I//' FRWW-
DESIGN OF A MARINE GAS TURBINE
by
McVey, Hector
Lieutenant, Chilean Navy
Chilean Naval Academy, 1940
\~ST.
Submitted in Partial Fulfillment of the Requirements
29 ~
~
for the Degree of
Lj~
;K*~
MASTER OF SCIENCE
in
NAVAL CONSTRUCTION AND MARINE ENGINEERING
from the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
I
Signature of Author:
r ?946
Signature redacted
Department of Naval Architecture and Marine Engineering, 1946
Signature of Professor
Signature redacted
in charge of Research:
Signature of Chairman
Department
Committee on Graduate Students:
r\
Signature redacted
Cambridge, Mass., U. S. A.
September, 1946
Professor J. S. Newell
Secretary of the Faculty
Massachusetts Institute of Technology
Cambridge, Massachusetts, U. S. A.
Dear Sir:
In accordance with the requirements for the Degree of Master
of Science in Naval Construction and Marine Engineering, I submit
herewith a Thesis entitled: Design of a Marine Gas Turbine.
(r'espectfuk.y,
Signature redacted
Hlector Mjvey
TABLE OF CONTFNTS
Table of Symbols
Page
1
I.
Summary
5
II.
Introduction
7
9
III. Procedure
IV.
Results
11
V.
Discussion of results
14
VI.
Conclusions and Recomendations
16
VII. Appendix A
17
B
49
C
147
D
217
243
Bibliography
282074
TABLE OF SYMBOLS
2
A
Area in ft.
a
Velocity of sound in gas, Ft./sec.,
b
Chord of blades in inches
bs
Shroud width of blades in inches
c
Absolute velocity, ft./sec.
C,
p
Specific heat of air, 0.24 Btu/(lb.)(OF)
d
Pitch diameter, inches
d
Tube diameter, inches
D
Shell diameter, inches
f
Friction coefficient
G
Mass rate of flow, lbs./sec.
Ga
Mass rate of flow of air
Gf
Mass rate of flow of gas
h
Isentropic enthalpy drop through turbine, Btu/lb.
h
Heat transfer coefficient, Btu/(lb.)(OF)(hr)
A h
Isentropic enthalpy drop through a stage, Btu/lb.
i
Enthalpy in Btu/lb.; used to denote a state of the gas.
A1
Small change in enthalpy in Btu/lb. for conditions other than at
constant entropy.
J
Mechanical equivalent of heat, 778 ft. x force pounds per Btu.
k
Ratio of specific heats, Cp/C
k
Ratio (1-P)/(l
k
Thermal conductiviey, Btu/(hr)(ft)(deg. F.)
k
A
Ratio of cross sectional area of gas flow to that of pir flow
ku
Ratio of film coefficient inside tubes to overall coefficient
k -
Ratio of flow of gas to flow of air * Ga Gf
; flow area
=
T
1.4 for air.
P), k. ratio at which stage is symmetrical.
G
a
1
Blade length, inches
L
Length of tubes, ft.
n
Speed in r.p.m.
N
Number of tubes
M
Mach number, u/a
M
Mass velocity, lbs./sec. x ft. 2
o
Opening of blades
p
Static pressure in psia
pr
Relative pressure
P
Prandtl number,
Q
3
Volume rate of flow; ft. /sec.
q
Heat input, Btu/lb.
r
Overall pressure ratio
rB
Combustion chambers pressure ratio
ry
I
Intercooler pressure ratio
Re
Reynolds number, (cb)/( trg);
S
Number of stages
T
Absolute temperature, 0 Fahs.
t
Thickness of tubes, inches
U
Overall transfer coefficient, Btu/(hr)(ft. 2 )(oF)
u
Peripheral velocity at pitch diameter, ft./sec.
v
Specific volume, ft. 3/lb.
w
Specific air consumption, lbs/(hr)(H.P.)
w
Relative velocity, ft./sec.
W
Work input or output, Btu/lb.
W
Weight of Regenerltor, lbs.
(Cp')/k
DM/p
GREEK LETTERS
L,(3
Angles
i
Density of metal
A
Small change
Clearance, inches
Partial derivative
Diagram ratio
F,
Loss coefficient
-'
Efficiency
e
Angle of turn of flow through b'ades.
Viscosity, lbs.-sec./ft2
Velocity ratio
Pi, 3.14159
Blade length-pitch diameter ratio
Density of gas or air
Summation
Pitch of blades
A function
Angular velocity, rads./sec.
SUBSCRIPTS
a
Stands for air
b
Stands for base of blades
c
Compressor
e
Exhaust
f
Fuel
g
Gas
i
Initial or entering state of gas; internal
1
Leakage
max
Maximum
r
Relative, relative pressure
s
Isentropic condition
st
Stage
t
Turbine; tip of blade
th
Theoretical
u
Component of absolute velocity in the transverse direction
w
Component of relative velocity in the transverse direction
x
Component of velocity in thexial direction
W,W.F. Friction and windage
o
Entering or initial state
1
Entering or initial state; stator
2
Final or leaving state; rotor
I
I
SUMMARY
The object of this Thesis is to investigate the thermodynamic problems involved in the design of a Marine Gas Turbine Plant.
The study will be devoted to the constant-pressure gas turbine cycle.
There are many possible arrangements of the elements in the cycle, and the
author has selected the arrangement suggested by C. R. Soderberg, which
consists of a low pressure compressor coupled to a high pressure turbine;
an intercooler, a high pressure compressor coupled to a low pressure turbine, a regenerator, and two combustion chambers.
The useful power is
obtained from the low pressure turbine, while the high pressure turbine
constitutes merely a gas generator.
The fuel is burned at essentially con-
stant pressure in the combustion chambers.
The author decided to design the plant for a useful power output of
7500 h.p., with a propeller speed of 375 r.p.m. for the power turbine, due
to considerations of the size of the reduction gears.
The maximum temperature of the gas at the inlet of the turbines was
fixed at 14000 F., which is a little high for the materials now available
for turbine construction.
It is expected that within a few years this tem-
perature will present no problems to the designer, on account of the rapid
developments of the recent years in the field of the metallurgy of steel
and its alloys.
The useful power obtained for the plant was 7960 h.p. with an overall
efficiency of 37% and an overall pressure ratio of 7.
Undoubtedly this is
an optomistic result, but considerirg the fact that the author did not take
into account parasitic and stray losses, then it is a reasonable result to
expect.
In any case, it can be said that the gas turbine plant can complete
successfully with the best existing steam power plants, which have an overall thermal efficiency of about 25%; and it is on an even if not advantageous
position with Diesel installations, for which efficiencies of 32 nnd 33%
have been attained.
A problem that the gas turbine plant presents in its use as a marine
prime mover is the case of reversing, which is a very important fact.
No
attempt has been made in the present work to present * solution to this pro-
blem.
The turbines are of the reaction type, and the compressors of the axial
flow type using airfoil type of blades.
II. INTRODUCTION
For a historical background of the development of the gas turbine the
author found sufficient information in references (2)*, (4), and (5), and
in the interest of brevity it will not be given here.
The information now available for the design proper of a gas turbine
is very little; and a great part of the principles, theory, and methods used
are based on steam turbine design procedure.
In the arrangement of the elements of the cycle the process is as
follows:
Air at standard atmospheric condtions enters the low pressure compressor
where it is compressed to a prescribed pressure, it then goes into the intercooler where the temperature of the air is lowered at practically constant
pressure, from there it proceeds to the high pressure compressor where it
is again compressed to a higher pressure; after that it enters the regenerator,
where it receives heat from the exhaust hot gases; this process is also done
at essentially constant pressure; then, the air enters the combustion chambers where fuel is
injected and burned at constant pressure too.
The hot
gases then enter the high pressure turbine where they are expanded a prescribed amount so that the work delivered by the high pressure turbine is
enough to supply the power required by its coupled low pressure compressor
and the losses that may occur in this complete branch.
From the exhaust
of the high pressure turbine the gas enters the second combustion chamber
where more fuel is injected and burned at constant pressure until the gas is
reheated to the initial turbine inlet temperature.
The gases proceed
through the low pressure turbine where they are expanded to atmospheric
* Numbers in brackets designate, from here on, the references in the
bibliography at the end of the thesis.
7
pressure; and finally they enter the regenerator, where they deliver heat
to the high pressure air flowing through the tubes; the gases are then
exhausted to the stack and to the atmosphere.
III. PROCEDURE
In the development of the design use will be made of the general and
derived relations of Thermodynamics and other necessary relations of mechanics.
The general procedure of the design is as follows:
(a.) Investigation of the cycle, to determine the most efficient overall pressure ratio and intercooler pressure ratio, that is, the pressure
ratio at which the low pressure compressor works.
The specific air con-
sumption will also be determined.
In connection with the study and calculations of the cycle, the following assumptions and considerations will be made, and are listed here only,
to be explained later when their use comes up:
(1) Standard atmospheric conditions at inlet of low pressure compressor.
(2) Compressors and turbines internal efficiency of 0.85.
(3) Intercooling to a temperature of 900 F.
(4) Maximum inlet temperature to turbines, 14000 F.
(5)
High pressure turbine supplies power only to the low pressure
compressor.
(6).Exhaust pressure of low pressure turbine equal to 14.7 psia.
(7) Regenerator efficiency of 0.65.
(8) Intercooling, regeneration, and combustion at constant pressure.
(9) Neglect parasitic losses.
(b.) Once the overall pressure ratio has been selected the mass rate of
flow is determined based on a required power output of 7500 h.p.
(c.) The next step is to make a preliminr;ry decision on the turbines
diameter, number of stages, and speed of rotation.
(d.) Investigation of the characteristics of blades at different angles
to the flow and at various degrees of reaction, and based on dimensionless
coefficients.
(e.) Preliminary design of the low and high pressure compressors.
Determination of the characteristics of the first stage.
(f.)Determination of the characteristics of the blades to use in the
design of the compressors.
The airfoil N.A.C.A. No. 4409 is to be used.
Influence of the Mach number.
(g.) Preliminary decision on compressor and turbine design.
(h.) Detail design of low pressure turbine.
through the stages.
Distribution of the work
Dimensions of the blades in each stage.
Performance
of the turbine at full power.
(i.) Same as (i) for the high pressure turbine.
(j.) Detail design of the compressors, number of stages, dimensions,
and performance.
(k.) Design of the regenerator.
(1.) Development of the turbine characteristics, in order to predict
the partial load characteristic.
(m.) Development of the characteristics of the compressors.
(a.) Full load performance and partial load characteristic.
10
IV. RESULTS
Overall pressure ratio, full power
7
Intercooler pressure ratio
2.65
Intercooler effectiveness
0.848
Turbines:
H.P.
L. P.
Pitch diameter, constant
26"
38.5"
Number of stages
7
14
3525
2700
Speed of rotation, r.p.m.
0.55
Gauging
800
Best inlet angle
Preipheral speed, ft./sec.
400
0.55
800
455
2
Flow area at inlet, ft.
2.74
3.40
Flow area at exhaust, ft.2
3.73
9.15
Blade length, first stage
4.79"
4.04"
Blade length, last stage
6.5"
10.74"
Axial width of blades, first stage
1.66"
1.35"
Clearance, first stage
0.05"
0.04"
0.06"
Clearance, last stage
180,000
Average Reynolds number
0.11"
290,000
Stage efficiency
0.891
0.891
Reheat factor
1.006
1.021
Internal efficiency
0.896
0.91
17.55"
28.13"
Compressors:
Pitch diameter, constant
Airfoil N.A.C.A. No.
Number of stages
Speed of rotation, r.p.m.
11
L-
4409
4409
29
29
3525
5830
45 0
Stagger angle
450
Mach number
0.4
0.4
Opening pitch ratio of blades
0.6
0.6
536
Peripheral speed, ft./sec.
519
Flow area at inlet, ft.2
1.34
3.45
Flow area at exhaust, ft.2
0.69
1.77
Blade length, first stage
3.51"
5.09"
Blade length, last stage
1.8"
2.88"
1.17"
1.870
Blades chord, last stage
0.6"
0.96"
Clearance, first stage
0.035"
0.056"
Clearance, last stage
0.018"
0.029"
Blades chord, first
stage
250,000
Average Reynolds number
170,000
Stage efficiency
0.868
0.868
Reheat factor
1.023
1.023
Internal efficiency
0.85
0.85
Regenerator:
20,000 lbs.
Weight
5760
Number of tubes
Tube diameter
3/8"
Wall thickness of tubes
0.02"
15.0 ft.
Length of tubes
2
7500 ft.
Heat transfer surface
Shell diameter
4.81 ft.
Shell thickness
9/32"
10.4 Btu/(hr)(ft2 )(OF)
Overall transfer coefficient
12
.
i
sfiftl=
I-1-
-I' , - I
--..
19,250
Reynolds number
Turbulent flow
7960 h.p.
Power output, full load
Mass air rate
64.7 lbs./sec.
Overall thermal efficiency
37%
1I
- -jq
V. DISCUSSION OF RESULTS
The overall pressure ratio of 7 was selected in spite of the fact that a
better efficiency is obtained with an overall pressure ratio of 19, because
it gives less chances of compressor difficulties in their performance.
With
a high compression ratio leakage problems arise, and consequently good adjustment of the clearances is necessary, which is not a favorable perspective.
The design of the turbines is perhaps the easiest problem to solve, and
the author is quite satisfied with the characteristips obtained.
The results
are in a fairly good agreement with the few existing gas turbines, and are
consistent throughout.
If we compare the gas turbines to steam turbines on the basis of number
of stages for the same power output, we see that this design has about the
same number of stages to a comparable steam turbine.
In the design of the compressors, many difficulties were encountered.
The author estimates that the number of stages is too large for the work the
compressors have to perform.
The author was not able to match the speed of the high pressure com-
pressor with the speed of the low pressure turbine to which it is coupled;
but it was possible to do so in the case of the low pressure compressor and
the hivh pressure turbine.
The Reynolds number originally assumed for the investigation of the airfoil characteristics did not agree with the Reynolds number obtained for the
low pressure compressor, but it
pressure compressor.
consequency,
was close enough in the case of the high
In any case, the effect of such discrepancy is of little
since by observing the data on Reference (15), it
can be noticed
that the variation in lift and drag coefficients is very small, practically
negligible, for a wide variation of the Reynolds number in the relative
A
neighborhood of the value assumed originally.
The results obtained for the regenerator are fairly good, since it was
obtained a value of approximately one square foot of heat transfer surface
per horse power output.
It can be possible to go to a higher effectiveness
for the regenerator, with a consequent gain in efficiency, and only a small
increase in weight and volume of the regenerator.
The turbine characteristics were developed for both turbines, and the
procedure is indicated in the corresponding part in the Appendix.
In regards
to the compressors, the author only attempted an approximation to their characteristics, since an exact evaluation of the variables involved would require
more time than what was available.
The fuel load performance was obtained by placing together the entire
design; and the partial load characteristic was not derived but the method
to obtain it was given in Appendix D, Section D-3, Part II.
1ly
~
I
-"~
VI. CONCLUSIONS AND RECOMM1ENDATIONS
The author considers that he did not have the time to do a more thorough
investigation of all the problems involved in the design.
What requires especial attention is the theory and design procedure of
the compressors.
The airfoil N.A.C.A. No. 2409 promised good characteristics too, but
the author did not have the chance to study them, and decided to take the
airfoil N.A.C.A. No. 4409 for which more complete data was available.
The future of the gas turbine as a marine power plant is indeed brilliant, even though it
has somewhat bulky accessories such as the compressors
and the regenerator in particular.
1
~
I
Vil . -
APPENDIX A
7
SECTION A-1
Part I
Selection of Cycle
The selection of the cycle under which the gas turbine plant is
to operate is of the utmost importance.
It is obvious that unless the
gas turbine plant can compete in efficiency with other types of power
plants, its field of application will be restricted.
The combination of compressors, turbines, intercoolers, regenerators,
and other parts in a gas turbine plant permits a variety of arrangements.
The author has selected the cycle suggested by C. R. Soderberg in
reference (2), because, as he states, that cycle lends itself well for
part-load performance.
This fact is a very important one in connection
with naval ships, since in their lifetime the requirements for develop-
ment of full power are small, and most of the time the speed developed
is about one-third of the maximum speed, requiring approximately about
one-quarter of the full power.
The cycle consists of:
two compressors, two turbines, an inter-
cooler, a regenerator, and two combustion chambers.
The air enters the Low Pressure Compressor and after being compressed
is passed through an intercooler which reduces the temperature, to follow
to the High Pressure compressor, from which the compressed air passes
through a regenerator to receive some heat of the exhaust gases.
Next
it enters the first combustion chamber where the temperature is raised
by burning fuel at constant pressure, to enter then the High Pressure
turbine and deliver part of the work, after which it enters the second
combustion chamber where it is reheated before entering the Low Pressure
turbine.
From the L. P. turbine the gas passes through the regenerator
1w
where it delivers heat to the high pressure air, and
finally goes to
the exhaust.
For a clear understanding refer to Figs. I and II.
The low pressure compressor is connected directly to the H. P. turbine, and this turbine delivers power only to move the L. P. compressor.
The H. P. compressor is connected to the L. P. turbine which besides
delivering power to that compressor it also delivers the useful power to
the propeller shaft.
In this arrangement the H. P. turbine is really a gas generator
only since it does not deliver useful power to the propeller shaft.
AIM INTAKE
C
C
-
LIP Co.wREwssoiR
--
I.IrReCOLER
w.~.CoMvEnssow
--
I-SME-RATOR
COMB. CHAM3E~IR
-YiRST
T
C,
T_. H.P. TRb INQ
~--
SECON.D. COMM. CHAOIABER
T2-
- -
r
-.
TulkiBNE
- -- -~n
[--LA2'vvv2
Wtet
Fie I.
D/AC,RAMATIC
IMX e
AKRANG 1WcMENT
1860
OF
ELEMENT.,
OF
CYC(e-
f4L.
6as
)-
8s
to
Zss
48
enE-ro/l
FIG
A0..E1
R
I4Alk-
20
I
Y.
OF
CYC L r=
5
Part II
Cycle Conditions and-Specifications
The power delivered to the propeller shaft will be fixed at 7500
H. P.
It is yet necessary to know better all the requirements of the
design of a gas turbine plant, to go to higher powers.
The maximum temperature, at inlet of turbines will be 14000 F.,
which is a little high considering the present available materials that
can withstand the stresses at such a temperature.
Nevertheless the
author has in mind possible development of special alloys which promise
good heat and corrosion-resisting materials.
The internal efficiency of the compressors and turbines will be
assumed to be
=
0.85 for the preliminary study of the cycle and the
preliminary design of turbine and compressors.
The turbine efficiency
is here taken at a rather low value, since there have been cases of an
internal efficiency of 0.92.
Nevertheless, our assumption is on the con-
servative side anyway.
The regenerator effectiveness will be assumed to be:
I
R
=
0.65.
Greater effectiveness can be obtained, but results in a very big regenerator, which is not desirable in naval practice.
This effectiveness
will be defined later.
The intercooler discharge temperature will be considered constant
and at 900 F.
This is a conservative assumption, since all types of
ships have the ocean temperature available, which can safely be considered
to have an average temperature of the order of 700 F.
The reheat will be done to the initial tur'ine inlet temperature,
that is, 14000 F. (18600 Fa.)
F
In connection with the fact that the compressors are coupled
directly to the turbines, it will be assumed that the mechanical effec* 0.98.
iencies of the turbine and compressor parts of the coupling is
M
In the preliminary computations of the cycle parasitic losses will
be neglected.
The initial state of the air entering the low pressure compressor
will be assumed to be standard atmospheric conditions that is:
p
t
or
T
. 14.7 psia
=
70 0 F.
= 5200 Fa.
With these basic assumptions and specifications, we can proceed
to the calculations of the cycle for different conditions.
SECTION A-2
Part I
Cycle Calculations
As stated in Section A-1, the state of the air entering the L. P. compressor is:
Ti
a 5200 Fa.
p1
a 14.7 psia
z.28.77 Btu/lb.
=l
pri
= 2.504
All calculations will be done using the Air Tables, Reference (10).
For the notation refer to Figs. I and II.
The variables in the cycle are:
p/p1 a P5/P8
over-all pressure ratio
r
intercooler pressure ratio r,
/
p2/pl
The pressure ratios of the turbines, rB, as will be shown later is
tied to the intercooler pressure ratio, once the latter has been fixed the
former becomes also fixed, since the H. P. turbine supplies all its power
to the L. P. compressor, thus being related by the enthalpy drop in the L.P.
compressor and the H. P. turbine.
The values that shall be used for the overall pressure ratio will be:
r
- 4,5,6,7,8,10, and 15.
The values of the intercooler pressure ratio will be varied from 1 to
the value of the overall pressure ratio for the particular case in condaideration.
Sample computation for:
r
rT
.4
z
1.5
23
Intercooler pressure ratio:
r,
: p2 /p1 = p2 /
1
: pr2 s/pr
1.5
= p2 6/14.7 a pr2 s/2.504
P2
= P2s
(1)
Therefore:
: 22.05 psia
and:
pr2 s : 3.756
from the air tables:
1 2s
44.11 Btu/lb.
consequently on fig. II, the isentropic enthalpy raise on the L. P. compressor is:
i2s
1
44.11 - 28.77
(2)
15.34 Btu/lb.
and the state of the air at the exhaust of the L. P. compressor is then
given by:
i2
* 1
+ (12s ' i1 )1/-L
(3)
L. P. compressor's work:
WC 1=
2-
'il)l/
Ic
15.34 x 1/0.85
: 18.04 Btu/lb.
where:
c = compressorts internal efficiency
therefore:
12
: 28.77 + 15.34/0.85
= 46.81 Btu/lb.
For an intercooler discharge temperature of 900 F., fixed as condition in
Section A-i:
T3
:460+90
34
S5500 Fa.
then:
35.98 B,9tu,/lb.
-
13
pr3
= 3.047
The intercooler efficiency can be defined as:
I
i
i
i -
i
2
-
(4)
1
46.81 - 35.98
46.81 - 28.77
-
0.601
Now:
: 4 1
p3 /pg
p4 /p3
14_
Spr4
1.5
pr 3
(5)
therefore:
L__ x 3.047
4
1.5
pr.4s
-
and:
from air tables:
8.124
P
p4
= p5
p
-
1_4_
Y P3
_4_
15
- x P2
x22.05
p5 = 58.8 psia.
i4 s = 78.62 Btu/lb.
Work done on the compressor:
"C2
' 4s - i
c
= 78.62 - 35.98
0.85
= 50.2 Bt/lb.
Consequently:
i
4
S3 + C2
86.18 Btu/lb.
The maximum temperature of the air at inlet to turbines has been
(6)
fixed at 14000 F.
Therefore:
T5
460
z 1400
a 18600 Fa.
then:
15
.
pr 5
: 269.1
370.92 Btu/lb.
Since the H. P. turbine is attached to the L. P. compressor, the
work done by the former must match the work required by the latter, and
as stated on Section A-1, with our regard to mechanical efficiencies of
the connection.
If it is assumed that the mechanical efficiency of both the L. P. Compressor and H. P. turbine is:
:0.98
m
then the mechanical efficiency of the connection is:
T
M-c
= 0.98 x 0.98
= 0.96
The work required by the L. P. compressor is:
WCi
18.0 18.8 Btu/lb.
a 18.04 Btu/lb.
:
W i
then:
0.96
but:
W
therefore:
= 5 i6
6
v 370.92 - 18.8
u
352.12 Btu/lb.
The isentropic enthalpy drop across the H. P. turbine is then:
(8)
i5 -i6
where
t is the internal efficiency of the turbine.
i5 *s
a 18.8/0.85 a 22.11 Btu/lb.
26
therefore:
i6s
22.11
-
x
: 370.92 - 22.11
=
348.81 Btu/lb.
from the air tables:
pr6s
so that:
p6
: 225.41
= p6s = P5
x
pr6 /pr 5
= 58.8 x 225.41/269.1
=
49.3 psia
The reheat is done to a temperature of 14000 F. again, as said in
Section A-1.
T7
:1400
-
17
for which:
460
1860 0 Fa.
u 370.92 Btu/lb.
but, as the reheat us done at constant pressure:
P7
pr 7
P6
a 49.3 psia
269.1
The combustion chambers pressure ratio, that is the pressure ratio of
the H. P. turbine is:
r3
: pr7 /pr 8 s a p7 /p8 x P5a 8
p5/p7
(9)
a 4/58.8/49.3
=3.35
hence:
pr 8 , v 269.1/3.35
a 80.4
ig8
for which:
: 238.29 Btu/lb.
The isentropic enthalpy drop in the L. P. turbine is then:
17
-
is
= 370.92 - 238.29
27
(10)
- 132.63 Btu/lb.
and the turbine work is:
Wt2
( 7
-
(11)
8 s)
132.63 x 0.85
: 112.75 Btu/lb.
i8
17 -Wt2
- 370.92 - 112.75
: 258.17 Btu/lb.
for which:
1440.40 Fa.
T8
t8- 980.40 F.
It is desired to have a regenerator with an effectiveness of:
'R
0.65
which can be defined, according to fig. II as:
R
i1
=i8
0
i8 -i4
(11)
and can also be in the form:
R
using form (1la):
and therefore:
x
i8
i
(1la)
4
ig-86.18 z0.65(258.17 - 86.18)
19
= 197.98 Btu/lb.
T9
-
12080 Fa.
From figs. I and II, the heat input will be:
5 - '9) +*(7
qi :
- '6)
(12)
= 370.92 - 197.98 +370.92 - 352.12
-
191.74 Btu/lb.
The net work output of the cycle is:
t .ne
28
-
WC
(13)
Wnet
: (18.8+112.75)
-
(18.04+50.2)
= 63.31 Btu/lb.
and is the useful work to produce the required power.
The over-all efficiency of the cycle is then:
Wnet1
(14)
191.74
3
0.33
The specific air consumption for the cycle and pressure ratios indicated is, in pounds of air per horse-power hours:
= 2545
et
(14a)
a 2545
63.31
v = 40.2
lbs./h.p.-hr.
Making the computations as in the foregoing, for various over-all and
intercooler pressure ratios, a thorough investigation of the cycle can be
done in order to obtain the conditions under which the cycle is most efficient, and this what now follows in tabular form.
29
TABLE I
ClLCLE CALCULATION
r -p4/pl - p/8 = 4;
P4 = P5 : 58.8 psia.
1.5
2
2.5
3
14.7
14.7
14.7
14.7
r
p
psia
T
0 Fa.
'1
Btu/lb.
520
pr1
pr 2 s
12s
2s
Cl
12
3
28.77
520
28.77
520
28.77
520
28.77
2.504
2.504
2.504
2.504
3.756
5.008
6.26
7.512
Btu/lb.
44.11
56. n
66.10
74.77
id.
15.34
27.34
37.33
46.0
id.
18.04
32.17
id.
46.81
60.94
72.71
82.92
id.
35.98
35.98
35.98
35.98
43.94
54.15
pr3
3.047
3.047
3.047
3.047
11
0.601
0.778
0.837
0.867
p4 /p pr4 /pr 3
2.667
2.0
1.6
1.333
pr 4
8.124
6.084
4.876
4.062
i4
Wi
45
WC2
T5
w
i5
-siI
Btu/lb.
78.62
64.79
54.94
47.28
id.
42.64
28.81
18.96
11.30
id.
50.2
34.0
22.3
13.30
id.
86.18
69.98
5''.28
49.28
0 Fa.
1860
Btu/,lb. 370.92
269.1
t
Btu/lb.
18.8
1860
1860
1860
370.92
370.92
370.92
269.1
269.1
269.1
33.5
45.8
56.4
30
I
-~
________
TABLE I CONTID
6
6s
5
337.42
325.12
314.52
22.11
39.4
53.9
66.4
348.81
331.52
317.02
304.52
225.41
195.11
172.14
153.99
49.3
42.6
37.6
33.65
370.92
370.92
37(.92
269.1
269.1
269.1
id.
352.12
id.
id.
16
pr s
6
P6 - F7
psia
Btu/lb. 370.92
I
7
269.1
,or
r77
2.9
3.35
rB
2.280
92.8
105.0
117.6
Btu/lb. 238.29
251.87
264.15
275.68
106.77
95.24
80.4
pr8s
8s
2. r6
I -i
8s
7
id.
132.63
119.05
wt 2
id.
112.75
101.2
90.8
81.0
id.
258.17
269.72
280.12
289.92
0.65
0.65
0.65
0.65
Btu/lb. 171.99
199.74
221. 84
240.64
8
18
-
i4
I -I
4
9
9
I
5
-I
9
17 -'6
q1
2 i
C
net
TV~
w'
id.
111.8
129.8
144.0
156.4
id.
197.98
199.78
202.28
205.68
id.
172.94
171.14
168.64
165.24
id.
18.8 0
33.50
45.80
r-6.40
id.
191.74
204.64
214.44
221.64
id.
131.55
134.70
136.60
137.40
id.
68.24
66.17
66.24
67.45
id.,
63.31
68.53
70.36
69.95
0.33
lb/IP-hr 40.2
0.335
37.16
31
0.328
36.2
0. 3154
36.4
I'm
-=44
TABLE II
CYCLE CALCULATION
p4/p1
r
p5/P8
=
5;
p:
p5 : 73.15 psia.
3
2.25
r I.5
14.7
p1
T
520
28.77
1
1
pr
pr2s
i
14.7
14.7
520
520
28.77
28.77
3.75
14.7
520
28.77
2.504
2.504
2.504
2.504
3.756
5.64
7.52
9.39
44.11
61.34
74.77
85.99
15.34
32.57
46.00
57.22
38.30
54.15
67.37
2s
i
- i1
2s
1
ial18.04
12
46.81
67.07
82.92
96.14
i
35.98
35.98
35.98
35.98
3
pr
3.047
3.047
3.047
3.047
0.601
0.812
0.867
0.893
2.23
1.667
1.332
6.765
5.08
4.061
p4 /p3=pr4 /pr 3 3.333
pr4s
10.150
14
90.07
69.74
56.73
47.27
54.09
33.76
20.75
11.29
63.64
39.72
24.40
13.28
99.62
75.70
60.38
48.26
i
4s
C2
4
T5
- 1.
3
1860
1860
1860
1860
370.92
370.92
370.92
370.92
pr 5
26.91
26.91
26.91
26.91
Wtl
18.8
39.9
56.4
70.18
16
352.12
331.02
314.52
300.74
1
5
32
TABLE II CONT'D
-
I
5
i
6s
16s
pr6s
7
p6
17
pr
22.11
47.0
66.4
82.5
348.81
323.92
304.52
288.42
225.41
182.79
153.99
132.79
49.9
42.0
36.23
370.92
370.92
370.92
370.92
269.1
269.1
269.1
269.1
61.55
7
4.19
rB
pr 8s
8s
7
i
8s
wt 2
~8
LR
2.86
3.4
2.47
64.2
79.25
94.1
109.0
217.67
23.94
253.31
267.92
153.25
133.98
117.61
103.0
130.3
113.9
100.0
240.62
257.02
270.92
283.35
0.65
0.65
0.65
o.65
87.57
I
141.0
181.32
210.54
235.09
i
91.7
117.8
136.8
152.9
191.32
193.5
197.18
201.16
S-
9
9
S-
i
179.6
177.42
173.74
169.76
i
6
18.8
39.9
56.4
70.18
7
q1
198.4
217.32
230.14
239.94
E wt
149.1
153.8
156.4
157.75
5
C
Wt
81.68
78,02
78.55
80.65
67.42
75.78
77.85
77.1
0.34
37.74
0.3485
0.338
32.7
33.6
33
0.321
33.0
TABLE III
CYCLE CALCULATION
r - p4 /p1
p5/p8 = 6;
ry
1.5
p1
14.7
T
520
28.77
i
p4
=p
88.2 psia
2.45
14.7
3.5
4.5
14.7
14.7
520
520
28.77
28.77
pr
2.504
2.504
2.504
pr 2 s
3.756
6.14
8.764
520
28.77
2.504
11.26
44.11
65.20
82.45
95.65
15.34
36.43
53.68
66.88
"Cl
18.04
42.90
63.18
78.70
12
46.81
71.67
91.95
107.47
1
35.98
35.98
35.98
35.98
i2s
S2s
1
i
3
pr3
p4/p3 =pr
pr
/pr
3.047
3.047
3.047
3.047
0.601
0.832
0.886
0.909
4.0
2.45
1.713
1.333
12.188
7.46
5.22
4.06
4s
1
100.03
74.49
57.91
47.25
4s
1 4s -i
64.05
38.51
21.93
11.27
%C2
75.35
45.31
25.80
13.25
111.33
81.29
61.78
49.23
ix
1860
T
1860
1860
1860
5
15
370.92
370.92
370.92
370.92
pr 5
269.1
269.1
269.1
269.1
44.7
65.8
82.00
352.12
326.82
305.12
288.92
22.11
52.6
77.4
96.5
vi t18.8
16
15 i
6-
34
TABLE III CONTtD
S6s
348.81
318.32
293.52
274.42
pr 6 s
225.41
174.11
139.25
116.17
73.8
57.03
45.61
38.08
370.92
370.92
370.92
370.92
269.1
269.1
269.1
269.1
P6
7
p7
pr
7
rB5.02
B
pr
3.88
3.10
2.59
53.65
69.40
86.90
103.88
202.41
224.80
245.64
263.08
168.51
146.12
125.28
107.84
't2
143.2
124.2
106.5
91.6
1
8
227.72
246.72
264.42
279.32
R
.65
0.65
0.65
0.65
116.39
165.43
202.64
230.09
75.8
107.5
131.6
149.5
187.13
188.79
193.38
198.73
183.79
1P2.13
177.54
172.19
18.80
44.70
65.80
82.00
202.59
226.83
243.34
254.19
162.00
168.90
172.30
173.60
7 WC
93.39
88.21
88.98
91.95
Wt
68.61
80.69
8.?2
81.65
S8s
i
7
-
I
8
-
i
9
1
9
I
-
Ss
4
i
4
-
5
I
i
1
9
-
7
q1
net
6
6
0.3385
37.1
0.3558
31.54
0.3420
30.55
36
0.3214
31.17
TABLE IV
CYCLE CALCULATION
/P
r -p
4 1
r
p/ P :7
5 8
1.5
14.7
p1
T
520
1
28.77
p:
4
55:102.9
2.65
14.7
520
psia.
3.5
5
14.7
14.7
520
28.77
28.77
520
28.77
pr1
2.504
2.504
2.504
2.504
pr2s
3.756
6.64
8.768
12.510
101.50
44.11
68.85
82.4.7
15.34
40.08
53.70
72.73
18.04
47.16
63.20
85.60
12
46.81
75.93
91.97
114.37
13
35.98
35.98
35.98
35.98
'2s
I
-
2s
W
i
1
1
pr3
3.047
3.047
3.047
3.047
0.601
0.848
0.886
0.916
p4 ,/p3 :pr4 /pr3 4.667
2.64
2.000
1.400
pr 4 s
14.21
8.05
6.094
4.265
108.75
78.16
64.87
49.27
72.77
42.18
28.89
13.29
WC 2
85.60
49.62
34.00
15.63
i4
121. 58
85.60
69.98
51.61
i4s
i
4s
- i
3
1860
T
1860
1860
1860
5
15
370.92
370.92
370.92
370.92
pr 5
269.1
269.1
269.1
269.1
wl
18.8
49.15
6r,.()
16
352.12
321.77
305.02
281.72
22.11
57.85
77.50
104.90
i
5
- i
6s
36
89.20
TABLE IV CONT'D
16s
348.81
313.07
293.42
266.02
pr6s
225.41
166.23
139.12
106.97
86.2
63.60
53.2
40.9
C7
370.92
370.92
370.92
370.92
pr,
269.1
269.1
269.1
269.1
p7
p6
5.86
rB
pr
.8s
4.322
3.62
2.780
62.28
74.4
96.90
215.18
231.11
256.18
181.26
155.74
139.81
114.74
154.0
132.3
118.8
97.60
216.92
238.62
252.12
273.32
0.65
0.65
0.65
0.65
- i
95.34
153.02
182.14
221.71
i9 - i
62.00
99.50
118.40
144.0
183.58
185.10
188.38
195.61
187.34
185.82
182.54
5
175.31
18.80
49.15
65.90
89.20
206.14
234.97
248.44
264.51
172.80
181.45
184.70
186.80
103.64
96.78
97.20
101.23
69.16
84.67
87.50
85.57
45.94
Ss189.66
17 - is
t2
i
'ZR
i
i
9
i
-
9
9
I - 1
5
7
q
z
6
net
0.3350
36.8
0.3603
0.3521
29.08
30.08
37
0.3231
29.73
TABLE V
CYCLE CALCULATION
p4 /p
r
ry
p5 /p8
1.5
14.7
p1
520
T1
28.77
i1
pr
8;
p
-
p-
117.6 psia.
2.83
14.7
4
5
14.7
14.7
520
520
28.77
28.77
28.77
2.504
2.504
2.504
2.504
3.756
7.095
10.016
12.520
1
pr
520
2s
1
2s
1
2s
- i
1
44.11
72.01
89.37
101.54
15.34
43.24
60.60
73.77
50.90
71.30
86.78
79.67
100.07
115.55
35.98
35.98
35.98
":' 118.04
46.81
'2
S35.98
3.047
3.047
3.647
3.047
0.601
0.858
0.899
0.917
p4 /p3=pr 4 s/pr3 5.333
2.830
2.00
1.600
6.094
4.875
pr
pr
16.24
8.62
i
116.73
81.60
64.87
54.93
80.75
45.62
28.89
18.95
wC2
95.00
53.70
34.00
22.30
14
130.98
89.68
69.98
58.28
4s
- i
1
4s
T
3
1860
5
i
pr
1860
1860
370.92
370.92
370.92
370.92
269.1
269.1
269.1
269.1
5
53.00
74.30
90.30
352.12
317.92
296.62
280.62
22.11
62.38
87.42
106.10
Tti18.80
'6
1
5
-i
6s
1860
38
TABLE V CQNT'D
i
6s
pr6s
348.81
308.54
283.50
264.82
225.41
159.64
126.75
105.71
P6 - P7
6
7
17
98.60
69.80
55.40
46.20
370.92
370.92
370.92
370.92
pr
269.10
269.1
269.1
269.1
7
6.70
4.74
3.76
3.14
40.18
56.78
71.55
85.15
179.05
207.22
227.56
243.70
191.87
163.70
143. 36
127.22
163.00
139.20
121.80
108.10
207.92
231.72
249.12
262.82
84
0.65
0.65
0.65
0.65
is -1i
76.94
142.04
179.14
204.54
50.00
92.40
116.40
132.90
180.98
182.08
186.38
191.18
189.94
108.34
184.54
179.74
18.80
53.00
74.30
90.30
208.74
241.84
258.84
270.04
181.80
192.20
196.10
198.40
113.04
104.60
105.30
10 .08
68.76
87.60
90.80
89.32
rB
pr8
1 8s
8s
7
t2
1
9
-
4
4
19
i1 -i
75
Tinet
ti
69
0. 3292
37.04
0.3624
29.05
0. 3503
28.03
39
0.3300
28.50
TABLE VI
CYCLE CALCULATION
r =p/p1
p 5/P 8
r
1.5
p1
4.7
Ti
=
520
11
28.77
10;
p
=
=
147.00 psia.
3.16
5
7
14.7
14.7
p5
14.7
520
520
28.77
28.77
520
28.77
pr1
2.504
2.504
2.504
2.504
pr2 ,
12s
3.756
7.924
12.520
17.530
44.11
77.38
101.54
122.15
15.34
48.61
72.77
93.38
WCl
18.04
57.24
85.58
109.80
'2
46.81
85.01
114.35
138.57
13
35.98
35.98
35.98
35.98
i
-
1
pr3
3.047
3.047
3.047
3.047
9
0.601
0.857
0.915
0.935
p4 /p3 pr4 s/pr 3 6.667
3.160
2.00
1.428
9.638
6.094
4.35
Pr4
20.32
5
130.73
87.22
64.87
50.16
94.75
51.24
28.89
14.18
WC2
111.40
60.30
33.97
16.68
14
147.38
96.28
69.95
52.66
'4s
145
T
5
15
pr 5
-
13
1860
1860
1860
1860
370.92
370.92
370.92
370.92
269.1
269.1
269.1
269.1
w
18.8
59.66
89.10
114.30
16
352.12
311.26
281.82
256.62
40
T
TABLE VI CONT'D
22.11
70.20
104.75
134.40
i6
348.81
300.72
266.17
236.52
pr 6
225.41
148.79
107.77
78.82
123.1
81.2
58.9
43.05
370.92
370.92
370.92
370.92
269.1
269.1
269.1
269.1
15
1 6s
=
P6
P7
17
pr
7
2.93
8.38
5.52
32.12
48.74
67.22
91.90
162.17
194.46
221.93
251.02
208.75
176.46
148.99
119.90
Wt2
177-40
149.95
126.60
1C 1. 82
18
193.52
220.97
244.32
269.10
nR
0.65
0.65
0.65
0.65
14
46.14
124.69
174.37
216.44
19 - 14
30.00
81.00
113.30
140.60
177.38
177.28
183.25
193.26
193.54
193.64
187.67
177.66
18.80
59.66
89.10
114.30
q,
212.34
253.30
276.77
291.96
z
196.20
209.61
215.70
216.12
SC
129.44
117.54
119.55
126.48
Tnet
66.76
92.07
96.15
89.64
rB
pr8s
i 8s
17
i8s
-
18 -
19
15
15
17
199
-6
0.3140
r
38.14
0.3633
27.64
4.005
0.3472
26.47
41
0.3068
28.38
-.
-~
____--
___
TABLE VII
CYCLE CALCULATION
p4/p
r
P5/P8 = 15;
r
1.5
p
14.7
520
T1
28.77
1
p4 = P5
=
220.5 psia.
3.874
5
14.7
14.7
520
520
28.77
28.77
pr
2.504
2.504
pr2s
3.756
9.700
2.504
10
14.7
520
28.77
2.504
12.52
25.04
44.11
87.68
101.54
144.60
15.34
58.91
72.77
115.83
WCl
18.04
69.32
85.60
136.20
1
2
46.81
98.09
114.37
164.97
13
35.98
35.98
35.98
35.98
S2s
2
-
2s
1
1
pr
3.047
3.047
3.047
3.047
0.601
0.896
0.915
0.947
3.874
3.00
1.500
9.141
4.570
p 4 /p3 pr4 /pr3lO.00
30.47
11.80
158.34
98.25
84.59
52.17
122.36
62.27
48.61
16.19
"C2
143.80
73.25
57.21
19.04
14
179.78
109.23
93.19
55.02
pr4 ,
i
i
4s
T5
- 1
3
1860
1860
1860
1860
15
370.92
370.92
370.92
370.92
pr5
269.1
269.1
269.1
269.1
VIt1
18.80
72.21
89.20
141.85
16
352.12
298.71
281.72
229.07
42
TABLE VII CONT'D
15
6s
85.00
104.95
166.90
348.81
285.92
265.97
204.02
225.41
129.69
106.92
54.69
184.8
106.25
87.60
44.81
370.92
370.92
370.92
370.92
269.1
269.1
269.1
269.1
,
16s
pr 6
22.11
P7
p6
17
pr7
rB
12.57
7.23
5.96
pr8
21.41
37.23
45.17
88.40
S8s
134.11
173.19
188.30
247.26
236.81
197.73
182.62
123.66
wt 2
201.3
168.00
155.20
105.10
18
169;62
202.92
215.72
265.82
0.65
0.65
0.65
0.65
-10.16
93.69
122.53
210.80
60.88
79.62
137.00
-
17
i8s
SR
14
-
19
3.047
4
19
to
.O
170.11
172.81
192.02
o 2
m
200.81
198.11
178.90
18.80
72.21
89.20
141.85
209.94
273.02
287.31
320.75
220.10
240.21
244.40
246.95
161.84
142.57
142.81
155.24
58.26
97.64
101.59
91.71
0 0 -H
-
15
1
7
q1
E wt
SC
w
net
i
166
0.2775
43.7
0.3575
26.06
0.3539
25.03
43
0.2693
27.72
0:56
0.15
~
w
/4-7 /t-i4
O~fl~i~
Xor7R'Oile~
-PRSSR
-
RT/
J~rr
0.45
Q YC~E
-$~~,RCON$~~
OV0R t.L
EsVR
e
'Ario
670
0.5
P~~orr~
Fb~t
P~w&#,.?
I
RC9Q4E1 9
E$R
i'iir
0.4
1)
U
'Ii
4
0.2
0
74
0.1
0
I.
-
1
to
O-VERis
RE~SSURE
-It
7
?ATio
-,
tic
-
V
N P~.It
IC
10
.
U
$
;
c
oQfpoI
45
40
ir=4
[
35
X
30
0r
26
to
P7,- v07*
/AI4~T COA9ZrT'0N
is
to
0
0
I
3
7
S2
/IvTE6,fOO4.g
46
a5
?*e-7v~eE -*'4TO -
9
r
/0
/1o:
SECTION A-3
Part I
Specific Air consumption
From the calculations on Section 2 it can be seen that the best efficiency
is obtained with an overall compression ratio p
/p-
10.
Nevertheless, this
is a rather high value for the compressors to work at, especially axial flow
Consequently a lower value must be chosen.
type.
r
=
It can be noticed that for
7, the cycle efficiency is only slightly less; therefore this will be the
value for which the design will be developed.
It can also be seen that the best efficiency in the cycle occurs with an
intercooler pressure ratio of r 1 :
/~T: 2.65.
The specific air consumption can now be found by:
3600 G(Wnet - Leaving Loss) - PF7.
hp
(15)
2545
Where:
specific air consumption in lbs./sec.
G
power loss, due to friction and windage, which shall be
WF
taken as 2% of the net shaft horsepower required.
It is
to be noted that this consideration will be considered valid
For partial loads this loss will vary
only at full power.
roughly with the square of the r.p.m.
From the computations for the cycle in Section 2:
2.65
-
p4/pl - 7 and r = p2/p1
wnet - 84.67 Btu/lb.
The leaving loss shall be considered, as a preliminary assumption, as equal
to 1.2% of the L. P. turbine enthalpy drop.
Leaving loss
hL : 0.012 x 84.67 . 1.02 Btu/lb.
From the statement of the problem, the power required is
P w 7500 h.p.
47
Therefore:
PFW
0.02 x 7500
* 150 h.p.
And:
7500
-
G
3600 G (84.67 - 1.02) - 150
2545
64.7 lbs. air/sec.
48
APPENDIX B
49
SECTION B-l
Preliminary Characteristics of L. P. Turbine
The first step is to determine the preliminary turbine mean diameter at
the exhaust.
Some limitations are necessary though; limitations which will be explained
as they are given.
(a) Blades tip speed limit of 500 ft./sec. for the hottest stage.
Since it is a well known fact that the strength of alloys used
in turbine construction decreases rapidly with increase in temperature, a limit must be set for the maximum stress that a
blade can be subjected to, in order as to assure working of the
turbine without mishaps.
This is one of the most important
topics in gas turbine design.
If alloys can be obtained that
show a good strength at high temperatures, the gas turbine can
be put in an advantageous position over other kinds of prime
movers.
(b) No appreciable inlet velocity to low pressure compressor.
(c) The ratio of blade length to pitch diameter will be 1/d = 0.28.
It can be shown that for a constant circulation stage, to obtain
a zero pressure change across the base of the rotor blade, the
ratio 1/d should not exceed 0.3; this is absolutely true for
isentropic conditions throughout, Pnd for k
=
1.4 (ratio of
specific heats).
(d) Axial flow at discharge of turbine, to be assumed.
(e) Pitch diameter shall be taken as constant throughout the turbine.
50
Part I
Pitch Diameter
The pitch diameter can be determined as a function of the leaving area,
or exhaust area, and from an estimate of the leaving loss.
The leaving loss can be expressed as:
AhL
Where:
C2
G
C eGv
2
2g
A2e
e)
1-;1
(16)
2gJ
axial component of flow at exhaust
= specific air consumption
specific volume of gas at the exhaust, which can be
v2e
determined from the computations of the cycle on section A-2, Appendix A, for the pressure ratio selected.
Consequently:
v 2
2e
RT
3
e
=
53.35 x 1360.5
P- 14.7 x 144
(17)
34.26 ft.3/lb.
Therefore:
Exhaust area:
Ae
Gv2 e
V2gJ hL
(16)
64.7 x 34.26
V64.4 x 778 x 1.02
(AhL
1.02, see section
A-3 Part I)
* 9.05 ft. 2
From fig. VI:
Ae
dl
-i
d
-td 2(l)
d
144
=
_14FEx A
w
x (d)
1
51
(18)
d
= /'jx
9.05 x(_
w
=
.
Substituting the corresponding values:
0.28
38.5 inches
F ic
tan.stage
nnVec's
i
52
-
I
Part II
Stage Characteristics
Before going into the determination of the number of stages required, it
is necessary to find some characteristics of typical reaction stages, based on
dimensionless factors, in order not to commit ourselves to particular dimensions.
A brief exposition of the derivation of the most important formulas to be
used in connection with the determination of the stage characteristics follows.
To simplify calculation and derication of the formulas a symmetrical stage
will be considered.
By defining stage efficiency as:
= Work output from stage
Work input to stage
(19)
and the following concepts, which can be easily understood by observing fig. IX
and X:
1
Stator velocity ratio
u
u
c1
1(20)
u.
where ul is the peripheral velocity
e
Stator diagram ratio:
1 (2cos
:
(21)
1
Similarly for the rotor:
S-
2
2 =
and
:u
U2
v2
(22)
'"2
2(2cos
2
(23)
2
It is necessary to define also a factor
reaction (
S, which depends on the degree of
) of the blade, and is equal to:
s5 -
- (
- f ) E 1(24)
The necessity of this factor
can be readily seen from fig. VII, where
2:
l -(1
2
2
s.C1
53
-
)2
CO
is a value that indicates the
degree of reaction of the blade.
-
ar,2
0 indicates impulse type of
blade
1 is a normal reaction blade
X
FcI,F 4
2 is an abnormal condition.
1,i.
Furthermore, two more concepts
From fig. VIII
shall be defined.
1.
the isentropic enthalpy drop across
.2
~
one stage,
is assumed to
&h,
~13
have a corresponding theoretical
velocity, which would be the velocity that a gas would develop if
allowed to drop in enrthalpy the
Fig. VIII
Ah shown.
amount
Correlatively, a new velocit y ratio can be defined for the stage, and it is called:
theoretical velocity ratio o C the stage.
=:u
(26)
cth
Where, as stated above:
2
cth = J-4h
2g
(27)
The theoretical velocity ratio can be further developed by studying fig. IX,
from which it can be easily derived that:
J-h
=
64
1~ ~9)c
2
2+(6 24
1l 42~ -f 2)
2g
=2[6 +
2
2g
54
(28)
jk
~
0
/-
tk
: C2
Cth
Then: th
u
Cth
And substituting eq. (20)
th
-
--
1
(E
+__
2 * 52)
(29)
'S
Fig. IX
and
g2
There:
are loss coefficients depending on:
(a) blade inlet angle
(b) blade aspect ratio
(c) Reynolds number
(d) lift coefficient
(e) angle of turn of the blade
(f) leakage
(g) friction and windage
Subscripts 1 and 2 refer to stator and rotor blades, respectively.
These loss coefficients can then be supposed to be formed of:
-
and
1: Sol'
2 ~
(30)
Ll " Wl
p2+SL2 + W2
(30)
pl and .p2 include effects from (a) to (e)
Ll is the leakage loss (f)
W, is the friction and windage loss.
These coefficients will be further developed or shown how they are obtained,
when an example of the computations for one set of conditions is given later.
55
CCo
4r
FiA.g X
Going back to the definition of stage efficiency eq. (19), and making use of
fig. IX and X, it can be shown that:
Stage output:
17
=
0
2
61C2 + E2 Or2
(31)
2
2g
Input to stage:
2
2
2
2
+ ( E
2g)c
(32)
2g
Therefore:
2
12 2
1)c
E'
(33)
2
-
1
+
( 22 +
2)vv2
2
Introducing eq. (25), and simplifying:
(34)
El
1+
d (P2 - 29
In the determination of the turbine stage characteristics the following conditions must be satisfied:
(1) For symmetrical stages, the blade foils have the following properties, determined experimentally, and given in Reference [11:
gauging
sin O 1
= sin /2
corresponding best inlet angle
0.25
700
0.30
750
0.40
0.50
800
800
56
(2) It has been found that for good performance, and for strength
purposes the blades should have approximately the following construction characteristics, which in this design will be assumed
they are so:
1 =
Aspect ratio
b
Clearance ratio
(3)
3
Z 0.01
Shroud ratio
b
Length ratio
1
d
d
= 0.1
= 0.28
The diagram ratio, El, in eq. (24), to be taken at the blade velocity ratio corresponding to t'e best inlet angle
It was found that, eq. (21):
E1
=
where
l (2 cos
1
-Al)
(21)
u
1
1
Plotting equation (21) a parabola is obtained, such as shown in
Fig. XI.
The blade velocity ratio for
the best inlet angle can be
found by taking the derivative
of eq. (21) and equating to
zero.
del
- 0 = 2(cosot 1 -
1)
Therefore, the optimum vel-
ocity ratio is:
Fig. XI
V.
Ort.
V
lopt. = cos
57
1
for which:
: cos
<k 1
(2 cos
1
- cosd
)
1
Scos2 41
(35)
(4) Use will be made of the basic loss coefficients of Figs. XII, XIII,
and XIV, which have been determined experimentally, and are taken
from Professor C. R. Soderberg's class notes, 2.211, [1.
(5) The value of
will be taken for the following degrees of reaction:
0.5
1.0
(normal reaction)
f : 1.5
(6) The friction and windage loss for the stage is given by the empirical relation, [1]:
P,
3
5
n ) , (_) 1)
CW(1000
Cw
0.042 [2(1- )
12
(36)
in h.p.
v
where:
14
1.25
b
.
14
(1 + d)
n
speed of rotation in r.p.m.
d
mean diameter of stage in inches
1
blade height in inches
(37)
bs Ishroud width in inches
v
specific volume in ft. 3 /lb.
The friction and windage loss for the stage, can also be expressed
as:
x550
G
(
P
2g
Where air flow:
G
2 A2w
v
=
wdlw 2 sin/ 2
144v
58
(39)
In equation (39)
the leakage area has been neglected.
Substituting eqs. (22), (39) and:
n = 12 x 60 x u
in eq. (38)
",d
and combining and simplifying e3 s. (38) and (36)
14
b
14
v2
d)
s(l+
1.25
+
d)
[2(1
12
w = 0.00033
I
d
1s in,32
d
Since in the sample calculation and tables that follows, the
computation of the stage characteristics is done separately for
the rotor and the stator blades, it will be assumed that this loss
coefficient given by (40) divides equally for each blade.
A sample computation for one set of conditions is given now, in
order to make clear the procedure followed in the tables.
Take:
sina,
=sin
=
2
0.25
best inlet angle
700
f
0.5
Then:
cosA
Cos
zcos (32
1
2
cos 2
2
l
O
0.968
0.937
14.5
2
'ROTOR
Fw
le
59
(40)
For optimum velocity ratio:
6:-cos2c
-
0.937
Therefore:
P
1
-
(1
1
-
(1 - 0.5)0.937
-
)
a:
= 0.5325
The purpose is to find the stage efficiency as a function of the
stage theoretical velocity ratio, which is done by assuming different values of blade velocity ratios; for example:
Take:
0.4
Then:
0.4(2 x 0.968 - 0.4)
41
= 0.6144
Z 2 =7E2
: 0.4
. 0.548
V.5325
2(2 cos (2
-
2)
-
0.548(2 x 0.968 - 0.548)
=
0.761
From the geometry of fig. XV:
tan
sin d,
-
cosoe
:
-
0.25
0.968
0.44
/3,
:23.750
.5-
60
-
0.4
Eq. (21)
best inlet angle
-
-l
700
Therefore, angle of turn:
I
8TATrOR
go,1 -180
- (ol+ 13)
180 - (14.5+ 70)
95.50
(
~
For which, from fig. XII
S ol = 0.093
Fig. XVI
This value ofJol is constant for all values of l, other conditions
remaining the same.
Correction for aspect ratio 1
* 3; from fig. XIII
b
correction
* 1.0
Correction for inlet angle:
Inlet angle to stator
IS- /30
2 or
S2
in:
/0
* 30.75 - 70
* -39.250
For which, from fig. XIV:
Correction:
1 +&P
1 -+ol
= 1.091
1 + pl
= 1.091(1+0.093)
- 0.192
Sp
The leakage losses can be obtained by the following relation,
which can be easily derived:
L1
61
- 1/d) ( J/1
(
singj - (1 - 1/d) (J/1)
coS CQ$Pj/,qIT
of B4.4ve's
0.08
Q-02
#:A
2o4
olo
6
NoIL
4Nr~
r7up
6
-
(1 - 0.28) 0.01
0.25 - (1 - 0.28)0.01
-
0.028
The friction and winda e loss for the stare is
0.00033
2
T1/d) sin#2
1.25 ds (1+1/d)
2(l - 1/d)
0.00033 -(0,/)
0.28 x 0.25
d
2(0.72) 41.25 x 0.1(1.28)4
0.001026
(41)
sin,2
0.0007
And as stated previously:
Swl x 0.0007 = 0.00035
2
The total loss coefficient is then, for the stator blade:
51=p+- SLl -'-
wl
0.192+ 0.028 +0.0003
0.2203
Consider now the rotor blades:
In a similar manner it can be found:
, = 0.4
. 0.6144
ROTOR
.
*
2
0.548
2
0.761
23.750
(61
2
=230.75
Angle of turn:
02
=
180 -
({2+ 0)
180 - (14.5
Fig. XVII
63
70)
A$PEC -R4TQ
CO
R6CT
ON
t.05
1$
e+J~
II.
I.o~
~.0
I..,
I
4
3
4
ASPQr lATIO
HA.M -/94W6
64
-
________--
95.5
And from fig. XII, then:
0.093
02
Correction for 1/6
=
3 is:1.0
Correction for inlet angle:
/o
I3-
:/31
-A 0
23.75 - 70
= -46.25
0
Correction:
1
+
p2
u 1.22
02
0.334
5 p2
Leakage loss coefficient:
t
L2
d)(iiD
(1+028)0.01
0.25 - (1+ 0.2 8)0.01
0.0487
Friction and windage loss coefficient:
Sw2
0.00035
Therefore, the total loss coefficient for the rotor blade is:
S2
3
p2
L2 " i w2
0.334+ 0.0487+ 0.0003
0.3830
The stage theoretical velocity ratio is then:
4th =
(29)
)
1vf 1*i 2 -r5
65
I
cooecrjo
Poo
I/Ler
A
4
'.31I
0,0
I.+f
It!
it F~
/.o
".0i
-40
~200
4
66
0*6
0.4
1 /.6144+
0.2203 + 0.5325(0.761f 0.323)
0.4
/1.4437
-
0.3325
The stage efficiency:
7st
:-1
2
'El+Sl +
_
(
2 +
2
- o.6144+0.5325 x 0.761
1.4437
- 0.707
Use will be made later, of the following functions, thus there
computation is shown now:
Circulation function:
1
-
1.0
= 0.832
i1.4437
Leaving loss function:
1
(1 - E2e)
-2
1 - 0.761
0.239
A tabulation follows for each value of sin ,, and for each value
.
of the degree of reaction P
67
TABLE VIII
TURBINE STAGE CHARACTERISTICS
sinc
cos
1
: sin/32
0.25
* cos/ 2
0.968
=2
?= 0.5
E1 opt
14.5
t
6
a1
Ott
01
se
=
0.5325
0.8
1.0
0.3
0.4
0 .4908
0.6144
0.8016
0.9088
0.9360
0.4113
0.548
0.822
1.095
1.370
0.627
0.761
0.9155
0.9210
0.755
0.3743
0.44
o . 680
1.488
-7.810
0.4r;75
0.595
1.712
-1.968
-0.622
0.6
20.52
23.75
34.2
56.1
97.3
24.58
30.75
59.7
116.9
148.1
84.5
84.5
84.5
84.5
84.5
95.5
95.5
95.5
95.5
95.5
0.093
0.093
0.0 q 3
0.093
0.093
1.0
1.0
1.0
1.0
1.0
-39.25
-10.3
46.9
78.1
-45.42
.4'
0.937
= 0.73
#0 - 70*
8.
=
1. -174
1.091
1.003
1.054
1.219
1.283
1.192
1.096
1.151
1.332
0.283
0.192
0.096
0.151
0.332
0.028
0.028
0.028
0.028
0.028
0.0001
0.0003
0.0011
0.0027
0.0052
0.3111
0.2203
0.1251
0.1817
0.3652
95.5
0.093
95.5
0.093
95.5
0.093
68
95.5
0.093
95.5
0.093
TABLE VIII CONT'D
1.0
1.0
-49.48
-46.25
ZA" 1
P-A
1.0
1.0
1.0
-35.8
-13.9
27.3
1.308
1.220
1.066
1.007
1.018
ASP2,
1.430
1.334
1.165
1.100
1.113
S%
0.430
0.334
0.1650
0.1000
0.1130
3L2
0.0487
0.0487
0.0487
0.0487
0.0487
vqaz
0.0001
0.0003
0.0011
0.0(27
0.0052
0.4788
0.3830
0.21-48
0.1514
0.1669
1.3909
1.4437
1.5287
1.6615
1.8032
0.8244
1.0194
1.2890
1.3991
1.3488
0.2542
0.3325
0.4855
0.620
0.744
0.707
0.844
0.842
0.748
0.848
0.832
0.809
0.775
0.744
0.373
0.239
0.085
0.079
0.225
0.4113
0.548
0.822
1.095
1.370
1*.
sz
}
,
LL e 0.593
6.0
TABLE IX
TURBINE STAGE CHARACTERISTICS
sin 32 = 0.3
cosc,41
E
A
.aI
1
0.5i
=1750; 3
0t= 0.91.,
0.9539;)32
s : 0.545
Js= 0.7385
0.3
0.5
0.7
0.8
0.9
1.1
0.482
0.724
0.845
0.886
0.907
0.889
0.407
0.677
0.948
1.093
1.218
1.489
0.610
0.833
0.910
0.890
0.840
0.624
0.459
0.671
1.181
1.948
5.560
-2.053
0.549
1.094
50.9
-2.156
-1.135
-0.561
24.7
33.9
49.8
62.8
79.8
116.0
28.8
47.6
88.9
114.9
131.4
150.7
92.5
92.5
92.5
92.5
87.5
87.5
87.5
87.5
92.5
+.
cos 2
;
87.5
92.5
87.5
0.085
0.085
0.085
0.085
0.085
0.085
1.0
1.0
1.0
1.0
1.0
1.0
-46.2
-27.4
13.9
39.9
56.4
75.7
j-( P3Q
1.2
1.032
1.005
1.038
1.083
1.193
1.302
1.120
1.090
1.126
1.175
1.294
er
0.302
0.120
0 .0)0
0.126
0.175
0.294
Li
0.0234
0.0234
0.0234
0.0234
0.0234
0 .0234
0.0001
0.0005
0.0015
0.0022
0.0031
0.0053
0.3255
0.1439
0.1149
0.1516
0.2015
0.3227
S4.
87.5
aO2,
2.
aI
87.5
F7. 5
87.5
87.5
87.5
0.085
0.085
0.085
0.085
0.085
0.085
1.0
1.0
1.0
1.0
1.0
1.0
70
TABLE IX CONT'D
-50.3
-12.2
-25.2
-41.1
4.8
41.0
}z 1.360
1.108
1.026
1.005
1.000
1.041
1.476
1.202
1.114
1.090
1.085
1.130
tL
0.4760
0.2020
0.1140
0.0900
0.0850
0.1300
L2
0.0410
0.0410
0.0410
0.0410
0.0410
0.0410
vz
0.0001
0.0005
0.0015
0.0022
0.0031
0.0053
0.5171
0.2435
0.1565
0.1332
0.1291
0.1763
1.4225
1.4549
1.5419
1.5956
1.6365
1.6477
3
s,
t
+ E2
.814 1.178
1. 341 1. 371
1.365
1.239
0.252
0.415
0.564
0.634
0.704
0.858
0.572
0.810
0.870
0.860
0.834
0.752
0.838
0.829
0.805
0.792
0.782
0.779
0.390
0.167
0.090
0.110
0.160
0.376
0.407
0.677
0.948
1.093
1.218
1.489
71
TABLE X
TURBINE STAGE CHARACTERISTICS
3 41 = 23.60
sinxi = sin / 2 = 0'4
cost
Peo
8 0%-
f
= cos /3 2 = 0.91 6 4; P2J
0.5
clo-pt
0.84
0.58
v/~"~'
0.762
0.3
0.5
0.7
0.8
0.9
1.2
0.459
0.666
0.793
0.826
0.840
0.760
I)
0.394
0.657
0.919
1.050
1.180
1.575
4.
0.567
0.773
0.840
0.822
0.7710
0.407
t-3,
0.649
0.961
1.846
3.437
taal
0.766
1.-4
-117.6
-2.
9 90
24.37
-1.410
-1.516
-0.608
A
33.0
43.9
61.6
73.8
87.7
125.3
OZ
37.5
57.0
90.5
108.5
123.4
148.7
.I+o
103.6
103.6
103.6
103.6
103.6
103.6
eol
76.4
76.4
76.4
76.4
76;4
76.4
5
/-s
wl
9,
0o2
0.076
0.076
0.076
0.076
0.076
0.076
1.0
1.0
1.0
1.0
1.0
1.0
-42.5
-23.0
10.5
28.5
43.4
68.7
1.123
1.021
1.004
1.019
1.044
1.140
1.208
1.099
1.080
1.096
1.123
1.227
0.2080
0.0990
0.0800
0.0960
0.1230
0.2270
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0.0001
0.0003
0.0010
0.0015
0.0020
0.0050
0.2257
0.1169
0.0986
0.1151
0.1426
0.2496
76.4
so2
j/16
76.4
76.4
76.4
76.4
76.4
0.076
0.076
0.076
0.076
0.076
0.076
1.0
1.0
1.0
1.0
1.0
1.0
72
TABLE X CONT'D
I's-
-47.0
-36.1
-18.4
-6.2
7.7
45.3
1.222
1.068
1.012
1.002
1.002
1.051
+9
1.315
1.150
1.089
1.079
1.079
1.131
Sp
0.3150
0.1500
0.0890
0.0790
0.0790
0.1310
ha
0.0310
0.0310
0.0310
0.0310
0.0310
0.0310
0.0001
0.0003
0.0(10
0.0015
0.0020
0.0050
0.3461
0.1813
0.1210
0.1115
0.1120
0.1670
1.2142
1.3367
1.4488
1.4823
1.4946
1.3424
0.787
1.114
1.280
1.303
1.287
0.996
0.272
0.433
0.582
0.657
0.736
1.035
0.648
0.834
0.885
0.879
0.862
0.742
908
0.865
0.830
0.820
0.817
0.863
0.433
0.227
0.160
0.178
0.229
0.593
0.394
0.657
0.919
1.050
1.180
1.575
s +
s
1;
73
TABLE XI
TURBINE STAGE CHARACTERISTICS
sino'
=sin/ 2
COSO
Es2
:0.5
; C73
At
X
S
W
+51
302.
0,
4Lib6
liopt = 0.75
COS/"c /2 0.866; 321 3o625
- =
0.791
0.3
0.5
0.7
0.8
0.9
1.2
0.43
0.616
0.723
0.746
0.749
0.638
0.38
0.632
0.885
1.011
1.138
1.516
0.513
0.695
0.749
0.729
0.676
0.338
0.884
1.366
3.01
7.58
1.028
2.135
ta" A
to n Q'kz
o=800, : 0.5
300
-26.3
-14.7
-3.447
-1.836
-1.496
-0.769
41.5
53.8
71.6
82.5
93.9
123.8
45.8
64.9
92.2
106.2
118.6
142.4
110
110
110
110
110
110
70
70
70
70
70
70
0.071
0.071
0.071
0.071
0.071
0.071
1.0
1.0
1.0
1.0
1.0
1.0
-34.2
-15.1
12.2
26.2
38.6
62.4
1.058
1.008
1.004
1.016
1.036
1.107
1.134
1.080
1.075
1.089
1.110
1.186
0.134
0.080
0.075
0.089
0.110
0.186
0.0142
0.0142
0.0142
0.0142
0.0142
0.0142
0
0.0001
0.0004
0.0009
0.0015
0.0036
0.1482
0.0943
0.0896
0.1041
0.1257
0.2038
70
70
70
70
70
70
0.071
0.071
0.071
0.071
0.071
0.071
1.0
1.0
1.0
1.0
1.0
1.0
74
TABLE XI CONT'D
A-/ 3
-38.5
-26.2
2.5
-8.4
13.9
43.8
}j+ 1.084
1.029
1.002
1.00
1.004
1.046
1.161
1.102
1.073
1.071
1.075
1.121
tk
0.161
0.102
0.073
0.071
0.075
0.121
L2.
0.0231
0.0231
0.0231
0.0231
0.0231
0.0231
wz.
0
0.0001
0.0004
0.0009
0.0015
0.0036
0.1841
0.1252
0.0965
0.0950
0.o996
0.1477
1.0634
1.2224
1.3406
1.365
1.359
1.1449
0.7504
1.05
1.191
1.201
1.171
0.849
0.291
0.452
0.604
0.685
0.772
1.121
0.706
0.859
0.889
0.881
0.862
0.742
0.97
0.905
0.863
0.855
0.858
0.935
0.487
0.305
0.251
0.271
0.324
0.662
0.38
0.632
0.885
1.011
1.138
1.516
14
se
-e
75
ST AQ a
c
4e R AgrE RsT rocs
4Z44
5rolo
~~t
0
:0s
7~.2Qy uV4C
.5
/( a '5.o
r~
1
ri,
e- o. 20
to
09
:40.8
f
5%
A
a7
0.6
o.e
04
HA'V
0.1
0.2
0.3
0.%
0.5
06
%?hk
76
0.7
0.8
0.9
/.0
1.1
-
0
A
TABLE XII
TURBINE STAGE CHARACTERISTICS
sin,3 2
sina,
cos
cos
3
2
0.25;
J,
0.968;
/2
= 700;
0
1
f
Clopt
:1.0
;
e:1.0
0.937-, fs:
1.0
0.3
0.5
0.7
0.8
0.9
1.2
0.491
0.718
0.865
0.909
0.932
0.883
0.3
0.5
0.7
0.8
0.9
1.2
0.491
0.718
0.865
0.909
0.932
0.,883
0.3744
0.5347
0.933
1.488
3.679
-1.068
0.3744
0.5347
0.933
1.488
3.679
-1.068
01
&+l*
A
20.5
20.5
28.1
28.1
43.0
56.1
43.0
56.1
74.8
74.8
133.1
133.1
84.5
84.5
84.5
84.5
84.5
84.5
95.5
95.5
95.5
95.5
95.5
95.5
01
t3,
.o,
i/6
to
'}
}
0.093
0.093
0.093
0.093
0.093
0.093
1.0
1.0
1.0
1.0
1.0
1.0
-49.5
-41.9
-27.0
-13.9
4.8
63.1
1.308
1.116
1.031
1.006
1.000
1.111
1.430
1.220
1.126
1.100
1.093
1.215
0.4300
0.2200
0.1260
0.1000
0.0930
0.2150
0.0280
0.0280
0.0280
0.0280
0.0280
0.0280
0
0.0002
0.0006
0.0010
0.0018
0.0040
0.4580
0.2482
0.1546
0.1290
0.1228
0.2470
95.5
95.5
95.5
95.5
95.5
95.5
0.093
0.093
0.093
0.093
0.093
0.093
1.0
1.0
1.0
1.0
1.0
1.0
77
TABLE XII CONT'D
/
3
-49.5
1.116
1.308
-13.9
-27.0
-41.9
4.8
63.1
1.031
1.006
1.000
1.111
.4 -o 1.430
1.220
%p
0.4300
0.2200
0.1260
0.1000
0.0930
0.2150
12.
0.0487
0.0487
0.0487
0.0487
0.0487
0.0487
vwz
0
0.0002
0.0006
0.0010
0.0018
0.0040
0.4787
0.2689
0.1753
0.1497
0.1435
0.2677
1.9187
1.9531
2.0599
2.0967
2.1303
2.2807
0.982
1.436
1.730
1.818
1.864
1.766
0.217
0.357
0.488
0.553
0.617
0.795
0.512
0.736
0.840
0.867
0.876
0.775
0.722
0.715
0.697
0.691
0.685
0.663
0.509
0.282
0.135
0.091
0.068
0.117
0.3
0.5
0.7
0.8
0.9
1.2
+
s
1.126
1.100
1.093
1.215
78
TABLE XIII
TURBINE STAGE CHARACTERISTICS
2sin/2
0.3;
sin
Cos
A
+
01,
Go,
0(1
0 : 750 ; P = 1.0
= 17.50
COS/32 =.9539; /2
elopt =
3
0.910;
S=
1.0
S': 1.0
0.3
0.5
0.7
0.8
0.9
1.2
0.482
0.724
0.845
0.886
0.907
0.850
0.3
0.5
0.7
0.8
0.9
1.2
0.482
0.724
0.845
0.886
0.907
0.850
0.459
0.671
1.181
1.948
5.560
-1.218
0.459
0.671
1.181
1.948
5.560
-1.218
24.7
33.9
49.8
62.8
79.8
129.4
24.7
33.9
49.8
62.8
79.8
129.4
92.5
92.5
92.5
92.5
92.5
92.5
87.5
87.5
87.5
87.5
87.5
87.5
Sol
0.085
0.085
0.085
0.085
0.085
0.085
* /6
1.0
1.0
1.0
1.0
1.0
1.0
-50.3
-41.1
-25.2
-12.2
4.8
54.4
+ ol
LI
1.360
1.108
1.026
1.005
1.000
1.076
1.476
1.202
1.114
1.090
1.085
1.167
0.4760
0.2020
0. 1140
0.0900
0.0850
0.1670
0.0234
0.0234
0.0234
0.0234
0.0234
0.0234
0
0.0002
0.0005
0.0009
0.0016
0.0038
0.4994
0.2256
0.1379
0.1143
0.1100
0.1942
87.5
S 0'2
S/6
87.5
87.5
87.5
87.5
87.5
0.085
0.085
0.085
0.085
0.085
0.085
1.0
1.0
1.0
1.0
1.0
1.0
-50.3
-41.1
-25.2
-12.2
'79
54.4
TABLE XIII CONT'D
}
1.360
1.108
1.026
1.005
1.000
1.076
1.476
1.202
1.114
1.090
1.085
1.167
,
0.4760
0.2020
0.1140
0.0900
0.0850
0.1670
L2
0.0410
0.0410
0.0410
0.0410
0.0410
0.0410
0
0.0002
0.0005
0.0009
0.0016
0.0038
r0.5170
0.2432
0.1555
0.1319
0.1276
0.2118
1.9804
1.9168
1.9834
2.0182
2.0516
2.1060
0.964
1.448
1.690
1.772
1.814
1.700
"
0.213
0,361
0.497
0.563
0.629
0.828
e
0.487
0.756
0.853
0.879
0.884
0.807
0.710
0.721
0.710
0.704
0.698
0.689
0.518
0.276
0.155
0.114
0.093
0.150
0.3
0.5
0.7
0.8
0.9
1.2
I+5,2
4
5
sS
6,-
'-ee
oze
TABLE XIV
TURBINE STAGE CHARACTERISTICS
sin o(I
=
0.4
sin 12
0.9164;
cosd 1 : cosA
2
%)1
;
I = 23.6
0 :
800
E1opt
/2
S
S : 1.0
0.84
1.0
Vs: 1.0
0.3
0.5
0.7
0.459
0.666
0.793
0.3
0.5
0.7
0.459
0.666
0.793
0.826
tan l
0.649
0.961
1.846
3.437
24.37
-1.410
tan c
2
0.649
0.961
1.46
3.437
)4.37
-1.410
901
.ol1
corr.for 1/b
0.826
0.9
1.2
0.840
0.760
0.9
1.2
0.840
0.760
33.0
43.9
6i.6
73.8
87.7
125.3
33.0
43.9
61.6
73.8
87.7
125.3
103.6
103.6
103.6
103.6
103.6
103.6
76.4
76.4
76.4
76.4
76.4
76.4
0.076
0.076
0.076
0.076
0.076
0.076
1.0
1.0
1.0
1 .0
1.0
1.0
-47.0
-36.1
-6.2
7.7
45.3
1
1.222
1.068
1.012
1.002
1.002
1.051
1.315
1.150
1.089
1.079
1.079
1.131
p1
0.3150
0.1500
0.0890
0.0790
0.0790
0.1310
Ll
0.0176
0.0176
0.0176
0.0776
0.0176
0.0176
Wi
0
0
0.0002
0.0008
0.0014
0.0031
0.3326
0.1676
0.1068
0.0974
0. 09F0
0.1517
1
4
1
-'
5
p1
S1
902
-02
corr.for 1/b
76.4
76.4
76.4
76.4
76.4
76.4
0.076
0.076
0.076
0.076
0.076
0.076
1.0
1.0
1.0
1.0
1.0
1.0
El
TABLE XIV CONT'D
A-A.
-47.0
-6.2
-36.1
7.7
45.3
1 4 p2
1.222
2.068
1.012
1.002
1.002
1.051
1-315
1.150
1.089
1.079
1.079
1.1-31
0.3150
0.1500
0.0890
0.0790
0.0790
0.1310
0.0310
0.0310
0.0310
0.0310
0.0310
0.0310
0
0
0.0002
0.0008
0.0014
0.0031
0.3460
0.1810
0.1202
0.1108
0.1114
0.1651
2) 1.5966
1.6806
1.8148
1.8602
1.9894
3.8368
0. 918
1.332
1.586
1.652
1.780
1.520
th
0.237
0. 386
0.520
0.587
0.638
0.8P6
7 st
0.575
0.793
0.875
0.889
0.895
0.792
0.772
0.743
0.733
0.709
0.738
0.541
0.334
0.207
0.174
0.160
0.240
0.3
0.5
0.7
0.9
1.2
1+
1
02
SP 2
L2
S2
2
s s (E 2
1
2
)
e 2+
- 2e
(
+
E
2e
TABLE XV
TURBINE STAGE CHARACTERISTICS
sin d 1
Cos 0
= cos/
0,
3
2
=
3
0
300
0.866; /2
80
Elopt
1.0
s-- 1.0
0.75
/s'- 1.0
0.3
0.5
0.7
0.8
0.9
1.2
0.430
0.616
0.723
0.746
0.749
0.638
0.3
0.5
0.7
0.8
0.9
1.2
0.430
0.616
0.723
0.746
0.749
0.638
0.884
1.366
3.010
7.580
-14.70
-1.496
0.884
1.366
3.010
7.580
-1.70
-1.496
ta n dz
ERl-
0.5
sin/2
41.5
53.8
71.6
82.5
93.9
123.8
41.5
53.8
71.6
82.5
93.9
123.8
110.0
110.0
110.0
110.0
110.0
110.0
70.0
70.0
70.0
70.0
70.0
70.0
So,
&2
'
a'
0.071
0.071
0.071
0.071
0.071
0.071
1.0
1.0
1.0
1.0
1.0
1.0
-38.5
-26.2
-8.4
2.5
13.9
43.8
1.084
1.029
1.002
1.000
1.004
1.046
1.161
1.102
1.073
1.071
1.075
1.121
0.1610
0.1020
0.0730
0.0710
0.0750
0.1210
0.0142
0.0142
0.0142
0.0142
0.0142
0.0142
0
0
0.0002
0.0003
0.0005
0.0020
0.1752
0.1162
0.0874
0.0855
0.0897
0.1372
70.0
70.0
70.0
70.0
70.0
0.071
0.071
0.071
0.071
0.071
0.071
1.0
1.0
1.0
1.0
1.0
1.0
/3% -38. 5
-26.2
2.5
13.9
02r-0 T
-.
Yb6
70.0
S3
43.8
TABLE XV CONT'D
1.084
1.029
1.002
1.000
1.004
1.046
1 rv
1.161
1.102
1.073
1.071
1.075
1.121
I'p'
0.1610
0.1020
0.0730
0.0710
0.0750
0.1210
0.0231
0.0231
0.0231
0.0231
0.0231
0.0231
(w-
0
0
0.0002
0.0003
0.0005
0.0020
5
0.1841
0.1251
0.0963
0.0944
0.0986
0.1461
1.2193
1.4733
1.6297
1.6719
1.6863
1.5593
0.860
1.232
1.446
1.492
1.498
1;276
0.272
0.412
0.548
0.619
0.693
0.962
0.706
0.837
0.888
0.893
0.888
0.820
S0.906
0.824
0.784
0.773
0.770
0.801
0.570
0.384
0.277
0.254
0.251
0.362
0.3
0.5
0.7
0.8
0.9
1.2
'4-
Lt
(
s
iUt
j85e.-
e
}
E,4
e4
A
!vro
DOC
'c7o
7
Sc$
eS diet*
,,.
4_
OkI04
Itf
/-0
0.9
3'
4,
5,7
-Y
0.8
I 'I
.9.
~~C
0,7
4
I.J
1'4,
4.
.9-
0.4
CS,
0.4
o.3
0.2
O.f
09
0
0.1
0.2
0.3
0.4
of
o.6
85
0.7
o.8
0.9
1.0
1.1
/.2
--------",--------
TABLE XVI
TURBINE STAGE CHARACTERISTICS
sin o1
0.25 ;
, A
= 700
P
: 1.5
S: 1.468
-
osin
cos
0
-COS /3'2
1
0.937
0.968; /3 2
Vs: 1.211
0.3
0.5
0.7
0.8
0.9
1.2
0.491
0.718
0.865
0.909
0.932
0.883
0.248
0.413
0.578
0.660
0.743
0.991
62.
0.419
0.614
0.785
0.842
0.886
0.936
4a-n ,
0.374
0.535
0.933
1.488
3.679
-1.068
0.455
0.707
1.365
1.984
3.048
7.815
-tao4
d
20.5
28.1
43.0
56.1
74.8
133.1
24.5
35.3
53.8
63.3
71.8
82.7
d,#/S0 84.5
84.5
84.5
84.5
84.5
84.5
95.5
95.5
95.5
95.5
95.5
95.5
d,.
Oo
0,
*OTp.)
Ak/
0.093
0.093
0.093
0.093
0.093
0.093
1.0
1.0
1.0
1.0
1.0
1.0
-45.5
-34.7
-16.2
-6.7
1.8
12.7
1. 176
1.059
1.009
1.002
1.000
1.004
1.285
1.157
1.102
1.095
1.093
1.097
Sp
0.2850
0.1570
0.1020
0.0950
0.0930
0.0970
9L/
0.0280
0.0280
0.0280
0.0280
0.0280
0.0280
Svv
0
0.001
0.004
0.0010
0.0018
0.0024
0.3130
0.1851
0.1304
0.1240
0.1228
0.1274
s,
34
U-z
95.5
95.5
95.5
95.5
95.5
95.5
0.093
0.093
0.093
0.093
0.093
0.093
1.0
1.0
1.0
1.0
1.0
1.0
*-49.5
-41.9
-27.0
-13.9
4.8
63.1
86
TABLE XVI CONT'D
e-*P7a
7
s
e
1.308
1.116
1.031
1.006
1.000
1.111
1.430
1.220
1.126
1.100
1.093
1.215
0.4300
0.2200
0.1260
0.1000
0.0930
0.2150
0.0487
0.0487
0.0487
0.0487
0.0487
0.0487
0
0.0001
0.0004
0.0010
0.0018
0.0024
0.4787
0.2688
0.1751
0.1497
0.1435
0.2662
1.8920
1.9731
2.1594
2.2340
2.3018
2.4654
0.986
1.462
1.816
1.929
2.006
2.017
0.218
0.356
0.477
0.535
0.593
0.765
0.521
0.741
0.842
0.864
0.866
0.818
0.727
0.712
0.681
0.669
0.659
0.637
0.581
0.386
0.215
0.158
0.114
0.064
0.248
0.413
0.578
0.660
0.743
0.991
-
87
TABLE XVII
TURBINE STAGE CHARACTERISTICS
0
sin cA
Cos
1
dl,
1
sin /3
0.3
;
0 ,30
2
-cos 2
1.5
= 75
17.5
0.9539;
1opt
S2
5
0.91 ;
s = 1.455
=-5 1.206
0.5
0.7
0.9
1.0
1.2
1.4
0.724
0.845
0.907
0.908
0.85
0.711
0.414
0.580
0.746
0.829
0.995
1.160
0.618
0.770
0.867
0.894
0.909
0.868
0.671
1.181
5.560
-1.218
-0.673
0.556
0.803
1.442
-7.320
-1.47
6,
-6.53
2.400
33.9
49.8
79.8
98.7
129.4
146.1
29.1
38.8
55.3
67.4
97.8
124.3
,+/o
92.5
92.5
92.5
92.5
92.5
92.5
Go,
87.5
87.5
87.5
87.5
87.5
87.5
So
/10
0.085
0.085
0.085
0.085
0.085
0.085
1.0
1.0
1.0
1.0
1.0
1.0
-45.9
-36.2
-19.7
-7.6
22.8
49.2
2I
1.190
1.069
1.014
1.002
1.012
1.061
1.291
1.160
1.100
1.087
1.099
1.151
0.291
0.160
0.100
0.087
0.099
0.151
0.0234
0.0234
0.0234
0.0234
0.0234
0.0234
0.0001
0.0003
0.0008
0.0013
0.0017
0.0025
0.3145
0.1837
0.1242
0.1117
0.1241
0.1769
)
I+
14 10,
87.5
goz
87.5
87.5
87.5
87.5
87.5
0.085
0.085
0.085
0.085
0.085
0.085
1.0
1.0
1.0
1.0
1.0
1.0
-41.1.
-25.2
23.7
54.4.
71.1
88
TABLE XVII CONT'D
1.108
1.026
1.00
1.013
1.076
1.157
1.202
1.114
1.085
1.099
1.168
1.256
0.202
0.114
0.085
0.099
0.168
0.256
0.041
0.041
0.041
0.041
0.041
0.041
0.0001
0.0003
0.0008
0.0013
0.0017
0.0025
0.2431
0.1553
0.1268
0.1413
0.2107
0.2995
2.293
2.375
2.477
2.526
2.602
2.588
s
1.623
1.965
2.168
2.208
2.174
1.972
k
0.33
0.454
0.572
0.629
0.743
0.87
0.708
0.827
0.875
0.875
0.835
0.763
.661
0.649
0.635
0.629
0.619
0.621
0.382
0.23
0.133
0.106
0.091
0.132
0.414
0.58
0.746
0.829
0.995
1.160
I
l
3
s
89
TABLE XVIII
TURBINE STAGE CHARACTERISTICS
sino
= cos
2
2
0.4
4
5
23.6
o.9164;
800
3
f
E1opt
2
: 1.5 ;
= 084
&
1.42
S1. 191
0.5
0.7
0.9
1.0
1.2
1.4
0.453
0.666
0.840
0.8.33
0.760
0.606
0.420
0.587
0.755
0.839
1.008
1.175
/,
3
A
0.593
0.732
0.814
0.844
0.832
0.773
,
coseL
sin
a
0.961
1.846
2.437
-4.786
-1.410
-0.827
0.806
1.214
2.478
5.170
-4.368
-1.546
o Sol
-+.
O'}
}
43.9
61.6
87.7
101.8
125.3
140.4
38.9
50.5
68.0
79.1
102.9
122.9
103.6
103.6
103.6
103.6
103.6
103.6
76.4
76.4
76.4
76.4
76.4
76.4
0.076
0.076
0.076
0.076
0.076
0.076
1.0
1.0
1.0
1.0
1.0
1.0
-41.1
-29.5
-12.0
-0.9
22.9
42.9
14,
Sb
1.109
1.039
1.005
1.000
1.012
1.045
1.193
1.117
1.081
1.076
1.089
1.124
0.1930
0.1170
0.0810
0.0760
0.0890
0.1240
0.0176
0.0176
0.0176
0.0176
0.0176
0.0176
0
0.0002
0.0007
0. 0010
0.0015
0.0022
0.2106
0.1348
0.0993
0.0946
0.1081
0.1438
01
ja0
76.4
76.4
76.4
76.4
76.4
76.4
0.076
0.076
0.076
0.076
0.076
0.076
1.0
1.0
1.0
1.0
1.0
1.0
90
TABLE XVIII CONT'D
-36.1
/&o
-18.4
21.8
7.7
45.3
60.4
}
1.068
1.012
1.002
1.011
1.051
1.099
io x,
1.150
1.090
1.078
1.089
1.131
1.183
Spx
0.1500
0.0900
0.0780
0.0890
0.1310
0.1830
Ll.
0.0310
0.0310
0.0310
0.0310
0.0310
0.0310
0
0.0002
0.0007
0.001C
0.0015
0.0022
0.1810
0.1212
0.1097
0.1210
0.1635
0.2162
1.7696
2.0118
2.2503
2.2980
2.2821
2.1548
1.301
1.706
1.995
2.032
1.941
1;704
0.376
0.494
0.600
0.660
0.794
0.955
0.736
0.848
0.887
0.885
0.851
0.791
0.752
0.705
0.667
0.660
0.661
0.681
E
0.407
0.268
0.186
0.156
0.168
0.227
Zte
0.420
0.587
0.755
0.839
1.008
1.175
S+jo
s
)
+
1, 7 -s(v4
91
TABLE XIX
TURBINE STAGE CHARACTERISTICS
: sin
2 =
0.5
/% - 80*
,
sin )
7 ' - 30
0.866;
cosO 1 : cos3
2
2 =
1.5j ,5
S 0.75J
1.375
: 1.173
0.5
0.7
0.9
1.0
1.2
1.4
0.616
0.723
0.749
0.732
0.638
0.465
0.426
0.597
0.767
0.853
1.023
1.194
0.556
0.678
0.740
0.750
0.726
0.642
1.366
3.01C
-14.70
-3.73
-1.496
-0.679
1.136
1.857
5.05
38.45
-3.182
-1.524
Z,
4
tzA4
ta -K A
A
v r. c,1
1a
53.8
71.6
93.9
105.0
123.9
145.8
48.6
61.7
78.8
88.5
107.5
123.3
110.0
110.0
110.0
110.0
110.0
110.0
70.0
70.0
70.0
70.0
70.0
70.0
0.071
0.071
0.071
0.071
0.071
0.071
1.0
1.0
1.0
1.0
1.0
1.0
-1.2
8.5
27.5
43.3
-31.4
-18.3
1+J
l }
/3-3l
34
1.046
1.012
1.000
1.002
1.018
1.045
1.120
1.085
1.071
1.073
1.090
1.120
0.1200
0.0850
0.0710
0.0730
0.0900
0.1200
0.0142
0.0142
0.0142
0.0142
0.0142
0.0142
0
0.0001
0.0003
0.0005
0.0010
0.0018
0.1342
0.0993
0.0655
0.0077
0.1052
0.1360
Z }
70.0
70.0
70.0
70.0
70.0
70.0
0.071
0.071
0.071
0.071
0.071
0.071
1.0
1.0
1.0
1.0
1.0
1.0
92
TABLE XIX CONT'D
A/.
-26.2
-8.4
25.0
13.9
43.9
65.8
1.029
1.002
1.004
1.015
1.046
1.124
+4,
1.102
1.073
1.075
1.087
1.121
1.205
gpz
0.1020
0.0730
0.0750
0.0870
0.1210
0.2050
51-Z
0.0231
0.0231
0.0231
0.0231
0.0231
0.0231
0
0.0001
0.0003
0.0005
0.0010
0.0018
0.1251
0.0962
0.0984
0.1i06
0.1451
0.2299
1.6872
1.8873
1.9675
2.0037
1.9412
1.800
1.381
1.656
1.766
1.762
1.636
1.348
0.385
0.510
0.641
0.707
0.861
1;044
0.819
0.877
0.897
0.880
0.843
0.749
0.770
0.724
0.713
0.707
0.717
0.745
0.444
0.322
0.260
0.250
0.274
0.358
0.426
0.597
0.767
0.853
1.023
1.194
vwz
e,
eZ
tk
'-6"e
93
0.128!
0.0
0..8
0.4
.,
0 .1
o.a
.
VwLocty
4
.
XmnbTO
94
.
0.7
l
oa
-
ve e
o.
~
.
0.1
Part III
Number of Stages
As stated previously, the tip speed has been limited to 500 ft./sec; limitation that accounts for the fact that the strength of metals decreases rapidly
with high temperatures.
With the materials available today, this speed limit will be used for the
hottest stage in the L. P. turbine, which is the first stage.
Since the design characteristics of the first stage have not yet been determined, and the limitation of 1/d
=
0.28 maximum is valid for the last stage, it
shall be assumed that here the ratio 1/d for the first stage is 0.1, consequently
if the peripheral velocity, at the pitch diameter is:
00
'--:
= 500
u
455 ft./sec.
1.1
then, the blade tip speed for the last stage is
amax = 455(1+1/d)
582 ft./sec.
455(1+- 0.28)
From the computations made in Part II, and from fig. XIX, it can be concluded that the best theoretical velocity ratio, for a degree of reaction of
= 1.0 (normal reaction), is:
th = 0.6
It can be shovm that the relation existing between the number of stages,
the mean peripheral velocity and the enthalpy drop across the turbine is given by:
~42
where:
Z X
th
th
is the sum of the wheel speeds at the ritch diameter which in this
case is the same value throughout the turbine, since the pitch diameter is constant.
$
is the number of sta,'es.
OWN
is the theoretical velocity ratio, as defined by equation (26).
th
is the theoretical velocity corresponding to the isentropic
C
th
enthalpy drop through the turbine, and given by eq. (27).
The isentropic enthalpy drop across the L. P. turbine, for conditions of
the cycle as selected previously is:
h
-
7
h
8s
- 155.74 Btu/lb.
therefore:
th
-v2 x 32.2 x 778 x 155.74
- 2790 ft./sec.
and:
vr
x7-
(4 55
0.6 x 2790
S
13.5
Since the result is a fractional number, then the number of stages for the L. P.
turbine must be:
3 = 14 stages.
96
Part IV
Speed of Rotation
Since the basis for the determination of the number of stages was the mean
peripheral velocity, a preliminary calculation of t'e speed of rotation of the
L. P. turbine can now be made, taking as a fixed value, the peripheral velocity
of the last stage, which is 582 ft./sec.; the diameter for this stage is also
known and the maximum value of the ratio l/d is well f'ived.
38.5 inches
d
O.28
0
1/d
therefore:
and:
38.5 x 1.28
dmax
n
-
49.3 inches
:
60
x
12 x umax
Sd
max.
720 xrS82
iT x 49.3
-
2700 r.D.m.
97
SECTION B-2
Preliminary Characteristics of H
Turbine
P
The same assumptions and limitations given for the L. P. turbine are valid
in this case.
Part I
Pitch Diameter
Assumptions and limitations
(a) Blades tip speed limit:
500 ft./sec. for the hottest stage.
(b) Approach velocity to H. P. turbine:
negligible
(c) Blade length-pitch diameter ratio = 0.25.
(d) Axial flow at exit of last stage.
(e) Leaving loss of 1.2% of the enthalpy drop across the turbine.
(f) Pitch diameter constant throughout the turbine.
From the computations of the cycle in Part 1, Section A-1, Appendix A, the
exhaust temperature of the H. P. turbine was
b6 = 63.6 psia
: 321.77;
h6
16800 Fa.
-:
therefore:
for this temperature and pressure the specific volume of the gas is:
T( 3.3 x 1680 9.79 ft. 3/lb.
v,
-
RTe
0
6
-
63.6 x 144
the enthalpy drop across the H. P. turbine was round to be:
5 -t
5
Btu./lb.
6 :49.15
by equation (16), the leaving loss is:
At L
and:
A
-
49.15 x 0.012
64.7 x 9,7
223.8
0.59
3.685 ft.~
98
0.59 Btu/lb.
consequently, tle pitch diameter is, by eq. (18):
d
- _4
w
x
3. 68 5
x 1
0.25
26 inches.
99
Part II
Speed of Rotation
Since the H. P. turbine is to operate at a higher average temperature
then the L. P. turbine, the tip speed limit of 500 ft./sec. will be the maximum value in any part of the H. P. turbine, and not in the hottest stage as
in the L. P. turbine.
The diameter at the tip of the blade is given by:
d(l
d
max
(43)
1/d)
-
26(1
-
32.5 inches
0.25)
and the speed of rotation is then:
v- 60 x 12 x u
Imax
60 x 12 x 500
7F x 32.5
3525 r.p.m.
100
Part III
Number of Stages
The maximum tip speed, A max
500 ft./sec., corresponds to the last stage,
since it is the one with the longest blade.
This last stage is the coolest stage in the H. P. turbine.
The wheel speed at the pitch diameter is:
max
A
(44)
T+i/d
500
140.25
400 ft./sec.
1.5
whence:
x (400)2
since tt is constant
The isentropic enthalpy drop for the H. P. turbine is:
: 57.85 Btu/lb.
6
its corresponding theoretical velocity is:
th
-
5 -
-223.8
-
57.835
1700 ft./sec.
As discussed previously, the best theoretical velocity ratio for the
stage was taken as:
3
th
- 0.6
0
therefore:
Sx (400)
:0.6 x 1700
S = 6.5
to use the',7 stages in the H. P. turbine.
101
(45)
SECTION B-3
Part I
Preliminary Design of L. P. Compressor
It is necessary to determine the speed of rotation and relevant properties of the first stage of both L. P. and H. P. compressors, before making a
decision on the detail design of the turbiaes.
The operating conditions of the L. P. compressor,
as seen on Part I,
Section A-2, Appendix A are:
Inlet conditions:
14.7 psia
py
Intercooler
pressure ratio
T,
:5200 Fa.
hl
- 28.77 Btu/lb.
r
-
2.65
~
p 1
A set of guide vanes, forward of the first stage rotor, is used to induce
the proper amount of pre-rotation.
- 90% will be assumed for these guiding vanes; it
An efficiency of
will also be assumed that the velocity of approach of the gas to guiding vanes,
Cox, is negligible, Fig. XXI.
The design of the compressor will
be made for symmetrical stages at
the pitch radius; and the follow-
-e
-
-4.-
-
L
ing assumptions are made:
(1) Axial component of flow is con-
---------
stant throughout the compressor.
(2) Stator and rotor blades have same
profile at pitch radius.
(3) Density of gas constant through
the stage.
Fig. XXI
102
Definitions:
Referring to fig. XXII:
Velocity ratio for stage:
ux
(46)
U
cot 1
+coto
(47)
and since stage is symmetrical;
also:
cot/3 2 + coto(
Fig. XXII
(48)
Stage diagram ratio:
u2/A
u22
(49)
where Ai
is the work done on
kL
Aplying momentum
the stage.
equation:
x U
CU
u /2g
-2
x Cu
U
From fig. XXIII, it cn
(50)
be shown
that the work done on the stage
is also equal to:
Fig. XXIII
)
E : 2(cot' 3 1 - cot 3
2
cotC
4+ cot /3
1
103
(51)
- .-
-
---------
The influence of the Mack number must be investigated now.
The Mack mumber limitation has to do with the constriction of the foils.
(see fig. XXII)
Let's call M =
w1ere
%,
(52)
is the corresponding vel-
ocity of sound
2
- kgRT
-
--
Since it has been specified a symmetrical velocity triangle at the
Fig. XXIV
pitch diameter, then for the tip and the base of the blades, respectively, we
have:
The length diameter ratio of the
blade can be expressed as:
(53)
F l/d
and let:
-
k
AA-
-
rb
r t(54)
b - base; t
(subscripts:
=
tip)
Therefore it can be found that:
k :1
-
(55)
1 +pf
A constant circulation type of
BASE
0
blades will be considered. Experience
shows that the maximum limit for
1/d is about 0.25, and therefore
k
-
0.6.
In this region, experience
shows that the Mack number limitations are approximately the same for
Fig. XXV
104
-~
-~
for the base and the tip of the blade.
Therefore:
it
(56)
2b
-
This means that, from fig. XXV:
C
+0C
U - Cul = C
ul
k
- ku
C
- CU
(57)
This settles the value of the horizontal projection of vI
Also:
u - Cu -
C
u
-k
-u
4
and C2b
u
1+k
1 + k
(58)
from fig. XXV
2
2
+ (u - C1
C
since:
(59)
)
2
Wit
+(
/
Therefore:
/
(1
(1 + k
)
(60)
Introducing R, and kR, outside an inside radius respectively:
u
R
w:
where:
W
(61)
angular velocity
then the volume of flow is:
Q
(62)
= 77 R2 (1 - k2)C
Having in mind that, as shown before:
6 : 2. Cu
u
and
Cx
T1hen in eq. (60), it can be easil, arrived at:
(63)
(1 + k)
and also that:
R 2 (1 - k2) x
x U
F (1 - k2). 0. u
2
105
(64)
-----------
It can be seen that the choice of speed, volume of flow are not independent.
By substituting in (64), and collecting terms:
2
A2)
3
x
(%) 2+
(65)
2
A:
where:
2)3/2
i+
(67)
k
The velocity ratio shall be now so chosen as to get maximum speed:
3
A)
k(2
-
x 0
2+ A2)3/2-3x 3/2 x 2 %)(,%)
2
(Q 2+ A2
2+ A2
Therefore:
2
0
A
1
F23.
1 +
(68)
k
With this relation we have really fixed only relation between
0 and 6 , which
means we have fixed certain angles:
C
x
C
x
-
l t
(69)
/v XV
)
(sin/ 3
but
(1k)
and substituting value for %)found in (68), and simplifying:
(70)
o: 3u- 13
Therefore:
(sin /3
C
-
..
1
x
-
)
St
C
x
- u -r
E
u
u
(71)
Consequently:
At-
350
Summary:
For maximum speed:
1
2
S+ 1
(68)
+
1
r2
Mo.
(70)
U
106
0
"
-
---
Combining (50) and (57):
Clu
k
-2
(72)
2
C1
From fig.
C 2 + C2
ul
x
therefore:
C1
C~
x
(Cx)
(73)
Speed of rotation in r.p.m.:
rt-:30 (a
IT
Substituting (65), (67), and (68):
',
1
-30
-
-
,
zmll -
-
-
1'
-
(sin/ )t
'
(74)
Q
27
1
(71)
from (46) and (70):
C
In addition, the following relations not derived before, but that can be easily
arrived at from fig. XXIV:
A, v
Flow area:
:
d
max.
and
d 1
= " d2 1/d a T d2
fP
(d 4- 1) a (1+ ? )d
max.
(75)
(76)
(77)
The radius at which the stage is syrmetrical, will be, say at:
(78)
r : kR
s
ratio for which stape is symmetrical.
k
The condition to have this degree of symmetry is that for the velocity triangle
of fig. XXVI:
Cu
-k
u
-
Cu
k5
-ul
ks
107
Sk2(80)
2
-k u -C
C
ul -s
u
-C ul
Therefore:
k
-
2C
s -9
ul
+0
u
(79)
U
Substituting (57) and
_
3
(72) and collecting
k
2k
-
terms:
1
Fig. XXVI
In this design this latter derivations will not be ta&en into account,
because its a refinement that does not contribute greatly to the precision
of the design; and it shall be considered that the stage is symmetrical at
the pitch radius, thus symplifying computations.
Nevertheless, it must be had in mind that the correct attack of this problem would be as stated in the foregoing.
The position at which the stage is symmetricaT will not differ very much
from that assumed, that is, it will be very near the pitch radius.
Using the above relations, the speed of rotation and relevant properties
of the first stage of the L. P. compressor will be computed in tabular form,
which are explicative by themselves.
Experience has shown that the best performance of compressors occurs at
a Mack number of the order of 0.75; this value will be tried in a preliminary
computation for the first stage characteristics.
The value of blade length-pitch diameter to use in compressors has been
found to be 1/d = 0.2; this ratio occurs in the first stage.
Additional tables are shown, to give an idea how the speed of rotation varies
with varying Mack numbers and bladelength-pitch diameter ratio.
'Y8
From section 2, Appendix B, the reauired speed of rotation for the L. P.
compressor is 3525 r.p.m.
109
TABLE XX
RELEVANT PROPERTIES OF THE FIRST STAGE
M
=0.75
F : 0.2
L. P. COMPRESSOR
-
:2/3
k
_3(2_
10i1'
e
2 Cu
0.1
0.2
0.3
0.4
S:
Cx/u
0.445
0.466
0.488
0.510
Cul/u
0.370
0.340
0.310
0.280
Cul/cx
0.832
0.732
0.635
0.550
C 1 /CX
1.30
1.24
1.182
1.141
o Fa.
T1
Btu/lb.
28.77
520
28.77
520
28.77
520
28.77
,
hi
520
2.504
Prl
0 Fa.
2.504
2.504
2.504
495
497
499
501
ft. /sec.
1090
1093
1095
1098
ft./sec.
818
820
822
823
ft. /sec.
472
473
474
475
C1
ft./sec.
614
587
561
542
C /2gJ
Btu/lb.
5.86
5.36
4.90
4.53
h(
Btn/lb.
22.91
23.41
23.87
24.24
Assume T(j
a(,
C
0)
0)
-
Ma(1
-
-
0)
x
- 0)
check T( 1
0)
0 Fa.
495
497
499
501
(C /2gJ)
Btu/lb.
6.54
5.96
5.45
5.03
0)s
Btu/lb.
2.074
2.109
2.147
2.168
Ig-v
h( 1
-
P( 1 - 0)s
V(l -
Q
A1
0)s
= G(A)(1-)s
psia
12.17
12.38
12.60
12.74
cu.ft.,/lb.
15.06
14.86
14.67
14.55
cu.ft./lb. 975
ft. 2
2.066
962
2.034
110
949
2.00
942
1.983
TABLE XX CONT'ID
d
in.
21.78
21.6
21.36
21.3
d
in
26.13
25.92
25.61
25.57
u
ft./sec.
1060
1016
973
932
r.p.m.
9300
8980
8700
8350
max
111
From Table XX it can be seen that the speed of rotation is too high in
comparison to the speed at which the H. P. turbine, which is the one attached
to the L. P. compressor, as shown on Fig. 1, should be run.
Condequently, new values will be tried for the Mack number and the blade
length-pitch diameter ratio.
Observing Table XX, the conclusion is obtained that the Mack number is
the characteristic that has a greater influence in the speed of rotation of
the L. P. compressor.
The new value selected will be MI
0.4.
As stated previously some more tables are included so as to show the
f
.
variation in speed with varying
112
- m'l
-
-
A
TABLE XXI
RELEVANT PROPERTIES OF FIRST STAGE - L. P. COMPRESSOR
=
p : 0.2
k
0.1
0.2
0.3
0.4
): cx/u
0.445
0.466
0.488
0.509
Cul/u
0.370
0.340
0.310
0.280
Cul/C
0.832
0.730
0.638
0.550
1.301
1.239
1.185
1.141
M = 0.4
6
2
Cu
0.667
3: 2+d
4.72
U
Cl/C
520
T1
28.77
hi
2.504
pr
520
2? .77
2.504
520
28.77
520
28.77
2.504
2.504
511
511
512
513
1108
1108
1109
1110
Ma(1-0)
443
443
444
444
cx
255
255
256
256
C
332
316
304
292
Assume T(1-0)
a(1-_0)
Cl/2gJ
2.20
2.00
1.84
1.71
h(1-0)
26.57
26.77
26.93
27.06
T (1-0)
. (C /2gJ)
511
511
512
513
2.44
2.22
2.05
1.90
26.33
26.55
26.72
26.87
g-v
h (1-0)s
pr(1-0)s
P(1-0)
4(1-0)
2.336
2.351
2.363
2.372
13.71
13.80
13.87
13.93
13.80
13.70
13.66
13.62
113
TABLE XXI CONT'D
Q
892
3.50
A
887
3.48
884
3.46
882
3.44
d
28.3
28.2
283.15
28.1
d
34.0
33.9
33.8
33.75
547
525
max
u
Tu
Ai
573
3860
0.653
3700
1.20
114
3560
1.65
503
3410
2.02
TABLE XXII
RELEVANT PROPERTIES OF FIRST STAGE - L. P ,a COMPRESSOR
M = 0.4
=
k a 0.695
3
0.1
0.2
0.3
0.4
0.438
0.457
0.480
0.500
cul/u
0.380
0.351
0.321
0.292
cu1 /Cx
0.868
0.768
0.669
0.584
c1/c
1.325
1.260
1.203
1.159
:
2, U
- 0.18
2 +e
4.80
I
lu
520
Ti
28.77
h1
2.504
pr1
T(1-0)
520
520
28.77
2.504
28.77
2.504
520
28.77
2.504
510
511
512
513
a(1-0)
1106
1108
1109
1110
Ma(1-a)
443
443
444
444
255
255
256
256
338
321
308
297
Assume
x
,1
Cl
C2/2gJ
h(1 -
2.06
1.89
1.75
26.49
26.71
26.88
27.02
)
0
2.28
T (1 -
510
511
512
513
)
0
2
1 t (C1/2gJ)
2.53
2.29
2.10
1.95
26.24
26.48
26.67
26.82
g-v
h (1-0) s
pr( 1 -O)s
P(1-0)
V(1-0)
Q
2.331
2.347
2.359
2.369
13.69
13.77
13.85
13.90
13.78
13.72
13.68
13.65
892
888
885
115
883
TABLE XXII CONTID
A
3.50
3.48
d
29.85
29.76
29.7
29.6
dmax
35.27
35.16
35.1
35.0
u
'L
Ai
3.46
3.45
583
558
533
512
3790
3630
3480
3350
0.679
1.24
116
1.72
2.08
TABLE XXIII
RELEVANT PROPERTIES OF FIRST STAGE - L, P. COMPRESSOR
=
-
k0.7392*
:O.15
M
0
0.4
4.92
2
0.1
0.2
0.3
0.4
3: C/u
0.427
0.447
0.468
0.488
Cul/u
0.394
0.367
0.339
0.310
c
/C
0.923
0.821
0.746
0.635
C1'/CX
1.361
1.294
1.246
1.185
En
c
U
uli
X
520
T
28.77
h
2.504
pr1
Assume T(1-0)
a(1-0)
Ma( 1
c
0
)
x
C
520
28.77
2.504
520
28.77
2.504
520
28.77
2.504
510
511
512
513
1106
1108
1109
1110
443
443
444
444
255
255
256
256
348
330
319
303
2
C 1/2gJ
2.41
2.17
2.04
1.84
0)
26.36
26.60
26.73
26.93
h(1
510
T(,-()
2
1 , (C /2gJ)
I-V
511
512
513
2.68
2.41
2.27
2.04
26.09
26.36
26.50
26.73
1
g-v
h(1-0)s
pr (1
)
2.320
2.338
2.347
2.364
P(1-0)
13.61
13.72
13.77
13.87
-X(1-0)
13.86
13.77
13.75
13.70
897
Al
3.52
891
3.49
117
889
3.47
886
3.46
TABLE XXIII CONT'D
d
d
max
32.8
32.65
32.55
32.5
37.7
37.5
37.4
37.35
u
597
571
547
525
Ii
3630
3490
3350
3220
0.711
1.30
118
1.79
2.20
r
It can be concluded now, that the characteristics to use in the design
of the L. P. compressor are those shown on Table XXI, that is:
FIRST STAGE L. P. COMPRESSOR CHARACTERISTICS
M
P
Q, ft. 3 /lb.
d, inches
u, ft./sec.
n, r.p.m.
Ai, Btu/lb.
-
0.4
0.2
0.1
0.2
0.3
0.4
0.445
0.466
0.488
0.509
892
28.3
887
28.2
884
28.15
882
28.1
573
547
525
503
3860
3700
3560
3410
0.653
1.2
1.65
These results are shown also in the curves of Fig. XXVII.
119
2.02
all
FS
rise
Irme AL 4 ha
4/10Y
5qpo
A= 0
,3.
38cc
039j
CLZ.
;fl6 0700j .mr
28.8
1.0I136ftI *a
0.7
12041 's] isod *,Is
Z614
[u'2.
1.0
3400 o 0
isool 0
tS.g
t.eI o 1t0
0
0.5-0
I.Ociry
AT';o
120
x
A)
%.l
O.So
. 4rg
SECTION B4
Part I
Preliminary Design of H. P. Compressor
To determine the speed of rotation and other relevant properties of
the first stage of H. P. compressor, the same principles will be used as
those employed in Section 3.
The flow of air after leaving the L. P. compressor passes through the
Intercooler, to follow then to the admission of the H. P. compressor.
It
will be assumed that the gas enters the H. P. compressor with a speed
Cox : 100 ft/sec., in that manner it has been amply accounted for losses
in the Intercooler.
From Table IV, Section 2,
Appendix A, the state of the
air entering the H. P. compressor is (point 3):
-L
h3
--
35.98 Btu/lb.
T 3 : 5500 Fa.
:;
pr3
14.7 x 2.65
p3
3-
Fig. XXVIII
3.047
-
39.0 psia
Again an efficiency of 90%
will be assumed for the guiding
OsX
vpne; and the computations for
the first stage will be done in a
similiar manner as that for the
Fz
L. P. compressor.
But first the
state of the air at point 3'
Fig. XXIX
be determined.
121
must
State (3) to state (3'):
-h
h
-
3
31
C2
ox
2gJ
-35.98 -
1002
64.4 x 778
35.78 Btu/lb.
From Air Tables:
0
T3
3'
pr3 ,P3 1
549 Fa.
3.031
39 x 3.031
:38.8 psia.
3.047
A table to determine the characteristics and properties of the first stage
of the H. P. compressor follows.
122
r
TABLE XXIV
RELEVANT PRCPERTIES OF FIRST STAGE - H. P. COMPRESSOR
M = 0.4
2 Cu/u
).
u
Cul/Cx
G1/ x
k a 0.667
0.1
0.2
0.3
0.4
0.445
0.466
0.488
0.509
0.370
0.340
0. 310
0.280
0.832
0.730
0.638
0.550
1.301
1.239
1.185
549
T 3'
35.78
07
549
35.78
549
35.78
549
35.78
'
h3
V - 2VE
f = 0.2
3.031
3.031
3.031
3.031
,
pr 3
38.8
p3
Assume T (;-0)
a (3-0)
38.8
38.8
539
540
541
542
1137
1138
1139
1141
455
455.5
456
456.4
262.2
262.7
263
263.2
342
325.5
312
300.5
)
Ma( 3 - 0
38.8
C
C
x
1
C 2 /2gJ)
2.33
2.12
1.94
1.80
h( 3 -0.)
33.45
33.66
33.84
33.98
539
I
(C3-2
h (3-0)s
pr(^,_O)s
Q
A
541
542
2.59
2.35
2.16
2.00
33.19
33.43
33.62
33.78
2.827
36.2
v(3-0)
540
5.51
356.4
1.358
2.845
2.861
2.874
36.38
36.60
36.80
5.50
5.47
5.45
355.7
1.353
123
354.0
1-345
352.3
1.337
TABLE XXIV CONT'D
d
d
17.65
17.60
17.55
17.50
21.18
21.11
21.07
21.0
max
u
590
564
579
517
n
6.390
6.110
5.860
5.650
Ni
0.695
1.27
1.74
2.13
124
VU
4
cl
Lk
StRgg- got('~ ?i!4ir.
pt~ssvrc
he&~
0.
S2.6 6300
690
18 2.0 6/09
-!7.7
0-7-
~0
590
0L2
. ...
6000
1.0 57.0
*S40
00.
17.3
)7~2~
QS66*OS24
AfMSb46I-
01o
aZ
o~k
07.2
0.45 0.4 ON'9
' 7Soo .0Vo0
VP-oCevy
IRATrio
125
V=/
From Table XXIV, it is seen, that the speed of rotation is high, compared with the speed of rotation of the L. P. Turbine, to which the H. P.
compressor is attached.
Nevertheless, the results obtained will be considered for the design
of the H. P. compressor, since any further reduction in the Mack number
would mean loss in the efficiency of the compressor.
The author suggests then, as a solution, the use of reduction gears between the L. P. turbine and the H. P. compressor.
It would be a matier of
studying possible arrangements of these reduction gears, in order to get the
least loss in power.
It looks like a good idea the possibility of arranging these reduction
gears together with the main propulsion reduction gears, since the L. P.
turbine is the one that supplies the power to the propeller.
In any case, the author leaves this question open, to be considered in
the case of an actual building of a Marine Gas Turbine Power Plant.
126
I
SECTIN B-5
Part I
Comvres or Airfoil Characteristics
A study of the characteristics and properties of airfoils, to be used
in the design of the compressors, shall be done now.
In addition to the formulas and derivations of Part I, Section B-3,
the following concepts and formulas are required.
Let's suppose a known foil has been fitted into a grid, of which two
foils are shown in fig. XXI.
For the purposes of this discussion it will be assumed that
F
the foils are sufficiently
apart so that no interference
--
exists between them.
Fig. XXXI
Symbols in Fig. XXXI and Fig. XXXII,
are self explicative.
Using Bernouilli's equation:
2
P2 -
on
1
f
x
2- 1
where:
/
2
_'l-2
(Sl)
,7g
It will be assumed that the den-
F/
sity
F
.
stage.
X
is constant throughout the
The force Y, is the force
due to momentum, and is equal to:
Y
Fig. XXXII
=
C-
x
(wul
-
\'-L) (82)
1 is the length of foil, or span.
127
The force X, due to pressure change, is:
x-
(p2
Pl
-
2
2
1 .Wul
-*u2
2
(83)
The resultant force F is then:
X 2 Y2
F
wl
vu29
-
2
+C cl
+
2
()
2
The radical in equation (84), shall be defined as the "mean velocity",
and denoted by W;
, thus:
/C 2
2
**
ul *u22
2
(85)
From the geometry of fig. XXXII:
sin
3.-
Cx
Wr"o
:
(86)
F
It can be concluded then that F is perpendicular to the mean velocity u..
By definition the lift coefficients::
F
CL-
(87)
bl
Therefore, substituting eq. (84), and eq. (85):
CL
2
t
ul - .ru2
(88)
AT
It is known that the circulation around the foil is given by:
r
:
fr
Is
(89)
In this case, as it can be observed in fig. XXXI:
(U
)
-
(90)
The general relation, for a single foil, for the force on the foil
per unit length is then:
1
vf, -
\r
)
128
(91)
From the grometry of fig. XXXIJ, the following retlations can be ob-
tained:
K/
ul
-
Wu2
C (cot 8 - cot!32 )
12
x
(92)
2C
2C
tan,/3
''~t :
(93)
Aul + \u2
2
-
cot /31
u,:
cot ,2
cot
1
C1
cotj$2
4
'
2
(94)
and:
C
-
cot
2
-
cot/3
2
L
-
+
co(95)
2
The work lost by drag can be found, from figs. XXXI and XXXII.
By
definition
-2
L
Lift force on foil
-
C W_
(96)
.bl
2
D
Drag force on foil
(97)
CD w 1fbl
C
2
Using common equation for work, it is obtained:
Work lost by drag:
(98)
L\WD
where:
: drag force on foil
D
uJ
and:
Therefore:
Q
WD
mean velocity of foil through gas
amount of gas flow
2
CD
x
b
1
fbl) x koo x
(CD kL
cfilg
C
(99)
_r_
2CYx
g
It will be assumed that this work lost by drag is a fraction of enthalpy
corresponding to the foil entrance velocity
,j such as:
7
2
(ico)
consequently, equatingto (99):
3
C
.
dOO
D
xl1
But from the geometry of fig. XXXII:
a 3
sin213
sin3 /3.
X1
(101)
Therefore:
C
D
2
sin 3
1
. 3
sin /3
(102)
In some literature, this loss coefficient
is found to be :
C
P
D
V(103)
but equation (102) is more exact.
The above theory applies best when in
fig. XXXII, the mean velocity f is close to both Wl and
No account will be made here for Constant's Rule for the deviation
angle effect, for which there are some formulas that approximate such deviation.
The way to determine the loss coefficient
is by means of a cascade
test, but as this is not availalbe, equation (102) will be used in its
determination.
This coefficient does not account for end effects in the
airfoils.
As stated previously, the design will be made fcr a symmetrical stage
at the pitch diameter.
By defining stage efficient as:
st
dory by the stage
Work
Work input to the stage
130
(19)
r
A
K
ell
3
Lj
LIQ 3
AL
~---(
A~
Li
Fig. XXXIII
Since the stage is symmetrical, it is necessary only, tc find the
efficiency for either the rotor or stator, since they will be both the
same.
Therefore, frr the rotor, from figs. XXXIII and XXII:
tj h 1 2
Ai
:
Jhi 2 - \
1
--
1
:-
JAi2
(104)
W.here 41.,is the loss in the rotor, and can be expressed as:
2
12g
with
(105)
a total loss coefficient, whichAaccurately given by:
+ 0.C2(b)+ o.o6 b
thus,
2 2
CL
(106)
includes the following effects:
(a)
5,
the same as in equation (30), and expresses the loss
due to drag.
(b) The second term expresses the loss due to friction surface
at ends of blade, since the blades are not of infinite
aspect ratio.
131
(c) The third term accounts for losses due to clearance between
blade and casing, and also losses produced by induced drag
and leakage other than that due to clearance.
The work input to the rotor is readily found to be:
2
2
1
(107)
2g
Consequently, equation (19) becomes:
1,
- 1
2
1-
It=1 - -7
~2
(108)
1 1
From geometry of fig. XXII, this can be expressed as:
lst
1
-
(sin 2/ 1 )
(s'n
In Part I, Section B-3 it
(109)
2
has been found that the diagram ratio for the
rotor, and at the same time of the stage is:
- cot/ 2
2(cot/
cot
)
Est :
1
- 'cot
3
(51)
and the velocity ratio:
cot 3 1 + coto,
It
must be kept in mind, that since the stage is symmetrical:
and:
21
132
(4S)
All the discussions in the foregoing are valid only for very small
values of Mack number; it is necessary therefore to consider it's effect.
Up to the value of velocity ratio corresponding to maximum efficiency,
the Mach number has little or no effect, but beyond that it decreases the
stage efficiency and also the value of the diagram ratio, md more so for
higher Mack numbers.
In connection with fig. XXXIV, representing two foils of a grid, it
has been observed that the pressure on the straight part of the foil, or
front, is greater than that on the back or curved part, when the grid is
rotating; in which case, applying Bernouilli's theorem, the velocity distribution would be as shown on
---
'fig.
XXXIV.
Consequently sonic
velocities will be reached first
on the back of the foil.
An ex-
pression for the ratio of the
areas A
2
to A
1
can be found, as
a function of Mach numbers, pssuming ideal gas flow, isentropic
Fig. XXXIV
conditions and stable flow; this
expression is:
A
-M
A
1
2
2(K - 1)
K+1
1
K +-1
.4K-i
(110)
2
For K = 1.4, the results obtained are as shown on fig. XXXV.
obtained is a curve of M
The curve thus
, for values of Mack number beyond this, the com-
pressor would simply refuse to perform any work of compression.
133
Fvi oxji
.
A 4 i.o #IrN
MACM
-it F
i.t
/.7
09
Uld
'U
0.8
0.7
0.6
A
.
.
0., .
o A- /A6
134
6.7
0.8
0.9
1. 0.
A critical value of Mack number is usually considered, and compressor
and M rit.
work thus in a region bounded by this two values, Ma
It is
usually difficult to predict where dg the values of Mrit lie.
In fig. XXXVI, the actual values of
IM
max
are plotted as a dotted line,
and by experimental work done Mcrit
has been found to be as shown, being
approximately about 0.4 below Mx'
In the present design it shall be
considered as lying a constant value
of 0.4 below M
fig. XXXV,
.
The curve of
max
will be used later to make
the computations for the foils.
Fig. X7XVI
The effect of the Mack number on the stage efficiency and diagram
ratio, has been approximated by Professor C. R. Soderberg by the following
formulas:
0
1
3
-0.78 (M - Mcrit 2
-M
(M
max
crit
(111)
31 - (M - Mcrit2
(112)
and:
&st
s
o
max
crit
where subscript 0 refers to the uncorrected values of
and Ist as given
by equations (51) and (109).
,
These corrections can be thus written
1 - 0.78 (M - Mrit) 2_
(M
135
M it)
(113)
and:
2
(M - Mrit)
;3
(M.x - Mrit)2
1
0.78 (M
M)rit)2
(Max
Mcrit)2
Thus:
E :
ktdt
=
The functions
4
.
-o
(115)
(116)
AZst - 0)
assume approximately the shape shown in fig.
and
XXXVII, and check fairly well with experimental data.
It must be kept in mind that neither curve nor formulas are exact,
,nd are only an approximate representation of the phenomena.
It is also important to mention that when M in considerationis less
than M1crit, the formulas do not apply, and:
4
-
-
1
(M
M
it)
M - -Mit
Fig. XXXVII
Other relations that will be used later, will also be given here.
The velocity of sound in a gas is given by:
at
-
gRT1
(117)
136
From fig. XXII, and using eqs.
z-
(52) and (69):
C
x
(& sin/
-
1
MAL
Cx
u
o-
1
sin>31
(118)
By definition:
s
jai
-.
u2
2g
Fig. XXXVIII
(49)
Therefore:
.
:
Jdh
(119)
2g
But using elementary thermodynamic principles it
can be shown also that,
between states (1) and (2):
k
m
JA h
RT 1
[
k-l
- 1
(p2 /P 1)C
(120)
k - 1
Substituting eq. (119) in (117) and collecting terms:
k-1
2
1
(121)
L
2+ 6
t (
k-~
0
The factor E
has been called sometimes "pressure rise coefficient",
and denoted by:
(122)
S ist
The pressure change from (1)
t-
(2) would then be:
)
M1
2
(123)
2
1
Equation (121) can be simplified by expanding it, and neglecting some terms,
137
and thus become:
2
P2/Pl
1 + k/2
02
1
st
(124
g2(124)
The deflection suffered by the gas passing through the stage is, referring
to fig. XXXII:
Q. eg2
(125)
The angle defining the position of the mean velocity of approach to
the foil, /co,
can be obtained knowing the relative position of the blades,
that is, their stagger angle.
If the blades havd one side straight, which will be the case, as
explained later; then:
/I,
= /9
-t (
(126)
Where / sis the stagger angle and 0
is the angle of attack to the foil.
From fig. XXXIV, it can be seen that:
'4A
A2A
1/,sAin
(127)
With all the relations in the foregoing, the compressor stage characteristics can be found, first for zero Mact number, and then for M Z 0.4 another
table will be shown.
The following design characteristics will be considered:
opening-pitch ratio
span-chord ratio
-C
1
0.6
: 3
b
stagger angle
/ 5 t = 450
: 1
pitch-chord ratio
b
symmetrical stage at pitch diameter.
From References (14) and (15), and after studying the shapes and properties of the different foils, ttme best airfoils to select would be
138
N. 2409-34 or N. 4409.
The author has selected the latter, since data on
that foil shape is more complete and of later issue.
After making the different corrections on the lift and drag coefficients, the following table is obtained for the airfoil N.A.C.A. 4409,
at an aspect ratio l/b : *and a Reynolds number
=
329000.
TABLE XXV
0(
-8
CL
-0.3
CD
0.02
12
16
20
24
0
4
8
0
0.3
0.6
0.89
1.16
1.35
1.25
1.0
0.01
0.013
0.028
0.055
0.09
0.135
0.191
0.360
-4
With all this information, the properties of the stage can be computed
in tabular form.
139
TABLE XXVI
CHARACTERISTICS OF AIRFOIL - N.A.C.A. No. 4409
M = 0
-4
-8
CL
-0.3
C
0.02
A
/ti
Rt
450
A t:
1/b : 3
0
4
8
0
0.3
0.6
0.89
1.16
1.35
1.25
1.0
0.01
0.013
0.028
0.055
0.09
0.135
0.191
0.360
49
53
:329,000
41
45
37
12
33
16
29
20
25
24
21
tan / 00
1.327
1.150
1.00
0.869
0.754
0.649
0.554
0.466
0.384
cot /Al+cot /32
1.508
1.739
2.00
2.30
2.653
3.08
3.607
4.29
5.212
(cot1
/-.+cotA) 0.754 0.87
1~2
f(cot / 4-cot /32 )
1*(cotAli-co/ 2)2
+ Cot )
L(b)
cot/3
-
cot
1.00
1.15
1.327
1.54
1.804
2.15
2.606
0.568
1.756
1.00
1.323
1.76
2.37
3.253
4.602
6.795
1.568
2.756
2.00
2.323
2.76
3.37
4.253
5.602
7.795
1.252
1.325
1.414
1.525
1.661
1.835
2.061
2.368
2.791
-0.15
0
0.15
0.30
0.445
0.58
0.675
0.625
0.500
-0.188
0
0.212
0.458
0.739
1.065
1.392
1.48
1.395
2 cot7 3
1.32
1.739
2.212
2.757
3.392
4.145
4.999
5.770
6.607
cot /31
0.66
0.869
1.106
1.379
1.696
2.073
2.449
2.885
3.303
56.6
49
42.1
36
30.5
25.7
21.8
19.1
16.9
TABLE XXVI CONT'D
2 cot/32
2
cot
2
2 -1
1.696
1.739
1.788
1.843
1.914
2.015
2.215
2.81
3.817
0.848
0.869
0.894
0.922
0.957
1.008
1.108
1.405
1.909
49.7
49
48.2
47.4
46.3
44.8
42.2
35.4
27.7
-6.9
0
6.1
11.4
15.8
19.1
20.4
16.3
10.8
sinl/
0.835
0.755
0.670
0.588
0.507
0.434
0.371
0.327
0.290
sin2
0.697
0,570
0.449
0.345
0.257
0.188
0.137
0.107
0.084
0.510
0.430
0.353
0.282
0.218
0.162
0.114
0.0756
0.0458
0.0273
0.0132
0.0165
0.0342
0.0648
0.1045
0.1623
0.2706
0.6610
0.02(b/1)
0.0067
0.0067
0.0067
0.0067
0.0067
0.0067
0.0067
0.0067
0.0067
0.06(b 2 /1)C2
0.0018
0
0.0018
0.0072
0.0158
0.0269
0.0364
0.0312
0.020
0.0358
0.0199
0.025
0.0481
0.0873
0.1381
0.2054
0.3085
0.6877
0.763
0.755
0.745
0.736
0.723
0.705
0.672
0.579
0.465
0.582
0.570
0.555
0.542
0.523
0.497
0.451
0.335
0.216
1.197
1.0
0.809
0.637
0.492
0.378
0.304
0.320
0.389
0
0.191
0.363
0.508
0.622
0.696
0.680
0.611
0.131
0.133
0.172
0.222
0.295
0.454
1.125
sin 3
O
sin /2
2
sin2
,3
2 2
2
sin2/ /sin2/ 2
1- (sin2 /31/sin 2 /22 )-0.197
1- sin 2 /3 1
sin2
2
-0.182
- .27
0
0.212
0.398
0.557
0.692
0.773
0.690
0.536
0.664
0.576
0.500
0.435
0.377
0.325
0.277
0.233
0.192
1.1815
- 00
0.869
0.867
0. '82g
0.778
0.705
0.546
-0.125
(St-0)
0
st-0
B
~I~4~Cr~i
AI ~
4/~C.4
#si~/
~
~
Mo449~?
o.8 t6
~
~o
0.5
B
j4
OA#
o~
0.2
B
0I
4
0*
0
Jo
to
3~
/3'
142
40
.fo
Q
era
a of
A. V.4. A(.. 4409
IVF'~s.
Aeyio,.as
,ric.
= 3R9,000
,e
1/A.
f=
0.2
A/1y= 3-0
450
1.0
09
.6
0.7
0.(
o .5
o.o
0.2
0./
0
\
-o. 2.
0
0.1
.2
O.4
0.3
V-R..CT
4yi A-
143
HM. -,996
0.5
o.6
The values obtained on Table XXVI, are to be corrected for the Mach
number to use in the design, which in Part I, Section B-3, was found to be:
M = u/a1 = 0.4
for both L. P. and H. P. compressors.
, fig. XXXV will be used.
For Mmax and M
crit
The value M = 0.4 does not enter directly on eqs. (113) and (114) for
ke
and kt
,
but must be modified first as shown in eq. (118), that is:
M :(u/a)
(128)
0
sin
1
Table XXVII, following, shows the corrections on the stage characteristics, according to the derivations given before.
144
TABLE XXVII
CORRECTIONS TO AIRFOIL CHARACTERISTICS
MACH NUIMBE
u/a 1 = 0.4
0.664
(
-0.249
(st-C)
0.500
0.435
0.377
0.325
0.272
0.233
0.192
0
0.212
0.398
0.557
0.692
0.773
0.690
0.536
1.181
-
0.869
0.867
0.828
0.778
0.705
0.546
-0.125
I(st-0)
sin 131
0.835
0.755
0.670
0.588
0.507
0.434
0.371
0.327
0.290
2/Ai2
0.719 0.795
Ssi
1
0.896 1.02
1.283
1.382 1.617
1.834 2.068
0.670
1.00
1.00
1.00
1.00
1.00
1.00
0.078
0.149
0.270
0.6
0.6
0.6
0.6
0.6
0.6
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.4
0.795
0.763
0.746
0.740
0.744
0.749
0.747
0.713
0.663
0.318
0.305
0.298
0.296
0.298
0.299
0.299
0.285
0.265
0.24
0.156
0.028
-0.304
-0.302
-0.301
-0.301
-0.315
-0.335
0.6
0.39
0.07
-0.76
-0.755
-0.752
-0.752
-0.787
-0.837
x2
0.36
0.152
0.005
-0.577
-0.570
-0.565
-0.565
-0.62
-0.7
0.78 x x2
0.28
0.118
0.004
-0.45
-0.444
-0.44
-0.44
-0.483
-0.546
1 - 0.78 x 2
0.72
0.882
0.996
1.45
1.444
1.44
1.44
1.483
1.546
1
0 6
0 . 84
0
1.577
1.570
1.565
1.565
1.62
1.7
merit
m
max
-m
crit
/sin3 1
IA
m
Mrit
X
114 - Mcrit
m
-
72
- M
.
0.549
.
0.478
max
c-n
0.576
9s
ke
*
TABLE XXVII CO NT ID
0.947
0.998
1.255
1.253
1.25
1.25
1.273
1.304
0.896
0.959
0.998
1.0
1.0
1.0
1.0
1.0
1.0
0.962
0.988
1.0
1.0
1.0
1.0
1.0
1.0
1.0
0
0.212
0.398
0.557
0.692
0.773
0.690
0.536
- 00
0.869
0.867
0.828
0.778
0.705
0.546
-0.125
*
kI
0.862
Est
-0.223
It st
1. 136
* For all values above
: 0.5, k.
and k,
are equal to 1.0, since M < Mcrit.
The above table could be repeated for other values of u/a1 difV erent from 0.4, but since in this design
that is the value we are interested in, further calculations are not necessary.
APPENDIX C
147
'rAC
3;Q
.
...
....
..
. ....
..
.......
.....
. .......... ......
..
....
...
..
.
. ...
...
.
......
...
...
..... . . . ....
..
..
......
. . . . . . . ..
...
..
...
........
......
.
.....
.
........
....
....
...
A /44
...
.....
.... ..
.
...
......
.......
-----------------...... ...
.........
.
..
.......
...
.......
...
/.0 6 .....
...
...
................
...
.......
o. a7f
..
. ..
....
.....
...
..
------
.
..................
............
LOS
--
-----..........
.
.. .......
..
.....
......
/2
RX
148
PA
NS
1 ON
7?4-rio
14.
SECTION C-1
Part I
Detail-Desian of L, P. Turbine
With the purpose of distributing the total heat drop across the turbine,
in the stages, a plot of specific volume against enthalpy is quite helpful,
and thus anticipate the stage conditions.
Experience shows that first stages should be loaded slightly higher,
consequently the enthalpy drop through the first stages is larger than through the last stages.
The type of blade to use is that given on table XV, with the following
characteristics:
: sinO
sin/3 2
1
- 0.5
800
Best inlet angle
1.0
S
1.0
0.75
Elopt
th
For:
:0.6
st
Is
0.891
Since the number of stages is known, the Reheat factor can be determined.
Using fig. XLI, taken from reference (1), it is obtained:
6
No. of stages
For:
14;
r
4.322;
(l+ R).,
r
For:
(1+ R),Therefore:
(1 R)
%
0.891
(-s in table)
1.022
4.322
-
:1.001
1.022
1.001
3
1.021
149
1.106
The state of the gas entering the L. P. turbine, fom Table IV, is:
T
18600 Fa.
h.
370.92 Btu/lb.
p
63.6 psia
269.1
pri
The required enthalpy drop across the turbine is:
- 132.3 Btu/lb.
Ai.
ie
The internal efficiency of the turbine is now:
i-(1+R)
-
st
s
1.021 x 0.891
0.91
Referring to fig. II, then:
Isentrovic enthalpy drop across turbine:
h
145.3
132.3
0.91
Therefore:
e
370.92
145.3
-
14-s
-
225.62 Btu/lb.
pr 1
70.04
p-
Pe-4
16.54 pdia.
The enthalpy drop across the turbine will now be distributed among
the stages.
If every stage would take equal enthalpy drop then:
132.3 = 9.45
A
14
Butas stated previously the first stages will take more of the load
than the rest.
age then:
If for the first,
Ai l
it
is assumed it
a 9.45 x 1.1
-
10.4
150
takes 10% 6ver the aver-
-
- -----
For the other stages it will be distributed in a gradually decreasing
manner, shown on Table XXVIII.
In this table, all symbols refer to fig. XLII.
ZeF
-
--
-
-
-
I.
Les
9
Fig. XLII
151
TABLE XXVIII
st
Stage
pre
Lu
Inlet
conditions
1
L, P. TURBINE - PROPERTIES OF GAS PER STAGE
o.891
0
370.92
210.14
9.8
Te
e
370.92
269.1
63.6
1860T
359.25
245.32
58.0
1822
11.62
349.08
225.92
52.8
1785
12.50
339.11
208.00
48.2
1748
13.42
329.32
191.47
43.9
1713
14.45
319.15
175.33
39.8
1676
15.59
310.26
162.13
36.2
1641
16.78
300.87
149.00
32.9
1606
18.07
291.57
136.76
29.9
1571
19.45
10.82
11.00
193.44
9.6
10.77
178.10
9.5
10.66
163.82
9.4
10.55
302.02
8
Pe
11.21
311.42
7
228.1
10.0
320.92
6
pres
11.44
330.92
5
24.78
10.2
340.32
4
ies
11.67
350.32
3
269.1
10.4
360.52
2
LAh
150.52
9.3
10.43
292.70
138.19
TABLE XXVIII CONT'D
9.2
9
10.32
283.52
10
126.78
9.1
116.17
265.42
106.34
256.47
97.18
247.52
273.32
114.94
24.5
1502
22.70
264.32
105.18
22.2
1468
24.50
255.37
96.10
20.1
1434
26.40
246.42
87.61
18.2
1400
28.60
237.52
79.74
16.5
1365
31.00
10.00
8.9
238.62
= 132.)
21.00
10.05
8.95
14
1536
10.05
8.95
13
27.1
10.10
9.0
12
125.41
10.20
274.42
11
282.38
80.67
E A h : 148.45
2Ah a 148.45 = 1.021, thus it checks Reheat factor already computed.
h
145.3
Turbine exhaust pressure
T 16.5, also checks closely enough with that obtained in the preceding page.
p14-e
Ace
30
s5
7.
080
6 7TfC
-,;k
- ve,'v
"o0N
The new value obtained for the exhaust state of the L. P. turbine does
not coincide with the value obtained in Table IV, in the original computation
of the cycle.
The difference is due to the fact that the internal efficiency
of the turbine was assumed to be:
0.85, and the value obtained after
the stage characteristics were determined and the Reheat "'actor known, is
=0.91.
The correct design procedure would be to recompute all the design characteristics up to this point with this new value of internal efficiency;
nevertheless the author will proceed with the characteristics as determined,
since in any case the design will be on the safe side; and as a matter of fact
the flow areas will be about 5% larger than required as seen from the values
of the originally assumed exhaust specific volume and the new one determined
two pages before.
The detail design, stage by stage can now be done, but first an explanation of some facts necessary for the comprehension of the tables that
follow is necessary.
The dimensions determined in the previous sections are summarized ina
table under the heading Dimensions.
The only symbols here that need ex-
planation are:
,
is the clearance between blade and casing, and between blade and
drum, and is defined previously as:
-0.01
o(0* and/
1*
are the best inlet angles to the type of blade selected.
AA is the leakage area and is given by:
A AR
As
w d(1+1/d) S
for the rotor blades
w d(l
for the stator blades.
-
l/d)c
155
LO4RAN4wrR
vecogiry4
61' '
tre
iam
M4O
Y4P4 l
MO
I.z
I.'
I.0
0.9
t4#
Ai
4.7
*.81
0!
0./
0
eao
400
boo
84.
AW
11
156
Iwoo
I/&0
2000
The flow area will be determined by means of the continuity equation:
...
C
Al~
G =AOCO
(129)
,
It will be assumed that the axial component of flow velocity, C0, C1
etc., is constant therefore:
A
A,
Lo
A2
.
(130)
*-2
The following assumptions are also considered:
(a) Inlet pressure for stage is the same throughout the stage, and
also volume and temperature.
(b) In each stage the enthalpy drop is divided equally for the rotor
and the stator blades.
(c) Stage blade characteristics, as determined or given on Table.
(d) Viscosity of the gas, as given by experimental curve on fig. XLIII
The second part, will be labeled "Performance", where the following considerations, appearing in that table, must be explained.
The process will be divided in each stage, for the rotor and for the
stator.
If the design wasn't for a symmetrical stage the quantities in each
column would differ, but in this case practically all of them are the same.
For a clear understanding of the tables and the explanations following,
refer to figs. IX and XLV.
Fga
XLI
AW2
-
S3
F ig. XLV
157
In the state of the gas entering a stage, besides the enthalpy of the
gas as given by its properties, it must be considered the energy due to the
velocity with which the gas enters the stage.
"carry-over".
This energy is called the
Condequently the true state of the gas entering the state is
given by the "stagnation point".
Carry-Over:
1
2(
)
jh 2
2gJ
2gJ
It will be assumed that for the first stage the carry-over is zero,
since it has bee-,n previously assumed that the gas entering the turbine has
negligible velocity.
The carry-over to the first stage rotor blades, pro-
duced by the leaving velocity of the stator blades will be neglected, in
order to get symmetry.
This compensates for the assumption that the enter-
ing velocity to the first stage is negligible.
The loss coefficients will be taken from table XV.
The velocity developed in the stator blade, C1 , iS given by:
2gJ
(132)
)
(h +
C1
2gJ
l +
For the rotor, the velocity is expressed by a similar equation to the above.
C2
0__
fixes the stagnation point, which as said above will be zero for
2gJ
the first stage.
The values of the velocity ratios and diagrams ratios as given by
eqs. (20) and (21), and found in Table XV.
The work done on the stator is:
J ig1
;Ci -2
-
(1 2g
2g
158
L)Cl
=
11
2g
(133)
And the work done on the rotor is:
2
2
2
2
W2 .
2
2
2
(134)
2g
2g
Total work for the stage:
A'W2
ZA iT -i ah+
(135)
The Reynolds number for the stage is given by:
-
C b
ge
p 1-g
(
R,
(136)
159
TABLE XXIX
p
14.7 psia; Ti
=
5200 Fa.;
G = 64.7 lbs./sec.; -: 2700 r.p.m.;
7
DIMENSIONS
STAGE
1
2
ITEM
UNITS
d
in.
u
ft./sec.455
A
ft.
STATOR
38.5
2
1/d
ROTOR
38.5
455
STATOR
38.5
455
3
ROTOR
38. 5
455
STATOR
38.5
455
ROTOR
38.5
455
3.39
3.39
3.65
3.65
3.92
3.92
0.105
0.105
0.113
0.113
0.121
0.121
1
in.
4.04
4.04
4.34
4.34
4.67
4.67
b
in.
1.35
1.35
1.45
1.45
1.56
1.56
0.5
0.5
0.5
0.5
0.5
0.5
0.04
0.04
0.04
0.04
0.05
0.05
sinso
: sin
2
in.
2
ft.
0.01
0.01
0.03
0.04
0.04
0.05
A AA
ft.2
3.40
3.40
3.68
3.69
3.96
3.97
0( 0* 0/1
Deg.
*
AA
80
80
160
80
0.91
TABLE XXX
STATOR
ITEM
6
5
4
STAGE
ROTOR
STATOR
ROTOR
STATOR
ROTOR
d
38.5
38.5
38.5
38.5
38.5
38.5
u
45.5
45.5
45.5
45.5
45.5
45.5
A
4.22
4.22
4.55
4.55
4.90
4.90
1/d
0.131
-. 131
C.141
0.141
0.151
0.151
1
5.03
5.03
5.42
5.42
5.83
5.83
b
1.67
1.67
1.80
1.80
1.94
1.94
0.5
0.5
0.5
0.5
0.5
0.5
0.05
0.05
0.05
0.05
0.06
0.06
0.04
0.05
0.04
0.05
0.04
o.o6
4.27
4.29
4.61
4.96
4.98
sin
sin
AA
A+& A1
1
80
80
80
*
IDo *
2
161
80
80
TABLE XXXI
DIMENSIONS - L. F. TURBINE
STATOR
d
38.5
u
455
ROTOR
38.5
STATOR
38.5
ROTOR
3..
ROTOR
STATOR
5 35
455
455
10
9
8
7
STAGE
38.5
STATOR
33.5
455
,5t-
ROTOR
38.5
455
A
5.28
5.28
5.68
5.68
6.13
6.13
6.63
6.63
1/d
0.163
0.163
0.175
0.175
0.190
0.190
0.205
0.205
1
6.29
6.29
6.76
6.76
7.3
7.3
7.9
7.9
b
2.09
2.09
2.25
2. 2
2.43
2.43
2.63
2.63
sin o(
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.06
0.06
0.07
0.07
0.07
0.07
0.08
0.08
0.04
0.06
0.05
0.05
0.07
O.05
0.08
5.32
5.34
5.73
6.10
6.20
6.63
6.71
siLnP2
AA
A
*
A+-
80
0
30
5.75
80
80
162
TABLE XXXII
DIMENSIONS - L, T, TURBINE
STAGE
11
STATOR
d
38.5
u
455
12
ROTOR
38.5
4Z5
STATOR
38.5
455
13
ROTOR
38.5
455
STATOR
38.5
455
ROTOR
38; 5
455
14
STATOR
38.5
455
ROTOR
38.5
455
A
7.15
7.15
7.71
7.71
8.35
8.35
9.05
9.05
1/d
0.221
0.221
0.238
0.238
0.258
0.258
0.28
0.28
1
8.52
8,52
9.18
9.18
9.95
9.95
10.78
10.78
b
2.84
2.84
3.06
3.06
3.32
3.32
3.66
3.66
sin( q.
sin 1~
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.09
0.09
0.09
0.09
0.10
0.10
0.11
0.11
0.06
0.09
0.06
0.09
0.06
0.10
0.06
0.10
7.21
7.24
7.77
7.80
8.41
8.45
9.11
9.15
AA
At.
A
0 * :p 1(* 80
80
80
80
163
80
TABLE XXXIII
PERFORMANCE - L. P. TURBINE
3
2
STAGE
ROTOR
STATOR
ROTOR
STATOR
ROTOR
STATOR
370.92
360.52
350.32
id.
11.67
11.44
11.21
p
psia
63.6
58.0
48.2
T
0 Fa.
Btu/lb.
h
1860
ft. 3 /lb.
10.82
L
1+
1785
11.62
1.088 x 10-6
lb.-sec.
p
1822
1.075
12.50
x
10
1.062 x 10-6
-66
x1
-6
0.0712
0.0712
0.0712
0.0712
0.0712
0.0712
0.0182
0.0226
0.0180
0.0228
0.0179
0.0229
0.0894
0.0938
0.0892
0.0940
0.0891
0.0941
0.918
0.915
0.918
0.914
0.919
0.914
Ah
Btu/lb.
1.85
1.9
1.8
1.79
1; 77
1.75
h
Btu/lb.
5.84
5.84
5.72
5. 72
5.605
5.605
Ah + h
id.
7.69
7.74
7.52
7.51
7.375
7.355
id.
7.03
7.07
6.90
6.87
6.78
6.72
2
(h +,A h)
cl
2cos
f-t./sec . 594
ur 2
1
2cos f
AiW2
587
583
581
1.732
1.732
1.732
1.732
1.732
1.732
0.767
0.77
0.773
0.775
0.78
0.784
0.740
0.740
0.741
0.742
0.742
0.743
0.260
0.260
0.259
0.258
0.258
0.257
5.20
5.20
5.11
5.09
5.03
4.98
2
92
Ai
589
595
Btu/lb.
164
TABLE XXXIII CONT'D
1
R
C2
ft. /sec. 297
300
297
299
296
294
e
A i;
I st
B tu/1b.
10.4
10.2
0.891
0.891
165
10.01
0.892
TABLE XXXIV
PERFORIANCE
-
L, P, TURBINE
6
5
4
STAGE
STATOR
ROTOR
STATOR
ROTOR
STATOR
ROTOR
340.32
330.92
320.92
Xh
11.0
10.77
10.66
p
48.2
43.9
39.8
v
13.42
14.45
1.05 x 10-6
L
Ah
h
ht
h
$2(h+
A h)
2
cl
2cost(
2
15.59
1.039 x 10-6
1.025 x 10-6
0.0712
0.0712
0.0712
0.0712
0.0712
0.0712
0.0177
0.0232
0.0165
0.0235
0.0173
0.0237
0.0889
0.0944
0.0877
0.0947
0.0885
0.0949
0.919
0.914
0.919
0.914
0.919
0.913
1.73
1.70
1.68
1.65
1.63
1.63
5.50
5.50
5.385
5.385
5.33
5.33
6.64
7.20
7.065
7.035
6.96
6.96
6.64
6.58
6.49
6.43
6.40
6.36
577
2cos
1676
1713
1748
T
574
570
568
566
564
1.732
1.732
1.732
1.732
1.732
1.732
1
2
0.789
0.793
0.798
0.801
0.804
0.807
.744
0.256
0.745
0.746
0.746
0.746
0.746
1- 6
2
1- E
0.255
0.254
0.254
0.254
0.254
4.94
4.89
4.84
4.78
4.76
4.74
1
2
Ali
-2
'L
2
292
287
290
166
286
285
284
TABLE XXXIV CONT'D
Re
176,000
175,000
A 17
st
178,000
9.84
9.62
9.50
0.893
0.892
0.891
167
TABLE XXXV
PERFORMANCE - L. P. TURBINE
7
STAGE
STATOR
8
ROTOR
9
STATOR
311.42
ROTOR
10
STATOR
ROTOR
STATOR
ROTOR
302.02
292.70
283.52
10.
10.43
10.32
10.20
36.2
32.9
29.9
27.1
ih
p
1641
T
1606
1.014 x 10-6
1536
1571
1.0 x 10-6
0.99 x 10-6
0.977 x 10-6
0.0712
0.0712 0.0712
0.0712
0.0712
0.0712
0.0712
0.0712
0.0170
0.0237 0.0168
0.0241
0.0165
0.0244
0.0162
0.0247
0.0882
0.0949 0.0880
0.0953
0.0877
0.0956
0.0874
0.0959
0.919
0.913
0.919
0.912
0.919
0.912
0.92
0.912
A h
1.615
1.602
1.584
1.58
1.565
1.558
1.543
1.54
h
5.275
5.275
5.215
5.215
5.16
5.16
5.10
5.10
6.89
6.877
6.799
6.795
6.725
6.718
6.643
6.64
6.33
6.28
6.25
6.19
6.18
6.12
6.11
6.05
p
/2
h+L
h
e2(h*&sh)
2y
W2
1
2
2cos
2cos
1
2
E1
62
2
1i
1
Re
Aiw
st
1-,2
2
563
561
560
557
556
554
553
551
1.732
1.732
1.732
1.732
1.732
1.732
1.732
1.732
0.808
0.811
0.813
0.817
0.818
0.821
0.822
0.826
0.747
0.748
0.747
0.747
0.748
0.748
0.748
0.748
0.253
0.252
0.253
0.253
0.252
0.252
0.252
0.252
4.72
4.69
4.66
4.64
4.62
4.58
4.57
4.53
2
283
282
179,000
281
280
180,300
279
278
179,400
278
276
183,000
9.41
9.3
9.2
9.1
0.892
0.891
0.891
0.891
168
TABLE XXXVI
PERFORMANCE - L, P, -TURBINE
STAGE
11
12
STATOR
ROTOR
STATOR
13
ROTOR
STATOR
14
ROTOR
STATOR
ROTOR
i
274.42
265.42
256.47
247.52
Zh
10.10
10.05
10.05
10.00
24.5
22.2
20.1
18.2
T
1468
1502
22.70
24.50
0.968 x 10-6
(L
1434
1400
26.40
0.955 x 10-6
28.60
0.942 x 10
-6
0.93 x 10-6
0.0712
0.0712
0.0712
0.0712
0.0712
0.0712
0.0712
0.0712
0.0158
0.0250
0.0155
0.0254
0.0151
0.0258
0.0146
0.0263
0.0870
0.0962
0.0867
0.0966
0.0863
0.0970
0.0858
0.0975
0.92
0.912
0.92
0.912
0.92
0.911
0.921
0.911
Ah
1.525
1.525
1.505
1.515
1.495
1.505
1.493
1.506
h
5.05
5.05
5.025
5.025
5.025
5.025
5.00
5.00
h+ A h
6.575
6.575
6.53
6.4
6.520
6.53
6.493
6.506
e 2(h4 &h)
6.05
5.99
6.01
5.96
6.00
5.95
5.98
5.93
C1
55 0
12
2
2cosot
2cos
1
2
1- E
1- a
1
~
1
Re
A iw
Ist
Ak
2
2
2
548
549
547
548
546
548
545
1.732
1.732
1.732
1.732
1.732
1.732
1.732
0.827
0.83
0.829
0.832
0.83
0.833
0.831
0.835
0.748
0.749
0.748
0.749
0.749
0.749
0.748
0.749
0.252
0.251
0.252
0.251
0.251
0.251
0.252
0.251
4.52
4.49
4.50
4.46
4.49
4.46
4.47
4.44
2 1.732
27 6
275
184,000
275
274
184,000
274
273
184,000
9.01
8.96
8.95
0.892
0.892
0.891
169
274
273
188,000
8.91
0.892
m
Tre results obtained are quite satisfactory, especially the turbine
stage efficiency, which came out to be 0.891 or over, thus checking the value
obtained in the preliminary design, when the stage characteristics were determined on the basis of dimensionless coefficients.
The work output of the L. P. turbine is then the sum of the work done
by each stage and is equal to:
(Woutput)
-
133.41 Btu/lb.
which is slightly higher than the work required from the L. P. turbine
(132.3 Btu/lb).
The leaving loss, as stated previously, is the carry-over from the last
stage which is:
(
-
2
2.
:(l6
2
)$2 (h A h)
2(h+
)
Ah
AhL
2gJ
- 0.251 x 5.93
- 1.49 Btu/lb.
If this value is compared with the value assumed in the preliminary
design, it will be seen that they are in relative agreement.
the per-cent leaving loss is:
- 1.49 x 100
133.41
and originally it was assumed 1.2%.
170
w
1.12%
In this case
SECTION C-2
Detail Design of H. P. Turbine
Using the same relations and procedure as for the L. P. turbine in
Section C-1:
State of the gas entering H. P. turbine:
T5
- 1860 0 Fa.
102.9 psia
-
p5
h
5
: 370.92 Btu/lb.
= 269.1
pr
5
V5
- 53,25 x 1860 = 6.
69
cu. ft. /lb.
144 x 102.9
2.65:
The expansion ratio through the H. P. turbine is for r
1.62
102,9
p /p6
63.6
5
The stage efficiency, using the same type of blades as for the L. P.
turbine is:
= 0.891
Ist
From fig. XLI, for the above stage efficiency and r
(1 + R )O
For
Jst
-1.007
1.62
0.891 and r
(11i R) .
1.07:
- 1.001
Therefore:
1.007 = 1.006
1.001
(1 t- R)
and the turbine internal efficiency is then:
i
=
-
+- R)
st x (1
0.891 x 1.006
0.896
171
(7 stages)
1.62:
The enthalpy drop across the H. P. turbine, is, from Table IV:
h
5
- h
6
49.15 Btu/lb.
-
The isentropic enthalpy drop would then be:
h
- h
5
6s
: 49.15 x
1
0.896
54.90 Btu/lb.
Therefore:
h6
s
- 370.92 - 54.90
316.02
for which, from the air tables:
170.6
p6s
=
T
- 1658
p
x P6 s
-p
5s
102.9 x 170.6
269.1
-
65.2 psia
consequently, the specific volume of the gas leaving the turbine is:
for
h6
u 370.92 - 49.15 = 321.77 Btu/lb.
and
T6
: 1679
v6
RT 6
p
-
Fa.
: 53.35 x 1679
65.2 x 14
9.55 cu.ft/lb.
For equal drop of enthalpy across the stages:
Ai
: 42.15 : 7.02 Btu/lb.
7
In the distribution through the stages, for the same reasons given in
Section C-1 the drop across the first stages will be made greater.
172
TABLE XXXVII
st = 0.891;
i e
STAGE
Inlet
conditions
- 0.896
pre
Ah
ies
pr
P
T
v
370.92
269.1
370.92
269.1
102.9
1860
6.69
363.62
254.1
362.72
252.2
96.4
1833
7.04
355.54
238.2
90.4
1807
7.40
348.45
224.73
84.7
1781
7.79
341.46
212.23
79.5
1755
8.17
334.57
200.21
1730
8.60
327.73
188.9
69.8
1704
9.03
320.94
178.14
65.2
1679
9.55
1 7.3
2
8.08
7.2
356.42
3
7.97
7.1
349.32
4
7.86
7.75
74.5
7.69
190.26
6.80
7.63
321.77
2Ai
201.64
6.85
328.57
7
213.65
6.9
335.42
6
226.34
7.0
342.32
5
239.82
49.15
179.42
-~6h = 55.18
2:,h = 5.18 = 1.006, which checks the value of the Reheat factor.
h
54.9
173
F1m
rH4~.PM
mj,
ePFe;Ffc
7
Vol-A
-Q9s
Af6 5T-. /I
10
IE
8
7
4
-5.
4
a
2
'I
40
to0
4 T4 A-,Y
174
-
k
11.1
r
TABLE XXXVIII
DIAENSIONS - H, F, TURBINE
UNITS
STATOR
3
2
1
STAGE
STATOR
ROTOR
ROTOR
STATOR
ROTOR
d
in.
26
26
26
26
26
26
u
ft./sec400
400
400
400
400
400
A
2.717
Z.717
2.857
2.857
3.009
3.009
1/d
0.184
0.184
0.194
0.194
0.204
0.204
1
in.
4.79
4.79
5.04
5.04
5.31
5.31
b
in.
1.66
1.66
1.68
1.68
1.77
1.77
0.5
0.5
0.5
0.5
0.5
0.5
in.
0.05
0.05
0.05
0.05
0.05
0.05
ft. 2
0.022
0.032
0.023
0.034
0.024
0.036
2.739
2.749
2.880
2.891
3.033
".045
sin (1= sin92
f t.
A+A A
0(
*
:
1*
2
Deg.
80
80
80
175
80
80
80
TABLE XXXIX
DIMENSIONS - H. P. TURBINE
5
4
STAGE
STATOR
ROTOR
ROTOR
STATOR
7
6
STATOR
ROTOR
STATOR
ROTOR
d
26
26
26
26
26
26
26
26
u
400
400
400
400
400
400
400
400
A
3.153
3.153
3.319
3.319
3.484
3.484
3.685
3.685
1/d
0.214
0.214
0.225
0.225
0.236
0.236
0.25
0.25
1
5.56
5.56
r,.85
5.85
6.14
6.14
6.50
6.50
b
1.85
1.85
1.95
1.95
2.05
2.05
2.17
2.17
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.05
0.05
0.06
0.06
0.06
0.06
0.06
0.06
AA
0.025
0.038
0.026
0.041
0.027
0.043
0.028
0.046
A4A A
3.178
3.191
3.345
3.360
3.511
3.527
3.713
3.731
c0 *
<1
sinf 2 0.5
*
sin
80
80
80
80
176
80
80
80
80
TABLE XL
PERFORMANCE - H, P, TIRBINE
STAGE
1
UNITS
STATOR
Btu/lb
h
2
ROTOR
STATOR
370.92
id.
3
ROTOR
STATOR
ROTOR
363.62
356.42
8.08
7.97
8.2
102.9
90.4
p
psia
T
0 Fa.
v
cu. ft/lb.
6.69
7.04
7.40
lb-sec,
1.088 x 10-6
1.079 x 10
1.07 x 10-6
96.4
1860
1833
1807
ft.
0.0712
0.0712
0.0712
0.0712
0.0712
0.0712
0.0166
0.0242
0.0164
0.0244
0.0162
0.0247
0.0878
0.0954
0.0876
0.0956
0.0874
0.0959
0.919
.0.913
0.920
0.913
0.920
0.913
Btu/lb. 1.100
1.237
1.265
1.24
1.224
1.206
h
id.
4.200
4.200
4.04
4.04
3.985
3.985
h+6 h
id.
5.30
5.437
5.305
5.28
5.209
5.191
id.
4.87
4.96
4.88
4.82
4.79
4.738
p
L
Ah
2(A h + h)
Cl
ft./sec49 6
"'
2cos
1
2cosh2
6
E2
1-f1
1- 62
A iwi.
Ai T,2 2
1
"I-,
C
2
499
495
491
490
487
1.732
1.732
1.732
1.732
1.732
1.732
0.807
0.802
0.809
0.815
0.816
0.821
0.746
0.745
0.746
0.746
0.748
0.748
0.254
0.255
0.254
0.254
0.252
0.252
3.69
3.64
3.59
3.57
3.54
Btu/lb. 3.63
ft/sec250
252
249
177
247
246
245
TABLE XL CONT'D
R
284,000
e
Ai
Ist
Btu/lb.
288,000
290,000
7.32
7.23
7.11
0.F93
0.895
0.892
178
TABLE XLI
PERFORVANCE - H, P. TURBINE
ROTOR
STATOR
ROTOR
STATOR
ROTOR
STATOR
ROTOR
STATOR
7
6
5
4
STAGE
349.32
342.32
335.42
328.57
Ih
7.86
7.75
7.69
7.63
p
84.7
T
79.5
7.79
v
1.061 x 10-6
1704
1730
1755
1781
69.8
74.5
8.17
8.60
9.03
1.051 x 10-6
1.044 x 1e-6
1.036 x 10-6
0.071 2
0.0712
0.0712
0.0712
0.0712
0.0712
0.0712
0.0712
0.16 0
0.0249
0 .0157
0.0251
0.155
0.0254
0.0152
0.0256
0.087 2
0.0961
0.0869
0.0963
0.0867
0.0966
0.0864
0.0968
0.92
0.912
0.92
0.912
0.92
0.912
0.921
0.912
Ah
1.194
1.188
1.170
1.165
1.154
1.154
1.145
1.145
h
3.930
3.930
3.875
3.875
3.845
3.845
3.815
3.815
h4A h
5.124
5.118
5.045
5.040
4.999
4.999
4.960
4.960
4.714
4.665
4.64
4.595
4.60
4.56
4.57
4.523
2
2
(h+Ah)
V2 486
01
484
482
480
480
478
479
475
2cosN1
2cos
2
1.732
1.732
1.732
1.732
1.732
1.732
1.732
1.732
1
2
0.823
0.826
0.83
0.833
0.833
0.836
0.835
0.842
1
2
0.748
0.749
0.749
0.749
0.749
0.749
0.749
0.749
1-e2
0.252
0.251
0.251
0.251
0.251
0.251
0.251
0.251
3.52
3.49
3.47
3.43
3.44
3.415
3.42
3.385
1-96
hivi MW2
1
12
2 244
243
242
240
240
179
239
240
238
TABLE XLI CONT'D
Re
Ai
st
291,000
291,000
291,000
291,000
7.01
6.9
6.855
6.805
0.892
0.891
0.891
0.891
180
Again the stage efficiency came out, for some stages, slightly higher than the
one predicted on Part II, Section B-1, and based on dimensionless characteristics; in any case, it can be concluded that the stage efficiency is:
st - 0.891
The work output of the H. P. turbine is the sum of the work done by each
stage, thus:
(W
output
) H.P.
49.23 Btu/lb.
And compared with the required work output, W = 49.15,
it
is seen that
it is larger by a negligible amount, and in any case on the favorable side.
The leaving loss is, as stated before the carry-over from the last stage:
2
(1 -
AhL
2
2gJ
0.251 x (238)2
64.4 x 778
-
1.135 Btu/lb.
that is:
1.135
0.023 or 2.3%, and in this case greater than
49.23
what was assumed in the preliminary design of the H. P. turbine. A correction
for this discrepancy should be made, but since the efficiency obtained is
higher than the one originally assumed, and the work done by the turbine is
also higher, the author estimates that this correction is not worth the trouble
of a re-computation.
181
.Wj
SECTION C-3
Part I
Detail Design of L. P. Compressor
From the properties of the first stage of the L. P. compressor, given
on Table XXI, and fig. XXVII, for a rotational speed of 3525 r.p.m. the
following coefficients are obtained:
V = 0.493
6: 0.325
For these coefficients, the airfoil characteristics given on Table
XXVII and Fig. XL, show a stage efficiency of:
Ist - 0.868
consequently the design will be done for these conditions.
In the rotation used in the tables following, subscript o refers to
entrance state of gas to rotor, subscript 1, to state leaving the rotor
and entering the stator, and subscript 2, to state leaving the stator and
entering next stage.
Following the same procedure as indicated on Table XXI, for the above
values of the velocity and diagram ratios selec t ed, the entering state of
the gas to the first stage is determined.
For the next stages the proced-
ure to follow is self-explanatory by the notation on the tables.
The pitch diameter will be constant throughout the turbine, and the
stages will be all symmetrical and will use N.A.C.A. airfoil No. 4409.
The raise in enthalpy
i will be the same for every stage, since it
depends on the peripheral velocity u and the diagram ratio
Ai
x u2_
2gJ
(49)
The computations for the first stage will be done next, for the values
of V and E above; this computation would not be necessary if in fig. XXVII
182
all the properties had been plotted, but since this is not possible to
do, a calculation must be done for this particular case.
For:
6 : 0.325
: _1
(1+0.325/2)
1+/2
0
1 +- k
C (1 + 0.667)
(68)
0.493
which checks the value obtained from fig. XXVII.
C
0,667
:k
u
0.325/2
-
1 +-0.667
1 + k
(72)
0.303
Cu1
C
CUl
V
y
C/u
0303
0.493
0.615
C1
1+ (Cu1/ 0 )2 :
1+ (0.615)2
(73)
- 1.174
X
The state of gas entering the guiding vanes is:
28.77 Btu/lb.
T1
:5200 Fa.;
hi
p1
= 14.7 psia pri
2.504
Let's assume To = 512 (T(1 -0 ) in Fig. XXIII)
it's corresponding sound velocity is:
L
:
kg RT0
:/1.4 x 32.2
x
53.35 x 512
(52)
: 1110 ft./sec.
The value of the Mach number found to be the one to use in this design
is:
M:
0.4
183
Therefore:
: 0.4 x 1110
MOt
-
444 ft/sec.
Consequently:
C
-M
(71)
X
= 256 ft./sec.
C1 /Cx x O,
C1
=
1.174 x 256
- 301 ft./sec.
2
C1 /2gJ
:
(301)
2
64.4 x 778
-
1.81 Btu/lb.
:
11 - Cj/2gJ
-
28.77 - 1.81
-
26.96 Btu/lb.
2
1
For which, from the air tables:
- 5120 Fa. thus checking the assumed value above.
T
i
= 1
08 1
-Ig/
)x (C2gJ
g-2gJ
:28.77 - 1
x 1.81
0.9
-
26.76 Btu/lb.
For which from the air tables:
: 2.365
pr
Therefore:
o
os
-
2.504
13.68
184
-
v
53.35 x ]12
RT 0
0p
-~
____--~
13. 68 x 144
0
13.65 ft.3/lb.
-
Volume flow:
x
S:G
v : 64.7 x 13.65
883 cu.ft./sec.
Flow area:
A
Q/C
= 883
301
: 3.45 ft.2
Since for the first stage:
1/d : 0.2
A
w d 1 = w d2
/d
then:
/
d
' x 0.2
= 28.13 inches, which also checks with the result
that can be obtained from fig. XXVII.
Consequently:
d
max
d
max
d(111/d)
=
28. 13(1+0.2)
a 33.76 inches.
but:
U
max
.Ma
:C/v
0.493
: 519 ft./sec.
(70)
and:
x 12 x
Utax
n
60
g 519 x 720
w x 33.76
a 3525 r.p.m., which checks with the value that can
be obtained from fig. XXVII for '2: 0.493, and is also the speed at which
the L. P. compressor is to run.
The raise in enthalpy across the stage is:
2gJ
0.325 x
(49)
(519)2
64.4 x 778
1.745 Btu/lb.
The change in pressure across the stage is given very approximately
by:
2
1+ k
P12/ 0
(u/ao)
(see eqs. (119) to (124))
But since the change in enthalpy is known accurately, the change in
pressure can be determined with more precision using the air tables:
Ai x
: 1.745 x 0.8 6 8
s
: 1.515 Btu/lb.
Therefore:
12s
:26.96 + 1.515
28.475 Btu/lb.
For which, from the air tables:
for 1 0
for 1
2s
26.96
: 28.475
pr 0
2.378
pr2s : 2.483
186
and then:
P2
-13.68
x
2.483
2.378
14.29 psia
Thus the rise in pressure is:
p
: 14.29 - 13.68
-
0.61 psi
As it will be seen later the pressure rise in the first stages is
In order to obtain a higher rise
feeble compared with the last stages.
in pressure the compressor would have to run very slow in which case the
compressor foils would go into the surge region and thus working on an
inefficient range.
From the conditions of the cycle, the L. P. compressor must raise the
pressure from 14.7 psia at the inlet to a pressure of 39 psia at the exhaust.
As for the turbine blades the Reynolds number is given by:
Re
- C1 x b
g1.
9 p(136)
The velocity Cl is fairly constant through the stages, since these
are symmetrical and all alike.
Therefore:
Re
x
201
32.2 x 12
b
vg
0.779 b/rg
187
TABLE XLII
STAGE CHARACTERISTICS - L. P. COMPRESSOR
) .- 0.493
STAGE
1
2
3
4
5
534.3
541.7
TO
o Fa.
i
Btu/lb.
26.96
28.705
30.450
32.195
p0
psia
13.68
14.29
14.90
15.53
16.18
ft. /lb
13.65
13.45
13.08
12.73
1?.38
0
512
pr
A
ft.2
d
in.
1/d
1
i
2s
Ap
Re
2.624
2.752
2.885
3.45
3.40
3.31
3.22
3.13
28.13
28.13
28.13
28.13
28.13
0.186
0.181
5.626
5.535
5.387
5.242
5.097
3
3
3
3
3
in.
1.875
1.845
1.796
1.747
1.699
in.
0.056
0.055
0.054
0.052
0.051
ft./sec .519
xAi
519
5 19
519
519
Btu/lb.
1.745
1.745
1.745
1.745
1.745
Btu/lb.
28.705
30.450
32.195
33.940
35.605
519.7
5 34.3
527
541.7
548.5
Btu/lb.
1.515
1.515
1.515
1.515
1.515
Btu/lb.
28.475
30.220
31.965
33.710
35.455
2.483
2.607
2.736
1.869
3.005
pr 2s
P2
2.50
0.1913
0 Fa.
' st
2.378
0.1965
1/b
u
33.940
0.2
in.
b
527
519.7
psia
14.29
14.90
15.53
16.18
16.85
psi
0.61
0.61
0.63
0.65
0.67
lb-sec,
0.59 x 10-.6 0.593x10-6 0.597x10-6 0.6x10-6
t
0. 603x10
2
181,400
180,200
188
1 79,200
178,300
177,300
TABLE XLIII
STAGE CHARACTERISTICS - L. P. COMFRESSOR
V: 0.493
STAGE
T
0
i0
6
7
548.5
555.7
35.605
16.85
12.04
8
563
37.350
17.55
11.70
9
10
570.2
577.5
39.095
40.840
42.585
18.26
18.00
19.71
11.40
11.10
10.83
pr-
3.017
3.159
3.307
3.46
3.616
A
3.04
2.96
2.881
2.807
2.738
d
28.13
28.13
28.13
28.13
28.13
1/d
0.1758
0.171
0.1666
0.1623
0.158
1
4.95
4.82
4.694
4.57
4.458
1/b
3
3
3
3
3
b
1.65
1.606
1.565
1.523
1.486
0.049
0.048
0.047
0.045
0.044
519
u
519
519
519
519
t1
1.745
1.745
1.745
1.745
1.745
:12
37.350
39.095
40.840
42.585
44.330
2
T2
Ist Xi
12
pr2s
P2
,&p
Re
555.7
563
570.2
577.5
584.8
1.515
1.515
1.515
1.515
1.515
37.120
38.865
40.610
42.355
44.100
3.14
3.287
3.438
3.594
3.756
17.55
18.26
18.99
19.71
20.48
0.70
0.71
0.73
0.72
0.77
0. 605x10-6
0.609 x 10-6
0.611 x 10- 6
0.615 x
0.618 x 10-6
176,600
175, 000
175,600
139
173,800
173,000
TABLE XLIV
STAGE CHARACTERISTICS
STAGE
T
0
0
p0
L. P. CO"PRESSOR
-
11
12
13
14
15
584.8
592
599.3
606.5
613.8
44.330
46.075
47.820
49.565
51.310
20.48
21.28
22.09
22.91
23.71
10.57
10.29
10.04
9.79
9.57
pro
3.777
3.944
4.116
4.296
4.477
A
2.671
2.6
2.538
2.474
2.42
d
28.13
28.13
28.13
28.13
28.13
1/d
0.1545
0.1504
0.1467
0.143
0.14
1
4.35
4.233
4.13
4.03
3.94
1/b
3
3
3
3
3
b
1.45
1.411
1.38
1.343
1.313
0.043
0.042
0.041
0.04
0.039
u
519
12
T
Ai
tX
0 st x0i
S2s
pr2s
P2
pAp
Re
519
519
519
1.745
1.745
1.745
1.745
1.745
46.075
47.820
49.565
51.310
53.055
592
2
519
606.5
599.3
613.8
621
1.515
1.515
1.515
1.515
1.515
45.845
47.590
49.335
51.080
52.825
3.922
4.093
4.271
4.452
4.641
21.28
22.09
22.91
23.71
24.58
0.80
0.81
0.82
0.80
0.87
0.621 x 10- 6
0.623
0.626 x 10-6
0.63 x
0.632x10-6
172,000
x
10-6
171,000
171,500
1.90
170,000
169,000
TABLE XLV
0.493
V
STAGE
T
0
STAGE CHARACTERISTICS - L
P. COMPRESSOR
16
17
18
19
20
621
628.2
635.5
642.7
650
53.055
54.800
56.545
58.290
60.035
p0
24.58
25.49
26.40
27.32
28.26
"e0
9.36
9.13
8.91
8.70
8.52
pr
4.666
4.86
5.059
5.266
5.478
A
2.37
2.308
2.251
2.2
2.153
d
28.13
0
28.13
28.13
28.13
28.13
1/d
0.137
0.1334
0.1302
0.1272
0.1245
1
3.86
3.758
3.669
3.584
3.509
1/b
3
3
3
3
3
b
1.287
1.253
1.223
1.195
1.169
0.038
0.037
0.037
0.036
0.035
12
T
519
519
u
i
2s
pr
2s
p2
Rp
Re
519
1.745
1.745
1.745
1.745
1.745
54.800
56.545
58.290
60.035
61.780
628.2
2
519
519
635.5
642.7
650
657.2
1.515
1.515
1.515
1.515
1.515
54.570
56.315
58.060
59.805
61.550
4.834
5.031
5.239
5.448
5.666
25.49
26.40
27.32
28.26
29.25
0.91
0.91
0.92
0.94
0.99
0.637 x 10 -6
0.639 S10- 6
0.641
0.644 x 10- 6
0.648x10-6
168,000
166,800
167,300
191
x10
166,000
165,000
TABLE XLVI
: 0.493
STAGE
T
0
STAGE CHARACTERISTICS - L. P COMPRESSOR
21
22
23
24
25
657.2
664.5
671.7
678.9
686.1
61.780
63.525
65.270
67.015
68.760
29.25
30.26
31.29
32.31
33.37
8.31
8.13
7.95
7.77
7.62
pro
5.694
5.92
6.148
6.385
6.627
A
2.1
2.055
2.01
1.964
1.925
d
28.13
0
p0
28.13
28.13
28.13
28.13
1/d
0.1215
0.1189
0.1163
0. 1136
0.1115
1
3.42
3.346
3.272
3.199
3.134
1/b
3
3
3
3
3
b
1.14
1.115
1.091
1.066
1.045
0.034
0.033
0.033
0.032
0.031
519
u
519
519
519
519
&i
1.745
1.745
1.745
1.745
1.745
12
63. 525
65.270
67.015
68.760
70.505
664.5
T
2
i
st x &
678.9
671.7
1.515
1.515
1.515
1.515
63.295
65.040
66.785
68.530
70.275
5.889
6.117
6.339
6.594
6.843
2s
p
A.p
30.26
31.29
32.31
33.34
34.39
1.01
1.03
1.02
1.03
1.05
0.65x10-6
0.652x10- 6
0. 657x10-6
0.66x16 6
0. 663x1&-6
164,400
162,300
163,800
*
R
693.4
1.515
2s
pr
686.1
192
161,800
161,000
TABLE XLVII
q: 0.493
STAGE
To
STAGE CHARACTERISTICS - L. P. COPRESSOR
26
693.4.
27
28
29
700.6
707.8
715
70. 505
72.250
73.995
75.740
34.39
35.48
36.6
37.78
7.46
7.31
7.16
7.01
pro
6.876
7.134
7.395
7.664
A
1.885
1.848
1.81
1.772
i
p0
%ro
d
28.13
28.13
28.13
28.13
1/d
0.109
0.107
0.1047
0.1025
1
3.07
3.009
2.949
2.885
1/b
3
3
3
3
b
1.023
1.003
0.983
0.962
0.031
0.03
0.029
0.029
s
u
519
519
519
519
Ai
1.745
1.745
1.745
1.745
i2
72.250
73.995
75.740
77.485
T2
Ist A
S2s
pr2s
p2
Sp
Re
700.6
707.8
722.2
715
1.515
1.515
1.515
1.515
72.020
73.765
75.510
77.255
7.099
7.358
7.627
7.903
35.48
1.09
1.12
1.18
0.665x10-6
0. 669x10- 6
0.671x10-
160,500
39.0
37.78
36.6
159,700
193
159,200
1.22
6
-6
0.673x10
158,900
Z
40
RCH
F-6
C TO
........
..
2
...........
455
.......
..
sta
.......
...
1.50
.........
.....
.......
.......
....
....
..
......
...
......
..
....
. ...
...
..
......... ...
120
0.70
...
......
...
......
......
......
.......
......
...
*.77
...........
......
..
...
.........
.....
...
.
..........
CLS.
7 5
----------
0-95
...
.......
----------
/.Do
193
a-
1 -100"""
__
-
-
" I
-_--
- . __--
- ------
__ "
- -
I
, .1 - -1 1
1
- ---
The number of stages is then 29, in order to obtain a discharge
pressure of 39 psia.
The Reynolds number varies from 181,400 to 158,900, and on Reference
(15), page 8, data is given for various Reynolds numbers, two of which
are
Re z 165,800 and Re = 329,000.
In the original determination of
the airfoil characteristics the values of CL and CD were taken for
Re = 329,000.
At first thought it would seem that there would be a diff-
erence in the values for CL and CD, but examining the curves, it can be
seen that the variation is for all practical purposes negligible.
The compressor's internal efficiency can be found now, using Fig.
XLVII, taken from 2.211 Advanced Problems in Gas Turbines, C. R. Soderberg.
Atage efficiency:
st
=0.868
-29
No. of stages:
r,
for
rI
for
(1 +-R)
(1 + R),= 1.023
:2.65
z
/ 2.65
1.023
= 1.037
(1 + R)*= 1.000
1.023
1.00
Internal efficiency:
= 0.868
Ist
(1 + R)
1.023
0.85, which is equal to the value assumed
originally for the calculations of the cycle.
This efficiency so defined does not account for the energy of the
gas leaving the compressor.
If it is desired to consider it, then the efficiency can be called
"over-all compressor efficiency" and would be given by:
2
0-a
: h + Ce/2gJ
C
194
-
- - __ - _A.
Where:
h is the isentropic enthalpy raise through the turbine.
Ce is the leaving velocity, and
WC is the work done on the compressor.
To determine this overall efficiency the air tables are used:
Entering state of air to compressor:
p0
:14.7 psia
0
To
w 520
Fa.
ho
x 28.77 Btu/lb.
pro
: 2.504
Since the air is compressed to 39 psia, then:
pr e
39 x 2.504
14.7
: 6.65
For which from the air tables:
ies
a 68.91
The leaving velocity is practically equal to Ci:
C2/2gJ
:
* 1.806 Btu/lb.
(301)2
64.4 x 778
Therefore:
if
1e
0-a
: 77.485 Btu/lb. (see Table XLVII):
-
1.806
(77.485 - 28.77)
(68.91 - 28.77)
0-
:0.861, which is higher than what has been
called "internal efficiency".
The internal efficiency will be used for later developments since
it is on the conservative side.
195
Part
Detail Design of H. P. Compressor
The question of the H. P. compressor's speed has been left open in Part
I, Section B-4; consequently a speed shall be chosen so that the highest
possible efficiency is obtained.
On Fig. XXX, it can be seen that the less the speed, the greater the
change in enthalpy, and consequently the greater the rise in pressure through
the stages.
But, from Fig. XL, for a value of the velocity ratio %?, slightly
greater than o.5, the surge limit is reached, therefore the necessity of not
going to a too low a speed which may give a value for the velocity ratio
greater than 0.5.
In this case a value of 0.49 will be selected, for which
the stage efficiency is:
st = 0.868
Proceeding in the same manner as in Section C-3, each stage dimensions
and characteristics can be obtained:
The pitch diameter will be constant, and the stages will be all symmetrical at that diameter, and they will use also the airfoil N.A.C.A. No.
4409.
Q:
For:
0.49
(68)
1 + k
#(1 + k) -2
V
22
2 f2~'x 0.49 (1
0.667) -2
w 0.312
which also checks with the value that can be obtained from Fig. XXX.
Cul
u1
k-
+ k
196
(72)
-
-~
-~
.0.667 - 0.15
1 0.667
Cu1
C
: 0.3064
l036
uu/
0.49
x
l 0.626
1T -(06
c
Cx
(73)
The state of the gas entering the guiding vanes, from Part I, Section
B-4, is:
13l
- 35.78 Btu/lb.
= 5490 Fa.
x 38.8 psia
'
p3
= 3.031
'
pr 3
Using the same notation indicated in Part I, Section C-3:
Let's aSsume:
: 5410 Fa.
TO
its corresponding sound velocity is:
Ao
kgR To
/y 1.4 x 32.2 x 53.35 x 541'
1138 ft./sec.
The Mach number previously selected is:
M
then:
so that:
a 0.4
= 455.2 Ft./sec.
Mk
z Ma
x
C
455.2
rT
x 262.7 ft./sec.
Cl
x.1.18 x 262.7
x 310 ft/sec.
2
2
2gJ
64.4 x 778
197
C12.
i
then:
: 1.92 Btu/lb.
=
i0
a
35.78 - 1.92
33.86 Btu/lb.
for which, from the air tables:
To
= 5410 Fa.
it thus checks the assumed value.
1
x(
y)2gJ
g-v
then:
1/0.9 x 1.92
2.13 Btu/lb.
2g
io5
a
35.78 - 2.13
33.65 Btu/lb.
and:
therefore:
pros u 2.863
p0
- p
a 38.8 x 2.863/3.031
36.65 psia
so that:
53.35 x 541
144 x 36.65
0
5.465 Ft.3 /lb.
volume flow:
Q
.
Gx
Q =64.7 x 5.465
353.6 ft. 3 /sec.
flow area:
A
a
Q/C
: 353.6/262.7
* 1.345 ft.2
Since for the first stage 1/d
d
=
0.2:
:)/Ax
/ r( /d)
1.345 x144
w x 0.2
17.55 inches.
198
d
max
: 17.55 (1+ 0.2)
: 21.07 inches.
U
: Ma//= CX/
u
: 262.7
0.49
536 ft./sec.
then:
: 536 x 720
w x 21.07
5830 r.p.m.
which checks with the value obtainable from Fig. XXX for
: 0.49.
The dimensions and characteristics of each stage follow in tabular
form.
199
TABLE XLVIII
STAGE CHARACTERISTICS - H. P. COMPRESSOR
Q: 0.49
T
2
1
STAGE
0 Fa.
0
io
Btu/lb
33.86
p0
psia
36.65
f t 3 /lb
V
pro
A
ft. 2
d
in.
39.8
41.41
41.02
43.12
5.465
5.31
5.175
5.035
4.9
2.88
3.021
3.167
3.318
3.475
1.345
1.307
1.274
1.239
1.207
17.55
17.55
0.18
3.515
3.414
3.33
3.236
3.153
3
3
3
3
3
in.
1.172
1.138
1.11
1.079
1.051
in.
0.035
0.034
0.033
0.032
0.032
Btu/lb
id
12
39.23
0.184
o Fa.
x Ai
37.44
0.19
id
Sst
571
0.195
Btu/lb
T2
563.5
0.2
ft./sec.
i2
556.1
17.55
1/b
u
38.22
5
17.55
in.
b
35.65
4
17.55
1/d
1
548.6
541
3
536
536
1.79
1.79
37.44
35.65
548.6
556.1
1.554
1.554
37.20
35.4
2.147
3.002
536
1.79
1.79
1.79
39.23
41.02
42.81
571
563.5
1.554
1.554
40.78
38.99
3.454
3.298
578.4
1.554
42.57
3.614
,
pr2
536
536
Re
psia
38.22
psi
1.57
lb-sec.
2
f t.
0.601x1269,400
41.41
43.12
44.86
1.58
1.61
1.71
1.74
0. 605x10-6
0. 609x10
0.612x10- 6
0.616x10
39.8
6
267,500
200
266,000
6
264,200
262,800
6
TABLE XLIX
STAGE CHARACTERISTICS - H. P. COMPRESSOR
V: 0.49
STAGE
6
578.4
T0
8
7
585.9
593.2
9
10
608.2
600.8
i0
42.81
44.60
46.39
48; 18
49.97
po
44.86
46.65
48.5
50.39
52.3
v
0
pr
0
A
d
4.774
4.646
4.527
4.415
4.305
3.636
3.803
3.974
4.153
4.337
1.175
1.144
1.114
1.086
1.067
17.55
17.55
17.55
17.55
17.55
1/d
0.175
0.17
0.166
0.162
0.159
1
3.07
2.988
2.908
2.836
2.785
1/b
3
3
3
3
3
b
1.023
0.996
0.969
0.945
0.928
0.031
0.03
0.029
0.028
0.028
U
536
u
536
536
536
536
Ai
1.79
1.79
1.79
1.79
1.79
12
44.60
46.39
48.18
49.97
51.76
585.9
T2
st
12s
1.554
44.36
3.78
1.554
46.15
3.952
600.8
1.554
47.94
4.129
615.6
608.2
1.554
1.554
49.73
51.52
4.5
4.312
,
pr 2
x Ai
593.2
p2
Ap
R
e
46.65
48.5
50.39
1.79
1.85
1.89
1.91
0. 618x10-6
0.622x10- 6
0.625x10-6
0. 629x10-
261,800
260,200
258,400
201
54.3
52.3
257,000
2.0
6
6
0. 631x10257,000
TABLE I
Q
STAGE CHARACTERISTICS - H. P. COMPRESSOR
0.49
STAGE
11
615.6
To
pio0
12
623
13
630.5
14
15
637.9
645.3
51.76
53.55
55.34
57.13
58.92
54.3
56.32
58.40
60.48
62.68
Vo
4.196
4.094
3.997
3.901
3.81
pr
4.525
4.72
4.921
5.127
5.341
A
1.033
1.008
0.984
0.961
0.939
17.55
d
17.55
17.55
17.55
17.55
1/d
0.154
0.15
0.146
0.143
0.14
1
2.697
2.63-
2.57
2.51
2.45
1/b
3
3
3
3
3
b
0.899
0.877
0.857
0.837
0.817
0.027
0.026
0.026
0.025
0.024
U
536
u
536
536
536
536
1.79
1.79
1.79
1.79
1.79
53.55
55.34
57.13
58.92
60.71
Si
12
623
T2
Ist xai
X
:122s
pr2s
P2
A p
R4-
1.554
\
53.31
4.693
630.5
1.554
55.10
4.894
637.9
1.554
56.89
5.1
645.3
1.554
58.68
5.312
652.8
1.554
60.47
5.531
56.32
58.40
60.48
62.68
2.02
2.08
2.08
2.20
2.22
0. 633x10-6
0. 638x10'-6
0. 64x10-6
0.642x10- 6
0.645x10-6
255.600
253,800
253,000
202
252,200
64.9
251,100
S: 0.49
STAGE
STAGE CHARACTERISTICS
16
652.8
To
i
0
p0
17
660.1
-
TABLE LI
H. P. COMPRESSOR
18
667.6
19
675
20
681.6
60.71
62.50
64.29
66.08
67.87
64.9
67.17
69.47
71.9
74.4
v
0
3.72
3.637
3.553
3.475
3.389
pro
5.561
5.785
6.018
6.256
6.502
A
0.916
0.896
0.875
0.856
0.835
17.55
d
17.55
17.55
17.55
17.55
1/d
0.136
0.133
0.13
0.127
0.124
1
2.39
2.34
2.284
2.234
2.18
1/b
3
3
3
3
3
b
0.797
0.78
0.761
0.745
0.727
0.024
0.023
0.023
0.022
0.022
s
536
u
536
536
536
536
Ai
1.79
1.79
1.79
1.79
1.79
12
62.50
64.29
66.08
67.87
69.66
660.1
T2
1.554
I st
p2
5
Pr2
P2
A p
Re
62.26
5.755
667.6
1.554
64.05
5.986
67.17
69.47
2.27
2.30
0.649x10- 6
0.651x10- 6
249,300
248,700
675
1.554
65.84
6.224
681.6
1.554
67.63
6.468
689.1
1.554
69.42
6.72
74.4
76.89
2.43
2.5
2.49
0.653x 10-6
0.657x106
0. 66x10- 6
71.9
247,600
203
246,500
245,400
--
-~
I
-
I
I
I
TABLE LII
STAGE CHARACTERISTICS - H. P. COfUPRESSOR
Y 0.49
STAGE
To
21
22
23
689.1
697.2
704.7
25
24
719.5
712
1
69.66
71.45
73.24
75.03
76.82
PO
76.89
79.4
82.07
84.7
87.41
VO
3.315
3.25
3.176
3.11
3.042
pro
6.754
7.014
7.281
7.553
7.834
A
0.816
0.801
0.782
0.766
0.75
d
17.55
17.55
17.55
17.55
17.55
1/d
0.121
0.119
0.117
0.114
0.111
1
2.13
2.09
2.04
2.0
1.958
1/b
3
3
3
3
3
b
0.71
0.693
0.68
0.667
0.653
0.021
0.021
0.02
0.02
0.019
u
Ai
12
T2
1 st x A 1
12s
pr2 s
P2
A p
Re
536
536
536
536
536
1.79
1.79
1.79
1.79
1.79
71.45
73.24
75.03
76.82
78.61
697.2
704.7
1.554
71.21
1.554
73.00
6.976
7.244
79.4
82.07
712
1.554
74.79
7.516
84.7
87.41
90.13
2.72
2.71
0.662x10-6
0.665x10- 6
0.668x10- 6
0.671x10
204
8.073
7.797
2.63
241,900
78.37
76.58
2.67
242,000
1.554
1.554
2.51
244,300
726.9
719.5
241,200
6
0.674x10
240,400
6
TABLE LIII
\-
0.49
STAGE
T0
STAGE CHARACTERISTICS - H. P.
COMPRESSOR
26
27
28
726.9
734.2
741.6
29
749
i
78.61
80.40
82.19
83.98
PO
90.13
92.93
95.9
98.89
VO
2.984
2.923
2.86
2.801
pr
8.123
8.417
8.72
9.033
A
0.736
0.721
0.705
0.69
17.55
d
17.55
17.55
17.55
1/d
0.109
0.107
0.105
0.103
1
1.92
1.88
1.84
1.8
1/b
3
3
3
3
b
0.64
0.627
0.613
0.6
0.019
0.019
0.018
0.018
536
12
x
pr2
1.79
1.79
80.40
82.19
83.98
85.77
1.554
80.16
8.379
1.554
81.95
83.74
8.68
8.99
98.89
95.9
2.97
2.80
238,200
749
741.6
1.554
0. 679x10
Re
536
1.79
92.93
Lp
536
1.79
734.2
T2
:122s
536
6
0. 681x10
237,400
205
6
756.4
1.554
85.53
9.309
102.3
2.99
3.21
0.684x10- 6
0.688x1-6
236,600
235,000
The average Reynolds number for the H. P. compressor is about 250,000
which is closer to the value at which the airfoil characteristics were determined, than for the case of the L. P. compressor.
As stated, before, the
ranges in which the Reynolds number varies in this design as compared to
the value selected in Part I, Section B-5, does not introduce any significant error in the calculations, and more so in the case of the H. P.
compressor where the difference in effects on CL and CD is negligible.
206
SECTION C-5
Part I
Design of the Regenerator
The regenerator plays a very important part in the overall efficiency of the cycle of the plant.
It is also a bulky and heavy acces-
ory; consequently great care must be taken in its design in order not
to make an exaggerated estimate of the required heating surface.
The design will be based on the original requirements of the cycle,
just as it has been done for the other parts of the plant.
The specifications for its design are:
(a) Operational:
Rate of flow of air
64.7 lbs/sec.
Inlet pressure of air
102.9 psia
Inlet temperature of air
749.0
*
Fo..
Maximum allowable pressure
2.0 psi
drop on air side
Inlet pressure of gas
Inlet temperature of gas
Exit pressure of gas
Turbine inlet temperature
16.5
r'k-
1365.0 *FA.
14.7 psia
1860 0 Fa.
(b) Material:
20,000 lbs.
Weight
3/8 inches
Diameter of tubes
Wall thickness of tubes
0.02 inches
Ratio of shell wall-thickness
to shell diameter
0.005
207
-
~ woo
--
.- "I ,
_
The following assumptions will be made:
1. High pressure air flows through the tubes.
2. Low pressure gas flows countercurrently through the shell.
3. Flow is turbulent.
This assumption will be checked later.
4. Heat transfer area is the same for air and gas.
5. Neglect radiation between fluids and walls.
6. Neglect the thermal resistance of the wall.
7. Air and gas shall be considered incompressible,
df
that is,
constant density.
For the explanation of the symbols and subscripts, refer to list
of
symbols and to figures I, II, and XLVIII.
In section A-1 it
was selected a regenerator effectiveness of
was defined as:
R = 0.65, which in Section A-2
: '8 - l0
i -i4
9-- 14
8 i 4i
i-o
Fig. XLVIII
208
(11)
From Table IV, and also from Table XXVIII:
ig : 238.62 Btu/lb.
14
= 85.6
Btu/lb.
Therefore:
0.65
i9-856
238.62 - 85.6
i 9: 185.1
Btu/lb.
This value has already been calculated in Table IV.
8
-
10
: 29 - '4
'10 :139.12
Btu/lb.
The amount of heat taken by the air going through the regenerator is
then:
Q
G x 3600(9 - i 4 )
(137)
64.7 x 3600(185.1 - 8.5)
23,140,000 Btu/hr.
and in terms of the mass velocity, it is:
q
=M m d2 jV (19 - 14)
(138)
4
with the mass velocity M equal to:
M
: 3600 x G
lbs. of air/hour
sectional area of flow-ft
A
2
(139)
The amount of heat taken by the air must be equal to that delivered by
the gas, which can be expressed as:
q
with At
m
(140)
: U(M d L N) Atm
which for counterflow is,
the log mean temperature difference,
for the overall transfer coefficient I= constant, and C
g
8T. T9).(1
ln
(T 8
(T1 0
209
constant:
T9
)
A tm
=
-T
(141)
With the corresponding values of the enthalpy on Table IV, and using
the Air Tables:
T8
1365.50 Fa.;
T4
= 755.70 Fa.;
=
11430 Fa.
T10 =
9740 Fa.
T9
Therefore:
391.5 - 218.3
M
Ltm
ln (391.5)
(218.3)
-
2970 F.
But as stated above, equation (137)
3600 G(i9 - 1 )
must be equal to (140), thus:
U(W d L N )Atm
or:
M Yrd2 N(i 9 - 14)
:U(T d L N) Atm
4
If we assume that the specific heat of the air is constant, then we can
express:
i9 - i
C (T9
-
T4)
(142)
Introducing this value, simplifying and collecting terms above, we get:
M Cp(T
- T4
)
L/d
4U At m
(143)
The overall heat transfer coefficient, neglecting thermal resistance of
the all of the tube and assuming that the tubes are clean, therefore no
scale effects to account for, is given by:
I 1
U
h1
-
1
=1
(1 + hi/ho)
:ku/hi
(144)
hi
ho
The heat transfer coefficient of the air film inside the tubes,
can be predicted, for Reynolds number greater than 2100, from eq. (4c)
page 168 of Reference (16):
h
xd
:0.023
kk
210
04
(145)
The quantity (C p/k) is the Prandtl number, and shall be desigP
r
.
nated henceforth by:
Solving for h1i in (145) and together with (144) substituted in
(143), and collecting terms we get:
: ku x (d)
L/d
0.2
x (Pr)
0.2
0.6
x(T 9 - T4 ) x M
0.092 ( )0.2
k
(146)
x tm
in equations (144) and (146) can be shown to be:
ku
U
: 1 + (ka/k
0 '.
)
(147)
kA is the ratio of cross-sectional area of gas flow to that of
where:
air flow.
k k is the ratio of flow of gas to flow of air.
and:
that is:
kA
D
Where:
and:
k
:Aa/Ag
wd2 N
SD4
d2 N
(148)
= diameter of shell.
:Ga+ Gf
G
For the combustion process:
:1 t G /G
(149)
a
1f2. t h* i- i5
G,/G a
(150)
The weight of the heat transfer surface of thickness t (of the tubes),
and dnsity ( is:
W':
ed L N) t x
(151)
-
Eliminating L and N by means of equations (137) to (147), we get:
t z ( t(d)0. 2 ku(p)0. 6 (T
0.023 (
)0.2 x (M)0'
-
T) Ga
(152)
x At
If we assume that the weight of the complete regenerator is twice
the weight of the heat transfer surface, thus accounting for the weight
211
of the shell, end plates, baffles, supports, etc., we get:
Regulator weight:
W :2 W'
(153)
The pressure drop for the air inside the tubes, assuming in compressible fluid, or much better to say constant density, and neglecting
head losses at the ends, is given by:
212
p
-M
fM (154)
2gx Fa x rh
-
Where the hydraulic radius rh is
stream cross section
wetted perimeter
rh
2
and the friction factor,
f/2
f,
is given by:
: 0.023 (Md/tL)-O.2
(for Re> 2100)
(156)
Combining (146), (154), (155), and (156):
P4 - P
.
2
ku x ( Pr)o.6 x (T9 - T) xM
g x fa x ttm
(157)
A similar expression can be obtained for the pressure drop outside
the tubes, noting that it occurs in an equivalent diameter equal to kAd,
)
that the flow of gas is equal to that of kjM, and that T9 - T4 = T8 - T10
1.8
0.6
x (T - T
(pr)
Pg - plo kuxk,
kA
x g x
(158)
x &tm
The weight of the heat exchanger may be expressed in terms of the
pressure drop in the tubes, by combining (152), (153), and (157), i.e.:
W
=
2 x t x ft x (d)0.2 x(k)
2
".023()
L(P4 -
1 .4
x(Pr 0.84 x Ga
gJ0.4
9)
a
4
T9 r4
Atm
The equation (159) has only one unknown ku, solving for ku, we obtain:
1.4
0.4
4
0.2
1.4
ku
- 0.023 W (Q)
[(pa- p)
g
(Im
2t x
0
x (d) *2 x (Pr)
84 x
Ga
T9 -T
If the material to use has a density of f - 489 lb/ft.
(159a)
and we take
the value of the Prandtl number as an average value of 0.7 for both air and
213
(159)
-
-A,
gas, and we take an average value of temperature in the regenerator of
9500 Fa. to find
1.4
0.4
x102.9x44
0.023 x20,000(0.73 x
-(2xl44)
x 32.2 x53.35x950 1
2 x (0.02/12) x 489 x (0.375/12)0.2 x 10.7) 0 . 8 4 x 64.7
(297/387.3)
'
(ku)
and Pa, we get:
x1 0
x
so that:
ku
: 5.1974
We could substitute this value in (152) and solve for M, but if first
we divide (152) by (159) and then solve for M, we get:
(P4 -P9)Px1 0.4+
M.9
(ku)4 x (r)
M0.8
0
'24 (T9 - T4 tm)0 4
[F(2 x V4) 102.9 x 14
x 32.21 0.4
53.35 x 950
(5.1974)0'4 x (0.7)0.24 (387/297)0.4
:
52,800
lbs./(hr.xft. 2
)
M
Substituting this value in equation (139) we get for the cross sectional
area:
A
G
.3600
1600 x 64.7
52,800
-
4.41 ft.2
This is the value of the cftoss sectional area of the tubes since all
calculations have been based on the air side.
The cross sectional area of flow is given by:
A
d2 x N
4
Therefore:
A x 4 x 144
r d2
214
(160)
x
Wil
4.41 x 4 x 14
i X (0.375/12)2
5760 tubes
Substituting the value of M in (145), and using for k a value of:
k
x 0.0265
hju0.023 52,800 x 0.375
)0.8
x (0.7)0.4 x(0.0265/0.375)x 12
0.73 x_10-6 x A2 x 12)
20.9 x 10- 6
)
54.0 Btu/(hr) x (OF.) x (ft.2
Consequently in (144):
= 5.1974/54.0
Btu/(hr.) x (0 F.) x (ft.2
)
10.4
V
Equating (137) to (140) and introducing
o10.4[
23,140,000
L
,
we get:
, ).375/12 L x 5760)3 x 297
: 13.25 Feet.
Therefore, the heat transfer area is:
d L N
S:
=
r x(0.375/12)x 13.25
x 5760
7500 ft.2
From equation (157) and (158), we can get:
: (p4 - P9 )/(Pg - P1 0) X
k
Substituting the corresponding values we get:
k
: 4.12
Consequently sectional area of shell is:
A5
x 4.12 x 4.41
= 18.16 ft.2
Therefore the shell diameter is:
D
:
18.16/*)x 4
215
a/rg
(161)
~D : 4.81 ft.
Shell thickness:
t5
3 0.005 x 4.81 x 12
v 0.289 inches
9/32 inches.
Reynolds number:
Md
Re
Z 52,800 x 0.375/12
From fig. XLIII:
for 9500 Fa.
= 0.74 x 10-6
lb.-sec.
ft. 2
:0.74 x l0 6
20.9 x 10-6
x 2.42
0.0857
Re
lb./(ft.)(hr)
52,800 x 0.375
0.0857 x 12
19,250
which checks our assumption of turbulent flow.
216
7
APPENDIX D
217
SECTION D-1
Partial Load Characteristics
n the development of the partial load characteristic, we first neod
to have the turbine and compressor characteristics; and in this connection
a few basic remarks on the theory that shall be employed is necessary.
First
we shall start with the turbine characteristic curves.
Part I
Flow Through the Stage of a Turbine
Let's take any one stage of a multistage turbine, for which the continuity equation must be satisfied at all points.
For a particular stage, the continuity equation is:
G
A1 C1
v1
(129)
But from equation (28):
Cth
E-' +
:Cix
C
1
{+sE+
1
2
2
Consequently, the flow G becomes:
G
Cth
A
e
s
%Jr[
+
(165)
+S1 2 2
The heat drop across this same stage, is according to eq. (28):
JAh
2
- Cth /2g
(28)
Using the First Law of Thermodynamics:
dQ
:dE +dW
dE
+
(166)
pdv + vdp
Since the processin the stage is assumed occur adiabatically:
dQ
= 0
(167)
dh
: dE + pdv
(168)
we also have that:
218
If we consider that the pressure drop across the stage is small, which
is practically the case, then the heat drop through the stage can be very
well approximated:
From (166),
(167),
and (168):
-v 1 Ap
Jdh
(169)
Therefore, from (28) in (169):
2
- -v 1Jth4 p = Cth
J Ah
2g
(170)
Substituting the value of Cth from (165) we get:
s' s( E2 2 )22
JA h
+
-
2g
xA
x Gv
2g
(171)
-vi dp
Let us define an inerement to be a certain function
2g
then:
G 2 4X
so that:
,,
A
(172)
-A'I
(173)
This differential equation is true within certain limitations, as it is
said above, for small pressure drop across the stage.
If we assume that all the stages are alike, which is our case, at least
at the pitch diameter,then this differential equation can be intergrated:
222
v
(174)
Under the conditions that:
pvn
-
constant
(175)
where n can be shown to be:
n
1
1-
t x k-1
st
k
219
(176)
then, equation (174) becomes:
1/n + 1
G
El+
1+ s ( E
2g
S2)
x pl1
,l/A2
(1n+1
v1
(177)
-(P 2 0 )
x
10
that varies through the turbine in equation (172) is 1/A
(178)
f
1=1
n
Let's call:
.
where it has been considered that for equal stages, the only expression
and from (176) it can be derived that:
x k-
2 -
N
(179)
k
Some authors use
forNI a value of 2, since the second part of equation
is very small.
(179)
Introducing (178) in (177), and rearranging we get:
G
x
1s (E2xJ
X
2
(180)
10
1 1
110
The first radical is a constant for the turbine.
The second radical is a function depending on the velocity ratio.
The third one, expresses the flow through any nozzle as it varies,
and depends on the inlet conditions.
The last radical is an elliptic variation and expresses the influence
of the pressure ratio.
Equation (180) can also be written:
x f(Pe/pi) x Pi
t
Where:
)2
th
2(E
x
l
N
(Pppi)
)(
2
x2
-
(P20 10
220
2)
R
(182)
(183)
and:
p/ F
comes from:
I
[
01
-
R T0 1 P0 1
10
1
0
-.-
R
0ol
(184)
01
In this discussion no attention has been paid to the fact that critical velocities may set in; nevertheless in the present design this condition
has not arisen, as seen on the detail design of the turbines.
The turbine characteristics are then given by the functions in equation
(180) or (181).
In addition the leaving loss function and either the turbine
internal efficiency or the stage efficiency as a function of the velocity
ratio.
It is preferable to have the internal efficiency function, since for
the power computation at part loads it simplifies those computations because in that way the necessity of dinding the Reheat factor in each case is
avoided, due t
the fact that is already included in the determination of the
internal efficiency function.
The leaving loss function can be derived from figs. VIII and IX, and
from the definitions given in equation (20) to (23), so that:
Continuity equation:
G
A20
e
2~
2
e(129)
For the last stage we have a velocity head of C2 /2gJ, therefore the leav2e
ing loss is:
C2
L
e
2
-2e
2gJ
Therefore we can think of (1
-
2e
(185)
E ) as the leaving loss function, depend2e
ing on the velocity ratio:
221
2e
Sa
-A e
(186)
2e
It has been shown previously that for one stage:
J A h
=
(28)
C2 t2g
but, by definition, Eq. (26):
(26)
u/C th
t hk
Substituting above:
u 2 /2g x 1/
J A h
h
(187)
If we add up all the heat drops across the stages, we cet for the
1/ 2
:
J - a h
th is constant throughout:
2: u2
2g
x
I
(188
)
whole turbine, considering that
The sum of the stages isentropic drop in enthalpy is greater than the
isentropic enthalpy of the turbine considered as a whole by the amount
(1 t R), so that:
/hx
U2 .J(1 +R)h
J
h
(189)
t
2g
Therefore the velocity ratio will be:
)th
:
I +'
:1/14 R'
X
u2/ 2gJh
(190)
u
C'
th
x
C1
is the corresponding theoretical velocity for the heat drop of the enth
tire turbine. In other words, the original velocity ratio must be corrected
for the Reheat factor by the amount l//y
.
This correction is indeed very
small, and can be neglected altogether.
For the turbine internal efficiency, we must have in mind that it depends on
the stage efficiency and the Reheat factor:
(1 4- R)x It(191)
222
where the stage efficiency depends on the velocity ratio, and the reheat
factor on the number of stages and the expansion pressure ratio.
For the internal efficiency function, Figs. XIX and XLI are used, and
for the leaving loss function, the results already computed on Table XV.
223
Part II
Low Pressure Turbine Characteristics
At full load condition, r = 7, the air flow is known, and it is:
G
The value of
f (p
= 64.7 lbs./sec.
pi) can be computed using the corresponding values,
taken from Table XXVIII:
pi
-63.6
pe
psia
16.5 psia
Therefore, f rom Eq. (183):
1
)
f (p'p
(pjp 1 ) N
-
where:
x k
2 -
N
stk
2- 0.891. 1.4 - 1
1.4
1.746
then:
f e/pi
/p.1.746
(p
(16.5/63.6)
0.952
The inlet condition function is:
1
63.6 x 144
pi/F
212.2 lbs./ft. 2 x "F2
Cl860
Consequently, from (181):
(Oth)
G
-
) x P
64.7
0.952 x 212.2
-
0.32
Since in equation (182) the only variable is the second radical, we can separate this value, so as to facilitate further computations, so that:
224
from Fig. XIX:
1
4-
0.775
(E%
2~
then:
1 x
2Z
N (l/A2)-1C
2+2
0.32 x 0.775
0.248
From Part I, Section C-1, we have the value of the internal efficiency, thus
giving us one point in this curve, for the conditions set above; and it was
found to be:
0.91
From Fig. XIX, the leaving loss function plotted against
0
,
we haye for
2e
2e - 0.835, mhich is the value of the velocity ratio for the last stage
in Table XXXVI:
1 - k2e = O.251
which checks the value determined in that same table.
We are in a position to plot the characteristic curves of the L. P. turbine, by using results obtained in Table XV and in Fig. XIX, and by computing
the other functions.
It must be noted though, that the function p
be plotted for the best stage efficiency.
225
p, will
TABLE LIV
VARIATION OF
r
WITH
x
th
0
th
-
L. P. TURBINE
N)
414
t 1, +
Sp
24- 52)
0.272
0.248
0.906
0.2248
0.412
0.248
0.824
0.2043
0.548
0.248
0.784
0.1950
0.619
0.248
0.773
0.1917
0.693
0.248
0.770
0.1910
0.962
0.248
0.801
0.1986
TABLE LV
LEAVING LWSS FUNCTION
32e
1 2e
- L. P. TURBINE
0.3
0.5
0.7
0.8
0.9
1.2
0.57
0.384
0.277
0.254
0.2 51
0.362
TABLE LVI
L. P. TURBINE INTERNAL EFFY. FUNCTION
th
0.272
0.412
0.548
0.619
0.6 93
0.962
St
0.706
0.837
0.888
0.893
0.8 88
0.820
(1 + R)
1.048
1.034
1.024
1.018
1.0 22
1.030
ii
0.74
0.865
0.908
0.909
0.9 07
0.844
0
226
--------
-----
TABLE LVII
INLET PRESSURE - EXHAUST PRESSURE FUICTION
L. P. TURBINE
(P,/p
k
N
)N
1-
/pi
0
0
1.0
1.0
0.1
0.0179
0.9821
0.992
0.2
0.0603
0.9397
0.970
0.3
0.1220
0.8780
0.937
0.4
0.2020
0.7980
0.894
0.5
0.2994
0.7006
0.837
0.6
0.4090
0.5910
0.769
0.7
0.5360
0.4640
0.681
0.8
0.6780
0.3220
0.568
0.9
0.8320
0.1680
0.41
1.0
1.00
0
0
227
14- (-r
E
TV
Ie
.
fr5S
4 -i
XL O#"
D sCt4.
CaMc
I.0
0.9
o.e L
0.7
0.6
r.
0.3
0-
0.
0.
0 .',
0.5
Pc
o.6
2ry 8rw
228
.7
e.G
o.9
/.o
1.1
t.
Part III
High Pressure Turbine Characteristics
Proceeding in the same manner as in Part II, we get:
Since the stage efficiency is the same then:
N : 1.746
so that, using the values from Table XXXVII:
l
(p,/pi)
-
(65.2/102.9)1746
: 0.741
Inlet condition function:
= 102.9 x 144
V 1860
then:
a 343.8 lbs./(ft.2 ) x (oFj)
6
0.7441 x 343.8
(D th)
u 0.254
and:
2g/x
S'R
N X(1/A
=
0.254 x 0.775
: 0.1967
The different functions can now be determined as in Par t
II, in
tabular form.
The elliptic function is the same as that for the L. P. turbine, since
in Table LVII it has been computed on a dimensionless basis.
reasoning applies for the leaving loss function.
229
The same
TABLE LVIII
VARIATION OF ' WITH Zth
lx
-
H. P. TURBINE
2g__
FN (1/A,)
1
2
2
0.272
0.1967
0.906
0.1783
0.412
0.1967
0.824
0.1621
0.548
0.1967
0.784
0.1542
0.619
0.1967
0.773
0.1520
0.693
0.1967
0.770
0.1515
0.962
0.1967
0.801
0.1576
TABLE LIX
H. P. TURBINE - INTERNAL EFFY. FUNCTION
0.272
0.412
0.548
0.619
0.693
0.962
0.706
0.837
0.888
0.893
0.888
0.820
(1 + R)
1.024
1.014
1.008
1.006
1.009
1.013
li
0.723
0.848
0.894
0.898
0.895
0.830
0
th
l st
230
'7
?
f4
1.0
e.8
. ....
.......
0.7
0.&,
0.9
.2.
.3
'l-
.5-. 0.~ 0.7
232.
0'a
a
1.0
1.9
I-
SECTION D-2
Part I
L. P. Compressor Characteristics
In connection with these characteristics, a short explanation of the
basis in which they are drawn is convenient.
In Part I, Section B-5, it was said that the pressure rise across a
stage is given by Eq. (121):
P2
a (1 + k-1 x
p1
a1
with:
u2 k
k-1
2j
(121)
(117)
: kgRT
The volume flow through the compressor is given by the continuity equation:
Q
. G v, a A C1
(192)
which can be expressed also as:
Q,
w Al. a1 ( u/a)(C/u)
Q
a A 1 x a1 x D x(u/a)
Substituting (46):
(193)
But, substituting expression for perfect gas in (192):
a G R Ti
(194)
p1
Therefore:
Q/N
= R G FT
p1
(195)
Also it can be readily seen that the value of the Mach number represented
by:
u/a1 , is proportional to:
u/a
Therefore:
n//f, since:
r 1 x n
2 VT(196)
u/a1 c-. n/Vr
1
232
(197)
Let's call:
0= T1/Tl*
(198)
(99)
(pi/p1*
and:
where subscript 1 stands for inlet state; and no asterisk (*) means initial
state for fuel load at normal conditions, while the symbol with asterisk
represents any initial state that is wished to investigate.
of Q /G,
or against G xif'/ci
.
We are now in a position to plot the pressure rise P2/P1 against values
It must be noted that in Table iXXVII the correction to the values of
efficiencies and diagram ratios for a Mach number of 1.0 is unity, consequently for any other Mach number less than 0.4 it is also unity, therefore the curves of Fig. XL are good for all Mach numbers below 0.4;
under
a closer examination of the tabulation it will be seen that the correction
is also unity for Mach numbers greater than 0.4, and up to about M a 0.9.
Consequently in the tabulation that follows the values of
the same for all Mach numbers.
233
and E are
TABLE LIX
COMPRE:SOR CHARACTERISTICS*
u/a1 = 0.44
n/C91 = 110%
q st = 0.85
ql 5 . 0.86
I st = 0.8
0.507
0.424
0.511
0.409
0.521
0.347
0.198
0.426
0.183
0.467
0..157
0.637
pQ/p
2.98
3.04
2.90
3.11
2.59
3.18
Q/ 6
1.073
1.056
1.095
1.035
1.13
1.00
u/a 1 = 0.4
Q / A,
n/le
: 100%
2.58
2.651
2.46
2.70
2.23
2.75
1.021
0.999
1.04
0.973
1.062
0.94
u/a 1 . 0.36
n/19 1
90%
Pe/pi
2.28
2.35
2.21
2.40
2.03
2.43
Q 1/1;7
0.967
0.94
0.99
0.914
1.01
0.88
u/a 1 = 0.3
p / 9
&
Q
2.15
2.03
2.09
1.97
2.13
0.906
0.88
0.924
0.855
0.95
0.821
1.61
1.88
u/a 1 = 0.28
pe/p
n/v/':- 80
n/ 'G:
70%
1.78
1.83
1.75
1.86
0.835
0.81
0.853
0.781
0.744
S
(F2/pl) =
* The values of p /pi are obtained from p2/Pi by the relation:
plow followthe
and
above
table
The
pe/p., where S is the number of stages.
since unless
characteristic,
compressor's
to
the
approximation
an
ing are only
with number
work
to
necessary
is
it
,
/p
p
determine
another method is used to
1
2
(29).
exponent
a
large
to
raised
1.0
to
very close
234
4re~T
OAd.F
-
Ao
speed ornt/ o4
S-N at* /ooa
,
A
-
C H
F1Pe~o
AY= o.4
/OQ*
NQte
ft'ced i-r-
6
ge
)
(see
3.o
907
9'
2.oo
S O- g
2.0
i0
0
Ofr
0!.
09
o.4
OS
O -.
235
0.7
o.8
O.9
~
4.0
..4
t
SECTION D-2
Partial Load Characteristic
It will be assumed that the power absorbing device has a cube power
speed characteristic, such as shown on Fig. where the speed and power are
in per-cent of the full load values.
This assumption represents fairly well the power demand of a ship in
normal conditions.
The full power condition will be investigated first.
Assuming standard atmospheric conditions, and that the turbines are
running at the designed speed for full load we have:
From Section C-3, the internal efficiency of L. P. compresFor is:
I.
Therefore, if:
p0
14.7 psia
T
5200 Fa.
i0
28.77 Btu/lb.
0
then:
= o.85
pr 0
2.504
The overall pressure ratio is 7, and each compressor's pressure ratio
is r=
2.65
Consequently:
Pe = 14.7 x 2.65
39 psia
pr
es
n_32
x 2.504
14.7
= 6.64
F6r which from the air tables:
i
then:
i
-
es
io
. 68.85 Btu/lb.
68.85 - 28.77
* 40.08
236
FiG
4
1.7
5~~
'F
Pe fa in4~V
A ^f 4vE
-3
1.'
1.5
04
1.4
0.7
8.2
.4
g
8
..
Pan e
r
P7 0#10J-Vt. LOAX0
ZOW&Aa
Therefore, the work required by the compressor is:
40,.08
L.P.
47.18 Btu/lb.
=
0.85
i
e
consequently:
Te
28.77
-
47.18 = 75.95 Btu/lb.
= 715.90 Fahs.
The intercooler must bring this temperature down to 900 F., with a cor-
responding i3
and
= 35.98
Btu/lb.
pr3
a 3.047
Therefore, the intercooler must have an effectiveness of:
I - 12= 75.9- 35.98
le
75.95 - 28.77
i - io
0.848
=
The H. P. compressor efficiency is also 0.85, and we then have:
pr4 s a 8.08
i
;
z 85.80 Btu/lb;
102.9 psia;
p4
x 78.35 Btu/lb.
148
T4
= 756.50 Fa.
(WC HP : 49.82 Btu/lb.
Inlet conditions for the H. P. turbine are:
p5
pr 5
-
102.9 psia;
: 269.1;
T5
i
18600 Fa.
- 370.92 Btu/lb.
under which conditions the work output is, from Section C-2:
WH.P. z 49.23 Btu/lb, with an internal efficiency of 0.897, therefore:
321.79 Btu/lb.;
16s = 326.54 Btu/lb.
-
'6
186.97;
pr 6 s
P6 x 71.5 psia
Reheating at constant pressure to T7 : 18600 Fa., and allowing a pressure
drop of 1.5 psi in the combustion chamber we get:
pr 7
=
18600 Fa.
70.0 psia;
T7
269.1
i7 z 370.92 Btu/lb.
-
p7
238
Expanding in the L. P. turbine to a pressure of 16.5 psia, we gets
pr8
269.1 x 16.
70.0
63.5
then:
i8
216.94
Then, the turbine work is, with
= 0.91
WL.P.
140.0 Btu/lb.
i8
:230.92
T8
a 13360 Fa.
With a pressure drop of 16.5 - 14.7 . 1.8 psi available in the gas side
of the regenerator, and an effectiveness of 0.65 we gets
0.65 (230.92 - 85.8)
19 - i 4
94.4 Btu/lb.
The leaving loss can be approximated by the value found on Section C-1,
though the conditions are now different; but to get an idea of the probable efficiency of the whole plant, that value will be used, then:
Ai L
--1.49 Btu/lb.
Then the net work is:
140.0 - 49.82 - 1.49
Inet
88.69 Btu/lb.
Heat inputs
q
:(370.92 - 321.79) + (370.9
94.4
-
85.8)
239.85 Btu/lb.
Therefore the cycle efficiency is:
S88.6
x 100
239.85
= 37.0
Which is higher than the efficiency obtained in the original calculations
239
of the cycle, and it is something to expect, since the turbine ef-
ficiencies came to be higher than what was assumed originally.
Assuming
the loss of 150 h.p. for friction and windage and other mechanical losses,
the output power is then:
P : 3600 x 64.7 (88.69) - 150
2545
7960 h.p.
It can be concluded that to develop the requiredpower of 7500 h.p. it
will be necessary to reduce the flow keeping efficiencies constant, in
which case the mass rate of flow would approximately be:
G :
4.7
7960
x 7500
* 56.9 lbs./sec.
240
-----------
Part 11
Development of the-Partial Load Characteristio
The partial load characteristic can be obtained by using the following procedure:
Known data:
(1) Turbine and compressor's characteristic curves.
(2) Design features:-
(a)
d
T
/-S x d, since the pitch diameter is constant.
(b)
uF
u:
-
S x u, againdue to the fact that the pitch
diameter is constant.
The value of u can be found from
the speed that is wished to be developed.
(c) Exhaust and leakage area:
A2 e and AA2e
(d) Gauging of the last stage:
sin /
2e
=
0.55
(e) Peripheral speed of the last stage:
u
e
=u
(3) Mechanical losses, including friction and windage.
They can
be assumed to vary with the square of the speed, and the
value assumed originally, 2% of the useful power output, can
be used as a starting value.
With the data of (1), (2), end (3) known, an overall pressure ratio
can be used first, and from that the pressure ratio of the low pressure turbine can be determined from fig. III for the best conditions of operation.
241
M-
, ___
-
- "I --
- "iAlq
__
The pressure ratio of the low pressure turbine determines the isentropic enthalpy drop of the low pressure turbine, noting that the other
necessary conditions are:
exhaust pressure known, assumed to be atmospheric
pressure, and inlet temperature to turbine also fixed at the maximum limit.
With h known, and by using eq. 127):
C2
th
2g
Jh
(27)
the theoretical velocity is found, which introduced in (26) gives the value
for the theoretical velocity ratio of the turbine:
9
th
- u/Cth
(26)
We have thus fixed one condition in the characteristic curves of the L. P.
turbine, from which we get:
(1) the internal efficiency
(2) the leaving loss function
(3) and the function for
V(
1 -
E 2e
Kth)
With the pressure ratio for the turbine the elliptic function can be determined from the same characteristic curves.
Thus using eq. (181), the mass rate of flow is found:
Introducing this value in equation (15), the power is obtained, for the
conditions desired:
P
3600
2545
G
Ci
x h
-A iL)
-
PF.W.
(15)
Repeating this procedure for other conditions, the curve representing the
partial load characteristic is found.
242
BIBLIOGRAPHY
(1.)
C. R. Soderberg; 2.213 Adv. Probl. in Gas Turbines, class notes,
M.I.T. Spring Term 1946.
(2.)
C. R. Soderberg and R. B. Smith; "The Gas Turbine as a Possible
Prime Mover", copy of a paper presented before the Society of Naval
and Marine Engineers in 1943.
(3.)
C. R. Soderberg, R. B. Smith, and Lt. Comdr. A. T. Scott, "A Marine
Gas Turbine Plant", paper presented before the Society of Naval Architects and Marine Engineers, November 1945.
(4.)
A.Stodola, "Steam and Gas Turbines", authorized translation from 6th
German edition by L. C. Loewenstein; Peter Smith, New York, 1945.
(5.)
R. T. Sawyer, "The Modern Gas Turbine"; Prentice-Hall Inc., New York,
1945.
(6.)
"Gas Turbine Operates at 1350 degrees F Temperature", Marine Engineering and Shipping Review, Vol. LI, No. 3, pages 133-164, March 1946.
(7.)
Charles H. Johnson, "A Marine Gas Turbine Installation"; Marine
Engineerin and Shipping Review, Vol. LI, No. 4, pares 108-116, April
1946.
(8.)
"Gas Turbine Tests at U. S. Naval Experiment Station"; Marine Engineering and Shipping Review, Vol. LI, No. 5, pages 111-119, May 1946.
(9.)
Lt. Comdr. C. F. Kottcamp, "Instrumentation and Techniques Used in
Testing Gas Turbines at Annapolis"; Marine Engineering and Shipping
Review, Vol. LI, No. 5, pages 120-121, May 1946.
(10.)
J. Keenan and J. Kaye, "Thermodynamic Properties of Air"; John Wiley
and Sons, Inc., New York, 1945.
(11.)
Lionel S. Marks, "Mechanical Engineers Handbook"; McGraw-Hill Book
Co., Inc., New York 1941.
(12.)
Fred K. Fischer, Charles A. Meyer, "The Combustion Gas Turbine Cycle";
Westinghouse Engineer, Reprint 4095, May 1944.
(13.)
F. W. Godsey, Jr., C. D. Flagle, "The Place of the Gas Turbine in
Aviation"; Westinghouse Engineer, July 1945, Reprint 4221.
(14.)
"Test of 16 Related Airfoils at High Speeds", Report no. 492, NationalAdvisory Committee for Aeronautics, 1934.
(15.)
"Airfoil Section Characteristics as Affected by Variations of the
Reynolds Number", Eastman N. Jacobs and Albert Sheuman, National Advisory Committee for Aeronautics, Report No. 586, 1937.
243
(16.)
W. H. McAdams, McGraw Hill Book Co., Inc.; New York 1942.
(17.)
J. H. Keenan, "Thermodynamics", John Wiley and Sons, Inc., New York,
1941.
244