NATIONAL AVIATION UNIVERSITY
Institute: The Aerospace Institute
Faculty: Aircraft Maintenance
Department: The Aero-engine department
Educational and Qualifications level: Bachelor
Training direction: 6.070103 “Aircraft Maintenance”
APPROVED BY
Head of the Department
__________M.S. Kulik
“__”___________2018 y.
Student’s Course Paper Assignment
Horbach Zakhar Yaroslavovich
Full name (Surname Name Patronymic)
1.
The project topic: «Thermodynamic and gas dynamic calculations of
aircraft turbofan engine»
2.
The project to be performed between 01.04.2018 y. and 20.05.2018 y.
3.
Initial data for the project: calculation holds in ground conditions at
normal ambiance conditions V=0 m/s, ТH=288К, РH=101325Pа
Initial data for thermodynamic and gas dynamic calculations
Parameter
Designation
Dimension
Value
R
kN
230
Trust
Fan Pressure ratio
 *fan
1.65
Pressure ratio in the compressor
𝜋𝑘∗
26
Bypass ratio
m
5,5
Gas temperature in the
combustion chamber outlet
𝑇г∗
K
1550
Pressure on an in inlet
𝑃н
Pa
101325
Temperature on an in inlet
𝑇н
K
288
Time-table
№
Task
Fulfillment term
01.04.20185.04.2018
2.
Literature review of materials for
course paper
Thermodynamic calculation
3.
Gas dynamic calculation
4.
Performance calculation
graphic notes
1.
and
Completion
mark
6.04.2018-21.04.2018
22.04.201814.05.2018
15.05.201820.05.2018
8. Assignment issue date: «___»: __________ 2018 y.
Diploma project supervisor:
_________________
Kirchu F. I.
(Professor signature)
Assignment is accepted for execution:
________________ Horbach Z. Y.
(Student’s Signature)
Content
Thermodynamic calculation of the real cycle of Turbofan engine ........................................................ 4
1.1 The air parameters determination at the entry to the fan ......................................................... 4
1.2 The air parameters determination behind the fan in the secondary flow ................................. 4
1.3 The air parameters determination at the exit from secondary flow jet nozzle .......................... 5
1.4 The air parameters determination behind the compressor ....................................................... 5
1.5 The gas parameters determination behind the combustion chamber ....................................... 6
1.6 The gas parameters determination behind the turbine ............................................................. 7
1.7 The gas parameters determination at the exit from the primary flow jet nozzle ...................... 7
1.8 Calculation of specific parameters ............................................................................................. 8
Gas – dynamic calculation of gas turbine engine ................................................................................... 9
2.1 The fan inlet diametric dimensions determination..................................................................... 9
2.2 Determination of fan stages number ........................................................................................ 10
2.3 Distribution of compression work between compressor spools and determination of the
number of high pressure turbine stages of turbofan engines ........................................................ 11
2.4 Determination of air parameters and diametric dimensions at the fan exit ............................ 12
2.4.1 Secondary flow ....................................................................................................................... 13
2.4.2 Primary flow ........................................................................................................................... 13
2.5 Determination of diametric sizes at the entry of the low-pressure compressor...................... 14
2.6 Determination of air parameters and diametric sizes at the low-pressure compressor exit ... 15
2.7 Determination of diametric sizes at the entry of the high-pressure compressor .................... 16
2.8 Determination of air parameters and diametric sizes at the high-pressure compressor exit .. 17
2.9 Determination of diametric sizes at the entry to the high-pressure turbine ........................... 18
2.10 Determination of diametric sizes at the high-pressure turbine exit ....................................... 19
2.11 Determination of low-pressure compressor stages number .................................................. 20
2.12 Determination of high-pressure compressor stages number ................................................. 21
2.13 Determination of low-pressure turbine number of stages and distribution of work between
them ................................................................................................................................................ 23
2.14 Determination of diametric sizes at the input to the low-pressure turbine........................... 23
2.15 Determination of diametric sizes at the exit to the low-pressure turbine ............................. 24
2.16 Determination of fan-turbine number of stages and distribution of work between them ... 26
2.17 Determination of diametric sizes at the exit of fan-turbine ................................................... 27
2.18 Determination of sizes of sections at the exit from jet nozzles of turbofan engine ............... 28
2.19 Determination of elaborated parameters of the projected engine ........................................ 29
Appendix 1 ...................................................................................................................................... 30
Appendix 2 ...................................................................................................................................... 31
Page
Content
№ of document
Signat.
Data
3
Thermodynamic calculation of the real cycle of Turbofan engine
The purpose of the thermodynamic calculation is to determine the parameters
of the working fluid in the typical flow-sections of the installation and specific
power, specific fuel consumption, the main efficiency gas turbine. Next
showing scheme of cross section when using in my calculation.
1.1 The air parameters determination at the entry to the fan
The temperature at the entrance to GTU is determined by the formula:
TH  288 К
The total air pressure is:
pci   ent  101325  100818.375
(Pa)
where σent – factor, which takes into account the total pressure losses in the absorption system of air (
before compressor); ent = 0,995
1.2 The air parameters determination behind the fan in the secondary flow
The work of air compression in the fan can be calculated as follows:
k 1
L fan
k
1

 R  TH  ( fank  1) 
k 1
 fan
Page
Thermodynamic calculation
№ of document
Signat.
Data
4
J
where: k =1.4, R = 287 kg∙K,
Lfan =
1.4∙287∙288
0.4
0.4
1
J
(1.651.4 − 1) 0.85 = 52407.277 ( kg )
where efficiency is:
η∗f = 0.85
The temperature at the outlet of the fan is determined by the formula:
𝑇𝑓𝑎𝑛𝑛 = 𝑇𝐻 +
Lfan
kR
k−1
,
52407.277
𝑇𝑓𝑎𝑛𝑛 = 288 + 1.4∗287/0.4 = 340.118 ( K )
The pressure at the outlet of the compressor is determined by the formula:
𝑝𝑓𝑎𝑛𝑛 = 𝑝𝑐𝑖 ∙ πf∗ ,
𝑝𝑓𝑎𝑛𝑛 = 100818.375∙ 1.65 = 166350 ( Ра )
1.3 The air parameters determination at the exit from secondary flow jet
nozzle
T fd 2  T fann  340.188( K )
p fd 2  p fann  166350( Pa)
The jet velocity of fan air at the jet nozzle is determined by the formula for full expansion:
𝑐𝑗𝑛2 = φn2 √2 ∗
k
PH k−1
∗ R ∗ 𝑇𝑓𝑑2 ∗ (1 − (
) k )
k−1
Pfd2
101325 1.4−1
1.4
𝑐𝑗𝑛2 = 0.985√2 ∗ 1.4−1 ∗ 287 ∗ 340.118 ∗ (1 − (166350)
where:
1.4
)=295.901(m/s)
 n 2 is velocity coefficient of the secondary flow jet nozzle,  n 2  0.985
Static pressure and static temperature at the secondary flow jet nozzle are:
p jn 2 st  pam  101325 (Pa);
T jn 2 st  T jn 2
2
k  1 c jn 2
295.9012


 340.118  0.286 
 296.535
k 2 R
2  287
(K);
1.4 The air parameters determination behind the compressor
Efficiency is determined by the formula:
k−1
η∗c
=
π∗c k − 1
k−1
kη∗st
∗
πc
−1
,
where η∗cs - efficiency of the compressor stage; η∗st = 0.91
Sheet
Thermodynamic calculation
Змн.
Арк.
№ докум.
Підпис
Date
5
1.4−1
η∗c
=
26 1.4 − 1
1.4−1
261.4∙0.91 − 1
1.2856
= 1.5071 = 0.863
The work of air compression in the compressor can be calculated as follows:
Lcomp =
Lcomp =
k−1
𝑘𝑅𝑇𝐻
(πc∗ k
k−1
1.4∙287∙288
0.4
− 1)
0.4
1
η∗c
,
1
J
(261.4 − 1) 0.863 = 515349 ( kg )
The total air temperature behind the compressor:
𝑇𝑐𝑑 = 𝑇𝐻 +
𝑇𝑐𝑑 = 288 +
Lc
kR
k−1
0.4∗515349
1.4∗287
,
= 801 ( K )
The total air pressure behind the compressor:
𝑝𝑐𝑑 = 𝑝𝑐𝑖 ∙ π∗c ,
𝑝𝑐𝑑 = 100818.375∙ 26= 2621277.75 ( Ра )
1.5 The gas parameters determination behind the combustion chamber
The temperature before turbine is determined by the formula:
𝑇𝑡𝑖 = 𝑇𝑔 = 1550 (К)
The pressure before turbine is determined by the formula:
𝑝𝑡𝑖 = 𝑝𝑐𝑑 ∙ σcc ,
σc=0.98
𝑝𝑡𝑖 = 2621277.75 ∙ 0.98 = 2568852.195 ( Ра )
Average specific heat of gases in combustion chamber is determined by the formula:
Сp =
878 + 0.208 (𝑇𝑡𝑖 + 0.48𝑇𝑐𝑑 ) ,
Сp = 878 + 0.208 (1550 + 0.48 ∙ 801) = 1280.376 (
J
)
kg ∙K
Relative fuel consumption in the combustion chamber is determined by the formula:
gf =
gf =
𝐶𝑝(𝑇𝑡𝑖 − 𝑇𝑐𝑑 )
Нu∙ηg
q
1
= Hu∙η
,
1280.376∗(1550 −801)
42∙106 ∙0.995
g
= 0.0229469
Where : Hu = 42 ∗ 106.
Арк.
Thermodynamic calculation
Змн.
Арк.
№ докум.
Підпис
Дата
6
Average excess air/fuel ration in combustion chamber:
α=
1
gf∗l0
=
1
0.0229469∗14.9
= 2.925
𝑙0 = 14.9
1.6 The gas parameters determination behind the turbine
Effective work of all stages of the turbine is determined by the following equation:
L
+m∗Lf
Lt = (1+gcomp
)(1− g
cool )ηm
f
,
where: gcool = 0.027 – relative consumption of air, which selected on the exit of the compressor, for
cooling of details of the turbine;
where: ηm – mechanical efficiency; ηm = 0,995.
Lt =
5.5∗52352.553+515348.555
(1+0.023)(1−0.027)∙0.995
J
= 811114.293 ( kg ).
Air temperature at the outlet of the turbine is determined by the formula:
𝑇𝑡𝑑 = 𝑇𝑡𝑖 –
where: kg = 1,33, Rg = 288
J
kg∙K
Lt (kg − 1)
kg Rg
,
;
𝑇𝑡𝑑 = 1550–
811114.293 (1.33−1)
1.33∙288
= 846.436 ( K ).
Air pressure at the outlet of the turbine is determined by the formula:
kg
𝑝𝑡𝑑 = 𝑝𝑡𝑖 ∙ (1 −
𝑇𝑡𝑖 − 𝑇𝑡𝑑 kg − 1
)
,
𝑇𝑡𝑖 η∗t
where: η∗t - efficiency of the turbine of compressor drive; η∗ct = 0.9 ,
𝑝𝑡𝑑 = 2568852.195∙ (1 −
1550−846.436 1.33
)0.33
1550∙0.9
=154715.807 ( Ра ).
1.7 The gas parameters determination at the exit from the primary flow jet
nozzle
Jet nozzle pressure ratio:
π∗nozzle =
𝑝𝑡𝑑
𝑝𝐻
,
154715.807
π∗nozzle = 100818.375 = 1.535
𝑐𝑗𝑛1 = φn1 √2 ∗
Kg−1
Kg
1
∗ Rg ∗ 𝑇𝑡𝑑 ∗ (1 − ( ∗
) Kg )
Kg − 1
πnozzle
Арк.
Thermodynamic calculation
Змн.
Арк.
№ докум.
Підпис
Дата
7
1.333
1.333−1
1.333
1
𝑐𝑗𝑛1 = φn1 √2 ∗ 1.333−1 ∗ 288 ∗ 846.436 ∗ (1 − (1.535)
)=433.869 (m/s)
where: φn1 = 0.975.
Air temperature at the outlet of the power turbine is determined by the formula:
𝑇𝑗𝑛1 = 𝑇𝑡𝑑 𝑇𝑗𝑛1 = 846.436–
(1.33−1)∗𝑐𝑗𝑛1 2
1.33∗2∗Rg
0.33∗433.8692
1.33∗2∗288
,
= 764.795 ( K ).
Air pressure at the outlet of the power turbine is determined by the formula:
𝑝𝑡𝑑 = 𝑝𝐻 .
1.8 Calculation of specific parameters
Specific thrust:
Psp1 = 𝑐𝑗𝑛1 ∗ (1 + g f ) = 433.869 ∗ (1 + 0.023) = 443.825
Psp2 = 𝑐𝑗𝑛2 = 295.901
Psp =
Psp1 + m ∗ Psp2 443.825 + 5.5 ∗ 296.056
=
= 318.659
1+m
1 + 5.5
Specific fuel consumption is determined by the formula:
Сsp =
Csp =
3600gf ∗(1−gcool )
,
Rg(1+m)
3600 ∙0.023(1−0.027)
318.659(1+5.5)
kg
= 0.03881 ( N∗hour ).
Mass air flow rate:
R
Ga = P ,
sp
230000
Ga = 318.659 = 721.776 (
kg
s
).
The mass flow rate of core engine air passing through engine:
Ga1 =
Ga
721.776
=
1+m
1+5.5
=111.042 (
kg
s
).
The mass of secondary airflow passing through the engine outer duct is:
Ga2 =
Ga∗m 111.042 ∗5.5
= 1+5.5 =610.733
1+m
(
kg
s
).
The total mass air flow rate is determined by the formula:
Ga = Ga1+ Ga2 = 721.776 (
kg
s
).
Арк.
Thermodynamic calculation
Змн.
Арк.
№ докум.
Підпис
Дата
8
The internal engine efficiency is determined by the formula:
ηin =
2
2
𝑃𝑠𝑝1
+m∗𝑃𝑠𝑝2
2gf ∗Hu ∗(1−gcool )
,
443.8252 +5.5∗295.9012
ηin = 2∗0.023∗42∗106 ∗(1−0.027) = 0.362 .
Gas – dynamic calculation of gas turbine engine
The purpose of gas-dynamic calculation is to determine the size of typical polarized-sections of flow
settings of the rotors and the frequency of their rotation, the number of compressor and turbine
stages, distribution of the compression (expansion) between the stages and degrees, clarify the
parameters of GTU.
2.1 The fan inlet diametric dimensions determination
Reduced velocity λ1a is determined by the formula:
λ1a =
С1a
Сcr
=
С1a
18.3√TH
,
where: c1a – circular speed at the inlet of compressor, c1a = 220
220
√288
λ1a = 18.3
m
s
,
= 0.708.
The function of relative density is determined by the formula:
k  1 k11
k  1 2 k11
q ( )  (
)    (1 
 )
2
k 1
,
q (1a )  0.898.
The area of flowing part at the inlet of compressor is determined by the formula:
Ffi 
Ga  TH
,
ma  pH  q (1a )
where mair = 0.040348.
Ffi 
722.128  288
 3.352 (m2).
0.040348  100818.375  0.898
Gas – dynamic calculation
Изм. Лист
№ докум.
Подпись Дата
Лист
9
The first-stage sleeve relative diameter 𝑑ℎ𝑢𝑏 = 0.45.
External diameter of the fan at the inlet in the first stage can be determined by the formula:
D1 ft 
4 Ffi
 (1  d hub 2 )
,
4  3.352
 2.313 (m).
3.14(1  0.452 )
D1 ft 
The diameter of the hub is determined by the formula:
D1sl  D12ft 
4  Ffi
4  3.352
 1.041 (m).
3.14
 2.3132 

Height of blade is determined by the formula:
hbl 
D1 ft  D1sl
2.313  1.041
 0.636
2
(m).
Ga 2
610.733
 3.352 
 2.837
Ga
721.776
(m2).

2
Height of blade should not exceed 15 mm.
F2  Ffi 
Diameter of imaginary cylinder:
D1  D12ft 
4  F2

 2.3132 
4  2.837
 1.32
3.14
(m).
2.2 Determination of fan stages number
Circumferential velocity on diameter
u1  u1 ft 
D1 is determined as:
D1
1.32
 500 
 285.12
D1 ft
2.313
(m/s).
Circumferential velocity near the sleeve diameter is calculated by the formula:
u1sl  u1 ft 
D1sl
1.041
 500 
 225 (m/s).
D1 ft
2.313
The air swirl in rotor blades on diameter
D1 and near the sleeve can be calculated by the formula:
Wu1  c1a 
1.55
t
1  1.5  ( )1
b
,
Gas – dynamic calculation
Изм. Лист
№ докум.
Подпись Дата
Лист
10
Wusl  c1a 
where:
1.55
t
1  1.5  ( ) sl
b

b
D
b
b
1.041
( ) sl  2.2 , ( )1  ( ) sl  1sl  2.2 
 1.736 ,
t
t
t
D1
1.32
The work on diameter
Wu1  220 
1.55
 182.937 ,
1  1.5  0.576
Wusl  220 
1.55
 202.757 
1  1.5  0.455
D1 and near the sleeve is calculated by Euler’s equation:
L1  u1  Wu1  285.128  182.937  52160.631 (J/kg),
L1sl  u1sl  Wusl  225  202.757  45620.27 (J/kg),
L fan1  0.5  ( L1  L1sl )  0.5  97780.901  48890.451 (J/kg),
z fan 
where:
z fan
L fan1
L fan

48890.451
 1,
52407.277
is number of fan stages.
2.3 Distribution of compression work between compressor spools and
determination of the number of high pressure turbine stages of turbofan
engines
The work of high pressure compressor can be determined by the formula:
Lhpc  0.5  Lcomp  0.5  515348.555  257674.277 (J/kg),
and of high pressure turbine:
Lhpt 
Lhpc
(1  g f )  (1  g cool ) m

257674.277
 260184.883
(1  0.023)  (1  0.027)  0.995
(J/kg),
where:  m
 0.995 is the mechanical efficiency.
Gas – dynamic calculation
Змн.
Арк.
№ докум.
Підпис
Дата
Арк.
11
Loading coefficient of high pressure turbine can be determined by the formula:
Y *  utmd
where:
z  hpt
2  Lhpt
,
z  1 is number of high pressure turbine stages;
utmd  400 is circumferential velocity on middle turbine radius;
hpt  0.89 is the high pressure turbine efficiency.
Y *  400
1  0.89
 0.523 .
2  260184.883
The high pressure turbine has one stage, then
Lst  Lhpt  260184.883 (J/kg).
The work of low pressure compressor:
Llpc  Lcomp  Lhpc  L fan1  515348.555  257674.277  48890.451  208783.827
(J/kg).
2.4 Determination of air parameters and diametric dimensions at the fan
exit
Pressure ration of air in the low-pressure compressor:
 lpc
 lpc  (1 
Llpc   fan kk1
 (1 
)
kR
 T fd
k 1
,
208783.827  0.85 3.5
)  4.324 .
1005.55  340.118
T he total air temperature is:
Tlpcd  T fd 
k 1
 Llpc  340.118  0.001 208783.827  547.967
kR
(K).
The total air pressure is:
plpcd   lpc  p fi  4.324  166350.319  719311.564 (Pa).
Gas – dynamic calculation
Изм. Лист
№ докум.
Подпись Дата
Лист
12
2.4.1 Secondary flow
The fan outlet axial air speed in the secondary flow zone:
cafan 2  c jn 2  10  295.901  10  285.901 (m/s).
Reduced velocity
afan 2 , relative density q(afan 2 ) , and the fan outlet area in secondary flow
Ffan 2 , can be determined from:
afan 2 
afan 2 
cafan 2
18.3  T fd
286.056
18.3  340.118
,
 0.847 ;
q(afan 2 )  0.972 ;
Ffan 2 
Ffan 2 
Ga 2  T fd
ma  p fd  q(afan 2 )
;
610.733  340.118
 1.727 (m2).
0.040348  166350.319  0.972
A diameter of imaginary cylinder dividing the primary and secondary airflows is found by the
formula:
D2  D 2fan 2 
where:
D fan 2
4

 Ffan 2  2.082 
4
 1.727  1.462
3.14
(m).
external diameter behind the fan:
D fan 2  0.9  D1 ft  0.9  2.313=2.082 (m).
2.4.2 Primary flow
The fan outlet axial air speed in the primary flow zone:
cafan1  c1a  10  220  10  210 (m/s).
Reduced velocity
afan1 , relative density q(afan1 ) , and the fan outlet area in secondary flow
Ffan1 , can be determined from:
afan1 
cafan1
18.3  T fd

210
 0.622 ;
18.3  340.118
Gas – dynamic calculation
Изм. Лист
№ докум.
Подпись Дата
Лист
13
q(afan1 )  0.831;
Ffan1 
Ga1  T fd
ma  p fd  q(afan1 )

111.042  340.118
 0.367 (m2).
6711.903  0.831
Sleeve diameter can be calculated by the formula:
D fansl  D22 
4
4
 Ffan1  2.094 
 0.367  1.275
3.14
3.14
(m).
The external diameter of the primary flow:
D fan1  D2    1.462-0.015  1.447
where:
(m)
  15  103 width of separating partition.
2.5 Determination of diametric sizes at the entry of the low-pressure
compressor
The fan outlet axial air speed in the LPC zone:
calpc  200 (m/s).
Reduced velocity
alpc
, relative density
q(alpc ) , and the fan outlet area in secondary flow Flpc1 ,
can be determined from:
alpc 
calpc
18.3  T fd

200
 0.593 ;
18.3  340.118
q(alpc )  0.804 ;
Flpci 
where:
Ga1  T fd
ma  p fd   icc  q(alpc )

111.097  340.118
 0.384 (m2).
6711.903  0.99  0.804
 icc  0.99 pressure recovery coefficient;
Gas – dynamic calculation
Изм. Лист
№ докум.
Подпись Дата
Лист
14
Circumferential diameter of the first axial compressor stage rotor is calculated by the equation:
4  Flpci
D1lpc 
where:
  (1  d )
2
1sl1

4  0.383
 0.978
3.14  0.51
(m)
d1ls1  0.7 , relative diameter of the first stage sleeve of the LPC;
The diameter section near the sleeve is calculated:
D1lpcsl  D12lpc 
4
4
 Flpci  0.978 
 0.383  0.685 (m).
3.14
3.14
2.6 Determination of air parameters and diametric sizes at the lowpressure compressor exit
The fan outlet axial air speed in the LPC exit equal to the LPC inlet:
calpc  calpcd  200 (m/s)
Reduced velocity
alpcd
, relative density
q(alpcd ) , and the fan outlet area in exit of LPC Flpcd ,
can be determined from:
alpcd 
calpcd
18.3  Tlpcd

200
 0.467 ;
18.3  548.017
q(alpcd )  0.671;
Flpcd 
Ga1  Tlpcd
ma  plpcd  q(alpcd )

111.097  547.967
 0.133 (m2).
29022.783  0.804
The diameter section near the sleeve in the inlet of LPC equal to the sleeve diameter at the exit of
LPC:
D1lpcsl  Dlpcsld  0.685 (m).
External diameter :
D1lpcd  D12lpcsl 
4
4
 Flpcd  0.467 
 0.133  0.799
3.14
3.14
(m).
Blade length:
hlpcd 
D1lpcd  D1lpcsl
2

0.799  0.133
 0.057
2
(m);
Лист
Explanatory note
Изм. Лист
№ докум.
Подпись Дата
15
D1lpcsl
dlpcsl 
0.133
 0.857
0.799

D1lpcd
(m).
2.7 Determination of diametric sizes at the entry of the high-pressure
compressor
The air velocity at the entry to the HPC
cahpc
is accepted more than air velocity at the LPC exit.
cahpc  clpcd  10  210 (m/s).
Reduced velocity ahpc and relative density
q(alpcd )
ahpc 
alpcd 
is calculated by the formula:
cahpc
18.3  Tlpcd
210
18.3  547.967
,
 0.49 ;
q(ahpc )  0.698 .
The area of section at the entry to the high-pressure compressor is calculated by the formula:
Fhpci 
Ga1  Tlpcd
ma  plpcd   icc  q(ahpcd )

111.042  548.017
 0.13 (m2).
29031.275  0.99  0.698
The first-stage sleeve relative diameter of the HPC:
d hpcsl  0.8 (m).
First stage axial compressor stage rotor is calculated by equation:
D1hpc 
4  Fhpci
  (1  d
2
hpcsl
)

4  0.13
 0.677 .
3.14  0.8
The diameter section near the sleeve is calculated:
D1hpcsl  D12hpc 
4
4
 Fhpci  0.677 
 0.13  0.542 (m)
3.14
3.14
Gas – dynamic calculation
Изм. Лист
№ докум.
Подпись Дата
Лист
16
2.8 Determination of air parameters and diametric sizes at the highpressure compressor exit
The temperature of air at the exit from HPC is determined by the formula:
Tcd  Tlpcd 
k  1 Lhpc
 801 (K).
k R
Air pressure in HPC is found by the formula:
 hpc 
 hpc 
pcd
plpcd   icc
;
2621277.75
 3.68 .
719311.564  0.99
For determination of sectional area at the high-pressure compressor exit speed
cacd
is selected
within the limits of 110-140 m/s :
cacd  140 (m/s)
Reduced velocity
acd
, relative density
q(acd ) , and the fan outlet area in exit of HPC Fcd , can
be determined from:
acd 
cacd
18.3  Tcd

140
18.3  801.04
 0.27 ;
q(acd )  0.414 ;
Fcd 
Ga1  Tcd
ma  pcd  q(acd )

111.042  801.04
 0.072 (m2);
105763.315  0.414
D1hpc  const ,
Dcsl  D12hpc 
hb 
4
4
 Fcd  0.672 
 0.072  0.598 (m);
3.14
3.14
D1hpc  Dcsl
2

00.67  0.598
 0.036
2
(m);
Gas – dynamic calculation
Изм. Лист
№ докум.
Подпись Дата
Лист
17
dcsl 
Dcsl
0.598

 0.892 (m).
D1hpc
0.67
2.9 Determination of diametric sizes at the entry to the high-pressure
turbine
Jet velocity of gas emission from the first nozzle diaphragm is determined by Euler equation, on the
assumption of axial outlet from the first turbine wheel of high-pressure turbine being:
c1 
Lhpt
utmd  cos 
where: the angle of a stream output from the ND 
c1 
Reduced velocity
.
 25
260184.883
 717.706 (m/s).
400  cos 25
1 is determined by the formula:
1 
c1
18.3  Tti

717.706
18.3  1550
 1.004 ;
qg (1 )  1 .
Mass gas flow rate through the first stages ND is determined bu the formula:
Ggti  Ga1  (1  g f )(1  gcool1 )
where
equal
g cool1 is
;
relative consumption of compressor-bleed air for cooling parts of HPT, which is
g cool1  0.027 .
Ggti  111.042  (1  0.023)(1  0.027)  110.462 (kg/s).
The section area of turbine air-gas channel at the exit from ND is calculated by the equation:
F1nd 
Ggti  Tti
ma  pti   nd  q(1 )  sin 

110.462  1550
 0.1043
0.0396  2568852.195  0.97  1  0.423
(m2),
where: the total pressure recovery coefficient in the ND  nd
The middle diameter of turbine
 0.97 .
Dtmd  1.6  D1hocd  1.072 (m).
Лист
Gas – dynamic calculation
Изм. Лист
№ докум.
Подпись Дата
18
The height of turbine wheel blade is calculated by the formula:
hlpcd 
F1nd
0.104

 0.031 (m).
  Dtmd 3.14  1.072
The turbine wheel external diameter:
Dt  Dtmd  h1  1.072  0.031  1.103 (m);
Dsl  Dt2 
Dtav 
4

 F1nd  1.1032 
4

0.1043  1.041 (m);
Dt  Dsl 1.103+1.041

 1.072 (m).
2
2
Tension due to centrifugal forces action is calculated by the formula:
h1
 106
Dtav
2
 ten  2    K f  utav
where  is density of blades material,
Kf
is coefficient of blades form,
;
  8100 kg/m3;
K f  0.5
I choose blades material ЖС6-К and limit of long durability  
t
 50
MPa
The safety factor determined by the formula :
 t
n
 ten
;
 ten  2  8.1 103  0.5  4002  0.031/ 1.072 106  37.437
n
The condition is satisfied
n  1.2  1.5
(MPa);
50
 1.336 .
37.437
.
2.10 Determination of diametric sizes at the high-pressure turbine exit
The total gas temperature at the exit from HPT is determined by the formula:
Thptd  Tti 
k g  1 Lhpt
kg
Rg
 1550 
1.333-1 260184.883

 1324.315 (K).
1.333
288
Gas – dynamic calculation
Изм. Лист
№ докум.
Подпись Дата
Лист
19
The total gas pressure at the exit of the GPT is:
phptd  pti (1 
Tti  Thptd
Tti hpt
kg
)
k g 1
 2568852.195  (1 
Reduced velocity of gas at the HPT exit
2a
is set
1550  1324.315 4.003
)
 11334728.293( Pa).
1550  0.965
2 a  0.5 .
Relative density:
qg (2 a )  0.712 .
The gas flow rate at the exit of the HPT is calculated by the formula:
Ggtd  Ga1  (1  g f )(1  gcool1 )  111.042  (1  0.023)(1  0.027)=110.524 (kg/s) .
The section area at the exit of the HPT is determined by the formula:
Fhpt 
Ggtd  Thptd
mg  phptd  qg (2a )

110.524  1324.315
 0.107 (m2).
0.04 1334728.293  0.712
The height of turbine blade on the target edge is calculated by the formula:
hbl 
Fhpt
  Dtmd

0.107
 0.0318 (m).
3.14 1.072
Diameters of the section at the exit of the HPT are calculated by the equations:
Dhptd  Dtmd  hbl  1.072  0.0318  1.104
2
Dslhptd  Dhptd

4

 Fhpt  1.1042 
4

(m);
 0.107  1.04 (m).
2.11 Determination of low-pressure compressor stages number
Circumferential speeds on periphery
of the last stage
u1c , near the sleeve of the first stage u1sl
and near the sleeve
u zsl are calculated by the formulas:
u1lpc  utmd 
D1lpc
Dtmd
 400 
0.978
 365.031 (m/s);
1.072
Gas – dynamic calculation
Изм. Лист
№ докум.
Подпись Дата
Лист
20
u1sllpc  utmd 
D1lpcsl
uzsllpc  utmd 
Dzlpcsl
The lattice density of the first stage
Wlpcu1sl  cahpc 
Wlpcuzsl  cacd 
Dtmd
Dtmd
 400 
0.685
 255.522 (m/s);
1.072
 400 
0.799
 298.227
1.072
b
( ) sl  1.8
t
1.55
and for the last stage
 140 
t
1  1.5  ( ) sl
b
1.55
t
1  1.5  ( ) slz
b
(m/s).
b
( ) slz  1.4 .
t
1.55
 169.091;
2.5  0.556
 200 
1.55
 149.655 ;
2.5  0.714
Determination of work of first and last stages:
Llpcst1  u1csl  Wu1sl  365.031  169.091  61723.429 ;
Llpcstz  uzsl  Wuzsl  298.227  149.655  44631.159 ;
Lavlpc 
1
1
 ( Lst1  Lstz )  (61723.429  44631.159)  53177.294 .
2
2
Number of stages:
Llpc
zlpc 
Lavlpc

208783.827
 4.
53177.294
2.12 Determination of high-pressure compressor stages number
Circumferential speeds on periphery
of the last stage
u1c , near the sleeve of the first stage u1sl
and near the sleeve
u zsl are calculated by the formulas:
u1c  utmd 
D1hpc
Dtmd
u1csl  utmd 
uzsl  utmd 
 400 
D1hpcsl
Dtmd
Dzhpcsl
Dtmd
0.677
 252.635 (m/s);
1.072
 400 
 400 
0.542
 202.08 (m/s);
1.072
0.598
 223.073 (m/s).
1.072
Gas – dynamic calculation
Изм. Лист
№ докум.
Подпись Дата
Лист
21
Choosing the lattice density of the first stage
b
( ) sl  2.2
t
we can find twisting of air in the first and last stage rotors
and for the last stage
Wu1sl , Wuzsl
b
( ) slz  1.651
t
and work of the first
and last stages:
Wu1sl  cahpc 
1.55
 210 
t
1  1.5  ( ) sl
b
Wuzsl  cacd 
1.55
t
1  1.5  ( ) slz
b
1.55
 193.541 ;
2.5  0.455
 140 
1.55
 113.699 ;
2.5  0.606
Lst1  u1csl  Wu1sl  202.08  193.541  39110.68
;
Lstz  u zsl  Wuzsl  223.073  113.699  25363.244 ;
Lav 
1
1
 ( Lst1  Lstz )  (39110.68  25363.244)  32236.962 .
2
2
Number of stages:
zhpc 
Lhpc
Lav

257674.277
8 .
32236.962
Power of HPT is determined
Nhpt  Ggtd  Lhpt  110.524  260184.883=28756557.301 (W);
Nhpc  Ga1  Lhpc  111.042  257674.277=28612774.515 (W),
N hpt m  Nhpc  0 .
Rotational speed is determined by the formulas:
nhpc  60 
u1csl
202.108
 60 
 7126.341 (rev/min);
  D1hpcsl
3.14  0.542
nhpt  60 
utmd
400
 60 
 7126.341 (rev/min).
  Dtmd
3.14  1.072
Gas – dynamic calculation
Изм. Лист
№ докум.
Подпись Дата
Лист
22
2.13 Determination of low-pressure turbine number of stages and
distribution of work between them
Mass gas flow rate in LPT is calculated by the formula:
Gg  Ga1  (1  g f )(1  gcoollpt ) ,
Gg  111.042  (1  0.023)  (1  0.02)  111.319
.
From powers balance condition we have:
Llpt 
Llpc

(1  g f )  (1  g cool ) m
208783.827
 209312.233 .
(1  0.023)  (1  0.027)  0.995
Circumferential speed on middle diameter LPT :
ulptmd  400 ,
zlpt hpt
Ylpt *  ulptmd
 400 
2  Llpt
1  0.9
 0.587 .
2  209312.233
z  1.
where: z number of turbine stages,
2.14 Determination of diametric sizes at the input to the low-pressure
turbine
Jet velocity we can determine by formula:
c2 
Llpt
ulptmd  cos 1
The angle 1 is accepted equal to
Reduced velocity
2
25

209312.233
 577.376 (m/s).
400  0.906
.
at the exit:
2 
c2
18.15 Thptd

578.121
 0.875 ,
18.15  1324.123
qg (2 )  0.982 .
The section area at the exit of the first ND is determined by the formula:
Flpti 
Gg  Thptd
ma  phptd   int   nd  q(2 )  sin 1
;
Gas – dynamic calculation
Изм. Лист
№ докум.
Подпись Дата
Лист
23
Flpti 
111.319  1324.315
 0.193(m2 ) ,
0.0396  1334728.293  0.97  0.985  0.982  0.423
where:  int and  nd are total pressure recovery coefficient in the intermediate case between HPT
and LPT, and total pressure recovery coefficient in the ND respectively;
 int  0.985 ;
Dtmd  Dlptmd  1.072 (m).
Height of rotor blade:
hbllpti 
Flpti
  Dlptmd

0.193
 0.057
3.14  1.072
(m);
Dtlpti  Dlptmd  hbllpti  1.072  0.057  1.129 (m);
2
Dsllpti  Dtlpti

4

 Flpti  1.1292 
4

 0.193  1.015 (m).
2.15 Determination of diametric sizes at the exit to the low-pressure
turbine
Parameters of gas at the exit of LPT are elaborated by the lowing equations:
Tlptd  Thptd 
(k g  1)  Llpt
k g  Rg
 1324.315 
0.333  209312.233
 1142.756(K);
1.333  288
plptd  phptd   int  (1 
plptd  1334728.293  0.985  (1 
Thptd  Tlptd
Thptd lpt
kg
)
k g 1
,
1324.315  1142.756 4.003
)
 678457.871 (Pa).
1324.315  0.9
Reduced velocity approximated:
2 d  0.6 ,
Flptd 
Gg  Tlptd
mg  plptd  qg (2 d )
,
Gas – dynamic calculation
Изм. Лист
№ докум.
Подпись Дата
Лист
24
Flptd 
111.319  1142.756
 0.172 (m2).
0.04  677348.113  0.813
Diameters at the LPT exit:
Dslplptd  Dtlpti  1.129
2
Dtlptd  Dsllptd

4

(m);
 Flptd  1.1292  0.22  1.223 (m).
The height of the last step blade:
hb 
Dtlptd  Dsllptd
2
Dtlptdav 

Dtlptd  Dsllptd
2
1.223  1.129
 0.047 (m);
2

1.223+1.129
 1.176 (m).
2
After final determination of LPT sizes, the last stage turbine blades are checked for durability. For
this purpose we calculate tension due to action of centrifugal forces by the formula:
2
 ten  2    K f  utav
h1
 106 .
Dtav
I choose blades material ЖС6-К and limit of long durability  
t
 ten  2  8.1  103  0.5  4002 
 65
MPa
0.047
 106  51.372 .
1.176
Safety factor:
n

65

 1.265 .
 ten 51.372
Condition is satisfied.
LPT and fan powers is determined by the formulas:
N lpt  Gglpt  Llpt ;
Nlpc  Ga1  Llpc
;
Nlpt  111.319  209312.233  23300362.54 (W);
Nlpc  111.042  209053.173  23183860.728 (W);
Nlpt m  23300362.54  0.995  23183860.728 (W).
Gas – dynamic calculation
Изм. Лист
№ докум.
Подпись Дата
Лист
25
Rotational speed of LPT is determined by the formulas:
nlpt  60 
ulptmd
  Dtlptdav
utlpc  ulptmd 
nlpc  60 
 60 
D1lpc
Dtlptdav
utlpc
 400 
 60 
  D1lpc
400
 6495.851 (rpm);
3.14  1.176
0.979
 332.736
1.176
(m/s);
400
 6495.851 (rpm).
3.14  0.978
2.16 Determination of fan-turbine number of stages and distribution of
work between them
Mass gas flow rate in fan-turbine is calculated by the formula:
Ggf  Ga1  (1  g f )  111.097  (1  0.023)  113.59 (kg/s).
Work of fan turbine:
(m  1)  L fan
L ft 
(1  g f ) m

6.5  52352.553
 334329.774 .
(1  0.023)  0.995
Circumferential speeds:
u1 ft  500 (m/s);
u1 ft
u ftmd 
where: fan-turbine middle diameter
D1 ft
 D ftmd  259.353 (m/s),
D ftmd  0.9 (m).
Loading coefficient of fan-turbine can be calculated by formula:
Y ft  u ftmd 
z ft  ft
2  L ft
 259.353 
where: z – number of stages of fan turbine,
4  0.9
 0.521 ,
2  334329.774
z  3.
Determination of work of fan-turbine stages:
L ft1  0.35  L ft  0.2  334329.774=117015.421 ;
L ft 2  0.2  L ft  0.2  334329.774=117015.421;
Gas – dynamic calculation
Изм. Лист
№ докум.
Подпись Дата
Лист
26
L ft 3  L ft  ( L ft1  L ft 2 )  334329.774  (117015.421  2)=100298.932 .
2.17 Determination of diametric sizes at the exit of fan-turbine
The total gas temperature at the exit of FT:
T ftd  Tlptd 
(k g  1)  L ft
k g  Rg
 1142.756 
0.333  334329.774
 852.757 (K).
383.904
The total gas pressure at the exit of FT:
4
p ftd  plptd   int
 (1 
p ftd  678457.871  0.9854  (1 
Tlptd  T ftd
Tlptd  fft
kg
)
k g 1
,
1142.756  852.757
)  154756.692 (Pa).
1142.756  0.851
Reduced velocity:
d  0.75 , qg (d )  0.926 ;
The section area at the exit of the fan-turbine is determined by the formula:
Fftd 
Ggf  T ftd
mg  p ftd  q(d )

113.646  852.757
 0.584
0.04 154756.692  0.926
(m2).
Diameters of the section at the exit of the fan turbine are calculated by equations:
Dslftd  D 2ftmd 
Dtftd  D ftmd 
4  Fftd

 1.22 
Fftd
  D ftmd
 1.2+
4  0.584
 0.834 (m);
3.14
0.584
 1.355
3.14  1.2
(m).
Height of the blades:
hblftd 
Dtdtd  Dslftd
2

1.355  0.834
 0.26
2
(m).
Tension due to centrifugal forces action is calculated by the formula:
2
 ten  2    K f  utav
h1
 106 .
Dtav
Gas – dynamic calculation
Изм. Лист
№ докум.
Подпись Дата
Лист
27
I choose blades material ЖС6-К and limit of long durability  
t
 ten  2  8.1 103  0.5  259.3532 
 175
MPa,
0.262
 106  118.21 MPa .
1.2
Safety factor:
n
 ten
175

 1.48 ;
  118.715
Power of the fan-turbine:
N fan  Ga  L fan
N ft  Gg f  L ft ;
;
N ft  113.59  334329.774  37976684.413 (W);
N fan  721.776  52352.553  37786800.991 (W);
N ft m  37976684.413  0.995  37786800.991 (W).
The condition
N fan  N ft m is satisfied.
Rotational speed is determined by the formulas:
n ft  60 
n fan  60 
The condition
n ft  n fan
u ftmd
  D ftmd
u1 ft
  D1 ft
 60 
 60 
259.353
 4127.735 ;
3.14  1.2
500
 4127.735 .
3.14  2.313
is satisfied.
2.18 Determination of sizes of sections at the exit from jet nozzles of
turbofan engine
Reduced velocities
 jn1 and  jn 2 are determined by the formulas:
 jn1 
 jn 2 
c jn1
18.15  Ttd
c jn 2
18.15  T fann

433.869
18.15  846.436

295.901
18.15  340.118
 0.822 ;
 0.877 ;
Gas – dynamic calculation
Изм. Лист
№ докум.
Подпись Дата
Лист
28
qg ( jn1 )  0.962 , qg ( jn 2 )  0.982 .
The areas of jet nozzle sections are found by the equations:
Fftd 
Fjn 2 
Ggf  T ftd
mg  p ftd   jn1  qg ( jn1 )

113.59 852.757
 0.592 (m2);
0.04  154756.692  0.95  0.962
Ga 2  T fann
mg  p fann   jn 2   2  qg ( jn 2 )

610.733  340.118
 1.8 (m2).
0.04  166350.319  0.95  1 0.961
Diameter of the primary flow jet nozzle can be determined by the formula:
D jn1 
4

4
 0.592  0.868 (m).
3.14
 Fjn1 
Diameter of the secondary flow jet nozzle can be determined by the formula:
D jn 2  Din2 
4

 Fjn 2  0.955 
4
 1.8  1.79 (m).
3.14
2.19 Determination of elaborated parameters of the projected engine
Specific thrust of the primary flow at the complete expansion of gas is determined by the following
equation:
Psp1  c jn1  (1  g f )  V  433.869  (1  0.023)  443.825
.
Specific thrust of the secondary flow:
Psp 2  c jn 2  V  295.901-0  295.901 .
Specific thrust of a turbofan engine:
Psp 
Psp1  m  Psp 2
1 m

443.825  5.5  295.901
 318.659 .
1  5.5
Thrust of turbofan engine:
P  Psp  Ga  318.503  722.128  230000 (N).
Specific fuel consumption is:
Csp 
3600  g f  (1  g cool )
Psp (1  m)

3600  0.023  (1  0.027)
 0.039 .
318.503
Page
Gas – dynamic calculation
№ of document
Signat.
Data
29
Appendix 1
3
2,5
Pressure MPa
2
1,5
1
0,5
0
External Fan inlet
Fan
behind
LPC
HPC
CC
HPC
LPT
FT
Nozzle External
Cross section
1800
1600
1400
Temperature K
1200
1000
800
600
400
200
0
External Fan inlet
Fan
behind
LPC
HPC
CC
HPC
LPT
FT
Nozzle External
Cross section
Page
Appendix 1
№ of document
Signat.
Data
30
Appendix 2
Parameters
Sections
dsleeve
1-1
2-2
1.041
1.275
dtip
2.313
2.313
2c-2c
1.462
3-3
3c-3c
4-4
4c-4c
5-5
6-6
0.685
0.598
1.03
1.015
1.129
1.102
0.978
0.67
1.103
1.129
1.223
1.28
7-7
0.868
Page
Appendix 2
№ of document
Signat.
Data
31
National Aviation University
Aviation engines department
COURSE WORK
from subject «Heat engines theory»
Theme: «Thermodynamic and gas-dynamic calculation of gasturbine engine»
Prepared by
Student FLA-305
Horbach Z. Y.
Checked by
F.I. Kirchu
Kyiv 2018