# Entropy Generation Analysis of Desalination Technologies

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Derivation of Performance Parameters for Desalination

#### 2.1. Work and Heat of Separation

_{0}, and subtracting from Equation (1) while noting that the specific Gibbs free energy is, g = h − T

_{s}(all evaluated at T = T

_{0}).

#### 2.2. Second Law Efficiency

#### 2.3. Energetic Performance Parameters

## 3. Analysis of Entropy Generation Mechanisms in Desalination

#### 3.1. Flashing

#### 3.2. Flow through an Expansion Device without Phase Change

#### 3.3. Pumping and Compressing

#### 3.4. Isobaric Heat Transfer Process

#### 3.5. Thermal Disequilibrium of Discharge Streams

#### 3.6. Chemical Disequilibrium of Brine Stream

## 4. Application of Entropy Generation Mechanisms to Seawater Desalination Technologies

#### 4.1. Multiple Effect Distillation

^{th}brine stream.

#### 4.2. Multistage Flash

^{th}feed heater are:

^{th}stage. An energy balance on the brine heater is written as ${\dot{m}}_{s}{h}_{,s}={\dot{m}}_{f}({h}_{b,0}-{h}_{h,0})$. The required conservation equations for the evaporators are mass, salinity, and energy, respectively given as:

#### 4.3. Direct Contact Membrane Distillation

^{3}/day) [22,23,24,25].

#### 4.4. Mechanical Vapor Compression

#### 4.5. Reverse Osmosis

_{product}-K, while ${\dot{S}}_{\mathrm{gen}}$ from the depressurization of the brine is only 1.0 kJ/kg

_{product}-K; the entropy change from compositional change is $-12.9$ J/kg

_{product}-K. Therefore, the diffusion of water through the RO membrane is the largest source of irreversibility, owing mainly to the large pressure drop (68 bar). Note that the high pressure pump handles the same flow rate of water through the same pressure difference, but does so at 85% efficiency and therefore generates substantially less entropy than the (zero efficiency) flow through the membrane.

^{3}) and the Second Law efficiency, per Equation (16), is 31.9%.

#### 4.6. Humidification-Dehumidification

## 5. Conclusions

- A Second Law efficiency is developed for desalination systems and is defined as the useful work output divided by the total work input to the system. The useful work output of a desalination system is the minimum least work of separation, since the useful output of the system is pure water, not pure hot water. Minimum least work of separation is defined such that all input and output streams with exception of the product stream are in thermal, mechanical, and chemical equilibrium with the environment (total dead state). The product stream is in thermal and mechanical equilibrium with the environment (restricted dead state). The exergy input to the desalination systems analyzed is either in the form of work or heat. See Equation (16).
- When considering the work input to be the minimum least work of separation plus lost work due to entropy generation, it is essential to consider entropy generated not only due to irreversibilities in the separation process, but also due to temperature disequilibrium of the discharge and the irreversible mixing of the brine with the ambient seawater. See Equation (14).
- The application of entropy generation analysis to various desalination technologies showed that thermal disequilibrium of the discharge streams results in a substantial portion of the entropy generated in thermal systems. Similarly, it was seen that entropy generation due to chemical disequilibrium is important only in systems with high recovery ratios. Depending on whether thermal or chemical disequilibrium is important, modifications to the systems can be implemented in order to capitalize on the potential differences between the discharge streams and the environment and reduce the required energy input.

## Acknowledgments

## Nomenclature

Symbols | ${\dot{\Xi}}_{\text{destroyed}}$ | exergy destruction rate [kW] | |

B | membrane distillation coefficient [kg/m^{2}-Pa-s] | $\dot{\Xi}$ | exergy flow rate [kW] |

c | specific heat [kJ/kg-K] | ${\xi}_{\text{destroyed}}$ | specific exergy destruction [kJ/kg] |

${c}_{p}$ | specific heat at constant pressure [kJ/kg-K] | ρ | denisty [kg/m^{3}] |

${D}_{i}$ | distillate from effect i [kg/s] | ||

${D}_{f,i}$ | distillate from flashing in effect i [kg/s] | Subscripts | |

${D}_{fb,i}$ | distillate from flashing in flash box i [kg/s] | atm | atmospheric |

${d}_{\mathrm{ch}}$ | flow channel depth [m] | b | brine |

g | specific Gibbs free energy [kJ/kg] | f | flashing |

h | specific enthalpy [kJ/kg] | p | product |

h_{fg} | latent heat of vaporization [kJ/kg] | F | feed |

h_{fg} | latent heat of vaporization [kJ/kg] | i | state |

L | length [m] | ref | reference |

$\dot{m}$ | mass flow rate [kg/s] | s | steam |

n | number of effects or stages [-] | $\mathrm{sw}$ | seawater |

p | pressure [kPa] | ||

$\dot{Q}$ | heat transfer [kW] | Superscripts | |

${\dot{Q}}_{\mathrm{least}}$ | least heat of separation [kW] | HX | heat exchanger |

${\dot{Q}}_{\mathrm{least}}^{\mathrm{min}}$ | minimum least heat of separation [kW] | IF | incompressible fluid |

${\dot{Q}}_{\mathrm{sep}}$ | heat of separation [kW] | IG | ideal gas |

R | ideal gas constant [kJ/kg-K] | s | isentropic |

r | recovery ratio [(kg/s product)/(kg/s feed)] | ′ | stream before exiting CV |

${\dot{S}}_{\mathrm{gen}}$ | entropy generation rate [kW/K] | ||

s | specific entropy [kJ/kg-K] | Acronyms | |

${s}_{\mathrm{gen}}$ | specific entropy generation per unit fluid [kJ/kg-K] | BH | brine heater |

${\mathcal{S}}_{\mathrm{gen}}$ | specific entropy generation per unit water produced [kJ/kg-K] | CAOW | closed air open water |

T | temperature [K] | CD | chemical disequilibrium |

${T}_{0}$ | ambient (dead state) temperature [K] | DCMD | direct contact membrane distillation |

${T}_{H}$ | temperature of heat reservoir [K] | ERI | Energy Recovery Inc. |

v | specific volume [m^{3}/kg] | FF | forward feed |

${\dot{W}}_{\mathrm{least}}$ | least work of separation [kW] | GOR | gained output ratio |

${\dot{W}}_{\mathrm{least}}^{\mathrm{min}}$ | minimum least work of separation [kW] | HD | humidification-dehumidification |

${\dot{W}}_{\mathrm{sep}}$ | work of separation [kW] | HP | high pressure |

w | width [m] | MED | multiple effect distillation |

w | specific work [kJ/kg] | MSF | multistage flash |

x | quality [kg/kg] | MVC | mechanical vapor compression |

y | salinity [g/kg] | OT | once through |

z | generalized compressibility [-] | PR | performance ratio |

PX | pressure exchanger | ||

Greek | RDS | restricted dead state | |

Δ | change in a variable | RO | reverse osmosis |

η | mole ratio of salt in seawater [-] | TD | temperature disequilibrium |

${\eta}_{e}$ | isentropic efficiency of expander [-] | TDS | total dead state |

${\eta}_{p}$ | isentropic efficiency of pump/compressor [-] | WH | water heated |

${\eta}_{\mathit{II}}$ | Second Law/exergetic efficiency [-] |

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## Appendices

## A. Least Work of Separation as a Function of Recovery Ratio

## B. Derivation of Entropy Generation Equations

#### B.1. Incompressible Fluid and Ideal Gas Approximations

#### B.2. Flashing

#### B.3. Flow Through an Expansion Device Without Phase Change (Expanders, Pipes, Throttles, Membranes, etc.)

#### B.4. Pumping and Compressing

#### B.5. Approximately Isobaric Heat Transfer Process

**Figure 1.**When the control volume is selected suitably far away from the physical system, all inlet and outlet streams are at ambient temperature and pressure. The temperature of the streams inside the control volume, denoted by ${T}_{i}^{\prime}$, might not be at ${T}_{0}$.

**Figure 2.**Addition of a high temperature reservoir and a Carnot engine to the control volume model shown in Figure 1.

**Figure 3.**Entropy is generated in the process of a stream reaching thermal equilibrium with the environment.

**Figure 5.**Entropy production in the various components of a 6 effect forward feed multiple effect distillation system.

**Figure 6.**Relative contribution of sources of entropy generation in a forward feed multiple effect distillation system. Irreversibilities in the effects dominate. Total specific entropy generation is 196 J/kg-K.

**Figure 9.**Relative contribution of sources of entropy generation in a once-through multistage flash system. Irreversibilities in the feed heaters dominate. Total specific entropy generation is 423 J/kg-K.

**Figure 11.**Relative contribution of sources of entropy generation in a direct contact membrane distillation system. Total specific entropy generation is 925.4 J/kg-K.

**Figure 13.**Relative contribution of sources of entropy generation in a mechanical vapor compression system. Total specific entropy generation is 98.0 J/kg-K. Contributions of the temperature disequilibrium of the distillate and brine streams are 0.5% and 0.2%, respectively.

**Figure 15.**Relative contribution of sources to entropy generation in the reverse osmosis system. Irreversibilities associated with product flow through the membrane dominates. Total specific entropy generation is 19.4 J/kg-K.

**Figure 16.**A schematic diagram of a closed air open water, water heated humidification-dehumidification desalination cycle.

**Figure 17.**Relative contribution of sources to entropy generation in the closed air open water, water heated humidification-dehumidification system. Irreversibilities in the dehumidifier dominate. Total specific entropy generation is 370 J/kg-K.

**Figure 18.**GOR versus Second Law efficiency for closed air open water humidification-dehumidification cycle configurations analyzed by Mistry et al. [6]. The original data, Figure 18a ([6], Figure 10), shows no correlation between GOR and the old definition of ${\eta}_{\mathit{II}}$. Figure 18b shows that using a minimum least work of separation based definition for Second Law efficiency results in a positive correlation between the energetic performance (GOR) and Second Law performance (${\eta}_{\mathit{II}}$) of the cycles.

**Figure 19.**Second Law efficiencies calculated for the systems modeled in this paper. Reverse osmosis has a substantially higher Second Law efficiency than the other desalination processes considered in this paper.

**Table 1.**Representative values of reference state constants for Equations (B.3), (B.4), (B.7), and (B.8).

Pure water and vapor constants, ${T}_{\mathrm{sat}}=50\phantom{\rule{0.166667em}{0ex}}\xb0\mathrm{C}\phantom{\rule{0.166667em}{0ex}}{p}_{\mathrm{sat}}=12.3$ kPa | |||

c | 4.18 kJ/kg-K | ${h}_{\mathrm{ref}}^{\mathrm{IG}}$ | 2590 kJ/kg |

${c}_{p}$ | 1.95 kJ/kg-K | ${h}_{\mathrm{ref}}^{\mathrm{IF}}$ | 209 kJ/kg |

R | 0.462 kJ/kg-K | ${s}_{\mathrm{ref}}^{\mathrm{IG}}$ | 8.07 kJ/kg-K |

v | $1.01\times {10}^{-3}$ m^{3}/kg | ${s}_{\mathrm{ref}}^{\mathrm{IF}}$ | 0.704 kJ/kg-K |

Seawater constants, 50 °C, 35,000 ppm | |||

c | 4.01 kJ/kg-K | ${h}_{\mathrm{ref}}^{\mathrm{IF}}$ | 200 kJ/kg |

v | $0.986\times {10}^{-3}$ m^{3}/kg | ${s}_{\mathrm{ref}}^{\mathrm{IF}}$ | 0.672 kJ/kg-K |

Output | Model Value | |

Performance ratio | PR | 4.2 |

Gained output ratio | GOR | 4.6 |

Top brine temperature | ${T}_{h}$ [°C] | 109 |

Steam flow rate | ${\dot{m}}_{s}$ [kg/s] | 91.1 |

Max salinity | ${y}_{n}$ [g/kg] | 47.3 |

Input | Value |

Seawater inlet temperature | 25 °C |

Seawater inlet salinity | 35 g/kg |

Product water salinity | 0 g/kg |

Discharged brine salinity | 58.33 g/kg |

Top brine temperature | 60 °C |

Pinch: evaporator-condenser | 2.5 K |

Recovery ratio | 40% |

Isentropic compressor efficiency | 70% |

Compressor inlet pressure | 19.4 kPa |

Output | Value |

Specific electricity consumption | 8.84 kWh/m^{3} |

Discharged brine temperature | 27.2 °C |

Product water temperature | 29.7 °C |

Compression ratio | 1.15 |

Second Law efficiency, ${\eta}_{\mathit{II}}$ | 8.5% |

© 2011 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/.)

## Share and Cite

**MDPI and ACS Style**

Mistry, K.H.; McGovern, R.K.; Thiel, G.P.; Summers, E.K.; Zubair, S.M.; Lienhard, J.H., V. Entropy Generation Analysis of Desalination Technologies. *Entropy* **2011**, *13*, 1829-1864.
https://doi.org/10.3390/e13101829

**AMA Style**

Mistry KH, McGovern RK, Thiel GP, Summers EK, Zubair SM, Lienhard JH V. Entropy Generation Analysis of Desalination Technologies. *Entropy*. 2011; 13(10):1829-1864.
https://doi.org/10.3390/e13101829

**Chicago/Turabian Style**

Mistry, Karan H., Ronan K. McGovern, Gregory P. Thiel, Edward K. Summers, Syed M. Zubair, and John H. Lienhard V. 2011. "Entropy Generation Analysis of Desalination Technologies" *Entropy* 13, no. 10: 1829-1864.
https://doi.org/10.3390/e13101829