SME
Hydrology · Problem type
Residence time of a lake
Use this for “A lake holds 2.4 × 10⁹ m³ and its outlet carries 30 m³/s. How long does water stay in it?”
Diagram
Each slice of the bar is one year’s use, 9.4673 × 10⁸ m³.
2.4 ×
10
9
9.4673
×
10
8
=
2.54:
the
reserve
lasts
2.54
years
if
use
stays
the
same.
This is an example. Type your own values to start a new problem.
τₛ (s)
=
V (m³)
÷
Q (m³/s)
Qₐ
Yearly outflow
calculated · m³/yr
τ
Residence time
calculated · yr
Formulas
τₛ
=
V
Q
8 ×
10
7
=
2.4 ×
10
9
30
Qₐ
=
Q
×
31,557,600
9.4673 ×
10
8
=
30
×
31,557,600
τ
=
V
Qₐ
2.535
=
2.4 ×
10
9
9.4673 ×
10
8
Assumptions
•
Steady state: inflow equals outflow, so the lake’s volume stays the same.
•
τ is the average stay of the water; a well-mixed lake renews itself in about that time.
•
A year is 365.25 days, or 31,557,600 s.
Step-by-step
WE KNOW
V = 2.4 × 10⁹ m³, Q = 30 m³/s
FIND
time in seconds (τₛ), yearly outflow (Qₐ), residence time (τ)
Step 1 · Find time in seconds (τₛ)
Use τₛ = V ÷ Q
The volume divided by the flow out each second: the seconds it takes to drain once.
τₛ
=
2.4 ×
10
9
30
τₛ
=
8 ×
10
7
s
Step 2 · Find yearly outflow (Qₐ)
Use Qₐ = Q × 31,557,600
The flow each second times the seconds in a year.
Qₐ = 30 × 31,557,600
Qₐ
=
9.4673 ×
10
8
m
3
/yr
Step 3 · Find residence time (τ)
Use τ = V ÷ Qₐ
How many years’ outflow the lake holds.
τ
=
2.4 ×
10
9
9.4673 ×
10
8
τ
=
2.535
yr
Check
Put the numbers back in. Both sides should match.
8 ×
10
7
=
2.4 ×
10
9
30
✓
9.4673 ×
10
8
=
30
×
31,557,600
✓
2.535
=
2.4 ×
10
9
9.4673 ×
10
8
✓
Topic check
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