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Fluid properties - Assume constant properties:
1.333:=... Ratio of specific heat
Cp 1147
J
kg K:= ... Specific heat of air standard at constant
pressure
kJ 1000J:=
kmol 1000 mol:=
... Gas constantR 287
J
kg K:=
P1 1.6bar 0.16 MPa=:=
r 18.5:=
ASSUMPTIONS
rc 2:= ... V3 per V2, cut off ratio
T1 300K 26.85 C=:= ... Temperature inlet of cylinder
11
r 1
rc
1
rc 1( )
0.569=:=
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STAGE 1:
P1 0.16 MPa=
T1 300 K =
u1 214.07kJ
kg:=
r1 621.2:=
1R T1
P1
0.538m
3
kg
=:=
11
11.858
kg
m3
=:=
STAGE 2:
r2r1
r33.578=:=
From table T-9
T2 900K r2 34.31
32.18 34.31920 900( ) K+ 906.87K=:=
h2 932.93kJ
kg
r2 34.31
32.18 34.31955.38 932.93( )
kJ
kg+ 940.641
kJ
kg=:=
P2 P1T2
T1 r 8.948MPa=:=
2R T2
P20.029
m3
kg=:=
21
234.379
kg
m3
=:=
STAGE 3:
T3 rc T2 1813.739373 K=:=
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CENTRIFUGAL COMPRESSOR - TURBOCHARGER
D 92mm:=
Stroke 93.8mm:=
N 4:=... Number of cylinders
V1_2 D
2
4 Stroke N 2494.183293 cm
3=:=
V2V1_2
r 1142.525 cm
3=:=
V1 V2 r 2636.708052 cm3
=:= 2494cm3
2.494L=
V1 V2 2494.183293 cm3
=
RPM 3000rpm:=
m_exhaust 4 V1 V2( )RPM
2 0.728
kg
s=:=
Given (request) data:
mf m_exhaust 0.728kg
s=:= ... The mass flow rate
PR13P1
1atm1.579=:= ... Pressure ratio of compresor
Svaneless 5mm:= ... Vaneless space (between impeller anddiffuser)
rdt 55mm:=... The diffuser throat radius
rdo 65mm:= ... The diffuser outlet radius
Ndif 15:= ... Number of channels of diffuser
Guess & suggestion data:
rperR 0.5:= ... (r/R) The ratio of hub to shroud diameter at theeye
t 0.82:= ... Overall efficiency
0.525:= ... Head coefficient, from Aungier Fig. 1-9
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1.04:= ... Power input factor
0.835:= ... The slip factor
m 0.95:= ... Mechanical efficiency of compressor
limitedness:
W3max 90m
s:= ... Maximum outlet velocities
a3max 11:= ... The maximum included angle of the vaneddiffuser passage
Umax 460m
s:= ... Maksimum tip speed
M1Wmax 0.8:= ... Suggested maximum Mach number at inletimpeller - W1 direction
Tmax 400K := ... Maximum static temperature
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T1To1
1 1( ) M1
2
2+
:= T1 280.673 K=
C1 M1 R T1:= C1 131.074m
s=
Ca1 C1:=
From the isentropic relationship at a point
1Po1
R To1
T1
To1
1
1
:= 1 1.132kg
m3
=
P1 Po1T1
To1
1
:= P1 91.203 kPa=
Therefore the eye tip diameter at inlet is:
R1tmf
1 k Ca1:= R1t 45.630499 mm=
D1t 2 R1t:= D1t 91.261 mm=
And the hub diameter at inlet is:
D1r D1t rperR:= D1r 45.63 mm=
Peripheral speed at the impeller eye tip (shroud) & 1t:
U1t D1tRPM
rev:= U1t 226.974699
m
s=
1t atan Ca1U1t :=
1t 30.006 =
Peripheral speed at the impeller eye root (hub) & 1r:
U1r D1rRPM
rev:= U1r 113.48735
m
s=
1r atanCa1
U1r
:= 1r 49.113096 =
1.b. The impeller outlet diameter
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From Eq (4-11) the stagnation temperature difference is
To3 To1To1
tPR13
1
1
+:= To3 330.632 K=
To3 To1 42.482 K=
From Eq (4-9)
U2To3 To1( ) Cp
:= U2 236.877
m
s=
And Tip flow coefficient (2) at exit of impeller
D2U2 rev
RPM:= D2 95.242 mm=
The new value of To3 & t,
U2 D2 RPM
rev:= U2 236.877
m
s=
To3U2
2
Cp
To1+:= To3 330.632 K=
tTo1 PR13
1
1
To3 To1
:= t 0.82=
Therefore the tip flow coefficient is:
r2D2
2:= r2 47.621175 mm=
2mf
o1 r22 U2
:= 2 0.352109=
The Number of Blades:
Z0.63
1 .83:= Z 11.642=
Z 12:=
10.63
Z:= 0.835= ... Will be used re-calculate
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1.c. Velocity at exit and the the losses in the impeller and diffuser are the same, The
axial depth of the impeller is:
First guess M2 = 0.8 and iteration. The overall loss is proportional to (1 - c) = (1 - 0.82). Half of
the overall loss is therefore 0.5(1 - 0.82) = 0.09 and therefore the effective efficiency of theimpeller in compressing from Po1 to Po2 is (1 - 0.09) ...
M2 1:=
c 1 0.5 1 t( ):= c 0.91=
From Eq (4-11) afer rearranging the subscripts, and To3 = To2:
To2 To3:= To2 330.632 K=
T2To2
1M2
2 R
2 Cp+
:= T2 283.373 K=
PR12 1 cTo2 To1( )
To1+
1
:= PR12 1.655=
P2 Po1T2
To2
1PR12
:= P2 90.456 kPa=
2P2
R T2:=
2 1.112kg
m3
=
Po2P2
T2
To2
1
:= Po2 94.009 kPa=
And TOTAL enthalphy rise (Hrev) at exit of impeller
To2s c To2 To1( ) To1+:= To2s 326.808 K=
H_adiabatic Cp To2s To1( ):= H_adiabatic 44341.039968m
2
s2
=
The head coefficient become ....
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1r 49.113096 =
At outlet section (2):
D2 95.242 mm=
b2 8.311479 mm=
2 atanU2 Cax2
Cr2
:= 2 8.443 =
W2Cr2
cos 2( ):= W2 266.099
m
s=
C2 329.257 ms
=
Cax2 197.808m
s=
Cr2 263.216m
s=
2 acosCr2
C2
:= 2 36.925 =
Z 12=
RPM 47500 rpm=
2 0.352=
0.835=
2. DIFFUSER
In the vaneless space between the impeller outlet and diffuser vanes the flow is that of a free
vortex which at any radius requires that Cx.r = constant.
At the diffuser vane leading edge the radius is (r2 + 10)mm = (47.62 + 10) mm = 57.62 mm
r2D2
2:= r2 47.621 mm=
r2v r2 Svaneless+:= r2v 52.621175 mm=
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Ar3i 2 r2v b2:= Ar3i 0.002748m2
=
Cr3imf
3i Ar3i:= Cr3i 302.407
m
s=
by ITERATION ...
Cr3i 302.407m
s:= ... change this value for iteration
C3i Cr3i2
Cx3i2
+:= C3i 424.543m
s=
T3i To2C3i
2
2 Cp:= T3i 252.063 K=
P3i Po1T3i
To2
1
PR12:= P3i 56.608 kPa=
3iP3i
R T3i:= 3i 0.783
kg
m3
=
Ar3i 2 r2v b2:= Ar3i 0.002748m2
=
Cr3imf
3i Ar3i:= Cr3i 338.5755428
m
s=
No further iterations are necessary. Thus at the inlet to the vanesCr,3,i = 246.2013186 m/s.
3i atanCx3i
Cr3i
:= 3i 41.350188 =
Moving to the radius at the diffuser throat, at the throat radius, 343 mm
Cx3tC2 r2
rdt:= Cx3t 285.084
m
s=
by ITERATION again to find propertes at diffuser throat section ...
Start with assuming Cr,3 = Cr,3,i
Cr3t Cr3i:= Cr3t( ) 338.576m
s=
Cr3t 1308.0100793m
s:= ... change this value for iteration
C3t Cx3t2
Cr3t2
+:= C3t 1.339 103
m
s=
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T3t To2C3t
2
2 Cp:= T3t 450.608 K=
P3t Po1T3t
To2
1 PR12:= P3t 579.136 5.464i+( ) kPa=
3tP3t
R T3t:= 3t 4.478 0.042i( )
kg
m3
=
Ar3t 2 rdt b2:= Ar3t 0.002872m2
=
Cr3tmf
3t Ar3t:= Cr3t 56.5986266 0.5339793i+( )
m
s=
It may be seen that there is no change in the new values sothe radial velocity at the diffuser
throat = 184.0268017 m/s
3t atanCx3t
Cr3t
:= 3t 78.770868 0.103249i( ) =
A3tAr3t Cr3t
C3t:= A3t 0.000121 0.000001i+( ) m
2=
As we have 15 diffuser vanes, the width of each throat is:
Throat_widthA3t
Ndif b2
:= Throat_width 0.974021 0.009189i+( ) mm=
2 r2v 105.242 mm=
rdt 2 110 mm=
Svaneless 5 mm=
Moving to the radius at the diffuser outler, at the outlet radius, 550 mm
Cx3C2 r2
rdo:= Cx3 241.225
m
s=
by ITERATION again to find propertes at diffuser outlet section . ..
Start with assuming Cr,3 = Cr,3,i
Cr3o Cr3t:= Cr3o 56.599 0.534i+( )m
s=
Cr3o 87.84762m
s:= ... change this value for iteration
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C3o Cx32
Cr3o2
+:= C3o 256.723m
s=
T3o To2 C3o2
2 Cp:= T3o 301.902 K=
P3o Po1T3o
To2
1
PR12:= P3o 116.559 kPa=
3oP3o
R T3o:= 3o 1.345
kg
m3
=
Ar3o 2
rdo
b2:=
Ar3o 0.003394m
2=
Cr3omf
3o Ar3o:= Cr3o 159.4398561
m
s=
It may be seen that there is no change in the new values sothe radial velocity at the diffuser
throat = 87.84762 m/s
3o atanCx3
Cr3o
:= 3o 56.536926 =
A3oAr3o Cr3o
C3o:= A3o 0.002108m
2=
As we have 15 diffuser vanes, the width of each throat is:
Throat_width_outletA3o
Ndif b2:= Throat_width_outlet 16.909629 mm=
Overall dimension of DIFFUSER:
At inlet:
r2v 52.621 mm=
3i 41.350188 =
At throat:
rdt 55 mm=
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3t 78.770868 0.103249i( ) =
Throat_width 0.974021 0.009189i+( ) mm=
At outlet:
rdo 65 mm=
3o 56.536926 =
Throat_width_outlet 16.909629 mm=
Ndif 15=
TURBINE - TURBOCHARGER
P4 0.474MPa=
T4 888.694 K=
From previous data - calculation:
Po1 P4 0.474 MPa=:= ... Stagnation pressure at inlet to nozzles
To1 T4 888.694 K =:= ... Stagnation temperature at inlet to nozzles
m_flow m_exhaust 0.728kg
s=:= ... The mass flow of exhaust gas available to the turbine
Assumptions :
P2 .725 Po1 0.344MPa=:= ... Static pressure at exit from nozzles
T2 0.9275 To1 824.264 K=:= ... Static temperature at exit from nozzles
P3 0.5 Po1 0.237MPa=:= ... Static pressure at exit from rotor
T3 0.86325 To1 767.165 K=:= ... Static temperature at exit from rotor
To3 1.002 T3 768.7K=:= ... Stagnation temperature at exit from rotor
r3av_r2 0.5:= ... Ratio r,ave / r1
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RPM 47500rpm= ... Rotational speed
Analysis:
a). The total-to-static efficiency is given by
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t_ts
1To3
To1
1 P3Po1
1
0.849=:=
b). The outer diameter of rotor, inlet diameter
Cx3 0:= Cx2 U2=
U2 Cp To1 To3( ) 370.99m
s=:=
D2U2 rev
RPM 149.166 mm=:=
c). The enthalphy loss coefficient for the nozzles and rotor rows
Nozzle loss coefficient:
T2s To1P2
Po1
1
820.093 K=:=
NT2 T2s
To1 T20.064736=:=
Rotor loss coefficient:
U3 U2 r3av_r2 185.495m
s
=:=
C3 2 Cp To3 T3( ) 59.328m
s=:= C3
23519.754213
m2
s2
=
W3 C32
U32
+ 194.752m
s=:= W3
237928.205778
m2
s2
=
h3_h3s Cp T3P3
P2
1( )
T2
18314.248869m
2
s2
=:=
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Rh3_h3s
r3av_r2 W32
0.966=:=
d). The blade outlet angle at the mean diameter 3,av, and
3av acotC3
U3
72.264 =:=
e). The total-to-total efficiency of the turbine
t_t1
1
t_ts
1
2r3av_r2 cot 3av( )( )
2
0.858558=:=
f). The volume flow rate at rotor exit
Po3P3
T3
To3
1
238.888 kPa=:=
ho1_ho3ss Cp To1 1Po3
Po1
1
160.353
kJ
kg=:=
3
P3
R T3 1.076
kg
m3=:=
Q3m_flow
30.676
m3
s=:=
g). The hub and tip diameters of the rotor at exit
Q3 r3t2
r3h2
( ) C3=
Q3 2 r3av h C3=
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- END of CALCULATION -
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