🍳Separation Processes Unit 2 – Thermodynamics and Phase Equilibria

Key Concepts and Definitions

• Thermodynamics studies the interrelationships between heat, work, and energy in systems
• Phase refers to a distinct, homogeneous part of a system with uniform physical and chemical properties (solid, liquid, or gas)
• Equilibrium describes a state where no net changes occur in the macroscopic properties of a system over time
• Intensive properties are independent of the amount of material in the system (temperature, pressure, density)
• Extensive properties depend on the size or extent of the system (volume, mass, energy)
• State variables define the current condition of a system and determine its thermodynamic properties (pressure, temperature, volume)
• Process variables describe how a system changes from one state to another (heat, work)
• Ideal gas assumes molecules have negligible volume and no intermolecular forces, follows the ideal gas law ($PV = nRT$)

Thermodynamic Laws and Principles

• Zeroth Law of Thermodynamics states that if two systems are in thermal equilibrium with a third system, they are in thermal equilibrium with each other
• First Law of Thermodynamics expresses conservation of energy, stating that energy cannot be created or destroyed, only converted from one form to another
  • Mathematically represented as $\Delta U = Q - W$, where $\Delta U$ is the change in internal energy, $Q$ is heat added to the system, and $W$ is work done by the system
• Second Law of Thermodynamics introduces the concept of entropy, stating that the total entropy of an isolated system always increases over time
  • Entropy quantifies the amount of disorder or randomness in a system
• Third Law of Thermodynamics states that the entropy of a perfect crystal at absolute zero is zero
• Gibbs Free Energy ($G$) predicts the spontaneity of a process at constant temperature and pressure, defined as $G = H - TS$
  • Processes with $\Delta G < 0$ are spontaneous, $\Delta G > 0$ are non-spontaneous, and $\Delta G = 0$ are at equilibrium
• Enthalpy ($H$) represents the total heat content of a system, defined as $H = U + PV$

Phase Equilibrium Fundamentals

• Phase equilibrium occurs when two or more phases of a substance have the same temperature, pressure, and chemical potential
• Chemical potential ($\mu$) is the molar Gibbs free energy and represents the driving force for mass transfer between phases
  • At equilibrium, the chemical potential of a component is equal in all phases
• Raoult's Law states that the partial vapor pressure of a component in an ideal solution is equal to the product of its mole fraction and its pure component vapor pressure
  • Mathematically expressed as $P_i = x_i P_i^*$, where $P_i$ is the partial vapor pressure of component $i$, $x_i$ is its mole fraction in the liquid phase, and $P_i^*$ is its pure component vapor pressure
• Dalton's Law states that the total pressure of a gas mixture is equal to the sum of the partial pressures of its components
  • Mathematically expressed as $P_{total} = \sum_{i=1}^n P_i$, where $P_{total}$ is the total pressure and $P_i$ is the partial pressure of component $i$
• Henry's Law describes the solubility of a gas in a liquid, stating that the partial pressure of a gas above a solution is proportional to its mole fraction in the solution
  • Mathematically expressed as $P_i = H_i x_i$, where $P_i$ is the partial pressure of component $i$, $H_i$ is its Henry's law constant, and $x_i$ is its mole fraction in the liquid phase

Vapor-Liquid Equilibrium (VLE)

• VLE describes the distribution of components between the vapor and liquid phases at equilibrium
• Relative volatility ($\alpha$) measures the ease of separation of two components in a mixture, defined as the ratio of their K-values
  • $\alpha_{ij} = \frac{K_i}{K_j}$, where $K_i$ and $K_j$ are the K-values of components $i$ and $j$, respectively
• K-value ($K_i$) is the ratio of a component's mole fraction in the vapor phase to its mole fraction in the liquid phase at equilibrium
  • $K_i = \frac{y_i}{x_i}$, where $y_i$ and $x_i$ are the mole fractions of component $i$ in the vapor and liquid phases, respectively
• Azeotrope is a mixture that boils at a constant temperature and has the same composition in the vapor and liquid phases
  • Azeotropic mixtures cannot be separated by simple distillation
• Bubble point is the temperature at which the first vapor bubble forms when a liquid is heated at a given pressure
• Dew point is the temperature at which the first liquid droplet forms when a vapor is cooled at a given pressure
• VLE data can be represented using xy, Txy, and Pxy diagrams

Equations of State and Activity Models

• Equations of state (EOS) describe the relationship between temperature, pressure, and volume of a substance
  • Examples include the ideal gas law, van der Waals equation, and Peng-Robinson equation
• EOS are used to predict thermodynamic properties and phase behavior of pure components and mixtures
• Cubic equations of state (van der Waals, Redlich-Kwong, Soave-Redlich-Kwong, Peng-Robinson) are widely used in the oil and gas industry
  • They account for the non-ideal behavior of gases and liquids by introducing parameters that consider molecular size and intermolecular attractions
• Activity coefficient models (Margules, van Laar, Wilson, NRTL, UNIQUAC) describe the non-ideal behavior of liquid mixtures
  • They relate the activity coefficients of components to their mole fractions and binary interaction parameters
• Activity coefficient ($\gamma_i$) is a correction factor that accounts for the non-ideal behavior of a component in a mixture
  • It is defined as the ratio of a component's fugacity in the mixture to its fugacity in an ideal solution at the same conditions
• Fugacity ($f_i$) is a corrected partial pressure that accounts for the non-ideal behavior of a component in a mixture
  • At equilibrium, the fugacity of a component is equal in all phases

Phase Diagrams and Their Interpretation

• Phase diagrams graphically represent the equilibrium relationships between the phases of a substance as a function of temperature, pressure, and composition
• Unary (single-component) phase diagrams show the regions of stability for solid, liquid, and vapor phases
  • Important features include the triple point (solid, liquid, and vapor coexist), critical point (liquid and vapor become indistinguishable), and sublimation curve (solid-vapor equilibrium)
• Binary (two-component) phase diagrams display the equilibrium behavior of mixtures, such as VLE, LLE (liquid-liquid equilibrium), and SLE (solid-liquid equilibrium)
  • Types of binary phase diagrams include ideal (zeotropic), azeotropic, and partially miscible systems
• Ternary (three-component) phase diagrams represent the equilibrium behavior of three-component mixtures using an equilateral triangle
  • Tie lines connect the compositions of coexisting phases at equilibrium
• Phase diagrams are essential for understanding and designing separation processes, such as distillation, extraction, and crystallization

Applications in Separation Processes

• Distillation separates components based on their relative volatilities, utilizing the difference in boiling points
  • VLE data and phase diagrams are crucial for designing and optimizing distillation columns
• Absorption and stripping involve the transfer of a solute between a gas and a liquid phase
  • Henry's law and activity models are used to predict the solubility and mass transfer of components
• Extraction separates components based on their relative solubilities in two immiscible liquid phases
  • LLE data and ternary phase diagrams are essential for designing and optimizing extraction processes
• Adsorption separates components by selective adhesion to a solid surface
  • Adsorption isotherms (Langmuir, Freundlich) describe the equilibrium relationship between the adsorbed amount and the concentration in the fluid phase
• Membrane separations utilize differences in permeability and selectivity of components through a semi-permeable barrier
  • Phase equilibrium and activity models are used to predict the driving force and separation efficiency
• Crystallization separates components based on their different solubilities in a solvent
  • SLE data and phase diagrams are essential for designing and optimizing crystallization processes

Problem-Solving Techniques

• Identify the relevant thermodynamic properties, state variables, and process variables for the given problem
• Determine the appropriate thermodynamic laws, principles, and equations applicable to the problem
• Recognize the type of phase equilibrium (VLE, LLE, SLE) and the corresponding data or models required
• Apply the suitable equations of state or activity models to predict the thermodynamic properties and phase behavior
• Interpret phase diagrams to understand the equilibrium relationships and feasibility of separation processes
• Use mass and energy balances to analyze and solve problems related to separation processes
• Employ iterative methods (e.g., bubble point, dew point, flash calculations) to solve phase equilibrium problems
• Validate the results using thermodynamic consistency tests and compare them with experimental data or literature values
• Perform sensitivity analyses to understand the impact of process variables and model parameters on the separation performance