Key Learning Objectives

• • Reviewing the basics of numerical integration

• • Knowing commonly used Newton–Cotes integration technique

• • Studying Richardson extrapolation for Romberg integration

• • Knowing Gauss quadrature for more efficient integration

• • Learning how to choose a suitable numerical integration method

INTRODUCTION

Here, our primary objective is to compute the integral I of the following form:

It basically stands for the integral of the function, f(x) with respect to x (independent variable) over a given interval [a, b]. Prior to doing integration, we need the fitted function, f(x). In the last chapter, we discussed various methods to determine this function based on the specified data points. Here we fix our subsequent goal to do the integration of f(x) having its polynomial form.

The integral I in Equation (9.1) geometrically represents the area under the curve y = f(x) for a certain range of x. As shown in Figure 9.1, it is basically the area of the region bounded by the curve y = f(x), the lines x = a = x0 and x = b = xn, and the x-axis. On the other hand, the definite integral I in Equation (9.1) can be evaluated algebraically having the function f(x) as a polynomial. This scheme is often called the numerical quadrature.

The numerical integration or quadrature2 formula is used to obtain approximate answers for definite integrals, which are either impossible or difficult to evaluate analytically. This is true particularly when the function is complicated, which is not so uncommon in realistic examples. Moreover, we employ the numerical integration approach to integrate an unknown function, for which, only a few sample points are available.

Numerical integration has various applications in several disciplines. A popular example includes the determination of enthalpy (H) at any temperature (T) from:

In which, Cp is the heat capacity that is usually available as a polynomial function of T. This apart, there are many other applications in mass transfer, reaction engineering, and so on.

Key Learning Objectives

• • Knowing various types of boundary conditions

• • Learning numerical methods to solve boundary value problems

• • Applying numerical methods on linear and non-linear systems

• • Modeling case studies for their numerical simulation

• • Analyzing simulated process behaviors at various conditions

INTRODUCTION

We have seen in the last chapter that for an initial value problem (IVP), the initial conditions need to be specified at the starting (i.e., at time t = 0). On the other hand, for a boundary value problem (BVP), one needs to specify the boundary conditions that hold for all times. Basically, the conditions that one imposes on the boundary of the domain are termed as the boundary conditions (BCs) and the resulting problems are called the boundary value problems (BVPs). For instance, for a second-order differential equation, one needs to specify two boundary conditions.

There are typically three common types of boundary conditions and they are highlighted in the following section.

A. Dirichlet BC

The Dirichlet boundary condition is perhaps the most common type of constraint that is concerned with the value of the dependent variable, y on the boundary. For example, in which, β is a constant. Note that this BC isspecificto the left boundary (i.e., at x0) and one can have a similar expression for the right boundary (i.e., at xM) as y(xM) = constant.

In physical systems, this BC is typically used in fixing the state variables, namely temperature, composition and so on, at a boundary.

B. Neumann BC

The Neumann boundary condition is concerned with the derivative of y on the boundary. For example,

Similar boundary condition one can have for the left boundary (i.e., at x0).

In physical systems, this BC is used in fixing the mass or energy flux. Note that for no-flux condition (when, say the boundary is impermeable or perfectly insulated), β= 0.

C. Robin BC

The Robin boundary condition involves a combination of the above two (i.e., y and its derivative on the boundary). Thus, it is also called as mixed BC.

Key Learning Objectives

• • Knowing the basics of data fitting

• • Learning various error criteria

• • Understanding the fundamental of method of least-squares

• • Applying least-squares on linear and non-linear examples

INTRODUCTION

The availability of data (in terms of dependent variable, y) at several discrete locations (values of the independent variable, x) is quite common. Often, we call it as input-output data set in different fields of our study. It may be laboratory-scale, pilot-scale, industrial-scale, or field-scale data. To deal with different scientific and engineering problems at various occasions, one may need to:

For these mathematical operations, one needs to first construct an analytical function, f based on the available y versus x data. Thus, prior to performing the interpolation, differentiation or any kind of integration, one has to know an appropriate technique to fit the data sets by function f. Accordingly, here we fix our goal to determine a formula, y = f(x) that best fits the available experimental data set. To do so, the most common technique used is method of least-squares or least-squares curve fitting. Actually, there is a class of formulas (depending on the process characteristics) and studying them will be the subject matter of this chapter.

At this point of time, let us know shortly the distinction between the interpolation and curve fitting. Any individual data point may be incorrect and thus, without intersecting every data point, we can derive a curve to follow the pattern (or general trend) of the points as a group. This is often called curve fitting (or sometimes regression). Accordingly, Figure 7.1(a) is drawn to show that the function determined through curve fitting does not necessarily pass through any data point. On the other hand, in case of interpolation [Figure 7.1(b)], the derived function typically passes through every data point (to be covered in Chapter 8) with the presumption that the data are known to be very precise.

Key Learning Objectives

• • Revising basics of linear systems and their different types of solutions

• • Learning various solution methods (direct and indirect)

• • Applying these methods to a wide variety of systems examples

• • Knowing how to choose a suitable solution method

INTRODUCTION

There are several systems whose physical operations are described solely by algebraic equations. The general form of such a system of n coupled linear1 algebraic equations for n unknowns (i.e., xi, i = 1, 2, …, n) is:

Here, a11, a12,………ann, are the coefficients of the system, and 12bb,………, bn, are the constant terms.

One can represent Equation (2.1) in matrix form as:

in which, the coefficient matrix (i.e., an n × n square matrix),

unknown vector (i.e., column vector),

and constant vector,

In Equation (2.3), aij denotes an element in the ith row and jth column of matrix A. Notice that here we use capital/uppercase letters to denote the matrices and small/lowercase letters for their entries/elements.

A linear system is said to be homogeneous when it is represented by a modified form of Equation (2.2) as:

AX = 0(2.6)

Here, 0denotes a null matrix (zero vector).

At this point we should note that aside from inherently linear systems of algebraic equations, these equations arise in the systems of differential equations (e.g., boundary value problems and partial differential equations) as well. Anyway, prior to reading the subsequent sections of this chapter, the reader is recommended to go through the Appendix 2A that covers the basics of vector and matrix algebra.

Types of Solution

For linear systems of algebraic equations, mainly three types of solutions exist:

• i) a unique solution

• ii) infinitely many solutions

• iii) no solution

Let us briefly discuss them.

To solve a system represented by,

AXB= (2.2)

instead of matrix A [Equation (2.3)], one can use the following augmented matrix (denoted by aug A) for the ease of computation:

Part IV of this book consists of:

• • Data fitting

• • Numerical integration

Data fitting

Functions are used to model many real world problems, for which, we have discrete data points that are either directly measured or derived quantities. To understand the behavior of these functions, they need to be continuous or differentiable. Constructing such a continuous function based on the discrete data is called data fitting. Here, we will cover two approaches for data fitting:

• • Least-squares data fitting (Chapter 7)

• • Polynomial interpolation (Chapter 8)

In the subsequent Chapter 9, we will discuss numerical integration approaches using the functions fitted in Chapter 8.

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)

Key Learning Objectives

• • Learning several numerical integration methods for initial value problems

• • Analyzing numerical stability of those methods

• • Example systems are solved to understand some important characteristics of the numerical methods

• • Learning how to choose a suitable method for a specific system

INTRODUCTION

We will discuss here a couple of elegant and advanced numerical methods, and their use to solve the systems of first-order ordinary differential equations (ODEs).1 At the beginning, let us first review a few basic concepts.

Order of an ODE

The order of the highest derivative appeared in a differential equation is typically termed as the order of that equation. For example,

is a second-order ODE since the highest derivative that appears in it is second-order. Note that here, y is the dependent variable and x the independent variable.

Degree of an ODE

The degree of a differential equation is the exponent/power of the highest order derivative in that equation. With this, Equation (4.1) has degree 1 since the exponent of the highest order derivative is 1.

Transforming Higher-Order ODE to a Set of First-Order ODEs

As stated, here we will learn how to solve the first-order ordinary differential equations (ODEs) numerically. For any higher-order ODE, however one can easily transform it to a set of first-order ODEs and that we are going to discuss. To understand this point, let us take an example system represented by a simple second-order ODE:

Let us consider,

Accordingly, the second-order Equation (4.2a) gets the form of a set of two first-order ODEs:

In a similar fashion, one can reduce an nth-order ODE to:

where, k=1, 2,……n. This issue is further elaborated below with a second example.

Transforming a third-order ODE to a set of first-order ODEs

Consider the following system of equation:

This third-order equation requires three initial conditions, given as: