Ind. Eng. Chem. Res. 2005, 44, 4139-4141
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RESEARCH NOTES Ternary Error Diagrams: Graphical Representation of Errors in Ternary System Compositions David C. Shallcross* and Binh Son Vo Department of Chemical and Biomolecular Engineering, University of Melbourne, Victoria 3010, Australia
A new technique is described for graphically presenting the errors between predicted and measured ternary compositions. The ternary error diagram is constructed using three axes, one for each of the errors associated with each component. Each of the three axes is rotated by 120° about the origin with respect to the other two axes. When plotted on the diagrams, each point represents the error in the ternary composition predicted by a model compared with the actual experimentally determined composition. The better the agreement is between the model predictions and the experimental data, the closer the data point will be to the diagram’s origin. Also presented is a new measure for quantifying the accuracies of a model’s predictions compared to observed compositions. The use of the graphical technique is illustrated in the context of modeling ion-exchange equilibrium behavior in three-component systems. Introduction Many semitheoretical models have been proposed in the past to predict ion-exchange equilibrium compositions involving exchange of three competing ions between two phases. One challenge for these models is to predict accurately the composition of one of the phases, say, the exchanger phase, that is in equilibrium with the other phase of known composition. A comparison is then made between the predicted exchanger-phase composition and the experimentally observed composition. To quantify the accuracies of the equilibrium model predictions, a number of measures have been proposed. Among these are the relative residue and the average absolute error. The relative residue, R, is defined as N
R)
[( M
∑ ∑ i)1 j)1
)]
χmodel - χexperiment χexperiment
2
j
i
NM - 1
(1)
where M is the number of sets of equilibrium data; N is the number of cationic species; and χ is the quantity of interest, namely, the equivalent ionic fraction either in the solution phase or in the exchanger phase. Thus, for a three-component system, a predicted composition will make three contributions to the sum of the relative residue, one for each of the components. The average absolute error, AAE, can be defined as N
AAE )
M
[∑abs(χmodel - χexperiment)j]i ∑ i)1 j)1 NM
(2)
* To whom correspondence should be addressed. E-mail:
[email protected]. Tel.: +61 3 8344 6614. Fax: +61 3 8344 4153.
Again, each predicted composition makes three contributions to the sum of the error. Comparing predicted and determined compositions of ternary systems is very difficult to do graphically. In this paper, the use of the ternary error diagram is proposed. This diagram is based on the assumption that the sum of the composition fractions of any threecomponent system, whether predicted or measured, must be equal to 1. If this condition is accepted, then the sum of the errors in each of the three components must be 0. The Ternary Error Diagram Figure 1 presents a simple example of a ternary error diagram. The diagram consists of three axes, one for the error associated with each component. Each axis is rotated by 120° with respect to the other two axes. Each point on these diagrams represents the error in the ternary composition predicted by a model compared with the actual experimentally determined composition. The better the agreement is between the model predictions and the experimental data, the closer the data point will be to the diagram’s origin. As an example, consider the case where the fractions of components A, B, and C in a system are measured experimentally and found to be 0.627, 0.221, and 0.152, respectively. A model can be proposed that predicts the composition with respect to A, B and C to be 0.632, 0.220, and 0.148. Thus, the errors in the model predictions for the fractions of components A, B, and C are 0.005, -0.001, and -0.004, respectively. It is assumed that the sum of the fractions for both the observed and predicted compositions will be equal to 1. A natural consequence of this is that the sum of the errors in the three components must come to 0. Point R in Figure 1 represents the error in the ternary composition for this hypothetical point. It lies on a line that cuts the component A error axis at 0.005, a line that cuts the component B error at -0.001, and a line that cuts the component C error axis at -0.004.
10.1021/ie050039i CCC: $30.25 © 2005 American Chemical Society Published on Web 04/21/2005
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Ind. Eng. Chem. Res., Vol. 44, No. 11, 2005
Figure 1. Simple ternary error diagram for a three-component system.
Figure 2. Ternary error diagram for the K+-Na+-H+ system at four different solution-phase concentrations. Data taken from Vo1.
If we accept the assumption that the closeness of a data point to the origin of the ternary error diagram reflects the qulaity of the agreement between the model predictions and the experimental data, then we can formulate another numeric expression for the agreement between two ternary compositions. This we might call the average ternary error, ATE. It is essentially the average radial distance from the origin of all of the points on a ternary error diagram. Thus
[x
ATE ) M
2
∑ j)1
]
[error(A)]2 + error(A) × error(B) + [error(B)]2 3
j
M
(3) where error(A) is the error in the composition of component A
error(A) ) χmodel - χexperiment
(4)
In eq 3, A and B can be any two of the three components present in the system. Unlike the relative residue, the average ternary error defined above takes no account of the relative magnitudes of the measured values for the fractions of A, B, and C. To illustrate the application of the diagram, consider the ion-exchange equilibrium behavior for the K+-Na+-H+ ternary system. Figure 2 presents the ternary error diagram for the system studied by Vo1 for 65 sets of equilibrium compositions. The predictions made for the exchanger-phase equilibrium composition by applying the model proposed by Mehablia et al.2 are compared to the compositions measured experimentally by Vo.1 The diagram allows visual confirmation that the model predicts the exchanger-phase equilibrium compositions equally well for all four solution concentrations and for all three components. The same observation cannot be made for the H+-Mg2+-Ca2+ ternary system. Again, the predictions
Figure 3. Ternary error diagram for the H+-Mg2+-Ca2+ system at four different solution-phase concentrations. Predictions made using the model of Mehablia et al.2 Data taken from Vo.1
made for the exchanger-phase equilibrium compositions by the model of Mehablia et al. have been compared with the compositions observed experimentally by Vo. The ternary error diagram of Figure 3 clearly shows that the model predicts far more accurately the fraction of H+ ions in the exchanger phase than it does the fractions of either the Mg2+ or Ca2+ ions. The classical graphical representation of composition error data is presented in Figure 4. In this diagram, each predicted exchanger-phase composition is represented by three points, one for each of the three exchanging ions, as each ion can be in error by a differing amount. As the diagram does not distinguish between the different ions, using this figure, it is not
Ind. Eng. Chem. Res., Vol. 44, No. 11, 2005 4141 Table 1. Comparison of Relative Residue (R), Average Absolute Error (AAE), and Average Ternary Error (ATE) for Three Equilibrium Predictions Models for the Na+-H+-Ca2+ System
Figure 4. Errors in the predictions for the equilibrium exchangerphase compositions for the H+-Mg2+-Ca2+ system at four different solution-phase concentrations. Predictions made using the model of Mehablia et al.2 Data taken from Vo.1
model
R
AAE
ATE
A B C
0.000 510 0.000 286 0.000 091
0.009 18 0.006 20 0.003 16
0.014 10 0.009 54 0.004 88
We clearly see, by the way in which the points are clustered about the diagram’s origin, that the predictions of model C are superior to those of either of the other two models. This is confirmed by the values for the three measures of agreement between ternary compositions as presented in Table 1. Not surprisingly, the values for the average absolute error and the average ternary error are closely related. For example, we note that both the AAE and the ATE for the predictions of model C are 34.5% of their respective values for the predictions of model A. Diagram Preparation A FORTRAN program was written that takes the composition errors for any two of the three components of a ternary system and converts them into Cartesian coordinates. When run, the FORTRAN program generates an output file containing instructions to prepare the diagram in the Postscript graphics language. The Postscript file can then be sent either to a suitable Postscript laser printer or through the Ghostscript utility to a non-Postscript printer. At present, the program can plot five series of data using up to six different symbols. The program is available at no cost from the author. Concluding Remarks A new method is proposed for graphically presenting the errors in ternary system compositions. The method is equally applicable whether the composition is based on mass fractions, moles fractions, or equivalent ionic fractions. Ternary error diagrams permit rapid and unambiguous visual comparison between ternary compositions.
Figure 5. Ternary error diagram for the H+-Mg2+-Ca2+ system with predictions made using three different models: model A, Melis et al.;4 model B, Vo and Shallcross;3 model C, Mehablia et al.2 Data taken from Vo.1
possible to see that the model more accurately predicts the fraction of H+ ions than those of the other two ions. The ternary error diagram is also a powerful tool for visually demonstrating the clear superiority of one model over others. Recently, Vo and Shallcross3 compared the performance of three multicomponent ionexchange equilibrium models. Model A was a model proposed by Melis et al.,4 model B was another proposed by Vo and Shallcross,3 and model C was a model developed by Mehablia et al.2 The ternary error diagram for the Na+-H+-Ca2+ system is presented in Figure 5.
Literature Cited (1) Vo, B. S. Ph.D. Thesis, University of Melbourne, Melbourne, Australia, 2003. (2) Mehablia, M. A.; Shallcross, D. C.; Stevens, G. W. Prediction of multicomponent ion exchange equilibria. Chem. Eng. Sci. 1994, 49, 2277-2286. (3) Vo, B. S.; Shallcross, D. C. Multi-Component Ion Exchange Equilibria. Chem. Eng. Res. Des. 2003, 81, 1311-1322. (4) Melis, S.; Cao, G.; Morbidelli, M. A new model for the simulation of ion exchange equilibria. Ind. Eng. Chem. Res. 1995, 34, 3916-3924.
Received for review January 11, 2005 Revised manuscript received April 5, 2005 Accepted April 13, 2005 IE050039I
