ChemicalProcess ControlmediumCourseworkPE / Advanced

Settling Time of a First-Order Process

A first-order process with time constant $\tau = 5.0\ \text{min}$ and no dead time is subjected to a step change in input.

Find the time required for the output to reach $90\%$ of its new steady-state value.

Given

τ	5 min	Process time constant
target	0.9 -	Fraction of final value

Hint 1

A first-order step response approaches its final value exponentially.

Hint 2

Set the response equal to 0.90 of final and solve for t.

Hint 3

e^(−t/τ) = 0.10, so t = τ·ln(10) ≈ 2.3τ.

Worked solution — try the problem first

The step response of a first-order process:

$y(t) = y_\infty\left(1 - e^{-t/\tau}\right)$

Set $y/y_\infty = 0.90$ :

$0.90 = 1 - e^{-t/\tau} \quad\Longrightarrow\quad e^{-t/\tau} = 0.10$

$-\frac{t}{\tau} = \ln(0.10) = -2.3026$

$t = 2.3026\tau = 2.3026(5.0) = 11.5\ \text{min}$

$\boxed{t = 11.5\ \text{min}}$

Useful landmarks for any first-order system:

Fraction of final value	Time
63.2%	$1\tau$
86.5%	$2\tau$
95.0%	$3\tau$
98.2%	$4\tau$
99.3%	$5\tau$

90% falls between $2\tau$ and $3\tau$ , so 11.5 min (2.3τ) is consistent.

The same mathematics governs RC and RL electrical transients, thermometer lag, and well-mixed tank dynamics — one exponential, three applications.

Concepts:First-order systemsProcess dynamicsTime constantStep response

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