SME
Compressible Flow
Isentropic flow
Use this for “Air flows at M = 2 with T = 220 K and p = 20 kPa. Find T₀, p₀ and ρ₀/ρ.”
Diagram
T₀
T
=
1
+
0.2
×
2
2
=
1.8;
p₀
p
=
7.824.
Drawn
to
scale:
A
A
∗
=
1.688,
so
the
duct
has
passed
a
throat
and
widens
again.
This is an example. Type your own values to start a new problem.
T₀ (K)
T (K)
=
1
+
0.2
×
M
2
p₀ (kPa)
p (kPa)
=
(1
+
0.2
×
M
2
)
3.5
T₀/T
Temperature ratio
calculated
p₀/p
Pressure ratio
calculated
ρ₀/ρ
Density ratio
calculated
Formulas
T₀
T
=
1
+
0.2
×
M
2
1.8
=
1
+
0.2
×
2
2
T₀
=
(
T₀
T
)
×
T
396 = 1.8 × 220
p₀
p
=
(
T₀
T
)
3.5
7.824
=
(1.8)
3.5
p₀
=
(
p₀
p
)
×
p
156.5 = 7.824 × 20
ρ₀/ρ
=
(
T₀
T
)
2.5
4.347
=
(1.8)
2.5
Assumptions
•
Adiabatic and frictionless (isentropic), so p₀ and T₀ stay constant along the flow.
•
Calorically perfect air, γ = 1.4: (γ − 1) ÷ 2 = 0.2, γ ÷ (γ − 1) = 3.5, 1 ÷ (γ − 1) = 2.5.
•
The stagnation state is the flow brought to rest without loss, as on the tip of a probe.
Step-by-step
WE KNOW
M = 2, T = 220 K, p = 20 kPa
FIND
temperature ratio (T₀/T), stagnation temperature (T₀), pressure ratio (p₀/p), stagnation pressure (p₀), density ratio (ρ₀/ρ)
Step 1 · Find temperature ratio (T₀/T)
Use T₀/T = 1 + 0.2 × M²
Bring
the
flow
to
rest:
its
kinetic
energy
heats
it
by
the
factor
1
+
0.2
M
².
T₀
T
=
1
+
0.2
×
2
2
T₀
T
=
1
+
0.2
×
4
T₀
T
=
1
+
0.8
T₀
T
=
1.8
Step 2 · Find stagnation temperature (T₀)
Use T₀ = (T₀/T) × T
Multiply the temperature by the ratio.
T₀
=
1.8
×
220
T₀
=
396
K
Step 3 · Find pressure ratio (p₀/p)
Use p₀/p = (T₀/T)3.5
Isentropic:
the
pressure
ratio
is
the
temperature
ratio
to
the
power
γ
γ − 1
.
p₀
p
=
(1.8)
3.5
p₀
p
=
7.824
Step 4 · Find stagnation pressure (p₀)
Use p₀ = (p₀/p) × p
Multiply the pressure by the ratio.
p₀
=
7.824
×
20
p₀
=
156.5
kPa
Step 5 · Find density ratio (ρ₀/ρ)
Use ρ₀/ρ = (T₀/T)2.5
The
density
ratio
is
the
temperature
ratio
to
the
power
1
γ − 1
.
ρ₀/ρ
=
(1.8)
2.5
ρ₀/ρ = 4.347
Check
Put the numbers back in. Both sides should match.
1.8
=
1
+
0.2
×
2
2
✓
396 = 1.8 × 220 ✓
7.824
=
(1.8)
3.5
✓
156.5 = 7.824 × 20 ✓
4.347
=
(1.8)
2.5
✓
Topic check
See what you know: 5 questions on this topic
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