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A new method for optimizing the structure of thermodynamic correlation equations

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Abstract

An optimization strategy is presented for optimizing the structure of empirical thermodynamic correlation equations. Based on a comprehensive functional expression for the physical dependence considered, which is called a “bank of terms,” the new procedure optimizes the structure and the length of the equation as well. The application of this method results in an equation which meets the quality wanted for representing the experimental data with the lowest number of fitted coefficients. The procedure can be used for the determination of the structure of any equation where the method of the linear least squares is applicable. A detailed description of the algorithm is given which includes values for the control parameters for different applications in the field of thermodynamics (vapor pressure equations, equations of state, etc.) and also for applications in other fields. The optimization steps are described using an equation which represents a relationship between variables in a general form. It is demonstrated how even the complex problem of the optimization of a fundamental equation for the Heimholtz energy can be written in terms of this general equation.

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Abbreviations

A :

Helmholtz energy

[A]:

Matrix

a i :

Element of the bank of terms

[B]:

Regression matrix

[BB]:

Working matrix

b ij :

Element of the matrix [B]

C :

Number of constraints

[C]:

Matrix containing constraints

c, i, k, m, n, p :

Serial numbers

F :

Fisher F statistic

I :

Number of elements in the bank of terms

K :

Number of independent variables

L :

Order of the matrix [B]

[IN]:

Vector

M :

Number of experimental data

N :

Maximum number of terms in an equation

N a :

Number of terms in an equation

NM :

Number of mutations

NP :

Number of equations of the population

NR :

Number of regression runs in each generation

NS :

Number of start attempts for the initialization

[N]:

Column matrix of coefficients n i

n i :

Adjustable coefficient

P F , P t , S :

Statistical probabilities

p :

Pressure

p n :

Position of a term in the bank of terms

p s :

Vapor pressure

[Q], [QC]:

Column matrices

R :

Gas constant

S :

Weighted sum of squares

T :

Temperature

t :

Student t statistic

V, V −1 :

Variance of an equation

X k :

“True” independent state variable

x k :

Independent state variable

Y :

“True” dependent state variable

y :

Dependent state variable

z :

Normally distributed random number

[0]:

Null matrix

X 2 :

Weigted sum of squares

δ=ρ/ρ c :

Reduced density

Γ :

Gamma function

λ,λ′ :

Lagrangian multipliers

v :

Degrees of freedom

ρ :

Density

Φ=A/(RT) :

Dimensionless Helmholtz energy

σ :

Standard deviation

σ 2 :

Variance

τ=T c/T:

Inverse reduced temperature

ξ :

State function

ζ :

Empirical relationship

c :

Constraint, critical

i, j, k, m, n :

Indices for terms in the matrices

n:

New

o:

Old

r :

Real part

T :

Transpose of the matrix

-:

Sign for a vector

o:

Ideal-gas state

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Authors and Affiliations

  1. Institut für Thermo- und Fluiddynamik, Ruhr-Universität Bochum, D-4630, Bochum, Germany

    U. Setzmann & W. Wagner

Authors
  1. U. Setzmann
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  2. W. Wagner
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Dedicated to Prof. Dr. phil. F. Kohler on the Occasion of His 65th Birthday

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Setzmann, U., Wagner, W. A new method for optimizing the structure of thermodynamic correlation equations. Int J Thermophys 10, 1103–1126 (1989). https://doi.org/10.1007/BF00500566

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  • DOI: https://doi.org/10.1007/BF00500566

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