Thermodynamics, Exergy, and Energy Efficiency in ...

Thermodynamics, Exergy, and EnergyEfficiency in Desalination Systems

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Citation J.H. Lienhard V, K.H. Mistry, M.H. Sharqawy, and G.P. Thiel,“Thermodynamics, Exergy, and Energy Efficiency in DesalinationSystems,” in Desalination Sustainability: A Technical,Socioeconomic, and Environmental Approach, Chpt. 4, H.A. Arafat,editor. Elsevier Publishing Co., 2017.

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Chapter 4

Thermodynamics, Exergy, andEnergy E�ciency inDesalination SystemsJohn H. Lienhard VDepartment of Mechanical EngineeringMassachuse�s Institute of Technology, Cambridge MA 02139, USA

Karan H. MistryDepartment of Mechanical EngineeringMassachuse�s Institute of Technology, Cambridge MA 02139, USA

Mostafa H. SharqawySchool of EngineeringUniversity of Guelph, Ontario N1G 2W1, Canada

Gregory P. ThielDepartment of Mechanical EngineeringMassachuse�s Institute of Technology, Cambridge MA 02139, USA

J.H. Lienhard V, K.H. Mistry, M.H. Sharqawy, and G.P. Thiel, “Thermodynamics, Exergy, and EnergyE�ciency in Desalination Systems,” in Desalination Sustainability: A Technical, Socioeconomic, andEnvironmental Approach, Chpt. 4, H.A. Arafat, editor. Elsevier Publishing Co., 2017. ISBN: 978-0-12-809791-5.

1

2 Thermodynam i c s o f De sa l i nat i on §4.0

AbstractDesalination is the thermodynamic process of separating fresh water from waterthat contains dissolved salts. This chapter introduces the concepts and methodsrequired for thermodynamic analysis of desalination systems. Thermodynamic lawsare summarized along with the chemical thermodynamics of electrolytes. Exergyanalysis is introduced. The work and heat of separation are de�ned, and the roles ofentropy generation and exergy destruction are identi�ed. Important sources of entropygeneration are discussed. Examples are given for the application of these methods toseveral representative desalination systems.

Keyword s : Thermodynamics, exergy, entropy generation, e�ciency, physicalproperties, desalination systems

Contents4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34.2 Thermodynamic Essentials . . . . . . . . . . . . . . . . . . . . . 4

4.2.1 Thermodynamic Analysis of Open Systems . . . . . . . . 54.2.2 Thermodynamic Properties of Mixtures . . . . . . . . . . 64.2.3 Activity Coe�cient Models for Electrolytes . . . . . . . . 114.2.4 Colligative Properties: Boiling Point Elevation, Freezing

Point Depression, Vapor Pressure Lowering and OsmoticPressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.3 Exergy Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3.1 Exergy Variation . . . . . . . . . . . . . . . . . . . . . . . 224.3.2 Seawater Exergy . . . . . . . . . . . . . . . . . . . . . . . 26

4.4 Thermodynamic Analysis of Desalination . . . . . . . . . . . . . 284.4.1 Derivation of Performance Parameters for Desalination . 284.4.2 Analysis of Entropy Generation Mechanisms in Desalination 38

4.5 Entropy Generation in Desalination Systems . . . . . . . . . . . 454.5.1 Multiple E�ect Distillation . . . . . . . . . . . . . . . . . . 454.5.2 Direct Contact Membrane Distillation . . . . . . . . . . . 494.5.3 Mechanical Vapor Compression . . . . . . . . . . . . . . . 514.5.4 Reverse Osmosis . . . . . . . . . . . . . . . . . . . . . . . 53

4.6 Second Law E�ciency for Cogeneration . . . . . . . . . . . . . . 574.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.A Seawater Properties Correlations . . . . . . . . . . . . . . . . . . 66

4.A.1 Speci�c Volume . . . . . . . . . . . . . . . . . . . . . . . . 664.A.2 Speci�c Enthalpy . . . . . . . . . . . . . . . . . . . . . . . 67

§4.1 I ntroduct i on 3

4.A.3 Speci�c Entropy . . . . . . . . . . . . . . . . . . . . . . . 684.A.4 Chemical Potential . . . . . . . . . . . . . . . . . . . . . . 694.A.5 Osmotic Coe�cient . . . . . . . . . . . . . . . . . . . . . . 704.A.6 Speci�c Heat Capacity at Constant Pressure . . . . . . . . 714.A.7 Tabulated Data . . . . . . . . . . . . . . . . . . . . . . . . 72

4.B Pitzer Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 734.C Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.1 IntroductionDesalination is an energy intensive process, and reduction of energy consumptionis central to the development and design of all desalination processes. At the heartof the process is the chemical energy of separating water and dissolved salts. Thisminimum amount of energy will always be required, no matter how desalinationis to be accomplished. The entire desalination system, however, brings additionalenergy consumption to the process of desalinating water, as a result of a varietyof ine�ciencies that are present in any real system. The total energy consumed isnormally several times greater than the minimum chemical energy of separation.Identifying and reducing this additional energy consumption requires thermodynamicanalysis of the desalination system.

The minimum separation energy can be characterized as work, in thethermodynamic sense of the word. Examples of thermodynamic work include thework done by high pressure pumps in moving water, the work done by the rotatingsha� of an electric motor, and the work done by current �owing in an electric �eld.So, we commonly speak of the minimum or least work of separation as de�ning thethermodynamic limit of performance for a desalination process. The least work isusually higher when the salinity of the source water is higher.

A real desalination system will require greater amounts of work, owing tofactors such as losses in pumps, hydraulic pressure behind membranes that greatlyexceeds osmotic pressure, or incomplete energy recovery from high pressure brines.From a thermodynamic viewpoint, the losses or irreversibilities of components in adesalination system can be measured in terms of the entropy they generate. Entropyproduction directly increases the energy requirements of a system that produces agiven amount of desalinated water. Entropy is produced whenever friction occurs andwhen heat is transferred through a di�erence in temperature.

Work is a form of energy transfer, but it is distinct from energy in the form ofheat. Most o�en, work is generated by heat transferred from a high temperaturesource (perhaps a burning fuel in a combustor), through some process (expansion ofsteam through a turbine perhaps), to a low temperature sink (the cooling water ina plant condenser). The number of joules of work generated (by the steam turbine)is substantially less than the number of joules of heat transferred from the hightemperature source, with the di�erence between them ending up as heat transferto the low temperature sink. This di�erence is greater when the high temperature


source is cooler. Any ine�ciencies in the system, perhaps from friction or from largetemperature di�erences in heat exchangers, will reduce the amount of work that canbe produced.

The di�erence between heat and work is essential to characterizing and evaluatingthermal desalination processes, and more so because many desalination systems thatare driven by heat also consume signi�cant electrical energy, which is a form of work.Heat and work cannot be added or compared directly. Heat gains its potential todo work from the presence of a signi�cant temperature di�erence between the hightemperature source and the low temperature sink. Consequently, thermodynamicmethods, in the form of exergy analysis, are essential in assessing the e�ciencyof a thermal desalination process, especially when comparing such a process to awork-driven desalination process. Heat transfer between two temperatures representsa source of exergy, as does any work transfer, but a given amount of heat transferprovides less exergy when taken from a lower temperature source. Thus, exergeticmethods are all the more important when thermal processes are to be driven by lowtemperature heat sources, such as common solar collectors. Exergy is destroyed byfriction or when heat is transferred to a lower temperature. In fact, any process thatgenerates entropy destroys exergy in the process.

In this chapter, we introduce the concepts and methods required for assessing thethermodynamic e�ciency of desalination systems. In Section 4.2, thermodynamiclaws for open systems (those through which �uids �ow) are given, and key results onthe chemical thermodynamics of electrolyte mixtures (salts dissolved in water) aresummarized, including the Pitzer model. Osmotic pressure and boiling point elevationare discussed. Section 4.3 introduces exergy analysis of desalination. With these toolsin hand, Section 4.4 proceeds to the thermodynamic analysis of desalination processes.The work and heat of separation are derived, and the role of entropy generationand exergy destruction are identi�ed. Important sources of entropy production aredescribed and equations for their evaluation are given. Finally, Section 4.5 applies thesemethods to give brief assessments of the causes of ine�ciency in several representativedesalination systems. The appendices to this chapter give some useful correlationsfor the thermodynamic properties of seawater and and further details of one of theelectrolyte models.

4.2 Thermodynamic Essentials

This chapter deals with the energy consumption of desalination systems: how it isevaluated, what its limits are, and how to push real systems closer to those limits.The key to answering these questions is a ground-up understanding of e�ciency, η –strictly, the Second Law E�ciency. Conceptually, η is the fraction of energy consumedthat must be consumed according to the laws of physics, or

η =Minimum Energy Input

Actual Energy Input(4.1)

§4.2 Thermodynam i c E s s ent i a l s 5

......

Heat Transfer, Q Work Transfer, W

InletStreams, i Outlet

Streams, o

Systemproperties

S, U, V, ...

BoundaryTemp. T0

Figure 4.1: In an open system, streams i enter the system at some state, undergo achange of state and exit as outlet streams o. A work transfer ÛW and/or a heat transferÛQ may accompany the change in state. The system has instantaneous properties, such

as internal energy U , entropy S , and volume V .

Conceptually, the actual energy input is

Actual Energy Input = Minimum Energy Input+ Energy to Overcome Losses (4.2)

A more rigorous de�nition of η and the interplay between the three terms aboveare described in the sections that follow; but �rst, in order to understand the detailsof each of these terms, we require some thermodynamic basics. In this section, usingthe Gibbs free energy as the fundamental thermodynamic function, we will provide abrief overview of the essential thermodynamic concepts and terms used throughoutthe remainder of the chapter.

4.2.1 Thermodynamic Analysis of Open Systems

An open system, or control volume (CV), is shown in Fig. 4.1: streams �ow into thesystem at some inlet state(s) i , undergo a change of state within the control volume,and exit at state(s) o. A work or heat transfer may occur across the boundary of thesystem to e�ect the streams’ change of state or as a consequence of their change ofstate. We �x the system boundary temperature at T0.

The First Law of Thermodynamics for this system reads

dU

dt= ÛQ + ÛW +

∑i

ÛHi −∑o

ÛHo (4.3)

where U is the internal energy of the system, ÛQ is the net heat transfer rate into thesystem, ÛW is the net rate at which work is done on the system, and ÛHi and ÛHo are theenthalpy in�ows and out�ows, respectively. The Second Law of Thermodynamics for


this system isdS

dt=ÛQ

T0+

∑i

ÛSi −∑o

ÛSo + ÛSgen (4.4)

where S is the entropy of the system, ÛQ/T0 is the rate of entropy transfer into thesystem; ÛSi and ÛSo are the entropy in�ows and out�ows, respectively; and ÛSgen is theentropy generation rate within the CV.

Multiplying Eq. (4.4) by the boundary temperature T0 and subtracting it fromEq. (4.3), we �nd that the heat transfer terms cancel; and at steady state we are le�with

0 = ÛW +∑i

( ÛHi −T0 ÛSi ) −∑o

( ÛHo −T0 ÛSo) −T0 ÛSgen (4.5)

If we de�ne the CV such that the streams enter and exit at T0, then H and S are alsoevaluated at T0, and the preceding reduces to

ÛW =∑o

ÛGo −∑i

ÛGi +T0 ÛSgen (4.6)

where the grouping H −TS is the Gibbs free energy, G.Equation (4.6) illustrates the fundamental variables involved in desalination system

energetic analysis. When ÛSgen = 0, the system is reversible, and ÛW becomes ÛWrev, thereversible work associated with the streams changing from their inlet to their outletstates:

ÛWrev =∑o

ÛGo −∑i

ÛGi (4.7)

Thus, we see that di�erences in Gibbs free energy determine a system’s reversiblelimits; consequently T0 ÛSgen is identically equal to the energy required to overcome thelosses that produce ÛSgen.

In cases where the outlet streams have di�erent temperatures than the inlet stream,a control volume analysis will not isolate Gibbs energy in the same way, and thereversible work would di�er because would it be possible to extract additional workfrom the di�erences in temperature relative to T0. In those situations, exergy isdiscarded with the leaving streams (see discussion in Sec. 4.4.1). Some cases of thistype are analyzed in Sec. 4.5. A formulation using �ow exergy (Sec. 4.3) would alsoaccount for these di�erences in outlet state.

The two groupings in Eq. 4.6, ∆G and T0 ÛSgen, are the building blocks forthermodynamic analysis of desalination systems: G determines the limits, and preciseidenti�cation of ÛSgen guides avenues for improvement. In addition, as we will seeshortly, when G is known for a substance as a function of temperature and pressure, itcontains all of the necessary information to compute e�ciency, including T0 ÛSgen.

4.2.2 Thermodynamic Properties of Mixtures

As discussed in Sec. 4.2.1, G serves two useful purposes in our analyses. First, theproperty itself de�nes the reversible work associated with any change in state. Second,because its conjugate variables are temperature and pressure, which are measurable


Table 4.1: Relationships betweenG = f (T ,p,Ni ) and several thermodynamic variables

Property Expression

Entropy S = −

(∂G

∂T

)p,Ni

Molar Volume v =1N

(∂G

∂p

)T ,Ni

Enthalpy H = G −T

(∂G

∂T

)p,Ni

= −T 2(∂(G/T )

∂T

)p,Ni

Heat Capacity Cp = −T

(∂2G

∂T 2

)p,Ni

Chemical Potential µi =

(∂G

∂Ni

)T ,p

and controllable thermodynamic variables, it provides a convenient basis with whichto correlate substance behavior. A model describing G for aqueous solutions (e.g.,seawater) is thus essential.

Gibbs Energy as a Fundamental Thermodynamic Function

Once a thermodynamic state is �xed, any thermodynamic property at that state canbe computed as a function of any other complete set of independent properties at thatstate. However, there are speci�c independent variables, called conjugate variables,that when used to formulate a property, yield all thermodynamic properties of thesubstance at any state.

The conjugate variables for G are temperature, T , pressure, p, and number ofmoles of species i , Ni (equivalently mole fraction, molality, or other measures ofconcentration). This can be shown as follows. With the de�nition of G = H −TS =U + pV −TS ,

dG = dU + p dV +V dp −T dS − S dT (4.8)

By the fundamental relationship of thermodynamics, dU = T dS − p dV +∑

i µi dNi ,and so

dG = −S dT +V dp +∑i

µi dNi (4.9)

Thus, knowledge of G = f (T ,p,Ni ) allows one to compute all thermodynamicproperties of the substance, as shown for several properties by the relationships inTable 4.1. With these properties, the actual energy consumption, the losses, and theenergy required to overcome the losses, T0 ÛSgen, can be computed [see Eqs. (4.3) and(4.4)].


Standard Formulations for Gibbs Energy and the Chemical Potential

By de�nition, any extensive mixture property X can be written as the weighted sumof partial molar properties over each species i:

X =∑i

(∂X

∂Ni

)T ,p,Nj,i

Ni (4.10)

where the partial molar property, ∂X/∂Ni , physically represents the change in mixtureX with an incremental addition of species i . Thus, the Gibbs free energy of the solutioncan be written as

G =∑i

(∂G

∂Ni

)T ,p,Nj,i

Ni (4.11)

But sinceT and p are the conjugate variables ofG [see Eq. (4.9)], the chemical potentialµi is the partial molar Gibbs free energy,

(∂G∂Ni

)T ,p,Nj,i

.

The chemical potential may be written as

µi ≡

(∂G

∂Ni

)T ,p,Nj,i

= µ◦i + RT lnai (4.12)

where the superscript ◦ denotes the standard (or reference) state, R is the molar(universal) gas constant, T is the absolute temperature in kelvin, and ai is the activityof species i in the solution. For solutes, the reference state is usually a hypotheticalsolution of in�nite dilution and unit concentration (i.e., 1 mol/L or 1 mol/kg, etc.) atthe same temperature. For solvents (water), the reference state is typically the pureliquid at the same temperature. Depending on the convention, the reference pressuremay or may not be �xed at 1 atm [1, 2].

Chemical activity is o�en termed a “thermodynamic concentration”, and is relatedto the change in energy of a component as it is added to a mixture, i.e., as itsconcentration changes. For solutes, it is modeled as the product of the activitycoe�cient, γ , and a measure of concentration, giving several possible formulations:

ax,i = γx,ixi , ab,i = γb,ibi/b◦, ac,i = γc,ici/c

◦ (4.13)

where xi is mole fraction, bi is molality, and ci is molar concentration (moles per unitvolume) of species i . The reference quantities c◦ and b◦ usually have a magnitude ofunity and thus o�en not written explicitly.

In the ideal solution model, the �rst building block in mixture thermodynamics,the rational activity coe�cient γx = 1. Physically, in an ideal solution, the introductionof a solute causes little change in the average interaction potential between all species.This can approximate real solution behavior when, e.g., the solution is very dilute1and solvent–solvent interactions are negligibly small. The model can also be suitablefor mixtures of two chemically similar components. For a mixture of toluene and

1In dilute systems, the di�erence between γx , γb , and γc is generally quite small, so in the ideal solutionmodel, all activity coe�cients are commonly taken as unity. Formulas for converting from one activitycoe�cient scale to another are straightforward to use and can be found in [3].


benzene, e.g., benzene–benzene interactions are like those of benzene–toluene andtoluene–toluene [4]. The activity coe�cient γ thus represents departures from idealsolution behavior, and is the lynchpin in computing G for electrolyte solutions.

For water, the solvent, deviations from ideality are o�en expressed as an osmoticcoe�cient, the form of which is also dependent on the concentration scale:

ϕx = −µ◦w− µw

RT lnxw(4.14a)

ϕb =µ◦w− µw

RTMw

∑i bi

(4.14b)

The water activity is related to the osmotic coe�cient by the relation µw − µ◦w=

RT lnaw . For an ideal solution, ϕx = ϕb = 1.The water activity, aw , is not independent of the solute activities, and it is usually

calculated from solute activity using the Gibbs-Duhem relationship. The latter is foundby equating Eq. (4.9) and the di�erential form of Eq. (4.11), dG =

∑i d(µiNi ):∑

i

Ni dµi = −S dT +V dP (4.15)

All models for activity must satisfy this relationship. At constant temperature andpressure, Eq. (4.15) can be restated on a mole fraction basis by dividing through by N :∑

i

xidµi = 0 (4.16)

This equation can be manipulated to �nd the solvent activity, aw as

d lnaw = −1xw

∑i,w

d(γixi )

γi(4.17)

Analogous expressions can be found for the other concentration scales.

Ideal Solutions and Deviations from Ideality as Functions of Activity

A common modeling approach for the activity coe�cient is to model the energeticcontributions that lead to deviations from ideality—the excess Gibbs free energy—andthen di�erentiate to compute an activity coe�cient. Because of the relationshipsbetween G = f (T ,p,Ni ) and the suite of thermodynamic properties, such models arealso related to deviations from ideality for all other thermodynamic properties: theexcess enthalpy, excess entropy, excess volume, etc. These relationships are discussedbrie�y later.

The Gibbs free energy can be written as the sum of ideal and excess components

G = G id +Gex (4.18)

Based on the de�nition of chemical potential (Eq. (4.12)) and the osmotic coe�cient(Eq. (4.14b)), and the condition for ideality, G id can be written on a mole fraction or


molal concentration scale, respectively as:

G id

N=

∑i

(µ◦x,i + RT lnxi

)xi (4.19a)

G id

mw

=©«µ◦w − RTMw

∑j ∈solutes

bjª®¬bw +

∑j ∈solutes

(µ◦b, j + RT lnbj

)bj (4.19b)

The excess component is similarly written on any of the concentration scales, andyields the following expressions for the activity coe�cient:

lnγx,i =∂(Gex/RT )

∂Ni, lnγb,i =

∂(Gex/mwRT )

∂bi(4.20)

Combining this ideal–excess breakdown with the relationships shown in Table 4.1,we can �nd the properties of ideal solutions and formulate deviations as a functionof the activities and their pressure and temperature derivatives. We will show theprocedure explicitly for entropy and enthalpy; several other properties are shown inTable 4.2.

For an ideal solution, the entropy and enthalpy are:

S id = −

(∂G id

∂T

)P,xi= −

∑i

Ni

(∂µ◦i∂T

��P+ R lnxi

)(4.21)

H id =∑i

Ni

(µ◦i −T

∂µ◦i∂T

��P

)(4.22)

When a solution undergoes an isothermal, isobaric change of state by the addition(or removal) of some species, the corresponding change in entropy and enthalpy areknown as the entropy of mixing and the enthalpy of mixing, respectively. For an idealsolution, we are le� with

∆S idmix = −R∆

(∑i

Ni lnxi

)(4.23)

∆H idmix = 0 (4.24)

Because ideal solutions have zero enthalpy of mixing, the isothermal, isobaric change inGibbs free energy—the reversible work associated with salt dissolution or desalination(cf. Eq. (4.7))—is identically equal to T∆S id

mix:

∆G idmix = RT∆

(∑i

Ni lnxi

)(4.25)

All desalination processes must overcome the entropy of mixing.


Table 4.2: Relationships between partial molar excess thermodynamic properties andthe activity coe�cient

Property Expression

Partial Molar Excess Entropy −R

(T∂ lnγi∂T

��P+ lnγi

)Partial Molar Excess Volume RT

∂ lnγi∂P

��T

Partial Molar Excess Enthalpy −RT 2 ∂ lnγi∂T

��P

Partial Molar Excess Heat Capacity −RT 2(∂2 lnγi∂T 2

��P+

2T

∂ lnγi∂T

��P

)Deviations from these ideal approximations are entirely contained within the

temperature and pressure dependence of the activity coe�cient. The excess entropy is

Sex = −

(∂Gex

∂T

)P,Ni

= −∑i

Ni R

(T∂ lnγi∂T

��P,xi+ lnγi

)︸︷︷︸

partial molar excess entropy

(4.26)

and the excess enthalpy is

H ex = Gex −T

(∂Gex

∂T

)P,Ni

= −∑i

Ni

(RT 2 ∂ lnγi

∂T

��P,xi

)︸︷︷︸

partial molar excess enthalpy

(4.27)

Relationships for the other properties are shown in Table 4.2, and mimic those shownin Table 4.1, with Gex replaced by µex = RT lnγi .

4.2.3 Activity Coe�cient Models for Electrolytes

Thus far, we have seen that knowledge of G allows us to predict the limits ofdesalination system performance; that knowledge of G = f (T , P ,Ni ) allows us topredict all properties of a solution; and that γ and ϕ re�ect departures from idealsolution behavior. In this section, we will review some common models for activitycoe�cients in aqueous electrolyte solutions: the Debye-Hückel Limiting Law, theDavies Equation, and the Pitzer Model. For a salt MX that dissociates like

MνM XνX ←−→ νM MzM + νX XzX (4.28)

it is typical to report values of a mean activity coe�cient, γ±, which is de�ned as

γ ν± = γνMM γ νXX (4.29)

where ν = νM + νX . Values of the mean molal activity coe�cient γb,± for NaCl areshown in Fig. 4.2: as model complexity increases from the Debye-Hückel Limiting Law,the models are more accurate over larger concentration domains.

0 0.5 1 1.5 2 2.5b1/2

0

0.5

1

1.5

Activ

ity C

oeffi

cien

t

Debye-HückelDaviesPitzerMeasurements

Figure 4.2: A comparison of three models for the activity coe�cient of NaCl showsthat the simpler Debye-Hückel and Davies models are limited to concentrations belowabout 0.01 and 0.5 mol/kg, respectively. Measured data is an average over severalvalues in literature provided in [5].

12


Debye-Hü el Theory For very dilute ionic solutions, the most important additionto mixture energy is that which derives from ionizing the salt as it dissolves, whichis re�ected in electrostatic interactions between ions. Through a combinationof electrostatics and statistical mechanics, Debye-Hückel theory [6] provides anexpression for the activity coe�cient that is accurate for solutions of ionic strength upto about 0.01 molal. Full derivations can be found in a variety of texts, e.g., [3] and theresulting expression for activity coe�cient is

lnγx,± = −A |zMzX |√I = − |zMzX |

[e3 (2NAρw)

1/2

8π (εr ε0kbT )3/2

]√I (4.30)

where zM and zX are the cation and anion charge numbers, e is the elementary charge,NA is Avogadro’s number, εr is the dielectric constant (relative permittivity) of thesolvent, ε0 is the vacuum permittivity, kB is Boltzmann’s constant, ρw is the density ofpure water, and I is the molal ionic strength. The molal ionic strength is de�ned as

I =12

∑i

biz2i (4.31)

Equation (4.30) is known as the Debye-Hückel Limiting Law, which has asquare-root dependence on ionic strength. To �rst order, most expressions for activitycoe�cient are characterized by a square-root dependence on ionic strength, re�ectingthe long-range electrostatic contributions to the excess Gibbs free energy that arethe �rst to appear as solute concentration increases from zero. The temperaturedependence of the Debye-Hückel activity coe�cient is not quite T −3/2 because thedielectric constant is also a function of temperature.

Davies Equation Several other equations exist that extend the Debye-HückelLimiting Law using mostly empirical or semi-empirical methods, and these can befound in, e.g., [3, 7]. The general approach is to add concentration and/or ionicstrength dependent terms to the Debye-Hückel expression to capture the curvature ofγ vs. concentration (Fig. 4.2) that arises from the increasing importance of short-rangeinteractions as solute concentration increases further. One particularly useful equationthat requires no adjustable parameters for a particular electrolyte is given by Davies [8]:

− logγx,± = 0.50|zMzX |

( √I

1 +√I− 0.20I

)(4.32)

which is approximate for ionic strengths up to about 0.1 and a temperature of 25◦ C.

Pitzer Model The Pitzer model [9–12] is based on a virial expansion of the excessGibbs free energy, and extends the Debye-Hückel model to account for short rangeinteractions between solute pairs and triplets. Detailed derivations are given inreferences [9, 10, 13]. For calculations beyond the dilute limit, the Pitzer model amongthose most widely used for single and mixed electrolytes. The model has been validatedand used for calculations in several mixed electrolytes, e.g., [7, 14]. Of the three models


discussed in this section, the Pitzer model is the most accurate for seawater and itsconcentrates.

Expressions are available for the mean molal activity coe�cient, but for added�exibility, we will give the single ion expressions here2. We also provide an expressionfor uncharged solutes, which may also exist in mixed-electrolytes or arise as a resultof ion-pairing in concentrated mixtures.

The activity coe�cient of an individual cation, M , is given by

lnγM = z2MF +

∑a

ba(2BMa + ZCMa)

+∑c

bc (2ΦMc +∑a

baΨMca)

+∑∑a<a′

baba′Ψaa′M

+ |zM |∑c

∑a

bcbaCca +∑n

bn(2λnM ) (4.33)

For an individual anion, X , the expression is analogous:

lnγX = z2X F +

∑c

bc (2BcX + ZCcX )

+∑a

ba(2ΦXa +∑c

bcΨXac )

+∑∑c<c ′

bcbc ′Ψcc ′X

+ |zX |∑c

∑a

bcbaCca +∑n

bn(2λnX ) (4.34)

The activity coe�cient of uncharged species N (e.g., aqueous CO2) is

lnγN =∑c

bc (2λNc ) +∑a

ba(2λNa) (4.35)

2Of course, as the activity of an individual ion cannot be measured explicitly, the physical meaning ofsuch expressions is unclear. However, as noted by [11], the combination of Eqs. (4.33) and (4.34) in theform of a measurable mean activity coe�cient produces the same equation as Pitzer [10], and is far moreconvenient for calculations in mixed electrolytes.


The molal osmotic coe�cient ϕb is calculated from the expression

(ϕ − 1)∑i

mi = 2[−Aϕ I 3/2

1 + 1.2√I

+∑c

∑a

bcba(Bϕca + ZCca)

+∑∑c<c ′

bcbc ′

(Φϕcc ′ +

∑a

baΨcc ′a

)+

∑∑a<a′

baba′

(Φϕaa′ +

∑c

bcΨaa′c

)+

∑n

∑a

bnbaλna +∑n

∑c

bnbcλnc

](4.36)

in whichZ =∑

i |zi |mi , Mw is the molar mass of water, and the remainder are functionsquantifying particular solute interactions, as de�ned below. Subscript c denotes cationsother thanM , a denotes anions other thanX , andn denotes uncharged (neutral) solutes.Summation over all i indicates a sum over all solutes; likewise summation over all c , a,and n denotes a sum over all cations, anions, and neutral solutes, respectively. Thesummation notations c < c ′ and a < a′ indicate that the sum should be performedover all distinguishable cation pairs and anion pairs, respectively. Equations for Aϕ ,Bij , B

ϕi j , F , Φi j , and Φ

ϕi j are given in Appendix 4.B.

4.2.4 Colligative Properties: Boiling Point Elevation, FreezingPoint Depression, Vapor Pressure Lowering and OsmoticPressure

Mixture properties that depend on the total mole numbers of dissolved substances, butnot the speci�c chemical species dissolved in a solvent, are called colligative properties.Colligative properties are truly independent of the chemical species dissolved onlywhen the solution is very dilute, so that the solution behaves as an ideal mixture.They must be corrected to some degree at higher concentrations, typically throughthe osmotic coe�cient.

Four colligative properties of great importance in desalination system analysisrelate to chemical equilibrium between two phases or two di�erent mixtureconcentrations: the boiling point elevation, δb ; the freezing point depression, δf ;the osmotic pressure, π ; and the vapor pressure lowering, ∆psat. In osmoticallydriven processes, such as reverse osmosis (RO) and forward osmosis (FO), the �uxof pure water across the membrane is a function of π . Likewise, in thermal systems,such as multistage �ash (MSF), multi-e�ect distillation (MED), and mechanical vaporcompression (MVC), the evaporative �ux is a function of δb ; and in freeze desalination,the ice formation rate is a function of δf . In membrane distillation (MD), the �ux ofwater vapor across the membrane is a function of ∆psat. Accurate values of δb , δf , π ,and ∆psat are thus needed for a wide range of salinities and temperatures.

45 50 55 60Temperature (°C)

10

12

14

16

18

Vapo

r Pre

ssur

e (k

Pa)

Pureb=0.5 mol/kg

1 mol/kg2 mol/kg

δb

∆Psat

Figure 4.3: The vapor pressure curve for sodium chloride solutions: the boiling pointelevation (δb ) and the vapor pressure lowering (∆psat) increase with increasing saltconcentration (b). The curves are computed from Eq. (4.43) using the equations ofPitzer [2] and Saul and Wagner [15].

16


Boiling point elevation

Adding salt to water increases its boiling temperature at a given pressure. The boilingpoint elevation, δb , is the di�erence between the saturation temperatures of a solution,Tsat, and of its pure solvent,T ◦sat at a �xed pressure (Fig. 4.3). It tends to be an increasingfunction of salt concentration and vapor pressure (equivalently, solvent saturationtemperature). As with all of the colligative properties, it is related to the activity of thesolvent (water), and can thus be written as a function of ϕb :

δb = Tsat −T◦sat =

RT ◦2sath◦fg

ϕb∑i

bi (4.37)

where R is the molar gas constant, and h◦fg is enthalpy of vaporization of pure water3.For low salinities, ϕb is close to one (see Fig. 4.4), and so δb has a nearly lineardependence on salinity. A good approximation for the boiling point elevation istherefore given by the linear equation

δb = Kbb (4.38)

where b is the total molality of the solute ions in moles/kg-solvent, the ebullioscopicconstant Kb is de�ned as

Kb =RT ◦2sath◦fg

(4.39)

and T ◦sat is in kelvin. For water at 1 atm, Kb = 0.513 K-kg/mol.

Freezing point depression

The boiling point elevation’s analog at the solid–liquid phase boundary is the freezingpoint depression, δf . The freezing point depression is the di�erence between thefreezing temperatures of a solution, Tf , and of its pure solvent, T ◦f . A mirror of δb , δfincreases with increasing salt concentration, and can be written similarly3:

δf = T◦f −Tf =

RT ◦2f

h◦sfϕb

∑i

bi (4.40)

where h◦sf is enthalpy of fusion of pure water. Like the boiling point elevation, we cande�ne a linear relationship for the freezing depression at low salinities, where ϕb isnear unity:

δf = Kf b (4.41)

The prefactor Kf is known as the cryoscopic constant:

Kf =RT ◦2f

h◦sf(4.42)

3 Eq. (4.37) is valid when δb � T ◦sat and h◦fg does not change signi�cantly between Tsat and T ◦sat, whichis true for seawater at typical thermal desalination operating conditions. Likewise, Eq. (4.40) is valid whenδf � T ◦f and h◦sf does not change signi�cantly between Tf and T ◦f .


Like the ebullioscopic constant, T ◦f is an absolute temperature (i.e., in K). For water at1 atm, Kf = 1.86 K-kg/mol.

Vapor pressure lowering

The boiling point elevation is considered at a �xed pressure. The equivalent e�ectat a �xed temperature is the vapor pressure lowering, ∆psat (Fig. 4.3). A solutionmaintained at �xed temperature will require a greater vacuum for solvent to evaporate.The vapor pressure of water in solution is related to the activity of water4 by

ln(psat

p◦sat

)= lnaw (4.43)

The vapor pressure lowering is the di�erence between the vapor pressures of thesolution, psat, and of the pure solvent, p◦sat, at �xed temperature. In terms of the osmoticcoe�cient, the vapor pressure lowering is thus

ln(∆psat

p◦sat+ 1

)= −ϕbMw

∑i

bi (4.44)

where Mw is the molar mass of the solvent. For su�ciently dilute solutions, where ϕbis near one, Eq. (4.44) reduces to Raoult’s Law for the vapor pressure of the solvent:

psat = p◦satxw =

p◦sat1 +Mw

∑i bi

(4.45)

Osmotic pressure

The osmotic pressure represents the pressure that must be applied to a solution tomaintain equilibrium with the pure solvent at a �xed temperature. Osmotic pressurerises as the solute concentration increases, and it is proportional to the absolutetemperature. For a non-ideal electrolyte solution such as seawater, the osmoticcoe�cient characterizes the deviation from ideal solution behavior. The osmoticpressure for a solution composed of multiple solutes may be expressed in terms of themolality of solutes as [3]:

π = −ρwMw

RT lnaw = ϕb RT ρwb (4.46)

where π is the osmotic pressure; ϕb is the molal osmotic coe�cient [Eq. (4.14b)]; R isthe molar gas constant; T is the absolute temperature in kelvin; ρw is the density ofthe solvent, in this case pure water; Mw is the molar mass of the solvent; and b is the

4Equation (4.43) is valid when the fugacity coe�cients and the Poynting correction factor are near unity;that is, when the vapor behaves ideally and the pressure is not too high. For a 2 mol/kg NaCl solution at80 ◦C, Eq. (4.43) overpredicts the vapor pressure by less than 0.001% relative to the general expression forvapor-liquid equilibrium that includes fugacity coe�cients and the Poynting correction factor; see [1] fordetails.

§4.3 Ex ergy Analy s i s 19

total molality of the solute ions in moles/kg-solvent5. The total molality of the ions inseawater can be written as a function of the salinity as:

b = 31.843ws

1 − ws(4.47)

where ws is the salt mass fraction of the solution in kg-salt/kg-solution, and 31.843 isa constant which takes into account the weighted average of the molecular weightof each dissolved solute of the seawater constituents which have the same relativecomposition at any salt concentration.

The osmotic coe�cient is a function of salinity and temperature (see Section 4.A.5).Using a piecewise �t for the osmotic coe�cient function [i.e., Eq. (4.A.11) and (4.A.13)],the osmotic pressure of seawater can be calculated from Eq. (4.46) for a range of salinityof 0–120 g/kg and a range of temperature of 0–200 ◦C.

In the literature, a linearized expression for the osmotic pressure modeled on thevan ’t Ho� equation is widely used for quick calculation of the seawater osmoticpressure. The van ’t Ho� equation itself applies to very dilute (or ideal) solutions(ϕb = 1):

π = iRTc (4.48)

where i is the van ’t Ho� factor (accounting for dissociation of the solute), and c is themolarity of the solution (mol/L). We can de�ne a modi�ed van ’t Ho� coe�cient tomake a linear approximation to the osmotic pressure function, Eqs. (4.46) and (4.47),as follows:

π = Cws (4.49)

The modi�ed van ’t Ho� coe�cient,C , is determined to be 73.45 kPa-kg/g for seawaterat 25 ◦C. The linear model represented by Eq. (4.49) can be used for a salinity rangeof 0 to 70 g/kg, which is a typical range for many desalination applications. For thisrange, the maximum deviation from the nonlinear osmotic pressure function, Eq. (4.46),is 6.8%. The osmotic pressure, Eq. (4.46), the osmotic coe�cient, Eqs. (4.A.11) and(4.A.12), and the linear osmotic pressure, Eq. (4.49), are shown as a function of salinityin Fig. 4.4.

4.3 Exergy Analysis

Exergy is the maximum amount of work obtainable when a thermodynamic system isbrought into equilibrium from its initial state to the environmental (dead) state. In thisregard, the state of the environment must be speci�ed. The system is considered to beat zero exergy when it reaches the environment state, which is called the dead state.The equilibrium can be divided into thermal, mechanical and chemical equilibrium.These equilibria are achieved when the temperature (T ), pressure (p) and concentration

5Eq. (4.46) is valid when water can be modeled as an incompressible �uid and when µ◦w

is pressuredependent (see Sec. 4.2.2). Strictly, therefore, aw should be evaluated at the reference pressure plus π ,but evaluating it at 1 bar only leads to small errors up to moderately high pressures (e.g., for a 4 mol/kgNaCl solution at 25 ◦C, lnaw at 200 bar is 0.8% higher than at 1 bar). Glasstone [1] provides an alternativeformulation of osmotic pressure for a pressure-independent reference state.

Figure 4.4: Seawater molal osmotic coe�cient and osmotic pressures versus salinityfor a �xed temperature shown on the le�- and right-hand axes. The osmotic coe�cientcurve and nonlinear osmotic pressure curves are extrapolated for salinities greaterthan 120 g/kg and these sections are shown as bold dashed lines. The linear osmoticpressure curve is solid.

20


(w) of the system reach the values found in the surrounding environment (T0, p0, andw0, respectively). Therefore, exergy consists of a thermomechanical exergy and achemical exergy. The thermomechanical exergy is the maximum work obtained whenthe temperature and pressure of the system changes to the environment temperatureand pressure (T0, p0) with no change in the concentration. In this case, we say thatthermomechanical equilibrium with the environment occurs. The chemical exergy isthe maximum work obtained when the concentration of each substance in the systemchanges to its concentration in the environment at the environment pressure andtemperature (T0, p0). In that case, chemical equilibrium occurs.

For a control mass (closed system), the exergy, e , can be mathematically expressedas [16, 17]

e = (u − u∗) + p0 (v − v∗) −T0 (s − s

∗) +

n∑i=1

wi(µ∗i − µi,0

) /Mi (4.50)

where u, s , v, µi , wi , and Mi are the speci�c internal energy, speci�c volume, speci�centropy, chemical potential of species i , mass fraction of of species i , and molar massof species i , respectively. Properties with superscript ∗ in the above equation aredetermined at the temperature and pressure of the environment (T0, p0) but at thesame composition or concentration of the initial state. This is referred to as therestricted dead state, in which only the temperature and pressure are changed tothe environmental values. However, the properties with subscript 0 in the aboveequation (i.e., µi,0) are determined at the temperature, pressure and concentration ofthe environment (T0, P0, w0), which is called the global dead state.

For a control volume (open system), the �ow exergy, ef , can be calculated by addingthe �ow work to the exergy in Eq. (4.50) which mathematically can be expressedas [16, 17],

ef = e + v (p − p0) (4.51)

Knowing that h = u + pv, and eliminating e in Eq. (4.51) using Eq. (4.50), the �owexergy can be rewritten as

ef = (h − h∗) −T0 (s − s

∗) +

n∑i=1

wi(µ∗i − µi,0

) /Mi (4.52)

If the system and the environment are both the same pure substance (e.g., purewater), the chemical exergy, which is the last term in Eqs. (4.50) and (4.52), will vanish.However, for a multi-component system (e.g., seawater, exhaust gases) the chemicalexergy must be considered. Ignoring it may lead to unrealistic and illogical results forthe exergy variation with the concentration.

The following section discusses the variation of exergy and �ow exergy withtemperature, pressure, and composition. In particular, we show that the [control mass]exergy is never negative, whereas the [control volume] �ow exergy can be negativeif the system pressure is less than the dead state pressure. Changes in temperatureor composition relative to the dead state values create the potential to do work bytransferring heat or mass between the system and the environment, leading to positiveexergy values in all cases.


4.3.1 Exergy Variation

The exergy of a control mass system, given by Eq. (4.50), and the exergy of acontrol volume system, which is the �ow exergy given by Eq. (4.52), are intensivethermodynamic properties which represent the maximum obtainable work per unitmass of the system. They are functions of the initial state as well as the environmentstate. However, if the environmental state is speci�ed (T0, p0, w0), the exergy is afunction only of the system initial state (T , p, w).

In this section we examine how the exergy (for a control mass system) and the�ow exergy (for a control volume system) change with the temperature, pressure,and mass concentration of the initial state with respect to the environmental deadstate, assuming the environmental dead state is at T = T0, p = p0 and w = w0, andassuming an ideal gas mixture that satis�es the ideal gas relation (pv = RT ) and whichhas equivalent mixture properties R, cp , and c v (ideal gas constant R = R/M , whereM is the mixture average molar mass; speci�c heat at constant pressure; and speci�cheat at constant volume, respectively).

Case 1: p = p0, w = w0 butT , T0. In this case, the chemical exergy, which is the lastterm in Eq. (4.50), vanishes and the exergy can be written for an ideal gas mixture as:

e = c v (T −T0) + p0

(RT

p−RT0

p0

)−T0

[cp ln

(T

T0

)− R ln

(p

p0

)](4.53)

For p = p0

e = c v (T −T0) + R (T −T0) −T0cp ln(T

T0

)(4.54)

Using cp = c v + R, the exergy will be

e = cpT0

[T

T0− ln

(T

T0

)− 1

](4.55)

Equation (4.55), considered for the case when the pressure and concentration areequal to the dead state, shows that the exergy is always positive at any temperatureother than the dead state temperature (see Fig. 4.5). If the system has a temperatureequal to the dead state (T /T0 = 1), the exergy is zero. The positive exergy is due tothe heat that can be transferred between the system temperature and the dead statetemperature, in one direction or the other as appropriate, to operate a heat enginecycle that can produce work. The same result, Eq. (4.55), can be obtained for a controlvolume system using the �ow exergy equation, Eq. (4.52). Therefore, as long as p = p0,w = w0, any temperature di�erence between the system state and the dead state willresult in positive exergy and positive �ow exergy.

Case 2: T = T0, w = w0 but p , p0. In this case, the chemical exergy again vanishes,and the exergy of a control mass is again given by Eq. (4.53). For T = T0, the exergywill be

e = RT0

[p0

p+ ln

(p

p0

)− 1

](4.56)


0 1 2 3 4 50

1

2

3

4

5

1ln000

−⎟⎟⎠

⎞⎜⎜⎝

⎛−=

TT

TT

Tcep

Temperature ratio, T/T0

Dim

ensi

onle

ss e

xerg

y, e

/cpT0 p = p0

w = w0

Figure 4.5: Dimensionless exergy as a function of temperature ratio

Equation (4.56) for the case when the temperature and concentration are equal tothe dead state, shows that the exergy of the control mass systemâ is always positiveat any pressure other than the dead state pressure (see Fig. 4.6). If the system has apressure equal to the dead state pressure (p/p0 = 1), the exergy is zero. The positiveexergy is due to the mechanical work that can be obtained by expansion (if p > p0) orcompression (if p < p0) of the system to reach the environment pressure.

For temperatures di�erent from the dead state (T , T0), one may show that theminimum of Eq. (4.53) with respect to pressure occurs for p/p0 = T /T0 and that thevalue at the minimum is positive. Thus, the control mass system has positive or zeroexergy for any temperature and pressure when w = w0.

For a control volume (open system), however, the signs on exergy can behavedi�erently. The �ow exergy, given by Eq. (4.52), can be written for an ideal gas mixture(with w = w0) as

ef = cp (T −T0) −T0

[cp ln

(T

T0

)− R ln

(p

p0

)](4.57)

For T = T0 the �ow exergy will be

ef = RT0 ln(p

p0

)(4.58)

It is clear from Eq. (4.58) that the �ow exergy of a control volume system may bepositive or negative depending on the pressure of the system. If the pressure of thecontrol volume system is higher than the dead state pressure (i.e., p > p0), a �owstream can be expanded reversibly (e.g., using a turbine) to the environment pressureand produce work resulting in a positive �ow exergy. However, if the pressure islower than the dead state pressure (i.e., p < p0), an external work should be applied


0 1 2 3 4 50

1

2

3

4

5

1ln0

0

0

−⎟⎟⎠

⎞⎜⎜⎝

⎛+=

pp

pp

RTe

Pressure ratio, p/p0

Dim

ensi

onle

ss e

xerg

y, e

/RT 0 T = T0

w = w0

Figure 4.6: Dimensionless exergy of a control mass system as a function of pressureratio

to compress the �ow stream (e.g., using a compressor) to the environment pressureresulting in a negative �ow exergy (see Fig. 4.7). Therefore, the exergy of the controlmass is positive at any pressure other than the dead state pressure; however the exergyof the control volume (�ow exergy) can be negative at pressures lower than the deadstate pressure.

Case 3: T = T0, p = p0 but w , w0. In this case, whenT = T0 and p = p0, the �rst twoterms in the exergy equation, Eq. (4.50), and �ow exergy equation, Eq. (4.52), vanish.The only remaining term is the last term which is the chemical exergy. The exergy or�ow exergy in this case can be written as follows

e = ef =n∑i=1

wi(µ∗i − µi,0

) /Mi (4.59)

For an ideal mixture model, the chemical potential di�erences are given, usingEqs. (4.12) and (4.13) with γx,i = 1, as

µi − µ◦i = RT ln (xi ) (4.60)

where xi is the mole fraction, µ◦i is evaluated at a hypothetical standard state for thecomponent i , and it is not equal to µi,0. Therefore, the chemical potential di�erencesin Eq. (4.59) can be written as

µ∗i − µi,0 =(µ∗i − µ

◦i)−

(µi,0 − µ

◦i)= RT ln

(xixi,0

)(4.61)


0 1 2 3 4 5-3

-2

-1

0

1

2

3

⎟⎟⎠

⎞⎜⎜⎝

⎛=

00

lnpp

RTef

Pressure ratio, p/p0

Dim

ensi

onle

ss fl

ow e

xerg

y, ef/RT 0

T = T0w = w0

Figure 4.7: Dimensionless �ow exergy as a function of pressure ratio

Substituting equation (4.61) into Eq. (4.59) yields, with Ri = R/Mi ,

e = ef =n∑i=1

wiRiT ln(xixi,0

)(4.62)

Assuming for simplicity that the mixture consists of two substances (1 and 2) andusing T = T0 yields,

e = ef = w1R1T0 ln(x1

x1,0

)+ w2R2T0 ln

(x2

x2,0

)(4.63)

We may eliminate wi in favor of xi using the following two relationships

w1R1 = x1R (4.64a)w2R2 = x2R (4.64b)

where R is the gas constant for the mixture. We also know that

x1 = 1 − x2 (4.65a)x1,0 = 1 − x2,0 (4.65b)

Substituting equations (4.64a)–(4.65b) into Eq. (4.63) and dropping the subscript 2yields

e = ef = RT0

[(1 − x) ln

(1 − x1 − x0

)+ x ln

(x

x0

)](4.66)

Now, we can prove mathematically that Eq. (4.66) is always positive at any molefraction other than x0 by taking the �rst derivative with respect to x :

∂e

∂x= RT0

[ln

(x

x0

)− ln

(1 − x1 − x0

)](4.67)


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟⎠

⎞⎜⎜⎝

⎛−−−=

000

ln11ln1

xxx

xxx

RTe

Mole fraction, x

Dim

ensi

onle

ss fl

ow e

xerg

y, e

/RT 0

T = T0p = p0

x0 = 0.3x0 = 0.3x0 = 0.5x0 = 0.5x0 = 0.7x0 = 0.7

Figure 4.8: Dimensionless �ow exergy as a function of concentration

From Eq. (4.66), at x = x0 the exergy (and �ow exergy) is zero. From Eq. (4.67) at x < x0the �rst derivative (slope) is negative, meaning that the exergy is decreasing. And atx > x0 the �rst derivative (slope) is positive, meaning that the exergy is increasing.Thus, the point x = x0 is a minimum.

The variation of exergy (or �ow exergy) with mole fraction is shown in Fig. 4.8,in which exergy is always positive, except for a value of zero at the dead stateconcentration. The positive value of exergy arises from the potential for a masstransfer process, which can be used to produce work by transferring a solute betweenthe high or low concentration of the system and the concentration of the environmentaldead state. Because the chemical exergy is additive in both Eqs. (4.50) and (4.52), thesame behavior will clearly occur for any selected dead state.

We have shown mathematically that, for an ideal mixture gas, the exergy ofthe control mass system is always positive at any temperature, pressure, and massconcentration other than the dead state, while the exergy of a control volume system(the �ow exergy) may have negative values if the pressure of the system is lower thanthe dead state pressure. The same conclusion can be obtained for real systems usingactual thermodynamic data. In the following section, the exergy and �ow exergy ofseawater are calculated to demonstrate the various trends.

4.3.2 Seawater Exergy

The correlations given in the Appendix for the thermodynamic properties of seawaterare used to calculate the �ow exergy of seawater. In this regard, the (environment)dead state should be speci�ed6. In seawater desalination systems, the intake seawatercondition of the desalination plant is usually taken as the environment dead state

6However, it is important to mention that the choice of the dead state does not a�ect the exergy di�erencebetween any two states.


Figure 4.9: Speci�c �ow exergy of seawater as a function of temperature

condition. This condition varies from place to place depending on the geographicallocation of the desalination plant (ambient temperature, altitude, salinity of theseawater source). In addition, the pressure of the intake seawater depends on the depthof the intake system which varies from 5–50 m. Therefore, the dead state pressuremay change from 1 to 5 atmospheres.

The e�ect of changing the environmental dead state as well as the initial state onseawater �ow exergy is shown in Figs. 4.9–4.11. Figure 4.9 shows the speci�c �owexergy of seawater as it changes with the initial state temperature when the pressureand salt concentration are equal to the dead state values. As shown in this �gure,the �ow exergy is zero at the dead state temperature. It is always positive at anytemperature other than the dead state temperature. This is true for any selected deadstate temperature, therefore whenever there is a di�erence in temperature betweenthe system and environment, there will be a thermal potential di�erence that makesthe �ow exergy positive.

Figure 4.10 shows the speci�c �ow exergy of seawater as it changes with the saltconcentration of the initial state temperature when the pressure and temperature areequal to the dead state values. As shown in this �gure, the �ow exergy is alwayspositive at any concentration other than the dead state concentration. This fact is truefor any selected dead state salt concentration: whenever a concentration di�erenceexists between the system and environment, the �ow exergy is positive. For instance,if the salt concentration of the �ow stream is higher than the salt concentration atthe dead state, pure water can �ow from the environment to the �ow stream througha semi-permeable membrane. This will increase the static head of the �ow streamand can produce work (exergy) though a hydropower turbine [18]. The same thingwill happen if the salt concentration of the �ow stream is lower than that of theenvironmental dead state, but the �ow of water in this case will be from the �owstream to the environment. This is clearly illustrated in Fig. 4.10, which is applicable


Figure 4.10: Speci�c �ow exergy of seawater as a function of salt concentration

for any selected dead state.The e�ect of changing the dead state pressure is shown in Fig. 4.11. As shown in

this �gure, the �ow exergy is zero at the dead state pressure. Flow exergy is positive atpressures higher than the dead state pressure and negative at pressures lower than thedead state pressure. In contrast, the exergy of a control mass system (closed system) isalways positive, irrespective of whether the pressure is higher or lower than the deadstate pressure, as shown in Fig. 4.12.

4.4 Thermodynamic Analysis of DesalinationProcesses

In this section, a consistent de�nition of Second Law e�ciency for desalination systemsbased on the least work of separation is presented [19]. Additionally, the required workof separation is decomposed into the least work of separation plus the contribution fromall signi�cant sources of irreversibility within the system, and methods of evaluatingthe entropy generation due to speci�c physical processes are derived. In Section 4.5,these methods are applied to four common desalination systems.

4.4.1 Derivation of Performance Parameters for Desalination

Work and Heat of Separation

Consider a simple black-box separator model for a desalination system, with a separatecontrol volume surrounding it at some distance, as shown in Fig. 4.13. The work ofseparation entering the system is denoted by ÛWsep and the heat transfer into the systemis ÛQ . Stream sw is the incoming seawater, stream p is pure water (product), and stream

Figure 4.11: Speci�c �ow exergy of seawater as a function of pressure

50 100 150 200 250 300 350 400 450 5000

0.1

0.2

0.3

0.4

0.5

Pressure, (kPa)

Spec

ific

exer

gy x

104 , (

kJ/k

g)

T = T0 = 25 °Cws = ws,0 = 0 g/kg

po = 50 kPapo = 50 kPapo = 100 kPapo = 100 kPapo = 200 kPapo = 200 kPapo = 300 kPapo = 300 kPapo = 400 kPapo = 400 kPapo = 500 kPapo = 500 kPa

Figure 4.12: Speci�c exergy of water as a function of pressure (closed system)

29


b is the concentrated brine. By selecting the control volume su�ciently far from thephysical plant, all the inlet and outlet streams enter and leave the control volume atambient temperature, T0, and pressure, p0. Additionally, the heat transfer, ÛQ , occurs atambient temperature.

ÛQ,T0 ÛWsep

T ′b

T ′pT ′sw Black Box

Separator

Product (p)Tp = T0Brine (b)Tb = T0

Seawater (sw)Tsw = T0

Figure 4.13: When the control volume is selected suitably far away from the physicalsystem, all inlet and outlet streams are at ambient temperature and pressure. Thetemperature of the streams inside the control volume, denoted by T ′i , might not be atT0.

The logic underlying this latter formulation is that the exergy of the outlet streamsattributable to thermal disequilibrium with the environment is not deemed useful. Inother words, the purpose of a desalination plant is to produce pure water, not purehot water. Consider separately the thermal conditions at the desalination systemboundary (solid box) and the distant control volume boundary (dashed box). Productand reject streams may exit the desalination system at temperaturesT ′p andT ′b , di�erentthan ambient temperature, T0. The exergy associated with these streams could beused to produce work that would o�set the required work of separation. However,if the exergy associated with thermal disequilibrium is not harnessed in this way,but simply discarded, entropy is generated as the streams are brought to thermalequilibrium with the environment. This entropy generation is analyzed in Section 4.4.2.Similarly, pressure disequilibrium would result in additional entropy generation [18]. Ingeneral, di�erences in concentration between the various streams represent a chemicaldisequilibrium which could also be used to produce additional work; however, sincethe purpose of the desalination plant is to split a single stream into two streams ofdi�erent concentrations, the outlet streams are not brought to chemical equilibriumwith the environment.

The least work and least heat of separation are calculated by evaluating the Firstand Second Laws of Thermodynamics for the distant control volume. The conventionthat work and heat input to the system are positive is used.

ÛWsep + ÛQ + ( Ûmh)sw = ( Ûmh)p + ( Ûmh)b (4.68a)ÛQ

T0+ ( Ûms)sw + ÛSgen = ( Ûms)p + ( Ûms)b (4.68b)

In Eqs. (4.68a) and (4.68b), Ûmi , hi , and si are the mass �ow rate, speci�c enthalpy andspeci�c entropies of the seawater (sw), product (p), and brine (b) streams. The Firstand Second Laws are combined by multiplying Eq. (4.68b) by ambient temperature, T0,

§4.4 Thermodynam i c Analy s i s o f De sa l i nat i on 31

and subtracting from Eq. (4.68a) while noting that the speci�c Gibbs free energy is,д = h −Ts (all evaluated at T = T0):

ÛWsep = Ûmpдp + Ûmbдb − Ûmswдsw +T0 ÛSgen (4.69)

Least Work and Heat of Separation

In the limit of reversible operation, entropy generation is zero and the work ofseparation becomes the reversible work of separation, which is also known as the leastwork of separation:

ÛWleast ≡ ÛWrev

sep = Ûmpдp + Ûmbдb − Ûmswдsw (4.70)

Equations (4.69) and (4.70) should be evaluated using seawater properties [20].7In order to gain better physical insight into the separation process, it is instructive

to consider how the least work varies with recovery ratio. The recovery ratio isde�ned as the ratio of the mass �ow rate of product water to the mass �ow rate of feedseawater:

r ≡Product WaterInlet Seawater

=Ûmp

Ûmsw(4.71)

Using a simple mass balance ( Ûmsw = Ûmp + Ûmb ) and normalizing Eq. (4.70) by theamount of water produced gives:

ÛWleast

Ûmp= дp +

Ûmsw − Ûmp

Ûmpдb −

Ûmsw

Ûmpдsw = дp +

(1r− 1

)дb −

1rдsw (4.72)

The Gibbs free energy of each of the streams in Eq. (4.72) is evaluated usingseawater properties [20], as a function of temperature and salinity. Provided the inletsalinity and the product salinity is known, then the brine salinity is found using a massbalance:

ws,b =Ûmswws,sw − Ûmpws,p

Ûmb=

ws,sw

1 − r−rws,p

1 − r(4.73)

Since the least work is evaluated assuming all streams leave the control volume atambient temperature, Eq. (4.72) is a function of temperature, inlet salinity, productsalinity, and recovery ratio.

Holding temperature constant at 25 ◦C, the least work of separation is plotted asa function of these variables in Fig. 4.14.8 It is seen that regardless of inlet salinityand product salinity, the least work is minimized as the recovery ratio approaches

7The least work should not depend on the thermodynamic frame of reference chosen for analysis. Indeed,using the Gibbs-Duhem relationship, we can show that the result we derive using a control volume (CV)analysis, Eq. (4.70), is identical to the result obtained from a control mass (CM) approach. As Ûmд = ÛG ,expanding Eq. (4.70) in terms of salt (s ) and water (w) using Eq. (4.11) yields: ÛWleast = ÛGp + ÛGb − ÛGsw =ÛNw,p µw,p + ÛNs,p µs,p + ÛNw,b µw,b + ÛNs,b µs,b − ÛNw,swµw,sw − ÛNs,swµs,sw. Rewriting the di�erences as

integrals, ∆ ÛN µ =∫d ( ÛN µ), and considering a pure product (d ÛNs = 0 and µw,p = µ◦w ), all remaining terms

like ÛNdµ sum to zero by Gibbs-Duhem, Eq. (4.15). The result, Wleast =∫ b

sw RT lnawdNw , is identical toEq. (12) in [21], which is obtained using a CM approach.

8These curves have been updated relative to those in [19], using the newer Gibbs energy correlation in[20] rather than the older one from [22]. This has changed the values of least work at the lowest salinities.


zero. This is true in general because, in the limit of zero recovery, the only stream thatexperiences an energy change is the product stream. At �nite recovery, work mustalso be provided to supply the chemical potential energy change of the brine streamdue to a change in salinity. Since the least work is de�ned per unit mass of product,the least work represents the amount of energy necessary to create 1 kg of pure waterplus the amount of energy necessary to change the chemical potential of the brinestream.

0 10 20 30 40 50 60 700

1

2

3

4

5

r = mp/msw [%]

Wleast

[kJ/kg]

ws,sw = 35 g/kg

w s,sw = 20 g/kg

w s,sw = 5 g/kg

ws,p = 0 g/kgws,p = 0.5 g/kg

Figure 4.14: The least work of separation is minimized when the recovery ratioapproaches zero.

From Fig. 4.14, it can be seen that the least work of separation is minimized as therecovery ratio approaches zero (i.e., in�nitesimal extraction).

ÛW minleast ≡ lim

r→0ÛWleast (4.74)

Using seawater properties [20] and assuming an inlet salinity of 35 g/kg, zero salinitywater product, and T = 25 ◦C, the least work of separation at in�nitesimal recovery is2.59 kJ/kg.

Equation (4.69) represents the amount of work required to produce a kilogram ofpure water. If heat is used to power a desalination system instead of work, the heat ofseparation is a more relevant parameter. Recalling that heat engines produce workand reject heat, the calculation of the heat of separation is straightforward. Figure 4.15shows the control volume from Fig. 4.13 but with a reversible heat engine providingwork of separation.


ÛQ ÛWsep

T ′b

T ′pT ′sw Black Box

Separator

Product (p)Tp = T0

Brine (b)Tb = T0

Seawater (sw)Tsw = T0

ÛQsep,TH

Figure 4.15: Addition of a high temperature reservoir and a Carnot engine to thecontrol volume model shown in Fig. 4.13.

If the heat is provided from a high temperature reservoir, then the First Law forthe heat engine is

ÛQsep = ÛWsep + ÛQ (4.75)Assuming a reversible heat engine operating between the high temperature reservoirat TH and ambient temperature T0 and considering work per unit mass produced,

ÛWsep

Ûmp=ÛQsep

Ûmp−ÛQ

Ûmp=ÛQsep

Ûmp

(1 −

T0

TH

)(4.76)

where the second equality holds as a result of the entropy transfer that occurs in areversible heat engine operating between two heat reservoirs. Therefore, the heat ofseparation is:

ÛQsep

Ûmp=

ÛWsep(1 − T0

TH

)Ûmp

=ÛW rev

sep +T0 ÛSgen(1 − T0

TH

)Ûmp

(4.77)

where the second equality holds by combining Eqs. (4.69) and (4.70). Note that Eq. (4.77)can also be derived from Eqs. (4.68a) and (4.68b) if ÛWsep is set to zero and the temperaturein the Second Law is set to TH [23]. Equations for the least heat of separation, ÛQleastand the minimum least heat of separation, ÛQmin

least can be obtained from Eq. (4.77) in thesame manner as the corresponding work equations.

In practice, the entropy generation term in Eqs. (4.69) and (4.77) o�en dominatesover the least work or least heat. Therefore, the parameter, ÛSgen/ Ûmp is of criticalimportance to the performance of desalination systems [23]. This term is referred toas the speci�c entropy generation, Sgen, and is a measure of entropy generated perunit of water produced:

Sgen =ÛSgen

Ûmp(4.78)

In the formulation described above, all streams enter and exit the system at ambienttemperature. Therefore, the speci�c exergy destroyed, ed , in the system is equal to theproduct of Sgen and the ambient temperature. This term is physically re�ective of thesame phenomenon that produces Eq. (4.78):

ed =T0 ÛSgen

Ûmp(4.79)


Least Work of Separation for Salt Removal

Thus far, we have considered the removal of water from a saline feed. But suppose wewished instead to remove salt from the feed, resulting in pure salt and pure water. Theresults are given for an NaCl solution, but the methodology can be applied to othermixtures.

Employing the same control volume formulation we’ve used up to this point [SeeSec. 4.2.1, Eq. (4.7) and Sec. 4.4.1, Eq. (4.70)], the inlet stream is the feed F , and theoutlet streams are the salt (product) p and the pure water stream d . When all threestreams are at the same temperature and pressure, Eq. (4.7) applies and:

ÛWrev = ÛGF − ÛGd − ÛGp (4.80)

Each ÛG can be expanded in terms of water and salt, ÛG = ( ÛN µ)w + ( ÛN µ)s , but becausethe d and p streams are pure, ÛNw,d = ÛNw,F and ÛNs,p = ÛNs,F , so:

ÛWrev = ÛNw,F (µw,F − µw,d ) + ÛNs,F (µs,F − µs,p ) (4.81)

When solid and aqueous salt are in equilibrium, i.e., at the solubility equilibrium, thechemical potential of salt in the solid and aqueous phases are equal. Thus µs,p =µ◦s + RT lnas,p , where as,p is the activity of the salt in a saturated solution. The dstream is pure, so aw,d = 1, and µw,d = µ

◦w,d . Dividing the LHS side of Eq. (4.81) by

RT ÛNs,p and the RHS by the numerically equal term RT ÛNs,F , Eq. (4.81) reduces to

ÛWrev

RT ÛNs,p=

1bs,FMw

ln(aw,F

)+ ln

(as,Fas,p

)(4.82)

At 25◦C and atmospheric pressure, the solubility of NaCl is 6.147 mol/kg, γb,± =1.006, and so as,p = (γb,±b)

2 = 38.24 (See Fig. 4.2 and [2]). For a 0.62 mol/kg NaClsolution, a rough approximation of seawater, aw,F = 0.9796 and as,F = 0.1734. Theleast work for salt removal is thus 307 kJ/kg-salt produced, or equivalently 11.1kJ/kg-water removed – over four times the least work required to separate saltwaterinto brine and water! This �gure is also a good approximation to the minimum energyrequired for zero-liquid-discharge (ZLD) seawater desalination. (In ZLD, all of thewater in the seawater is converted to fresh water, so the recovery ratio is the massfraction of water in the seawater, or about 96.5%.)

Second Law E�ciency

The Second Law (or exergetic) e�ciency is employed as a measure of thethermodynamic reversibility of a desalination system. Unlike First Law e�ciency,which measures the amount of an energy source that is put to use, Second Lawe�ciency, ηII , measures the extent of irreversible losses within a system. A completelyreversible system will have a Second Law e�ciency of 1 even though the First Lawe�ciency is likely to be lower. Bejan et al. [24] de�ne the exergetic e�ciency as theratio of the exergy of the process products to the process fuel. In other words, the


exergetic e�ciency is the ratio of the useful exergy of the outputs of the process( ÛΞout,useful) to the exergy of the process inputs ( ÛΞin):

ηII ≡ÛΞout,usefulÛΞin

= 1 −ÛΞdestroyed + ÛΞlost

ÛΞin(4.83)

The second equality in Eq. (4.83) is valid since the useful exergy out is equal to theexergy minus the sum of the exergy destroyed ( ÛΞdestroyed) and the exergy lost ( ÛΞlost).Exergy destroyed represents lost available work due to irreversibilities within thesystem. Exergy lost represents lost available work due to discarding streams to theenvironment that carry exergy. Note that when the material inputs to the systemare taken to be at equilibrium with the environment, Ξin equals Ξfuel, Ξ ÛWsep

, or Ξ ÛQsep,

depending on the energy input. Additionally, Eq. (4.83) is equivalent to the de�nitionused by Kahraman and Cengel [25].

Prior to applying Eq. (4.83) to desalination systems, it is important to understandthe di�erences between the three de�nitions of work that are presented. The work ofseparation, ÛWsep, is the actual amount of work necessary to produce a given amountof water from a �xed feed stream using a real separation process. The least workof separation, ÛWleast, represents the amount of work necessary to produce the sameamount of product water from the feed stream while operating under reversibleconditions. Finally, the minimum least work, ÛW min

least, is the minimum required work ofseparation in the limit of reversible operation and in�nitesimal extraction. As a result,the following relation will always hold:

ÛWsep > ÛWleast(r > 0) > ÛW minleast(r = 0) (4.84)

In a desalination process, puri�ed water is considered to be the useful product.The useful exergy associated with pure water is the minimum least work (or heat) ofseparation that is required to obtain puri�ed water from feed water of a given salinity(i.e., in�nitesimal extraction of pure water with inlet and outlet streams at ambienttemperature). The minimum least work (at zero recovery), rather than the least work(at �nite recovery), is used since it represents the actual exergetic value of pure water.To further illustrate, when analyzing a unit of pure water, it is impossible to knowthe process that was used to produce it. Therefore, the minimum energy required toproduce it must be the exergetic value and ÛΞout,useful = ÛW

minleast(r = 0).

Since the control volume is de�ned so that the inlet stream is at the dead state,the only exergy input to the system comes in the form of either a work ( ÛWsep) or heat( ÛQsep) input (exergy of the feed stream is zero). The work of separation is equivalent tothe useful work done within the system plus the exergy destroyed within that system.

In order to calculate the work of separation, two equivalent processes may beconsidered. The �rst involves a separation process where the products are broughtto thermal and mechanical equilibrium with the environment, whereas the brine isalso brought into chemical equilibrium (total dead state, TDS). The reversible workrequired to achieve this process corresponds to the least work at zero recovery. Thetotal work of separation is given by the sum of the reversible work required plus theexergy destruction associated with entropy generated in the separation and run-down


to equilibrium processes:

ÛWsep = ÛWmin

least(r = 0) +T0 ÛSTDSgen (4.85)

The second process involves a separation process where the products are only broughtto thermal and mechanical equilibrium with the environment (restricted dead state,RDS). The reversible work required to achieve this process corresponds to the leastwork at �nite recovery. The total work of separation again is given by the sum of thereversible work required plus the exergy destruction associated with entropy generatedin this process:

ÛWsep = ÛWleast(r > 0) +T0 ÛSRDSgen (4.86)

It can be shown that Eqs. (4.85) and (4.86) are equivalent.9 Note that the work ofseparation for a system can also be directly evaluated using a First Law analysis.

As result, when Eq. (4.83) is applied to a desalination system that receives bothwork and heat input, it should be written as:

ηII =least exergy of separation

exergy input=

ÛW minleast

ÛWsep + ÛQsep

(1 −

T0

TH

) (4.87)

For the case of a purely work driven system, such as reverse osmosis desalination, thisbecomes:

ηII =ÛW min

leastÛWsep

=ÛW min

leastÛW min

least +T0 ÛSTDSgen=

ÛW minleast

ÛWleast +T0 ÛSRDSgen

(4.88)

For a heat driven system, Equation (4.83) can be written in terms of the least heat ofseparation:

ηII =ÛQmin

leastÛQsep

=ÛQmin

least

ÛQminleast +

(1 − T0

TH

)−1T0 ÛS

TDSgen

=ÛQmin

least

ÛQleast +(1 − T0

TH

)−1T0 ÛS

RDSgen

(4.89)

Clearly, the two de�nitions of Second Law e�ciency presented in Eqs. (4.88)and (4.89) are bounded by 0 and 1 because ÛWsep > ÛWleast and ÛQsep > ÛQleast. Observethat ÛWleast and ÛQleast are functions of feed salinity, product salinity, recovery ratio, andT0. Additionally, ηII will only equal 1 in the limit of completely reversible operation, asexpected. Note that the selection of the control volume suitably far away such that allstreams are at thermal and mechanical equilibrium allows for this bounding.

Three relevant Second Law based performance parameters for desalination systemshave been discussed thus far: speci�c entropy generation, Eq. (4.78); speci�c exergydestruction, Eq. (4.79); and Second Law e�ciency, Eqs. (4.88) and (4.89). This sectionwill focus on speci�c entropy generation and Second Law e�ciency.

9Substitution of ÛW minleast from Eq. (4.124) into Eq. (4.86) while noting that ÛSTDS

gen = ÛSRDSgen + ÛS

brine RDS→TDSgen

exactly gives Eq. (4.85).


Energetic Performance Parameters

Three o�en-used parameters are key to describing the energetic performance ofdesalination systems. The �rst, called gained output ratio (GOR), is the ratio of theenthalpy required to evaporate the distillate (or equivalently, the energy release incondensation) and the heat input to the system, or

GOR ≡Ûmphfg(T0)

ÛQsep(4.90)

In essence, GOR is a measure of how many times the latent heat of vaporizationis captured in the condensation of pure water vapor and reused in a subsequentevaporation process to create additional pure water vapor from a saline source. By theFirst Law of Thermodynamics, a thermal desalination system that has no such heatrecovery requires at least the latent heat of vaporization multiplied by the mass of purewater produced as its energy input: its GOR is approximately one (or less when feedheating and heat losses are taken into account). It is important to note that Eq. (4.90)is valid as written only for a desalination system driven by heat; that is, a thermaldesalination system. A work-driven desalination system, in contrast, uses electricityor sha� work to drive the separation process. Normally, this work is produced by athermal process, such as a heat engine. Thus, to evaluate the heat input required for awork-driven desalination system, a First Law e�ciency of the process that producesthe work of separation must be known.

The second parameter, known as the performance ratio (PR), is de�ned as the ratioof the mass �ow rate of product water to that of the heating steam:

PR ≡Ûmp

Ûms(4.91)

For a thermal desalination system in which the heat input is provided by condensingsteam, as is typical of large-scale thermal processes such as MED and MSF, the valuesof PR and GOR are quite similar. In that case, the two parameters di�er only by theratio of the latent heat of vaporization at the distillate and heating steam temperatures.That is, GOR = PR× hfg(T0)

hfg(Tsteam). (Some authors interchange these de�nitions of GOR and

PR.)The third parameter, speci�c electricity consumption (SEC) is best suited to

work-driven desalination systems. It is de�ned as the ratio of the work of separation(or work input) to the mass �ow rate of product water, or

SEC ≡ÛWsep

Ûmp(4.92)

As was the case with GOR, because thermal and electrical energy are not directlycomparable, numerical values of SEC cannot be compared between thermal- andwork-driven systems. SEC as de�ned by Eq. (4.92) should only be used for desalinationsystems driven by work.


4.4.2 Analysis of Entropy Generation Me�anisms in Desalination

Several common processes in desalination systems result in entropy generation,including heat transfer, pressure di�erentials, and non-equilibrium conditions. Byutilizing the ideal gas and incompressible �uid models, simple expressions are derivedto show the important factors in entropy generation for various physical processes.Physical properties, evaluated at a representative reference state of 50 ◦C, are providedin Table 4.3 for pure water [26] and seawater [22]. Proper selection of the referencestate is discussed below. In all equations in this section, states 1 and 2 are the inlet andoutlet states, respectively, for each process.

Table 4.3: Representative values of reference state constants for Eqs. (4.94a), (4.94b),(4.96a), and (4.96b).

Pure water and vapor constants, Tsat = 50 ◦C, psat = 12.3 kPa

c 4.18 kJ/kg-K h◦IG 2590 kJ/kgcp 1.95 kJ/kg-K h◦IF 209 kJ/kgR 0.462 kJ/kg-K s◦IG 8.07 kJ/kg-Kv 1.01 × 10−3 m3/kg s◦IF 0.704 kJ/kg-K

Seawater constants, 50 ◦C, 35,000 mg/kg

c 4.01 kJ/kg-K h◦IF 200 kJ/kgv 0.986 × 10−3 m3/kg s◦IF 0.672 kJ/kg-K

Before analyzing the entropy generation mechanisms, the ideal gas andincompressible �uid models are discussed. By de�nition, the density of anincompressible �uid does not vary, and the speci�c heat capacities at constant pressureand constant volume are the same (cp = c v = c). As a result, an incompressible �uid isone which satis�es the following equations:

dhIF = c dT + v dp (4.93a)

dsIF = cdTT

(4.93b)

Integrating Eqs. (4.93a) and (4.93b) from an arbitrary reference state to the state ofinterest while assuming constant speci�c heat (c) yields the following expressions:

hIF = c(T −T◦) + v(p − p◦) + h◦IF (4.94a)

sIF = c lnT

T ◦+ s◦IF (4.94b)

Similarly, an ideal gas follows the equation of state, pv = RT , and is governed bythe following equations:

dhIG = cpdT (4.95a)

dsIG = cpdTT− R

dpp

(4.95b)


Integrating Eqs. (4.95a) and (4.95b) from an arbitrary reference state to the state ofinterest while assuming constant speci�c heat at constant pressure, cp , yields thefollowing expressions:

hIG = cp (T −T◦) + h◦IG (4.96a)

sIG = cp lnT

T ◦− R ln

p

p◦+ s◦IG (4.96b)

For increased accuracy, the generalized compressibility model, pv = ZRT can be usedinstead if R is replaced with ZR in Eqs. (4.95a), (4.95b), (4.96a), and (4.96b) and all futureequations.

When evaluating Eqs. (4.94a), (4.94b), (4.96a), and (4.96b), the physical properties(speci�c heat, volume, compressibility factor, etc.) and reference values of enthalpy andentropy should be evaluated at a suitable reference state. The reference state shouldbe selected as the saturated state corresponding to the average temperature betweenthe inlet and outlet streams. Representative values of these constants, evaluated forpure water [26] at 50 ◦C, are provided in Table 4.3. For seawater, the average salinityshould be used. Representative values of these constants, evaluated for seawater [22]at 50 ◦C and 35 g/kg, are also provided in Table 4.3. It should be noted that the speci�cheat of seawater is signi�cantly lowered with increasing salinity. Therefore, theseapproximations should not be used for processes in which composition substantiallychanges. Instead, Gibbs free energy should be used (see Section 4.4.2).

Flashing

When liquid water near saturation conditions passes through a throttle, a portion willvaporize as a result of the pressure drop through the device. The exiting �uid is thus amixture of liquid and low pressure vapor and can be modeled as an incompressible�uid and ideal gas, respectively. Application of the First and Second Laws to the �ashbox (throttle) control volume reduces to:

h1, IF = h2 = (1 − x)h2, IF + xh2, IG (4.97a)

s�ashinggen = s2 − s1 =

[(1 − x)s2, IF + xs2, IG

]− s1, IF (4.97b)

Substitution of Eqs. (4.94a), (4.94b), (4.96a), and (4.96b) into Eqs. (4.97a) and (4.97b)with simpli�cation gives the quality and entropy generation due to �ashing.

The entropy generated in this process is

s�ashinggen = c ln

T2

T1+ x

{(cp − c) lnT2 − R lnp2

+[s◦IG − s

◦IF − (cp − c) lnT

◦ + R lnp◦]} (4.98)

where the quality, x , is given by:

x =c(T1 −T2) + v(p1 − p2)

(cp − c)T2 − vp2 + [h◦IG − h

◦IF − (cp − c)T

◦ + vp◦](4.99)


and cp is the speci�c heat at constant pressure, c is the speci�c heat of an incompressible�uid, R is the ideal gas constant for steam, v is the speci�c volume of the liquid, h◦IGand s◦IG are the enthalpy and entropy for steam at the reference state, and h◦IF and s◦IFare the enthalpy and entropy for liquid water at the reference state.

Flow through an expansion device without phase �ange

Although the physical causes for pressure drops di�er when considering �ow throughexpanders, pipes, throttles, membranes, and other �ow constrictions, the controlvolume equations that govern the entropy generated remains constant. As with theanalysis of the �ashing case, the First and Second Laws for an isenthalpic processsimplify to:

w =ÛW

Ûm= h2 − h1 (4.100a)

sgen = s2 − s1 (4.100b)

For an expansion device, the isentropic e�ciency, ηe , is de�ned as:

ηe ≡w

ws =

h2 − h1

hs2 − h1(4.101)

where w is the work produced per unit mass through the device and ws is the work

produced assuming isentropic expansion.For entropy generation in the expansion of an incompressible �uid, Eq. (4.94b)

shows that for an isentropic expansion from p1 to p2, T s2 = T1. Combining this result

with Eqs. (4.94a), (4.100a), and (4.101) and solving for T2 gives

T2 = T1 +v

c(p1 − p2) (1 − ηe ) (4.102)

Substitution of Eqs. (4.94b) and (4.102) into Eq. (4.100b) yields the entropy generateddue to irreversible expansion of an incompressible �uid:

sexpansion,IFgen = c ln

[1 +

v

cT1(p1 − p2) (1 − ηe )

]≈

v

T1(p1 − p2) (1 − ηe ) (4.103)

In the limit of a completely irreversible pressure drop (such as through a throttle) inwhich no work is generated, ηe = 0 and (4.103) reduces to:

s∆p, IFgen = c ln

[1 +

v

cT1(p1 − p2)

]≈

v

T1(p1 − p2) (4.104)

For entropy generation in the expansion of an ideal gas, Eq. (4.96b) shows that foran isentropic expansion from p1 to p2,

T s2 = T1

(p2

p1

)R/cp


Combining this result with Eqs. (4.96a), (4.100a), and (4.101) and solving for T2 gives

T2 = T1

{1 + ηe

[(p2

p1

)R/cp− 1

]}(4.105)

Substitution of Eqs. (4.96b) and (4.105) into Eq. (4.100b) yields the entropy generateddue to irreversible expansion of an ideal gas:

sexpansion,IGgen = cp ln

{1 + ηe

[(p2

p1

)R/cp− 1

]}− R ln

p2

p1(4.106)

In the limit of a completely irreversible pressure drop (such as through a throttle) inwhich no work is generated, ηe = 0 and Eq. (4.106) reduces to:

s∆p, IGgen = −R ln

p2

p1(4.107)

Based on Eqs. (4.104) and (4.107), for an incompressible �uid, entropy generationis determined by the pressure di�erence, whereas for an ideal gas, it is determined bythe pressure ratio.

Pumping and compressing

Application of the First and Second Laws to a pump (or compressor) control volumeyields Eqs. (4.100a) and (4.100b). For pumping and compressing, the isentropice�ciency, ηp , is de�ned as:

ηp ≡ws

w

=hs2 − h1

h2 − h1(4.108)

For entropy generation in pumping, assume that the liquid can be modeled as anincompressible �uid. Equation (4.94b) shows that for an isentropic expansion from p1to p2,T s

2 = T1. Combining this result with Eqs. (4.94a), (4.100a), and (4.108) and solvingfor T2 gives

T2 = T1 +v

c(p2 − p1)

(1ηp− 1

)(4.109)

Substitution of Eqs. (4.94b) and (4.109) into Eq. (4.100b) yields the entropy generateddue to irreversible pumping:

spumpinggen = c ln

[1 +

v

cT1(p2 − p1)

(1ηp− 1

)]≈

v

T1(p2 − p1)

(1ηp− 1

)(4.110)

The entropy generated due to irreversible pumping can also be derived by noticingthat the di�erence between the actual work and the reversible work is simply theexergy destruction. Since irreversibilities during the compression process of anincompressible �uid will result in only minor changes in temperature (i.e., T2 ≈ T1),


the entropy generation can be determined by dividing the exergy destruction by theinlet temperature in accordance with Gouy-Stodola theorem [16]:

spumpinggen =

Ξd

T1=

w − ws

T1=h2 − h

s2

T1=h(T2,p2) − h(T1,p2)

T1

=v

T1(p2 − p1)

(1ηp− 1

)(4.111)

Note that Eq. (4.111) is simply the Taylor series expansion of the second term ofEq. (4.110). This alternate derivation is only appropriate because the pumping processis nearly isothermal.

For entropy generation in vapor compression, assume that both the inlet and outletvapor can be modeled as an ideal gas that follows the generalized compressibility form.Equation (4.96b) shows that for an isentropic expansion from p1 to p2,

T s2 = T1

(p2

p1

)R/cpCombining this result with Eqs. (4.96a), (4.100a), and (4.108) and solving for T2 gives

T2 = T1

{1 −

1ηp

[1 −

(p2

p1

)R/cp ]}(4.112)

Substitution of Eqs. (4.96b) and (4.112) into Eq. (4.100b) yields the entropy generateddue to irreversible compression:

scompressiongen = cp ln

{1 −

1ηp

[1 −

(p2

p1

)R/cp ]}− R ln

p2

p1(4.113)

Note that unlike in the incompressible �uid case, Eq. (4.113) cannot be derivedthrough the use of the Gouy-Stodola theorem since the compression of a gas is not anisothermal process.

Approximately isobaric heat transfer process

Actual heat exchangers always have a pressure drop associated with viscous forces.However, without knowledge of speci�c �ow geometry or the local temperature andpressure �elds, it is impossible to partition entropy generation according to particulartransport phenomena. For example, Bejan [27] has shown that for a simple, single-�uidheat exchanger, comparing the trade o� between entropy generation due to heattransfer across a �nite temperature di�erence and pressure drop across a �nite �owvolume yields a thermodynamically optimal heat exchanger geometry.

In heat exchangers within typical desalination processes, however, the e�ect ofpressure drop on physical properties is insigni�cant. Thus, entropy generation may becalculated as a function of terminal temperatures alone. For the range of temperaturesand �ow con�gurations encountered in the present analysis, this approximation holdsfor �uids that may be modeled as both ideal gases and incompressible �uids.


The entropy generation equation for a heat exchanger is

ÛSHXgen = [ Ûm(s2 − s1)]stream 1 + [ Ûm(s2 − s1)]stream 2 (4.114)

In the case of a device that transfers heat at a relatively constant pressure, anapproximate expression may be developed for entropy generation as a function ofinlet and outlet temperatures alone. Entropy may be written as:

ds =1T

dh −v

Tdp (4.115)

Integrating Eq. (4.115) at constant pressure gives:

s2 − s1 =

∫ 2

1

1T

dh (4.116)

For an ideal gas, Eq. (4.115) is written as Eq. (4.95b) which can be integrated at constantpressure to give:

s2 − s1 = cp lnT2

T1(4.117)

For an incompressible �uid, entropy is not a function of pressure as seen in Eq. (4.93b).Therefore, the entropy di�erence is given by:

s2 − s1 = c lnT2

T1(4.118)

If it is now assumed that the heat exchanger is adiabatic with respect to the environmentand that there is no work, then the above equations can be substituted into Eq. (4.114).

For an isobaric phase change from a saturation state (either liquid or vapor), theentropy change is

s2 − s1 = xsfg = x(sIG − sIF) for evaporation (4.119a)= (x − 1)sfg = (x − 1)(sIG − sIF) for condensation (4.119b)

where x is the quality at the exit of the process.

Thermal disequilibrium of dis�arge streams

Referring again to Fig. 4.13, the entropy generated in bringing outlet streams from thesystem control volume to the ambient temperature reached at the exit of the distantcontrol volume may be calculated. Consider a stream that is in mechanical, but notthermal equilibrium with the environment (Fig. 4.16). The environment acts as a heatreservoir, and through an irreversible heat transfer process, the stream is brought tothermal equilibrium.

The First and Second Laws for this control volume give:

ÛQ = Ûmi (hi − h′i ) (4.120a)

ÛSgen = Ûmi

[(si − s

′i ) −

ÛQ

T0

]= Ûmi

[(si − s

′i ) −

hi − h′i

T0

](4.120b)


Heat Reservoir

(i ′) Discharge StreamT ′i , T0

(i) Discharge StreamTi = T0

ÛQ

T0

Figure 4.16: Entropy is generated in the process of a stream reaching thermalequilibrium with the environment.

For incompressible �uids at mechanical equilibrium with the environment, si −s ′i =ci ln T0

Tiand hi − h′i = ci (T0 − Ti ). Substituting into Eq. (4.120b) gives the entropy

generated in bringing a stream of incompressible �uid to thermal equilibrium with theenvironment:

ÛST disequilibriumgen = Ûmici

[ln

(T0

Ti

)+TiT0− 1

](4.121)

Chemical disequilibrium of concentrate stream

When considering a desalination system, the concentrate is typically considered to bewaste and is discharged back to the ocean. Since the concentrate is at higher salinitythan the ocean, entropy is generated in the process of restoring the concentrate tochemical equilibrium (also called distributive equilibrium) with the seawater. Thisentropy generation can be calculated in one of two ways.

First, consider the addition of the concentrate stream at the restricted dead state toa large reservoir of seawater at the total dead state. An exergy balance governing themixing of the concentrate stream with the seawater reservoir is written as follows:

ÛΞmixingdestroyed = −[( Ûmc + Ûm

reservoirsw )дout − Ûmcдc − Ûm

reservoirsw дsw] (4.122)

where ÛΞmixingdestroyed is the exergy destroyed as a result of irreversible mixing. In the limit

that Ûmc/ Ûmreservoirsw → 0, the mixed state дout approaches дsw and the concentrate stream

is brought to chemical equilibrium with the environment. Using the Gouy-Stodolatheorem [16], the exergy destroyed due to irreversible mixing can be used to evaluatethe entropy generated as the concentrate stream runs down to chemical equilibrium:

ÛSconcentrate RDS→TDSgen =

ÛΞmixingdestroyed

T0(4.123)

The mixing process described by Eq. (4.122) is analogous to the separation processshown in Fig. 4.13 performed in reverse.

A second method to evaluate the entropy generation due to chemical disequilibriumof the concentrate stream is based on the least work of separation. When consideringthe control volume given by Fig. 4.13 and the minimum least work of separation,there is an in�nitesimally small product stream of pure water along with a stream

§4.5 Entrop y Generat i on in De sa l i nat i on Sy s t em s 45

of concentrate of salinity that is in�nitesimally above that of seawater. Therefore,the concentrate stream is in thermal, mechanical, and nearly chemical equilibriumwith the environment. If, however, there is a �nite recovery ratio, the concentratestream salinity is greater than that of seawater. Additionally, as the recovery ratioincreases, the �ow rate of the concentrate stream decreases and �ow rate of the productwater increases (assuming �xed input feed rate). Since the concentrate stream is notat equilibrium with the environment, there is a chemical potential di�erence thatcan be used to produce additional work. This additional work is exactly equal to thedi�erence between the least work of separation, Eq. (4.70), and the minimum leastwork of separation, Eq. (4.74). When the concentrated concentrate is discarded tothe ocean, this work potential is lost. Therefore, entropy generation due to chemicaldisequilibrium of the concentrate stream can also be evaluated through the use of theGouy-Stodola theorem as follows:

T0 ÛSconcentrate RDS→TDSgen = ÛWleast(r > 0) − ÛW min

least(r = 0) (4.124)

Evaluation of entropy generation using Eqs. (4.123) and (4.124) gives equivalent results.

4.5 Entropy Generation Me�anisms in SeawaterDesalination Te�nologies

Using the methods developed in preceding section, the component and systemlevel entropy production and Second Law e�ciency of several common seawaterdesalination technologies are now evaluated. Four simple examples of commonsystems are considered: forward feed, multiple e�ect distillation (MED); direct contactmembrane distillation (DCMD); single e�ect mechanical vapor compression (MVC); andsingle-stage reverse osmosis (RO). These examples serve to illustrate the applicationof the models and methods to both thermally driven and work driven desalinationprocesses. These analyses may alternatively be done using the �ow exergy functiondescribed in Sec. 4.3.

4.5.1 Multiple E�ect Distillation

A very simple model based on approximations from Mistry et al. [28], El-Sayed andSilver [29], Darwish et al. [30], and El-Dessouky and Ettouney [31] is used to generateall the temperature pro�les and mass �ow rates within a forward feed (FF) multiplee�ect distillation (MED) cycle (Fig. 4.17).

Several common approximations are made: The temperature drop between e�ectsis assumed to be constant, ∆T = (Tsteam − Tlast e�ect)/n. Additionally, the drivingtemperature di�erence between condensing vapor and evaporating brine and thetemperature rise across feed heaters are both taken to be ∆T . The temperature rise inthe condenser is set to 10 ◦C. The distillate is approximated as pure water, and it isassumed that distillate is produced in each e�ect (Di ) at a rate of 99% of that producedin the previous e�ect (i.e., Di+1 = 0.99Di ) to approximate the e�ect of increasing latentheat with decreasing e�ect temperature. Distillate produced from �ashing in each

Stea

mlin

eE�ec

tFl

ash

box

Feed

heat

er

Dis

tilla

te

Brin

e

Feed

Figu

re4.

17:A

typi

cal�

owpa

thfo

rafo

rwar

dfe

edm

ultip

lee�

ectd

istil

latio

nsy

stem

.

46


e�ect is given by Df ,i = Ûmb,i−1cp,i∆T /hfg,i wheremb,i−1 is the brine from the previouse�ect which becomes the feed to the current e�ect. The remainder of the distillate isproduced from boiling in the e�ect. There is no �ashing in the �rst e�ect. Distillateproduced from �ashing in the �ash boxes is given by Df b,i =

∑i−1j=1 D jcp,i∆T /hfg,i , for

i ≥ 2. The quality of the distillate leaving the feed heater is calculated using an energybalance on the heater, ÛmFcp,i∆T = (Di + Df b,i )(1 − xi )hfg , where ÛmF is the mass �owrate of the feed seawater.

Water and salinity mass balances for the e�ects are:

Ûmb,i−1 = Di + Ûmb,i (4.125)Ûmb,i−1ws,b,i−1 = Ûmb,iws,b,i (4.126)

where ws,b,i is the salinity of the ith brine stream.An energy balance on the �rst e�ect gives the required amount of heating steam:

Ûmshfg,s = D1hD,1 + Ûmb,1hb,1 − ÛmFhF . Accurate properties for seawater [22] andsteam [26], including enthalpies, entropies, speci�c heats, etc., are used and evaluatedat each state.

The inputs to the simpli�ed MED FF model with 6 e�ects include: 1 kg/s of distillate,seawater salinity of 42 g/kg, maximum salinity of 70 g/kg, steam temperature of 70 ◦C,last e�ect temperature of 40 ◦C, and seawater (and environment) temperature of 25 ◦C.

Using the above approximations and inputs, all thermodynamic states for the MEDFF system are found. Entropy generation in each component is computed by usinga control volume for each component. Pumping work and entropy generated due to�ashing in e�ects are evaluated using Eqs. (4.98) and (4.110), respectively.

Figure 4.18 shows the entropy generated in each component, whereas Fig. 4.19shows the percentage of entropy generated in each type of component. Entropygenerated during pumping is not included in the �gure since it is much less than 1% ofthe overall amount. Looking at Fig. 4.19, it is clear that heat transfer is the dominantsource of entropy generation in MED systems since most of the generation occursin the heat exchange devices (e�ects, feed heaters, and condenser). It was found thatentropy generated due to �ashing in the e�ects was very small.

Although the e�ects result in the greatest portion of the entropy generated, it isimportant to note that the condenser is the single greatest source of irreversibility,as seen in Fig. 4.18. The condenser is such a large source of entropy generation asa result of the large temperature di�erence between the condensed vapor and thecooling water or feed stream.

Many modern MED plants operate using a thermal vapor compressor (TVC). TheTVC is used to entrain the vapor from the �nal e�ect and re-inject it into the �rst e�ect.MED-TVC plants have much higher performance ratios than non-TVC plants and theyreduce the size of the �nal condenser, thus reducing this large source of irreversibilities.It is important to note, however, that the TVC is also a highly irreversible device sothat total entropy production may not be as much reduced.

Finally, it is seen that for this MED plant, entropy generated as a result of thenon-equilibrium discharge of the brine and distillate corresponds to approximately 8.7%of the plant’s overall losses. The Second Law e�ciency, accounting for disequilibriumof the discharge, is ηII = 5.9%. Additionally, PR = 5.2 and GOR = 5.4.

1 2 3 4 5 6 Cond. TD CD0

5

10

15

20

25

30

35

40

45Sgen/mp

[J/kg-K]EffectFeed HeaterFlashboxCondenserTemperature Disequilibrium - BrineTemperature Disequilibrium - DistillateChemical Disequilibrium - Brine

Figure 4.18: Entropy production in the various components of a six e�ect forward feedmultiple e�ect distillation system.

Effects: 56.5%

Feed Heaters: 12.3%

Flashboxes: 0.6%

Condenser: 21.8%

Temperature Disequilibrium: 6.2%

Chemical Disequilibrium: 2.5%

Figure 4.19: Relative contribution of sources of entropy generation in a forward feedmultiple e�ect distillation system. Irreversibilities in the e�ects dominate. Total speci�centropy generation is 196 J/kg-K.

48


4.5.2 Direct Contact Membrane Distillation

Direct contact membrane distillation (DCMD) is a membrane-based thermal distillationprocess in which heated feed passes over a hydrophobic microporous membrane [32].The membrane holds back a meniscus of water near the pores. On the opposing side,cooled fresh water passes over the membrane. The temperature di�erence betweenthe water streams induces a vapor pressure di�erence that drives evaporation throughthe pores. This can be described in terms of a vapor pressure di�erence multipliedby a membrane distillation coe�cient B, which represents the di�usion resistancethrough the pores. It is based on material properties and pore geometry, and dependsweakly on temperature and is assumed to be constant for this calculation. On the feedside, boundary layers in concentration, temperature, and momentum are present, withcorresponding di�usional transport of heat and mass. On the cooler fresh water side,there is condensation of vapor and warming of the fresh water, with boundary layerprocesses similar to those on the feed side. Direct contact membrane distillation hasbeen successfully used to produce fresh water at small scale (0.1 m3/day) [33–36].

A transport process model for DCMD based on validated models by Bui et al. [37]and Lee et al. [36] was implemented to obtain the permeate �ux, and outlet temperaturesof a DCMD module. The calculation of system performance used heat transfercoe�cients calculated from correlations based on module geometry [38]. While the Buiet al. [37] model used a hollow-�ber membrane con�guration, the present calculationsare done for a �at-sheet con�guration. Membrane geometry and operating conditionsare taken from some pilot-sized plants the literature [39, 40]. Seawater enters thesystem at 27 ◦C and 35,000 mg/kg total dissolved solids at a mass �ow rate of 1 kg/s,The inlet feed is preheated to a constant temperature of 85 ◦C, and the required heat isprovided by a 90 ◦C source. The permeate side contains fresh water with an inlet �owrate of 1 kg/s. The resulting recovery ratio for this system is 4.4%. The regeneratoris a liquid-liquid heat exchanger with a terminal temperature di�erence of 3 K. Thepressure drop through the thin channel in the membrane module was found to be thedominant pressure drop in the system and was the basis for calculating the entropygeneration due to pumping power. Properties for seawater [22] were used in thecalculation. A schematic diagram of the system is shown in Fig. 4.20, with modulegeometry and constants shown.

Entropy generation was calculated for each component in the system by using acontrol volume analysis. Figure 4.21 shows the breakdown of entropy generation ineach component.

The greatest source of entropy generation is the module. This is owed mostly todi�usion through the pores and to a lesser extent heat conduction losses, as onlya thin membrane separates the cold and hot streams in the module. The smallpore size contributes substantially to the di�usion resistance; the pore diameteris usually on the order of 1000 times less than the membrane thickness. Recentwork [41] has also shown that equating the inlet �owrate of one stream with theoutlet �ow rate of the other stream instead of equating the module streams’ inlet�ow rates, as done here, achieves better thermal balancing in DCMD. Such thermalbalancing reduces the entropy generation [42] in the module and can result in a10–20% improvement in e�ciency [41]. The heater contributes substantially due to

Regenerator

TTD = 3 K

Seawater InletTsw, in = 27 ◦CÛm = 1 kg/s

Ûm = 1 kg/s

ÛQin T = 90 ◦C

Brine Reject

Pure Water

Tf , in = 85 ◦C

L = 10 mw = 0.7 m

dch = 4 mmB = 16 × 10−7 kg/m2Pa s

MDModule

Heater

Figure 4.20: Flow path for a basic direct contact membrane distillation system.

Module: 34.5%

Heater: 26.3%

Regenerator: 16.3%

Temperature Disequilibrium: 22.9%

Figure 4.21: Relative contribution of sources of entropy generation in a direct contactmembrane distillation system. Total speci�c entropy generation is 925.4 J/kg-K.

50


Regenerator

Compressor

Feed

Product

Reject Brine

Pre-heated FeedVapor

Compressed Vapor

Reject Brine

Evaporator/Condenser

Figure 4.22: Single e�ect mechanical vapor compression process.

the large amount of heat transferred, and the large temperature di�erence betweenthe source temperature (usually a steam saturation temperature) and the heater inlet.The regenerator has lower entropy generation as it transfers energy through a lowertemperature di�erence, which remains constant throughout its length. The dischargetemperature disequilibrium entropy generation is low compared to other thermalsystems, as the brine reject temperature is lower. Additionally, since the recovery ratiois low, the chemical disequilibrium of the brine is also negligible (entropy generationdue to brine disequilibrium is approximately three orders of magnitude smaller thanfrom other sources). Like most other systems discussed here, the entropy generationcontributed by low pressure-rise circulation pumping is negligible.

Reducing the top temperature, Tf , in, results in a net increase in speci�c entropygeneration. This is primarily due to the heater, as a lower top temperature gives riseto a higher temperature di�erence in the heater. Speci�c entropy generation in themodule goes down slightly, as evaporation happens at a lower temperature; however,this is negated by an increase in speci�c entropy generation in the regenerator, aswater production decreases faster than the temperature gradient in the regenerator.Entropy generation to temperature disequilibrium goes up primarily owing to thelower recovery ratio and additional brine reject.

Given the MD’s low recovery ratio and high discharge temperature, entropygeneration is high when compared to other desalination systems, and as a resultηII = 1.0%, as calculated with Eq. (4.88) and taking account all sources of entropygeneration. Other con�gurations of MD, such as conductive-gap MD (CGMD), can cutoverall energy consumption in half relative to DCMD [43].

4.5.3 Me�anical Vapor Compression

A simple single e�ect mechanical vapor compression (MVC) model is considered. Aschematic diagram of the process is shown in Fig. 4.22. The design values chosen forthe process are guided by those reported for single stage MVC plants analyzed byVeza [44] and Aly [45] and are listed in Table 4.4.

The inlet pressure to the compressor is taken to be the average of the saturation


Table 4.4: MVC design inputs.

Input Value

Seawater inlet temperature 25 ◦CSeawater inlet salinity 35 g/kgProduct water salinity 0 g/kgDischarged brine salinity 58.33 g/kgTop brine temperature 60 ◦CPinch: evaporator-condenser 2.5 KRecovery ratio 40%Isentropic compressor e�ciency 70%Compressor inlet pressure 19.4 kPa

Table 4.5: MVC model outputs.

Output Value

Speci�c electricity consumption 8.84 kWh/m3

Discharged brine temperature 27.2 ◦CProduct water temperature 29.7 ◦CCompression ratio 1.15Second Law e�ciency, ηII 8.5%

pressure of seawater at a salinity corresponding to the average of the feed andreject salinity. The regenerating heat exchanger is thermally balanced and thus thetemperature di�erence is taken to be constant between the rejected brine and the feedstream and also between the product water and the feed stream. By employing energyconservation equations for each component, the unknown thermodynamic states maybe computed (an explicit model for energy consumption is given in [46]). Knowing thethermodynamic states at each point, the entropy generated within each componentmay be calculated along with the entropy generated when the discharged brine isreturned to a body of water with the same composition and temperature as the feed.The key outputs from the model are reported in Table 4.5. The breakdown of entropygeneration among components is indicated within Fig. 4.23.

The majority of entropy generation may be attributed to heat transfer across a�nite temperature di�erence from the condensation process to the evaporation process.Entropy generation within the regenerator is less signi�cant, primarily because thesensible heat transferred in the regenerator is substantially smaller than the largeamount of latent heat recovered in the evaporator-condenser. Entropy generation dueto irreversibility within the compressor is important and depends upon the compressionratio and its isentropic e�ciency. Entropy generated in returning concentrated brineto a body of seawater is considerable as the recovery ratio is high (40%). Entropygenerated in returning product streams to the temperature of inlet seawater is smallas the regenerator is e�ective in bringing these streams to a temperature close to that


Evaporator-Condenser: 57.2% Compressor: 28.1%

Regenerator: 10.9%

Salinity disequilibrium - brine: 3.1%Temperature disequilibrium: 0.7%

Figure 4.23: Relative contribution of sources of entropy generation in a mechanicalvapor compression system. Total speci�c entropy generation is 98.0 J/kg-K.Contributions of the temperature disequilibrium of the distillate and brine streams are0.5% and 0.2%, respectively.

of the inlet seawater.The MVC system modeled above is a simple single e�ect system, satisfactory for

demonstrating the distribution of entropy generation throughout MVC plants. Detailedthermoeconomic models with multiple e�ects have been analyzed in literature [47].Research has also been undertaken on improving the heat transfer coe�cientswithin the evaporation and condensation processes of phase change. Lara et al. [48]investigated high temperature and pressure MVC, where dropwise condensation canallow greatly enhanced heat transfer coe�cients. Lukic et al. [49] also investigatedthe impact of dropwise condensation upon the cost of water produced. Suchimprovements in heat transfer coe�cients reduce the driving temperature di�erencein the evaporator-condenser leading to a lower compression ratio and thus reducedcompressor work requirements per unit of water produced. As the present analysisshows, reduction of entropy generation within the evaporator-condenser and thecompressor are crucial if exergetic e�ciency is to be improved.

4.5.4 Reverse Osmosis

A typical �ow path for a single stage reverse osmosis (RO) plant with energy recoveryis shown in Fig. 4.24 [50]. Since RO is a mechanically driven system and thermale�ects are of second order to pressure e�ects, reasonably accurate calculations can beperformed while only considering pressure work. The following approximations aremade:

Feed seawater is assumed to enter at ambient temperature and pressure (25 ◦C,1 bar) and at standard seawater salinity (35 g/kg). Pure water (0 g/kg salinity) isassumed to be produced at a recovery ratio of 40%. Further, it is assumed that 40%of the feed is pumped to 69 bar using a high pressure pump while the remaining60% is pumped to the same pressure using a combination of a pressure exchangerdriven by the rejected brine as well as a booster pump. The high pressure, booster,


and feed pump e�ciencies are assumed to be 85%. The concentrated brine loses 2bar of pressure through the RO module while the product leaves the module at 1 bar.Energy Recovery Inc. [50] makes a direct contact pressure exchanger that features asingle rotating part. The pressure exchanger pressurizes part of the feed using workproduced through the depressurization of the brine in the rotor. Equations (4.94a),(4.101), and (4.108) are used to match the work produced in expansion to the workrequired for compression. Assuming the expansion and compression processes are98% e�cient [50], the recovered pressure is calculated as follows:

precovered = pfeed − ηexpansionηcompression

(ρfeed

ρbrine

)(pbrine − patm) (4.127)

and the pressure exchanger e�ciency is evaluated using ERI’s de�nition [50]:

ηPX =

∑out Pressure × Flow∑in Pressure × Flow

(4.128)

Density of seawater is evaluated using seawater properties [22].Using the above assumptions, approximations, and inputs, the entropy generated

in the various components can be directly calculated using equations derived inSection 4.4.2. The entropy generated in the high pressure pump, booster pump, and thefeed in the pressure exchanger is evaluated using Eq. (4.110). The entropy generatedthrough the expansion of the pressurized brine in the pressure exchanger is evaluatedusing Eq. (4.103).

Additional consideration is necessary for the entropy generation in the RO modulebecause both the mechanical and chemical state of the seawater is changing. Thepure product stream’s pressure is 68 bar less than the feed, and the brine is 2 barless with an outlet salinity of 58.3 g/kg. To capture these e�ects, the Second Lawof Thermodynamics may be applied to a control volume surrounding the module,accounting for entropy �ow in and out, entropy generation, and heat transfer, ÛQmod,into the control volume boundary. Heat transfer is necessary if we evaluate the outletstreams at the inlet temperature, and to �nd the heat transfer we must also use theFirst Law on the same volume:

( Ûmh)p + ( Ûmh)b − ( Ûmh)sw = ÛQmod (4.129a)

( Ûms)p + ( Ûms)b − ( Ûms)sw = ÛSgen,mod +ÛQmod

T0(4.129b)

To evaluate the enthalpy and entropy changes, the physical properties of seawaterare needed as function of temperature, pressure, and salinity. Since seawater is nearlyincompressible, its entropy is essentially independent of p and could be evaluatedwithout accounting for pressure change (e.g., using the package of Sharqawy etal. [22]). The e�ect of pressure on enthalpy cannot be ignored, although it can alsobe approximated using an incompressible substance model. Instead, the pressuredependent property package of Nayar et al. [20] can be used. The property changes

∆ ÛH = Ûmp h(T0,patm,ws,p) + Ûmb h(T0,pbrine,ws,b) − Ûmf h(T0,pHP,ws, f )

∆ ÛS = Ûmp s(T0,patm,ws,p) + Ûmb s(T0,pbrine,ws,b) − Ûmf s(T0,pHP,ws, f )

RO Module Ûmp = 0.4 kg/spatm = 1 bar

ηHP = 85%

ηbooster = 85%

Pressure ExchangerηPX = 96%

Ûmbpbrine = 67 bar

Ûmb = 0.6 kg/spatm = 1 bar

Ûmbpfeed = 2 bar

Ûmbprecovered = 64.14 bar

ÛmfpHP = 69 bar

Ûmppfeed = 2 bar

∆ploss = 2 bar

ηfeed = 85%

Ûmf = 1.0 kg/spatm = 1 bar

Figure 4.24: A typical �ow path for a single stage reverse osmosis system.

55


RO module: 58.4%

High pressure pump: 19.4%

Feed pump: 0.7%Booster pump: 2.1%

Brine through PX: 3.2%

Feed through PX: 3.1%

Chemical Disequilibrium: 13.0%

Figure 4.25: Relative contribution of sources to entropy generation in the reverseosmosis system. Irreversibilities associated with product �ow through the membranedominates. Total speci�c entropy generation, omitting the chemical disequilibrium, isSTDS

gen = 19.9 J/kg-K.

are found to have the values ∆ ÛH = −2.70 kW and ∆ ÛS = −4.42 W/K. From these,the First and Second Laws give ÛQmod = −2.70 kW (out of the control volume) andÛSgen,mod = 4.65 W/K.

Regarding the heat transfer, if the system were adiabatic, the outlet streams wouldbe warmer (by roughly 0.7 K if they had the same temperature) but the value of ÛSgen,modwould be essentially the same. The energy dissipated by pump ine�ciency results insmall increases in the high pressure feed temperature (around 0.4 K), which is alsoassumed to be removed by heat transfer to the environment. Only negligible entropygeneration results from these heat transfers out of the system because the temperaturedi�erences above T0 are so small [see Eq. (4.121)].

Mistry et al. [19] used the incompressible �uid model to �nd the module entropygeneration as the sum of the entropy change of composition (at �xed T and p), theentropy generation from depressurizing the product (∆p = 68 bar), and the entropygeneration from depressurizing the brine (∆p = 2 bar). Algebra shows that thatapproach is also consistent with the First and Second Laws.

Figure 4.25 is a pie chart showing the relative amounts of entropy generationwithin the single-stage RO system. The greatest irreversibility occurs within the ROmodule, and the di�usion of water through the very high pressure drop of the ROmembrane is the principal source of this irreversibility. Note that the high pressurepump moves this same mass of water through the same pressure di�erence, but doesso at 85% e�ciency and therefore generates substantially less entropy than the zeroe�ciency �ow through the membrane.

For these conditions, the minimum least work is found to be ÛW minleast(r = 0) = 2.59

kJ/kg, and the total entropy generation is STDSgen = 19.9 J/kg-K. Therefore, from Eq. (4.85)

the required work of separation is 8.53 kJ/kg (2.37 kWh/m3) and the Second Lawe�ciency, per Eq. (4.88), is 30.4%.

Since RO systems tend to operate at higher Second Law e�ciency than thermalplants, the irreversibility due to discharge disequilibrium of the brine stream has a

§4.6 S e cond Law Ef f i c i enc y for Cogenerat i on 57

larger contribution to the total entropy generation. As seen in Fig. 4.25, the highsalinity of the brine accounts for 13% of the plant’s total irreversibility. The onlyway to reduce this e�ect is to lower the recovery ratio or to implement an osmoticpower recovery device (such as a pressure-retarded osmosis system) on the reject brinestream.

Based on these conditions, the minimum least work is found to be ÛW minleast(r = 0) =

2.59 kJ/kg and the total entropy generation is STDSgen = 18.9 J/kg-K. Therefore, from

Eq. (4.85) the required work of separation is 8.23 kJ/kg (2.28 kWh/m3) and the SecondLaw e�ciency, per Eq. (4.88), is 31.4%.

Since RO systems tend to operate at higher Second Law e�ciency than thermalplants, the irreversibility due to discharge disequilibrium of the brine stream has alarger contribution to the total entropy generation. As seen in Fig. 4.25, the highsalinity of the brine accounts for 13% of the plant’s total irreversibility. The only wayto reduce this e�ect is to lower the recovery ratio or to implement an osmotic powerrecovery device on the reject brine stream.

When trying to improve RO systems, designers target the irreversibilities in themodule. The simplest way to improve the performance of the system is to use a two(or more) stage RO system (e.g., as described in [51]). In a two stage system, wateris extracted at a lower recovery ratio from the �rst stage, resulting in a lower brineconcentration. Since the required pressure of the feed is dependent on the osmoticpressure, which itself is a function of the feed concentration, a lower recovery ratiomeans that lower pressures are needed in the �rst stage. Next, the brine from the �rststage is then further pressurized to the top pressure and additional water is extractedin a second stage. Even though the same top pressure is reached, since the �rst stageoperates with a lower pressure across the membranes, less total entropy is generatedin the two stage system.

Batch processing of seawater, in which a charge of seawater is slowly pressurized aspermeate is removed through membranes, can maintain a relatively smaller di�erencebetween hydraulic and osmotic pressure throughout the process [42]. This reduces theentropy generation and lowers the energy consumption per unit permeate. Semi-batchprocesses have been commercialized, e.g., by Desalitech Ltd. [52], and report signi�cantreductions in energy. True batch processes, such as those recently invented at MIT,could cut the energy requirements even further [53].

4.6 Second Law E�ciency for a Desalination SystemOperating as Part of a Cogeneration Plant

Many large-scale desalination processes use a cogeneration scheme, in which lowtemperature steam and electricity from a power plant are used to drive a desalinationplant. Additional primary energy (fuel), beyond that needed for electricity production,must be provided to drive the desalination plant. Thus, it is useful to consider theamount of additional heat energy that must be provided to the power plant in order togenerate the heat and work needed to power the desalination plant. In order to do so,consider a cogeneration system in which a power plant is connected to a desalination


Black BoxSeparator

Productws,p < ws,f

Concentratews,c > ws,f

Feedws,f

ÛQH ,TH︸︷︷︸ÛQpp+ ÛQd

ÛQ0,T0 ÛWpp

ÛWsep

ÛQsep,Ts

ÛQ0,T0

ηpp

PowerPlant

Figure 4.26: The power plant converts heat input into work output, work for thedesalination plant (represented as an unspeci�ed black box separator), and heat forthe desalination plant. It is assumed that the power plant operates at a Second Lawe�ciency of ηpp .

plant as shown in Fig. 4.26. In this system, a heat input ( ÛQH ) is provided to a powerplant. This heat input is equal to the amount of heat necessary to drive the powerplant ( ÛQpp ) plus the additional amount necessary to generate steam and electricity forthe desalination plant ( ÛQd to produce ÛQsep and ÛWsep). The power plant produces a netamount of work equal to the desired plant work production ( ÛWpp ) plus the amount ofwork necessary to drive the desalination plant ( ÛWsep). Note that typically, fuel, ratherthan heat, is the primary energy input to cogeneration systems and therefore, theanalysis should be done in terms of the amount of fuel required to drive the power plantplus the additional amount of fuel required to produce the heat and work necessaryto drive the desalination plant ( Ûmfuel = Ûmpp + Ûmd ). However, for simplicity and withthe goal of highlighting the di�erence between work and heat driven systems, thecontrol volume for this analysis is drawn under the assumption that heat, and not fuel,is transferred into the system. The e�ect of including the combustor in the analysis isnow discussed brie�y.

The following derivation is based on the work of Mistry and Lienhard [54] andEl-Sayed and Silver [29]. The First and Second Laws of Thermodynamics are writtenabout the power plant control volume:

ÛQpp + ÛQd = ÛQsep + ÛQ0 + ÛWpp + ÛWsep (4.130a)

ÛSgen +ÛQpp

TH+ÛQd

TH=ÛQsep

Ts+ÛQ0

T0(4.130b)

Multiplying the Second Law byT0 and substituting into the First Law to eliminate heattransfer to the ambient environment ( ÛQ0) gives:

ÛWpp + ÛWsep =(ÛQpp + ÛQd

) (1 −

T0

TH

)− ÛQsep

(1 −

T0

Ts

)−T0 ÛSgen (4.131)


In order to deal with the irreversibilities within the system, the rate of entropygeneration is assumed to be proportional to the amount of work produced by areversible power plant operating within the same heat transfer loads. That is

T0 ÛSgen ∝(ÛWpp,rev + ÛWsep,rev

)=

(ÛQpp + ÛQd

) (1 −

T0

TH

)− ÛQsep

(1 −

T0

Ts

)(4.132)

Letting the constant of proportionality be (1−ηpp ), where ηpp = η/ηCarnot is the SecondLaw e�ciency of the power plant, η is the First Law (energy) e�ciency of the plant,and ηCarnot is the Carnot e�ciency of a power plant operated between TH and T0,

T0 ÛSgen =

[ (ÛQpp + ÛQd

) (1 −

T0

TH

)− ÛQsep

(1 −

T0

Ts

)] (1 − ηpp

)(4.133)

Substituting T0 ÛSgen into Eq. (4.131) gives:

ÛWpp + ÛWsep =(ÛQpp + ÛQd

) (1 −

T0

TH

)ηpp − ÛQsep

(1 −

T0

Ts

)ηpp (4.134)

Since the goal is to determine how much additional heat is necessary to drive thedesalination system, ÛQd must be independent of the amount of work produced by thepower plant. Consider the same power plant in which the desalination system is notoperating and the power plant is producing a net output of ÛWpp . Then, setting ÛQd ,ÛQsep, and ÛWsep to zero, ÛQpp is found to be:

ÛQpp =ÛWpp(

1 − T0TH

)ηpp

(4.135)

which is consistent with the de�nition of the First Law e�ciency, η. Substituting thisback into the above equation results in ÛQpp and ÛWpp canceling out. Solving for ÛQd ,

ÛQd =ÛWsep(

1 − T0TH

)ηpp+ ÛQsep

(1 − T0

Ts

)(1 − T0

TH

) (4.136)

In order to evaluate the Second Law e�ciency of the desalination plant, one mightthink to use the exergetic value of the work and heat inputs to the plant,

ÛΞsep = ÛWsep +

(1 −

T0

Ts

)ÛQsep, (4.137)

for the denominator in Eq. (4.83) since it represents the total exergy input to thedesalination system. While this would be correct for a stand-alone system, ÛWsep andÛQsep do not represent the true exergy inputs for entire separation system which is

contained within the dashed border in Fig. 4.26. Instead, the exergy input to drive thedesalination system is that due to the extra heat transfer provided to the power plant,


ÛQd . Therefore, the Second Law e�ciency should be evaluated based on this quantity.Substituting Eq. (4.136) into Eq. (4.83) gives:

ηII =ÛΞmin

leastÛΞsep

=ÛW min

least

ÛQd

(1 − T0

TH

) = ÛW minleast

ÛWsep

ηpp+ ÛQsep

(1 − T0

Ts

) (4.138)

The important di�erence between using Eq. (4.137) and Eq. (4.136) is the fact thatthe work input ( ÛWsep) is divided by the Second Law e�ciency of the power plant. Thise�ectively accounts for the fact that the work is not produced reversibly from the heatsource, and therefore cannot be directly compared with the thermal exergy input. Ifthere is no work input, then ÛWsep = 0 and Eq. (4.138) correctly reduces to the ratio ofthe least heat of separation (based on Ts ) to the actual heat input to the desalinationsystem itself. Similarly, if there is no heat input, then ÛQsep = 0 and Eq. (4.138) reducesto:

ηII = ηppÛW min

leastÛWsep

(4.139)

In the limit of reversible operation for the power plant (i.e., ηpp = 1), Eq. (4.139) reducesto Eq. (4.88). The Second Law e�ciency of the power plant is present in Eq. (4.139)since the work used to power the desalination plant is produced irreversibly. Had thelosses in the combustor been included in this analysis, both ÛWsep and ÛQsep in Eq. (4.138)would be divided by the Second Law e�ciency of the combustor, ηII,combustor. Thiswould have the e�ect of reducing the Second Law e�ciency of the desalination processin proportion to the Second Law e�ciency of the combustor. Both the heat and workterms would be a�ected equally since the losses occur prior to the power generationprocess.

In order to better understand the energetic behavior of both membrane and thermalsystems, a parametric study of Eq. (4.138) is conducted in the following two sectionsfor systems using standard seawater as the feed source (35 g/kg, 25 ◦C). Under theseconditions, the minimum least work of separation of seawater per kilogram of productis 2.71 kJ/kg [19].10 The Second Law e�ciency is evaluated under two di�erentconditions: (1) work is the only input to the desalination system; and (2) heat at 100 ◦Cis the primary input and the amount of pumping work is varied.

Desalination Powered by Work

For desalination systems that are powered entirely using work, ÛQsep = 0 and Eq. (4.138)reduces to Eq. (4.139). As a result, it is clear that unless the power plant operatesreversibly, a work powered desalination system can never achieve 100% Second Lawe�ciency, even if the desalination process is conducted reversibly. This result isunavoidable because the primary energy source in the cogeneration scheme is heat tothe power plant, not electricity to the desalination plant. For the following study, thepower plant is assumed to be a representative combined cycle plant operating between

10More recent values of seawater thermodynamic properties reduce this value by 4.4%, to 2.59 kJ/kg. Forconsistency with [19] we retain the older value for the calculations that follow.


1400 and 298.15 K with a First Law e�ciency of η = 52.8% and a Second Law e�ciencyof ηpp = 67.2% [55]. The Second Law e�ciency of a work-driven desalination plant isshown in Fig. 4.27 as a function of ÛWsep starting at a minimum value of ÛWsep = ÛWleast.

TypicalRO

Wsep/mp [kJ/kg]

ηII [%]

0 5 10 15 20 25 300

10

20

30

40

50

60

70

80

90

1000 1 2 3 4 5 6 7 8

Wsep/Vp [kWh/m3]

Figure 4.27: The Second Law e�ciency of a work-driven desalination system operatingin a cogeneration scheme can never reach 100% unless the power plant operatesreversibly. Typical values for current reverse osmosis systems are highlighted. Feedwater is at T0 = 25 ◦C and ws,f = 35 g/kg.

All work-driven systems in this cogeneration scheme pay an energetic penaltyon e�ciency since the initial energy source (heat) must go through a conversionprocess (power plant) that operates irreversibly, and therefore, the limiting SecondLaw e�ciency as ÛWsep → ÛWleast is ηpp , not 1. If the primary source of energy wasconsidered to be mass of fuel, then the limiting Second Law e�ciency would be equalto the product of the Second Law e�ciencies of the combustor and the power plant.That is, ηII,combustorηpp . The typical range of operation for current RO technologies isin 2.5–5 kWh/m3 and is highlighted in Fig. 4.27. RO systems with energy recoverytend to be on the lower end of this range while systems without energy recovery tendto be on the higher end of this range. Exact values are a function of system design andfeed water characteristics [56–61]. This corresponds to Second Law e�ciency valuesranging from approximately 10% to 20%.


Desalination Powered by Co-Generated Heat and Work

Nearly all large-scale thermal desalination systems are connected to a power plantsince large quantities of steam are required to provide heating to the feed. The SecondLaw e�ciency of a thermal desalination plant operating using steam at 100 ◦C withpump work requirements ranging from 0 to 4 kWh/m3 is shown in Fig. 4.28. In the caseof zero pump work, 100% Second Law e�ciency is theoretically possible. However,once pump work is required, the possible Second Law e�ciency drops substantially(e.g., approximately 50% for 0.5 kWh/m3 of electrical work to drive pumps). Clearly,regardless of what the required heat of separation is, the additional requirement ofpump work results in a decrease in the Second Law e�ciency.

TypicalMED

TypicalMSF

Qsep/mp [kJ/kg]

ηII [%]

Ts = 100 ◦C

Wpump = 0 → 4.0 kWh/m3

0 50 100 150 200 250 300 350 4000

10

20

30

40

50

60

70

80

90

100100 50 30 20 15 10 8

GOR [-]

Figure 4.28: Second Law e�ciency for a thermal desalination plant requiring workfor pumping. Lines for pump work are in increments of 0.5 kWh/m3. As the pumpwork increases, the Second Law e�ciency decreases. Feed water is at T0 = 25 ◦C andws,f = 35 g/kg.

The results shown in Fig. 4.28 are generated based on the assumption that allenergy provided to the desalination system originally comes from a common energysource, ÛQd . This value of heat input is then substituted into Eqs. (4.83) and (4.137) toget Eq. (4.138). Should a desalination plant have energy inputs from multiple primaryenergy sources, then the analysis to derive the correct form of ηII will change slightly.All energy inputs should be traced to their primary sources (as was done for ÛWsep andÛQsep from ÛQd ) and then each primary input should be combined based on the exergetic

§4.7 Summary 63

value as done in Eq. (4.137).Based on Figs. 4.27 and 4.28, when a desalination plant operates as part of

a cogeneration scheme, the work-driven systems (based on currently availabletechnology) always behave in an exergetically more favorably manner than the thermaldriven systems (i.e., higher ηII ). This is true even when accounting for the energypenalty that comes from converting the source heat to work, and it is even furtherexempli�ed when considering that thermal systems typically require large amounts ofelectrical work for pumping (these are sometimes as high as the work requirementsfor an RO system by itself).

The present results are based on a Second Law e�ciency of ηpp = 67%,representative of a combined cycle plant. If a less e�cient Rankine cycle plantwere considered, having a First Law e�ciency of 30–35%, a representative ηpp mightbe 45–55%, depending on the top steam temperature. The di�erence between thework-driven and heat systems would decrease somewhat, but work driven (membrane)systems would remain much more e�cient than thermal systems.

Although current membrane systems are more e�cient from a Second Lawviewpoint, it should not be concluded that there is no role for thermal systems.Ultimately, several factors are considered when selecting a desalination technologyincluding capital and operating costs, quality of feed water, and existing expertise andinfrastructure. Although the work systems are favored energetically, these other factorscan lead to thermal systems being more desirable for a given location or application.

4.7 Summary

Reducing energy consumption is a key tool for minimizing the environmental footprintand increasing the sustainability of desalination. Using thermodynamic analysis,we have shown how to benchmark systems and process designs against physicallimits, how to model the thermodynamic properties of saline waters, and how feedwater properties and environmental conditions can a�ect actual and minimum energyconsumption. Entropy generation has been shown to characterize the energetic de�citrelative to reversible systems, and equations were developed to quantify the degreeof irreversibility, or ine�ciency, in individual processes and desalination systemcomponents. Identifying components with relatively large entropy production focusese�ciency-driven (re-)design onto those components that will have the greatest impacton overall system performance.

In Section 4.5, we applied these tools to analyze a suite of desalination systems,including both established and emerging technologies. The results of this analysis areshown in Fig. 4.29, where we see that work-driven technologies operate closest to thereversible limit, and that one or two components in each technology stand out as thelargest sources of irreversibility in each system. Tow et al. [62], reported the SecondLaw e�ciency of a range of real systems operating at various salinities, as shown inFig. 4.30. The highest e�ciencies are found for seawater reverse osmosis plants.11

11These results are based on the exergy entering the desalination system itself, in contrast to the resultsof Section 4.6 which refer exergy inputs to the power plant of a coproduction system.


0%

20%

40%

60%

80%

100%

RO MVC MED DCMD

Per

cent

Net

Ene

rgy

Con

sum

ptio

n Losses

MinimumEnergy

η = %

6% 1%

Effe

cts

Eva

pora

tor-

Con

dens

er

RO

Mod

ule

MD

Mod

ule

32%

9%

Figure 4.29: The net energy consumption for each technology assessed in Sec. 4.5 isdivided into minimum energy (or e�ciency) and losses on a percentage basis. Onlythe primary loss mechanism is labeled. The two work-driven technologies, MVC andRO, have the highest exergetic e�ciency. In each system, the core component doingthe separation is the greatest source of ine�ciency, but auxiliary components alsocontribute signi�cantly to overall irreversibility.

Other studies that capitalize on thermodynamic methods of this chapter haveconsidered: high salinity brine desalination [46]; balancing of forward osmosis (FO)mass exchangers [62]; energy e�ciency of FO relative to RO for seawater desalination[63]; performance optimization of humidi�cation-dehumidi�cation desalination [23,64–67]; e�ciency of desalination driven by waste heat [68]; energy requirements of avariety of hybrid desalination systems, e.g., [69, 70]; and even Second Law e�cienciesthat incorporate the costs of electricity and heat [71].

Population growth is increasingly straining our limited supply of renewable freshwater, and associated fossil energy emissions threaten our atmosphere. Projectionsof rising climate variability necessitate greater resilience in our water systems. Forall these reasons, the need for more e�cient and sustainable desalination is urgent.The concepts and methods developed here provide a framework for assessing andimproving the e�ciency of both established and emerging desalination technologies.

BW

RO

SWR

O

Ther

mal

FO 1

FO 2

FO 3

FO 4

1

2

3

2

1

2

1

3

2 1

3

Key

: Fe

ed

Co

nce

ntr

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nd

:

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1 F

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[2,1

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FO

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[2

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Pro

cess

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EDR

0%

5%

10

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15

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20

%

25

%

30

%

35

%

40

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0%

1%

10

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Exergetic efficiency

Salin

ity

5

0%

5%

10

%

15

%

20

%

25

%

30

%

35

%

40

%

0%

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Efficiency

0

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Salin

ity

Fig

ure

2:E

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dw

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Eqs.

4an

dA

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sin

gd

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rep

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s.[2

,11

,14

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esar

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pre

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dw

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that

beg

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and

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66 Thermodynam i c s o f De sa l i nat i on §4.A

APPENDICES

4.A Seawater Properties Correlations

Seawater is a complex electrolyte solution of water and salts. The salt concentration,ws , is the mass fraction of all dissolved solids present in a unit mass of seawater. It isusually expressed by the salinity (on reference-composition salinity scale) as de�nedby Millero et al. [72] which is currently the best estimate for the absolute salinityof seawater. In this appendix, correlations of seawater thermodynamic propertiesnamely speci�c volume, speci�c enthalpy, speci�c entropy, speci�c heat, chemicalpotential, and osmotic coe�cient to be used in thermodynamic analysis calculationsof this chapter are given. In this regard, the thermodynamic properties of seawater arecalculated using the correlations provided by Sharqawy et al. [22]. These correlations �tthe data extracted from the seawater Gibbs energy function of IAPWS 2008 [73]. Theyare polynomial equations given as functions of temperature and salinity at atmosphericpressure (or saturation pressure for temperatures over normal boiling temperature).In these correlations, the reference state for the enthalpy and entropy values is takento be the triple point of pure water (0.01 ◦C) and at zero absolute salinity.

For other correlations of seawater thermophysical properties, including pressuredependence, the equations provided by Nayar et al. [20] are recommended. A fullset of codes for calculating seawater thermophysical properties are available withoutcharge at: http://web.mit.edu/seawater.

4.A.1 Speci�c Volume

The speci�c volume is the inverse of the density as given by Eq. (4.A.1). Both areintensive properties however in thermodynamics literature it is preferred to use thespeci�c volume instead of the density because it is directly related to the �ow work.The density of seawater is higher than that of pure water due to the salts; consequentlythe speci�c volume is lower. The seawater density can be calculated by using Eq. (4.A.2)given by Sharqawy et al. [22] which �ts the data of Isdale and Morris [74] and that ofMillero and Poisson [75] for a temperature range of 0–180 ◦C and salt concentrationof 0–0.16 kg/kg and has an accuracy of ±0.1%. The pure water density is given byEq. (4.A.3) which �ts the data extracted from the IAPWS [76] with an accuracy of±0.01%.

vsw = 1/ρsw (4.A.1)ρsw = ρw + ws

(a1 + a2T + a3T

2 + a4T3 + a5ws T

2) (4.A.2)ρw = 9.999 × 102 + 2.034 × 10−2T − 6.162 × 10−3T 2

+ 2.261 × 10−5T 3 − 4.657 × 10−8T 4 (4.A.3)

where vsw is the speci�c volume of seawater in m3/kg, ρsw and ρw are the density ofseawater and pure water respectively in kg/m3, T is the temperature in ◦C, ws is the

http://web.mit.edu/seawater

§4.A S eawater Prop ert i e s Corr e lat i on s 67

Figure 4.A.1: Seawater speci�c volume variations with temperature and saltconcentration

salt concentration in kgs/kgsw and

a1 = 8.020 × 102, a2 = −2.001, a3 = 1.677 × 10−2

a4 = −3.060 × 10−5, a5 = −1.613 × 10−5 (4.A.4)

Figure 4.A.1 shows the speci�c volume of seawater calculated from Eq. (4.A.1)as it changes with temperature and salt concentration. It is shown that the speci�cvolume of seawater is less than that of pure water by about 8.6% at 0.12 kg/kg saltconcentration and 120 ◦C. It is important to mention that for incompressible �uids (e.g.,seawater) the variation of the speci�c volume with pressure is very small and can beneglected in most desalination practical problems. The error in calculating the speci�cvolume is less than 1% when the pressure is varying from the saturation pressure tothe critical pressure in the compressed liquid region. Therefore, Eq. (4.A.1) can be usedat pressures higher than the atmospheric pressure (up to the critical pressure) and atpressure lower than the atmospheric pressure (up to the saturation pressure) with anegligible error (less than 1%).

4.A.2 Speci�c Enthalpy

The speci�c enthalpy of seawater is lower than that of pure water since the heat capacityof seawater is less than that of pure water. It can be calculated using Eq. (4.A.5) givenby Sharqawy et al. [22] which �ts the data extracted from the seawater Gibbs energyfunction of IAPWS [73] for a temperature range of 10–120 ◦C and salt concentrationrange of 0–0.12 kg/kg and has an accuracy of ±0.5%. The pure water speci�c enthalpyis given by Eq. (4.A.6) which �ts the data extracted from the IAPWS [76] with an


Figure 4.A.2: Seawater speci�c enthalpy variations with temperature and saltconcentration

accuracy of ±0.02%. It is valid for temperature range of 5–200 ◦C.

hsw = hw − ws(b1 + b2ws + b3w

2s + b4w

3s + b5T + b6T

2 + b7T3

+ b8wsT + b9w2s T + b10wsT

2) (4.A.5)hw = 141.355 + 4202.070 ×T − 0.535 ×T 2 + 0.004 ×T 3 (4.A.6)

where hsw and hw are the speci�c enthalpy of seawater and pure water respectively in(J/kg), 10 ≤ T ≤ 120 ◦C, 0 ≤ ws ≤ 0.12 kg/kg and

b1 = −2.348 × 104,b2 = 3.152 × 105,b3 = 2.803 × 106,b4 = −1.446 × 107

b5 = 7.826 × 103,b6 = −4.417 × 101,b7 = 2.139 × 10−1,b8 = −1.991 × 104

b9 = 2.778 × 104,b10 = 9.728 × 101(4.A.7)

Figure 4.A.2 shows the speci�c enthalpy of seawater calculated from Eq. (4.A.5) asit changes with temperature and salt concentration. The speci�c enthalpy of seawateris less than that of pure water by about 14% at 0.12 kg/kg salt concentration and 120 ◦C.

The in�uence of pressure on the speci�c enthalpy has been analyzed and correlatedby Nayar et al. [20].

4.A.3 Speci�c Entropy

The speci�c entropy of seawater is lower than that of pure water. It can be calculatedusing Eq. (4.A.8) given by Sharqawy et al. [22] which �ts the data extracted from theseawater Gibbs energy function of IAPWS [73] for a temperature range of 10–120 ◦Cand salt concentration range of 0–0.12 kg/kg and has an accuracy of ±0.5%. The purewater speci�c entropy is given by Eq. (4.A.9) which �ts the data extracted from the


Figure 4.A.3: Seawater speci�c entropy variations with temperature and saltconcentration

IAPWS [76] with an accuracy of ±0.1%. It is valid for T = 5–200 ◦C.

ssw = sw − ws(c1 + c2ws + c3w

2s + c4w

3s + c5T + c6T

2 + c7T3

+ c8wsT + c9w2sT + c10wsT

2) (4.A.8)sw = 0.1543 + 15.383 ×T − 2.996 × 10−2 ×T 2

+ 8.193 × 10−5 ×T 3 − 1.370 × 10−7 ×T 4 (4.A.9)

where ssw and sw are the speci�c entropy of seawater and pure water respectively in(J/kg K), 10 ≤ T ≤ 120 ◦C, 0 ≤ ws ≤ 0.12 kg/kg and

c1 = −4.231 × 102, c2 = 1.463 × 104, c3 = −9.880 × 104, c4 = 3.095 × 105

c5 = 2.562 × 101, c6 = −1.443 × 10−1, c7 = 5.879 × 10−4, c8 = −6.111 × 101

c9 = 8.041 × 101, c10 = 3.035 × 10−1

(4.A.10)Figure 4.A.3 shows the speci�c entropy of seawater calculated from Eq. (4.A.8)

as it changes with temperature and salt concentration. It is shown that the speci�centropy of seawater is less than that of fresh water by about 18% at 0.12 kg/kg saltconcentration and 120 ◦C. It is important to mention that for incompressible �uids(e.g., seawater) the variation of speci�c entropy with pressure is very small and can beneglected in most practical cases [17]. Therefore, Eq. (4.A.8) can be used at pressuresdi�erent than the atmospheric pressure.

4.A.4 Chemical Potential

The chemical potentials of water in seawater and salts in seawater may be calculatedusing the equations given by Nayar et al. [20]. Figure 4.A.4 shows the chemical potential


Figure 4.A.4: Chemical potential of water in seawater

of water in seawater calculated as it changes with temperature and salt concentration.Figure 4.A.5 shows the chemical potential of salts in seawater as it changes withtemperature and salt concentration. It is seen in Fig. 4.A.4 that the chemical potentialof water in seawater decreases with both temperature and salt concentration, whilethe chemical potential of salts in seawater increases with both temperature and saltconcentration as seen in Fig. 4.A.5.

4.A.5 Osmotic Coe�cient

The molal osmotic coe�cient of a solution, Eq. (4.14b), can be determined from vaporpressure, boiling point elevation, and freezing point measurements. Sharqawy etal. [22] reviewed the literature for osmotic coe�cient data and provided a correlationthat is based on the data of Bromley et al. [77] due its wide parameter range of 0–200 ◦Cin temperature and 10–120 g/kg in salinity with a maximum deviation of ±1.4% and acorrelation coe�cient of 0.991. This correlation is copied here:

ϕb = a1 + a2T + a3T2 + a4T

4 + a5ws

+ a6wsT + a7wsT3 + a8ws

2 + a9ws2T + a10ws

2T 2 (4.A.11)

where

a1 = 8.9453 × 10−1,a2 = 4.1561 × 10−4,a3 = −4.6262 × 10−6

a4 = 2.2211 × 10−11,a5 = −1.1445 × 102,a6 = −1.4783 × 100

a7 = −1.3526 × 10−5,a8 = 7.0132 × 106,a9 = 5.696 × 104

a10 = −2.8624 × 102

(4.A.12)

Equation (4.A.11) has a validity of 0 ≤ T ≤ 200 ◦C and 0.010 ≤ ws ≤ 0.120 g/kg,with an accuracy of ±1.4%. However, this correlation is limited to a salinity of


Figure 4.A.5: Chemical potential of salts in seawater

10 g/kg and cannot be extended to lower salinities (i.e. for dilute solutions). Itis possible for a seawater stream to become diluted to a salinity below 10 g/kg incertain osmotically-driven processes. Literature values and correlations of the osmoticcoe�cient for diluted seawater with a salinity of 10 g/kg and below which adhere tothis proper physical limit are di�cult to �nd. As described by Debye-Hückel theory,the osmotic coe�cient for a mixture approaches a value of 1 with decreasing salinityand does so independently of temperature. Therefore, an extension of the correlationprovided by Sharqawy et al. [22] 2010 was developed by Sharqawy et al. [78] in 2013.By using the theoretical expression for the osmotic coe�cient of dilute solutions givenby Brønsted [79], the correct behavior as b → 0 can be obtained

ϕb = 1 − κ√b + λb (4.A.13)

where b is the molality of the solution given by Eq. (4.47), and κ and λ are two �ttingparameter constants. To �nd the value of these constants, Eq. (4.A.13) and its �rstderivative with respect to salt concentration are set to equal the value of ϕb given byEq. (4.A.11) its �rst derivative with respect to salt concentration at 0.010 kg/kg, formingtwo equations with the two constants as unknowns. At 25 ◦C, the two constants arefound to be κ = 0.3484 and λ = 0.3076. (For the complete correlation of κ and λas function of temperature, please see [20].) The �nal osmotic coe�cient functionis now set to be a piece-wise function with Eq. (4.A.13) forming the function for0 ≤ ws < 0.010 kg/kg and Eq. (4.A.11) forming the 0.010 ≤ ws ≤ 0.120 kg/kg section.

4.A.6 Speci�c Heat Capacity at Constant Pressure

The speci�c heat capacity at constant pressure for seawater is less than that offreshwater which reduces the amount of sensible heat that can be transferred atthe same temperature di�erence. The speci�c heat capacity can be calculated by using


Figure 4.A.6: Speci�c heat of seawater

Eq. (4.A.14) given by Jamieson et al. [80] which �ts the experimental measurementswith an accuracy of ±0.3%. Equation (4.A.14) is valid for temperatures of 0–180 ◦C andsalt concentration range of 0–0.18 kg/kg.

cp,sw = A + B (T + 273.15) +C (T + 273.15)2 + D (T + 273.15)3 (4.A.14)

where cp,sw is in kJ/kg K,T in ◦C, ws in g/kg (not kg/kg for this particular correlation),and

A = 5.328 − 9.76 × 10−2ws + 4.04 × 10−4

w2s (4.A.15)

B = −6.913 × 10−3 + 7.351 × 10−4ws − 3.15 × 10−6

w2s (4.A.16)

C = 9.6 × 10−6 − 1.927 × 10−6ws + 8.23 × 10−9

w2s (4.A.17)

D = 2.5 × 10−9 + 1.666 × 10−9ws − 7.125 × 10−12

w2s (4.A.18)

Figure 4.A.6 shows the speci�c heat of seawater calculated from Eq. (4.A.14) as afunction of temperature and salt concentration. It is shown that the speci�c heat ofseawater is less than that of fresh water by about 16% at a salt concentration of 0.16kg/kg.

It is important to mention here that the last coe�cient in Eq. (4.A.16) was printedwith a positive sign in the original paper [80]. However, the correlation matches theexperimental data given in the original paper only if a negative sign is used. Webelieve that there is a typographical error in the original paper, and we have adopted anegative sign here.

4.A.7 Tabulated Data

Tabulated data for seawater thermodynamic properties are given in Table 4.A.1 usingthe equations presented in this Appendix, and the equations for chemical potential

§4.B P i t z e r Parameter s 73

Table 4.A.1: Seawater Thermodynamic Properties: T0 = 25 ◦C, p = p0 = 101.325 kPa,ws = ws,0 = 0.035 kg/kg

T v u h s µw µs ef◦C m3/kg kJ/kg kJ/kg kJ/kg·K kJ/kg kJ/kg kJ/kg

10 0.000974 40.0 40.1 0.144 −3.12 64.57 1.7115 0.000975 59.8 59.9 0.214 −4.10 66.54 0.7720 0.000976 79.7 79.8 0.282 −5.45 68.50 0.2025 0.000977 99.7 99.8 0.350 −7.15 70.46 0.0030 0.000978 119.6 119.7 0.416 −9.20 72.41 0.1435 0.000980 139.6 139.7 0.482 −11.60 74.38 0.6240 0.000982 159.6 159.7 0.546 −14.33 76.35 1.4245 0.000984 179.7 179.8 0.610 −17.40 78.35 2.5350 0.000986 199.7 199.8 0.672 −20.80 80.36 3.9455 0.000989 219.8 219.9 0.734 −24.52 82.41 5.6460 0.000991 239.9 240.0 0.795 −28.56 84.49 7.6265 0.000994 260.0 260.1 0.855 −32.92 86.61 9.8770 0.000997 280.1 280.2 0.914 −37.58 88.77 12.3775 0.000999 300.3 300.4 0.972 −42.55 90.98 15.1380 0.001003 320.4 320.5 1.029 −47.82 93.25 18.1485 0.001006 340.5 340.6 1.086 −53.38 95.57 21.3990 0.001009 360.7 360.8 1.142 −59.23 97.95 24.86

given by Nayar et al. [20]. The properties include speci�c volume, speci�c internalenergy, speci�c enthalpy, speci�c entropy, chemical potentials, and speci�c �ow exergy.These are given at temperature of 10–90 ◦C, salt concentration of 0.035 kg/kg (absolutesalinity 35 g/kg) and pressure of 101.325 kPa. However, the equations presented inthe appendix can be used up to temperature of 120 ◦C. In this case for temperatureshigher than the normal boiling temperature, the pressure is the saturated pressure andthe state of the seawater is the saturated liquid state. For the �ow exergy values givenin Table 4.A.1, the environment dead state is selected at T0 = 25 ◦C, p0 = 101.325 kPaand ws,0 = 0.035 kg/kg.

As previously noted, a full set of codes for calculating seawater thermophysicalproperties are available without charge at: http://web.mit.edu/seawater.

4.B Pitzer Parameters

This appendix discusses the terms in the Pitzer equations for the activity and osmoticcoe�cients (Eqs. (4.33)–(4.36)) in aqueous solutions. The term F is based on an extendedDebye-Hückel function [6], re�ecting the characteristic �rst-order-square-root


74 Thermodynam i c s o f De sa l i nat i on §4.B

dependence on ionic strength caused by long-range electrostatic interactions:

F = −Aϕ

( √I

1 + 1.2√I+

21.2

ln(1 + 1.2√I )

)+

∑c

∑a

bcbaB′ca +

∑∑c<c ′

bcbc ′Φ′cc ′

+∑∑a<a′

baba′Φ′aa′ (4.B.1)

The parameter Aϕ is related to the Debye-Hückel limiting law, and is given by

Aϕ =13

[e3 (2N0ρw)

1/2

8π (ϵrϵ0kbT )3/2

](4.B.2)

where ρw is the density of pure water. Data for the relative permittivity of pure wateras a function of temperature can be obtained from [81].

Interactions between cations and anions are represented by the functions Bi j , B′i j ,Bϕi j , and Ci j :

BMX = β(0)MX + β

(1)MXд(αMX

√I ) + β (2)MXд(12

√I ) (4.B.3a)

B′MX = β(1)MXд

′(αMX√I )/I + β (2)MXд

′(12√I )/I (4.B.3b)

BϕMX = β

(0)MX + β

(1)MX exp (−αMX

√I ) + β (2)MX exp (−12

√I ) (4.B.3c)

CMX =CϕMX

2|zMzX |1/2(4.B.3d)

where αMX = 2.0 for j-1 electrolytes and αMX = 1.4 for 2-2 and higher electrolytes.The parameters β (i)MX are tabulated for a given ion pair, and β (2)MX is associated withcomplex formation and generally only non-zero for 2-2 electrolytes. The functionsд(x) and д′(x) are

д(x) = 2(1 − (1 + x)e−x )/x2 (4.B.4a)

д′(x) = −2x2

[1 −

(1 + x +

x2

2

)e−x

](4.B.4b)

Interactions between like-charged pairs are represented by Φi j and Φ′i j :

Φi j = θi j +Eθi j (I ) (4.B.5a)

Φ′i j =Eθ ′i j (I ) (4.B.5b)

Φϕi j = θi j +

Eθi j (I ) + IEθ ′i j (I ) (4.B.5c)

Here, the only adjustable parameter for a given ion pair is θi j . The terms Eθi j (I ) andEθ ′i j (I ) represent excess free energy arising from electrostatic interactions between

§4.C Nomenc lature 75

asymmetric electrolytes (i.e., ions with charge of like sign and unlike magnitude), andare functions of ionic strength only:

Eθi j =zizj

4I

(J0(xi j ) −

12J0(xii ) −

12J0(x j j )

)(4.B.6a)

Eθ ′i j =zizj

8I 2

(J1(xi j ) −

12J1(xii ) −

12J1(x j j )

)−

Eθi j

I(4.B.6b)

where

J0(x) =14x − 1 +

1x

∫ ∞

0

[1 − exp

(−x

ye−y

)]y2 dy (4.B.6c)

J1(x) =14x −

1x

∫ ∞

0

[1 −

(1 +

x

ye−y

)× exp

(−x

ye−y

)]y2 dy (4.B.6d)

andxi j = 6zizjAϕ

√I (4.B.6e)

The integrals in Eqs. (4.B.6c) and (4.B.6d) can be calculated numerically.In summary, the adjustable parameters are as follows. There are 3 to 4 per

unlike-charged pair, β (0)MX , β (1)MX , β (2)MX , and CϕMX ; one per like-charged pair, θi j ; and

one per cation-cation-anion and anion-anion-cation triplet, Ψi jk . The values of theseparameters can be found in a variety of sources, some of which contain slightly di�erentvalues. Tables of values can be found in, e.g., [7, 10, 12, 82–84].

In principle, each of the adjustable binary and ternary parameters (β (i)MX , CϕMX , θi j ,and Ψi jk ) are functions of temperature. Unfortunately, a complete set of these dataas a function of temperature over the range of interest are generally unavailable inopen literature (although some signi�cant collections are available, e.g., [10, 83, 84]).However, Silvester and Pitzer have noted that the temperature derivatives of theseparameters are o�en small [85], and much of the temperature variation in activitycoe�cient is con�ned to Aϕ (Eq. (4.B.2)) both in the parameter’s explicit temperaturedependence, as well as implicitly through variations in the dielectric constant [86].In addition, solubility computations by DeLima and Pitzer [87] were not impairedby using room temperature values for the mixing parameters (θi j and Ψi jk ) up to473 K—well outside the temperature range of typical desalination systems.

4.C NomenclatureSymbolsAϕ Modi�ed Debye-Hückel parameter [kg1/2/mol1/2]a ActivityBi j , B

ϕi j Pitzer parameter, second virial coe�cient [kg/mol]

B′i j Pitzer binary interaction parameter [kg2/mol2]B Membrane distillation coe�cient [kg/m2-Pa-s]b Molality [mol/kg]C Modi�ed van ’t Ho� coe�cient [kPa-kg/g]

76 Thermodynam i c s o f De sa l i nat i on §4.C

Ci j , Cϕi j Pitzer parameter, unlike-charged interactions [kg2/mol2]

Cp Heat capacity at constant pressure [J/K]CV Control volumec Concentration [mol/L]; speci�c heat capacity [J/kg-K]cp Speci�c heat capacity at constant pressure [J/kg-K]c v Speci�c heat capacity at constant volume [J/kg-K]Di Distillate from e�ect i [kg/s]Df ,i Distillate from �ashing in e�ect i [kg/s]Df b,i Distillate from �ashing in �ash box i [kg/s]dch Flow channel depth [m]e Elementary charge [C]; speci�c exergy [J/kg]ed Speci�c exergy destroyed [J/kg]ef Speci�c �ow exergy [J/kg]F Extended Debye-Hückel function, Eq. (4.B.1)G Gibbs free energy [J]д Speci�c Gibbs free energy [J/kg]H Enthalpy [J]h Speci�c enthalpy [J/kg]hfg Latent heat of vaporization [J/kg]hsf Latent heat of freezing [J/kg]I Ionic strength [mol/kg]i van ’t Ho� factor [-]Kb Ebullioscopic constant [K-mol/kg]Kf Cryoscopic constant [K-mol/kg]kB Boltzmann’s constant [J/K]L Length [m]Mi Molar mass of species i [g/mol]M Mixture average molar mass [g/mol]m Mass [kg]Ûm Mass �ow rate [kg/s]NA Avogadro’s number (6.022140 × 1023) [mol−1]Ni Amount of species i [mol]n Number of e�ects or stages [-]p Pressure [kPa]ÛQ Heat transfer rate [W]ÛQleast Least heat of separation [W]ÛQmin

least Minimum least heat of separation [W]ÛQsep Heat of separation [W]R Ideal gas constant [J/kg-K]R Molar gas constant, 8.31446 [J/mol-K]r Recovery ratio [(kg/s product)/(kg/s feed)]S Entropy [J/K]ÛSgen Entropy generation rate [W/K]s Speci�c entropy [J/kg-K]sgen Speci�c entropy generation per unit �uid [J/kg-K]

§4.C Nomenc lature 77

Sgen Speci�c entropy generation per unit water produced [J/kg-K]T Temperature [K]T0 Ambient (dead state) temperature [K]TH Temperature of hot-side reservoir [K]U Internal energy [J]u Speci�c internal energy [J/kg]V Volume [m3]v Molar volume [m3 mol-1]v Speci�c volume [m3/kg]ÛW Work transfer rate [W]ÛWleast Least work of separation [W]ÛW min

least Minimum least work of separation [W]ÛWrev Reversible work [W]ÛWsep Work of separation [W]w Speci�c work [J/kg]; mass fraction [kg/kg or g/kg]; width [m]ws Mass fraction of salts [kg/kg or g/kg]x Quality [kg/kg]; mole fractionZ Pitzer function,

∑i bi |zi | [mol/kg]; generalized compressibility [-]

z Charge number

Greekα Pitzer parameter [kg1/2/mol1/2]β Pitzer parameter [kg/mol]γx , γb , γc Rational, molal, and molar activity coe�cientδb Boiling point elevation [K]δf Freezing point depression [K]∆ Change in a variable∆psat Vapor pressure lowering [Pa]ε0 Vacuum permittivityεr Relative permittivityη Mole ratio of salt in seawater [-], e�ciencyηe Isentropic e�ciency of expander [-]ηp Isentropic e�ciency of pump/compressor [-]ηII Second Law/exergetic e�ciency [-]θ Pitzer parameter [kg/mol]κ Constant in Eq. (4.A.13)λ Constant in Eq. (4.A.13)λi j Pitzer parameter, uncharged interactions [kg/mol]µ Chemical potential [J/mol]ν Stoichiometric coe�cientÛΞdestroyed Exergy destruction rate [W]ÛΞ Exergy �ow rate [W]π Osmotic pressure [Pa]ρ Density [kg/m3]Φi j , Φ

ϕi j Pitzer parameter, like-charged interactions [kg/mol]

78 Thermodynam i c s o f De sa l i nat i on §4.C

Φ′i j Pitzer parameter, like-charged interactions [kg2/mol2]ϕ Osmotic coe�cientΨi jk Pitzer parameter, ternary interactions [kg2/mol2]

Subscripts0 Environment, or global dead statea, X Anionatm Atmosphericb Brine, molal basisc , M Cationd Desalination, diluentf FlashingF Feedi ith species, inlet staten, N Neutral specieso Outlet statep Productpp Power plants Steam; saltsat Saturated statesw Seawaterw Water

Superscriptsid Ideal solutionex Excess propertys Isentropic′ Stream before exiting CV◦ Standard (reference) state∗ Restricted dead state

AcronymsBH Brine heaterCAOW Closed air open waterCD Chemical disequilibriumERI Energy Recovery Inc.FF Forward feedGOR Gained output ratioHP High pressureHX Heat exchangerIF Incompressible �uidIG Ideal gasMED Multiple e�ect distillationMVC Mechanical vapor compressionOT Once throughPR Performance ratio

Re f e r enc e s 7 9

PX Pressure exchangerRDS Restricted dead stateRO Reverse osmosisTD Temperature disequilibriumTDS Total dead stateWH Water heated

References[1] S. Glasstone, Thermodynamics for Chemists. D. Van Nostrand Company, Inc., 1947.

8, 18, 19

[2] K. S. Pitzer, J. C. Peiper, and R. H. Busey, “Thermodynamic properties of aqueoussodium chloride solutions,” Journal of Physical and Chemical Reference Data,vol. 13, no. 1, pp. 1–102, 1984. 8, 16, 34

[3] R. A. Robinson and R. H. Stokes, Electrolyte Solutions. Dover, 2nd revised ed.,2002. 8, 13, 18

[4] P. Atkins and J. de Paula, Atkins’ Physical Chemistry. W. H. Freeman and Company,8th ed., 2006. 9

[5] W. Hamer and Y. Wu, “Osmotic coe�cients and mean activity coe�cients ofuni-univalent electrolytes in water at 25 ◦C,” Journal of Physical and ChemicalReference Data, vol. 1, no. 4, pp. 1047–1099, 1972. 12

[6] P. Debye and E. Hückel, “Zur theorie der elektrolyte,” Physikalische Zeitschri ,vol. 24, pp. 185–206, May 1923. 13, 73

[7] J. F. Zemaitis, D. M. Clark, M. Rafal, and N. C. Scrivner, Handbook of AqueousElectrolyte Thermodynamics. Wiley-AIChE, 1986. 13, 75

[8] C. W. Davies, “The Extent of Dissociation of Salts in Water. Part VIII. An Equationfor the Mean Ionic Activity Coe�cient of an Electrolyte in Water, and a Revisionof the Dissociation Constants of Some Sulphates.,” Journal of the Chemical Society,pp. 2093–2098, 1938. 13

[9] K. S. Pitzer and J. J. Kim, “Thermodynamics of Electrolytes. IV. Activity andOsmotic Coe�cients for Mixed Electrolytes,” Journal of the American ChemicalSociety, vol. 96, pp. 5701–5707, September 1974. 13

[10] K. S. Pitzer, “A thermodynamic model for aqueous solutions of liquid-like density,”Reviews in Mineralogy and Geochemistry, vol. 17, pp. 97–142, 1987. 13, 14, 75

[11] C. E. Harvie and J. H. Weare, “The prediction of mineral solubilities in naturalwaters: the Na-K-Mg-Ca-Cl-SO4-H2O system from zero to high concentration at25 ◦C,” Geochimica et Cosmochimica Acta, vol. 44, pp. 981–997, 1980. 14

8 0 Chapt er 4 : Thermodynam i c s o f De sa l i nat i on

[12] C. E. Harvie, N. Møller, and J. H. Weare, “The prediction of mineral solubilitiesin natural waters: The Na-K-Mg-Ca-H-Cl-SO4-OH-HCO3-CO3-CO2-H2O systemto high ionic strengths at 25 ◦C,” Geochimica et Cosmochimica Acta, vol. 48,pp. 723–751, 1984. 13, 75

[13] K. S. Pitzer, “Thermodynamics of Electrolytes. I. Theoretical Basis and GeneralEquations,” Journal of Physical Chemistry, vol. 77, no. 2, pp. 268–277, 1973. 13

[14] K. H. Mistry, H. A. Hunter, and J. H. Lienhard V, “E�ect of composition andnonideal solution behavior on desalination calculations for mixed electrolytesolutions with comparison to seawater,” Desalination, vol. 318, pp. 34–47, 2013. 13

[15] A. Saul and W. Wagner, “International equations for the saturation propertiesof ordinary water substance,” Journal of Physical and Chemical Reference Data,vol. 16, pp. 893–901, 1987. 16

[16] A. Bejan, Advanced Engineering Thermodynamics. Hoboken, New Jersey: JohnWiley & Sons, Inc., 3rd ed., 2006. 21, 42, 44

[17] M. J. Moran, Availability Analysis: A guide to e�cient energy use. New York, NY:ASME Press, corrected ed., 1989. 21, 69

[18] M. H. Sharqawy, J. H. Lienhard V, and S. M. Zubair, “On exergy calculationsof seawater with applications in desalination systems,” International Journal ofThermal Sciences, vol. 50, no. 2, pp. 187–196, 2011. 27, 30

[19] K. H. Mistry, R. K. McGovern, G. P. Thiel, E. K. Summers, S. M. Zubair, and J. H.Lienhard V, “Entropy generation analysis of desalination technologies,” Entropy,vol. 13, no. 10, pp. 1829–1864, 2011. 28, 31, 56, 60

[20] K. G. Nayar, M. H. Sharqawy, L. D. Banchik, and J. H. Lienhard V, “Thermophysicalproperties of seawater: A review and new correlations that include pressuredependence,” Desalination, vol. 387, pp. 1–24, July 2016. http://web.mit.edu/seawater/. 31, 32, 54, 66, 68, 69, 71, 73

[21] R. W. Stoughton and M. H. Lietzke, “Calculation of some thermodynamicproperties of sea salt solutions at elevated temperature from data on NaClsolutions,” Journal of Chemical and Engineering Data, vol. 10, pp. 254–260, July1965. 31

[22] M. H. Sharqawy, J. H. Lienhard V, and S. M. Zubair, “Thermophysical propertiesof seawater: A review of existing correlations and data,” Desalination and WaterTreatment, vol. 16, pp. 354–380, April 2010. http://web.mit.edu/seawater. 31,38, 39, 47, 49, 54, 66, 67, 68, 70, 71

[23] K. H. Mistry, J. H. Lienhard V, and S. M. Zubair, “E�ect of entropy generationon the performance of humidi�cation-dehumidi�cation desalination cycles,”International Journal of Thermal Sciences, vol. 49, no. 9, pp. 1837–1847, 2010.33, 64

http://web.mit.edu/seawater/

http://web.mit.edu/seawater/



[24] A. Bejan, G. Tsatsaronis, and M. Moran, Themal Design & Optimization. NewYork: John Wiley & Sons, Inc., 1996. 34

[25] N. Kahraman and Y. A. Çengel, “Exergy analysis of a MSF distillation plant,”Energy Conversion and Management, vol. 46, no. 15-16, pp. 2625–2636, 2005. 35

[26] W. Wagner and A. Pruss, “The IAPWS formulation 1995 for the thermodynamicproperties of ordinary water substance for general and scienti�c use,” Journal ofPhysical and Chemical Reference Data, vol. 31, no. 2, pp. 387–535, 2002. 38, 39, 47

[27] A. Bejan, “General criterion for rating heat-exchanger performance,” InternationalJournal of Heat and Mass Transfer, vol. 21, no. 5, pp. 655–658, 1978. 42

[28] K. H. Mistry, M. A. Antar, and J. H. Lienhard V, “An improved model for multiplee�ect distillation,” Desalination and Water Treatment, vol. 51, no. 4-6, pp. 807–821,2013. 45

[29] Y. M. El-Sayed and R. S. Silver, Principles of Desalination, vol. A, ch. 2:Fundamentals of Distillation, pp. 55–109. New York, NY: Academic Press, 2nd ed.,1980. 1980. 45, 58

[30] M. Darwish, F. Al-Juwayhel, and H. K. Abdulraheim, “Multi-e�ect boiling systemsfrom an energy viewpoint,” Desalination, vol. 194, no. 1-3, pp. 22–39, 2006. 45

[31] H. T. El-Dessouky and H. M. Ettouney, Fundamentals of Salt Water Desalination.Amsterdamn, The Netherlands: Elsevier, 2002. 45

[32] K. W. Lawson and D. R. Lloyd, “Membrane distillation,” Journal of MembraneScience, vol. 124, no. 1, pp. 1–25, 1997. 49

[33] J. Rodriguez-Maroto and L. Martinez, “Bulk and measured temperatures in directcontact membrane distillation,” Journal of Membrane Science, vol. 250, no. 1-2,pp. 141–149, 2005. 49

[34] L. Martinez-Diez and F. Florido-Diaz, “Desalination of brines by membranedistillation,” Desalination, vol. 137, no. 1-3, pp. 267–273, 2001.

[35] L. Song, B. Li, K. K. Sirkar, and J. L. Gilron, “Direct contact membranedistillation-based desalination: Novel membranes, devices, larger-scale studies,and a model,” Industrial & Engineering Chemistry Research, vol. 46, no. 8,pp. 2307–2323, 2007.

[36] H. Lee, F. He, L. Song, J. Gilron, and K. K. Sirkar, “Desalination with a cascadeof cross-�ow hollow �ber membrane distillation devices integrated with a heatexchanger,” AIChE Journal, vol. 57, no. 7, pp. 1780–1795, 2011. 49

[37] V. Bui, L. Vu, and M. Nguyen, “Modelling the simultaneous heat and mass transferof direct contact membrane distillation in hollow �bre modules,” Journal ofMembrane Science, vol. 353, no. 1-2, pp. 85–93, 2010. 49


[38] J. H. Lienhard IV and J. H. Lienhard V, A Heat Transfer Textbook. Mineola, NY:Dover Publications, 4th ed., 2011. http://ahtt.mit.edu. 49

[39] H. E. Fath, S. M. Elsherbiny, A. A. Hassan, M. Rommel, M. Wieghaus,J. Koschikowski, and M. Vatansever, “PV and thermally driven small-scale,stand-alone solar desalination systems with very low maintenance needs,”Desalination, vol. 225, no. 1-3, pp. 58–69, 2008. 49

[40] F. Banat, N. Jwaied, M. Rommel, J. Koschikowski, and M. Wieghaus, “Desalinationby a “compact SMADES” autonomous solar-powered membrane distillation unit,”Desalination, vol. 217, no. 1-3, pp. 29–37, 2007. 49

[41] J. Swaminathan, H. W. Chung, D. M. Warsinger, and J. H. Lienhard V, “Simplemethod for balancing direct contact membrane distillation,” Desalination, vol. 383,pp. 53–59, 2016. 49

[42] G. P. Thiel, R. K. McGovern, S. M. Zubair, and J. H. Lienhard V, “Thermodynamicequipartition for increased second law e�ciency,” Applied Energy, vol. 118,pp. 292–299, 2014. 49, 57

[43] J. Swaminathan, H. W. Chung, D. M. Warsinger, F. A. Al-Marzooqi, H. A. Arafat,and J. H. Lienhard V, “Energy e�ciency of permeate gap and novel conductivegap membrane distillation,” Journal of Membrane Science, vol. 502, pp. 171–178,2016. 51

[44] J. M. Veza, “Mechanical vapour compression desalination plants — a case study,”Desalination, vol. 101, no. 1, pp. 1–10, 1995. 51

[45] S. E. Aly, “Gas turbine total energy vapour compression desalination system,”Energy Conversion and Management, vol. 40, no. 7, pp. 729–741, 1999. 51

[46] G. P. Thiel, E. W. Tow, L. D. Banchik, H. Chung, and J. H. Lienhard V, “Energyconsumption in desalinating produced water from shale oil and gas extraction,”Desalination, vol. 366, pp. 94–112, 2015. 52, 64

[47] A. Nafey, H. Fath, and A. Mabrouk, “Thermoeconomic design of a multi-e�ectevaporation mechanical vapor compression (MEE-MVC) desalination process,”Desalination, vol. 230, no. 1-3, pp. 1–15, 2008. 53

[48] J. Lara, G. Noyes, and M. Holtzapple, “An investigation of high operatingtemperatures in mechanical vapor-compression desalination,” Desalination,vol. 227, no. 1-3, pp. 217–232, 2008. Issue 1 First Oxford and Nottingham Waterand Membranes Research Event 2-4 July 2006, Oxford, UK. 53

[49] N. Lukic, L. Diezel, A. Fröba, and A. Leipertz, “Economical aspects of theimprovement of a mechanical vapour compression desalination plant by dropwisecondensation,” Desalination, vol. 264, no. 1-2, pp. 173–178, 2010. 53

[50] Energy Recovery Inc, “ERI power model.” Online http://www.energyrecovery.com/resource/power-model/, 2010. San Leandro, CA. 53, 54

http://ahtt.mit.edu

http://www.energyrecovery.com/resource/power-model/

http://www.energyrecovery.com/resource/power-model/


[51] M. Elimelech and W. A. Phillip, “The future of seawater desalination: Energy,technology, and the environment,” Science, vol. 333, no. 6043, pp. 712–717, 2011.57

[52] Desalitech Ltd, “Doing away with RO energy-recovery devices,” Desalination &Water Reuse, vol. 20, no. 2, pp. 26–28, 2010. 57

[53] D. M. Warsinger, E. W. Tow, K. G. Nayar, L. A. Maswadeh, and J. H. Lienhard V,“Energy e�ciency of batch and semi-batch reverse osmosis desalination,” WaterResearch, 2016. under review. 57

[54] K. H. Mistry and J. H. Lienhard V, “Generalized least energy of separation fordesalination and other chemical separation processes,” Entropy, vol. 15, no. 6,pp. 2046–2080, 2013. 58

[55] M. J. Moran and H. N. Shapiro, Fundamentals of Engineering Thermodynamics.New Jersey: John Wiley & Sons, Inc., 6th ed., 2007. 61

[56] Global Water Intelligence, “Desaldata.com.” Online http://desaldata.com,2013. 61

[57] J. E. Miller, “Review of water resources and desalination technologies,” Tech. Rep.SAND 2003-0800, Sandia National Laboratories, Livermore, CA, March 2003.

[58] G. J. Crisp, “Actual energy consumption and water cost for the swro systems atperth,” in Desalination: An Energy Solution, no. IDA_HB2010-Crisp, (HuntingtonBeach, CA, USA), International Desalination Association, November 2–3, 2010.

[59] R. L. Stover, “Isobaric energy recovery technology — history and futureopportunities,” in Desalination: An Energy Solution, no. IDA_HB2010-Stover,(Huntington Beach, CA, USA), International Desalination Association, November2–3, 2010.

[60] R. L. Stover, “Seawater reverse osmosis with isobaric energy recovery devices,”Desalination, vol. 203, no. 1–3, pp. 168–175, 2007. EuroMed 2006: Conference onDesalination Strategies in South Mediterranean Countries.

[61] C. Fritzmann, J. Löwenberg, T. Wintgens, and T. Melin, “State-of-the-art of reverseosmosis desalination,” Desalination, vol. 216, no. 1âŁ“3, pp. 1–76, 2007. 61

[62] E. W. Tow, R. K. McGovern, and J. H. Lienhard V, “Raising forward osmosis brineconcentration e�ciency through �ow rate optimization,” Desalination, vol. 366,pp. 71–79, 2015. 63, 64, 65

[63] R. K. McGovern and J. H. Lienhard V, “On the potential of forward osmosis toenergetically outperform reverse osmosis desalination,” Journal of MembraneScience, vol. 469, pp. 245–250, 2014. 64

[64] R. K. McGovern, G. P. Thiel, G. P. Narayan, S. M. Zubair, and J. H. Lienhard V,“Performance limits of single and dual stage humidi�cation dehumidi�cationdesalination systems,” Applied Energy, vol. 102, pp. 1081–1090, Feb 2013. 64

http://desaldata.com


[65] G. P. Narayan, K. M. Chehayeb, R. K. McGovern, G. P. Thiel, S. M. Zubair, and J. H.Lienhard V, “Thermodynamic balancing of the humidi�cation dehumidi�cationdesalination system by mass extraction and injection,” International Journal ofHeat and Mass Transfer, vol. 57, pp. 756–770, February 2013.

[66] F. Al-Sulaiman, G. P. Narayan, and J. H. Lienhard V, “Exergyanalysis of a high-temperature-steam-driven, varied-pressure,humidi�cation-dehumidi�cation system coupled with reverse osmosis,”Applied Energy, vol. 103, pp. 552–561, March 2013.

[67] K. M. Chehayeb, G. P. Narayan, S. M. Zubair, and J. H. LienhardV, “Thermodynamic balancing of a �xed-size two-stage humidi�cationdehumidi�cation desalination system,” Desalination, vol. 369, pp. 125–139, 2015.64

[68] D. M. Warsinger, K. H. Mistry, K. G. Nayar, H. W. Chung, and J. H. Lienhard V,“Entropy generation of desalination powered by variable temperature waste heat,”Entropy, vol. 17, pp. 7530–7566, October 2015. 64

[69] R. K. McGovern, S. M. Zubair, and J. H. Lienhard V, “The bene�ts of hybridizingelectrodialysis with reverse osmosis,” Journal of Membrane Science, vol. 469,pp. 326–335, November 2014. 64

[70] J. Swaminathan, K. G. Nayar, and J. H. Lienhard V, “Mechanical vaporcompression-membrane distillation hybrids for reduced speci�c energyconsumption,” Desalination and Water Treatment, vol. 57, no. 55, pp. 26507–26517,2016. 64

[71] K. H. Mistry and J. H. Lienhard V, “An economics-based second law e�ciency,”Entropy, vol. 15, pp. 2736–2765, 2013. http://www.mdpi.com/1099-4300/15/7/2736. 64

[72] F. J. Millero, R. Feistel, D. G. Wright, and T. J. McDougall, “The composition ofstandard seawater and the de�nition of the reference-composition salinity scale,”Deep Sea Research Part I: Oceanographic Research Papers, vol. 55, pp. 50–72, 2008.66

[73] “Release on the IAPWS formulation for the thermodynamic properties ofseawater,” tech. rep., International Association for the Properties of Water andSteam, 2008. 66, 67, 68

[74] J. D. Isdale and R. Morris, “Physical properties of sea water solutions: density,”Desalination, vol. 10, pp. 329–339, 1972. 66

[75] F. J. Millero and A. Poisson, “International one-atmosphere equation of stateof seawater,” Deep Sea Research Part A: Oceanographic Research Papers, vol. 28,pp. 625–629, 1981. 66

http://www.mdpi.com/1099-4300/15/7/2736

http://www.mdpi.com/1099-4300/15/7/2736


[76] “Release on the IAPWS formulation 1995 for the thermodynamic properties ofordinary water substance for general and scienti�c use,” tech. rep., InternationalAssociation for the Properties of Water and Steam, 1996. 66, 67, 69

[77] L. A. Bromley, D. Singh, P. Ray, S. Sridhar, and S. M. Read, “Thermodynamicproperties of sea salt solutions,” AIChE Journal, vol. 20, pp. 326–335, 1974. 70

[78] M. H. Sharqawy, L. D. Banchik, and J. H. Lienhard V, “E�ectiveness-mass transferunits (ε-MTU) model of an ideal pressure retarded osmosis membrane massexchanger,” Journal of Membrane Science, vol. 445, pp. 211–219, 2013. 71

[79] K. S. Pitzer, Thermodynamics. New York: McGraw-Hill, Inc., 3rd ed., 1995. 71

[80] D. T. Jamieson, J. S. Tudhope, R. Morris, and G. Cartwright, “Physical propertiesof sea water solutions: heat capacity,” Desalination, vol. 7, pp. 23–30, 1969. 72

[81] M. Uematsu and E. U. Franck, “Static dielectric constant of water and steam,”Journal of Physical and Chemical Reference Data, vol. 9, no. 4, pp. 1291–1306, 1980.74

[82] H.-T. Kim and W. J. Frederick, “Evaluation of Pitzer Ion Interaction Parameters ofAqueous Electrolyte Solutions at 25 ◦C. 2. Ternary Mixing Parameters,” Journalof Chemical Engineering Data, vol. 33, pp. 278–283, 1988. 75

[83] R. T. Pabalan and K. S. Pitzer, “Thermodynamics of concentrated electrolytemixtures and the prediction of mineral solubilities to high temperatures formixtures in the system Na-K-Mg-Cl-SO4-OH-H2O,” Geochimica et CosmochimicaActa, vol. 51, pp. 2429–2443, 1987. 75

[84] F. J. Millero and D. Pierrot, “A chemical equilibrium model for natural waters,”Aquatic Geochemistry, vol. 4, pp. 153–199, 1998. 75

[85] L. F. Silvester and K. S. Pitzer, “Thermodynamics of Electrolytes X. Enthalpyand the e�ect of Temperature on the Activity Coe�cients,” Journal of SolutionChemistry, vol. 7, no. 5, pp. 327–337, 1978. 75

[86] L. F. Silvester and K. S. Pitzer, “Thermodynamics of Electrolytes. 8.High-Temperature Properties, Including Enthalpy and Heat Capacity, withApplication to Sodium Chloride,” The Journal of Physical Chemistry, vol. 81,no. 19, pp. 1822–1828, 1977. 75

[87] M. P. de Lima and K. S. Pitzer, “Thermodynamics of saturated electrolyte mixturesof NaCl with Na2SO4 and with MgCl2,” Journal of Solution Chemistry, vol. 12,no. 3, pp. 187–199, 1983. 75

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