SME
Fluid Mechanics · Problem type
A filling tank: how fast the level moves
Use this for “Water flows into a 2 m² tank at 0.02 m³/s and out at 0.012 m³/s. How fast does the level rise?”
Table
Flow out (Qₒᵤₜ)
(m³/s)
Level rate (dh/dt)
(mm/s)
This is an example. Type your own values to start a new problem.
A (m²)
×
dh/dt (mm/s)
=
Qᵢₙ (m³/s)
−
Qₒᵤₜ (m³/s)
Formulas
A
×
(dh/dt)
=
Qᵢₙ
−
Qₒᵤₜ
2 × 0.004 = 0.02 − 0.012
Formulas are worked in m/s. The step-by-step shows the conversions.
Assumptions
•
The liquid is incompressible, so the volume stored changes by Qᵢₙ − Qₒᵤₜ each second.
•
The tank has straight sides: its plan area A is the same at every level.
•
dh/dt above 0 means the level rises; below 0, it falls.
Step-by-step
WE KNOW
Qᵢₙ = 0.02 m³/s, Qₒᵤₜ = 0.012 m³/s, A = 2 m²
FIND
level rate (dh/dt)
Step 1 · Find level rate (dh/dt)
Use A × (dh/dt) = Qᵢₙ − Qₒᵤₜ
Mass
balance
on
the
tank:
the
volume
gained
each
second,
Qᵢₙ
−
Qₒᵤₜ,
spread
over
the
area
A
.
dh/dt
=
Qᵢₙ − Qₒᵤₜ
A
dh/dt
=
0.02 − 0.012
2
dh/dt
=
0.008
2
dh/dt = 0.004 m/s
Step 2 · Convert the answers
Change each answer back to the unit you chose.
dh/dt = 0.004 m/s = +4 mm/s (1 m/s = 1000 mm/s)
Check
Put the numbers back in. Both sides should match.
2 × 0.004 = 0.02 − 0.012 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.