Inferential Conversion Model

10.1.2.1 Model Basis

One of the advantages of the one-point control scheme was that it provided some implicit control of the isobutene conversion. This was achieved via the control of a temperature within in the stripping section, which effectively governed the entire temperature profile. A favourable composition was maintained on the stages just below the reactive section by preventing the temperature from becoming too high (a condition that corresponds to a deficiency of isobutene) or too low (corresponding to a deficiency of ethanol). Thus, the temperatures in the reactive and stripping sections might provide a means of inferring the isobutene conversion. Unfortunately, no single measurable variable correlates adequately with conversion.

This result is, perhaps, not surprising given the complexity of the system and the range of competing influences on the reaction zone conditions. For example, a decreased reaction zone temperature has a favourable effect on the reaction equilibrium constant but reduces the reaction rate constant. It also indicates a greater availability of isobutene, which promotes the reaction (ethanol has a retarding effect on the reaction rate: Jensen and Datta, 1995), and a higher concentration of non-reactive butenes, which reduces the maximum conversion due to dilution effects. This suggests that any conversion prediction method should reflect the values of at least two (and possibly more) independent variables.

Two approaches to the development of an inferential conversion model appear promising. Firstly, the reactive distillation system could be represented by a simplified model (e.g. using short-cut methods to calculate the feed split and fractionation under given input conditions) to predict the column behaviour to measured disturbances. These results could then be used to calculate the expected conversion. However, no such short-cut methods are available for reactive distillation and the known complexity of the process is contrary to the applicability of such a model.

The second approach is to utilise the statistical analysis of a quantity of data that contains a range of measurable variables and the dependent variable (i.e. the isobutene conversion). The data could either be from actual operation of the process or generated by simulations. In this case, simulation data was used. This eliminates process noise and measurement uncertainty from the inferential model but results in the inclusion of model uncertainty. It is desirable to base the model on sound engineering principles but regression analysis can be an invaluable tool in developing a suitable model that is both effective and robust. The statistical approach was applied here to develop an inferential model for the ETBE column. The actual implementation of any such model requires process noise to be filtered in order to be effective. However, tools are readily available for this end, and the requirement for an on-line filter is not a barrier to this approach.

The first stage of this process was the selection of candidate variables that would appropriately reflect the reaction environment and the column operating conditions. The ease of measurement (both the cost and the probable accuracy) was considered important, and temperatures and flows were preferred on that basis. Composite variables were also considered to incorporate possible non-linear behaviour. A greater representation of stripping section temperatures was specified to reflect the larger temperature differences and the greater sensitivity to operating conditions compared with the reactive and the rectifying sections. The full list of candidate variables is given in Table 10.1.

Table 10.1 - Candidate Variables for Inferential Conversion Model

Variable Name

Description

Condenser temperature

x:

Temperature at the top of the reaction zone (stage 3)

Temperature at the bottom of the reaction zone (stage 5)

x4

Stage 7 temperature

X5

Stage 8 temperature

Stage 9 temperature

X7

Reboiler temperature

X8

Reflux rate

x9

Reboiler duty

X|o

Bottoms rate (yield)

Xll

Overhead pressure

X|2

Bottoms composition (via a process analyser)

X,3

AT across the reaction zone

Model identification can be achieved using either simulation data or real plant data. Simulation data will contain modelling errors but will be free of instrument errors and noise that may be present in plant data. The choice of data source (simulation or plant data) should be made on a case-by-case basis but if a good model of the process exists, the collection of data that spans the entire significant operating range can be achieved more efficiently using a simulation. The use of a simulation was preferred here although some experimental data was also available. The significant operating range was taken to include reboiler duties of 7.3-9.3 kW, reflux rates of 2.0-3.0 L/min and stoichiometric excesses of ethanol in the feed of 0-10%. A constant overhead pressure was assumed here but if significant perturbations in pressure were expected it could be added to the analysis. The data could have been generated using a factorial experimental design but since there was essentially no cost involved in the data generation, a less rigorous approach was used. The three primary variables (reboiler duty, reflux rate and ethanol excess) were each incremented around the optimum operating point to produce 25 observations which effectively spanned the significant operating range.

10.1.2.2 Two-Term Regression Models

Initially, two-term models were considered using various combinations of the candidate variables listed in Table 10.1. Some of the more promising combinations of variables are summarised in Table 10.2 with calculated regression coefficients (R2). Variables were judged to be statistically significant at the 95% confidence level.

Table 10.2 - Candidate Two-Term Regression Models

Standard

Statistically

Model

Variables in

Rz

Error of the

Significant

Designation

Model

Prediction

Variables

2A

x, & x3

0.996

0.6 mol%

x, & x2

2b

X, & x3

0.970

1.8 mol%

X| & x3

2C

Xi & x7

0.976

1.6 mol%

X, & x7

x2 & x3

0.898

3.3 mol%

x2 only

2e

x2 & x4

0.965

1.9 mol%

x2 & x4

2F

x2& x5

0.985

1.3 mol%

x2 & x5

x2 & x6

0.990

1.0 mol%

x2& x6

2H

x2 & x7

0.986

1.2 mol%

x2 & x7

2'

x2 & x8

0.900

3.3 mol%

x2 only

2J

x2 & Xg

0.915

3.0 mol%

x2 only

2K

x2 & X,o

0.952

2.3 mol%

x2 & x10

2L

X2 & xl2

0.982

1.4 mol%

x2 & x12

JM

x3 & x7

0.888

3.5 mol%

x3 & x7

2N

x4& Xg

0.427

7.8 mol%

x4 only

xl0& XM

0.674

5.9 mol%

xn only

The best model (2A) was rejected because of sensitivity between the condenser temperature and the conversion estimate. The calculated regression coefficient was 18.7 which indicates that a 1°C change in the condenser temperature (for whatever reason) would change the conversion estimate by 18.7 mol%. This level of sensitivity is too high and would significantly reduce the robustness of the model to instrument error and/or modelling errors. Similarly, other models which included X[ as a manipulated variable were also excluded as the calculated regression coefficient for x, was usually greater than 10.

Three models (2F, 2C and 2H) were found to have a R2 value greater than 0.98, which is highly acceptable. Interestingly, the model which uses the bottoms composition directly

(model 2L), is not as good as models 2F, 2° or 2H, although the differences are small. The vector forms of each of these models are given by equations (10.1)-(10.3), where y is the conversion estimate:

The constants in each model have similar values that suggest an underlying relationship between the reaction zone and the stripping zone temperatures, and the isobutene conversion. A temperature increase of 1°C at the top of the reaction zone (stage 3) should decrease conversion by 3-4 mol% while an increase of 1°C in temperatures near the bottom of the column should increase the conversion by 0.15-0.20 mol%. Using model 2° as an example, the engineering significance of the terms is possibly that a higher reaction zone temperature implies a decreased availability of isobutene and a less favourable reaction equilibrium constant while an increase in the stage 9 temperature implies a higher reaction rate as ETBE is the highest boiling point component.

Interestingly, the temperature at the top of the reaction zone (stage 3) shows a much stronger correlation with the conversion than the temperature at the bottom of the reaction zone (stage 5) which shows a much greater variation. This can be explained with reference to the ethanol concentration in the distillate product. A high ethanol concentration would increase the VLE temperature on the uppermost stages in the column and also indicates a low isobutene conversion as the reaction is equimolar in isobutene and ethanol. Conversely, a low ethanol concentration would result in low temperatures on the uppermost stages and would indicate a high conversion of ethanol (and isobutene).

Fortunately, the best two-term models exclude manipulated variables (x8, x9 and x10). This is convenient as it simplifies the implementation of a closed-loop inferential controller. If a manipulated variable had been included in the model, there would be potential for a feedback loop to form. The inferential controller would have had a tendency to drive that variable to 0% or 100% of range (depending on whether the variable had a positive or negative effect on the conversion).

10.1.2.3 Three-Term Regression Models

The correlation coefficient of model 2G is already high but could possibly be increased further by including additional terms in the model. Several three-term models were evaluated and have been summarised in Table 10.3. None of these models are significantly better than the two-term models, 2F, 2° and 2H. Therefore, there is no incentive to further complicate the model by adding extra terms. A check of the residuals also shows that none of the available independent variables should be added to the model as the relationships between these variables and the residuals is essentially random in all cases.

Table 10.3 - Candidate Three-Term Regression Models

Standard

Statistically

Model

Variables in

R1

Error of the

Significant

Designation

Model

Prediction

Variables

3A

X2» X6» X8

0.992

0.9 mol%

all

3B

X2> X6> X10

0.990

1.0 mol%

x2 & x^ only

3C

x2> x6> XI1

0.990

1.0 mol%

x2 & x6 only

3d

X2> X7i Xg

0.987

1.2 mol%

x2 & x7 only

3e

X2j X3> X6

0.990

1.0 mol%

x2 & x6 only

3F

X2> X5> X7

0.990

1.0 moi%

all

X3, X4, X7

0.945

2.5 mol%

x3 & x4 only

There is insufficient justification for differentiating between models 2F, 2° and 2H with respect to the statistical fit but model 2H might be preferred to minimise equipment costs as it is likely that the reboiler temperature will already be measured for other reasons. Furthermore, model 2H shows the least sensitivity to the reaction zone temperature and this might contribute to the model's robustness. This model is sufficiently accurate to be used for closed-loop control and is simple enough to implement easily and to operate robustly.

This model is independent of pressure and would be suitable for a system where the pressure was tightly controlled and not susceptible to significant disturbances. If pressure fluctuations were anticipated to have a significant effect on the column operation two options exist for modifying the model. Firstly, the temperature measurements could be corrected for pressure using known VLE relationships. This is common in many operating facilities. Secondly, a new inferential model could be built using simulation data generated over the relevant pressure range. This type of model should be able to reflect the effect of pressure on temperature and also the effect pressure has on the overall column performance

(including its direct impact on the reaction via changes in the reaction equilibrium constant and the stace-to-stagc compositions), and might, therefore, be preferred.

There is clear potential to significantly improve the operation of ETBE reactive distillation columns through the use of inferential conversion models that could be applied to the simultaneous control of the isobutene conversion and the ether product purity. Separation considerations often predominate in the reactive distillation of MTBE, as the reaction can be expected to proceed at an acceptable rate oveT a wide range of conditions, but an inferential conversion model would be equally applicable and could be implemented with similar advantages. As well as the potential for closed-loop control, the additional process information that would be provided from a conversion prediction model would be highly valuable for routine process optimisation. An inferential model could also be developed for MTBE systems using the techniques demonstrated here.

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