6. Algorithm of the Calculation in AsTher

6. 1. Behaviour of the System
When the critical data Tc, Pc are known , the State functions can be calculated using real gases rules. The behaviour of the system can be selected separately in any AsTher application.
General Redlich-Kwong is set as default.
Menu System -> Behaviour the required selection can be altered.
By activating the check box „Extrapolation“, the state functions are calculated with extrapolate data beyond the existing range, if the calculation requires.

6.1.1. Ideal

6.1.2. Van der Waals

a=27 .R2 .Tc2 /(64 .Pc ) and b=R .Tc /( 8 .Pc )

6.1.3. Redlich-Kwong with

a=9.42748 .R2 .Tc2.5 / Pc and b=0.08664 .R .Tc / Pc

6.1.4. General (extended Redlich-Kwong) Equation

The enthalpy is calculated according to

There is residual enthalpy given by

The entropy is calculated according to

Residual-values are given by

The free energy is defined by

The Cp –Function is defined as

Remark: The values for the State variables as calculated by AsTher may differ from those of other sources, if those sources refer to unusual reference states. For example in ASME Steam tables, 4th. edition, The Am. Soc. Mech. Eng. New York 1979, App. I, pp. 11-29, the Enthalpy and Entropy of H2O (liq.) under 1 bar and 273,15 K is set to 0.

6.2. Extrapolation: By activating the checkbox „Extrapolation“ AsTher can be advised to extrapolate data beyond the existing range, if the calculation requires.

 
6.3. Calculation of the Equilibrium in the Application Equilibrium

The state of equilibrium is defined by following relations:

Free enthalpy of the pure substance at temperature and pressure of the equilibrium
a_i: Activity of the component i within equilibrium
f_i : Fugacity of the component i within equilibrium
R: Gas-constant T: Temperature
ve(i): Number of atoms of the element e within the component i
le: Lagrange-Coefficient of the element e within equilibrium

For the mass balance, following relations is valid:

Ne: Number of atoms of the element e
ni: Number of molecules of the component i
ve(i): Number of atoms of the element e in the component i

All substances in the system will be treated within a mixed phase, if the activity coefficient of the substance is not set to (1) (in the a.c.-column). Consistency of a calculation is reached, when two criteria are fulfilled: - The mass balance can be closed. -
For any reaction the criteria of the equilibrium have to be fulfilled.

Argumentation of the Correctness and Consistency:
 Take any possible reaction within a given system:
a A(l) + b B(g) = c C(s)

First calculate the equilibrium constant K via the partial pressures and activities:

then calculate the equilibrium constant K via the Gibb’s (free) energy:

Compare the K values from both equation

The given Gibb’s free energies are referring to to the pure substances at pressure and temperature of the system.
The calculation can be carried out with the AsTher Reaction.
The results should match within the selected accuracy. Abs. 3.3. shows an example for the argumentation of consistency.