Mathematical modelling

Overview

At the heart of this dry-sounding topic is the fact that, in the end, almost all behaviour can to some degree be described by mathematics. Some would say that an elegant mathematical proof or equation is a ‘thing of beauty’.

On a practical level the term ‘digital twin’ has recently appeared and is taken to mean an accurate representation or prediction of the behaviour or performance of an actual piece of equipment or of an entire plant. This capability has become possible thanks to the rapid (and ongoing) increases in computing power over some 4 decades.

The first stage is a model of the plant or equipment when it is running under ‘steady state’ or normal conditions. The effect of changing conditions and determination of the new steady state can be generally found using any one of the several commercial steady state simulation programs that can now run on a modern laptop computer.

A second stage is to predict in real time how the conditions of the equipment or plant will change with time in response to a disturbance. This involves dynamic modelling of the process and can again generally be performed on a reasonably powerful laptop computer. Dynamic modelling is less widely used than steady state modelling and requires significant additional input data, but is a useful extension of the top commercial simulation software.

Process Simulation

Using a process simulator is an important skill for process engineers / chemical engineers.  The ability to model parts or most of a chemical process constitutes a vital powerful tool in design, operations and troubleshooting.  A 'process' is  complex combination of equipment like heaters and compressors coupled together by piping to achieve the desired modification of the fluids passed through it.  A mathematical model is built to reproduce as closely as practicable how a real process reacts as conditions are altered.  This enables ideas to be tested, processes to be designed, and for operators to be trained, before the real plant or process is built or started up.

Generic simulation tools provide a palette of common unit operations, such as pumps, separator vessels, heat exchangers and many more, for which the main parameters are specified and which can be coupled together by conceptual or (‘real’) piping elements to form the desired process.

Steady State modelling

The feed(s) compositions and operating conditions are various points are defined and the energy and materials balance for the entire process is then calculated by the mathematical model.  This is is commonly called a ‘steady state process simulation.

The term – 'steady state' means that although fluids are continuously flowing through the process and equipment items, the compositions and the conditions of temperature and pressure at each point in the process are assumed to remain steady.

Dynamic modelling.

A more challenging situation arises when some of the process conditions or feed compositions change with time.  The model must now have significant additional information about capacitance and flow resistances (such as valves) to be able to predict how disturbances propagate through the process and whether it remains stable and whether the proposed control scheme and controller settings will work satisfactorily.

Dynamic models have also been used for some decades as operator training simulators (OTS), but these were often based on empirical shortcut relationships limited by the computing power of the day and therefore without high fidelity of today’s dynamic models.

Summary

In some cases, smaller specific equipment items or sections of plant can be conveniently modelled using the powerful spread-sheeting tool MS Excel, both in steady state and dynamic or transient situations.

Both steady state and dynamic simulator and spreadsheet approaches have their place in the toolkit of today’s process engineers.  They will be explored further with case studies in some future posts.

Process simulation guidance

Developing a simulation model using HYSYS, UniSim etc or Excel, here are a few recommendations to help maintain sanity for the author and other users:

1.      Save the file regularly

2.      Use a unique sequential reference number for each version of the file

3.      Keep a logbook of changes and results ,when developing a model or adapting for new circumstances.

4.      Sensitivity checks on assumed variables are useful if clearly recorded

5.      Prepare user documentation and troubleshooting tips before sharing / publishing.  EG how to reset an Excel model after a crash.

When building a simulation model, try to make the flowsheet neat and simple – easier to understand.  Only specify values for the minimum required number of process variables.  You can always add.  Get used to thinking about the interdependencies between things in the model.

If a Process simulation model is complex and hard to converge, break one of the recycles and converge approximately manually.  The reconnect the recyle operation.

Perform rough standalone calcs to see what ballpark answers to expect – with an approximate mass balance as a minimum.

For troublesome distillation modules within a larger model converge column separately, using simple specs.  Get a converged solution even with the wrong product split or purity and work towards the desired result.  Try different solvers. Then connect back into the main model.