mathrm{opof}}$ | Osmotic pressure of feed |

P | Actual pressure |

p_{o} | Standard pressure |

p^{s} | Product of the occurrence probability |

$\mathrm{PTC}$ | Parabolic trough collector |

${\mathrm{Q}}_{\mathrm{acc}}$ | Accumulated thermal power in the tank from preceding iterations |

${\mathrm{Q}}_{\mathrm{Boiler}}$ | Thermal power output of the boiler rate |

${\mathrm{Q}}_{\mathrm{collector}\to \mathrm{ambient}}$ | Total thermal power that loss from a collector to ambient |

${\mathrm{Q}}_{\mathrm{collector}\to \mathrm{fluid}}$ | Thermal power that transferred from a collector to a fluid |

${\mathrm{Q}}_{\mathrm{collector}\to \mathrm{reciever}}$ | Thermal power that absorbed by the receiver tube of a collector loop |

${\mathrm{Q}}_{\mathrm{htffe}}$ | Thermal power emitted by condensing distilled water into the tubes of the horizontal-tube falling film evaporator flow capacity |

${\mathrm{Q}}_{\mathrm{in}}$ | Inlet thermal power |

${\mathrm{Q}}_{\mathrm{LFP}}$ | Thermal power that loss from the headers (pipes) |

${\mathrm{Q}}_{\mathrm{LFV}}$ | Thermal power that loss from the expansion tank (vessel) |

${\mathrm{Q}}_{\mathrm{loss}}$ | Thermal power loss |

${\mathrm{q}}_{\mathrm{MED}}$ | Thermal energy requirements of MED |

${\mathrm{Q}}_{\mathrm{out}}$ | Outlet thermal power |

${\mathrm{Q}}_{\mathrm{solar}\text{}\mathrm{field}\to \mathrm{final}\text{}\mathrm{demand}}$ | Useful thermal power that produced by the solar field |

${\mathrm{Q}}_{\mathrm{sun}\to \mathrm{collector}}$ | Solar thermal power that produced by the solar field |

${\mathrm{Q}}_{\mathrm{TES}}$ | Net thermal power inside the tank |

Q | Flow capacity |

${\mathrm{r}}_{\mathrm{i}}$ | Inside radius of fibers |

${\mathrm{r}}_{\mathrm{o}}$ | Outside radius of fibers |

${\mathrm{R}}_{\mathrm{SL}}$ | Row shadow loss |

R | Relative humidity (%) |

RO | Reverse osmosis plant |

ROI | Return on investment |

${\mathrm{T}}_{\mathrm{amb}}$ | Ambient temperature |

${\mathrm{t}}_{\mathrm{c}}^{\mathrm{s}}$ | Supply temperature |

${\mathrm{T}}_{\mathrm{CT}}$ | Cold tank temperature |

${\mathrm{t}}_{\mathrm{c}}^{\mathrm{t}}$ | Target temperature |

${\mathrm{T}}_{\mathrm{h}}^{\mathrm{H}}$ | Temperature for each heating utility |

${\mathrm{T}}_{\mathrm{HT}}$ | Hot tank temperature |

${\mathrm{T}}_{\mathrm{in}}$ | Temperature at the inlet of the turbine |

${\mathrm{T}}_{\mathrm{ms}}$ | Temperature of the molten salt |

${\mathrm{T}}_{\mathrm{rec}}$ | Mean receiver pipe temperature |

${\mathrm{T}}_{\mathrm{sat}}^{\mathrm{in}}$ | Saturation temperature at the inlet of a turbine |

${\mathrm{T}}_{\mathrm{SH}}$ | Superheat temperature |

${\mathrm{T}}_{\mathrm{vapor},\mathrm{avg}}$ | Average temperature of the vapor |

${T}_{\mathrm{u}}^{\mathrm{s}}$ | Inlet (supply) temperature of a hot stream |

${T}_{\mathrm{u}}^{t}$ | Outlet (target) temperature of a hot stream |

T | Average maximum temperature |

TDS | Total dissolved content |

${t}_{\mathrm{v}}^{\mathrm{s}}$ | Inlet (supply) temperature of a cold stream |

${t}_{\mathrm{v}}^{t}$ | Outlet (target) temperature of a cold stream |

${\mathrm{U}}_{\mathrm{htffe}}$ | Overall heat transfer coefficient |

${\mathrm{U}}_{\mathrm{rec}}$ | Overall heat transfer coefficient of the receiver pipe |

${\mathrm{W}}_{\mathrm{c}}$ | Width of the collector aperture |

$\mathrm{W}$ | Watt |

${\mathrm{x}}_{\mathrm{B}}$ | Salt fraction in bine flow rate |

${\mathrm{x}}_{\mathrm{D}}$ | Salt fraction in distillate flow rate |

${\mathrm{x}}_{\mathrm{F}}$ | Salt fraction in total flow rate |

Subscript and Superscript Symbols | |

Acc | Accumulation |

Amb | Ambient |

Avg | Average |

B | Brine |

C | Cold |

C | Collector |

Cap | Capacity |

Cogen | Cogeneration |

CU | Cooling utilities |

CT | Cold tank |

CO | Column |

D | Distillated |

EL | End loss |

EQ | Equipment |

EX | Heat exchanger |

F | Factor |

H | Hot |

HU | Heating utilities |

HT | Hot tank |

HE | Heat exchanger |

Is | Isotropic |

In | Inelt |

K | Node |

LFP | Loss from pipes |

LFV | Loss from vessel |

Ms | Molten salt |

Opt | Optical |

Opof | Osmotic pressure of feed |

Out | Outlet |

P | Pump |

PR | Process |

Rec | Receiver |

S | Supply |

S | Scenario |

S | Salt |

S | Shaft power |

Sat | Saturation |

SC | Solar collector |

SCA | Single collector assembly |

SG | Steam generator |

SL | Shadow loss |

T | Target |

TW | Treated water |

TES | Thermal energy storage |

TR | Trays |

U | Hot stream |

V | Cold stream |

Ww | Wastewater |

RNG | Raw natural gas |

Greek Symbols | |

$\Delta {\mathrm{H}}_{\mathrm{c},\mathrm{avg}}$ | Latent heat of condensation |

${\mathsf{\u019e}}_{\mathrm{boiler}}$ | Efficiency of the boiler |

${\mathsf{\u019e}}_{\mathrm{is}}$ | Isentropic efficiency of the steam turbine |

${\mathrm{a}}_{\mathrm{Y}}$ | Annual operation time |

${\mathrm{v}}^{\mathrm{Chemicals}}$ | Value of produced chemicals |

${\mathrm{v}}^{\mathrm{Fuel}}$ | Value of produced Fuel |

${\mathrm{v}}^{\mathrm{L}}$ | Cost of labor |

${\mathrm{v}}^{\mathrm{MED}}$ | Value of produced water from MED |

${\mathrm{v}}^{\mathrm{RNG}}$ | Cost of raw natural gas |

${\mathrm{v}}^{\mathrm{RO}}$ | Value of produced water from RO |

${\mathsf{\beta}}^{\mathrm{MED}}$, ${\mathsf{\beta}}^{\mathrm{RO}}$ | Recovery fraction |

$\forall \mathrm{s}$ | For every scenario |

$\forall \mathrm{t}$ | For every time period |

$\Delta {\mathrm{h}}_{\mathrm{is}}$ | Isentropic enthalpy change |

ɳ _{opt} | Peak optical efficiency of a collector |

$\mathsf{\u019f}$ | Solar incidence angle |

$\mathcal{R}$ | Feasible region of the design |

$\mathsf{A}$ | Absorptivity of the receiver pipe |

Γ | Intercept factor |

Δ | Declination |

ΔT | Difference between inlet and outlet of the oil |

Θ | Vector of uncertain parameters |

Ρ | Reflectivity |

Τ | Glass transmissivity |

Ω | Hour angle |

$\mathsf{{\rm M}}$ | Viscosity |

## Appendix A

Equation | Description | ||
---|---|---|---|

${\mathrm{Q}}_{\mathrm{sun}\to \mathrm{collector}}$ = $\mathrm{DNI}.\text{}\mathrm{cos}\mathsf{\theta}.{\text{}\mathrm{W}}_{\mathrm{c}}$ | (52) | Thermal power (W/m), which can be produced by the solar collection process when the direct normal irradiance (DNI) hits the collector aperture | [61] |

$\mathrm{cos}\mathsf{\theta}=\sqrt{{\mathrm{cos}}^{2}{\mathsf{\theta}}_{\mathrm{z}}+{\mathrm{cos}}^{2}\mathsf{\delta}.{\text{}\mathrm{sin}}^{2\text{}}\mathsf{\omega}\text{}}$ | (53) | Incidence angle for the north–south orientation | [61] |

${\mathrm{Q}}_{\text{}\mathrm{collector}\to \mathrm{reciever}}=\mathrm{DNI}.\mathrm{cos}\mathsf{\theta}.{\text{}\mathrm{W}}_{\mathrm{c}}.{\text{}\mathsf{\eta}}_{\mathrm{opt}}.\mathrm{K}\left(\mathsf{\theta}\right).{\mathrm{F}}_{\mathrm{f}}.{\mathrm{R}}_{\mathrm{SL}}.{\mathrm{O}}_{\mathrm{EL}}$ | (54) | Thermal power (W/m), which can be absorbed by a receiver tube of a collection system loop | [13] |

${\mathsf{\eta}}_{\mathrm{opt}}=\text{}\mathsf{\rho}.\text{}\mathsf{\gamma}.\text{}\mathsf{\tau}\text{}.\text{}\mathsf{\alpha}$ | (55) | Peak optical efficiency of a collector when the incidence angle on the aperture plane is 0^{o} | [60] |

$\mathrm{K}\left(\mathsf{\theta}\right)=1-2.23073\text{}\times \text{}{10}^{-4}.\mathsf{\theta}-1.1\text{}\times \text{}{10}^{-4}$.${\mathsf{\theta}}^{2}+3.18596\text{}\times \text{}{10}^{-6}.{\text{}\mathsf{\theta}}^{3}\text{}-4.85509\text{}\times \text{}{10}^{-8}.{\text{}\mathsf{\theta}}^{4}\text{},$${0}^{0}\le \text{}\mathsf{\theta}\le {80}^{0}$$\mathrm{K}\left(\mathsf{\theta}\right)\text{}$ = 0 $\text{}\mathsf{\theta}{80}^{\mathrm{o}}$ | (56) | Incidence angle modifier for a LS-3 collector | [60] |

${\mathrm{R}}_{\mathrm{SL}}=\mathrm{min}\left[\mathrm{max}\left(0.0,\text{}\frac{{\mathrm{L}}_{\mathrm{spacing}}}{{\mathrm{W}}_{\mathrm{c}}}\text{}.\frac{\mathrm{cos}{\mathsf{\theta}}_{\mathrm{z}}}{\mathrm{cos}\mathsf{\theta}}\right);1.0\right]$ | (57) | Row shadow factor | [62] |

${\mathrm{O}}_{\mathrm{EL}}=1-\frac{\mathrm{f}.\mathrm{tan}\mathsf{\theta}}{{\mathrm{L}}_{\mathrm{SCA}}}$ | (58) | Optical end loss | [62] |

${\mathrm{Q}}_{\mathrm{collector}\to \mathrm{ambient}\text{}}={\text{}\mathrm{U}}_{\mathrm{rec}}\text{}.\text{}\mathsf{\pi}\text{}.{\text{}\mathrm{d}}_{\mathrm{o}}$.(${\mathrm{T}}_{\mathrm{rec}}-{\text{}\mathrm{T}}_{\mathrm{amb}})$ | (59) | Total thermal power (W/m), which may be lost from a collector represents the combination of the radiative heat loss from the receiver pipe to the ambient environment ${\text{}\mathrm{Q}}_{\mathrm{reciever}\to \mathrm{ambient}}$ and convective and conductive heat losses from the receiver pipe to its outer glass pipe ${\mathrm{Q}}_{\mathrm{receiver}\to \mathrm{glass}}$ | [60] |

${\mathrm{U}}_{\mathrm{rec}}=\mathrm{a}+\mathrm{b}\text{}\left({\mathrm{T}}_{\mathrm{rec}}-{\mathrm{T}}_{\mathrm{amb}}\right)+\mathrm{c}\text{}{({\mathrm{T}}_{\mathrm{rec}}-{\mathrm{T}}_{\mathrm{amb}})}^{2}$ | (60) | Overall heat transfer coefficient of a collector is found experimentally depending on a receiver pipe temperature | [60] |

${\mathrm{Q}}_{\mathrm{collector}\to \mathrm{fluid}\text{}}\text{}={\text{}\mathrm{Q}}_{\text{}\mathrm{collector}\to \mathrm{receiver}}-{\mathrm{Q}}_{\mathrm{collector}\to \mathrm{ambient}}$ | (61) | Thermal power (W/m), which can be transferred from a collector to a fluid | [13] |

${\mathrm{Q}}_{\mathrm{LFP}}=0.0583\text{}.{\text{}\mathrm{W}}_{\mathrm{c}}\text{}.\text{}\left({\mathrm{T}}_{\mathrm{rec}}-{\text{}\mathrm{T}}_{\mathrm{amb}}\right)$ | (62) | Thermal power (W/m), which may be lost from the headers (pipes) | [61] |

${\mathrm{Q}}_{\mathrm{LFV}}=0.0497\text{}.{\text{}\mathrm{W}}_{\mathrm{c}}.\text{}\left({\mathrm{T}}_{\mathrm{rec}}-{\mathrm{T}}_{\mathrm{amb}}\right)$ | (63) | Thermal power (W/m), which may be lost from the expansion tank (vessel) | [61] |

${\mathrm{Q}}_{\mathrm{PTC}\to \mathrm{final}\text{}\mathrm{demand}}={\text{}\mathrm{Q}}_{\mathrm{collector}\to \mathrm{receiver}}-{\text{}\mathrm{Q}}_{\mathrm{collector}\to \mathrm{ambient}}\text{}-{\mathrm{Q}}_{\mathrm{LFP}}-{\text{}\mathrm{Q}}_{\mathrm{LFV}}$ | (64) | Net useful thermal power (W/m), which can be produced by the solar collection process | [13] |

Equation | Description | ||
---|---|---|---|

${\mathrm{Q}}_{\mathrm{in}}={\mathrm{m}}_{\mathrm{ms}}\text{}.{\text{}\mathrm{C}}_{\mathrm{Pms}}\text{}.\text{}\left({\mathrm{T}}_{\mathrm{HT}}-{\mathrm{T}}_{\mathrm{CT}}\right)={\text{}\mathsf{\eta}}_{\mathrm{EX}}\text{}.{\text{}\mathrm{m}}_{\mathrm{oil}}\text{}.{\text{}\mathrm{C}}_{\mathrm{P},\mathrm{oil}}\text{}.\text{}\left(\mathsf{\Delta}\mathrm{T}\right)$ | (65) | Inlet thermal power (W) of the thermal storage (charge process) | [13] |

${\mathrm{Q}}_{\mathrm{out}}={\mathrm{m}}_{\mathrm{oil}}\text{}.{\text{}\mathrm{C}}_{\mathrm{Poil}}\text{}.\text{}\left(\mathsf{\Delta}\mathrm{T}\right)={\mathsf{\eta}}_{\mathrm{EX}}\text{}.{\text{}\mathrm{m}}_{\mathrm{ms}}\text{}.{\text{}\mathrm{C}}_{\mathrm{Pms}}\text{}.\text{}\left({\mathrm{T}}_{\mathrm{HT}}-{\mathrm{T}}_{\mathrm{CT}}\right)$ | (66) | Outlet thermal power (W) of the thermal storage (discharge process) | [13] |

${\mathrm{C}}_{\mathrm{Pms}}$ = $1443+0.172{\text{}\mathrm{T}}_{\mathrm{ms}}$ | (67) | Specific heat of the molten salt | [93] |

${\mathrm{Q}}_{\mathrm{TES}}={\text{}\mathrm{Q}}_{\mathrm{acc}}+{\mathrm{Q}}_{\mathrm{in}}-{\mathrm{Q}}_{\mathrm{out}}-{\mathrm{Q}}_{\mathrm{loss}}$ | (68) | Net thermal power (W) inside the tank | [13] |

${\mathrm{Q}}_{\mathrm{loss}}\text{}=\text{}0.00017.{\text{}\mathrm{T}}_{\mathrm{ms}}\text{}+0.012$ | (69) | thermal power loss (kW/m^{2}) of the cold and heat tanks | [93] |

Equation | Description | ||
---|---|---|---|

${\mathrm{T}}_{\mathrm{sat}}=112.72\text{}.{\text{}\mathrm{P}}_{\mathrm{sat}}^{0.2289}$ | (70) | Saturated temperature as a function of pressure (can be used at the outlet of a condenser or at the inlet of a boiler), Error = ±0.64% | [21] |

${\mathrm{h}}_{\mathrm{sat}}^{\mathrm{f}}=0.2674\text{}.{\text{}\mathrm{T}}_{\mathrm{sat}}^{1.2127}\text{}$ | (71) | Saturated liquid enthalpy (can be used at the outlet of a condenser or at the inlet of a boiler),$\text{}\mathrm{P}\le 2500\text{}\mathrm{psi},$ Error = ±3% | [21] |

${\mathrm{s}}^{\mathrm{v}}=\left(-0.5549\text{}.\mathrm{ln}\left({\mathrm{T}}_{\mathrm{sat}}\right)\text{}+3.7876\right)\text{}.{\text{}\mathrm{T}}^{0.1001\text{}.\mathrm{exp}\left(0.0017\text{}.{\text{}\mathrm{T}}_{\mathrm{sat}}\right)}$ | (72) | Entropy of steam (can be used at the inlet of a turbine), $\mathrm{P}\text{}\le \text{}2500\text{}\mathrm{psi},$ $\mathrm{T}\text{}\le \text{}1500\xb0\mathrm{F},$ Error = ±3.5% | [21] |

${\mathrm{h}}^{\mathrm{v}}=0.2029\text{}.{\text{}\mathrm{T}}_{\mathrm{sat}}\text{}.\text{}{({\text{}\mathrm{s}}^{\mathrm{v}})}^{3.647}+817.35\text{}$ | (73) | Enthalpy of steam (can be used at the inlet of a turbine or at the outlet of a turbine),$\text{}14.7\text{}\le \text{}\mathrm{P}\text{}\le \text{}2000\text{}\mathrm{psi},$ Error = ±0.6% | [21] |

$\Delta {\mathrm{h}}_{\mathrm{is}}={\text{}\mathrm{h}}^{\mathrm{v}}-{\text{}\mathrm{h}}_{\mathrm{is}}^{\mathrm{v}}$ | (74) | Isentropic enthalpy difference | [21] |

${\mathrm{h}}_{\mathrm{act}}^{\mathrm{v}}={\text{}\mathrm{h}}^{\mathrm{v}}-{\text{}\mathsf{\eta}}_{\mathrm{is}}\text{}.\text{}\Delta {\mathrm{h}}_{\mathrm{is}}$ | (75) | Actual enthalpy at the outlet of a turbine | [21] |

$\mathrm{m}=\frac{{\mathrm{Q}}_{\mathrm{process}}}{{\mathrm{h}}_{\mathrm{act}}^{\mathrm{v}}-{\mathrm{h}}_{\mathrm{sat}}^{\mathrm{f}}}$ | (76) | Mass flow rate in term of the required heat of the process (condenser) | [21] |

$\mathrm{T}=\text{}\frac{{\mathrm{h}}_{\mathrm{act}}^{\mathrm{v}\text{}}-\mathrm{B}\left(\mathrm{s}\right)}{\mathrm{A}\left(\mathrm{s}\right)}$ | (77) | Outlet temperature of a turbine | [21] |

$\mathrm{A}\left(\mathrm{s}\right)=\text{}-0.7918\text{}.\text{}{({\mathrm{s}}^{\mathrm{v}})}^{3}+3.4575\text{}.\text{}{({\mathrm{s}}^{\mathrm{v}})}^{2}+4.5513\text{}.{\text{}\mathrm{s}}^{\mathrm{v}}+2.1267\text{}$ | |||

$\mathrm{B}\left(\mathrm{s}\right)=710.22\text{}.\text{}{({\mathrm{s}}^{\mathrm{v}})}^{3}+3910.6\text{}.\text{}{({\mathrm{s}}^{\mathrm{v}})}^{2}+7117.3\text{}.{\text{}\mathrm{s}}^{\mathrm{v}}-3253.5$ | |||

${\mathrm{Q}}_{\mathrm{Boiler}}=\mathrm{m}\text{}\left({\mathrm{h}}^{\mathrm{v}}-{\mathrm{h}}_{\mathrm{sat}}^{\mathrm{f}}\right)$ | (78) | Thermal power output of a boiler | [21] |

${\mathrm{Q}}_{\mathrm{Boiler}}={\text{}\mathrm{Q}}_{\mathrm{Fuel}}\text{}.{\text{}\mathsf{\eta}}_{\mathrm{Fuel}}$ | |||

${\mathrm{m}}_{\mathrm{F}}=\frac{{\mathrm{Q}}_{\mathrm{Boiler}}}{{\mathrm{H}}_{\mathrm{v},\mathrm{F}}\text{}.{\mathsf{\eta}}_{\mathrm{boiler}}}\text{}$ | (79) | Mass flow rate of fuel is provided to a boiler | [21] |

${\mathrm{E}}_{\mathrm{Turbine}}=\mathrm{m}\text{}\left({\mathrm{h}}^{\mathrm{v}}-{\mathrm{h}}_{\mathrm{act}}^{\mathrm{v}}\right)$ | (80) | Turbine shaft power output | [21] |

Equation | Description | ||
---|---|---|---|

${\mathsf{\eta}}_{\mathrm{is}}=\frac{6}{5\text{}\mathrm{B}}\left(1-\frac{3.41443\text{}\times \text{}{10}^{6\text{}}\times \text{}\mathrm{A}}{\Delta {\mathrm{h}}_{\mathrm{is}}\text{}\times {\mathrm{m}}^{\mathrm{max}}\text{}}\right)\left(1-\frac{{\mathrm{m}}^{\mathrm{max}}}{6\text{}\times \text{}\mathrm{m}}\right)$ | (81) | Isentropic efficiency for a turbine ${\mathrm{a}}_{\mathrm{o}}$, ${\mathrm{a}}_{1}$, ${\mathrm{a}}_{2}$, and ${\mathrm{a}}_{3}$ are turbine regression coefficient [73] | [22] |

$\mathrm{A}={\mathrm{a}}_{\mathrm{o}}+{\mathrm{a}}_{1}\text{}.{\text{}\mathrm{T}}_{\mathrm{sat}}$ | |||

$\mathrm{B}={\mathrm{a}}_{2}+{\mathrm{a}}_{3}\text{}.{\text{}\mathrm{T}}_{\mathrm{sat}}$ | |||

${\mathsf{\eta}}_{\mathrm{is}}=\frac{1}{\mathrm{B}}\left(1-\frac{3.41443\text{}\times \text{}{10}^{6\text{}}\times \text{}\mathrm{A}}{\Delta {\mathrm{h}}_{\mathrm{is}}\text{}\times {\mathrm{m}}^{\mathrm{max}}\text{}}\right)$ | (82) | Isentropic efficiency for a turbine when m = ${\mathrm{m}}^{\mathrm{max}}$ at design condition | [22] |

Equation | Description | ||
---|---|---|---|

${\mathrm{Q}}_{\mathrm{Total}}={\text{}\mathrm{Q}}_{\mathrm{htffe}}\text{}.\text{}\mathrm{N}$ | (83) | Total thermal power loads (W) of all evaporators (assumed an equal thermal load of all evaporators) | [40] |

${\mathrm{Q}}_{\mathrm{Total}}=\text{}\Delta {\mathrm{H}}_{\mathrm{c},\mathrm{avg}}\text{}.{\text{}\mathrm{m}}_{\mathrm{D}}$ | |||

$\Delta {\mathrm{H}}_{\mathrm{c},\mathrm{avg}}=2.7532\text{}.{\text{}\mathrm{T}}_{\mathrm{vapor},\mathrm{avg}}+3278.8$ | (84) | Latent heat of condensation | [40] |

${\mathrm{T}}_{\mathrm{vapor},\mathrm{avg}}=\text{}\frac{{\mathrm{T}}_{\mathrm{vapor},0}+{\mathrm{T}}_{\mathrm{vapor},\mathrm{N}}}{2}$ | (85) | Average temperature of the vapor | [40] |

${\mathrm{Q}}_{\mathrm{htffe}}={\text{}\mathrm{U}}_{\mathrm{httfe}}\text{}.{\text{}\mathrm{A}}_{\mathrm{httfe}}\text{}.\text{}\Delta {\mathrm{T}}_{\mathrm{vapor}}$ | (86) | Thermal power (W) emitted by condensing distilled water into the tubes of the horizontal-tube falling film evaporator (HTFFE) | [40] |

${\mathrm{U}}_{\mathrm{htffe}}=0.8552+4.7\text{}\times {10}^{-3}\times {\mathrm{T}}_{\mathrm{vapor},\mathrm{avg}}$ | (87) | Overall heat transfer coefficient | [40] |

$\Delta {\mathrm{T}}_{\mathrm{vapor},\mathrm{avg}}=\text{}\frac{{\mathrm{T}}_{\mathrm{vapor},0}-{\text{}\mathrm{T}}_{\mathrm{vapor},\mathrm{N}}}{\mathrm{N}}$ | (88) | An average temperature driving force of evaporators by assuming an equal vapor temperature drop for each MED evaporator | [40] |

${\mathrm{m}}_{\mathrm{F}}={\text{}\mathrm{m}}_{\mathrm{D}}+{\text{}\mathrm{m}}_{\mathrm{B}}$ | (89) | Overall balance for the MED plant | [40] |

${\mathrm{m}}_{\mathrm{F}}\text{}.{\text{}\mathrm{x}}_{\mathrm{F}}={\text{}\mathrm{m}}_{\mathrm{D}}\text{}.{\text{}\mathrm{x}}_{\mathrm{D}}+{\text{}\mathrm{m}}_{\mathrm{B}}\text{}.{\text{}\mathrm{x}}_{\mathrm{B}}$ | (90) | Overall salt balance for the MED plant | [40] |

$\frac{{\mathrm{m}}_{\mathrm{D}}}{{\mathrm{m}}_{\mathrm{F}}}=1-\text{}\frac{{\mathrm{x}}_{\mathrm{F}}}{{\mathrm{x}}_{\mathrm{B}}}$ | (91) | Recovery ratio at ${\mathrm{x}}_{\mathrm{D}}=0$ | [40] |

${\mathrm{m}}_{\mathrm{D}}={\text{}\mathsf{\beta}}^{\mathrm{MED}}\text{}.{\text{}\mathrm{m}}_{\mathrm{F}}$ | (92) | Flow rate of distillate in term of the recovery fraction | [40] |

${\mathrm{m}}_{\mathrm{B}}=\left(1-{\mathsf{\beta}}^{\mathrm{MED}}\right).{\text{}\mathrm{m}}_{\mathrm{F}}$ | (93) | Flow rate of brine in term of the recovery fraction | [40] |

$\mathrm{GOR}=\mathrm{N}\text{}\times \text{}{9.8}^{\mathrm{N}}=\text{}\frac{{\mathrm{m}}_{\mathrm{D}}}{{\mathrm{m}}_{\mathrm{s}}}$ | (94) | Gained output ratio (performance metric of MED) | [40] |

Equation | Description | ||
---|---|---|---|

${\mathrm{F}}_{\mathrm{F}}={\text{}\mathrm{F}}_{\mathrm{D}}+{\text{}\mathrm{F}}_{\mathrm{B}}$ | (95) | Overall balance of the module | [40] |

${\mathrm{F}}_{\mathrm{F}}\text{}.{\text{}\mathrm{C}}_{\mathrm{F}}={\text{}\mathrm{F}}_{\mathrm{D}}\text{}.{\text{}\mathrm{C}}_{\mathrm{D}}+{\text{}\mathrm{F}}_{\mathrm{B}}\text{}.{\text{}\mathrm{C}}_{\mathrm{B}}$ | (96) | Overall solute (salt) balance of the module | [40] |

${\mathrm{m}}_{\mathrm{D}}={\text{}\mathsf{\beta}}^{\mathrm{RO}}\text{}.{\text{}\mathrm{m}}_{\mathrm{F}}$ | (97) | Flow rate of distillate in term of the recovery fraction | [40] |

${\mathrm{m}}_{\mathrm{B}}\text{}=\left(1-{\mathsf{\beta}}^{\mathrm{RO}}\right)\text{}.{\text{}\mathrm{m}}_{\mathrm{F}}$ | (98) | Flow rate of brine in term of the recovery fraction | [40] |

${\mathrm{F}}_{\mathrm{F},\mathrm{Total}}={\text{}\mathrm{F}}_{\mathrm{F}}\text{}.\text{}\mathrm{n}$ | (99) | Total flow rate when (n) modules are in parallel | [40] |

${\mathrm{J}}_{\mathrm{water}}=\mathrm{A}\text{}\left(\Delta \mathrm{P}-\text{}\frac{{\mathrm{P}}_{\mathrm{opof}}}{{\mathrm{C}}_{\mathrm{F}}}{\text{}\mathrm{C}}_{\mathrm{S}}\right){\text{}\mathsf{{\rm Y}}}_{\mathrm{RO}}$ | (100) | Water flux | [40] |

${\mathsf{{\rm Y}}}_{\mathrm{RO}}=\text{}\frac{\mathsf{\eta}}{1+\text{}\frac{16\text{}.\text{}\mathrm{A}\text{}.\text{}\mathsf{\mu}\text{}.{\text{}\mathrm{L}}_{\mathrm{f}}\text{}.{\text{}\mathrm{L}}_{\mathrm{S}}\text{}.{\text{}\mathsf{\eta}}_{\mathrm{RO}}\text{}}{1.0133\text{}\times \text{}{10}^{5}\text{}.{\text{}\mathrm{r}}_{\mathrm{i}}^{4}}}$ | (101) | Module properties | [40] |

${\mathsf{\eta}}_{\mathrm{RO}}=\text{}\frac{\mathrm{tan}{\mathsf{\theta}}_{\mathrm{RO}}}{{\mathsf{\theta}}_{\mathrm{RO}}}\text{}$ ${\mathsf{\theta}}_{\mathrm{RO}}={(\frac{16\text{}.\text{}\mathrm{A}\text{}.\text{}\mathsf{\mu}\text{}.{\text{}\mathrm{r}}_{\mathrm{o}}}{1.0133\text{}\times \text{}{10}^{5}\text{}.{\text{}\mathrm{r}}_{\mathrm{i}}^{2}})}^{\frac{1}{2}}\text{}.\text{}\frac{{\mathrm{L}}_{\mathrm{f}}}{{\mathrm{r}}_{\mathrm{i}}}$ | |||

$\Delta \mathrm{P}\approx \text{}\frac{{\mathrm{P}}_{\mathrm{F}}+{\text{}\mathrm{P}}_{\mathrm{B}}}{2}-{\text{}\mathrm{P}}_{\mathrm{D}}={\text{}\mathrm{P}}_{\mathrm{F}}-\left(\frac{\mathrm{shell}\text{}\mathrm{side}\text{}\mathrm{pressure}\text{}\mathrm{drop}\text{}\mathrm{per}\text{}\mathrm{module}}{2}+{\text{}\mathrm{P}}_{\mathrm{D}}\right)$ | (102) | Pressure drop across the membrane | [40] |

${\mathrm{C}}_{\mathrm{S}}\approx \text{}\frac{{\mathrm{C}}_{\mathrm{F}}+{\text{}\mathrm{C}}_{\mathrm{B}}}{2}$ | (103) | Average solute (salt) concentration | [40] |

${\mathrm{J}}_{\mathrm{solute}}=\left(\frac{{\mathrm{D}}_{2\mathrm{M}}}{\mathrm{K}\mathsf{\delta}}\right)\text{}.{\text{}\mathrm{C}}_{\mathrm{S}}$ | (104) | Solute (salt) flux | [40] |

${\mathrm{F}}_{\mathrm{D}}={\text{}\mathrm{A}}_{\mathrm{S},\mathrm{m}}\text{}.{\text{}\mathrm{J}}_{\mathrm{solute}}$ | (105) | Volumetric flow rate of the distillate per module | [40] |

${\mathrm{C}}_{\mathrm{D}}\approx \text{}\frac{{\mathrm{J}}_{\mathrm{solute}}}{{\mathrm{J}}_{\mathrm{water}}}$ | (106) | Solute (salt) concentration in the distillate | [40] |

${\mathrm{A}}_{\mathrm{S},\mathrm{m}}.\text{}\mathrm{A}.\text{}\frac{{\mathrm{P}}_{\mathrm{opof}}}{2{\text{}\mathrm{C}}_{\mathrm{F}}}\text{}.{\text{}\mathsf{{\rm Y}}}_{\mathrm{RO}}.{\text{}\mathrm{C}}_{\mathrm{B}}^{2}+\left[{\mathrm{F}}_{\mathrm{F}}-{\text{}\mathrm{A}}_{\mathrm{S},\mathrm{m}}\text{}.\mathrm{A}\text{}\left(\Delta \mathrm{P}-\text{}\frac{{\mathrm{P}}_{\mathrm{opof}}}{2\text{}}\text{}\right)\text{}.{\text{}\mathsf{{\rm Y}}}_{\mathrm{RO}}\right]{\mathrm{C}}_{\mathrm{B}}-{\text{}\mathrm{F}}_{\mathrm{F}}\text{}.{\text{}\mathrm{C}}_{\mathrm{F}}=0$ | (107) | To determine the value of brine (rejection) concentration | [40] |

Equation | Description | |
---|---|---|

${\mathrm{F}}_{\mathrm{F},\mathrm{Total}}.{\text{}\mathrm{C}}_{\mathrm{F},\mathrm{Total}}={\text{}\mathrm{F}}_{\mathrm{F},\mathrm{MED}}\text{}.{\text{}\mathrm{C}}_{\mathrm{F},\mathrm{MED}}+{\text{}\mathrm{F}}_{\mathrm{F},\mathrm{RO}}\text{}.{\text{}\mathrm{C}}_{\mathrm{F},\mathrm{RO}}$ | (108) | Overall salt balance on feed streams |

${\mathrm{F}}_{\mathrm{D},\mathrm{Total}}\text{}.{\text{}\mathrm{C}}_{\mathrm{D},\mathrm{Total}}={\text{}\mathrm{F}}_{\mathrm{D},\text{}\mathrm{MED}}\text{}.{\text{}\mathrm{C}}_{\mathrm{D},\mathrm{MED}}\text{}+{\text{}\mathrm{F}}_{\mathrm{D},\mathrm{RO}}\text{}.{\text{}\mathrm{C}}_{\mathrm{D},\mathrm{RO}}$ | (109) | Overall salt balance on distillate streams |

${\mathrm{F}}_{\mathrm{B},\mathrm{Total}}\text{}.{\text{}\mathrm{C}}_{\mathrm{B},\mathrm{Total}}={\text{}\mathrm{F}}_{\mathrm{B},\mathrm{MED}}\text{}.{\text{}\mathrm{C}}_{\mathrm{B},\mathrm{MED}}+{\text{}\mathrm{F}}_{\mathrm{B},\mathrm{RO}}\text{}.{\text{}\mathrm{C}}_{\mathrm{B},\mathrm{RO}}$ | (110) | Overall salt balance on brine streams |

Equation | Description | ||
---|---|---|---|

${\mathrm{AFC}}_{\mathrm{Cogen}}=\left({\mathrm{AFC}}_{\mathrm{Boiler}}+{\text{}\mathrm{AFC}}_{\mathrm{Turbine}}\right)\text{}.{\mathrm{k}}_{\mathrm{f}}$ | (111) | Annualized fixed capital cost of the cogeneration process | |

${\mathrm{AFC}}_{\mathrm{Boiler}}=3\text{}.{\text{}\mathrm{k}}_{\mathrm{f}}\text{}.{\text{}\mathrm{N}}_{\mathrm{P}}\text{}.{\text{}\mathrm{N}}_{\mathrm{T}}\text{}.{\text{}\mathrm{Q}}_{\mathrm{Boiler}}^{0.77}$ | (112) | Annualized fixed capital cost of the boiler | [21] |

${\mathrm{AFC}}_{\mathrm{Turbine}}=475\text{}.{\text{}\mathrm{k}}_{\mathrm{f}}\text{}.{\text{}\mathrm{E}}_{\mathrm{Turbine}}^{0.45}$ | (113) | Annualized fixed capital cost of the turbine | [21] |

${\mathrm{AOC}}_{\mathrm{Cogen}}=1.3\text{}.{\text{}\mathrm{F}}_{\mathrm{P}}\text{}.{\text{}\mathrm{C}}_{\mathrm{Fuel}}\text{}.{\text{}\mathrm{k}}_{\mathsf{\gamma}}$ | (114) | Annualized operating cost of the cogeneration process | [21] |

${\mathrm{C}}_{\mathrm{Fuel}}={\text{}\mathrm{a}}_{\mathrm{Fuel}}\text{}.{\text{}\mathrm{Q}}_{\mathrm{f}}\text{}.\text{}{10}^{-6}.{\mathrm{k}}_{\mathsf{\gamma}}$=${\mathrm{k}}_{\mathrm{f}}\text{}.\text{}({\mathrm{Q}}_{\mathrm{Boiler}}/{\mathsf{\eta}}_{\mathrm{f}})\text{}.\text{}{10}^{-6}$.${\text{}\mathrm{k}}_{\mathsf{\gamma}}$ | (115) | Fuel cost based on the selected type and amount of fuel | [71] |

${\mathrm{AFC}}_{\mathrm{PTC}}={\text{}\mathrm{C}}_{\mathrm{PTC}}\text{}.{\text{}\mathrm{A}}_{\mathrm{PTC}}\text{}.{\text{}\mathrm{k}}_{\mathrm{f}}$ | (116) | Annualized fixed capital cost of the parabolic trough collectors | [13] |

${\mathrm{AOC}}_{\mathrm{PTC}}={\text{}\mathrm{OC}}_{\mathrm{PTC}}\text{}.{\text{}\mathrm{Q}}_{\mathrm{PTC}\to \mathrm{final}\text{}\mathrm{demand}}\text{}.{\text{}\mathrm{k}}_{\mathsf{\gamma}}$ | (117) | Annualized operating cost of the parabolic trough collectors | [13] |

${\mathrm{AFC}}_{\mathrm{TES}}={\text{}\mathrm{C}}_{\mathrm{TES}}\text{}.\text{}\mathrm{SCH}\text{}.{\text{}\mathrm{Q}}_{\mathrm{TES}}\text{}.{\text{}\mathrm{k}}_{\mathrm{f}}$ | (118) | Annualized fixed cost of the thermal energy storage | [13] |

${\mathrm{AOC}}_{\mathrm{TES}}={\text{}\mathrm{OC}}_{\mathrm{TES}}\text{}.{\text{}\mathrm{Q}}_{\mathrm{TES}}\text{}.{\text{}\mathrm{k}}_{\mathsf{\gamma}}$ | (119) | Annualized operating cost of the thermal energy storage | [13] |

${\mathrm{AFC}}_{\mathrm{MED}}=13.0\text{}\times \text{}{10}^{6}+2227\text{}.\text{}{\left({\mathrm{F}}_{\mathrm{F},\mathrm{MED}},\frac{{\mathrm{m}}^{3}}{\mathrm{day}}\right)}^{0.7}$ | (120) | Annualized fixed capital cost of the MED plant | [81] |

${\mathrm{AFC}}_{\mathrm{RO}}=2.0\text{}\times \text{}{10}^{6}+1166\text{}.\text{}{\left({\mathrm{F}}_{\mathrm{F},\mathrm{RO}},\frac{{\mathrm{m}}^{3}}{\mathrm{day}}\right)}^{0.8}$ | (121) | Annualized fixed capital cost of the RO plant | [81] |

${\mathrm{AOC}}_{\mathrm{MED}}=\text{}0.24\text{}.\text{}\left({\text{}\mathrm{F}}_{\mathrm{F},\mathrm{MED}},\text{}\frac{{\mathrm{m}}^{3}}{\mathrm{hr}}\right).{\text{}\mathrm{k}}_{\mathsf{\gamma}}$ | (122) | Annualized operating cost of the MED plant | [81] |

${\mathrm{AOC}}_{\mathrm{RO}}=\text{}0.18\text{}.\text{}({\mathrm{F}}_{\mathrm{F},\mathrm{RO}},\frac{{\mathrm{m}}^{3}}{\mathrm{hr}})\text{}.{\text{}\mathrm{k}}_{\mathsf{\gamma}}$ | (123) | Annualized operating cost of the RO plant | [81] |

${\mathrm{AFC}}_{\mathrm{PR}}=\left({\mathrm{AFC}}_{\mathrm{EQ}}+{\mathrm{AFC}}_{\mathrm{S}}\right)\text{}.{\text{}\mathrm{k}}_{\mathrm{f}}{\mathrm{AFC}}_{\mathrm{EQ}}={\mathrm{C}}_{\mathrm{CO}}.{\mathrm{N}}_{\mathrm{CO}}$+${\mathrm{C}}_{\mathrm{TR}}.{\mathrm{N}}_{\mathrm{TR}}$+${\mathrm{C}}_{\mathrm{HE}}.{\mathrm{N}}_{\mathrm{HE}}$ | (124) | Annualized fixed capital cost of an industrial process | |

${\mathrm{AOC}}_{\mathrm{PR}}=\left({\mathrm{v}}^{\mathrm{RNG}}.{\mathrm{F}}_{\mathrm{RNG}}+{\mathrm{N}}_{\mathrm{P}}\text{}.{\text{}\mathsf{\eta}}_{\mathrm{P}}\text{}.{\text{}\mathrm{E}}_{\mathrm{S}}+{\mathrm{v}}^{\mathrm{L}}\text{}.{\text{}\mathrm{N}}_{\mathrm{L}}\right).{\text{}\mathrm{k}}_{\mathsf{\gamma}}{\mathrm{E}}_{\mathrm{S}}=\frac{{\mathrm{E}}_{\mathrm{C}}}{{\mathsf{\eta}}_{\mathrm{P}}}=\frac{\mathrm{q}.\text{}\mathsf{\rho}\text{}.\text{}\mathrm{g}\text{}.\text{}\mathrm{h}}{3.6\times {10}^{6}}$${\mathrm{N}}_{\mathrm{L}}={\left(6.29+31.7\text{}.{\text{}\mathrm{N}}_{\mathrm{PS}}^{2}+0.23\text{}.{\text{}\mathrm{N}}_{\mathrm{EQ}}\right)}^{0.5}$ | (125) | Annualized operating cost of an industrial process | [74] [74] |

${\mathrm{ANI}}_{\mathrm{Cogen}}=\frac{{\text{}\mathrm{a}}_{\mathrm{e}}\text{}.{\text{}\mathrm{E}}_{\mathrm{Turbine}}\text{}.{\text{}\mathsf{\eta}}_{\mathrm{g}}}{3.413}\text{}.{\text{}\mathrm{k}}_{\mathsf{\gamma}}$ | (126) | Annualized income of the cogeneration process (electric power generation) | [21] |

${\mathrm{ANI}}_{\mathrm{TW}}=\left({\mathsf{\nu}}^{\mathrm{RO}}\text{}.{\text{}\mathrm{F}}_{\mathrm{D},\mathrm{RO}}+{\text{}\mathsf{\nu}}^{\mathrm{MED}}\text{}.{\text{}\mathrm{F}}_{\mathrm{D},\mathrm{MED}}\right).{\text{}\mathrm{k}}_{\mathsf{\gamma}}$ | (127) | Annualized income of the treated water | [13] |

${\mathrm{ANI}}_{\mathrm{WW}}={\text{}\mathrm{C}}_{\mathrm{WW}}\text{}.{\text{}\mathrm{F}}_{\mathrm{B},\mathrm{Total}}\text{}.{\text{}\mathrm{k}}_{\mathsf{\gamma}}$=(${\mathrm{C}}_{\mathrm{FW}}+{\text{}\mathrm{C}}_{\mathrm{TR}}+{\text{}\mathrm{C}}_{\mathrm{DS}})\text{}.{\text{}\mathrm{F}}_{\mathrm{B},\mathrm{Total}}\text{}.{\text{}\mathrm{k}}_{\mathsf{\gamma}}$ | (128) | annualized value of avoided cost of discharging wastewater | [13] |

${\mathrm{ANI}}_{\mathrm{PR}}=\left({\mathsf{\nu}}^{\mathrm{Fuel}}\text{}.{\text{}\mathrm{eF}}_{\mathrm{FP}}+{\text{}\mathsf{\nu}}^{\mathrm{Chemicals}}\text{}.{\text{}\mathrm{F}}_{\mathrm{CP}}\text{}\right).{\text{}\mathrm{k}}_{\mathsf{\gamma}}$ | (129) | Annualized income of processing facilities (midstream) productions |

Characteristics | RO | MED |
---|---|---|

Outlet Salt Content (ppm) | 200 | 80 |

Water Recovery (m^{3} Desalinated Water/m^{3} Feed Seawater) | 0.55 | 0.65 |

Value of Desalinated Water ($/m^{3} Desalinated Water) | 0.88 | 0.82 |

Thermal Energy Consumption (kWht/m^{3} Desalinated Water) | - | 65 |

Electric Energy Consumption (kWhe/m^{3} Desalinated Water) | 4 | 2 |

**Table A10.**Cost of treatment, fresh water, transportation, and disposal of flow-back and produced water (FPW).

Type | PST | Fresh Water | Transportation | Disposal |
---|---|---|---|---|

Cost ($/barrel) | 0.34 | 0.24 | 0.89 | 0.05 |

Component | Capital Cost ($/m^{2}) | Component | Capital Cost ($/m^{2}) |
---|---|---|---|

Receivers | 43 | Electronic and Control | 14 |

Mirrors | 40 | Header Piping | 7 |

Concentrator Structure | 47 | Civil Works | 18 |

Concentrator Erection | 14 | Spares, HTF, Freight | 17 |

Drive | 13 | Contingency | 11 |

Piping | 10 | Structure and Improvement | 7 |

Sky Condition | ${\mathbf{k}}_{\mathbf{t}}$ |
---|---|

Cloudy | <0.3 |

Partly cloudy | 0.3 ≤ ${\mathrm{k}}_{\mathrm{t}}$≤ 0.5 |

Sunny | >0.5 |

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**Figure 7.**Cumulative distribution function of direct normal irradiance (DNI; W/m

^{2}), solar incidence angle (°), and dry bulb temperature (°C).

Parameter | Continuous Distribution | Discrete Approximation |
---|---|---|

Direct normal irradiance (W/m^{2})(Months: November–April) | θ$~\mathrm{N}\left(\mathsf{\mu},\mathsf{\sigma}\right)~\mathrm{N}\left(321.1,\text{}159.3\right)$ | Points: (59.3, 323.7, 555.2) Probabilities: (0.185, 0.630, 0.185) |

Direct normal irradiance (W/m^{2})(Months: May–October) | θ$~\mathrm{N}\left(\mathsf{\mu},\mathsf{\sigma}\right)~\mathrm{N}\left(356.0,\text{}196.4\right)$ | Points: (109.8, 356., 605.1) Probabilities: (0.185, 0.630, 0.185) |

Solar incidence angle (Months: November–April) | θ$~\mathrm{N}\left(\mathsf{\mu},\mathsf{\sigma}\right)~N\left(29.4,15.1\right)$ | Points: (4.4, 29.5, 53.1) Probabilities: (0.185, 0.630, 0.185) |

Solar incidence angle (Months: May–October) | θ$~\mathrm{N}\left(\mathsf{\mu},\mathsf{\sigma}\right)~N\left(12.6,9.8\right)$ | Points: (−2.2, 12.7, 29.7) Probabilities: (0.185, 0.630, 0.185) |

Dry bulb temperature (Months: November–April) | θ$~\mathrm{N}\left(\mathsf{\mu},\mathsf{\sigma}\right)~N\left(14.1,5.0\right)$ | Points: (5.7, 14, 22.4) Probabilities: (0.185, 0.630, 0.185) |

Dry bulb temperature (Months: May–October) | θ$~\mathrm{N}\left(\mathsf{\mu},\mathsf{\sigma}\right)~N\left(23.8,5.1\right)$ | Points: (15.4, 23.9, 32.3) Probabilities: (0.185, 0.630, 0.185) |

Natural gas price (Months: November–April) | θ$~\mathrm{N}\left(\mathsf{\mu},\mathsf{\sigma}\right)~N\left(4.3,2.2\right)$ | Points: (0.91, 4.3, 8.0) Probabilities: (0.185, 0.630, 0.185) |

Natural gas price (Months: May–October) | θ$~\mathrm{N}\left(\mathsf{\mu},\mathsf{\sigma}\right)~N\left(4.2,2.1\right)$ | Points: (0.5, 4.2, 7.7) Probabilities: (0.185, 0.630, 0.185) |

Description | Number of Trays | Reboiler Duty (kW) | Inlet Temperature of Reboiler (°C) | Outlet Temperature of Reboiler (°C) | Condenser Duty (kW) | Inlet Temperature of Condenser (°C) | Outlet Temperature of Condenser (°C) |
---|---|---|---|---|---|---|---|

Deethanizer | 19 | 5587.1 | 189.7 | 246.6 | |||

Debutanizer | 19 | 735.5 | 228.2 | 244.3 | −861.55 | 72.5 | 61.4 |

Depropanizer | 19 | 247.99 | 75.3 | 77.6 | −255.13 | 23.2 | 22.7 |

Butane Splitter | 30 | 185.92 | 63.2 | 65.2 | −190.74 | 30.5 | 29.1 |

Stream Change | Flowrate X Specific Heat (kW/°K) | Supply Temperature (°K) | Target Temperature (°K) | Enthalpy (kW) |
---|---|---|---|---|

H1 | 78.32 | 346 | 335 | −861.55 |

H2 | 255.13 | 297 | 296 | −255.13 |

H3 | 95.37 | 304 | 302 | −190.74 |

HU | ? | 525 | 522 | ? |

C1 | 98.01 | 463 | 520 | 5587.1 |

C2 | 45.97 | 501 | 517 | 735.5 |

C3 | 82.66 | 348 | 351 | 247.99 |

C4 | 92.96 | 336 | 338 | 185.92 |

CU | ? | 291 | 292 | ? |

(%RO,%MED) * | (25% Solar Energy, 75% Fossil Fuel) ** | |||

TAC | TAP | ROI | PBP | |

(MMUSD) | (MMUSD) | % | (year) | |

30 RO, 70 MED | 76.4 | 100 | 18.6 | 5.1 |

50 RO, 50 MED | 73.6 | 99 | 18.4 | 4.4 |

70 RO, 30 MED | 70.9 | 97.6 | 18.3 | 4.5 |

(%RO,%MED) * | (50% Solar Energy, 50% Fossil Fuel) ** | |||

TAC | TAP | ROI | PBP | |

(MMUSD) | (MMUSD) | % | (year) | |

30 RO, 70 MED | 86.6 | 97.5 | 17 | 4.9 |

50 RO, 50 MED | 75.2 | 97.9 | 17.1 | 4.8 |

70 RO, 30 MED | 71.1 | 95.2 | 17.3 | 4.8 |

(%RO,%MED) * | (75% Solar Energy, 25% Fossil Fuel) ** | |||

TAC | TAP | ROI | PBP | |

(MMUSD) | (MMUSD) | % | (year) | |

30 RO, 70 MED | 89.2 | 101 | 15.5 | 5.3 |

50 RO, 50 MED | 84 | 100 | 16.1 | 4.9 |

70 RO, 30 MED | 78.8 | 98.4 | 16.3 | 5.1 |

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**MDPI and ACS Style**

Al-Aboosi, F.Y.; El-Halwagi, M.M.
A Stochastic Optimization Approach to the Design of Shale Gas/Oil Wastewater Treatment Systems with Multiple Energy Sources under Uncertainty. *Sustainability* **2019**, *11*, 4865.
https://doi.org/10.3390/su11184865

**AMA Style**

Al-Aboosi FY, El-Halwagi MM.
A Stochastic Optimization Approach to the Design of Shale Gas/Oil Wastewater Treatment Systems with Multiple Energy Sources under Uncertainty. *Sustainability*. 2019; 11(18):4865.
https://doi.org/10.3390/su11184865

**Chicago/Turabian Style**

Al-Aboosi, Fadhil Y., and Mahmoud M. El-Halwagi.
2019. "A Stochastic Optimization Approach to the Design of Shale Gas/Oil Wastewater Treatment Systems with Multiple Energy Sources under Uncertainty" *Sustainability* 11, no. 18: 4865.
https://doi.org/10.3390/su11184865