Model C selection

Based on the results obtained in Chapter 5, the research hypothesis defined in Chapter 4 requires adjustment, i.e. Model A is not deemed appropriate for water purification exergy analyses, and thus, is not considered further for UPW plant exergy analyses. However, the need for an appropriate UPW exergy model remains. Following a refocused literature review of the broader chemical exergy and solution thermodynamics literature, two potentially suitable alternative approaches have been identified, (1) a chemical exergy model based on the Szargut reference environment, and (2) a chemical exergy model based on the solution thermodynamics literature.

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Szargut developed standard chemical exergy tables, which provide a very sound basis for computing the chemical exergy of elements and compounds and thus evaluating the earth’s natural capital. However, the use of standard chemical exergy tables, and indeed the overall Szargut model approach, has both advantages and disadvantages. First, although standard chemical exergy values are straightforward to use and greatly simplify the use of exergy analysis, they are only strictly valid for dead states at standard temperature and pressure.

The limitations of the standard exergy tables particularly regarding standard temperature are problematic. Not all real life processes occur at standard values and it is often necessary to account for other thermodynamic states. Research has been, and is currently being, carried out to address this issue, and these developments are discussed in greater detail in Chapter 7. Without the use of the standard exergy tables, the fundamental calculation of chemical exergy values is quite complex. There are also uncertainties associated with the chemical exergy of species, whose chemical exergy uses the lithosphere as a reference datum level, as discussed previously in Chapter 2.

Second, and this can be seen as both an advantage or a disadvantage depending on one’s viewpoint, the use of the standard chemical exergy approach takes a broader perspective of the natural environment, and importantly, considers possible interactions between process streams and this wider environment that may not necessarily be relevant to the process being analysed [93]. Some authors [125] advocate the need for practical limitations to be factored into the dead state definition. In the case of water purification, is this broader perspective necessary or valuable to understand and quantify process flows? This presents an interesting research question: how does the global perspective differ from the local

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perspective for desalination or water purification exergy analyses? This question in essence considers whether it is more appropriate to reference the chemical exergy of every species in the process streams to the final, most likely, form of that species as it interacts with the environment or if it is more appropriate, as in the case of Model A and Model B, to just consider the difference in concentration between the process stream and the defined dead state. Model A and Model B adopt the latter approach albeit using a different underlying assumption regarding the separation of the streams. The use of the Szargut approach, and thus, the calculation of exergy in desalination or water purification applications from the global natural capital perspective, is rare in the literature. The Szargut model is discussed in greater detail with regards to the exergy analysis of electrolytic solutions for desalination purposes in Chapter 7.

The remainder of this chapter considers the development of Model C from the solution thermodynamics perspective, and this relates only to the difference in concentrations between the process stream and the dead state. In this respect, it is the same approach as Model A and Model B. Importantly however, the aqueous solution is not treated as an ideal mixture but as an electrolytic solution. The various approaches of the four considered models are compared in Figure 6-1 on the basis of several criteria such as the chemical exergy calculation approach, solution modelling and separation assumptions.

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Figure 6-1: Breakdown of desalination exergy approaches

Careful consideration of the physical and chemical exergy definitions is also required.

Importantly, the ability to differentiate clearly between chemical and physical exergy is necessary for understanding process flows and sites of exergy destruction. The physical exergy model is relatively straightforward: the assumptions of an incompressible fluid did not result in significant differences in physical exergy calculation according to the results obtained in Chapter 5. Constant specific heat capacity is often assumed thus facilitating the simple integration of (2.7). These assumptions are re-examined for the relevant applications later in this chapter. Therefore, for the development of Model C, the breakdown of total

132 according to Pitzer and Brewer’s revision of the classic text Thermodynamics by Lewis and Randall [175] has led to many misunderstandings,

“Of all the applications of thermodynamics to chemistry, none has in the past presented greater difficulties, or been the subject of more misunderstanding, than the one involved in the calculation of what has rather loosely been called the free energy of dilution, namely the difference in the chemical potential or partial molar free energy of a dissolved substance at two concentrations”.

The chemical exergy at constant temperature and pressure arises due to the difference in chemical potential between the state of thermo-mechanical equilibrium (restricted dead state) and the dead state. This difference in chemical potential is determined by the change in Gibbs energy as the concentration changes at constant temperature and pressure, see (6.1). The Gibbs energy is a function of temperature, pressure and concentration (number of moles), G = f T P N( , , ).

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At constant temperature and pressure, the Gibbs energy is solely a function of the change in concentration of the relevant species. Utilising the TdS relations or Maxwell’s equations, and the definition of chemical potential of constituent i as the change in molar Gibbs energy at constant temperature, pressure, composition of other mixture constituents j

where j≠i, _i ( , , _j constant)

At constant temperature and pressure (6.2) simplifies to (6.3).

i i

i

dG=

∑

µdN ^(6.3)

Equations (6.1) to (6.3) define chemical potential, and it is this difference in chemical potential between the restricted dead state and the dead state that provides the driving force to perform chemical work, see Figure 6-2. Chemical work is analogous to the more common forms of work such as electrical and hydraulic work. Power or work can be thought of as the product of a relevant flowing current and a potential difference, i.e.

electric power is the product of a potential difference and electric current, hydraulic power is the product of a potential difference arising from a pressure difference at two ends of a conduit and a volumetric flow rate [106]. Then chemical work is the product of the difference in chemical potentials at two states and the current, in this case the molar flow rate [106].

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Figure 6-2: Restricted dead state to dead state via theoretical semi-permeable membrane

The chemical work potential of a substance as it comes into equilibrium with its environment can be derived theoretically by undertaking an energy and entropy balance of the mass interaction and mass transfer between the restricted dead state and the dead state through a semi-permeable membrane [125]. If Nɺ is the molar flow rate of constituent i and _i

µi is the chemical potential of constituent i then an energy and entropy balance can be undertaken to derive the chemical exergy that results from the mass transfer that occurs between the restricted dead state and the dead state. First, consider the energy balance, see (6.4), noting that Qɺ into the system is considered positive and Wɺ out of the system is positive.

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DS RDS

i i i i

DS RDS

Q W^ɺ− =^ɺ

∑

N h^ɺ −

∑

N h^{ɺ (6.4)}

The entropy balance is given by (6.5) where sɺ is the rate of entropy generation. _gen

0

Eliminating Qɺ between (6.4) and (6.5) results in (6.6).

0 0 0 obtainable from the transfer of mass from the restricted dead state to the dead state is given by (6.7).

( )

Ch RDS DS

i i i

E^ɺ =

∑

N^ɺ µ −µ ^(6.7)

The preceding analysis was based on a derivation in the cited reference [125].

To calculate the chemical exergy of each constituent in a solution or mixture under consideration from (6.7), all that is needed is a means to evaluate the difference between the chemical potentials for each constituent. This is relatively straightforward for an ideal gas in an ideal gas mixture (a mixture of ideal gases behaves as an ideal gas). For example, consider the waste products of combustion, which can be modelled as an ideal gas mixture, the potential to do work exists due to the difference in partial pressures of the gases in the environment between the restricted dead state and the dead state. The change in chemical potential depends on both the change in enthalpy and the change in entropy between the restricted dead state and the dead state. By definition, there is no temperature difference between the restricted dead state and the dead state. For an ideal gas the change in enthalpy between two states is a function of temperature alone, and therefore, at constant

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temperature, there is no change in enthalpy between the ideal gas at two states of differing partial pressures. The change in entropy for an ideal gas in an ideal gas mixture is a function of temperature and pressure. Thus, at constant temperature the change in entropy is solely a function of pressure, and can be derived from the TdS relations, see (6.8) where

dh=0.

Tds=dh vdP− (6.8)

Using the ideal gas equation of state and Dalton’s law of partial pressures, the change in entropy can be calculated as follows, where P_i0^DS and P_i0^RDSare the partial pressures of the ideal gas in the atmosphere at the dead state and the restricted dead state respectively.

0

The molar chemical exergy of an ideal gas is obtained by multiplying the change in entropy by the dead state temperature T , see (6.10). ₀

Hence, the molar chemical exergy of an ideal gas is a function of the dead state temperature and its concentration difference between two states.

Equation (6.10) moves ahead slightly in an effort to demonstrate how chemical exergy is calculated. In the general case the calculation of the chemical exergy of an ideal gas should really be thought of as a two-step process. First, the difference between the chemical potential of an ideal gas at each relevant concentration and the chemical potential of the

137 exergy resulting from the difference in concentration values between two states, for example, between the restricted dead state and the dead state, is calculated by subtracting the values for each state obtained in step 1. The chemical potential of the pure substance

This chemical potential term is the unitary chemical potential and is defined as the standard state concentration for ideal gases, although the standard state concentration can change depending on the system of interest.

For the purposes of UPW plant analysis, the use of (6.10) presents a problem, the derivation was based on the ideal gas equation of state, and although this equation is relevant for flow streams which can be modelled as an ideal gas mixture, it does not apply to non-ideal gas systems. What is the alternative if the ideal system approach is not appropriate?

First, the various types of solutions must be considered: a solution is defined as “a condensed phase (liquid or solid) containing several substances [115]”. There are several

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classifications of solutions, perfect, ideal and non-ideal. One definition of a perfect solution is that it obeys (6.11) over the complete range of concentrations (Raoult’s law). An ideal solution obeys Raoult’s law for dilute concentrations only. However, there is a special class of ideal solution, the “ideal dilute solution [115]”, where the change in chemical potential of the main solvent can be modelled according to (6.11), analogous to the “extremely dilute solutions [94]” referred to by Bejan. According to Pitzer and Brewer’s revision of Thermodynamics [175], this approach is deemed valid for a liquid or a solid that may act as a solvent in infinitely dilute solutions.

Finally, for non-ideal solutions, an activity coefficient γ is required to adjust the mole fraction for the departure from ideal gas, perfect solution or ideal solution behaviour case, see (6.13).

( , , ) ( , , 1) ln ( , , 1) ln

i T P xi i T P xi RT ixi i T P xi RT ai

µ =µ = + γ =µ = + (6.13)

The product of the activity coefficient and the mole fraction of constituent i is termed the activity a . _i

Based on the concept of the “ideal dilute solution [115]”, one would expect that the difference in the chemical potential of water, as the main solvent in an ideal dilute solution, could be calculated according to Raoult’s law. However, this simplification may not be valid for an aqueous solution of ions because an electrolytic solution behaves quite differently from a non-electrolytic solution and special consideration must be given to this deviation in behaviour.

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An electrolyte can be defined as a substance that dissociates into ions in an aqueous solution. Arrhenius was the first to discover that certain solutes dissociate into electrically charged ions. The behaviour of these solutes at low concentrations was expected to follow ideal solution behaviour at low concentrations but it was observed that their behaviour was fundamentally dissimilar to non-electrolytic solutions due to long range electrostatic forces between the ions [175]. Based on these findings, electrolytes should not be modelled according to (6.10) but according to (6.13).

Hence, applying the same approach to calculate the difference in chemical potential between two concentration levels of an electrolyte in an aqueous solution as that used to derive (6.12), one would expect that the change in chemical potential could be calculated from (6.14).

There are two extra complications however, when dealing with electrolytic solution thermodynamics. The first is relatively straightforward: the definition of the mole fraction of an electrolyte can lead to ambiguities [176]. Take NaCl for example, the mole fraction of NaCl in an ideal mixture of NaCl and water was defined earlier for Model B using a

However, NaCl is a strong electrolyte at the temperatures under consideration in this thesis (i.e. NaCl dissolved in water does not exist as NaCl molecules in water but as Na⁺ and Cl^-

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ions). Hence, the more traditional definition of the mole fraction for a strong electrolyte used in solution chemistry is (6.16) [176], where β is the number of ions generated on dissociation of the electrolyte (two for NaCl), N is the number of moles and the subscripts NaCl and w refer to the dissociated electrolyte NaCl and water respectively.

NaCl

The difference between the two approaches can be illustrated by considering a 1 M NaCl solution, i.e. one mole of NaCl in one kilogram of water (55.5 moles of water). In the first case, using (6.15), the mole fraction of NaCl is calculated to be 0.0177. Adopting the approach of (6.16) results in a mole fraction of 0.0348, almost twice that of the first case.

Pitzer termed the latter approach the “mole fraction on an ionized basis [177]”. In a similar manner, the mole fraction of water can be calculated using (6.17).

w

It is almost universal to use the molality scale when dealing with electrolytes due to the fact that molality is independent of the temperature and pressure of the solution, and also, equations tend to be simpler using the molality scale (concentrations in electrolytic solutions are generally dilute, leading to very small solute mole fraction values) [176].

Molality is defined as the number of moles of the solute in one kilogram of the solvent i.e.

water. Conversion from solute concentration or mass fraction to molality is carried out as follows;

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• To convert from the concentration of the solute (g/l, i.e. grams of solute per litre of solution) to the mass fraction of the solute (g/g, i.e. grams of solute per gram of solution) divide by the density of the solution;

• To convert from the mass fraction of the solute to molality divide the solute mass fraction by the product of the molar mass of the solute (g/mol) and the mass fraction of water (g/g, i.e. grams of water per gram of solution);

• Finally, multiply by one thousand to change from grams of water to kilograms.

It should be noted that the molality of the solute is undefined for the pure state of the solute [176]. The change in chemical potential of an electrolyte between one concentration and the concentration in the standard state, written in the molality scale, is shown in (6.18) where C is the concentration of species i , i M is the molality and _i γ_i is the activity coefficient of species i, and C^° is the standard state concentration of electrolyte species i .

( , , ) ( , , ) ln ( , , ) ln order to mirror the simple relationship between the chemical potentials in (6.11) and (6.18) the value of M_iγ_i in the standard state must be unity. To explain; if (6.11) is revisited, it is evident that the use of the pure state (i.e. x_i =1) as the standard state simplifies matters

greatly and that the final term is essentially ln 1 xi

RT . To maintain the same relationship in

142 activity coefficient γ_{H i} is the Henryan activity coefficient.

0 0 0 0 0 0

Presently, the solvent water is considered. The standard state for water is analogous to the ideal gas example, i.e. the standard state of water is defined as pure water, except in this case, the activity is used rather than the mole fraction to correct for the departure from ideal behaviour. Again, to replicate the chemical potential relationship of (6.11) and (6.18), the activity of pure water in the standard state is set equal to unity, x_iγ_i =a_i therefore in the standard state when x_i =1 the activity is equal to unity and thus γ_i =1. The activity coefficient used when the standard state is the pure substance is termed the Raoultian activity coefficient. The difference in chemical potential resulting from the differences in the concentration of water between the restricted dead state and the dead state can be calculated using (6.20) where the activity coefficient γ_{R w} is the Raoultian activity coefficient of water.

143 electrolytic solutions [116, 176], for this reason the activity coefficient of water is assumed to be unity in this research. Thus, the exergy of water is a function of the mole fraction ratios. An alternative approach is to use the osmotic coefficient, see the cited reference [176]. The activity of water can change dramatically at high electrolytic solution concentrations, temperatures and pressures (solutions close to saturation, temperatures of 500 to 900°C and pressures up to 15,000 bar [176]). However, these parameters are not relevant to the concentrations, temperatures and pressures considered in this research.

Marin and Turegano [112] outlined a methodology for calculating the chemical exergy of aqueous electrolytic solutions whereby the chemical exergy of each species was calculated using activities rather than mole fractions according to (6.14) albeit based on a different derivation approach than that used here. Equation (6.14) has also been used for calculating the exergy of ionic species in river water [178, 179], seawater [157] and proposed for calculating the concentration exergy of a body of water [180]. However, in the author’s opinion, the chemical exergy term proposed in [180] is confusing; the concentration exergy term in the main body of the journal article is changed from a ratio of activities to a ratio of concentrations in the appendix section. The concentration ratio is not necessarily equivalent to the activity ratio and the point of using the activity is that it accounts for a number of standard states and it maintains the relationship between the chemical potential “for any component i in any system under any condition [176]”. Several attempts by email were

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made to discuss this issue with the corresponding author of the cited reference [180], however, to date, no response has been forthcoming.

Equation (6.19) appears relatively straightforward; unfortunately, calculating activity coefficients of electrolytic solutions can be complex and is based on theories of the

Equation (6.19) appears relatively straightforward; unfortunately, calculating activity coefficients of electrolytic solutions can be complex and is based on theories of the

In document A detailed study of desalination exergy models and their application to a semi-conductor ultra-pure water plant (Page 155-177)