Capstone Project in Mathematical Sciences III
Semester 2 2024
Applied Mathematics Project
Title: Heat and mass transport problems in industry
Transport of matter/heat energy due to molecular motion (diffusion/conduction) and with a flow (advection) arises in many industrial processes. This project will focus on such processes.
You will first consider the derivation of
(1) the diffusion/conduction equation and
(2) the advection-diffusion/conduction equation
to obtain an understanding of the physics and assumptions behind them, and their application to both mass and energy transport processes. For this you will make use of the text book (see above).
You will then concentrate on a particular class of energy or mass transport problems in the context of different industrial processes. These problems can be written in terms of one or two partial differential equations in one spatial dimension (x) and time (t). In a sufficiently long time the solution approaches a steady state which can be found by seeking a time-independent solution. This reduces the problem to one or two ordinary differential equations in the spatial variable (x). You should have considered the solution of these equations in Differential Equations II or an equivalent course.
Applications involving transport of heat energy include (1) ground-source heating, (2) solar heating, (3) lake-source cooling, (4) continuous pasteurisation (e.g. of milk), and (5) heat exchangers more generally.
Some mass transport problems include (6) blood oxygenation in a heart-lung machine, and (7) haemodialysis.
You will undertake a small project enabling you to learn in the context of one of these ap-plications, or something similar. You will derive, non-dimensionalise (scale), simplify (where possible), and solve the relevant equations to obtain solutions which you interpret for the appli-cation you are considering. You will present your work in a written project report.
Goals
1) To understand the derivation of mathematical models from physical conservation laws.
2) To understand the difference between diffusive, conductive and advective transport.
3) To appreciate the need to make assumptions in order to render an industrial problem tractable.
4) To apply mathematical techniques you have learned in the context of real-world applications.
5) To interpret results, with regard for the assumptions, made in the context of an application.
6) To gain experience in oral and written presentation of mathematics.
Assumed knowledge
You should have passed one of MATHS 2102 Differential Equations II, MATHS 2106 Differential Equations for Engineers, MATHS 2201 Engineering Mathematics IIA, or equivalent. Ability to write basic MATLAB programs would be beneficial.
Outline
You will, in general, meet each week with your supervisor for both tuition and to discuss and obtain help with your work. In some weeks, these meetings will be replaced by a meeting of all groups to work on the development of writing skills needed for the project work. In addition to the project report and poster, you will complete a number of written assignments, covering both aspects of writing skills and material relevant to the project.