SME
Engineering Thermodynamics · Problem type
Compressor with an isentropic efficiency
Use this for “An air compressor takes in air at 290 K and raises its pressure 6 times with an isentropic efficiency of 0.82. Find the exit temperature and the work per kilogram.”
Diagram
The ideal compression ends at 2s (T = 543.4 K), straight up; the real one (dashed) ends at 2, T = 586.4 K, hotter and to the right: friction makes entropy.
Work in per kilogram w = 287.8 kJ/kg
This is an example. Type your own values to start a new problem.
T₂s (K)
=
T₁ (K)
×
rₚ
0.4
÷
1.4
T₂ (K)
=
T₁ (K)
+
(
T₂s (K)
−
T₁ (K)
)
÷
ηc
w (kJ/kg)
=
1.005
×
(
T₂ (K)
−
T₁ (K)
)
Formulas
T₂s
=
T₁
×
rₚ
0.4 ÷ 1.4
543.4342
=
300
×
8
0.4 ÷ 1.4
T₂
=
T₁
+
T₂s −
T₁
ηc
586.3932
=
300
+
543.4342 − 300
0.85
w
=
1.005
×
(
T₂
−
T₁
)
287.8251 = 1.005 × (586.3932 − 300)
Assumptions
•
Air is an ideal gas with constant cₚ = 1.005 kJ/(kg·K) and k = 1.4.
•
Steady flow and adiabatic; kinetic and potential energy changes neglected.
•
The ideal compressor is isentropic (s₂s = s₁) and reaches the same exit pressure.
•
cₚ stays near 1.005 kJ/(kg·K) from about 250 K to 1,000 K.
Step-by-step
WE KNOW
T₁ = 300 K, rₚ = 8, ηc = 0.85
FIND
ideal exit temperature (T₂s), real exit temperature (T₂), work in per kilogram (w)
Step 1 · Find ideal exit temperature (T₂s)
Use T₂s = T₁ × rₚ0.4 ÷ 1.4
An
isentropic
compression:
raise
the
pressure
ratio
to
k − 1
k
,
then
multiply
by
T₁
.
T₂s
=
300
×
8
0.4 ÷ 1.4
T₂s
=
300
×
8
0.28571
T₂s = 300 × 1.8114
T₂s = 543.4342 K
Step 2 · Find real exit temperature (T₂)
Use T₂ = T₁ + (T₂s − T₁) ÷ ηc
The
real
compressor
needs
more
work:
divide
the
ideal
rise
by
ηc,
then
add
it
to
T₁
.
T₂
=
300
+
543.4342 − 300
0.85
T₂
=
300
+
243.4342
0.85
T₂
=
300
+
286.4
T₂
=
586.3932
K
Step 3 · Find work in per kilogram (w)
Use w = 1.005 × (T₂ − T₁)
The work in is cₚ times the real temperature rise.
w
=
1.005
×
(586.3932
−
300)
w
=
1.005
×
286.3932
w
=
287.8251
kJ/kg
Check
Put the numbers back in. Both sides should match.
543.4342
=
300
×
8
0.4 ÷ 1.4
✓
586.3932
=
300
+
543.4342 − 300
0.85
✓
287.8251 = 1.005 × (586.3932 − 300) ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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