Question

A perturbation in the temperature of a stream leaving a chemical reactor follows a decaying sinusoidal variation, according to the following mathematical equation,

T(t)=5e^(-at ).sin⁡(wt)
where a and w are constant.
Then show that the temperature T (t) is maximum at time “t”. Also verify that t= 1/w.〖tan〗^(-1) (w/a)
Determine the maximum temperature value at t= 1/w.〖tan〗^(-1) (w/a).

Answer 1

Derivating the function of the temperature and equating to 0, we can find the critical points:


`T'(t)=-a\cdot5e^(-at)\cdot\sin(wt)+w\cos(wt)\cdot5e^(-at)\\\\ 0=5e^(-at)(-a\sin(wt)+w\cos(wt))`

As
`5e^(-at)\\eq0`:


`0=-a\sin(wt)+w\cos(wt)\iff a\sin(wt)=w\cos(wt)\iff \\\\\sin(wt)=(w)/(a)\cos(wt)\iff \tan(wt)=(w)/(a)\iff wt=\tan^(-1)\left((w)/(a)\right)\iff \\\\\boxed{t=(1)/(w)\tan^(-1)\left((w)/(a)\right)}`

Replacing:


`T(t)=5e^(-at)\cdot\sin(wt)=5e^{-(a)/(w)\tan^(-1)\left((w)/(a)\right)}\cdot\sin(\tan^(-1)\left((w)/(a)\right))`

We can reach:
`\sin(\tan^(-1)\left((w)/(a)\right))=(w)/(√(w^2+a^2))`

Hence:


`T(t)=(5w)/(√(w^2+a^2))e^{-(a)/(w)\tan^(-1)\left((w)/(a)\right)}`