Mathematics Question

Overview

Task 1

Two complex numbers, expressed by Eq. (1) and Eq. (2), represent the time-dependent signals measured by two temperature sensors installed in a combustion chamber.

𝑍1=𝐴𝑒𝑖𝜔𝑡

Eq. (1)

𝑍2=𝐵𝑒𝑖(𝜔𝑡+𝜑)

Eq. (2)

Where 𝐴 & 𝐵 denote the amplitudes of the temperature oscillations (in K), 𝜔 is the angular frequency of the oscillations (in rad/s), 𝑡 is time (in s). 𝜑 is the phase difference (in rad) between the two sensors.

For the analysis, parameter(s) can be assumed based on reasonable considerations. Compute the following (i to iv) and plot the results on the Argand diagram.

i. 𝑍1+𝑍2 ii. 𝑍2−𝑍1 iii. 𝑍1×𝑍2 iv. 𝑍2𝑍1

v.

Apart from the described scenario, explain the applications of complex numbers in fire engineering. Critically analyse a specific example and provide its mathematical representation.

Task 2

A dataset was obtained from the experimental work, as shown in Table 1 of Appendix A1.

i.

Determine the equation that accurately represents the data presented in the Table. 1 of Appendix A1. and provide a detailed explanation of the equation.

ii.

Using the graph, examine the function values immediately before and after the turning points. Discuss your observations.

iii.

Analyse the nature of the function at the turning points indicated in Eq. (3) using the derivative test. You can reasonably assume coefficients like 𝐶, 𝐷 and 𝐸.

𝑦=𝐶𝑥2+𝐷𝑥+𝐸

Eq. (3)

iv.

Apart from the discussed example, provide an example from fire engineering where the concept of maxima and minima can be applied, and support your explanation with a mathematical example.

Task 3

The temperature (𝑇) (in Kelvin) of a fire chamber varies with time (𝑡) (in seconds) and is modelled by Eq. (4),

𝑇(𝑡)=𝑋𝑡5+𝑌𝑡2+𝑍𝑡

Eq. (4)

Where X, Y, and Z are coefficients. You can assume the essential parameter(s) /Coefficient(s).

i.

Plot the graph of temperature (𝑇)versus time (𝑡) provide a critical discussion of the results.

ii.

Determine the instantaneous rate of temperature change at three different time (𝑡) points. You may reasonably assume the necessary parameters and discuss your results with supporting arguments

iii.

Calculate the temperature increase over a 24-hour period and discuss your findings.

iv.

Describe other scenarios in fire engineering, beyond the given example, where the concepts of accumulation and rate of change can be applied