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Tank residence time and rinsing

Residence time of a tank (τ = V/Q), rinsing time for a target reduction, or flow for a given τ.

Parameters

Volume in litres, flow in litres per minute.

Result

Rinsing time

46.052 min

Calculation
t = τ × ln(1 / 0.01) = 4.6052 τ
Residence time
10 min
Turnovers per hour
6
Tank volumes to sweep
4.6052
01234560 %25 %50 %75 %100 %volumes swept (τ)old content (%)4.61 τ · 46.1 min
Notes (2)

The axis counts swept volumes; the dimension line converts them into minutes at this flow.

Reducing the old content by 99% takes 4.6052 tank volumes, that is 46.052 min at this flow.

Calculation, assumptions and sources

OverviewResidence time τ = V/Q of a tank swept by a flow, turnovers per hour, time to reach a target reduction, or flow needed to hold a residence time.

Residence time τ = V/Q of a tank swept by a flow, turnovers per hour, time to reach a target reduction, or flow needed to hold a residence time. In a stirred tank, rinsing is exponential: one tank volume removes 63% of the old content, three remove 95%, five 99.3%.

Method & formulasτ = V / Q
τ = V / QC(t) / C₀ = e^(−t/τ)t = τ × ln(C₀ / C)

τ is the time the flow takes to fill the volume. In a stirred tank the content continuously dilutes into what enters, hence the exponential decay; the equation is that of a discharging capacitor.

Residence time (τ)
the average time a molecule spends in the tank, volume divided by flow. Some leave at once, others linger.
Stirred tank (CSTR)
an ideal reactor whose content is homogeneous at every instant; what leaves has its composition.
Plug flow
an ideal with no mixing, where the new pushes the old ahead of it.
Turnover
one tank volume passed through, the usual unit of rinsing, pools and ventilation.
Validity domainThe model assumes a perfectly stirred tank and a constant flow.

The model assumes a perfectly stirred tank and a constant flow. Plug flow (a long pipe) rinses in a single volume; a tank with dead zones or short-circuits rinses more slowly than the exponential. A dye tracer measures the gap through the residence time distribution.

Rinsing a stirred tankFrom the first second, what leaves is a blend of old and new content, and the share of clean water thrown out grows as rinsing goes on.

From the first second, what leaves is a blend of old and new content, and the share of clean water thrown out grows as rinsing goes on. Each volume removes 63% of what remains. The workshop rule "three volumes to rinse" corresponds to ln(20): three τ leave 5%.

Reading the resultTurnovers per hour equal 60/τ.

Turnovers per hour equal 60/τ. Diluting the old product a thousandfold takes ln(1000) τ, that is 6.9 tank volumes.