RELAXATION OF THE AIR IN TURBINE
Dr Mohammed Saleh AlAnsari
Bahrain Centre for Studies and Research
April 2010
The paper presents the practical case of the
irreversible adiabatic pressure drop, when the
presence of friction leads to a mechanical deficit of
energy, and that the heat of friction resulted from
this energy produces the irreversible increase in the
entropy of air. This study contains also the energy
and exergy assessment, as calculates it sizes
specific to the process of the expansion.
1.
THE ADIABATE IRREVERSIBLE
(NOT ISENTROPIQUE)
2.
I=m·cpT [J]
p1
1
nd
γ
p2
Qfd·mcp
0
ΔSirev.d
S [J/K]
Fig. 1. The adiabatic (reversible and
irreversible) of air.
n d 1

Consider a mass flow ( m ) to look perfect
which expands adiabatically in a turbine.
The gas is considered stabilized if any section of
the stream, its thermal parameters (p, v, T) and
speed (w) remain unchanged during the observation
period.
In Figure 1 are represented:
• 1-2s - the reversible adiabatic (S = ct.)
• 1-2 - the irreversible adiabatic expansion (with
friction);
On note:
d 
p2
p1
- degree of gas
p 
T2 s  T1  2 
 p1 
 1


 1

 T1 d
 p 2  nd

T 2  T 1 

 p1 
n 
dT
n 1
(3)
(2)
n d 1
nd
 T1   d
 T 2 s (4)
The frictional heat accumulated by the gas is:
Q fd  L fd  mcv
nd  
T2  T1   0
nd  1
(5)
The irreversible increase of entropy gas is:
S irev.d 
T2

T1
expansion.
The study has been realized to a quantity
m
of
ideal
gas.
In the case of the reversible adiabatic expansion (12s) we can calculate the mechanical work received
technical and final temperature:
L12 s  L sd  mc p T1  T 2 s  0
(1)

The adiabatic and friction can be likened to a
transformation of nature polytropic named
"pseudopolytrope. The difference between a 1-2
and a polytropic "pseudopolytrope" is that, while
relations between the thermal parameters of state
(p, v, T) follow the rules of the polytrope, the
energy exchange is different . For a polytropic
process, the elementary exchange of heat is:
where: n -adiabatic exponent;
γ -adiabatic exponent;
cv [J/kg·K]-specific heat at constant volume.
For relaxation 1-2, dTd<0,and because-as δQfd>0,
and that the gas receives the heat, it follows that the
exponent of the adiabatic and friction will be 1-2
nd<γ, that is to say nd(1,γ). The temperature at the
end of the trigger will be:
2
2s
This frictional heat will determine the increase in
irreversible
entropy
gas.
It will be shown later that the trigger 1-2 is a nonisentropic adiabatic change.
This frictional heat will determine the increase
of
irreversible
entropy
gas.
It will be shown later that the trigger 1-2 is a nonisentropic adiabatic transformation.
Qn  m  c n dT  m  c v
Lsd
Lrd
For the adiabatic irreversible (1-2), we lose by
frottment mechanical energy is transformed into
heat of friction, Qfd,, which will be fully received
by the gas if the turbine is completely isolated from
the thermal point of view, that is to say that the
enclosure is adiabatic.
Q fd
T
 mcv
n d   T2
ln
 0 (6)
n d  1 T1
The friction heat, Qfd, results from a loss of
mechanical work polytropic technique, so that the
mechanical work real trigger is:
 
Lt12  Lrd  Lt12
 Q fd  n d
pol
 1
nd  1
 Q fd  n d Lext12 
For any reversible polytropic expansion (1-2),
with the introduction of heat through the walls, the
mechanical work is technical:
m  c v T1  T2  
n 
mc T  T 
T2  T1   v 1 2  (7)
 mcv d
nd  1
nd  1
 n d (  1)  n d     mcv  T1  T2  
 mc p T1  T2   I 1  I 2  m(i1  i 2 )  0
We see that this mechanical work, Lrd, is adiabatic;
if in the first principle of thermodynamics (QLt=ΔI) we put Q=0, we obtain:
L t   I  mc p  T1  T 2   I 1  I 2
(8)
We define an internal efficiency of the trigger, or
yield adiabatic by the formula:
 d   ad
2
( L t 12 ) pol  E 1  E 2 

T  T0
1
T
Q d 
 E 1  E 2  Q 12  T 0 S 12
(12)

for pseudopolytrope 1-2 (adiabatic with
friction), where Q12=Qfd, mechanical technique will
work:
L rd  L t 12  ( L t 12 ) pol  Q fd 
 E 1  E 2  T 0 S irrev . d

(13)
where T 0 S irrev . d
thermodynamic system.
2
L
I  I2
T  T2
(9)
 rd  1
 1
Lsd I 1  I 2 s T1  T2 s

1
anergy
represents
(8)
the
T  T0
Q d
T
Figure no.2 shows the graph of energy balance for
adiabatic irreversible.
 L t  pol
12
L t pol  L pol
Q fd
Q fd
E1
12
E2
a).
L t12  L rd
T0ΔSirev.d
I1
Lrd
I2
E1
E2
Fig. 2. The energy balance for the adiabatic
irreversible
2. EXERGY ANALYSIS
For an open thermodynamic system, the
exergy is:
E  I  I 0  T0 ( S  S 0 )
Fig. 3. Exergy analysis in the case of the
polytropic expansion (a) and in the case of
pseudopolytrope.
3. LABORATORY STAND
(10)
where T0 – the temperature of the atmosphere.
In the case of gas expansion, mechanical work
is technical:

• for a reversible adiabatic-2s, the mechanical
work will be technical ( S 12  0 )
L sd  E 1  E 2 s  I 1  I 2 s 
 mc p ( T1  T 2 s )
b).
(11)
For the practical study was made a stand (Fig. 4)
which contains a compressed air reservoir R, a
centripetal radial turbine T and a breakdown of
adjusting the pressure at the inlet of the turbine.
We measured pressures and temperatures at the
inlet and outlet of the turbine. For each value of the
pressure p1 is the inlet temperature in a turbine is
constant:
We measured pressures and temperatures at the
inlet and outlet of the turbine. For each pressure
value p1 is the inlet temperature in a turbine is
constant T1=293 K (t1=200C);; cella was possible
because the evidence were made at long intervals
of time and therefore compressed air tank has had
time to cool to ambient laboratory (200C). For air
were considered:

specific heats: cp=1004,5 J/kgK; cv=717,5
J/kgK;

the adiabatic exponent: =1,4 ;

the pressure at the outlet of the turbine: p2=1
bar.
Air Compressor
T1
M
obtained by measurements and
2
245,4
241,3
0,9206
1,34
6,026
47,81
22,44
4
205
197,36
0,9201
1,34
11,14
88,39
45,2
6
185
175,83
0,921
1,34
13,67
108,49
58,21
8
172
161,75
0,917
1,34
15,32
121,54
67,39
p1
1
140
T
Qfd[KJ/KG]
m
2
T2
p2
300
T2[K]
5
24
120
250
5
22
100
IT12[KJ/KG]
lt12
sirev.d[KJ/KG]
VR
R
Table 1. Results
calculated.
p1
bar
T2
K
T2s
K
ηad
ηd
qfd
kJ/kg
lt12=lrd kJ/kg
Δsirev.d kJ/kg·K
5
20
5
19
5
18
8
17
80
2
17
200
150
60
100
40
50
20
0
0
2
3
4
5
6
7
8
p1[bar]
M
Atmosphere
Fig. 4. Laboratory Stand: R-compressed air tank;Tturbine; VR- breakdown of control; T1, T2thermistor thermometers; M- manometres.
4. LES RESULTATS OBTENUS
The rolling process of the air in RVs and relaxation
in the turbine T are shown in the diagram i-s (Fig.
5) VR calculation is done for the amount of air m =
1 kg.
i=cp·T
p1=6 bar p1=4 bar
p2=1 bar
1'
1
nd
lsd
γ
lrd
2
2s
2s
0
sirev.d[KJ/KG]
IT12[KJ/KG]
T2[K]
Fig. 6. The variations l t12 , q fd et s irrev . d No
function of pressure at the entry into turbine (p1).
5. CONCLUSIONS
Because the process of friction during the real
adiabatic (1-2), on receives less energy than in the
case of the reversible adiabatic (1-2s).
receives less energy than in the case of the
reversible adiabatic:
L d  L12 s  L t 12  mc p ( T1  T 2 s ) 
(14)
 mc p ( T1  T 2 )  mc p ( T 2  T 2 s )
and is equal to the heat (module):
p1=8 bar
i1=cp·T1
Qfd[KJ/KG]
2
s [J/kg·K]
Q 2 2 s  mc p ( T 2 s  T 2 )
(15)
required for the return of the thermodynamic
system of the state 2 to state 2s by cooling down at
constant pressure (p2).
For various values of p1 we obtain almost constant
values for the yield and the adiabatic exponent of
the adiabatic irreversible (nd).
The value of the adiabatic efficiency is
relatively small because the turbine was conducted
in the laboratory by non-performing technical
procedures
Increasing pressure p1 is proportional to the degree
of irreversibility of the process of relaxation, that is
to say qfd and sirrev.d increase both
Fig. 5. The diagram i-s; 1-1 ': Rolling VR.
The results are presented in Table 1 is in Figure 6.
SYMBOLES
cv [J/kg·K]- the specific heat at constant volume
E[W]- exergy of a thermodynamic system openadiabatic exponent of n
- Q fd [J]- frictional heat accumulated by the gas
- Qn [J]- Basic heat exchange
-p1[Pa]- departure of the pressure relaxation
-p2[Pa]- the pressure at the end of the trigger [Pa]
- S irev.d [J/kgK - irreversible increase of entropy
of gas]
-T1[K]- the temperature departure from the
relaxation
-T2[K]- the temperature at the end of the trigger
-  d - degree of gas expansion
-  d - internal efficiency of relaxation or adiabatic
efficiency
-γ- adiabatic exponent
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Notes.