Derive the governing equations for heat transfer under steady-state conditions (for x and y directions). Apply the finite difference method to find the algebraic representation of the governing equations for each node. Pick a temperature for each side of the sample (last integer ending with the last digit of your student ID) within the given range and calculate the temperatures at the internal nodes. Use two different methods for solving simultaneous eqiations in MATLAB.

Engineering Mathematics 2

Introduction
The main aims of this mathematical analysis and computer modelling tasks are twofold; firstly, to get a deeper insight and increase your understanding of the mathematical theory and the various methods used in engineering analysis. Secondly, to gain experience in computer modelling techniques by using an industry–leading mathematical simulation package such as MATLAB The coursework consists of Tasks 1 and 2. The first one involves the application of ODEs in chemical reaction engineering. The second one deals with the use of numerical methods to solve PDEs obtained for 2d conductive heat transfer. Use the template provided on Moodle to prepare your submission.

TASK 1: Application of ODEs in chemical reaction engineering (50%)

Reactions in series are commonly occurring in chemical engineering. This task is an open–ended assignment regarding the application of advanced mathematics in analyzing the reaction kinetics in a series of reactions. To keep the flow of the solution at a consistent level, use the below set of reactions in a batch reactor and find the changes of concentration over time for all the species via the general mole balances approach. Solve the resulting ODEs using three methods stated below and compare the results via MATLAB:
𝐴 𝑘1
→ 𝐵 𝑘2
→ 𝐶
a) The analytical solution using the exponential functions
b) Laplace transformation pathway with presumed initial conditions
c) Numerical methods and approximation of derivations
d) Using parameter tuning in your programming, obtain various trends under which B concentration peaks and degrades. What parameters influence this behaviour and how?

TASK 2: Applications of PDEs in 2d conductive heat transfer (50%)
For a two–dimensional heat transfer in the figure below, find the temperature at the four internal nodes while addressing the questions:
a) Derive the governing equations for heat transfer under steady–state conditions (for x and y directions).
b) Apply the finite difference method to find the algebraic representation of the governing equations for each node.
c) Pick a temperature for each side of the sample (last integer ending with the last digit of your student ID) within the given range and calculate the temperatures at the internal nodes. Use two different methods for solving simultaneous eqiations in MATLAB.
d) Create 9 internal nodes (instead of 4) and find the temperature distributions in that case. Research the suitable approaches to build the resulting equation system and solve it using