thermodynamics

1. Thermodynamics Summary
&
Polytropic Efficiency

2. Thermodynamics Summary
-In this section a review of fundamentals of thermodynamics is reported.
-The following concepts are necessary to understand the course contents.

3. Definitions:
We allow two important interactions between
the system and the surroundings:
or:

4. Definitions:

5. Properties and state of a substance

6. Properties and state of a substance

7. Properties and state of a substance

8. Thermodynamic processes

9. First Law of Themodynamics
2) dQ-dLshaft=dh
1) dQ-dLtd=dU
Q +
L +

15. 1
1
2
2
L term
L shaft
L term= Lshaft+Lp L term= pdv L p= d(pv)
L shaft= L term -L p= pdv- pdv-vdp=-vdp
L e= Lshaft=Wshaft

16. Shaft Work for isothermal process
(Compression)

17. Shaft Work for isentropic process
(Compression)
s:isentropic

18. dQ-dLshaft=dh
dQ-dLtherm=dU

19. Shaft Work for real process
his=(h2s-h1)/(h2-h1)= (T2s-T1)/(T2-T1)
Isentropic Efficiency

26. Polytropic Efficiency

27. Polytropic Efficiency:
A more appropriate way to compare
the performance of different
compressors and expanders.

28. Two Stage Compressor: real process
T2’s
T1
T2
T2s
T3s
T3

29. 1° Stage
T1=298.15 K b=5
T2 s =472.21 K
his =(T2s-T1)/(T2-T1)=0.9 T2=491.55 K
2° Stage
T2 =491.55 K b=4
T3s=730.45 K
his =(T2s-T1)/(T2-T1)=0.9 T3=756.99 K
T2’s
T1
T2
T2s
T3s
T3

30. But calculating the Compressor Isentropic Efficiency as a whole:
T1=298.15 K T3= 756,99 K b=20 T2’s=701,71 K
his =(T2s-T1)/(T2-T1)=0.879
whereas:
his1 Stage = his2 Stage =0.9
T2’s
T1
T2
T2s
T3s
T3

31. The higher the compressor ratio is, the larger the difference between the
actual and isentropic temperature at the outlet of the compressor is .
So, compressors with high compression ratio are «doomed» to have a low
isentropic efficiency.
Is Isentropic efficiency a proper way to define turbomachinery perfomance?
Why?

32. Real work:
T0=T1 so h0=h1 and h2-h1=h2-h0
The area A02C represents
Tds=Q = cp T2- cpT0 (the pressure is
constant on the process 0-2)
so Tds=Q= cpT2-cpT0 =h2-h0=h2-h1=Lr
Lr=h2-h1= Area (A02C)

33. We can approximate the real process with a
reversible polytropic process y: the process is not
adiabatic but the heat exchange is reversible
In the real adiabatic process the heat release is due
to irreversibilities.
r
s: isentropic
y: polytropic
r: real, actual
Isentropic vs Polytropic Process

34. = AreaB2s2C
= AreaB12C
l irr =Area B12C but Area B12C < AreaB2s2C
Real Work
Isentropic Work
What is the difference?
r
Lw=l irr

35. = AreaB2s2C
= AreaB12C
l irr =Area B12C but Area B12C < AreaB2s2C
Real Work
Isentropic Work
What is the difference?
r
Lw=l irr

36. Area 12S2 Work due to the increased specific volume
(compared to adiabatic process)
The specific volume is higher than expected for an adiabatic process
due to irreversibilities

37. Polytropic Efficiency:
Polytropic Efficiency does not consider the work increase due to higher
specific volume.
Polytropic Efficiency only takes into account the irreversibilities
Isentropic Efficiency takes into account both specific volume
increase and irreversibilities

38. R=cp-cv
k=cp/cv
Ly
Lr
Lr

39. = f(b)

41. n=1.45
v1=0.8445 m3/kg
v2=0.2784m3/kg
The actual discharge temperature based
on an isentropic path can be estimated by:
Similarly, the actual discharge temperature
based on a polytropic path can be
estimated by:
Or knowing the actual discharge
temperature n can be calculated as:
T2’
s
T1
T2
T2s
T3s
T3

42. From the previous example
= 0.92
n=1.45
= 0.9

43. Polytropic Efficiency: Turbine

44. Actual Work=Area C0’1B
Isentropic work=Area A01B

46. Isentropic work-Actual Work=Area A00’C≈Area B2s2D
l irr=Area B12D>Area B2s2D
Area 12S2 Work due to increased specific volume
‘Recovered Work’

47. Polytropic Efficiency: Turbine
= f(e)

49. l i : Actual shaft work
l w : work related to irreversibilities
Turbine

50. l i : Actual shaft work
l w : work related to irreversibilities
l is : Isentropic shaft work
l i= lis
l is = RT1