Thermocalculator

How Thermocalculator Works

Overview

This calculator uses a two-part approach to determine thermodynamic properties of real gases and liquids:

  1. Ideal Gas Properties: Temperature-dependent correlations that describe how properties behave in the ideal (low-pressure) limit
  2. Departure Functions: Corrections from the Peng-Robinson Equation of State (PR-EOS) that account for real gas behavior at elevated pressures

The final property is calculated as: Real Property = Ideal Property + Departure Function

This approach is standard in chemical engineering because it separates temperature effects (which are well-described by polynomial correlations) from pressure effects (which require an equation of state).

Core Model: Peng-Robinson Equation of State

The Peng-Robinson (PR) Equation of State is a cubic equation that describes how real gases deviate from ideal behavior. It's particularly accurate for hydrocarbons and is widely used in the chemical industry.

What it Does

The PR-EOS relates pressure (P), temperature (T), and molar volume (v) through a cubic equation. Instead of solving for volume directly, we solve for the compressibility factor (Z), which is a dimensionless measure of how much a gas deviates from ideal behavior:

  • Z = 1: Ideal gas behavior (low pressure, high temperature)
  • Z < 1: Gas is more compressible than ideal (attractive forces dominate)
  • Z > 1: Gas is less compressible than ideal (repulsive forces dominate)

Key Parameters

The PR-EOS requires three critical properties for each compound:

  • Critical Temperature (Tc): The temperature above which a gas cannot be liquefied
  • Critical Pressure (Pc): The pressure at the critical point
  • Acentric Factor (ω): A measure of molecular asymmetry and polarity

These are stored in the database for each compound and used to calculate:

  • a: Attraction parameter (related to intermolecular forces)
  • b: Repulsion parameter (related to molecular volume)
  • κ: Temperature correction factor (depends on acentric factor)

Compressibility Factor Calculation

The compressibility factor is found by solving a cubic equation:

Z³ + (B - 1)Z² + (-3B² - 2B + A)Z + (B³ + B² - AB) = 0

Where:

  • A and B are dimensionless parameters that depend on pressure, temperature, and the compound's critical properties
  • The equation can have 1 or 3 real roots:
    • Gas phase: Use the largest root (highest Z)
    • Liquid phase: Use the smallest root (lowest Z)

Temperature-Dependent Properties

These properties are calculated using empirical correlations (polynomial or exponential equations) that are fitted to experimental data. Each correlation has a specific temperature range where it's valid.

1. Vapor Pressure

What it is: The pressure at which a liquid and its vapor are in equilibrium at a given temperature.

Equations Used:

  • Equation 10 (Antoine-type): P = exp(A - B/(C + T))
  • Equation 101: P = exp(A + B/T + C·ln(T) + D·T^E)
  • Equation 200: A more complex form that includes critical temperature scaling

Limitations: Each compound has a specific temperature range (Tmin to Tmax) where the correlation is valid, typically from triple point to near critical temperature.

2. Heat Capacity (Ideal Gas and Liquid)

What it is: The amount of heat required to raise the temperature of one mole of substance by 1 Kelvin.

Ideal Gas Heat Capacity (used for gas phase):

  • Equation 3: Cp = A + B·T + C·T²
  • Equation 4: Cp = A + B·T + C·T² + D·T³
  • Equation 16: Cp = A + exp(B/T + C + D·T + E·T²)
  • Equation 100: Cp = A + B·T + C·T² + D·T³ + E·T⁴
  • Equation 107: Cp = A + B·((C/T)·sinh(C/T))² + E·((D/T)/cosh(D/T))²

Liquid Heat Capacity (used for liquid phase):

  • Similar polynomial forms (Equations 3, 4, 16, 100)

Limitations: Each correlation has temperature limits. Ideal gas correlations typically work from ~50K to 1500K, while liquid correlations have narrower ranges (often 75K to 300K).

3. Heat of Vaporization

What it is: The energy required to convert one mole of liquid to vapor at a given temperature.

Equation Used: Equation 106 (Watson correlation):

ΔHvap = A·(1 - Tr)^(B + C·Tr + D·Tr² + E·Tr³)

Where Tr = T/Tc is the reduced temperature.

This correlation accounts for the fact that heat of vaporization decreases as temperature approaches the critical point (where it becomes zero).

Limitations: Valid from triple point to critical temperature.

Pressure-Dependent Properties (From PR-EOS)

These properties are calculated using departure functions that correct ideal gas values based on the compressibility factor and PR-EOS parameters.

1. Enthalpy (H)

Calculation:

H = H_ideal(T) - H_ideal(T_ref) + H_departure(P, T)

Ideal Enthalpy: Calculated by integrating heat capacity from reference temperature to T:

H_ideal(T) = ∫[T_ref to T] Cp_ideal(T) dT

Enthalpy Departure: Correction for real gas behavior:

H_departure = R·T·(Z - 1 - (A/(√8·B))·(1 + κ·√(Tr/α))·ln[(Z + (√2 + 1)·B)/(Z - (√2 - 1)·B)])

This accounts for:

  • Work done against intermolecular forces (the Z - 1 term)
  • Temperature-dependent attraction effects (the logarithmic term)

2. Entropy (S)

Calculation:

S = S_ideal(T) + S_departure(P, T)

Ideal Entropy: Includes temperature integration and pressure correction:

S_ideal(T) = ∫[T_ref to T] (Cp_ideal(T)/T) dT - R·ln(P/P_ref)

Entropy Departure:

S_departure = R·ln(Z - B) - (A·R)/(√8·B)·κ·√(Tr/α)·ln[(Z + (√2 + 1)·B)/(Z - (√2 - 1)·B)]

3. Heat Capacity at Constant Pressure (Cp)

Calculation:

Cp = Cp_ideal(T) + Cp_departure(P, T)

Departure Function: Includes derivatives of the compressibility factor with respect to temperature, accounting for how intermolecular forces change with temperature.

4. Density (ρ)

Calculation:

ρ = M / v = M·P / (Z·R·T)

Where:

  • M = Molecular weight
  • v = Molar volume = Z·R·T/P (from ideal gas law, corrected by Z)

5. Fugacity (f)

What it is: An "effective pressure" that accounts for non-ideal behavior. For ideal gases, fugacity equals pressure.

Calculation:

f = P·exp(Z - 1 - ln(Z - B) - (A/(√8·B))·ln[(Z + (√2 + 1)·B)/(Z - (√2 - 1)·B)])

Fugacity is crucial for phase equilibrium calculations and chemical reaction equilibria.

6. Internal Energy (U)

Calculation:

U = H - P·v = H - Z·R·T

The difference between enthalpy and the P·v work term.

7. Gibbs Free Energy (G)

Calculation:

G = H - T·S

Used for determining phase equilibria and chemical reaction directions.

8. Helmholtz Free Energy (A)

Calculation:

A = U - T·S

Used in closed-system thermodynamics and statistical mechanics.

Limitations and Validation

Temperature Limits

Each temperature-dependent correlation has specific validity ranges:

  • Vapor Pressure: Typically from triple point (~50-100K) to near critical temperature
  • Ideal Gas Heat Capacity: Usually 50K to 1500K
  • Liquid Heat Capacity: Narrower ranges, often 75K to 300K
  • Heat of Vaporization: From triple point to critical temperature

Why limits exist:

  • Correlations are fitted to experimental data within specific ranges
  • Extrapolation beyond these ranges can produce physically unrealistic or inaccurate results
  • Near critical point, properties change rapidly and correlations may break down

Pressure Limits

Maximum Pressure: Limited to 10 × Critical Pressure

  • Beyond this, the PR-EOS becomes less accurate
  • Very high pressures may require more sophisticated equations of state
  • The cubic equation may not have physically meaningful roots

Minimum Pressure: Must be greater than 0

  • Negative pressures are unphysical
  • At very low pressures, ideal gas behavior is recovered (Z → 1)

Model Limitations

  1. PR-EOS Accuracy:

    • Best for non-polar and slightly polar compounds (hydrocarbons, noble gases)
    • Less accurate for highly polar compounds (water, alcohols) or associating molecules
    • Accuracy decreases near critical point

  2. Single-Phase Assumption:

    • The model calculates properties for either gas OR liquid phase
    • It does not automatically detect phase transitions
    • User must specify the phase

  3. Pure Component Only:

    • Calculations are for pure substances
    • Mixture properties would require mixing rules (not implemented)

  4. No Chemical Reactions:

    • Properties are for stable molecular forms only
    • Does not account for decomposition or reaction at extreme conditions

Calculation Flow

  1. Load Compound Data: Critical properties, correlation coefficients, and temperature limits from database
  2. Calculate PR-EOS Parameters: a, b, κ from critical properties
  3. Calculate Compressibility Factor: Solve cubic equation for Z
  4. Calculate Temperature-Dependent Properties:
    • Vapor pressure, heat capacity, heat of vaporization using correlations
  5. Calculate Ideal Gas Properties:
    • Integrate heat capacity for enthalpy/entropy
  6. Calculate Departure Functions:
    • Correct ideal values for real gas behavior
  7. Calculate Composite Properties:
    • Density, fugacity, internal energy, Gibbs/Helmholtz free energy

Units

  • Temperature: Kelvin (K)
  • Pressure: Bar (bar) - input and output
  • Enthalpy/Internal Energy: kJ/mol
  • Entropy: kJ/(mol·K)
  • Heat Capacity: kJ/(mol·K)
  • Density: g/cm³
  • Fugacity: Bar (dimensionless coefficient, multiplied by pressure)
  • Vapor Pressure: Bar (converted from Pascal in database)

References

The equations and methodology follow standard chemical engineering thermodynamics textbooks, particularly:

  • Peng-Robinson (1976) equation of state
  • Standard correlations for vapor pressure (Antoine, extended forms)
  • Watson correlation for heat of vaporization
  • Polynomial correlations for heat capacity (Shomate, etc.)