A quality discussion of the obtained Bender equation fit

Introduction:
Every least square fit is subjected to some degree of inavoidable arbitrariness which comes from different possible choices of the least square sum and from unequally distributed existing data. If the functions to fit are complicated, multiple possible solutions are possible and some luck is necessary to find the best starting values in order to come to the lowest possible least square sum. Therefore, fitting of complicated functions like the Bender equation can be a longer task especially if the functions to fit are not analytical but numerical iterated functions. If the starting values are not appropriate the iteration can lead to values where the algorithm breaks down.
That are the main problems one encounters, we are interested now here only in the results.

Method:
The experimental phase equilibrium data used to fit the Bender equation are given by [21].
As least square sum we summed up the relative errors (Pex - Ptheor)/Pex and (xgex - xgtheor)/xgtheor squared of pressure and gaseous molar ratio of the dew line of the phase diagram. Furthermore, in the end phase of a least square run, we added residuals which contained "reconstructed theoretical" xfl and vfl values and which were compared to the "experimental" theoretical value.
This "dirty" practical reconstruction method is done as follows:
We define the "experimental" fugacities

which contain something but not all of experimental information .
The  -values are calculated theoretically on the dew line from the Bender equation of state (Pex,xigex.) by using the inversion routine V_PTX, which gives the v-value for the subroutine EQOFSTATE. The dew line data have to be taken because there the experimental fugacity coefficients does not deviate too much from the theoretical data normally.
Using the inversion subroutine VX_PF of the Bender equation the "theoretical" values of xifl, vfl are reconstructed on the boiling line.Therefore, in sum we obtain the least square sum S
where the terms in the second line are included only for iterations in the neighbourhood of the least square minimum.

Results:
The fit constant of the most successful run showed the most less deviation from the trivial value 1 for all fit constants.
We used all points of the experimental phase equilibrium points at high pressure and 178 K. In the end phase of the iteration we included the reconstructed values in the least square sum, but had to drop the experimental values at the two highest pressures at 178 K, because then the algorithm would break down. Nevermind, afterwards the iteration converged until the critical point.
In Fig. 1 the measured values are shown compared to the calculated optimized solution from the Bender equation. Table 1 gives the numerical values exactly including all of the relevant material data.
Fig.2 shows the liquid volumes at 170 K calculated with a standard Handinson-Brobst-Thomson (HBT) estimation method [17] compared to the Bender equation. The estimation relation breaks down in the neighborhood of the critical point contrary to the Bender equation.

Discussion:
For our purposes the accuracy is sufficient. Probably it so good that other values as enthalpies entropies and so on can be taken as reasonable estimation values from the equation. It is clear that the accuracy of the fit is higher at lower emperature because there exist more data.

We remember that our equation is one solution to describe the mixture. It is not the only or the best solution, because if more and more exact data exist the mathematical modeling can to be refined too.

Table 1:
Relevant Data of the Least-Square Iteration
_______________________________________________________________________

System: Argon -Methan
Material-file: a:ar_ch4.mix   range of experimental data: 150 - 180 K
Date: 12-07-1999     Time: 20:00:28

Material data:
Argon                  Methane
mol. weight                   39.948                 16.043
crit. pressure                4865300                4598800
crit. volume                  7.452985075e-5         9.9030865D-05
crit. temperature             150.69                 190.56
Omega                         -.00234                .0086
Stiel factor                  .004493                .00539

ideal spec. heat polynom acc. to [17]
CPA                          20.80                  19.25
CPB                          0                      .05213
CPC                          0                      .05213
CPD                          0                      -1.132D-8
CPE                          0                      0
reference enthalpy            0                      0
reference entropy             154.729                186.3384

HBT - parameter
crit. temperature            150.8                  190.58
Omega                        -.0092                 .0074
crit. volume                 7.5452985075D-05       9.930865D-5
ZR                           .2922                  .2892

reference state at
pressure                     101325
temperature                  278.2

Remarks to the least-square-iteration

Calculated least-square fit parameter of the mixture:
kij= .9977865068023702   Chiij=1.033181352446963   EtaM=2.546215505163194

Accuracy:       single points 1.00D-07  Least-Squares-Iteration  1.00D-05

experimental values

No.   T/K     Xi/%   P/bar      Xg/%
____________________________________________________________________________

1    150.8    2.5   1.174D+01   10.5
2    150.8    6.2   1.294D+01   19.0
3    150.8   14.3   1.553D+01   34.6
4    150.8   19.1   1.696D+01   41.8
5    150.8   22.3   1.798D+01   46.2
6    150.8   30.4   2.061D+01   56.3
7    150.8   41.4   2.417D+01   65.4
8    150.8   57.4   2.977D+01   75.2
9    150.8   66.5   3.305D+01   79.7
10    150.8   81.0   3.894D+01   87.4
11    150.8   85.0   4.068D+01   89.5
12    150.8   92.0   4.416D+01   94.2
13    150.8   93.5   4.512D+01   95.2
14    164.0    4.1   2.043D+01   11.3
15    164.0    9.8   2.288D+01   22.2
16    164.0   17.5   2.618D+01   33.3
17    164.0   19.6   2.736D+01   36.2
18    164.0   26.8   3.069D+01   44.5
19    164.0   36.8   3.514D+01   53.5
20    164.0   37.4   3.564D+01   54.2
21    164.0   44.1   3.897D+01   59.0
22    164.0   50.5   4.209D+01   63.1
23    164.0   56.2   4.505D+01   66.8
24    164.0   63.0   4.876D+01   70.0
25    164.0   66.5   5.099D+01   69.6
26    178.0    2.4   3.210D+01    5.3
27    178.0    4.6   3.309D+01    9.0
28    178.0   10.9   3.648D+01   18.5
29    178.0   16.5   3.985D+01   25.3
30    178.0   20.4   4.229D+01   29.2
31    178.0   22.7   4.366D+01   31.4
32    178.0   26.8   4.629D+01   35.5
33    178.0   32.2   4.976D+01   38.7
34    178.0   33.3   4.995D+01   37.6

errors between experiment and theory
No.   T/K     Xi/%   P/bar      dP/%   Xg/%   dXg/%     Vfl         Vg
____________________________________________________________________________

1    150.8    2.5   1.165D+01    0.8    8.6   18.5   4.520D-05   8.834D-04
2    150.8    6.2   1.298D+01   -0.3   18.9    0.4   4.521D-05   7.846D-04
3    150.8   14.3   1.579D+01   -1.6   35.4   -2.3   4.522D-05   6.298D-04
4    150.8   19.1   1.740D+01   -2.6   42.6   -2.0   4.523D-05   5.630D-04
5    150.8   22.3   1.845D+01   -2.7   46.8   -1.2   4.524D-05   5.253D-04
6    150.8   30.4   2.110D+01   -2.4   55.5    1.3   4.529D-05   4.472D-04
7    150.8   41.4   2.468D+01   -2.1   64.9    0.8   4.542D-05   3.672D-04
8    150.8   57.4   3.008D+01   -1.1   75.6   -0.6   4.583D-05   2.811D-04
9    150.8   66.5   3.337D+01   -1.0   80.9   -1.5   4.629D-05   2.413D-04
10    150.8   81.0   3.915D+01   -0.5   88.8   -1.6   4.784D-05   1.846D-04
11    150.8   85.0   4.091D+01   -0.6   91.0   -1.6   4.865D-05   1.693D-04
12    150.8   92.0   4.424D+01   -0.2   94.8   -0.7   5.115D-05   1.410D-04
13    150.8   93.5   4.501D+01    0.2   95.7   -0.5   5.206D-05   1.341D-04
14    164.0    4.1   2.061D+01   -0.9   10.5    6.8   4.935D-05   4.850D-04
15    164.0    9.8   2.341D+01   -2.3   22.1    0.5   4.961D-05   4.171D-04
16    164.0   17.5   2.710D+01   -3.5   34.0   -2.0   5.004D-05   3.481D-04
17    164.0   19.6   2.809D+01   -2.7   36.7   -1.3   5.017D-05   3.324D-04
18    164.0   26.8   3.147D+01   -2.5   44.8   -0.7   5.071D-05   2.857D-04
19    164.0   36.8   3.617D+01   -2.9   53.9   -0.7   5.176D-05   2.338D-04
20    164.0   37.4   3.645D+01   -2.3   54.4   -0.3   5.184D-05   2.310D-04
21    164.0   44.1   3.965D+01   -1.8   59.4   -0.7   5.288D-05   2.017D-04
22    164.0   50.5   4.277D+01   -1.6   63.7   -0.9   5.430D-05   1.759D-04
23    164.0   56.2   4.562D+01   -1.3   67.0   -0.4   5.624D-05   1.535D-04
24    164.0   63.0   4.902D+01   -0.5   70.3   -0.4   6.069D-05   1.246D-04
25    164.0   66.5   5.060D+01    0.8   71.0   -2.0   6.618D-05   1.052D-04
26    178.0    2.4   3.219D+01   -0.3    4.8    9.6   5.645D-05   2.825D-04
27    178.0    4.6   3.363D+01   -1.6    8.8    2.5   5.686D-05   2.663D-04
28    178.0   10.9   3.767D+01   -3.3   18.4    0.4   5.826D-05   2.258D-04
29    178.0   16.5   4.119D+01   -3.4   25.3    0.1   5.997D-05   1.951D-04
30    178.0   20.4   4.358D+01   -3.1   29.2   -0.1   6.158D-05   1.755D-04
31    178.0   22.7   4.497D+01   -3.0   31.3    0.3   6.281D-05   1.642D-04
32    178.0   26.8   4.738D+01   -2.4   34.3    3.3   6.588D-05   1.438D-04
33    178.0   32.2   5.006D+01   -0.6   36.2    6.5   7.532D-05   1.113D-04
34    178.0   33.3   5.038D+01   -0.9   35.8    4.9   8.004D-05   1.019D-04

Captions:

Fig.1: Comparison between experimental given data from [21] and theoretical calculation for pressure versus molar ratio (Argon) data

Fig.2: The liquid volumes at 170 K calculated with a Handinson-Brobst-Thomson estimation relation and with the Bender equation

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