This document provides an overview of numerical analysis methods and their applications in chemical engineering. It begins with introductions to numerical analysis and its history. Literature review sections define numerical analysis and discuss why it is used in engineering and examples of its applications in chemical processes. The methodology section describes how numerical methods are applied and examples of specific methods. Results and discussion evaluate accuracy of numerical methods. The conclusion states that numerical methods are necessary for complex problems that cannot be solved analytically. References are provided.

1. Numerical Analysis
Application of Numerical Analysis Methods
in Chemical Process Engineering
Instructor
Mr. Ribwar K. Abdulrahman
Prepared by:
Safeen Yaseen Ja’far
Submitted on
7/4/2023
Koya University
Faculty of Engineering
Department of Chemical Engineering
Fourth Stage/2022-2023

2. Table of Content
Abstract........................................................................................................................................................1
1. Introduction.............................................................................................................................................2
1.1 Introduction to Numerical Analysis ................................................................................................2
1.2 A Brief Historical Background of Numerical Analysis .................................................................2
2. Literature Review ...................................................................................................................................4
2.1 Definition of Numerical Analysis?...................................................................................................4
2.2 Why are Numerical Methods Used in Engineering? (Advantages)..............................................5
2.3 Applications of Numerical Analysis in Chemical Engineering (with Examples) ........................6
3. Methodology (Procedure).......................................................................................................................8
3.1 How Numerical Analysis Methods Applied? – Procedure ............................................................8
3.2 Numerical vs Analytical Methods in Math.....................................................................................9
3.3 Numerical Analysis Different Methods.........................................................................................11
3.4 Example on the Numerical Analysis Methods..............................................................................12
4. Results & Discussion.............................................................................................................................13
5. Conclusion .............................................................................................................................................13
6. References..............................................................................................................................................14

3. 1 | P a g e
Abstract
Nowadays. numerical methods play a crucial role in chemical engineering applications, enabling
the modeling and simulation of complex chemical processes that would be difficult or impossible
to solve analytically. So, in this report we focus on that and we will mention many applications in
various chemical engineering factories.

4. Page | 2
1. Introduction
1.1 Introduction to Numerical Analysis
Since the beginning of this century, digital computers have become more powerful and widely
available, which has increased the use of realistic mathematical models in science and engineering.
To solve these more intricate mathematical representations of the world, numerical analysis has
become more complex. Numerical analysis is a formal academic discipline that ranges from
computer science difficulties to very theoretical mathematics research. In the 1980s and 1990s, a
field known as scientific computing or computational science began to emerge as a result of the
increasing significance of using computers to carry out numerical operations in order to solve
mathematical models of the world. This area looks at the use of numerical analysis from a
computer science perspective; see. It is concerned with using the most powerful tools of numerical
analysis, computer graphics, symbolic mathematical computations, and graphical user interfaces
to make it easier for a user to set up, solve, and interpret complicated mathematical models of the
real world. The branch of mathematics and computer science known as numerical analysis
develops, examines, and applies techniques to solve continuous mathematics problems
numerically. These issues arise in a variety of fields, including the natural sciences, social sciences,
engineering, medicine, and business, and are typically the result of real-world applications of
algebra, geometry, and calculus (Kendall & Weimin, 2007).
1.2 A Brief Historical Background of Numerical Analysis
Nearly as old as human civilization itself are numerical algorithms. Archimedes of Syracuse (287-
212 BC) developed a great deal of new mathematics, including the "method of exhaustion" for
computing the lengths, areas, and volumes of geometric shapes. The Rhind Papyrus (about 1650
BC) of ancient Egypt explains a root finding method for solving a simple equation; see. It is quite
similar to modern numerical integration when used to determine approximations, and it was a key
step in the creation of the calculus by Isaac Newton and Gottfried Leibnitz. The creation of the
calculus by Newton and Leibniz served as a major catalyst for the development of numerical
methods since it allowed for the creation of precise mathematical representations of physical
reality, initially in the physical sciences and later in other fields such as engineering, medicine,
and commerce. Typically, these mathematical models cannot be solved explicitly; therefore,
approximate solutions must be obtained using numerical approaches. The invention of logarithms
by Napier (1614) and others, which provided a considerably simpler technique of doing the
arithmetic operations of multiplication, division, and exponentiation, was another significant part
of the evolution of numerical methods (Kendall & Weimin, 2007).

5. Page | 3
Figure 1: History of numerical methods employed in modeling rockbursts (Jun Wang,
Mohammadali, Derek Apel, 2020).

6. Page | 4
2. Literature Review
2.1 Definition of Numerical Analysis?
Numerical analysis is a discipline of mathematics concerned with the development of efficient
methods for getting numerical solutions to complex mathematical problems. There are three
sections to the numerical analysis.
So, there are some notes about numerical methods as follow:
A person will become more comfortable with the process of creating new numerical methods
by studying different numerical techniques and assessing them. This is essential when current
solutions to a problem are insufficient or ineffective.
There are usually several ways to solve a problem. Therefore, following the correct process is
essential to getting a precise response quickly.
With a strong foundation, one can successfully implement techniques (especially when a
method has its own limitations and/or drawbacks in certain situations) and, more importantly,
examine what went wrong when outcomes fell short of expectations (BYJUS, 2023).
Figure 2: Wind direction in blue, true trajectory in black, Euler
method in red.
What happens if a feather is dropped into the wind when 100 fans are set up to circulate air from
one end of the room to the other, as shown in the above figure? The feather will travel with the air
currents, which might be extremely intricate. One method of approximating the wind speed is to
advance the simulated feather as though it were travelling in a straight line at that pace for one
second before measuring the wind speed once more. The process used to solve an ordinary
differential equation is known as the Euler method.

7. Page | 5
2.2 Why are Numerical Methods Used in Engineering? (Advantages)
Numerical methods are used in engineering for several reasons (advantages):
▪ Complex Equations: The equations used to represent the system in many engineering issues
are too intricate to be resolved analytically. These equations can have approximations of their
solutions thanks to numerical methods.
▪ Large-Scale Systems: Engineering problems often involve large-scale systems with many
variables. Analytical solutions to these problems are difficult or impossible to obtain, while
numerical methods can be used to handle such large-scale problems.
▪ Accuracy: Engineering problems can be solved with extreme accuracy using numerical
approaches. Numerical approaches can give findings that are more accurate than analytical
solutions by increasing the number of iterations or improving the discretization of the system.
▪ Design and Optimization: Numerical methods can be used to design and optimize
engineering systems. They can be used to simulate the behavior of a system under different
conditions, allowing engineers to identify the best design or operating parameters.
▪ Experimental Validation: Numerical methods can also be used to validate experimental data.
By comparing the results of a numerical simulation with experimental data, engineers can
verify the accuracy of their models and identify any discrepancies that need to be addressed.
Overall, numerical methods are an essential tool for engineers, enabling them to solve complex
problems, optimize designs, and develop new technologies (Kalra, I., 2020).
Figure 3: In the past few decades, more attention has
been given to the designing of spacecraft. Numerical
analysis and simulations through user-friendly
computer software make it simpler to modify the
dynamics of the system.

8. Page | 6
2.3 Applications of Numerical Analysis in Chemical Engineering (with
Examples)
Chemical engineers frequently utilize numerical analysis to build and improve processes, simulate
the behavior of chemical systems, and solve difficult mathematical issues.
Some specific applications of numerical analysis in chemical engineering are:
▪ Simulation of Chemical Plant, Chemical Processes and Equipment Design
Numerical methods are used to simulate the behavior of chemical processes, such as chemical
reactions, separation processes, and transport phenomena. These simulations can be used to
optimize the design and operation of chemical plants, as well as to predict the performance of new
processes.
▪ Process Optimization
Numerical methods can be used to optimize chemical processes, by identifying the best operating
conditions, reactor configurations, and catalysts. These optimizations can improve the efficiency
and profitability of chemical processes, as well as reduce waste and environmental impact.
Figure 4: Schematic diagram of SCAHP by applied numerical analysis

9. Page | 7
▪ Fluid Dynamics
In chemical processes, such as those that take place in reactors, pipelines, and distillation columns,
numerical methods are employed to simulate the fluid flow. These simulations can be used to
forecast the behavior of complex flows, such as turbulent and multiphase flows, and to improve
the design of these systems.
Figure 5: Viscous Flow in A Pipe (using ANSYS with numerical method application) (MIT
OpenCourseWare., 2022)
▪ Thermodynamics
Complex thermodynamic equations that explain the behavior of chemical systems are solved using
numerical techniques. Phase equilibria, reaction rates, and thermodynamic properties of chemical
systems can all be predicted using these models.
Figure 6: Mathematical modeling and thermodynamics of Prandtl–Eyring fluid with radiation
effect: a numerical approach (Ferziger, J.H., 1998).

10. Page | 8
3. Methodology (Procedure)
3.1 How Numerical Analysis Methods Applied? – Procedure
Numerical analysis methods can be applied to solve mathematical problems in various fields,
including physics, engineering, finance, and computer science. Here are some general steps to
apply numerical analysis methods:
Step 1: Formulate the problem: Clearly define the problem to be solved and its constraints.
Determine the mathematical model or equation that represents the problem.
Step 2: Discretize the problem: Break down the problem into smaller, more manageable parts or
steps. This may involve dividing the domain into smaller subdomains, approximating functions
using polynomials or other mathematical functions, or discretizing time or space.
Step 3: Choose a numerical method: Select an appropriate numerical method that can solve the
problem efficiently and accurately. This may involve selecting a numerical integration method,
root-finding method, optimization algorithm, or iterative method.
Step 4: Implement the method: Implement the numerical method using a computer program or
software package. This may involve writing code in a programming language such as Python,
MATLAB, or C++, or using a specialized numerical analysis software package.
Step 5: Test and validate the results: Test the numerical method on known problems with exact
or approximate solutions to validate the accuracy of the results. If necessary, refine the method or
adjust the parameters to improve accuracy.
Step 6: Apply the method: Apply the numerical method to solve the problem at hand. This may
involve analyzing data, simulating a physical system, or optimizing a process.
Step7: Interpret the results: Interpret the numerical results in the context of the original problem.
Analyze the sensitivity of the results to changes in input parameters, identify any limitations or
assumptions of the method, and draw conclusions about the behavior of the system or process
being analyzed.

11. Page | 9
3.2 Numerical vs Analytical Methods in Math
Some issues in mathematics can be resolved analytically and quantitatively. An analytical
approach entails formulating the precise remedy and articulating the issue in an understandable
way. A numerical solution entails speculating about the answer and determining if the issue is
sufficiently resolved to end. The square root, for instance, can be solved in both directions
(Dehghani, H., Noll, I., Penta, R., Menzel, A. and Merodio, J., 2020).
Here are some key points summarizing the differences between numerical and analytical methods
in mathematics:
Analytical Method:
• Involves solving equations or systems of equations using algebraic or calculus techniques to
obtain exact or closed-form solutions.
• Provides insight into the behavior of the system and can be used to make predictions about
future behavior.
• Suitable for solving simple problems with well-defined equations.
• Cannot always be used for more complex problems.
• Results in exact or closed-form solutions.
Numerical Method:
• Uses computational techniques to obtain approximate solutions to mathematical problems.
• Can be very accurate and is often the only feasible way to solve complex problems.
• Involves breaking down the problem into smaller, more manageable steps, and using
algorithms to perform the necessary computations.
• Accuracy and reliability depend on the chosen algorithm, the quality of the input data, and the
computational resources available.
• Results in approximate solutions.
• Necessary for solving more complex problems that cannot be solved analytically.

12. Page | 10
Example: Consider Duffing oscillator with cubic nonlinear term in the following form:
Figure 7: Difference between analytical and numerical solution for example above (Khan, Y.
and Mirzabeigy, A., 2014).

13. Page | 11
3.3 Numerical Analysis Different Methods
Some calculations cannot be solved using algebra or other Mathematical methods.
Newton method: also known as Newton Raphson Method, for this we need to use numerical
methods. Newton's method is one such method and allows us to calculate the solution of f (x) = 0.
Simpson Law: Other significant ones cannot be evaluated in terms of integration principles or
fundamental operations. Simpson's law is a formula that determines the value of a direct
combination in numbers.
Trapezoidal law: A mathematical technique that determines the numerical value of a direct
combination is the trapezoidal rule. The other significant ones cannot be evaluated using
integration guidelines or fundamental operations.
Figure 8: “Numerical Methods of Analysis” Tree

14. Page | 12
3.4 Example on the Numerical Analysis Methods
As we mentioned earlier, this method, also known as Newton’s method of tangents, is one of the
fastest iterative methods. This approach starts out with an approximation. Here, one needs to be
careful while choosing the starting approximation because it is extremely sensitive to it. Once the
first approximation is properly chosen (Khan, Y. and Mirzabeigy, A., 2014).
Given the one root of the non-linear equation:
𝑥3
− 4𝑥 − 9 = 0
Lies between 2.625 and 3.0. Find the root correct to four significant digits.
Solution Given: 𝑓(𝑥) = 𝑥3
− 4𝑥 − 9 , 𝑓(𝑥) = 3𝑥2
− 4

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4. Results & Discussion
Now, we knew that we have many methods to solve mathematical problems and usually many
methods that used not accurate such in numerical methods. The accuracy and reliability of
numerical method results depend on several factors, including the complexity of the problem, the
chosen numerical method, the quality of the input data, and the computational resources available.
In general, numerical methods provide increasingly accurate results as the number of computations
increases, but this comes at the cost of increased computational time and resources. One common
way to evaluate the accuracy of numerical method results is to compare them to analytical or exact
solutions, when available. This can be done by calculating the error or difference between the
numerical and analytical solutions. The smaller the error, the more accurate the numerical method
result.
Another important consideration when evaluating numerical method results is the stability of the
method. A numerical method is said to be stable if small changes in the input data or parameters
do not cause large changes in the output. Unstable numerical methods can produce wildly
inaccurate or unpredictable results.
In summary, numerical methods have several advantages over analytical methods:
✓ Including flexibility
✓ Accuracy, efficiency
✓ Visualization
✓ Optimization
✓ Verification.
5. Conclusion
As we mentioned earlier, we have many methods to solve equations and apply methods on very
large scale of our life also use in industrial manufacturing.
So, in summary, we can say that analytical methods are valuable for solving simple problems with
well-defined equations and obtaining exact or closed-form solutions, while numerical methods are
necessary for solving more complex problems that cannot be solved analytically. Both methods
have their strengths and weaknesses, and the choice of method depends on the nature of the
problem and the resources available.

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6. References
1. Atkinson, K. and Han, W., 2007. Theoretical Numerical Analysis. Springer New York: New
York Available at:
https://www.google.iq/books/edition/Theoretical_Numerical_Analysis/6TpGAAAAQBAJ?hl
=en&gbpv=0 [Accessed 24 Mar. 2023].
2. Wang, J., Apel, D.B., Pu, Y., Hall, R., Wei, C. and Sepehri, M. (2021). Numerical modeling
for rockbursts: A state-of-the-art review. Journal of Rock Mechanics and Geotechnical
Engineering, [online] 13(2), pp.457–478. Available at:
https://www.sciencedirect.com/science/article/pii/S1674775520301645 [Accessed 5 Jul.
2022].
3. BYJUS. 2023. Numerical Analysis | Introduction to Numerical Analysis. [online] Available
at: https://byjus.com/maths/numerical-analysis/. [Accessed 25 Mar. 2023].
4. Kalra, I. 2020. Why Do We Need Numerical Analysis In Everyday Life?» Science ABC.
[online] Science ABC. Available at: https://www.scienceabc.com/eyeopeners/why-do-we-
need-numerical-analysis-in-everyday-life.html. [Accessed 7 Apr. 2023].
5. MIT OpenCourseWare., 2022. Lecture Notes and References | Numerical Fluid Mechanics |
Mechanical Engineering. [online] Available at: https://ocw.mit.edu/courses/2-29-numerical-
fluid-mechanics-spring-2015/pages/lecture-notes-and-references/ [Accessed 6 Apr. 2023].
6. Ferziger, J.H., 1998. Numerical Methods for Engineering Applications. Wiley-
Interscience.[eBook] https://www.wiley.com/en-
us/Numerical+Methods+for+Engineering+Applications%2C+2nd+Edition-p-
9780471116219 [Accessed 24 Mar. 2023].
7. Dehghani, H., Noll, I., Penta, R., Menzel, A. and Merodio, J. (2020). The role of microscale
solid matrix compressibility on the mechanical behaviour of poroelastic materials. European
Journal of Mechanics - A/Solids, 83, p.103996. Available at:
https://doi.org/10.1016/j.euromechsol.2020.103996. [Accessed 7 Apr. 2023].
8. Khan, Y. and Mirzabeigy, A., 2014. Improved accuracy of He’s energy balance method for
analysis of conservative nonlinear oscillator. Neural Computing and Applications, 25(3-4),
pp.889–895. Available at: https://doi.org/10.1007/s00521-014-1576-2. [Accessed 7 Apr.
2023].
9. VEDANTU. 2023. Numerical Analysis. [online] Available at:
https://www.vedantu.com/maths/numerical-analysis. [Accessed 24 Mar. 2023].