Key Learning Objectives

• • Knowing the difference between analytical and numerical solution

• • Learning the basics of process modeling and simulation

• • Necessity of numerical simulator in process engineering

• • What is an error and what role it plays in numerical simulation

INTRODUCTION TO NUMERICAL METHOD

Numerical methods are widely used to solve mathematical problems that arise in natural sciences, social sciences, engineering, medicine, and business. These methods can provide approximate solutions of sufficient accuracy for engineering purposes. Numerical analysis of the engineering problems had started in the mid-1960s when advancement in high-speed digital computing was made. Since then, the numerical approaches have gained attention for solving practical problems, for which, closed form analytical solutions are either intractable or do not exist.

Why numerical method?

We typically follow two ways to solve mathematical equations. They include:

• • Analytical method

• • Numerical method

The former approach is usually preferred to solve the linear equations, whether they are in algebraic or in differential form. On the other hand, the numerical methods are very common in solving the non-linear equations, which constitute most of our natural systems. Moreover, the numerical methods are also used for linear systems when analytical solution is complicated and time-consuming.

Let us compare the merits and demerits between the analytical and numerical methods (Table 1.1) to understand why and when the numerical method gets preference over the analytical one.

INTRODUCTION TO PROCESS SIMULATION

PROCESS MODELING

Prior to process simulation, let us first know how to define the mathematical model in a simple way.

Definition of ‘mathematical model’: The mathematical description of a real-life system or phenomenon (whether physical, sociological, or even economic) is referred to as mathematical model.

The way and the purpose of developing a mathematical model for a process (i.e., a process model) are described step-by-step in the following section.

• Step 1: Make necessary assumptions or hypotheses

• Step 2: Formulate the mathematical equations for a process to represent its various phenomena

• Step 3: Solve the modeling equations (which is referred as process simulation)