SME
Fluid Mechanics
Control-volume analysis
Use this for “A 20 m/s water jet of 0.002 m² is turned 120° by a fixed vane. Find the force on the vane.”
Table
Turning angle (θ)
(°)
Force along the jet (Fₓ)
(N)
This is an example. Type your own values to start a new problem.
ṁ (kg/s)
=
ρ (kg/m³)
×
V (m/s)
×
A (m²)
Fₓ (N)
=
ṁ (kg/s)
×
V (m/s)
×
(1
−
cos(
θ
°))
Fᵧ
Force across the jet
calculated
R
Force on the vane
calculated
Formulas
ṁ
=
ρ
×
V
×
A
40 = 1,000 × 20 × 0.002
Fₓ
=
ṁ
×
V
×
(1
−
cos(
θ
))
1,200 = 40 × 20 × (1 − cos(120°))
Fᵧ
=
ṁ
×
V
×
sin(
θ
)
692.8 = 40 × 20 × sin(120°)
R
=
√
Fₓ² +
Fᵧ
2
1,386
=
√
1,
200
2
+
692.8
2
Assumptions
•
Steady flow; the vane is fixed and the jet keeps its speed across it (no friction).
•
Atmospheric pressure all round, so only momentum crosses the control volume.
•
Fₓ is along the incoming jet, Fᵧ across it; the jet’s weight is ignored.
Step-by-step
WE KNOW
V = 20 m/s, A = 0.002 m², ρ = 1,000 kg/m³, θ = 120°
FIND
mass flow rate (ṁ), force along the jet (Fₓ), force across the jet (Fᵧ), force on the vane (R)
Step 1 · Find mass flow rate (ṁ)
Use ṁ = ρ × V × A
The mass crossing the nozzle each second: density times speed times area.
ṁ
=
1,000
×
20
×
0.002
ṁ
=
20,000
×
0.002
ṁ
=
40
kg/s
Step 2 · Find force along the jet (Fₓ)
Use Fₓ = ṁ × V × (1 − cos(θ))
Momentum
in
along
x
is
ṁV
and
out
is
ṁV
cos
θ
;
the
vane
takes
the
difference.
Fₓ = 40 × 20 × (1 − cos(120°))
Fₓ = 40 × 20 × (1 − (−0.5))
Fₓ = 40 × 20 × 1.5
Fₓ = 800 × 1.5
Fₓ = 1,200 N
Step 3 · Find force across the jet (Fᵧ)
Use Fᵧ = ṁ × V × sin(θ)
The
jet
leaves
with
ṁV
sin
θ
across;
the
vane
is
pushed
the
other
way
by
as
much.
Fᵧ = 40 × 20 × sin(120°)
Fᵧ = 40 × 20 × 0.866
Fᵧ = 800 × 0.866
Fᵧ = 692.8 N
Step 4 · Find force on the vane (R)
Use R = √(Fₓ² + Fᵧ²)
The two components are square to each other: add their squares and take the root.
R
=
√
1,
200
2
+
692.8
2
R
=
√
1,440,000 + 479,971.84
R
=
√
1
,919,971.84
R
=
1,386
N
Check
Put the numbers back in. Both sides should match.
40 = 1,000 × 20 × 0.002 ✓
1,200 = 40 × 20 × (1 − cos(120°)) ✓
692.8 = 40 × 20 × sin(120°) ✓
1,386
=
√
1,
200
2
+
692.8
2
✓
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