What is a „mathematical model“?

What does the term of „mathematical model“ represent in chemical engineering and engineering in general? A broad definition is: a model is a „virtual version of reality“. In literature, there can be found definitions such as "an image of reality from a particular viewpoint". Or a more precise one: A model is a simplified representation of those aspects of an actual process that are being investigated (Kafarov & Kuznetsov 1976). A mathematical model of a real chemical process is a mathematical description combining experimental facts and establishing relationships between the process variables (Babu).
Definitions are differing almost as the models: in viewpoint, in the level of details and in the goal of development.

Model in practice

From practical point of view, these 3 points are important to have in mind while developing a model and analyzing the relationship between the model and the reality:

• Model always has a certain deviation from the real process,
• Definition of the model and a mathematical tool is dependent on the problem it is exploring,
• Characteristics of the model are dependent on the engineers who work on the development: the level of their knowledge, their experience and their vision of the reality.

There is no way that two people who work independently could develop the model which would look the same and function the same, no matter what tool they would use. As we perceive the colors differently, every person looks into a problem differently with the different knowledge level and different previous experience and it is inevitable that the developed model is different too. 
When developing a model of a particular operation, such as a distillation column, the different approach is employed when model is developed with the purpose to define the sizing parameters of the column or when the column has to be analyzed to explore the control strategy or product quality. Dependent on the purpose, different mathematical method and different level of details have to be applied.

Answer these questions before building the mathematical model

Preparation for any project that involves the process simulation and employment of process models requires answers to these questions:

• Define the system and define the modeling subject: what operations and process equipment need to be included for modeling, define the system boundaries,
• What is the goal of the model?  Is it process design? Is it optimization? Is it analysis of control strategy? Is it training? Is it safety? Etc... Answers to this questions can help you to define the level of modeling details and with it, the model complexity,
• What data need to be known for a defined system and purpose and are all those data available? Often, this is the point where certain assumptions have to be made with clear awareness that those are not in the conflict with defined goal of the model
• What software tools are needed to complete the task? Is it only a programming tool needed or a professional process simulator? Is it available or does it mean a new cost? Is this expensive or affordable and are there any free tools to complete the task?

Accuracy

Dependant on the model type, the simulation tool and adequacy of the data – one can build more or less accurate model. It will never be a 100% accurate. So to say: a perfect model doesn't exist, but having one that is close enough can be of a lot of help and a problem solution. 
Is it a model then an art of science? In a way, it can be accepted as a creative perspective of science and engineering. It's purpose is to solve, but to solve a problem one can use different approaches towards the solution and be as creative as possible. 
The model is corresponding to reality through the flowsheets, P&I diagrams, and all the data.The model needs to take into account properties of all the materials and other physical characteristics defined by temperature, flows, pressures and composition.

However, the value of a good model is often priceless.

A good model

A good model should reflect the important factors affecting a process and must not be crowded with minor, secondary factors that will complicate the mathematical analysis and might render the investigation difficult to evaluate. Depending on the process under investigation, a mathematical model may be a system of algebraic or differential equations or a mixture of both. It is important that the model should also represent with sufficient accuracy both quantitative and qualitative properties.

Application of models

Models are used for a variety of applications, such as the study of the dynamic behavior, process design, model-based control, optimization, controllability study, operator training, and prediction. These models are usually based on physical fundamentals, conservation balances, and additional equations.
 

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