In: P. Jansen, H.J.M. Kramer and M. Roelands (ed), Proceedings of BIWIC 2001, 8th International Workshop on Industrial Crystallization, Delft, The Netherlands, 19-20 September, 2001, pp 118-125.
A predictive model framework for design purposes: data quality (required) for parameter estimation Sean K. Bermingham, Andreas M . Neumann, Peter J.T. Verheijen and Herman J.M . Kramer Delft University of Technology, Delft, The Netherlands
[email protected] This contribution investigates the statistical significance of parameter estimates obtained from (1) perfect, simulated measurements, (2) the same measurements with added, randomly distributed noise, and (3) true experimental data. Parameter estimation is performed using CSD transients (L10, L50, and L90) a rigorous crystallisation model including detailed kinetics and a full population balance, and formal, mathematically -based methods. It is shown that parameter estimation can be of great value before any experiments have been done. Firstly, using perfect measurements one can identify strong cross-correlations between the various parameters. Secondly, addition of randomly distributed noise can be used to provide a good indication of the measurement accuracy required to obtain parameters with a certain variance. These insights are crucial to determine the usefulness of an experiment a priori. Introduction Reliable design and scale-up of crystallisers requires a model framework that is valid for a wide range of equipment and operating conditions without the adjustment of model parameters, e.g. simply apply parameters estimated using data from a laboratory scale stirred vessel to the optimal design and operation of an industrial scale forced circulation crystalliser. It is generally accepted that such a framework should entail the simultaneous, but not lumped, description of kinetics and hydrodynamics in order to provide good predictive capabilities as regards the relationships between model inputs (e.g. crystalliser geometry and operating conditions) and model outputs (e.g. supersaturation and CSD). Failure to do so will result in kinetic parameters that are not intrinsic, and, as a consequence, of limited use for scale-up. Another requirement for a model framework aimed at design purposes is the observability of the unknown model parameters in this framework, i.e. what number and type of experiments and measurements need to be performed to obtain adequate estimates for these parameters? And what does adequate mean in this context? These questions may seem rather trivial at first, but, in literature and probably also in industry, they are often not raised at all or at a too late a stage. This contribution aims to address these questions by studying the statistical significance of parameter estimates obtained from (1) perfect, simulated measurements, (2) the same measurements with added, randomly distributed noise, and (3) true experimental data. Model framework The crystallisation model framework used for this study is primarily aimed at solution crystallisation processes. It is a combination of a fundamental kinetic model for attrition, secondary nucleation, growth and dissolution, and a compartmental approach to account for the effects of spatially distributed process variables (Neumann et al., 1999). An extensive motivation for compartmental modelling and a methodology for subdividing a crystalliser into multiple compartments are given by Kramer et al. (1999). For the combination of model
system and crystalliser scale used in this study, a simple two-compartment model is used: one zero volume compartment for the propeller (attrition) and one finite volume compartment for the bulk of the crystalliser (crystal growth and dissolution). The kinetics of nucleation, growth, dissolution and attrition are described with the model of Gahn and M ersmann (1999). The kinetics of these phenomena are calculated on the basis of the detailed impeller geometry, impeller frequency, material properties, mechanical properties, CSD, solute concentration and local energy dissipation. The model consists of three sub-models. (1) A procedure to determine the collision rate of crystals with the impeller and to calculate the crystal size dependent impact energy. (2) A relation between the impact energy and the size distribution of attrition fragments produced due to a single collision of a crystal corner with the impeller. (3) A relation to derive the growth rate of the fragments formed by the attrition process. Attrition fragments resulting from a collision of a crystal corner with an impeller contain a certain amount of internal strain, which results in an increased solubility, c*real, in comparison to the solubility, c*, of ideal, stress free crystals: S * creal ( L) c* exp R T L
[1]
thus reducing the driving force for crystal growth, c(L). Growth rates of attrition fragments will therefore be lower than those of larger crystals, and may even be negative. As a result, attrition fragments may dissolve under macroscopic growth conditions. Assuming a combined diffusion and second order surface reaction controlled growth, the size dependent growth rate of a crystal, G(L), can be obtained as follows: 2
k ( L) kd ( L) c( L) G( L) c( L) kd ( L) d 2kd ( L) cs 2kr cs kr cs cs 2kr cs
[2]
M ost parameters in this model framework can be found in literature or can be determined by isolated experiments. There are however two parameters that need to be derived from crystallisation experiments: the condition of deformation, S, and the surface reaction coefficient, k r. Experimental work The crystallisation of ammonium sulphate from water is the model system used in this paper. An experiment performed on the 22-litre DT crystalliser depicted in Figure 1 is used for this work. This crystalliser is operated continuously, in an evaporative mode and at a constant temperature of 50 °C. A small slurry stream is continuously removed from the down-comer of the crystalliser by means of a peristaltic pump and recycled to the crystalliser. At regular time intervals the stream is injected into the dilution stream by switching the position of the pneumatic valves and the diluted product sample is thereafter transported through an on-line M alvern 2600c laser diffraction instrument (equipped with a 1000 mm lens). Note that dilution of the product is required in order to reduce the original product solid concentration of approximately 11 vol.% to a solid concentration of approximately 1.5 vol.%; the maximum allowable solid concentration to avoid multiple scattering and to enable an accurate measurement using the laser diffraction instrument.
In: P. Jansen, H.J.M. Kramer and M. Roelands (ed), Proceedings of BIWIC 2001, 8th International Workshop on Industrial Crystallization, Delft, The Netherlands, 19-20 September, 2001, pp 118-125.
Estimation problem Gahn and M ersmann (1999) used the steady state median size and supersaturation to estimate values for the two unknown parameters in their model, i.e. S and k r. Unfortunately, their approach is infeasible for (1) continuous crystallisation processes where the supersaturation is difficult to measure and (2) batch crystallisation processes, which by definition have no steady state.
Feed
Heat exchanger
Dilution
Recycle loop
Malvern 2600c
Pneumatic valves
Peristaltic pump
Figure 1: The 22-litre DT evaporative crystalliser.
Parameter estimation with dynamic data
Parameter estimation with steady state data The ammonium sulphate crystallisation experiments used in this work fall under the first category, i.e. difficult to obtain accurate supersaturation measurements. Neumann et al. (1999) showed that, as expected, an unambiguous determination of the kinetic parameters is not possible on the basis of the steady state median size alone. Subsequently, they used other characteristics of the steady state CSD, such as quantiles and the quartile ratio, as additional information. However, this did not lead to a less unambiguous determination.
An alternative approach, which can be used to compensate for the absence of a steady -state supersaturation measurement, is dynamic parameter estimation. As a matter-of-fact, even if the steady state median size and supersaturation are available for a certain continuous experiment, dynamic parameter estimation will still be of significant value. M easurements of dynamic process behaviour, if available, typically contain more information than steady state data alone, and as such, dynamic parameter estimation will improve the quality of the estimates. Finally, most laboratory scale crystallisation experiments are performed batch wise. Clearly, the only option for the systematic disclosure of information contained in batch experiments is dynamic parameter estimation. The following paragraphs address two issues that need to be considered before dynamic parameter estimation can be performed: data window and initial conditions. Selecting data window The first issue is selection of CSD characteristics and data window (time horizon) that will be used. The median size (L50) and two other quantiles (L10 and L90) were chosen as measures for the location and spread of the CSD. For determination of the data window, the following criteria were used: 1. The initial phase of the experiment involves two nucleation mechanisms: primary and secondary nucleation. As primary nucleation is not included in the model framework, it was decided to not use the CSD data from the first four hours of the experiment. After this time, just over three residence times, it is expected that practically no primary nuclei will be present in the crystalliser. 2. Due to fouling, the noise in the measurements tends to increase towards the end of the experiment. When the noise rises above a certain, subjective level the data is not used. The data window of the experiment conducted with a propeller speed of 775 rpm is shown in Figure 2. Data Window
1200 1000
Crystal Size [mu]
The crystalliser was operated at a residence time of 75 minutes, a specific heat input of 120 kWm-3 and impeller frequencies of respectively 775 and 910 rpm. During the experiment the product CSD was measured at two-minute intervals. The trends of three quantiles (L 10, L50 and L90) associated with the CSD of the experiment conducted with a propeller speed of 775 rpm are shown in Figure 2.
800 600 400 200 0 0
5
10
15
20
25
30
Time [hrs]
Figure 2: L10, L 50 and L90 and data window used for parameter estimation.
Initial condition The second issue is the initial condition. Transient behaviour always depends on the initial condition (starting point). Parameter estimation using dynamic data therefore requires a good description of the initial condition. The importance of including the initial conditions in the parameter estimation procedure was recently confirmed by Bermingham et al. (2001).
In: P. Jansen, H.J.M. Kramer and M. Roelands (ed), Proceedings of BIWIC 2001, 8th International Workshop on Industrial Crystallization, Delft, The Netherlands, 19-20 September, 2001, pp 118-125.
L L ln 2 ln 2 L0 g ,2 L0 g ,1 1 1 v L exp 2 1 exp 2 L ln g ,1 2 L ln g ,2 2 2ln g ,1 2ln g ,2
1200 crystal size [mu]
simulation settings to create perfect measurements
[-] L0g,2 [m]
g,2 [-]
[-]
750
850
600
950
1.4
1.25
1.2
1.6
375
450
250
550
2.5
2
1.5
3.5
0.5
0.3
0.0
1.0
The first parameter estimation result is promising in the sense that the values estimated using the perfect measurements are almost identical to the values used to create these measurements. Furthermore, all standard deviations except for the initial supersaturation are less than 0.1 % of the optimal value. However, an elongated confidence ellipsoid indicates a strong crosscorrelation between the two kinetic parameters (Figure 4).
1000 800 600 400 200 0 0
5
10
15
20
parameter estimation settings initial lower upper guess bound bound
2E-5
1E-4
1E-6
4E-4
2E-4
4E-5
1E-5
4E-3
2E-4
1E-3
0
1E-2
parameter estimation results using perfect using perfect measurements measurements with added noise (estimate; (estimate; standard dev.) standard dev.) 2.0023E-5 1.99E-5 0.0003E-5 0.06E-5 1.9992E-4 2.00E-4 0.0001E-4 0.02E-4 7.74E-4 4.7E-3 0.13E-4 2.4E-3
749 11 1.40 0.05 387 75 2.50 0.20 0.50 0.09
perfect measurements
1.9994E-04 1.9993E-04 1.9992E-04 1.9991E-04 1.9990E-04 1.9989E-04 1.9988E-04 2.0012E-05 2.0016E-05 2.0020E-05 2.0024E-05 2.0028E-05 2.0032E-05 Kr
Figure 4: 95% Confidence ellipsoid for the parameter pair k r-s, when using perfect measurements.
Next, normally distributed noise with a constant relative variance of five percent is added to the perfect measurements. As can be seen from Figure 5, a good fit can still be obtained after the addition of this noise, but the estimates are now slightly further away from the original values and the standard deviations have on average risen to approximately 10 % (Table 1). perfect measurements with randomly added noise 2.12E-04
1400 1200
2.08E-04
1000
2.04E-04
800 s
Figure 3: perfect L 10, L50 and L 90 measurements.
749.93 0.05 1.3997 0.0002 374.98 0.36 2.4997 0.0011 0.5003 0.0005
1.9995E-04
time [hrs]
Table 1: Parameter values used to obtain perfect measurements and estimated values for the same parameters using the perfect measurements without and with added noise. model parameters and initial condition parameters kr [ms-1 ] S [Jmmol-1 ] C/Csat [-]
g,1
crystal size [mu]
Observability of model parameters In order to determine whether the available measurements (L10, L50, and L90) can in principle be used to provide adequate estimates for the unknown model parameters, these parameters will be successively estimated from (1) perfect, simulated measurements and (2) the same measurements with added, randomly distributed noise. The perfect measurements were obtained from a simulation with the parameter values in column two of Table 1.
[3]
L0g,1 [m]
s
The initial condition is given by the CSD, solids concentration, solute concentration and temperature at time zero (lower bound of the data window). These quantities are usually known with varying degrees of certainty: high certainty for the temperature (accurate measurements), high certainty for the solids concentration (at steady state after 3 hours and can be calculated from mass and energy balances alone), low certainty for the CSD (measurement technique) and even lower certainty for the solute concentration (no supersaturation measurement). As a result, Bermingham et al. managed to significantly improve the quality of fit, by performing a simultaneous estimation of the unknown model parameters and uncertain initial conditions. For this purpose, they parameterised the initial CSD as a bimodal distribution consisting of two lognormal distributions on a volume density basis.
600 400
2.00E-04 1.96E-04
200
1.92E-04
0 0
5
10
15
20
time [hrs]
Figure 5: Measured and predicted L 10, L50 and L90, when using perfect measurements with normally distributed noise added to them.
1.88E-04 1.7E-05
1.9E-05
2.1E-05
2.3E-05
Kr
Figure 6: 95% Confidence ellipsoid for the parameter pair k r-s,when using perfect measurements with normally distributed noise added to them.
A similarly shaped confidence ellipsoid, but with much larger dimensions, indicates a strong cross-correlation between the two kinetic parameters over a wider range of values.
In: P. Jansen, H.J.M. Kramer and M. Roelands (ed), Proceedings of BIWIC 2001, 8th International Workshop on Industrial Crystallization, Delft, The Netherlands, 19-20 September, 2001, pp 118-125.
2. Parameter estimation with experimental data Now the experimentally determined L10, L50 and L90, of the 22-litre DT crystalliser run with a propeller speed of 775 rpm are used to determine the surface reaction coefficient, k r, and the condition of deformation, S. Table 2: Parameter values estimated using experimental data. model & initial condition parameters kr [ms-1 ] S [Jmmol-1 ] C/Csat [-] L0g,1 [m]
initial guess
1E-4 4E-5 1E-3 850
estimation results (estimate; stand.dev.) 1.7E-5 0.6E-5 1.5E-4 0.2E-4 0 684 10
model & initial condition parameters
g,1
initial guess
estimation results (estimate; stand.dev.) 1.32 0.04 436 176 2.40 0.30 0.75 0.12
1.25
[-] L0g,2 [m]
450
g,2
2
[-]
0.3
[-]
The values of the estimated parameters are listed in Table 2. Note that the standard deviations have again increased and for some of the parameters to the extent that they are of the same order of magnitude as the estimated value. The fit obtained with these parameters is shown in Figure 8, whereas the confidence ellipsoid for the two kinetic parameters is shown in Figure 9. Experimental data 2.50E-04
1000
Acknowledgements STW, AKZO-Nobel, BASF, Bayer A.G., Dow Chemicals, DSM , DuPont de Nemours and Purac Biochem for supporting the UNIAK research program. List of symbols kr rate constant for surface integration L0g location parameter of log-normal distribution v(L) initial CSD expressed in volume density terms relative weighting of the two log-normal distributions condition of deformation S g
spread parameter of log-normal distribution
[ms-1] [m] [m-1] [-] [Jmmol-1] [-]
2.00E-04
800 1.50E-04
600
Gs
crystal size [mu]
1200
Adding noise with a known variance to these perfect measurements gives a good indication of the increase in the standard deviations of the estimated parameters, and thus inversely, of the measurement accuracy required to obtain parameters with a certain accuracy. These insights may lead to the conclusion that planned experiments would be of limited value. An important outcome when experiments are lengthy and/or costly. The question whether the kinetic parameter estimates obtained with the presented experimental data are adequate has not been answered fully yet. The strong cross-correlation indicates it would be unwise to use one of these parameters in isolation, e.g. inserting the estimated value of k r in another kinetic model than the Gahn model. Future work will focus on (1) the validation of the obtained parameters by using experimental data from the same crystalliser at different operating conditions and from crystallisers of a significantly different scale, and (2) going through the same procedure with a reparameterisation that reduces the cross-correlation between the kinetic parameters.
1.00E-04
400 200
5.00E-05
0 0
5
10
15
20
time [hrs]
Figure 7: Measured and predicted L 10, L50 and L 90, when using experimental data.
0.00E+00 -1.0E-05
0.0E+00
1.0E-05
2.0E-05
3.0E-05
4.0E-05
5.0E-05
Kr
Figure 8: 95% Confidence ellipsoid for the parameter pair k r-s,when using experimental data.
Discussion and conclusions In general, parameter estimation is a model-based activity that should not only be performed after experimentation, but also, equally important, before experimentation. Parameter estimation can provide the following insights: 1. Using perfect, simulated measurements strong cross-correlations can easily be identified.
References Bermingham, S.K., A.M . Neumann, P.J.T. Verheijen and H.J.M . Kramer (2001). Parameter estimation on the basis of measured CSD transients. Proceedings of ICCG-15, Kyoto, Japan. Gahn, C. and A. M ersmann (1999). Brittle fracture in crystallisation processes. Chem Engng Sci., 54, 1273-1292. Kramer, H.J.M ., S.K. Bermingham and G.M . van Rosmalen (1999). Design of industrial crystallisers for a required product quality. J. Crystal Growth, 198/199, 729-737. Neumann, A.M ., S.K. Bermingham, H.J.M . Kramer and G.M . van Rosmalen (1999). M odeling industrial crystallizers of different scale and type. Proceedings of the 14th Symposium on Industrial Crystallisation, Cambridge, United Kingdom, 12 th-16th.
