# Model validation and model-based data analysis

## Using gPROMS’ state-of-the-art parameter estimation facilities

A gPROMS® advanced process model is constructed from first-principles equations describing the physical and chemical phenomena occurring in the system.

These equations usually involve parameters that can be adjusted to make the model predictions match observed reality – for example, reaction kinetic constants or heat transfer coefficients.

For example, in the reaction scheme on the right, the rate equations for r1, r2 and r3 depend on values of the Arrhenius parameters ki0 and Ei, the activation energy. The more accurate these parameters, the more accurate the models’s predictive capabilities.

## Parameter estimation

The process of fitting model parameters to laboratory or plant data is called parameter estimation.

gPROMS contains powerful, state-of-the art parameter estimation capabilities that have been applied successfully to a wide range of problems. Key features are:

• Multiple parameters occurring in dynamic or steady-state models may be estimated simultaneously. Nonlinear models of arbitrary size and complexity – including multi-unit flowsheets – may be used.
• Data from both dynamic and steady-state experiments may be used.
• The results of the estimation include extensive statistical analysis using the information contained in the models themselves as a basis. This provides a measure of the uncertainty in the calculated parameter values.

## Parameter estimation in gPROMS

gPROMS estimation techniques include Least Squares and the Maximum Likelihood formulation.

The latter provides simultaneous estimation of parameters in both:

• the physical model of the process
• the variance model of the measuring instruments, which can be:
• constant variance (e.g. a thermocouple with an accuracy of x K)
• constant relative variance (e.g. a composition analyser with an error of x %)
• heteroscedastic variance, combining
• both of the above.

Detailed model-based data analysis of results includes residual and overlay plots (above), confidence ellipsoidscorrelation matrix and model adequacy tests ## Example

The example below shows how parameter estimation is performed in gPROMS to determine reaction kinetic constants.

### Step 1 Construct a model of the process for which the measurements are being taken. Models are typically constructed from gPROMS libraries such as the AML:FBCR.

Typically for reaction kinetics, this is a laboratory system which is small enough to ensure that extraneous effects – such as mixing phenomena – do not obscure the measurements. Of course, parameters can also be estimated from pilot plant or real plant data.

This example uses a simple stirred-tank reactor.

### Step 2 Define the parameters to be estimated and the variance model to be used for each measuring instrument.

In this example, the heteroscedastic variance model is selected for the composition analysers, meaning that gPROMS will determine automatically the optimal proportion of constant and relative variance to be taken into account.

### Step 3 Enter the experimental data sets. You may include as many experiments as required, and these may contain steady-state or dynamic data sets.

### Step 4 Execute the estimation run.

### Step 5 Check the results in the detailed analysis (4) and the confidence ellipsoid plots (5).  