♨️Thermodynamics of Fluids Unit 2 – Properties of Pure Substances

Key Concepts and Definitions

• Pure substance consists of a single chemical species with a fixed chemical composition and distinct physical properties
• Phase refers to a particular state of matter (solid, liquid, or gas) with uniform physical properties
• Saturation condition occurs when a pure substance coexists in two or more phases in equilibrium (e.g., liquid-vapor equilibrium)
• Critical point represents the highest temperature and pressure at which liquid and vapor phases can coexist
  • Beyond the critical point, the substance exists as a supercritical fluid
• Triple point is the unique state where all three phases (solid, liquid, and gas) coexist in equilibrium
• Equation of state is a mathematical relationship that relates the pressure, volume, and temperature of a substance
• Ideal gas is a hypothetical gas that perfectly follows the ideal gas law ($PV = nRT$)
• Real gas deviates from the ideal gas behavior due to intermolecular forces and molecular size effects

Phase Diagrams and State Changes

• Phase diagram graphically represents the equilibrium states of a pure substance as a function of pressure and temperature
• Sublimation is the direct transition from solid to gas phase without passing through the liquid phase (e.g., dry ice)
• Vaporization (boiling) occurs when a liquid transforms into a gas at its saturation temperature and pressure
• Condensation is the reverse process of vaporization, where a gas transforms into a liquid
• Melting is the transition from solid to liquid phase at the melting point temperature
• Freezing is the reverse process of melting, where a liquid transforms into a solid
• Solid-solid phase transitions involve changes in the crystal structure without passing through the liquid phase (e.g., graphite to diamond)
• Triple point represents the unique combination of pressure and temperature where all three phases coexist in equilibrium
  • For water, the triple point occurs at 0.01°C and 611.73 Pa

P-v-T Behavior of Pure Substances

• P-v-T surface is a three-dimensional representation of the relationship between pressure, specific volume, and temperature for a pure substance
• Isotherms are lines of constant temperature on the P-v diagram
  • For an ideal gas, isotherms appear as hyperbolas ($PV = constant$)
• Isobars are lines of constant pressure on the v-T diagram
• Isochores are lines of constant specific volume on the P-T diagram
• Saturated liquid line represents the states where the liquid is in equilibrium with its vapor
• Saturated vapor line represents the states where the vapor is in equilibrium with its liquid
• Critical isotherm passes through the critical point and separates the liquid and vapor regions
• Compressed liquid region lies between the saturated liquid line and the critical isotherm

Ideal Gas Law and Its Limitations

• Ideal gas law relates the pressure, volume, temperature, and amount of an ideal gas: $PV = nRT$
  • $P$ is the absolute pressure, $V$ is the volume, $n$ is the number of moles, $R$ is the universal gas constant, and $T$ is the absolute temperature
• Ideal gas law assumes that gas molecules have negligible volume and no intermolecular forces
• Ideal gas law is most accurate at low pressures and high temperatures, where intermolecular forces are less significant
• Compressibility factor ($Z$) quantifies the deviation of a real gas from ideal behavior: $Z = \frac{PV}{nRT}$
  • For an ideal gas, $Z = 1$; for real gases, $Z$ deviates from unity
• Ideal gas law fails to accurately describe the behavior of gases at high pressures and low temperatures due to intermolecular forces and molecular size effects

Equations of State for Real Gases

• Van der Waals equation of state accounts for the finite molecular size and intermolecular attractions of real gases: $\left(P + \frac{a}{V_m^2}\right)(V_m - b) = RT$
  • $a$ is a measure of the attractive forces between molecules, and $b$ is the volume occupied by the molecules themselves
• Redlich-Kwong equation of state is another cubic equation that improves upon the Van der Waals equation: $P = \frac{RT}{V_m - b} - \frac{a}{\sqrt{T}V_m(V_m + b)}$
• Peng-Robinson equation of state is widely used in the oil and gas industry for its accuracy in predicting the properties of hydrocarbons: $P = \frac{RT}{V_m - b} - \frac{a\alpha}{V_m(V_m + b) + b(V_m - b)}$
• Virial equation of state is a power series expansion in terms of the molar density or inverse molar volume: $\frac{PV_m}{RT} = 1 + \frac{B}{V_m} + \frac{C}{V_m^2} + \cdots$
  • $B$ and $C$ are the second and third virial coefficients, respectively, which depend on temperature and the specific substance

Thermodynamic Properties and Relations

• Thermodynamic properties are measurable characteristics of a system that describe its state and energy content (e.g., pressure, temperature, volume, internal energy, enthalpy, entropy)
• Extensive properties depend on the size or extent of the system (e.g., volume, mass, internal energy)
• Intensive properties are independent of the system size (e.g., pressure, temperature, density)
• Specific properties are extensive properties per unit mass (e.g., specific volume, specific enthalpy)
• Fundamental thermodynamic relation expresses the change in internal energy as a function of changes in entropy, volume, and composition: $dU = TdS - PdV + \sum_i \mu_i dN_i$
  • $U$ is the internal energy, $T$ is the temperature, $S$ is the entropy, $P$ is the pressure, $V$ is the volume, $\mu_i$ is the chemical potential of component $i$, and $N_i$ is the number of moles of component $i$
• Maxwell relations are derived from the fundamental thermodynamic relation and relate the partial derivatives of thermodynamic properties (e.g., $\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V$)

Applications in Engineering Systems

• Refrigeration cycles utilize the phase changes of a working fluid to transfer heat from a low-temperature source to a high-temperature sink
  • Vapor-compression refrigeration is the most common type, used in household refrigerators and air conditioners
• Power cycles convert heat into mechanical work by exploiting the thermodynamic properties of a working fluid
  • Rankine cycle is the basis for steam power plants, where water undergoes phase changes to drive a turbine
  • Brayton cycle is used in gas turbines and jet engines, where a gas (usually air) is compressed, heated, and expanded to generate power
• Heat exchangers facilitate the transfer of heat between two fluids without direct mixing
  • Shell-and-tube heat exchangers are widely used in chemical processing and power generation
  • Plate heat exchangers offer high heat transfer efficiency in a compact design
• Compressors and pumps are used to increase the pressure of gases and liquids, respectively
  • Centrifugal compressors are common in large-scale industrial applications, such as natural gas processing and air separation units
  • Positive displacement pumps (e.g., reciprocating, gear, and screw pumps) are used for high-pressure and high-viscosity applications

Problem-Solving Techniques

• Identify the system and surroundings, specifying the system boundaries and interactions
• Determine the process type (e.g., isothermal, isobaric, adiabatic) and the relevant thermodynamic properties
• Apply the appropriate equations of state, thermodynamic relations, and conservation laws (e.g., mass, energy, entropy balances)
• Use property tables, charts, or equations to obtain the necessary thermodynamic data for the specific substance
  • Steam tables provide properties of water and steam at various states
  • Refrigerant property tables are used for common refrigerants (e.g., R-134a, R-410A)
• Solve the governing equations analytically or numerically, depending on the complexity of the problem
• Interpret the results in the context of the physical system and validate them using appropriate assumptions and approximations
• Perform sensitivity analyses to assess the impact of uncertainties in input parameters on the solution
• Communicate the findings effectively using appropriate graphs, tables, and diagrams to visualize the thermodynamic processes and states