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Modeling of Phase Equilibria in Reactive Mixtures

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Modeling of Phase Equilibria in Reactive Mixtures Sumitomo Chemical Co., Ltd. Modeling of Phase Equilibria in Industrial Technology & Research Laboratory Reactive Mixtures Tetsuya SUZUTA Most chemical processes contain at least two of the three phases of solid, liquid and gas. Therefore it is essen- tial to fully understand phase equilibria for the logical design of processes and apparatus. The basic theory of phase equilibrium has already been established, but there are some exceptions which are not compliant with the theory. Many of those are influenced by chemical reactions within a certain phase or between different phases, so consideration of the reaction is necessary for precise reproduction of such unusual behavior. In this paper, some models for phase equilibria in reactive mixtures and several examples of their application in industrial plants are presented. This paper is translated from R&D Report, “SUMITOMO KAGAKU”, vol. 2018. Introduction1), 2) Modeling of Phase Equilibria Involving Chemical Reaction There are three states of matter: solid; liquid; and gas. Phase equilibrium is defined as a state in which 1. Chemical Reaction in Phase Equilibria matter and heat are balanced between at least two During a chemical reaction, a substance recombines phases and each phase is present stably. The phase- the atoms and atomic groups by itself or mutually with equilibrium relationship of the temperature, pressure other substances and generates a new substance. Some and the component composition is one of the impor- chemical reactions involve the cleavage or formation of tant properties in chemical processes. Numerous covalent bonding, and some occur due to the changes measured data of phase equilibrium had been accumu- in ion bonding or hydrogen bonding. The former lated and released in the DETHERM published by the requires an activation energy to progress with a limited DECHEMA (Deutsche Gesellschaft für Chemisches reaction rate, whereas the latter reaches chemical equi- Apparatewesen, Germany)3) and Chemistry WebBook4) librium (the state in which forward and reverse reac- published by the NIST (the National Institute of Stan- tions balance each other) within a short period of time. dards and Technology, USA). Furthermore, various Either reaction affects phase equilibrium. In the former models that can express the phase-equilibrium rela- case, as the composition of the phase in which a chemi- tionship of temperature, pressure and composition in cal reaction is occurring changes with time, the compo- a broad range have been developed based on the sitions of other phases change as well. However, in the measured data. Additionally, computer simulations latter case it will reach equilibrium in short time, where- using such models are used for designing and analyz- by the composition of each phase will become constant. ing the chemical process. In the next section we will briefly introduce the basic This paper introduces an example that can express phase equilibrium model (physical model) first, and sub- actual phase equilibrium with high accuracy by model- sequently introduce modeling of the phase equilibrium ing a system involving a chemical reaction, of which involving chemical reactions. behavior cannot be expressed merely by a physical model generally used for expressing an equilibrium 2. Fundamental Equations for Phase Equilibria5) relationship (e.g., state equation, activity coefficient As described in standard physical chemistry text- equation), taking into account the reaction concerned. books, the ideal gas means the total absence of inter- SUMITOMO KAGAKU 2018 1 Modeling of Phase Equilibria in Reactive Mixtures molecular force (assuming that molecules don’t have position. Thus far, we have introduced the fundamental any size). On the other hand, the ideal solution means equation of the vapor-liquid equilibrium relationship by that in a binary system of molecules A and B for exam- order from top to bottom (i.e., from the ideal system to ple, A and B have the same size and the intermolecular equation (3)). The left side and right side of equation forces between A and A, B and B, and A and B are all (3) represent the fugacity of the vapor phase and that of the same. In this case, the tendency that molecule A the liquid phase, respectively. As described in standard enters into the vapor phase (indicated by the partial physical chemistry textbooks, the phase-equilibrium pressure of A in the vapor phase) is proportional to xA, conditions will be fulfilled when the temperature T, pres- the molar fraction of A. Particularly, it is said that when sure p and chemical potential of components are the the factor of proportionality is equivalent to the vapor same in all phases. Equation (4) represents the relation- pressure pA° of the pure substance A at a specific tem- ship between the fugacity f and chemical potential μ, perature, it conforms to Raoult’s law. The solution in showing that the equal fugacity and equal chemical which all its compositions conform to the Raoult’s law potential means the same. is defined as an ideal solution (° represents a pure sub- stance.) [dμi = RTdlnfi]T (4) pA = pA°xA (1) In many cases, under the low-pressure condition, the vapor phase fugacity coefficient can be considered to be In the ideal gas the partial pressure is the product of 1 (the exceptional case is the main theme of this paper, the total pressure p and the molar fraction in the vapor and it will be discussed later). Under the high-pressure phase yA. Consequently, the vapor-liquid equilibrium condition, the vapor-liquid equilibrium relationship will relationship of the system in which the gas phase is an be obtained by calculating the fugacity of both vapor and ideal gas and the liquid phase is an ideal solution can liquid phases using the thermodynamic relation equa- be expressed through the following equation: tion (5) (which is a state equation applicable to both vapor and liquid phases) instead of equation (3). How- pyA = pA°xA (2) ever, we will not discuss its details here. Up to around a moderate pressure, the vapor phase Therefore, in order to calculate the vapor-liquid equilib- fugacity coefficient can be calculated using the thermo- rium relationship of the so-called ideal system made of dynamic relationship given in equation (5) as well as the ideal gas and ideal solution, the pure substance the virial-state equation (6) with the second virial coef- vapor pressure (the function of the temperature alone) ficient. of each component is required. ∞ ∂p RT Because the intermolecular interaction acts on a real RTlnφi =∫ – dV – RT lnZmix (5) V ∂niT,V,nj≠i V gas (molecules having considerable size) and the molec- B ular size and intermolecular force are not consistent in Z = 1 + mix p (6) mix RT a real solution, the correction from the ideal system is required. Thus the fugacity coefficient φ will be intro- In these equations Zmix and Bmix represent the com- duced to the vapor phase, and the activity coefficient γ pressibility factor of the mixture and the second virial will be introduced to the liquid phase to express the coefficient, respectively. deviation from the ideal system. The fundamental equa- Modeling of the activity coefficient is a main chal- tion of the vapor-liquid equilibrium of component i in lenge in the low-pressure phase equilibrium. For a general multicomponent systems is as follows: “physical model” created in consideration of intermole- cular interaction, a model that can be extended to the φipyi = γi p°i xi (3) multicomponent system using a constant for the binary system (a parameter in the activity coefficient equation When calculating the vapor-liquid equilibrium in a real obtained by correlating the vapor-liquid equilibrium rela- system, it also requires another process, which is to tionship of the binary system) (i.e., a model by which understand the fugacity coefficient and activity coeffi- the vapor-liquid equilibrium relationship of the binary cient as functions of the temperature, pressure and com- system from the binary system data can be estimated) SUMITOMO KAGAKU 2018 2 Modeling of Phase Equilibria in Reactive Mixtures ∑ ∑ is used. There is an activity coefficient model, i.e., the δG = i j μAiBj δnAiBj (10) NRTL equation6). This equation can be applied to the ∑ ∑ ∑ ∑ liquid-liquid equilibrium of a system that is separated = μA1 i j idnAiBj + μB1 i j jdnAiBj into two liquid phases. This model is often built into process simulators and widely used. = μA1 δnA + μB1 δnB (11) 3. Phase Equilibria of the System Involving However, if the macroscopically viewed chemical Reaction7),8) potential of the solution is used, the Gibbs energy (1) Fundamental Equation change will be as follows: It is difficult to accurately express the vapor-liquid equilibrium relationship of systems containing compo- δG = μA δnA + μB δnB (12) nents that strongly associate or dissociate in the liquid phase, or those that associate in the vapor phase, merely From equations (11) and (12), the extremely important by a physical model. In such systems the actual phase- theorems independent of association pattern can be equilibrium relationship can often be explained ration- obtained as follows: ally by handling it in consideration of a new “chemical species” generated by the “chemical reaction.” This type μA = μA1 (13) of model, in which a new chemical species has been con- sidered, is referred to as a “chemical model” versus a μB = μB1 (14) “physical model.” The following description is also based on the con- Additionally, this relationship can be established for dis- cept of a
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