# MATH123 Mathematical Modelling Assessment

## Introduction

MATH123 Mathematical Modelling assessment is a concept in which real life problems are solved mathematically by making a mathematical model. In this, problems are expressed with the help of mathematical equations. These equations are solved and then translated back to results in real life (Ale, 1981).  The model is constructed keeping in mind that it is consistent throughout and addresses the situation it is designed for. The mathematical modeling is used in fields like psychology, operations research, biology, meteorology etc. Some other advantages of using models are:

• More reliable than pure intuition
• It simplifies the analysis.
• It reduces the cost of operation.

Here we take the case of corrosion that is a big problem in industries and how mathematical modeling is helpful in giving solution for the problem.

### Issue/Problem

Corrosion is one of the problems that are faced by modern day industries. An industry looses millions of dollars due to corrosion. This loss is because it decreases life span of the equipments. Due to corrosion, numerous pipelines have been ruptured, oil spillages have occurred. And these situations have resulted in other issues like environmental problems, ecological damages and various resources have been lost while cleaning this mess.  The chemical, mechanical and petroleum industries are the one which are most affected by corrosion problems.  The corrosion affects many industries in following ways:

• In chemical industries, corrosion can affect lot of chemical reactions and produces unexpected results and products. (Jones, 1967)
• In petroleum industries, perforated trays are used for the fractional distillation process. These trays can get affected by corrosion because plants get aged as days goes by.
• In mechanical industries, refined and crude oils pipelines get affected by corrosion.
• The various factors like temperature, change in pressure can also cause corrosion in the containers and oil reservoirs.

Corrosion also has some drastic effects like stress cracking. Stress cracking is a phenomenon in which material looses all its brittleness prematurely.  It occurred because the material is subjected to high temperature and corrosion rate can change significantly in these conditions.  Because of all the reasons stated above the need to study corrosion is need of the hour.

### Solution Offered

The corrosion problem can be solved by help of mathematical models. These models provide various tools which help in analyzing and studying the various reactions like chemical kinetics and also thermo chemistry of various compounds. They also provide solutions for petroleum products by studying material sciences of their containers and helps in storing, transporting and producing of their products. These solutions are qualitative as well as quantitative (Berry, 1990).They help to determine how much corrosion will contaminate the product over its life cycle and what effective steps we can take to reduce or finish its effect.  The models which provide solution to corrosion are:

#### Mass-Heat Transfer Model:

As one of the basic reasons for the corrosion is the flow of heat transfer across the metallic wall and hence this model is considered. This model analyzes what is the heat loss per unit area by making, analyzing and then solving the mathematical equations.With these we calculate plant’s life expectancy and the theory involved is called perturbation theory.The mathematical model gives us following results:

• Life expectancy of the plant.
• Suggests which factors can increase life expectancy.
•  The formula for an acceptable working plant.
• Will give an indicator if plant is not working in acceptable condition.

#### Zhimz-Hoffman (ZH) Model:

When there is a deposition of corrosive fluid on the solid surface then there arises a problem of phase transition.Phase transition occurs generally in the cracking process. The problem of phase transition can be handled by this model.This method produces a highly accurate result.This model analyzes various properties of the surface viz. conductivity, latent heat etc. and relates them to obtain mathematical equations.This model by using the temperatures produces in different phases computes the expected time a plant can live. It also helps in determining which material has best expectancy life. (Hoffman et al, 1994)

### Social & Environmental Implications

The results and finding of these models of real life problems have various implications on society, environment etc. These models help in the betterment of these things. Some of the implications are listed below:

• These models help in saving lot of money of the concerned industries. This money can be put in other things which will help the society.
• Oil spillages can be avoided by detecting corrosion of containers at earlier stages. This in turn will help in saving marine life and thus the environment.
• As these models determine the life expectancy of the equipments, company can dispose them at right time which will help in avoiding any major accident in the future. Hence saving human life and many other resources in the process.

### References

1. Berry John (1990). “Mathematical modeling: A Source Book of Case Studies”. Edited by I.D Huntley and D.J.G James, Oxford University Press London, pp 81 – 96.
2. Jones G.D. (1967). “Chemistry and Industry: Applications of Basic principles in Research and process Development”. Clarendon press, oxford.
3. Zhimz and Hoffman (1994). Institute of Mathematics and Applications Journal Numerical analysis. 14, 243 – 255.
4. ALE S.O. (1981) “Curriculum development in modeling process”. Trieste, Italy, International Centre for Theoretical Physics occasional publication.