Phase Behavior of Biomass Components in Supercritical Water

Abstract

The importance of thermodynamic and phase behavior is fundamental to supercritical water (SCW) technologies of biomass treatment. Considering the extremely large number of biomass components, it is obvious that there is need for developing theoretically sound methods of the prompt estimation of their phase behavior in aquatic media at supercritical conditions. A local mapping concept is introduced to describe thermodynamically consistently the saturation curve of water and biomass components. The global phase diagram studies of binary mixtures provide some basic ideas of how the required methods can be developed to visualize the phase behavior of biomass decomposition products in supercritical aqueous media. The mapping of the global equilibrium surface in the parameter space of the equation of state (EoS) model provides the most comprehensive system of criteria for predicting binary mixture phase behavior. Analytical expressions for selection the azeotropic states in binary mixtures are given. Results of calculations of phase equilibria and critical curves for main biomass components in supercritical water are described.

Success depends upon previous preparation, and without such preparation there is sure to be failure.

Confucius

1 Introduction

Biomass is the prevailing form of renewable energy which can provide the conversion to heat and power, biofuels and biobased chemicals as the replacement of conventional fossil. Three key macromolecules: lignin, cellulose, and triacylglycerides can generate the wide opportunities to transform biomass into goal products – energy and desired materials [1]. Proteins are not primary components of biomass and account for a lower proportion than do the previous three macromolecules.

Protein properties depend on the kinds and ratios of constituent amino acids, and the degree of polymerization. The amounts of the other organic components vary widely depending on specie, but there are also organic components. Biomass comprises organic macromolecular compounds, but it also contains inorganic substances (ash) in trace amounts. The primary metal elements include Ca, K, P, Mg, Si, Al, Fe, and Na.

There are many classical technologies that ensure the fulfillment of biomass conversion problem but modern challenges of sustainable development require the emergence of environmentally friendly green technologies. Water in supercritical state i.e. at temperature and pressure high enough to vanish differences between liquid and vapor phases, above 647 K and 22 MPa, is recognized as green solvent. Thermodynamic and phase behavior of SCW are differed qualitatively from the “normal” state (for example it can act as a quasi-organic solvent). Not only the chemistry, but also the physics can change drastically. For example, density variation with a factor of 7 for 25 MPa around a few Kelvin range in the supercritical region across the Widom line representing the maximum of response functions can change heat capacity and compressibility with several order of magnitudes, making almost impossible to use traditional flow- and thermal transport calculations. Development of new SCW technologies is impeded by the lack of fundamental understanding of many aspects of the supercritical fluid state itself, and particularly of their thermodynamic and phase behavior. Properties and phase equilibria of fluid mixtures define the scientific platform for developing green chemical processes and predetermine the success of emergent technologies [2].

The main aim of this Chapter is to review existing experimental data and theoretical models for analyzing, correlating, and predicting the phase behavior of biomass components in supercritical water.

The Chapter is organized as follows. In the first section, main biomass components are selected. Critical (or pseudocritical) parameters are considered as fundamental characteristics of a fluid by themselves and also often as the input from which the other physical properties can be generated. The quantitative structure-property relationships (QSPRs) to estimate the critical properties of biomass components are discussed. The boundaries of the Widom line for supercritical water where reaction mechanisms accelerate the biomass conversion maximally are given. In the second part, a theoretical analysis of the topology of phase diagrams as useful tool for understanding the phenomena of phase equilibrium that are observed in biomass components – supercritical water mixtures in vicinity of the Widom line is given. We review the global phase behavior of binary mixtures and derive an analytical expression for determining the boundaries of azeotropy in terms of the critical parameters of mixture components and the binary interaction parameter k ₁₂ for the one fluid models of the equation of state. The knowledge of binary interaction gives possibility to predict all types of phase behavior in supercritical water – biomass component mixtures. The results of simulation of phase equilibria for binary mixtures are illustrated using examples of various classes of biomass components.

2 Main Biomass Components

The estimation of the properties of main biomass components is an important prerequisite for the design of processes and equipment, environmental impact assessment and other major chemical engineering activities related to phase equilibria. The main components of biomass resources are typically 40–45 wt% cellulose, 25–35 wt%, hemicellulose, 15–30 wt% lignin and up to 10 wt% for other compounds [3]. Elemental composition of conventional biomass includes following components: C, H, O, N, and S. Chemical formula C_aH_bO_cN_dS_e simulates the various types of biomass, e.g., sewage sludge C_2.83H_4.86O_1.25N_0.34S_0.04, microalga Spirulina C_3.66H_6.81O_2.16N_0.47S_0.01, the zoomass CH_2.06 O_0.52N_0.12S_0.01, and others [4]. Possible reaction products depend on goals of biomass conversion and thermodynamic conditions of SCW treatment processes. The gasification of biomass and organic wastes generates the gas components such as H₂O, CO₂, CO, N₂, N₂O, NO, NO₂, SO₂, SO₃, H₂, CH₄, and hydrocarbons (HC). Biomass products produced from different materials and by different external conditions may differ greatly from one another. As a result, the compositions of different reaction products usually vary in wide ranges.

To establish the thermodynamic properties of great variety of compounds under SCW treatment the simple models of the EoS P = P (T,V, X ), e.g., the Peng-Robinson (PR) [5] or the Redlich-Kwong-Soave (SRK) [6, 7] models are a better choice to link key characteristics of biomass components and target properties. Critical parameters of biomass components can be considered as their information characteristics which could generate a set of target properties for designed SCW system. This class of EoS has simple relationships between their model parameters and critical constants derived from critical conditions. During the many years of super critical research, a variety of fluids have been investigated, and accordingly, both their critical temperature and pressure have been determined. An exposure of these physical properties is condensed in Table 2.1.

Table 2.1 Critical properties of selected chemical compounds (sorted by critical temperatures)

To define more exactly equation of state parameters for pure substances, especially for water, concept of local mapping is developed. Standard approach to determination of the EoS parameters X = X (a, b, α) is derived from critical conditions. But for phase equilibria prediction this parameter set is not sufficient to satisfy the limiting conditions for pure components (x → 1 and x → 0). To provide a reliable thermodynamic consistency between exact and model equations of state the equalities of pressures, isothermal compressibility’s and internal energies should be applied. As an example, a refinement of the SRK EoS parameters is considered. To satisfy thermodynamic consistency conditions at given point of P – V – T surface the set of equations (2.1) should be solved:

$$ \begin{aligned}[b] {P}_{\it exact}\left( T, V\right)-{ P}_{\it mod}\left( X, T, V\right)&= 0,\\ \frac{\partial { P}_{\it exact}\left( T, V\right)}{\partial T}-\frac{\partial { P}_{\it mod}\left( X, T, V\right)}{\partial T}&= 0,\\ \frac{\partial { P}_{\it exact}\left( T, V\right)}{\partial V}-\frac{\partial { P}_{\it mod}\left( X, T, V\right)}{\partial V}&= 0\end{aligned} $$

(2.1)

where Р _ехасt – fundamental EoS for H₂O [8], Р _mod – the SRK model [7], Х – vector of the SRK model parameters, α(T/Tc,ω) as function of reduced temperature and acentric factor (ω) is the attractive term in the original Redlich–Kwong equation. Results of calculations, which exactly describe data along saturation curve, are presented in Table 2.2.

Table 2.2 Parameters of the SRK EoS for water

More sound consideration of problem should be guided on quantitative “structure – property” relationships (QSPR). The basic idea of QSPR is to find a relationship between the structure of compound expressed in terms of constitutional, geometrical, topological, and other quantum-chemistry descriptors and target property of interest.

The QSPR employs two databases – the critical property database and structure database. The correlations between databases are established in the form of the property model – M(P), the parameters of which are determined by minimizing the “distance” between the experimental property P _j and its model Mj. For the most of compounds experimental data are available via on-line databanks e.g., NIST, KDB, CHEMSAFE, BEILSTEIN, GMELIN et~al. If direct measurement results are not in option, thermodynamic models from process simulators like ASPEN PLUS can be partly used to estimate the missing properties.

There are many group contribution methods estimating the critical properties of possible biomass compounds from molecular structure and success of any model depends on the amount of data used in determining the contribution of independent variables (molecular descriptors). The start point of group contribution technique is a decomposition of the molecular structure into particular groups and the counting of atoms in those groups. Increments are assigned to the groups by regression of known experimental data for the chosen property. The molecular structure can be retrieved by summation of the contributions of all groups. The least sophisticated atom count technique was proposed by Joback [9]. More general cases for molecular descriptors determination are described elsewhere, e.g. [10, 11]. At present time reliable correlations are established by heuristic, Multi-Linear Regression techniques, e.g. [12], by nonlinear techniques like Genetic Function Approximation [13] or by Artificial Neural Networks, e.g. [14]. The choice of appropriate technique to predict critical properties from molecular structure depends on the statement problem and should correspond to the final aim of molecular design. Distribution of critical points for hydrocarbons and key biomass components versus saturation curve of water are given in Fig. 2.1.

Fig. 2.1

Distribution of critical points of hydrocarbons () and selected chemical compounds (▲) versus location of H₂O saturation curve

Critical temperature and pressure data for the hydrocarbons were taken from correlation Wakeham et~al. [15]. Data for selected chemical compounds were taken from Table 2.1. Critical (or pseudocritical) parameters are considered as fundamental characteristics of a fluid by themselves and also often as the input from which the other physical properties can be generated. The chain starting from descriptors of molecular structure to target property of biomass components should be constructed through critical parameters of technological fluid and equation of state model, accordingly.

3 SCW Thermodynamic Behavior

The critical point defines the vertex of the vaporization boundary; hence, geometric considerations ensure that isothermal compressibility, isobaric expansion and other specific partial derivatives will diverge to infinity at critical conditions. This inherent divergence of derivatives effectively controls not only the thermodynamic, electrostatic, and transport properties of water, but they also influence on transport and chemical processes in SCW systems. The behavior of water as solvent, with some simplification, one can describe by following way while the subcritical water is good solvent for inorganic and bad solvent for organic materials, it turns into the opposite in the SC region; it could dissolve several organic materials but could not dissolve some inorganic ones. By using pure SCW, after days or months various corrosion products can still be dissolved into it.

Both compressibility and thermal expansion behaviour are very important factors in order to understand the shifts of the equilibrium constant in supercritical solvents as a system response on both thermal and mechanical perturbations. The coefficient of isothermal compressibility is defined as:

$$ \beta =\frac{1}{\rho}{\left(\frac{\delta \rho}{\delta p}\right)}_T, $$

(2.2)

and diverges sharply (critical exponent ∼ 1.241) from ideal gas values to infinity at the critical point. It is an interesting fact that the β divergence depends on pathways of approaching to the critical state along the critical isochore (sharpest slope) or along any other pathway in the p-T plane. There are three major regions in which it is possible to mark the p-T plane (Fig. 2.2):

Fig. 2.2

β – p – T – surface of supercritical water

• the low-density region along an isobar at T > 723 K where the isothermal compressibility decreases with both pressure and temperature;

• a region to the low-density side along an isotherm where the isothermal compressibility increases with pressure and decreases with temperature;

• a region to the high-density side along an isobar where the isothermal compressibility decreases with pressure and increases with temperature.

In supercritical fluid, the isothermal compressibility in the low-density region is typically several times larger than those in the high-density region. The coefficient of isobaric expansion is defined as:

$$ {\alpha}_T=-\frac{1}{\rho}{\left(\frac{\delta \rho}{\delta T}\right)}_p. $$

(2.3)

The qualitative behaviour of the α _T (p,T) and β(p,T) surfaces is similar, but relative maxima of α _T along the isotherms and isobars are shifted to larger densities relative to β. Isochoric heat capacity behaviour, like isothermal compressibility and isobaric expansion, changes from ideal gas values to infinity at the critical point.

In contrast to the strong divergence of the isothermal compressibility and isobaric expansion, the C _V has a near logarithmic divergence. Supercritical maxima of isochoric heat capacities in the pressure – temperature diagram tend to go to the side of decreasing density. At pressures less than 30 MPa, and temperatures less than 673.15 K, the surface C _V (p,T) looks like to the isothermal compressibility β. Isobaric heat capacity from thermodynamic relationships is

$$ { C}_P={ C}_V+ T\frac{\alpha_T^2}{\rho \beta}. $$

(2.4)

The near-critical region has the similar asymptotic behaviour for all fluids and theoretically is described by the universal critical exponents and scaling functions, and is based on the scaling concept. The power laws, that represent asymptotic behaviour along the iso-lines on thermodynamic surfaces, are described in numerous literature sources. The main conclusion is that – H₂O behaviour is not unusual in critical region and has the same anomalies as all other fluids.

4 Local Extrema of Thermodynamic Properties. The Widom Line

Thermal effects and volume changes of reactions are linearly dependent on compressibility behavior of fluids. Local extrema in α _T , β and ρ have a significant influence on the nature of both heat and mass transport processes. As an example, for the design of high-pressure units for SCW biomass treatment the local rate of convection heat transfer is proportional to both fluid flux and C _P . Usually fluid flux is also proportional to α _T and ρ. Therefore, the maximum flow rates occur at state conditions intermediate between α _T and ρ maxima. Because C _P is a function of α _T , its extreme projections are nearly coincident. As a result, convection heat transfer rates in supercritical conditions attain local maxima in the vicinity of α _T extremes. Near the critical point, the extrema lines for response functions are merged. An asymptotic line as continuation of vapour pressure curve to the one phase region is often named – the Widom line often referred as pseudo-critical or pseudo-spinodal line. The extrema lines (isobaric heat capacity – C _P , compressibility coefficients – α _T and β, speed of sound – a, inflection points – IP, the Joule – Thomson coefficient – λ) were calculated on the basis of the Wagner and Pruss equation of state [8]. A quantitative presentation of extreme behavior in the properties of H₂O in the P – T diagram is illustrated in Fig. 2.3.

Fig. 2.3

Extrema of response functions in the P – T diagram

The pressure range, where all extrema lines are merged, is located within 22 … 25 MPa. Similar picture giving the same lines for water with fittings based on IAPWS EoS is considered in [16]. The main conclusion from these calculations is as follows. No universal curve as continuation of vapor pressure curve exists.

In practical design of SCW units, we need to consider non-purity of water. It leads to a shift of extrema lines in the pressure – temperature plane. The typical impurity in supercritical water is a small concentration of NaCl. To study the changes in behaviour along the Widom line, we have calculated the shift of the max C _P line at different concentrations (100, 200, and 300 ppm) [17]. We suggested that at small concentrations of NaCl in water the correspondence state principle is working. The NaCl additives displace the position of critical point along critical line of aqueous solution (Fig. 2.4). Detailed analysis of data was given by Shibue [18]. Critical temperatures and critical pressures of aqueous NaCl solutions also were taken from [18].

Fig. 2.4

Shift of critical point and max C _P lines in aqueous solution of NaCl [17]

To those versed in the art, it is known that enhanced solubility is able to accelerate reaction rates, particularly supercritical water can be a medium for either ionic or single-radical chemistry. A variety of compounds are hardly soluble, if at all, in a medium at room temperature, however they could become soluble in super critical media. Contrary to this observation, those compounds that are soluble in media at room temperature could become less soluble at supercritical conditions.

Solubility growth in water – salt systems at supercritical conditions plays a principal role for the SCW biomass treatment and other industrial applications. Experimental data on solubility of solids in supercritical water have testified that pressure increases the solubility of most of the solids above the H₂O critical temperature (systems H₂O–SiO₂, H₂O–Na₂SO₄, H₂O–Li₂SO₄, etc. are examples confirming this observation). At the present time, there are many computer programs for modelling salt/water systems, taking into account the irreversible dissolution of the reactants and the reversible precipitation of secondary products. From a mathematical point of view, the most successful approach to calculate the equilibrium assemblage is based upon a solution of a set of non-linear mass-action equations, involving equilibrium constants for all relevant equilibrium and auxiliary sets of linear mass-balance and charge balance equations. The primary aim of computer programs is to calculate the speciation of aqueous solutions and their equilibrium to salts (e.g., http://h2o.usgs.gov/software). Most comprehensive review of experimental data on aqueous phase equilibria and solution properties at elevated temperatures and pressures is given by Valyashko [19].

5 Phase Equilibria in Binary Mixtures. Global Phase Diagram

Theoretical analysis of the topology of phase diagrams is a very useful tool for understanding the phenomena of phase equilibrium that are observed in biomass components – supercritical water mixtures in vicinity of the Widom line. The pioneering work of van Konynenburg and Scott [20] demonstrated that the van der Waals one-fluid model has opened opportunities of qualitative reproducing the main types of phase diagrams of binary fluids. The proposed classification was successful, and is now used as a basis for describing the different types of phase behavior in binary mixtures. At present, the topological analysis of equilibrium surfaces of binary fluid systems contains 26 singularities and 56 scenarios of evolution of the p–T diagrams [21]. The introduction of liquid–solid equilibria into classification schemes makes it possible to outline a strategy of continuous topological transformation for constructing complete phase diagrams. The mapping of the global surface of a thermodynamic equilibrium onto the space of parameters of an equation of state gives the possibility to obtain the most comprehensive system of criteria for predicting of the binary mixture phase behaviour. The influence of critical parameters of compounds on phase topology is visualized via global phase diagrams. Such diagrams are presented not in the pressure – temperature variables but in the space of the equation of state parameters, e.g. the van der Waals geometric – b and energetic – a parameters or critical parameters.

For understanding and classifying a great number of phase diagrams, the determination of critical points is an important practical and theoretical problem. A mixture of a given composition can have one, more than one, or none critical point. The types of phase behavior that are of interest for the SCW technologies are characterized as follows (Fig. 2.5).

Fig. 2.5

Types of phase diagrams: C₁ are critical points of pure components; C_E1 mean the critical end points; the continuous line denotes the equilibrium curves of pure components; the dashed lines correspond the critical lines; and the dash-dotted lines denote the three-phase equilibrium curves

• (I) The simplest type that has a continuous critical curve between the two critical points C₁ and C₂. It can shape of the critical curve and the position of the azeotropic line.

• (II) This type is characterized by the presence of an immiscibility zone at temperatures below the critical temperature of a more volatile component, by a critical curve that connects two critical points of pure components, and by a critical line that starts from the critical end point where the liquid–liquid equilibrium line ends.

• (III) This type comprises two different critical curves. One of them starts from the critical point of a pure component with a higher critical temperature and goes to the range of high pressures. The other critical curve starts from the critical point of the other component and leads to the critical end point at the end of the three-phase line. The type is divided into five subtypes. The main subtypes differ in the arrangement of the three-phase line at pressures above the saturation pressure of components and the azeotropic line that is bounded by the azeotropic end point from below and by the critical azeotropic point from above (subtype III–A). A distinctive feature for subtype III–H is the occurrence of azeotropy.

• (IV) Type IV is characterized by two curves of the liquid–liquid–gas equilibrium. The high temperature three-phase line is bounded by two critical end points (lower (LCEP) and upper (UCEP)). In the vicinity of the UCEP, the solution becomes immiscible with decreasing temperature. In the region of the LCEP, an immiscibility zone appears with increasing temperature.

• (V) This type resembles type IV which has no liquid–liquid critical line and three-phase line at low temperatures. For this type, the occurrence of azeotropic states and multiextremal critical lines is possible.

• (VI) This type of phase behavior is characterized by the liquid–vapor critical line that connects two critical points of pure components and by the liquid–liquid critical line with a pressure peak that connects the UCEP and the LCEP in the three-phase line.

• (VII) This type, unlike I–VI, has not been confirmed experimentally and differs from type VI by the behavior of the liquid–vapor critical curve, which is divided into two lines that start at the critical points of pure components and end at the LCEP and UCEP in the second three-phase line.

• (VIII) This type is characterized by three critical lines. One critical line starts at the critical point of one of the pure components and goes towards the range of high pressures as in type III. The other critical curves start at the LCEP and UCEP in the three-phase line and end in the region of infinitely high pressures and the critical point of the other pure component, respectively.

The systems of interests, for example, H₂O + n-alkanes, H₂O + CO₂ belong to the type III phase diagrams according to van Konynenburg and Scott [20], with three phase equilibrium and interrupted critical line, of which the lower branch connects the critical point of n-alkane and the upper critical end point and the upper branch runs from the critical point of pure water to high pressures (valid to water – n – hexacosane). Other systems include H₂, CH₄, CO₂ and etc. Figs. 2.6 and 2.7 illustrate collection of experimentally determined critical curves of aqueous systems.

Fig. 2.6

Experimental P – T data along critical curve for the SCW – key biomass component binary mixtures

Fig. 2.7

Collection of experimentally determined critical curves of aqueous systems

The system SCW – O₂. The critical curve, an envelope of the isopleths, begins at the critical point of water (647 K), has a temperature minimum (639 K) at about 75 MPa and proceeds to 250 MPa at 659 K. Excess volume, V ^E, values have been calculated for 673 K from 30 to 250 MPa. All V ^E values are positive. The maximum is 57 cm³ mol⁻¹ at 70 mol % of H₂O at 30 MPa and about 2 cm³ mol⁻¹ at 40 mol % H₂O and 250 MPa [22].

The water-carbon dioxide system exhibits type III phase behavior in the classification of van Konynenburg and Scott [20] with a discontinuous vapor–liquid critical curve, a wide region of liquid–liquid coexistence below the critical temperature of CO₂, and very limited mutual solubility in the regions of two- and three-phase equilibria. This system was studied by Takenouchi and Kennedy [23] to pressures of 160 MPa and at temperatures of 383–623 K. The critical curve of the system trends toward higher pressures at lower temperatures and departs strongly from the critical point of pure water. At low pressures the CO₂ rich phase is the light phase, but at higher pressures this phase is the denser fluid phase. In a natural system of H₂O–CO₂ complete miscibility will not exist below 538 K; at higher temperatures a completely mixed supercritical fluid may exist, but at lower temperatures this fluid will segregate into two fluid phases. Fluid-fluid equilibria and critical curves in the system H₂O–CO₂ have been studied, among others by Tödheide and Franck [24], Heidemann and Khalil [25] and Blencoe et~al. [26].

The water – nitrogen system has been investigated by Tsiklis and Maslennikova [27], Prokhorova and Tsiklis [28], Japas and Franck [29]. Despite some differences in p – T – x conditions, the phase diagram topology of the three systems is not changed. The available data indicate that immiscibility is generally restricted to temperature below about 673 K. The critical curves consistently exhibit temperature minima. These minima occur at 60–70 MPa and ∼ 638 K in the H₂O–N₂ system [29], and in vicinity 155–190 MPa and ∼ 538 K in the H₂O–CO₂ system [25].

The system SCW – H₂ has been studied isochorically from 0.5 to 90 mol-% H₂ and up to 713 K and 250 MPa pressure using an autoclave containing two sapphire windows through which phase transitions could be observed at elevated temperatures and pressures. The system was found to exhibit so-called “gas-gas” immiscibility with a critical curve proceeding to higher temperatures and pressures from the critical point of pure water. Within the range of these experiments, the critical temperature of H₂–H₂O mixtures does not change any noticeable from that of pure water (e.g. T _c = 654.5 K at p _c = 25.2 MPa for 38 mol % H₂) [30].

The system water – ethanol at temperatures up to 673.15 K, including the saturation curve and the critical and supercritical regions, and at pressures up to 50 MPa for ethanol concentrations of 0.2, 0.5, and 0.8 mole fractions was studied particularly by Abdurashidova et~al. [31]. The data of p, ρ, T, x – measurements are used to determine the critical parameters of mixtures. The thermal decomposition of ethanol molecules is observed at a temperature above 623.15 K [31–34]. Both systems water – ammonia [53–55] and water – ethanol [31–34] are a classic example of simplest type I of phase behavior. We should emphasize that, although presented in the context of key components, the topics tackled could apply to thermodynamic models of other biomass components.

Theoretical comprehension of the topology of fluid phase equilibrium has proven to be very useful for the description of complex fluid phase relations, which are observed in multiple-component systems. Indeed, whereas the phase diagram of a one-component system is very simple, at least eight different types of phase diagrams have already been observed for binary systems. Many studies have shown that equations of state are able to generate the different kinds of fluid phase equilibrium (liquid-vapour, liquid-liquid, gas-gas and liquid-liquid-gas). The first pioneering work of this type was the study of van Konynenburg and Scott [20]. They have shown that the simplest van der Waals model reproduces qualitatively the major types of phase diagrams of binary fluids. A classification was proposed and is currently used as a basis, for the discrimination of different kinds of phase diagrams, in binary systems. Following Varchenko’s approach [35], generic phenomena encountered in binary mixtures when the pressure p and the temperature T change, correspond to singularities of the convex envelope (with respect to the x variable) of the “front” (a multifunction of the variable x) representing the Gibbs potential G(p,T,x). Pressure p and temperature T play the role of external model parameters like k ₁₂ . Although there are no theoretical limits in the extension of this approach for three- or multiple-component mixtures, classification studies – even for ternaries – are just at their beginning because of the amount of work that has to be carried out.

Our aim is to recognise a wide variety of phase diagrams from analysis of variations in geometry and energy characteristics (e.g., in critical volume and critical temperature) of mixture components. The influence of these two parameters on the phase diagram topology could be conveniently visualised on the master diagram, called a “global phase diagram”. Such a diagram shows the different areas of occurrence of the possible phase diagrams as a function of the geometry and energy factors of the compounds used. Ever since the work of van Konynenburg and Scott [20], numerous studies have been carried out on other, more realistic EoS [36, 37]. The global phase diagrams of such different models, as the one-fluid EoS of binary Lennard-Jones fluids [36] and the Redlich-Kwong model [37], are almost identical in their main features including such a sensitive phenomenon as the closed-loops of liquid-liquid immiscibility. Therefore, most of the considerations and conclusions made on the basis of one of the realistic models can be transferred vice versa.

The boundaries, between the various types in the global phase diagram, can be calculated directly using the thermodynamic description of the boundary states (tri-critical line, double critical end-points, etc.). The dimensionless co-ordinates that are used for the representation of the boundary states, depend on the equation of state model, but normally they are designed similar to those proposed by van Konynenburg and Scott for the van der Waals model [20]. In this case, the global phase diagrams of all realistic models have a very similar structure, in particular for the case of equal sized molecules. We consider here the classical Redlich-Kwong model as an example.

The Redlich-Kwong EoS [6] is used in its classical, non-modified, form:

$$ p=\frac{ RT}{\left( V- b\right)}-\frac{ a}{{ T}^{0.5} V\left( V- b\right)}, $$

(2.5)

where R is the universal gas constant, and parameters a and b of the mixture EoS depend on the mole fractions x _i and x _j of the components i and j and on the corresponding parameters a _ij and b _ij for different pairs of interacting molecules:

$$ \begin{array}{l} a={\displaystyle \sum_{i=1}^2{\displaystyle \sum_{j=1}^2{ x}_i{ x}_j{ a}_{i j}}}\\ {} b={\displaystyle \sum_{i=1}^2{\displaystyle \sum { x}_i{ x}_j{ b}_{i j}}}.\end{array} $$

(2.6)

The set of dimensionless parameters for the Redlich-Kwong model is as follows [37]:

$$ \begin{aligned}[b] { Z}_1&=\frac{{ d}_{22}-{ d}_{11}}{{ d}_{22}+{ d}_{11}},\\ {}{ Z}_2&=\frac{{ d}_{22}-2{ d}_{12}+{ d}_{11}}{{ d}_{22}+{ d}_{11}},\\ {}{ Z}_3&=\frac{{ b}_{22}-{ b}_{11}}{{ b}_{22}+{ b}_{11}},\\ {}{ Z}_4&=\frac{{ b}_{22}-2{ b}_{12}+{ b}_{11}}{{ b}_{22}+{ b}_{11}}\end{aligned} $$

(2.7)

where

$$ { d}_{ij}=\frac{{ T}_{ij}^{*}{ b}_{ij}}{{ b}_{ii}{ b}_{jj}},\kern0.5em { T}_{ij}^{*}={\left(\frac{\Omega_b{ a}_{ij}}{ R{\Omega}_a{ b}_{ij}}\right)}^{\raisebox{1ex}{$ 2$}\!\left/ \!\raisebox{-1ex}{$ 3$}\right.},\kern0.62em {\Omega}_a={\left[ 9\left({ 2}^{\raisebox{1ex}{$ 1$}\!\left/ \!\raisebox{-1ex}{$ 3$}\right.}{-} 1\right)\right]}^{- 1},\kern0.5em {\Omega}_b{=}\frac{{ 2}^{\raisebox{1ex}{$ 1$}\!\left/ \!\raisebox{-1ex}{$ 3$}\right.}{-} 1}{ 3}. $$

(2.8)

It should be brought to the attention that the dimensionless parameter Z ₁ represents the difference of the critical pressures of the components, and that the dimensionless parameter Z ₃ represents the difference of the critical volumes. So, there is a direct correlation of the global phase behaviour between mixtures and critical properties, i.e. geometry and energy parameters of real binary fluids.

Recently, Cismondi and Michelsen [38] introduced a procedure to generate different type of phase diagrams classified by van Konynenburg and Scott. Their strategy does not take into account an existence of solid phase. At present time Patel and Sunol [39] developed an automated and reliable procedure for systematic generation of global phase diagrams for binary systems. The approach utilizes equation of state, incorporates solid phase and is successful in generation of type VI phase diagram. The procedure enables automatic generation of GPD which incorporates calculations of all important landmarks such as critical endpoints (CEP), quadruple point (QP, if any), critical azeotropic points (CAP), azeotropic endpoints (AEP), pure azeotropic points (PAP), critical line, liquid–liquid–vapor line (L1L2V, if any), solid–liquid–liquid line (SL1L2, if any), solid–liquid–vapor line (SLV) and azeotropic line. The proposed strategy is completely general in that it does not require any knowledge about the type of phase diagram and can be applied to any pressure explicit equation of state model. Figure 2.8 shows the global phase diagram for binary mixtures of equal-sized molecules, plotted in the two-dimensional (Z ₁ – Z ₂ ) space [40].

Fig. 2.8

Global phase diagram of the Redlich-Kwong model

The simplest boundary is a normal critical point when two fluid phases are becoming identical. Critical conditions are expressed in terms of the molar Gibbs energy derivatives in the following way:

$$ {\left(\frac{\partial^2 G}{\partial { x}^2}\right)}_{p, T}={\left(\frac{\partial^3 G}{\partial { x}^3}\right)}_{p, T}= 0. $$

(2.9)

Corresponding critical conditions for the composition – temperature – volume variables are:

$$ \begin{aligned}[b] {}& { A}_{xx}- W{ A}_{xV}=0;\\ {}&{ A}_{xx x}-3 W{ A}_{xx V}+3{ W}^2{ A}_{xV V}-3{ W}^3{ A}_{VVV}=0;\end{aligned} $$

(2.10)

where A is the molar Helmholtz energy,

$W=\frac{{ A}_{xx}}{{ A}_{VV}}$, ${ A}_{m Vnx}={\left(\frac{\partial^{n+ m} A}{\partial { x}^n\partial { V}^m}\right)}_T$ is a contracted notation for differentiation operation which can be solved for V _C and T _C at given concentration x.

Without exception, one of the most important boundaries is visualised by the tri-critical points (TCP). This boundary divides the classes I and V, II and IV, or III and IV. The tri-critical state is a state, where the regions of the liquid-liquid-gas immiscibility shrink to one point, which is named the TCP. Three phases become identical at a TCP. Another important boundary in the global phase diagrams is the locus of double critical end-points (DCEP) that divides types III and IV, or II and IV. Type IV is characterised by two liquid-liquid-gas curves. One is at high temperatures and is restricted by two critical end points [one lower critical end point (LCEP) and an upper critical end point (UCEP)]. At the upper critical end point, the solution (compounds) become(s) immiscible as the temperature is lowered. At a lower critical end point, the solution separates into two phases as the temperature is increased. A DCEP occurs in a type IV when LCEP high-temperature three-phase region joins the UCEP of low-temperature three-phase region. A DCEP is produced in a type III system when the critical curve cuts tangentially the three-phase line in a pressure – temperature diagram. The types I and II, or IV and V, differ in the existence of a three-phase line, which goes from high pressures to an UCEP. For this case, the boundary situation is defined by the zero-temperature end point (ZTEP). Thermodynamic expressions and mathematical tools are given in the literature [37, 38].

Azeotropy in binary fluids can be easily predicted in the framework of global phase diagrams. The corresponding boundary situation is called the degenerated critical azeotropic point (CAP) and represents the limit of the critical azeotropy at x _i  → 0 or at x _i  → 1. This results in solving the system of thermodynamic equations for a degenerated critical azeotrope.

One may obtain the relationships for azeotropy boundaries from the global phase diagram [shaded A (Azeotropy) and H (Hetero-azeotropy)] regions in Fig. 2.8. The above azeotropic borders are straight lines in the (Z ₁ , Z ₂ )-plane that cross at a single point in the vicinity of the centre for equal sized molecules. It opens the opportunity for obtaining the series of inequalities to separate azeotropic and non-azeotropic regions of the global phase diagram. For the Redlich – Kwong EoS a corresponding relationship was obtained in an analytical form [41]:

$$ { Z}_2=\mp { Z}_1-\left( 1\pm { Z}_1\right)\left(\frac{ 1-{ Z}_4}{ 1\pm { Z}_3}- 1\right) 0.6731. $$

(2.11)

Global phase diagrams of binary fluids represent the boundaries between different types of phase behaviour in a dimensionless parameter space. In a real P – T – x space, two relatively similar components usually have an uninterrupted critical curve between the two critical points of the pure components. An upper branch extends from the critical point of the higher boiling component to higher pressures, sometimes passing through a temperature minimum as in the C₆H₆–H₂O system.

A general simplification of the theoretical description of ternary mixtures has been achieved by the consideration of the water + salt system as a single component mixture. This component has interaction parameters (or EoS parameters) dependent on the concentration of the salt. The details of the water-salt interactions have to be taken as averaged by an effective, spherically symmetric, interaction potential. This approach is justified by the phase behaviour of the system water + sodium chloride that belongs to the simplest type II of phase behaviour and has a very simple critical curve in the vicinity of the critical point of water. Three main types of phase topologies have been known in binary water-electrolyte systems. These are type-III (H₂O–HCl) system where hydrogen chloride is more volatile than water and a negative azeotropic line for water-rich compositions, at pressures below the saturation pressure of the water and the liquid-liquid-gas equilibrium, occur at pressures near saturation pressures of HCl, type-II (H₂O–NaCl, H₂O–NaOH), and type-IV phase diagrams (H₂O–Na₂WO₄, H₂O–K₂SO₄, H₂O–UO₂SO₄).

The topological predictions on the basis of a global phase diagram will become a convenient method in the analysis of supercritical water mixtures of scientific and industrial interest. Topologically, there is no difference in isoproperty behaviour for any pure fluid. This fact allows us to find the parameters of the equation of state model, which can reproduce thermodynamic properties of an arbitrary substance in a local region of a phase diagram.

To simulate the behavior of different biomass components in the SCW media, models based on equations of state are more than sufficient. Considering a huge array of reaction products, which wait in queue to be destroyed or recycled, it is obvious that there is need to develop theoretically sound methods for reliable assessment of their thermodynamic and phase behaviour, especially, at supercritical water conditions.

If combination rules are known, then it is possible to determine the global phase diagram via the ratio of critical parameters of pure components only [42]. The boundaries between different types of phase behavior taken from their paper and distribution of reduced critical temperatures and volumes for aqueous solutions of hydrocarbons are given in Fig. 2.9.

Fig. 2.9

Allocation of phase behavior types for hydrocarbon – water and key biomass components – water mixtures in reduced variables. ▲ – selected aqueous solutions of hydrocarbons performing III type of phase behavior: (1) cyclohexane; (2) benzene; (3) n-decane; and (4) n-hexane [43]; ♦ – selected aqueous solutions of key biomass components

Critical parameters of hydrocarbons were taken from correlation [15]. The simple estimation of transition from III type of phase behaviour to II type (T _r  = T _CH2O /T _C  ≈ 0.5) in terms of pure components for the one fluid model of the Redlich-Kwong EoS [44] and Carnahan-Starling – van der Waals EoS [42] with the Lorentz – Berthelot combination rules (k ₁₂  = 0) shows the serious discrepancies with experimental evidence of III type phase behaviour for systems water – alkanes [45, 46, 43]. Experimental values of k ₁₂ for water–alkanes systems show significant deviations from zero. Therefore, for a correct classification of phase behavior types, we should shift the lines DCEP to DCEP* (Fig. 2.9), which is achieved due to variations in the interaction parameters k ₁₂ and l ₁₂ . The DCEP* lines in Fig. 2.9 classify experimentally observed types of phase behavior more correctly for the SRK model with parameters corresponding the values k ₁₂  = 0.1 and l ₁₂  = 0.01. The Lorentz – Berthelot model is not working also for ammonia water and ethanol – water systems.

The traditional classification of fluid phase behavior can easily be discussed with the aid of the p-T projections of fluid phase diagrams. There are two kinds of phase diagrams. Phase diagrams of types I, V, and VI have the vapour pressure curves that are started and ended in nonvariant points with equilibria where the no solid phase exist. In the case of types II, III and IV, some critical curves, starting in high-temperature nonvariant points, are not ended by the nonvariant points from the lower temperature side, where the solid phase should exist. A solid phase is absent in calculations of fluid phase diagrams using previously discussed equations of state and the nonvariant equilibria with solid could not be obtained even at 0 K. Therefore the monovariant curves remain incomplete on the theoretical p – T projections. As a result these diagrams can be considered as the ‘derivative’ versions. It demonstrates not only the main types of fluid phase behavior but also the fluid phase diagrams which appear when the heterogeneous fluid equilibria are bounded not only by another fluid equilibrium but also by the equilibrium with solid phase that is usually observed in the most real systems [47].

To give an exhaustive description of phase behavior in binary systems, the systematic classification of the main types of complete phase diagrams that considers any equilibria between liquid, gas and/or solid phases in a wide range of temperatures and pressures was proposed by V. Valyashko [47].

In the new nomenclature, suggested by Valyashko [48] for systematic classification of binary complete phase diagrams, four main types of fluid phase behavior are designated in order of their continuous topological transformation as a, b, c and d types. In order to simplify the construction of phase diagrams he accepted the following limitations for the main types of binary complete phase diagrams:

• 1. The melting temperature of the pure nonvolatile component is higher than the critical temperature of the volatile component.

• 2. There are no solid-phase transformations such as polymorphism, formation of solid solutions and compounds, and azeotropy in liquid-gas equilibria in the systems under consideration.

• 3. Liquid immiscibility is terminated by the critical region at high pressures and cannot be represented by more than two separated immiscibility regions of different types.

Complete phase diagram presentation is very sophisticated and we refer the readers to the cited original Valyashko’s papers.

The design and development of SCW technology depend on the ability to model and predict accurately the solid-supercritical fluid equilibrium. It has been known for more than 100 years that a supercritical fluid can dissolve a substance of low volatility and that the solubility is dependent on the pressure. The ability to control solubility by means of pressure as well as temperature has brought about the use of supercritical fluids in different applications. For SCW technologies, knowledge of the solubility of substances in supercritical fluids is very important and a large number of experimental data has become available in recent years.

Hence, a more correct estimation of binary interaction parameters is needed to predict phase behaviour of biomass components in supercritical water. The best source of information for model parameter determination to restore critical curves in the SCW – biomass component mixtures is experimental data in vicinity of solution critical line. In dilute near-critical solutions, the partial molar properties of solutes, the coordinates of the critical lines of binary mixtures, and the temperature variations of the vapor–liquid distribution and Henry’s constants, are controlled by the critical value of the derivative ${ A}_{Kr}=-{\left(\frac{\partial A}{\partial V\partial x}\right)}_{T, x= 0}^c={\left(\frac{\partial P}{\partial x}\right)}_{T, V, x= 0}^c$ which is called the Krichevskii parameter.

Here A is the Helmholtz energy of the binary system, x is the mole fraction of a solute, P is the pressure, V is the volume of the system, and the superscript c indicates evaluation at the solvent critical point. The Krichevskii parameter is finite and has the dimension of pressure. The physical meaning of the Krichevskii parameter is that it gives the change in pressure at the critical point of the solvent when an infinitesimal amount of solvent molecules is replaced by solute at constant volume and temperature: for most neutral solutes the pressure will rise, and for solutes with strong attractions to the solvent (for example, ions in water) the pressure of the system will decrease [49]. The review of Fernández-Prini et~al. [50] presents data for 14 gases (He, Ne, Ar, Kr, Xe, H₂, N₂, O₂, CO, CO₂, H₂S, CH₄, C₂H₆, SF₆) in water with the accuracy of 2–6 % at high temperatures; similar uncertainties are expected for derived AKr results. Most comprehensive review of the significance of the Krichevskii parameter for the analysis of the phase equilibria, including the solubility, in binary systems are presented by Plyasunov [49]. Recommended values of the Krichevskii parameters for selected biomass components from [49] and [50] are given in Table 2.3.

Table 2.3 Recommended values of the Krichevskii parameters for biomass component solutes

6 Uncertainties and Conflicts in the Parameter Estimation for Thermodynamic Models

The reliable estimation of model parameters from experimental data in phase equilibria simulation is an important requirement in many applications. Such models offer the useful tools of aggregating large amounts of data, allow both data for interpolation of data and extensions beyond regions in which measurements have been made, and provide insight into physical and chemical phenomena. Transition from real phenomenon to its model entails the appearance of uncertainty caused by the statistical pattern of experimental information, inadequacy and ambiguity of used models, etc. Experimental data, which are generated by different experimental units, have as a rule different dimensions, different physical meaning, and different statistical distribution. It results to conflict situation when the set of parameters restored according to the one category of data does not correspond to parameters from other data sources. Therefore, the conflict appears in model parameter estimation and it is desirable to reduce an arising uncertainty by the simultaneous consideration of all data fitness criteria for each property.

Advanced methods of probability theory consider uncertainty as some specific value, characterizing an emergence of predetermined chance outcomes. Statistical methods interpret all variety of uncertainty types in the framework of the randomness concept. Nevertheless, there are ill-structured situations, which have not any strictly defined boundaries and cannot be accurately formulated. Examples of such situation are standard statements in the problem of model parameter estimation: “mean-square deviation of calculated values from measured data must be in neighborhood of feasible deviations δ” or “to provide the best fit to measured data”. The expressions “in neighborhood” or “best fit” have not exact boundaries which separate one class of objects from others. The vague verbal models such as “equation of state is valid in critical point neighborhood”, “high temperature approximation”, “adequate model”, and etc. are fuzzy formulated targets, which depend on biased assessment of boundaries for used approximations.

The problem of optimum parameter estimation in models of thermodynamic and phase behavior under the uncertainty is a multicriteria problem of mathematical programming. The first step in solution of multicriteria problem is a search of the Pareto set, i.e. such compromise domain in the permissible parameter space X where the value of any criterion cannot be improved without the value of the others criteria being worsened. The diverse computational methods of the Pareto-optimum parameter estimation and different (crisp and fuzzy) convolution schemes to reduce a vector criterion into the scalar one should be used. As examples of case study, we demonstrate the Pareto-optimum estimation of parameters in the Soave–Redlich–Kwong equation of state (EoS), using different conflicting data sets (simultaneous description of the phase equilibria and critical line data in binary mixtures).

The last two decades have been characterized by a growing comprehension of the fact that the ideas of uncertainty should not be neglected in the fluid phase equilibria modeling. Since one of the meanings of uncertainty is randomness, a conventional approach is utilized via probability and random process theories. However, as recognized recently, the probabilistic methods are accompanied by serious problems during its implementation in practice of phase equilibria modeling and lead to the unreliable estimation of parameters.

Development of reliable models for thermodynamic and phase behavior description of binary mixtures in most cases is associated with the estimation of the binary interaction parameters from the restricted set of VLE data. Although there are rigorous relationships to obtain from EoS model the different derivatives of thermodynamic values, for instance, the prediction of critical lines from the EoS with parameters restored from VLE data is questionable due to diverse sources of uncertainty both in the used models and experimental data.

To illustrate conflict between phase equilibria and critical line description we consider the water – carbon dioxide system in the vicinity of critical point of water. Experimental data on p – V – T – x properties have been taken from [51, 44] data on phase equilibria and critical curve have been extracted from [52, 24]. The critical lines have been calculated using algorithm developed in [25]. We performed phase equilibria and critical line calculations for the SRK EoS with parameters binary interaction parameters k ₁₂ and l ₁₂ from standard mixing rules,

$$ \begin{array}{l} b={ b}_{11}{ x}^2+{ l}_{12} x\left( 1- x\right)\left(\frac{{ b}_{11}+{ b}_{2 2}}{ 2}\right)+{ b}_{2 2}\left( 1- x\right){}^2\\ {} a={ a}_{11}{ x}^2+{ k}_{12} x\left( 1- x\right)\sqrt{{ a}_{11}{ a}_{2 2}}+{ a}_{2 2}\left( 1- x\right){}^2\end{array} $$

(2.12)

restored from different crisp convolution schemes.

For simplicity the results of calculation for phase equilibria and critical line are presented in Figs. 2.10 and 2.11 only for additive compromise scheme. The parameters restored from P-V-T-x data [51, 44] are not suitable for phase equilibria description and completely unusable for critical curve description. These parameters are skipped here. There are many speculations about problem of EoS singular behavior near critical point. Here we do not discuss the possible approaches to consistent description of regular and singular behavior of thermodynamic functions, but point out only the fact that classical EoS models, which are widely used in phase equilibrium calculations, cannot describe simultaneously phase equilibria and critical curves. The problem of conflict between parameters restored from different sections of thermodynamic surface is remained valid for more sophisticated EoS with singular behavior also. As a whole, it is observed the general conflict situation: if the cross-interaction parameters are restored from the one category of properties the prognosis of other properties is dubious in spite of indubitable validity of thermodynamic relationships. The final solution has subjective nature and depends on the statement of problem and decision maker experience.

Fig. 2.10

Critical lines of Н₂О–СО₂ system. – experimental data [24]. – best fit of VLE data. ∘∘∘∘ – compromise solution. --- --- – best fit of critical curve

Fig. 2.11

Phase equilibria in Н₂О–СО₂ system. – experiment T = 548.15 K. – best fit of VLE data. ∘∘∘∘ – сompromise solution. --- --- – best fit of critical curve

7 Conclusion

This study is one of the first attempts to establish and demonstrate multiple links existing among the phase equilibria phenomena in supercritical aqueous mixtures with biomass components and their models. From the very beginning of these efforts, the global phase diagrams have been a very useful tool for scientists and engineers working in the field of emerging SCW technologies. There is no doubt that extension of our knowledge about global phase behaviour of two- and multicomponent fluids will lead to the creation of reliable engineering recipes for solving the actual problems of SCW technology applications in biomass conversion. Unfortunately, there are no rigorous mathematical methods and sound physical concepts to construct the adequate model without enlisting of subjective point of view due to inaccuracy of our knowledge. To minimize this source of unavoidable uncertainty the different compromise schemes of criteria convolution are considered and their selection is strongly depended on decision maker experience. Here, by way of illustration, it has been applied to estimate the Redlich-Kwong-Soave EoS parameters for simultaneous description of the phase equilibria and critical line data in binary mixtures.