Exergy analysis of water purification and desalination: A study of exergy model approaches

Desalination 359 (2015) 212–224

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Desalination journal homepage: www.elsevier.com/locate/desal

Exergy analysis of water purification and desalination: A study of exergy model approaches Lorna Fitzsimons a,⁎, Brian Corcoran a, Paul Young a, Greg Foley b a b

School of Mechanical and Manufacturing Engineering, Dublin City University, Glasnevin, Dublin 9, Ireland School of Biotechnology, Dublin City University, Glasnevin, Dublin 9, Ireland

H I G H L I G H T S • • • • •

Several desalination exergy model approaches are examined, both theoretically and practically. The assumptions and limitations of these models are assessed. The models are compared using a desalination plant dataset. Significant differences were found in the results of the exergy analyses obtained. Certain models may not be suitable for desalination plant exergy analyses.

a r t i c l e

i n f o

Article history: Received 6 October 2014 Received in revised form 21 November 2014 Accepted 21 December 2014 Available online xxxx Keywords: Exergy analysis Desalination Water purification Energy efficiency Membranes

a b s t r a c t In the literature, several exergy analysis approaches have been proposed to investigate desalination processes. It is not clear, however, which approach is the most appropriate or indeed whether all approaches are valid. The objective of this paper is to review the various methods and to critically assess their assumptions, limitations, advantages and disadvantages. The main focus of this work is the chemical exergy term. Several exergy calculation models were examined and compared using a dataset from the literature. In addition, an accurate approach to calculate the chemical exergy of electrolyte solutions, based on the Pitzer equations, was proposed. The models assessed were: (1) the ideal mixture model (NaCl and water), (2) the ideal mixture model (seawater salt and water), (3) the Sharqawy seawater functions, and (4) the electrolyte solution model (Pitzer equations, NaCl and water), (5) the model used by Drioli et al. and (6) the dissociated ion approach (NaCl and water). Four of the six approaches produced very similar results. Moreover, one other exergy calculation method was found to have serious limitations. The findings presented here show that the choice of exergy model can have a significant impact on the results obtained and that considerable care must be taken to select the most suitable approach. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Water and energy are inextricably linked; energy is required to treat water and water is required to source and convert primary energy. The water–energy nexus is receiving due consideration as pressures on both energy and water resources increase. According to Olsson [1], water and energy systems and operations should be planned together. The close association is very true of water treatment processes, which are becoming increasingly necessary to meet various water demands including potable, industrial and agricultural water requirements. Water purification technologies are manifold, for example, multi-stage flash distillation (MSF), multi-effect distillation (MED) and reverse osmosis (RO), and the application of these technologies ranges from large seawater ⁎ Corresponding author. E-mail addresses: Lorna.fi[email protected] (L. Fitzsimons), [email protected] (B. Corcoran), [email protected] (P. Young), [email protected] (G. Foley).

http://dx.doi.org/10.1016/j.desal.2014.12.033 0011-9164/© 2014 Elsevier B.V. All rights reserved.

desalination processes to ultra-pure water (UPW) applications in the pharmaceutical, semiconductor or power generation industries. The energy requirements vary significantly between applications, and from process to process. For example, high-end applications such as UPW have a greater energy footprint than potable water treatment, and, in general, thermal processes such as distillation are more energy intensive than membrane processes such as reverse osmosis or nanofiltration (NF). Table 1 collates data from several sources in the literature and illustrates the specific operating energy requirements for various purification processes. It is evident that the MSF and MED technologies consume significantly more specific energy than the seawater reverse osmosis process. According to Table 1, however, there is also a large deviation between the highest and lowest values of specific energy for the MSF and MED processes, which may signify potential scope for improvement. The energy requirements for high purity applications such as semiconductor UPW vary between 9.55 and 10.24 kWh/m3 respectively [2,3].

L. Fitzsimons et al. / Desalination 359 (2015) 212–224 Table 1 A comparison of water purification process specific operating energy requirements. Water purification technology

Energy (kWh/m3)

Reference

Brackish water RO (core process) Seawater RO with energy recovery (core process) Seawater RO (all auxiliary requirements) MSF MSF (all auxiliary requirements) MED MED (all auxiliary requirements) Ultra-pure water RO (all auxiliary requirements)

1 2.2 to 2.7 5 to 7 16 to 20 38.5 to 125 14 32 to 122.5 9.55 to 10.24

[4] [4,5] [5,6] [7,8] [6] [8] [4] [2,3]

Energy research in desalination processes is well-established and has contributed to lowering the water treatment energy footprint. In relation to RO processes, the energy footprint had dropped from approximately 20 kWh/m3 in the 1970s to a value of less than 2 kWh/m3 by 2004 [9]. According to reports, a more recent value of 1.58 kWh/m3 has been achieved under ideal conditions (new membrane, low water flux at 42% recovery) [10]. It should be stated, however, that this low specific energy value was obtained at the expense of permeate quality. Several factors have contributed to these significant achievements including improved membranes [9,11]; pump and motor efficiency improvements [9,12] and the use of variable speed drives (VSD); studies in RO system optimisation [13–18]; and the implementation of energy recovery devices such as pressure exchangers, and Pelton and Francis turbines to harness wasted throttling valve energy [19,20]. One approach that has been widely accepted as a useful analytical tool for the characterisation and optimisation of water purification and desalination systems is exergy analysis. However, researchers in this area have used a variety of exergy model approaches to determine the exergy of aqueous solutions such as seawater and brackish water. Primarily, the various approaches differ in three ways: 1. The dead state definition 2. The modelling of the aqueous solution 3. The exergy calculation equation. Regarding the dead state definition, the majority of researchers define the dead state as the ambient temperature, pressure and salinity of the seawater/brackish water in question. In contrast, other research groups have defined the dead state as that of pure water [5,21,22]. The impact of dead state selection has been investigated by several researchers [23,24]. In relation to the thermodynamic chemical exergy model of seawater and other relevant aqueous solutions, various assumptions have included both ideal behaviour [25–29] and non-ideal mixture behaviour [30–33], and even pure water [34]. Where ideal mixture behaviour has not been assumed, activity coefficient calculation models such as the Debye–Huckel limiting equation have been proposed and used [33,35,36]. Finally, the equations used to calculate the exergy rates differ significantly between research groups. The majority of researchers [5,26,30,31,33,35,37,38] split the exergy rate calculation of aqueous solutions into physical and chemical exergy terms, whereas others [25,39] couple the physical and chemical exergy rates implicitly. In addition, some researchers have used thermodynamic properties of seawater [38,40] to calculate the exergy rates. To date, limited comparisons between the various approaches have been presented in the literature. Sharqawy et al. [38] found differences of up to 80% between the exergetic efficiency values calculated for a seawater MSF desalination plant using updated thermodynamic properties of seawater in comparison with the exergy calculation model proposed by Cerci et al. [25,39]. Other researchers compared the Cerci and Drioli approaches and found differences of up to 30% between the exergy destruction rates calculated. In that case, the major differences between the two approaches occurred in the key separation technologies, indicating that the source of the difference was the calculation of chemical exergy [24]. Sharqawy [38] showed mathematically that the flow exergy of a binary ideal mixture was always positive; however,

213

Kahraman et al. [25], while purporting to use an ideal mixture model, reported negative values of flow exergy. Furthermore, as noted by many authors, seawater and other electrolyte solutions are not ideal mixtures. It is evident that it is not a trivial exercise to tease out the different approaches. With this in mind, the objectives of this work are: (1) to examine and assess the current exergy calculation models for desalination exergy analyses; (2) to propose a general approach applicable to electrolyte solutions, including an assessment of activity calculation methods; and (3) to compare and assess the impact of the various approaches on the exergy analysis of a desalination plant using a dataset from the literature. 2. Exergy analyses of desalination/water purification plants Exergy analysis considers energy in terms of both quantity (First Law of Thermodynamics) and quality (Second Law of Thermodynamics). Generally, in desalination analyses, the thermodynamic property exergy is broken down into physical and chemical exergy contributions. One key exception is the approach proposed by Cerci [39,41], where the physical exergy and chemical exergy are integrated, i.e. the chemical/ concentration exergy is implicitly included in the entropy of mixing differences. Velocity and elevation contributions are included if applicable, but this is not generally the case for desalination plants. Practically, water purification technologies such as MSF or RO plants are typically modelled as a series of processes, where each process is modelled as a control volume in steady state. The exergy rates are calculated at the relevant process stages and then an exergy balance is used to determine: (1) the exergy rates, the exergy destruction rates and the key sites of exergy destruction; and (2) the exergetic efficiency of individual process components and the overall process. The exergy balance identifies and quantifies the main sources of thermodynamic irreversibilities in the sequence of processes. However, unlike entropy balances, which are a function of the system alone, exergy balances are a function of both the system and the defined dead state. Therefore, the choice of dead state is important to gain insight into the availability at each process stage; this is particularly important in the assessment of waste streams which are rejected to, and mix with, the environment. Many researchers have applied exergy analyses to characterise desalination/water purification processes, and several exergy models have been utilised, see Table 2. Of the models presented, the first is based on the specific thermodynamic properties of seawater; others are based on more general modelling approaches that model seawater or other aqueous solutions as a combination of pure water and salts. Until recently, however, exergy analyses have been carried out in relative isolation. The various models in Table 2 have been applied by different research groups, but the results obtained using these models have not been assessed comprehensively, or more importantly, the limitations or validity of these models has not been examined. The choice of exergy models proposed in the literature poses a significant challenge to potential exergy researchers: do these models give similar results? Or, if not, which is the most appropriate exergy model to use for desalination/water purification purposes? The electrolyte solution model presented in Table 2 has been previously used in the literature for both aqueous solutions and desalination exergy analyses [31,33,35]. However, according to Spiegler and ElSayed, the use of the electrolyte solution model is problematic because “most of the activities of salt species are either unknown, uncertain or difficult to evaluate [31]”. Consequently, not only does this research assess and compare the various modelling approaches, but it also evaluates the important considerations associated with modelling electrolyte solutions. 3. Desalination/water purification exergy models When a system is in physical and chemical equilibrium with the dead state the opportunity to do work no longer exists and the exergy

214

L. Fitzsimons et al. / Desalination 359 (2015) 212–224

Table 2 Exergy models used in desalination/water purification applications. Relevant exergy or specific exergy equation variations

Source

1) Exergy calculations based on • Leyendekker, thermodynamics of seawater, part 1, 1976 (Romero-Ternero et al.) • Exergy calculations based on updated seawater thermodynamic properties (Sharqawy et al.) 2) Model used by Drioli et al.

[38,40]

C
 1000−∑ i T P−P 0 ρ e ¼ c T−T 0 −T 0 ln −N solution RT 0 ln xsolution where N solution ¼ and xsolution ¼ þ T0 MW solution ρ

[5,26,42]

N solution N solution þ ∑

βi C i ρMW i

3) Model proposed by Cerci and used by others (and variations of the following equation)

[25,27,29,39,41,43]

e = h − h0 − T0(s − s0) where h = wshs + wwhw and s = wsss + wwsw − Rim(xs ln xs + xw ln xw) [32,33,35]

4) Electrolyte solution model
 T a þ vðP−P 0 Þ þ ∑xi ðμ−μ i0 Þ where the chemical energy term ∑xi ðμ−μ i0 Þ ¼ RT 0 ∑xi ln i e ¼ c T−T 0 −T 0 ln T0 ai0

The activity values for the solution species can subsequently be calculated using a number of approaches 5) Spiegler and El-Sayed [31] Seawater desalination model for both finite and infinitesimal product water recoveries, including the chemical exergy of the concentrated brine for finite recovery given below — an adaptation of the electrolyte solution model.  e ¼ h−h0 −T 0 ðs−s0 Þ þ RT 0

ϕxs xs;0 xs ln −ϕxs;0 xs −xs;0 xs;0



of the system is zero. Generally, for desalination or water purification exergy analyses, the calculation of exergy is broken down into physical exergy (thermal, pressure, velocity and elevation) and chemical exergy thus facilitating greater understanding of plant exergy flows. Physical exergy relates to differences between the thermodynamic state under consideration and the defined, thermomechanical dead state, or what has been termed the restricted dead state (RDS) in the literature [44]. Simply put, physical exergy relates to differences in temperature, pressure, velocity and elevation between the thermodynamic state under consideration and the dead state at constant composition. The physical exergy of a solution ĖPh (kW) can be calculated using Eq. (1),





EPh ¼m ½h−h0 −T 0 ðs−s0 Þ

ð1Þ

where ṁ is the mass flow rate (kg/s); h and s are the specific enthalpy and entropy of the solution at the thermodynamic state under consideration (units: kJ/kg and kJ/kg·K respectively); the subscript 0 denotes the dead state temperature and pressure, and thus h0 and s0 are the enthalpy and entropy of the solution at the restricted dead state (units: kJ/kg and kJ/kg·K respectively) and T0 is the dead state temperature (K). Chemical exergy relates to differences between the thermomechanical dead state and the dead state. Regarding desalination and water purification analyses, it relates to differences in concentration at dead state temperature and pressure. Thus, the chemical exergy can be calculated as the difference in chemical potential between the thermodynamic state under consideration (at dead state temperature and pressure) and the dead state. Chemical potential is a thermodynamic property and can be derived from the total differential of the Gibbs function at constant temperature and pressure G = G(T, P, N1, N2…Ni), see Eq. (2),

 X
∂G  ∂G ∂G dT þ dP þ dN dG ¼ ∂T P;N j ∂P T;N j ∂Ni P;T;Nði≠ jÞ i

indicates that the number of moles of all other species except i remain constant. At constant temperature and pressure, the partial derivatives in the third term of Eq. (2) are defined as the chemical potential of the various solution species and are given the symbol
 ∂G . μi, i.e. μ i ¼ ∂Ni P;T;Nði≠ jÞ The differences in chemical potential can be calculated for ideal mixtures as follows,   x μ i ðT 0 ; P 0 ; xi Þ−μ i T 0 ; P 0 ; xi;0 ¼ RT 0 ln i xi;0 and for non-ideal mixtures as,   a μ i ðT 0 ; P 0 ; xi Þ−μ i T 0 ; P 0 ; xi;0 ¼ RT 0 ln i ai;0

where G is the Gibbs energy; T is the absolute temperature (K); P is the absolute pressure (Pa); N is the number of moles (mol); Nj denotes that the composition of the solution remains constant, the subscript i denotes each of the relevant species under consideration and N(i ≠ j)

ð4Þ

where μi is the chemical potential of species i (kJ/kmol); R is the universal gas constant (kJ/kmol·K); x denotes the mole fraction of the species; and a is the activity of the species under consideration. The asymmetrical nature of the activity coefficient calculation for the solvent and the solutes is discussed in depth elsewhere; see the following references [45,46]. Essentially, the activity of the ionic species in electrolyte solutions is calculated as the product of the molality (moles of solute per kilogramme of water) and the Henryan activity coefficient γH, i.e. ai = miγHi. The Pitzer equations, discussed in depth later, can be used to calculate the Henryan activity coefficient and the activity of water in electrolyte solutions. Using thermodynamic property relations for practical plant analyses, the physical exergy and chemical exergy can be combined into Eq. (5),   X mi γH i ðP−P 0 Þ T þ Ni RT 0 ln þ E ¼ m cv ðT−T 0 Þ−cp T 0 ln mi;0 γ H i;0 ρ T0 i a þ Nw RT 0 ln w aw;0


ð2Þ

ð3Þ










ð5Þ

where Ṅi and Ṅw denote the molar flow rates of the solutes and water respectively (kmol/s); cv is the specific heat capacity at constant volume and cP is the specific heat capacity at constant pressure (kJ/kg·K); ρ is

L. Fitzsimons et al. / Desalination 359 (2015) 212–224

the density of the solution (kg/m3). The approach using Eq. (5) is well established. The chemical exergy is a function of the natural logarithm of the ratio of the activities at dead state temperature and pressure. Specific exergy (kJ/kg) can be calculated using Eq. (6), T ðP−P 0 Þ þ e ¼ cv ðT−T 0 Þ−cp T 0 ln T ρ 0 " # X mi γH i a xi ln þ xw ln w þ Rsolution T 0 mi;0 γH i;0 aw;0 i

ð6Þ

where in addition to the Eq. (5) terms, xi and xw refer to the mole fractions of species i and water respectively; and Rsolution denotes the specific gas constant of the solution and is given by Rsolution ¼
 R= ∑ xi MW i þ xw MW w . i

Variations of Eqs. (1), (3)–(6) relate directly to approaches 1, 4 and 5 in Table 2. However, the exergy calculation models of Cerci and Drioli are quite different. Furthermore, several approaches to calculate the mole fractions and the activity of the various seawater salts have been proposed in the literature. 3.1. Model used by Drioli et al. The exergy calculation models seawater as a solution of various ionic species and calculates the exergy rates at the process stages under consideration using Eqs. (7) to (9). 
  T P−P 0 þ −Nsolvent RT 0 ln xsolvent E ¼ m cðT−T 0 Þ−cT 0 ln ρ T0





where Nsolvent


XC  i 1000− ρ ¼ MW solvent

Nsolvent and xsolvent ¼  X
β C  : i i Nsolvent þ ρMW i

ð7Þ

ð8Þ

ð9Þ

The first three terms within the brackets in Eq. (7) relate to thermal and pressure exergy and the final term relates to chemical exergy. The constant specific heat capacity and the incompressible fluid model are assumed. According to the relevant literature (and where feasible using the symbols proposed by the relevant authors [5,26]), Ė is the exergy rate (kJ/h); ṁ is the mass flow rate (kg/h); c is the specific heat capacity (kJ/kg·K); T is the absolute temperature (K) at the process stage under consideration; T0 is the absolute temperature (K) at the dead state; P and P0 refer to the pressures (kPa) at the process stage and the dead state respectively; ρ is the density of the solution (kg/l); Nsolvent is the number of moles of the solvent per kilogramme of the solution (molsolvent/kgsolution); R is the universal gas constant (kJ/mol·K); xsolvent is the mole fraction of the solvent (dimensionless); Ci is the concentration in unit mass per litre (gsolute/lsolution) of solute species i; MWsolvent is the molar mass of the solvent (gsolvent/molsolvent), pure water in this case; βi is the number of particles of solute species i generated on dissociation and MWi is the molar mass (gi/moli) of solute species i. The use of Eq. (8) to calculate the number of moles of the solvent per unit mass of solution requires further consideration. In order for the N solvent term to have the dimensions defined by the relevant authors, i.e. mol solvent/kg solution, the numerator of the term on the right of Eq. (8) must equal the mass fraction of the solvent divided by 1000, see Eq. (10) where w solvent denotes the mass fraction of the solvent.

Nsolvent MW solvent

molsolvent g g w ¼  solvent ¼ solvent ¼ solvent kgsolution molsolvent kgsolution 1000

On examination of the units of the ∑ Ci/ρ term in Eq. (8) it is clear that this term calculates the sum of the solute mass fractions in the solution. For dimensional accuracy, the units should be the same for the concentration of the salt species and the density of the solution. However, they are not and this may lead to confusion. It might be clearer to rewrite Eq. (8), based on a more consistent dimensional analysis, as follows in Eq. (8a).
X  Ci Þðρ; C i in kg=l 1− ρ Nsolvent ðkmolsolvent =kgsolution Þ ¼ MW solvent ðkgsolvent =kmolsolvent Þ

ð8aÞ

The chemical exergy is calculated as a function of the mole fraction of pure water in the solution. This will always be a negative value except in the case of pure water where the value is zero. The negative sign of the last term ensures that the final chemical exergy term is always positive for a solution. This could be seen as an advantage in that the flow exergy is always positive, however, two important questions remain: 1) whether this is an appropriate dead state definition for the purpose at hand, and 2) whether this is an appropriate equation to calculate the chemical exergy of a solution at various process stages. The first question becomes pertinent when one considers the exergy losses (i.e. external losses to the environment) for desalination or water purification plants. The definition of exergy is the maximum theoretical work that a system at a specific thermodynamic state could do as it comes into equilibrium with its environmental dead state. It would be fair to say that the natural environment of a desalination plant is not generally a reservoir of pure water and therefore one could argue that this does not represent the chemical exergy for the applications under consideration. Regarding the second question, the chemical exergy term is in fact the equation used to calculate the chemical exergy of pure water obtained from desalination plants in the Szargut reference environment model [47]. In the literature the equation used by Drioli et al. to calculate the chemical exergy has been shown to be a special case based on certain, very specific assumptions: the separation of 1 mol of pure water from a large quantity of a mixture, and the underlying assumption that the remaining mixture does not change in composition as a result of this separation [48]. Therefore, one could further argue that there is an implicit assumption that the brine has the same mole fraction of water as the incoming seawater, and consequently, there should be no change in chemical exergy. It does not follow that the model should be used to calculate the chemical exergy at each process stage of a desalination plant, including the chemical exergy of the brine, using an equation, which is based on the premise that practically no change in the mixture takes place following the separation. The preceding analysis presents a potential flaw with this approach. On a practical level, when compared with the Cerci model using data for a reverse osmosis desalination plant taken from the literature, the approaches used by Cerci and Drioli were seen to differ by up to 30% for the key separation processes [24]. 3.2. The Cerci model The Cerci exergy model equations, which model seawater as an ideal solution of solid NaCl and water have been adapted from the literature and are shown in Eqs. (11) to (13). 2

3 ½ðws hs Þ þ ðww hw Þ−½ðws hs Þ þ ðww hw Þ 0 5 E ¼ m 4 −T 0 ½ws ss þ ww sw −½ws ss þ ww sw 0 ð11Þ þT 0 ½Rim ðxs ln xs þ xw ln xw Þ−½Rim ðxs ln xs þ xw ln xw Þ0





where xs ¼

MW w  1 MW s −1 þ MW w ws 

MW s  : 1 −1 þ MW s ww



and xw ¼ ð10Þ

215

MW w

ð12Þ

ð13Þ

216

L. Fitzsimons et al. / Desalination 359 (2015) 212–224

In the preceding equations h is the specific enthalpy (kJ/kg) and s is the specific entropy (kJ/kg·K) at the relevant process stages; Rim is the specific gas constant of the ideal mixture (kJ/kg·K); w and x are the mass and mole fractions of the relevant species respectively; the subscript 0 refers to the dead state; MW is the molar mass of the species under consideration (kg/kmol); and the subscripts s and w refer to salt and water respectively. The Cerci model assumes ideal mixture behaviour on the basis that seawater is a solution dilute with an approximate salt mass fraction or salinity of 3.5% [39]. This assumption is not valid because it is widely known that electrolyte solutions such as seawater do not behave ideally. Researchers have commented on the negative exergy rates obtained using this approach [24,38,39], where it was found that negative values occurred when the salinity at various process stages was greater than the defined dead state salinity. The positive dimensionless exergy and dimensionless flow exergy results presented by Sharqawy [38] for various ideal mixture mole fractions do not concur with the negative exergy rates reported by Cerci despite the fact that both the calculation of dimensionless flow exergy and the Cerci approach consider an ideal mixture model. This would suggest that it is not solely the assumption of ideal mixture mole fractions to calculate chemical exergy that leads to these reported negative exergy results. Sharqawy et al. [38] suggest that Cerci [39] neglects chemical exergy in his proposed method. This is not strictly true because Cerci does consider the entropy of mixing, which is one of the principal bases of chemical exergy. Even if the ideal mixture chemical exergy term was added to the model proposed by Cerci the resulting equation differs from the breakdown of total exergy into its physical and chemical components. What Cerci does appear to do, however, is to calculate the chemical exergy (in his case termed the entropy of mixing) by considering changes in the ideal mixture gas constant and the mole fractions between each process stage and the dead state. So rather than making the distinction that physical exergy is calculated at constant mole fraction concentrations (i.e. the mole fraction at the relevant process stage under consideration) but at different temperatures and pressures, and chemical exergy is calculated at constant dead state temperature and pressure but at different concentrations, Cerci appears to combine these physical and chemical exergy calculations. If one considers only the changes between the natural logarithm term mole fractions, Eq. (11) simplifies to Eq. (14). 3 ½ðws hs Þ þ ðww hw Þ−½ðws hs Þ þ ðww hw Þ 0 6 −T 0 ½ws ss þ ww sw −½ws ss þ ww sw 0 7 7 6 " # E¼m 6 7 xs xw 5 4 þ xw ln þT 0 Rim xs ln xs;0 xw;0

would theoretically be more appropriate. However, putting aside the chemical exergy versus entropy of mixing discussion it remains unclear whether the use of the ideal mixture model has a significant impact on the chemical exergy rates for a typical desalination/water purification plant. 3.3. The Sharqawy method Sharqawy et al. [38] have developed a number of functions to calculate the thermodynamic properties of seawater using updated 2008 IAPWS seawater properties (http://web.mit.edu/seawater/), including MATLAB files to calculate the chemical potential of seawater salts and water. These are very user-friendly tools and are accurate for a wide range of salinity and temperature values. The approach may not be appropriate for more general water purification exergy analyses (for example, brackish water or other electrolyte solutions) because the thermodynamic properties are specific to seawater. 3.4. The electrolyte solution method (mole fraction and activity calculation) The mole fraction of completely dissociated electrolytes under ambient conditions is calculated in a different way than the usual method of calculating the mole fractions of components in mixtures [46,50]. Generally, the mole fractions of components in a binary solution are calculated using Eqs. (15) and (16). In these equations, by way of example, the mixture consists of sodium chloride and water. xNaCl ¼

xw ¼




ð14Þ

It transpires that the above equation is almost identical to that discussed previously for ideal and non-ideal mixtures; see Eqs. (1) and (3). This approach is aligned with the Gibbs energy in Eq. (2) and the decoupling of the restricted dead state and the dead state discussed previously. (Note that the Cerci equations use the gas constant of the ideal mixture versus molar flow rate and the universal gas constant, however, these are simple conversions linked by the molar mass.) Although Sharqawy et al. [38] compared the flow exergy and the exergetic efficiency obtained using seawater thermodynamic properties with the approach proposed by Cerci [39,49], based on the preceding paragraphs, that work did not fully assess the differences between ideal behaviour and non-ideal behaviour. Where does the ideal mixture model differ from the electrolyte solution model in practical terms? First, ideal mixture behaviour assumes that the enthalpy of mixing is zero. Second, the chemical exergy can be calculated using a ratio of mole fractions as opposed to the ratio of the activities of the solutes and the solvents. Whereas Cerci considers the thermodynamic properties of solid salt and pure water, the thermodynamic properties of a non-ideal electrolyte solution

Nw NNaCl þ Nw

ð15Þ

ð16Þ

At ambient conditions, however, the sodium chloride electrolyte dissociates fully into sodium and chloride ions. Thus, in solution chemistry, the mole fractions of sodium chloride and water are calculated using Eqs. (17) and (18), where β is the number of ions generated on dissociation, i.e. two in the case of the NaCl electrolyte. xNaCl ¼

2




NNaCl NNaCl þ N w

xw ¼

βNNaCl βNNaCl þ N w

Nw βNNaCl þ Nw

ð17Þ

ð18Þ

The difference between the two approaches can be illustrated by considering a 1 m NaCl electrolyte solution, that is, 1 mol of NaCl in 1 kg of water (55.5 mol of water). In the first case, i.e. using Eq. (15), the mole fraction of NaCl is calculated to be 0.0177. Adopting the approach of Eq. (17) results in a mole fraction of 0.0348, almost twice that of the first case. The corresponding mole fraction values of water are 0.9823 and 0.9652 respectively. Pitzer termed the latter approach the “mole fraction on an ionized basis [50]”. Spiegler and El-Sayed account for this with a factor ϕ that adjusts for ionic composition [31]; the method used by Drioli et al. uses a dissociation factor β; see Table 2. The Cerci method does not suggest using a dissociation factor to calculate the mole fractions of sodium chloride. The use, or not, of a dissociation factor to calculate the mole fractions of the solution components has an impact on the calculation of chemical exergy. The activity of electrolyte solutions also requires consideration. It is well known that electrolyte solutions do not behave ideally [45,51,52], however, the simplification of ideal solution behaviour has often been made in the literature [26,27,29,39], presumably to reduce the complexity of chemical exergy calculations. Indeed, even when ideal mixture behaviour is not assumed, there are several approaches to calculate the activity of the electrolyte under consideration. Some researchers

L. Fitzsimons et al. / Desalination 359 (2015) 212–224

[33,35] have suggested using the Debye–Huckel models to calculate the activity of the solutes in the chemical exergy term to take account of the deviation of electrolyte solutions from ideal behaviour. It is important to note however, that the appropriateness of the activity calculation model is a function of the ionic strength of the solution under consideration. The ionic strength of a solution, I, is calculated using Eq. (19), where m is the molality of the ionic species and z is the valence of the ions. ð19Þ

Stumm and Morgan [53] discussed the applicability of activity calculation models as a function of the ionic strength of solutions; their assessment has been collated and is presented in Table 3. Based on the approximate ionic strength of seawater, which is 0.7 m [54,55], none of the activity calculation models in Table 3 is suitable. Nevertheless, the Debye–Huckel models have been proposed in the literature for seawater desalination exergy analyses. This is further supported by Millero, who stated that even the Debye–Huckel extended model is unsuitable and “serves as a limit in dilute solutions; however it fails at the high ionic strength of seawater [56]”. The Pitzer equations are suitable; they are specific interaction models and are reliable for the calculation of activity coefficients in various electrolyte solutions including seawater. They are reliable far beyond the ionic strength of seawater. Depending on the model used, the Pitzer equations can be used over the entire concentration range [57,58]. The equations are semi-empirical and consist of a Debye– Huckel term, which accounts for the long-range interionic effects, and several virial terms to account for short-range ionic interactions typical of electrolyte solutions. The calculation of these virial terms involves the use of several parameters including specific ion interaction terms that are fitted to measured values of electrolyte solutions. For a single electrolyte of cation M and anion X (e.g. NaCl) the activity coefficient can be calculated using Eq. (20) [59]. "

# pffiffi 2  pffiffi þ ln 1 þ b I 1þb I b ( " #) ! ð1Þ pffiffi α 2 I −αpffiI 2β 2v v ð0Þ þ m M X 2βMX þ 2MX 1− 1 þ α I− e v 2 α I " # 3m2 2ðvM vX Þ3=2 ϕ C MX þ 2 v ϕ

ln γ ¼ −jzM zX jA

pffiffi I

ð20Þ In Eq. (20) γ± is the activity coefficient of the electrolyte; z is the valence of the relevant anions and cations; m is the molality of the electrolyte; I is the ionic strength of the solution; Aϕ is a coefficient that is a function of temperature, for NaCl at 20 °C, 1 bar, and ranging in concentration from 0 to 6 m, this value is reported as 0.3882; b and α are fixed parameters, for 1–1, 2–1 and 3–1 electrolytes these values are 1.2 and 2,

Table 3 Suitability of activity calculation models as a function of ionic strength. Activity calculation model

Equation

Debye–Huckel limiting law

pffiffi log γi ¼ −Az2i I

Debye–Huckel extended model Guntelberg model Davies model

(1) ϕ respectively [59,60]; the quantities β(0) MX, βMX and CMX are empirical parameters that are specific to the electrolyte under consideration, for NaCl at 20 °C, 1 bar, and ranging in concentration from 0 to 6 m, these values are reported as 0.0714, 0.2723 and 0.002, respectively [60]. The activity of the solvent, water, can be calculated as a function of the osmotic coefficient ϕ, see Eq. (21),

ln aw ¼ −ϕ

1X 2 mi zi I¼ 2 i

log γi ¼ −Az2i log γi ¼ −Az2i log γi ¼ −Az2i

Ionic strength suitability (molality)

pffiffi I

!

pffiffi 1 þ Bα I pffiffi ! I pffiffi 1þ I ! pffiffi I pffiffi−0:3I 1þ I

I b 10−2.3 (≈0.005 m) I b 0.1 m I b 0.1 m I b 0.5 m

217

vm 55:51

ð21Þ

where v is the number of ions generated on dissociation of the electrolyte; m is the molality of the electrolyte. The osmotic coefficient ϕ can be calculated using Eq. (22), where the parameters are identical to Eq. (21). ϕ

pffiffi I

pffiffi 1þb I pffi i 2vM vX h ð0Þ ð1Þ −α I þm βMX þ βMX e "v # 3=2 2 2ðvM vX Þ ϕ C MX þm v

ϕ−1 ¼ −jzM zX jA

ð22Þ

The various approaches are now considered in terms of practical plant analyses. 4. Plant exergy analyses The prevalent approaches in the literature for desalination exergy analyses have been presented and assessed. The calculation of mole fractions for ionic species in electrolyte solutions has been discussed. In addition, an appropriate approach for calculating the activity coefficients, and subsequently, the chemical exergy of electrolyte solutions of varying ionic strengths has been presented. The key question is how do these approaches compare when undertaking an exergy analysis. In practical terms, does it matter which model is selected? This study will be carried out using a dataset from the literature, the same as that originally presented by Kahraman and Cengel [49] and used by Sharqawy et al. [38] to undertake the aforementioned comparative assessment. In this case, however, the following approaches will be compared: 1. Ideal mixture model (NaCl and water) 2. Ideal mixture model (seawater salt and water — using a molar mass of 31.4 kg/kmol for seawater salt) 3. Dissociated mole fraction (NaCl and water) 4. Model used by Drioli et al. 5. Electrolyte solution model using the Pitzer equations (NaCl and water) 6. Thermodynamic properties of seawater (using the MATLAB seawater thermodynamic function files developed in MIT http://web.mit.edu/ seawater/). As the main focus is on the calculation of the chemical exergy term, the physical exergy is calculated in the same manner for each of the above, i.e. using the relevant MATLAB seawater functions for specific internal energy, density, specific entropy, specific volume and the reported desalination plant pressures and temperatures. The dead state is defined as ambient temperature and pressure, and incoming seawater salinity (308 K, 1.013 bar, 4.65% respectively). At this temperature and pressure the Pitzer parameters required to calculate the activity and osmotic coefficients are: Aϕ = 0.3982; b and α are 1.2 (1) ϕ and 2 respectively [59,60]; the parameters β(0) MX, βMX and CMX are 0.3982, 0.0815, 0.2846 and 0.0006 respectively. These values were obtained by fitting curves to data provided in [60]. The parameter CϕMX proved the least satisfactory to model because the reported values changed significantly between 30 °C and 40 °C, i.e. the tabulated value 103CNaCl (the ϕ is obtained, changes from 0.44 kg/mol relevant value from which CMX at 30 °C to − 0.02 kg/mol at 40 °C). (It turned out subsequently that

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L. Fitzsimons et al. / Desalination 359 (2015) 212–224

ϕ CMX could be omitted in the activity coefficient calculations without significant error, maximum error b 0.15%, at the relevant seawater concentrations.) The plant schematic is shown in Fig. 1 and the plant operating data is presented in Table 4. One of the issues with using the ideal mixture and electrolyte activity calculations is the fact that the natural logarithm terms are not defined for zero salinity. In the majority of water treatment applications zero salinity is unusual, even the highest grade UPW may contain trace ions. Conceptually, in thermodynamic mixing terms, a solution is not a solution if it is a pure substance, however, that does not mean that pure water cannot perform work by mixing with a more concentrated solution (chemical potential and osmosis). To combat the issue of zero salinity at the seawater separation processes a very low salinity value (0.01%) was chosen in place of zero salinity. For the purposes of this comparison the chemical exergy of process stages 11 and 12, the brine heater, is deemed not applicable because no change in composition takes place and it is assumed that process water rather than seawater is used. The physical exergy at these locations was calculated using steam tables. The mole fractions, activity coefficients and activities of salts and water at the various plant locations are shown in Table 5. Note the relatively high ionic strength of seawater at the dead state, 0.834 m, when compared with the typical seawater ionic strength of 0.7 m. Seawater in the dataset has high salinity (salinity 4.65%). When a typical salinity value of 3.5% was used the ionic strength dropped to 0.62 m, indicative of the simplified monovalent sodium chloride electrolyte. The value calculated for the activity of water is very similar to the ideal mixture mole fractions of water when calculated using the molecular weight of seawater salt, 0.972 versus 0.973 respectively. The activity of the solute changes significantly at the various process stage concentrations. In contrast, the activity of water differs by a maximum percentage difference of 4.3% between the highest and lowest salt concentrations. The total specific physical exergy values were calculated using the MATLAB functions; the specific chemical exergy values were calculating using the MATLAB seawater chemical potential functions. In addition,

the chemical exergy values were calculated using the ideal mixture model (NaCl and water); the ideal mixture model (seawater salt and water); the model used by Drioli et al.; the dissociated mole fraction approach (NaCl and water); and the proposed electrolyte solution model (Pitzer equations; NaCl and water). The results are shown in Table 6. A comparison of the specific chemical exergy and specific total exergy values using the various approaches are plotted for the significant process streams in Figs. 2–4 respectively. Note that Figs. 3 and 4 exclude the approach used by Drioli et al. for the purpose of clarity. The specific chemical exergy values calculated using the approach used by Drioli et al. are very different to the other model values. One major reason for this is the different dead state adapted in this approach (pure water). Fig. 2 shows that, contrary to the other models, the specific chemical exergy is significant at the incoming seawater salinity and increases in magnitude as the concentration of the process streams increase. At process streams 5, 13 and 16 the specific chemical exergy for the other models is significant when compared with the approach used by Drioli et al., which is at a minimum value due to the low salt concentration in the streams. When added to the physical exergy values, it is a clear that there will be a significant difference in the proportionality of the specific chemical exergy to the specific exergy in comparison to the other approaches. In reality, it makes little sense to directly compare models with different dead states, at least in terms of specific exergy values. Where one should be able to compare the models more appropriately is in the calculation of exergy destruction and/or exergetic efficiency, and this will be assessed later. Figs. 3 and 4 show that the approaches assessed here result in very similar values of both specific exergy and specific chemical exergy, with the exception of the ideal mixture model (NaCl and water). Table 7 reports the percentage differences in specific chemical exergy calculated using the various approaches relative to the Sharqawy seawater functions. Negative values of exergy have been reported in the literature previously [25,39,40]. The negative values of exergy reported by Kahraman and Cengel [49], using the Cerci approach, have been discussed by

Fig. 1. Desalination plant schematic [49].

L. Fitzsimons et al. / Desalination 359 (2015) 212–224 Table 4 Desalination plant data.a Adapted from [49]. Process stream

P (kPa)

T (°C)

ws (%)

m (kg/s)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

101.3 168 115 115 9 9 9 9 635 635 635 97.4 97.4 578 292 101.3 101.3 101.3

35 35 43.3 43.3 43.3 41.4 43.3 43.3 43.3 85 90.8 98.9 98.9 41.5 43.3 35 35 35

4.65 4.65 4.65 4.65 4.65 0 7.01 6.48 6.48 6.48 6.48 0 0 0 7.01 7.01 0 4.65

2397 2397 2397 808 808 272 536 3621 3621 3621 3621 34.9 34.9 272 536 536 272 1589

a It is noted that process stages 8, 9 and 10 have the same pressure; in reality there would be a pressure drop. However, the analysis is undertaken using the reported data [47].

Sharqawy et al. [38], where those authors demonstrated that an ideal binary solution cannot result in negative dimensionless exergy values. However, it is crucial to recognise that seawater and other electrolyte solutions are not ideal solutions; the osmotic potential and activity coefficients have been developed to address this deviation from non-ideal behaviour. Although the exergy values may differ in terms of relative and percentage difference, the overarching objective of an exergy analysis is to focus efficiency improvement efforts. The various approaches were compared on the basis of exergy destruction (at two of the key process stages); overall plant simple exergetic efficiency: the ratio of ‘exergy out’ to ‘exergy in’; and the minimum work of separation, similar to the work of Kahraman and Cengel [49] and Sharqawy et al. [38], but also including the additional approaches addressed here. The results are presented in Table 8. The calculation equations are detailed in [49].

219

Some important results in Table 8 stand out, some expected and some less expected. First, the exergy destruction is considered. The heat exchanger exergy destruction is the same for all approaches, which is expected because 1) there is no change in concentration of the streams in the heat exchangers, and 2) the physical exergy is calculated using the same thermodynamic seawater functions. The exergy destruction in the flash chambers is very similar for four of the six approaches. However, the exergy destruction values calculated using the ideal mixture model and the model used by Drioli et al. result in higher exergy destruction values, i.e. percentage differences of 5.1% and 11.2% respectively, when compared with the seawater properties. Regarding the other approaches, the differences are less than 0.6%. The minimum separation work and plant exergetic efficiency obtained using the ideal mixture model is the same as that presented in [25,38], a minor variation of 4.11% versus 4.2% exergetic efficiency. Despite the negative values obtained by Kahraman and Cengel [49] when calculating the specific exergy using the Cerci method, the exergetic efficiency and minimum separation work are identical to those calculated using the ideal mixture model (NaCl and water) presented here. The overall lower values obtained using the ideal mixture model are in keeping with the lower chemical exergy values presented in Table 6. The minimum separation work and the second law efficiency for the other four approaches are closely aligned. Again, comparing results with the seawater properties, there is a maximum 5.2% difference: percentage differences of 0.8%, −0.8% and 5.2% respectively for the ideal mixture; seawater salt and water; the electrolyte solution; NaCl and water; and the dissociated ions; NaCl and water. It is the model used by Drioli et al. that provides the least expected results, both in terms of the minimum separation work and the second law efficiency, values of −6402 kW and −37.5% respectively. The minimum separation work for the MSF desalination plant can be defined as minimum theoretical work required to separate seawater with a mass fraction of 0.465 into two outgoing streams, one concentrated brine and one product stream. Mathematically, the minimum separation work is defined in Kahraman and Cengel [49] as the difference between the exergy rates of the outgoing and the incoming streams, that is, the sum of the exergy rates of process streams 15 and 16 minus the exergy rate of process stream 0. These process streams are at the dead state temperature and pressure, and thus, the only differences between

Table 5 Relevant mole fractions, activity coefficients, and activities of salts and water. Process stream

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

ws (kg salt/kg seawater)

0.0465 0.0465 0.0465 0.0465 0.0465 0.0001 0.0701 0.0648 0.0648 0.0648 0.0648 n/a n/a 0.0001 0.0701 0.0701 0.0001 0.0465

ww (kg water/kg seawater)

0.9535 0.9535 0.9535 0.9535 0.9535 0.9999 0.9299 0.9352 0.9352 0.9352 0.9352 n/a n/a 0.9999 0.9299 0.9299 0.9999 0.9535

Ideal mixture (NaCl & water)

Ideal mixture (seawater salt & water)

Electrolyte solution (NaCl & water)

Dissociated mole fractions and model used by Drioli et al. (NaCl and water)

xs

xw

xs

xw

ms (mole salt/kg of water)

γs

as

aw

xs

xw

0.01481 0.01481 0.01481 0.01481 0.01481 0.00003 0.02271 0.02091 0.02091 0.02091 0.02091 n/a n/a 0.00003 0.02271 0.02271 0.00003 0.01481

0.98519 0.98519 0.98519 0.98519 0.98519 0.99997 0.97729 0.97909 0.97909 0.97909 0.97909 n/a n/a 0.99997 0.97729 0.97729 0.99997 0.98519

0.02721 0.02721 0.02721 0.02721 0.02721 0.00006 0.04145 0.03823 0.03823 0.03823 0.03823 n/a n/a 0.00006 0.04145 0.04145 0.00006 0.02721

0.97279 0.97279 0.97279 0.97279 0.97279 0.99994 0.95855 0.96177 0.96177 0.96177 0.96177 n/a n/a 0.99994 0.95855 0.95855 0.99994 0.97279

0.83421 0.83421 0.83421 0.83421 0.83421 0.00171 1.28950 1.18525 1.18525 1.18525 1.18525 n/a n/a 0.00171 1.28950 1.28950 0.00171 0.83421

0.66216 0.66216 0.66216 0.66216 0.66216 0.95356 0.65826 0.65783 0.65783 0.65783 0.65783 n/a n/a 0.95356 0.65826 0.65826 0.95356 0.66216

0.5524 0.5524 0.5524 0.5524 0.5524 0.0016 0.8488 0.7797 0.7797 0.7797 0.7797 n/a n/a 0.0016 0.8488 0.8488 0.0016 0.5524

0.9723 0.9723 0.9723 0.9723 0.9723 0.9999 0.9567 0.9603 0.9603 0.9603 0.9603 n/a n/a 0.9999 0.9567 0.9567 0.9999 0.9723

0.02918 0.02918 0.02918 0.02918 0.02918 0.00006 0.04440 0.04096 0.04096 0.04096 0.04096 0.00006 0.00006 0.00006 0.04440 0.04440 0.00006 0.02918

0.97082 0.97082 0.97082 0.97082 0.97082 0.99994 0.95560 0.95904 0.95904 0.95904 0.95904 0.99994 0.99994 0.99994 0.95560 0.95560 0.99994 0.97082

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Table 6 Specific physical and chemical exergy. Process stream

Physical exergy (kJ/kg)

Specific chemical exergy (kJ/kg) Seawater properties

Ideal mixture; NaCl and water

Ideal mixture; Seawater salt and water

Electrolyte solution; NaCl and water

Model used by Drioli et al.; NaCl and water

Dissociated mole fraction; NaCl and water

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

0 0.0648 0.4419 0.4419 0.3385 0.1798 0.3314 0.3328 0.9352 14.798 18.082 412.4 23.1 0.762 0.6027 0 0 0

0 0 0 0 0 3.972 0.4203 0.2675 0.2675 0.2675 0.2675 n/a n/a 3.972 0.4203 0.4203 3.972 0

0 0 0 0 0 2.0895 0.2489 0.1537 0.1537 0.1537 0.1537 n/a n/a 2.0895 0.2489 0.2489 2.0895 0

0 0 0 0 0 3.8634 0.4564 0.2820 0.2820 0.2820 0.2820 n/a n/a 3.8634 0.4564 0.4564 3.8634 0

0 0 0 0 0 3.9262 0.4640 0.2869 0.2869 0.2869 0.2869 n/a n/a 3.9262 0.4640 0.4640 3.9262 0

4.014 4.014 4.014 4.014 4.014 0.009 6.004 5.560 5.560 5.560 5.560 n/a n/a 0.009 6.004 6.004 0.009 4.014

0 0 0 0 0 4.147 0.457 0.284 0.284 0.284 0.284 n/a n/a 4.147 0.457 0.457 4.147 0

the models are the differences in concentration between the process streams. A minimum separation work of −6402 kW implies that work is an output of the separation process, which potentially could be true if each of the streams was allowed to mix with pure water. However, this is unlikely in a desalination plant. As mentioned previously, and as shown in Fig. 2, the different salinity dead (pure water) in the model used by Drioli et al. has a significant effect on the results obtained and may not be suitable for seawater exergy analyses. The basis of the chemical exergy calculation is the difference in chemical potential at dead state temperature and pressure between the different process streams and the defined dead state; however, based on the analysis presented here, the dead state should represent an intuitive, physical dead state. All the models are essentially developed from the Gibbs Eq. (2); however, the model used by Drioli et al. is based on a specific seawater separation model assumption, and the validity of that assumption requires consideration for desalination analyses.

5. Summary of approaches The majority of models under consideration, except, perhaps, the Sharqawy seawater functions, can be used for general electrolyte solutions, that is, they do not relate specifically to the thermodynamic

properties of seawater. Differences between the various approaches can be summarised as follows. The model used by Drioli et al.: • Seawater modelled as an ideal mixture of pure water and solutes. However, the use of the dissociation constant to calculate the solvent mole fraction results in the ionised mole fractions of Eqs. (17) and (18). • Reservations regarding the clarity of the equation to calculate Nsolvent in the chemical exergy term have been outlined previously. • The chemical exergy calculation term is based on a very specific separation assumption that may not be applicable for desalination analyses. • Decoupled physical exergy and chemical exergy. • The choice of pure water as the dead state has both advantages and disadvantages. The main advantage is that the chemical exergy rate is never negative. However, seawater salinity is the more intuitive dead state and better reflects the maximum theoretical work available at thermomechanical equilibrium as the brine is rejected: or, as was the case in this analysis the calculation of the minimum separation work.

Specific chemical exergy Specific chemical exergy (kJ/kg)

7 Chemical exergy (Seawater properes: kJ/kg)

6

Chemical exergy (Ideal mixture; NaCl and water: kJ/kg)

5 4

Chemical exergy (Ideal mixture; Seawater salt and water: kJ/kg)

3

Chemical exergy (Electrolyte soluon; NaCl and water: kJ/kg)

2 1

Chemical exergy (Ideal mixture; Dissociated NaCl and water: kJ/kg)

0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Process stream

Chemical exergy (Drioli model: kJ/kg)

Fig. 2. Comparison of specific chemical exergy values including the model used by Drioli et al.

L. Fitzsimons et al. / Desalination 359 (2015) 212–224

Specific chemical exergy

4.5 Specific chemical exergy (kJ/kg)

221

4

Chemical exergy (Seawater properes: kJ/kg)

3.5 3 2.5 2 1.5 1 0.5 0 5

6

7

8

9 10 Process stream

13

14

15

16

Chemical exergy (Ideal mixture; NaCl and water: kJ/kg) Chemical exergy (Ideal mixture; Seawater salt and water: kJ/kg) Chemical exergy (Electrolyte soluon; NaCl and water: kJ/kg) Chemical exergy (Dissociated NaCl and water: kJ/kg)

Fig. 3. Comparison of specific chemical exergy values excluding the model used by Drioli et al.

• Concerns have been raised about the particular assumptions underpinning the chemical exergy term.

separation etc. the values were found to be identical to the ideal mixture model (NaCl and water). The Sharqawy functions using the thermodynamic seawater properties:

The Cerci model: • Seawater/brackish water modelled as an ideal mixture. • Intuitive dead state defined at ambient conditions: temperature, incoming salinity, pressure. • Mole fraction calculation is not the “mole fraction on an ionized basis [50]” more traditional of electrolyte solutions. • The coupling of the physical and chemical exergy (entropy of mixing) does not facilitate clear understanding of exergy flows. • Does not delineate between the physical exergy calculated at constant composition and the chemical exergy calculated at thermomechanical equilibrium (restricted dead state and dead state). • Results in negative specific exergy when the salinity at a process stage is greater than the dead state salinity, however, when calculating the exergy destruction, minimum work of

• Specifically facilitates calculation of seawater thermodynamic properties as a function of salinity and temperature. • Physical exergy and chemical exergy readily de-coupled. • Intuitive dead state defined at ambient conditions: temperature, incoming salinity, and pressure. The ideal mixture model (NaCl and water; seawater salt and water): • Physical exergy and chemical exergy de-coupled. • Intuitive dead state defined at ambient conditions: temperature, incoming salinity, and pressure. • Seawater is not an ideal mixture. • Good alignment with seawater functions and electrolyte solution model when using the molecular weight of seawater salts despite the fact that seawater is not an ideal mixture. This is an

Specific exergy 20 18

Specific exergy (kJ/kg)

16 Specific exergy (Seawater properes: kJ/kg)

14 12

Specific exergy (Ideal mixture; NaCl and water: kJ/kg)

10

Specific exergy (Ideal mixture; Seawater salt and water: kJ/kg)

8 6

Specific exergy (Electrolyte soluon; NaCl and water: kJ/kg)

4

Specific exergy (Dissociated NaCl and water: kJ/kg)

2 0 5

6

7

8

9

10

13

14

15

16

Process stream Fig. 4. Comparison of total specific exergy values excluding the model used by Drioli et al.

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Table 7 Percentage differences relative to the seawater properties: specific chemical exergy. Process stream

Ideal mixture; NaCl & water (% diff.)

Ideal mixture; seawater salt & water (% diff.)

Electrolyte solution; NaCl and water (% diff.)

Dissociated mole fractions; NaCl and water (% diff.)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

n/a n/a n/a n/a n/a 47.4 40.8 42.5 42.5 42.5 42.5 n/a n/a 47.4 40.8 40.8 47.4 n/a

n/a n/a n/a n/a n/a 2.7 −8.6 −5.4 −5.4 −5.4 −5.4 n/a n/a 2.7 −8.6 −8.6 2.7 n/a

n/a n/a n/a n/a n/a 1.2 −10.4 −7.3 −7.3 −7.3 −7.3 n/a n/a 1.2 −10.4 −10.4 1.2 n/a

n/a n/a n/a n/a n/a −4.4 −8.8 −6.2 −6.2 −6.2 −6.2 n/a n/a −4.4 −8.8 −8.8 −4.4 n/a

•

•

•

•

interesting finding and will be considered further in future work. • Thermomechanical exergy at constant concentration and chemical exergy at thermomechanical equilibrium. The electrolyte solution model: • The electrolyte model acknowledges that ionic solutions do not behave as ideal solutions. • Physical exergy and chemical exergy decoupled. • Intuitive dead state defined at ambient conditions: temperature, incoming salinity, and pressure. • The “ionized” mole fraction. • However, the calculation of the activity coefficient, and hence, the activity of the solutes is not trivial for solutions with high ionic strengths, such as seawater. For example, the use of the Debye–Huckel limiting law, which has been proposed in desalination applications, is not suitable for seawater. The Pitzer calculations are recommended.

Based on the previous summary, the following approach is recommended: • The dead state for the chemical exergy term should represent likely mixing conditions; for seawater, this is the local seawater salinity, i.e. incoming seawater salinity. The physical exergy dead state is ambient temperature and pressure. • The choice of exergy model is a function of the ionic strength of the solution; see Table 3. Seawater is not an ideal mixture, and note that the commonly used electrolyte model approaches presented in Table 3 are not suitable for seawater. • As per the Gibbs equation, the physical exergy of electrolyte

solutions should be calculated at constant chemical potential and the chemical exergy should be calculated at constant temperature and pressure; see the Cerci model discussion. The approach used by Cerci et al. is not recommended due to the ideal mixture assumption and the coupling of physical and chemical exergy discussed previously. Concerns have been highlighted regarding the model used by Drioli et al. — the dead state is not intuitive for desalination exergy analyses and concerns have been raised about the specific assumptions underpinning the chemical exergy equation. The Sharqawy et al. functions can be used to calculate the thermodynamic properties of seawater over a wide range of temperatures and salinity. However, they are specific to seawater and may not be suitable for other electrolyte solutions. The Pitzer et al. equations are suitable for a wide range of ionic strength concentrations and can be adapted to different electrolytes.

Chemical exergy can also be thought of in more general terms as having both an intrinsic chemical exergy component and a concentration chemical exergy component. The intrinsic chemical exergy component can be calculated using the approaches of Szargut and others [47, 61–64], where the intrinsic chemical exergy of each element and substance is quantified using a defined reference environment and a series of suitable reference reactions. Szargut discusses the chemical exergy of solutions, which are a function of the intrinsic chemical exergy and the mole fraction/activity of each species in the solution. Intrinsic chemical exergy and the Szargut approach for electrolyte solutions are left for future work. Additional and interesting further work would be to carry out a similar assessment for membrane desalination plants, which typically operate at dead state temperature. It is expected that the chemical exergy differences would become more significant for isothermal processes. 6. Conclusions The objective of this research was to undertake a detailed study of desalination exergy analysis approaches. Each was assessed from a theoretical perspective, and then, using several models, from a practical perspective using a dataset from the literature. Concerns were raised about the appropriateness of three of the models, in particular, the approaches proposed and/or used by Cerci et al., Drioli et al. and the ideal mixture model. It was shown that the Cerci approach and the ideal mixture model are very similar save for consideration of the distinction between the restricted dead state and the dead state. A suitable approach to calculate the chemical exergy of electrolyte solutions at high ionic strength, based on the Pitzer equations, was presented. An exergy analysis of a seawater desalination plant was undertaken in order to compare the models from a practical perspective: the Sharqawy seawater functions; the ideal mixture model (NaCl and water); the ideal mixture model (seawater salt and water); the dissociated ion approach (NaCl and water); the model used by Drioli et al.

Table 8 Comparison of exergetic efficiency (second law efficiency) and exergy destruction. Processes

Seawater properties

Ideal mixture; NaCl & water

Ideal mixture; Seawater salt & water

Electrolyte solution; NaCl & water

Drioli et al.; NaCl and water

Dissociated ions; NaCl and water

Flash chambers exergy destruction (kW) Heat exchangers exergy destruction (kW) Minimum separation work (kW) Second law/exergetic efficiency (%)

11,910 1695 1306 7.64

12,514 1695 702 4.11

11,920 1695 1295 7.58

11,899 1695 1317 7.71

13,239 1695 −6402 −37.47

11,842 1695 1373 8.04

L. Fitzsimons et al. / Desalination 359 (2015) 212–224

(NaCl and water); and the electrolyte solution model (Pitzer equations). The analysis resulted in very similar values of specific exergy, exergy destruction and second law efficiency for four of the six approaches. Similar to findings previously reported in the literature, the ideal mixture model resulted in significant differences in second law efficiency. Three of the other models (Sharqawy seawater functions; seawater salt and water; Pitzer equations) differed by as little as 1%. The chemical exergy values differed more significantly across the four approaches, however, when incorporated into the total exergy rates and used to calculate exergy destruction and exergetic efficiency, the magnitude of these differences diminished. Nomenclature

223

Subscripts and superscripts Ch H i im 0 Ph R s solvent solution w

chemical Henryan (refers to activity coefficient) denotes the relevant chemical species ideal mixture denotes the dead state physical Raoultian (refers to activity coefficient) salt the solvent (water) the solution (seawater) water

Symbols a A Aϕ b B c cv cP Cϕ C e Ė G h ṁ m M MW N Ṅ Nsolvent P R

s T v v w Ẋ X x z

activity Debye–Huckel parameter Pitzer model parameter Pitzer model parameter Debye–Huckel parameter specific heat capacity (kJ/kg·K) specific heat capacity at constant volume (kJ/kg·K) specific heat capacity at constant pressure (kJ/kg·K) Pitzer parameter concentration (kg/l) specific exergy (kJ/kg) exergy rate (kW or kg/h when specified) Gibbs energy (kJ) specific enthalpy (kJ/kg) mass flow rate (kg/s or kJ/h when specified) molality (moles of solute per kg of solvent) refers to the relevant cation (meaning should be clear from the context) molar mass (kg/kmol or g/mol when specified) number of moles (mol or kmol when specified) molar flow rate (kmol/s) number of moles of the solvent per mass unit of the solution (mol/kg) absolute pressure (Pa or bar when specified) universal gas constant (kJ/kmol·K or kJ/mol·K when specified), when accompanied by the subscripts im or solution refers to the specific gas constant of the ideal mixture or solution respectively specific entropy (kJ/kg) absolute temperature (K) specific volume (m3/kg) stoichiometric coefficient (meaning should be clear from the context) mass fraction rate of exergy destruction, refers specifically to Ẋdes (kW) refers to the relevant anion mole fraction ionic valence

Greek symbols α β β(n) MX ϕ ρ γ μ

parameter for Debye–Huckel model, dependent on ion size number of ions/particles generated from dissociation Pitzer equation parameter osmotic coefficient density (kg/m3 or kg/l when specified) activity coefficient chemical potential (kJ/kmol)

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