ENGR201 2025 Q2(c) (ENGR5001)

ENGR201 2025 Q2(c)Past paperOld spec ENGR2012:228 marks25 min

ENGR201 Engineering Analysis 2024-25 Q2(c)[PARTIAL]

The differential equation below represents a temperature regulation system in a heat exchanger. $u(t)$ is a step (Heaviside) function denoting the power supplied to the heating element, and $y(t)$ is the temperature output, with initial conditions $y(0) = 0$ and $\dfrac{dy(0)}{dt} = 0$ .

$\frac{d^2y(t)}{dt^2} + 5\frac{dy(t)}{dt} + 6y(t) = u(t)$

(Parts (a) and (b) of the original question, on parametric curves and gradients, are not on the ENGR5001 syllabus.)

Formulas you may need

$\mathcal{L}[y'] = sY - y(0)$ , $\mathcal{L}[y''] = s^2Y - sy(0) - y'(0)$ ; unit step $\to 1/s$ (learn this)
$e^{-\alpha t} \leftrightarrow \dfrac{1}{s + \alpha}$ (learn this)
Final value theorem $\lim_{t \to \infty} y(t) = \lim_{s \to 0} sY(s)$ , or steady state gain = TF at $s = 0$ (learn this)

(i)
Use the Laplace transform to find the temperature $Y(s)$ in the s-domain.
[13]Third
(ii)
Use the inverse Laplace transform to find the temperature $y(t)$ .
y at t = 1 =-
[12]
(iii)
What is the stabilised temperature of the heat exchanger?
y(infinity) =-
[3]Third