Process Modeling and Linearization

The chemical industry is dependent on our ability to manufacture any substance of commercial value. Before we start production, we need first to understand and model the process in an effective and cost-efficient manner, as the industry relies on low-cost methods. Thus process modeling and linearization techniques are used to model and simulate the process flow in an easy and efficient method. It requires knowledge of the properties of the chemicals involved in the process and the physical and chemical properties of the components. Linear methods are the simplest way to represent a process flow diagram. Linearization of a nonlinear model is essential for us to achieve this goal.

Process Modeling:

It is a computer model-based simulation of the industrial scale’s physical, chemical, and biological processes. To construct such models, we need the basic physical and chemical properties of the pure components, mixtures, and mathematical models that will help us integrate and simulate them inside a computer. Flow diagrams are used to represent the models, and mass and energy equations are solved by the software to try and find stable optimal parameters for the process.

The advantages of Process Modeling are:

• It can save us a lot of time.

• Since there is no human intervention, human error can be avoided in the results.

• Let’s picture and model complex methods that are not possible otherwise.

• Different scenarios can be tested and retested as per the needs of the process at a meager cost and maximum efficiency.  

Linear Models:

Linearization is a technique that is often employed to solve complex ordinary differential equations (ODEs). Since it takes many computational and mathematical tools to solve nonlinear models, engineers often convert it into a linear differential equation. A linear equation is named such as it represents straight lines in the Cartesian coordinates. A simple linear equation consisting of two variables x and y can be stated as:

y=mx+c

This is the equation of a straight line with the y-intercept at c. Even though most of the processes found in the industry are nonlinear, generally, engineers will convert them into approx linear models. They are easier to solve, and even though they are not precisely accurate, it allows us to witness the behavior of the process and get a gross idea about it. Its applications can be found in a continuously stirred tank reactor, heat exchangers, or biological cell culture. 

Some of the advantages and disadvantages of linearization can be stated as:

• Advantages:

  • Provides a simple and effective way to solve ordinary differential equations.

  • It helps us in understanding the behavior of the process.

  • We can use any type of order of ODE.

• Disadvantages:

  • Since it is an approximation it only works for a short range of values and not for the system as a whole.

  • Even though it is a quick process it takes time initially to learn the process. 

Non-Linear Models:

In simple mathematical terms, we can understand that nonlinear models are models in which the change in output is not proportional to the change in input. Since most of the systems found in nature and industry are of this manner, it becomes essential for us to try and understand how to study them effectively. Nonlinear systems can appear chaotic, unpredictable, and more difficult to understand than linear systems. Some properties of nonlinear models are:

• They do not follow the superposition principle.

• They have multiple equilibrium points

• They exhibit properties such as a limit cycle, bifurcation, chaos, etc.

• They may not exist for all time scales.

 Thus process modeling and linearization techniques are used to study nonlinear models effectively. Some methods to understand and solve nonlinear ordinary differential equations are:

• Bifurcation theory

• Linearization via Taylor series.

• Perturbation methods etc.

Some examples of nonlinear models can be stated as follows:

• Algebraic Riccati Equation 

• Boltzmann equation

• General relativity

• Nonlinear Schrodinger equation

• Sine-Gordon equation

• Vlasov equation. 

Conclusion:

Computers have become essential in all walks of our life. The ability to perform highly time and effort-intensive tasks at the click of a button seems magical. Thus, it is no surprise that industries are utilizing these tools to build chemical process models and use them for parameter optimization and stress testing. Still, the engineer holds all the keys and makes important decisions. Thus based on our needs, we can use these process modeling and linearization techniques to simplify complex nonlinear problems. As the computing powers are improving yearly, so is our ability to construct better models that solve more complex problems.