Filling a Tank That Leaks

Pour water into a tank with a hole in the bottom. The faster it leaks the higher it gets (Torricelli: outflow grows like √h), so the level creeps toward a balance point. The time to reach a level is an integral with a square root in it, and substituting u = √h turns it into a fraction you can integrate.

How it works

Integrals involving roots

In one sentence

When an integrand contains $\sqrt[n]{ax + b}$ , substitute $u = \sqrt[n]{ax + b}$ ; then $x$ and $dx$ are polynomials in $u$ and the integral becomes rational, ready for long division or partial fractions.

General formula

{\htmlData{sym=0}{t(h)}} = {\htmlData{sym=1}{A}}\int_0^h \frac{dh}{{\htmlData{sym=2}{Q}} - {\htmlData{sym=3}{k}}\sqrt{h}} \;\overset{{\htmlData{sym=4}{u}}=\sqrt{h}}{=}\; {\htmlData{sym=1}{A}}\int_0^{\sqrt h}\frac{2{\htmlData{sym=4}{u}}\,d{\htmlData{sym=4}{u}}}{{\htmlData{sym=2}{Q}} - {\htmlData{sym=3}{k}}{\htmlData{sym=4}{u}}} = {\htmlData{sym=1}{A}}\left[-\frac{2\sqrt h}{\htmlData{sym=3}{{k}}} - \frac{2{\htmlData{sym=2}{Q}}}{{\htmlData{sym=3}{k}}^2}\ln\!\left(1 - \frac{{\htmlData{sym=3}{k}}\sqrt h}{\htmlData{sym=2}{{Q}}}\right)\right]

The root is the awkward part, so make it the variable; then everything is a polynomial or a ratio of polynomials.

What the symbols mean

$t(h)$
Time to fill
How long until the water reaches level h.
$A$
Tank area
Floor area of the tank: more area, slower rise.
$Q$
Inflow
Volume of water poured in per second.
$k$
Leak constant
Hole area times √(2g): how fast it leaks at a given depth.
$u$
Substitution
u = √h, which turns the root into a polynomial.

Water flows in at rate $Q$ and leaks out at $k\sqrt{h}$ , so the level rises at $\frac{dh}{dt} = \frac{Q - k\sqrt h}{A}$ .
Flip it: time per unit of height is $\frac{A}{Q - k\sqrt h}$ , and the time to reach level $h$ is its integral.
The root is the awkward part, so make it the variable: $u = \sqrt h$ , $h = u^2$ , $dh = 2u\,du$ . Now it's a ratio of polynomials.
Divide, integrate, and the log term blows up as $h \to (Q/k)^2$ : the tank approaches its balance level but never quite reaches it.

Learn the math

Integrals Involving RootsCalc II · Lesson 5, Integration Techniques Integration StrategyCalc II · Lesson 7, Integration Techniques

Filling a Tank That Leaks