performance analysis of a turbo-type energy ... - core
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PERFORMANCE ANALYSIS OF A TURBO-TYPE
ENERGY ABSORBER FOR AN AIRCRAFT
CARRIER ARRESTING GEAR
Leo Stanley Rolek
United StatesNaval Postgraduate School
"H "S« I 'HL_J kZ? -1 I.
PERFORMANCE ANALYSIS 01 1 A TURBO--TYPE ENERGY
ABSORBER FOR AN AIRCRAFT CARRIER ARRESTING GEAR
^y
Leo Stanley Rolek, Jr t
/ /y<r' 7\5fJune 1971
II
App/ioued &0A public. <%zle.aA2.; diAtnlbntlcn un&uruXzd.
ADUATE SCHOOLMOJT:
, CALI£... 93240
Performance Analysis of a Turbo-Type Energy
Absorber for an Aircraft Carrier Arresting Gear
by
Leo Stanley Rolek, Jr.
Ensign, United 'States NavyB.S.A.E., United States Naval Academy, 1970
Submitted in partial fulfillment of therequirements for the degree of
MASTER OF SCIENCE IN AERONAUTICAL ENGINEERING
from the
NAVAL POSTGRADUATE SCHOOLJune 1971
ft
c.t
SCHOOL"
93940
ABSTRACT
The increasing weight and speed of carrier based aircraft are
taxing the limit of conventional piston type energy absorbers which
are used for arresting gear. Turbo-type absorbers have been proposed
as an alternative. This study investigates a turbo-type energy
absorber for an aircraft arresting gear under development by the
Naval Air Engineering Center, A mathematical analysis, computer
simulation and performance prediction are given for each mode of
operation. It was initially expected that the absorber could operate
in just two modes, forward and reverse, but the analysis shows that
four distinct modes are possible, since not only shaft rotation but
the path of fluid flow can be reversed. Theoretical performance
predictions are also compared with test data from an existing smaller
scale version of the absorber. The agreement is excellent. An
approximation of scaling effects on power absorption is also
presented. It is concluded that the turbo-type absorber is basically
adequate to meet the demands of carrier operation for the forseeable
future
.
TABLE OF CONTENTS
I. INTRODUCTION 12
II. DESIGN VALUES AND CONSIDERATIONS 15
III. ANALYSIS OF THE PROBLEM 21
A. MODES OF OPERATION 21
B. DIMENSIONAL ANALYSIS 29
C. ENERGY BALANCE AT EQUILIBRIUM 31
D. MODEL TESTING 34
E. ASSUMPTIONS AND LIMITATIONS OF THE ANALYSIS 35
IV. RESULTS AND DISCUSSION . 38
V. CONCLUSIONS AND RECOMMENDATIONS 47
APPENDIX A Detailed Analysis of the Energy Absorber v.'ith theFluid Circulating in the Normal Direction, Mode 1
and 2 48
APPENDIX B Detailed Analysis of the Energy Absorber with theFluid Circulating in the Reverse Direction,Modes 1 and 2 82
APPENDIX C Program NORMAL 88
APPENDIX D Program REVERSE 110
LIST OF REFERENCES 129
INITIAL DISTRIBUTION LIST 1 30
FORM LD 1473 131
LIST OF TABLES
Table Page
I Parameters of the Absorber -------------- 18
II Modes of Operation ------------------ 21
III Detailed List of Results for All Modes of theAbsorber -__--------__----__-_ 42
IV Flow Chart for Computer Programs NORMAL andREVERSE 90
- LIST OF FIGURES
Figure Page
1, Overall Schematic, Programmed Drum Arresting Gear - - --j/
2, Schematic of Energy Absorber _____________^g
3, Diagram of Absorber Bladings ---------.__-_-jY
4» Velocity Diagrams for Mode 1 - - _______23
5» Velocity Diagrams for Mode 2 _____________24
6. Velocity Diagrams for Mode 3 -------------25
7# Velocity Diagrams for Mode 4 ------------- 26
8, Total Enthalpy-Entropy Diagram for Modes 1 and 2 _ _ -27
9. Total Enthalpy-Entropy Diagram for Modes 3 and 4 -28
10, Energy Balance for Equilibrium for Normal Flow - - - - 30
11, Energy Balance for Equilibrium for Reverse Flow - _ _ -z-z
12, Power Coefficient versus Rotational Speed _______ ^9
13» Power versus Shaft Speed __--__-_--_---_^q
14. Maximum Power Coefficient versus Maximum ReynoldsNumber _______________________ a^
15. Estimate of Loss Coefficient ? ------------ yc
16. Estimated Channel Wall Friction Factor \~ ------- nn
TABLE OP SYMBOLS
Symbol FORTRAN
Latin
a
b B
Cf
CF
ci
CL
cP
CP
C CU
Definition Units
u
D.h
h - •
H H
H^ -
*RIR
S IS
^ KB
L~„ LL
u
m
M,
LU
SP
"E
Opening at blade exit
Blade height
Local skin friction coefficient
Pressure loss coefficient forlower passage
Specific heat of fluid
Pressure loss coefficient forupper passage
Hydraulic diameter
Static enthalpy
Total head
Total enthalpy
Rotor incidence angle
Stator incidence angle
Blockage factor
Length of meridional contour oflower passage
Length of meridional contour ofupper passage
Mass flow rate
Frictional moment
Static pressure
Equivalent relative total enthalpy
ft
ft
BTU/lb/°R
ft
ft-lb/slug
ft-lb/slug
ft-lb/slug
degree
degree
ft
ft
slugs/sec
lb-ft
lb/ft
ft /sec
Symbol FORTRAN Definition Units
Q Power lb-ft/sec
Q> POWRKP Reference power lb-ft/sec
2Total pressure lb/ft
Radius ft
Radius of curvature of innercontour of passage in
Effective Reynolds number - - - -
Reynolds number lower passage - - - -
Reynolds number rotor - - - -
Reynolds number stator -. — --.
Reynolds number upper passage - - - -
Blade spacing ft
Radius of curvature of outercontour of passage in
Torque exerted on fluid by rotor lb-ft
Reference torque lb-ft
Torque exerted on fluid by stator lb-ft
Internal energy ft-lb/slug
Peripheral velocity of blade ft/sec
Absolute velocity ft/sec
Useful absolute velocity ft/sec
Meridional component ofabsolute velocity ft/sec
Peripheral component of
absolute velocity ft/sec
Relative velocity ft/sec
y _ _ _ _ Useful relative velocity ft/sec
PT
TP
R R
r.1
RI
^e.
*H1REL
*m RER
\z RES
^u REU
As .
ro
.
TR
TORQUE
TRref
TORREF
TS
u
U •
V V
V _ _ _ ,
\ VM
Vu
.vu
V ' w
t
Symbol FORTRAN Definition Units
W.1
wu
2B
\
wu
\ YR
Ys
YS
Greek
a ALPHA
a3B
A3B
*kAAR
As
ARS
4BA4B
3 BETA
Pl-D B1B
B2B
DEL
ZETAL
ZETAU
ETAMU"
ETAML
Relative velocity lost due toincidence
Peripheral component ofrelative velocity
Rotor blade loss factor
Stator blade loss factor
Absolute flow angle
Tangent to stator blade meancamber line at station 3
Aspect ratio of rotor
Aspect ratio of stator
Tangent to stator blade meancamber xine a^ Suasion ^
Relative flow angle
Tangent to rotor blade meancamber line at station 1
Tangent to rotor blade meancamber line at station 2
Change
Channel bending loss coefficientfor lower passage
Channel bending loss ' coefficientfor upper passage
Friction efficiency for upperpassage
Friction efficiency for lowerpassage
Rotor efficiency
ft/sec
ft/sec
degree
degree
degree
degree
degree
degree
Symbol FORTRAN Definition Units
li
u
h — — — .
Vl& PIPOWR
\ PITORQ
nCO
PIOM-
P RHO
T_ TAURR
CO
CO
ref
LMBD Wall friction factor in straightducts
LAMDAL Friction coefficient for lowerpassage
LAMDAU Friction coefficient for upperpassage
NTT Kinematic viscosity
Dimensionless head
Reference power
Dimensionless Torque
Reference rotational speed
Fluid density
Dimensionless torque coefficient
- - - - Wall shear stress
PHI Flow coefficient used in ProgramNORMAL
PHI Convenient flow coefficient used. . in program REVERSE
PSI Velocity coefficient
OM Rotational speed
OMREF Reference rotational speed
ft /sec
lb-ft/sec
slugs/ft'
J.D/1X
rad/sec
rad/sec
Subscripts
R Rotor
S Stator
I Lower passage
u Upper passage
o Outer passage
i Inner passage
TH Theoretical
MAX Maximum value
ref Reference value
1 Rotor inlet
2 Rotor discharge
3 Stator inlet
4 Stator discharge
Supersscript
t
Useful component
10
ACKNOWLEDGEMENTS
The author gratefully acknowledges the assistance provided in this
investigation by the following persons:
Professor T. H. Gawain of the Naval Postgraduate School, the
thesis advisor, whose understanding and patience made the learning
process most gratifying. His insight and guidance insured the worth of
this thesis.
Professor Michael H. Vavra of the Naval Postgraduate School who
developed the original analysis of the absorber. Without his help and
inspiration this thesis would not have been possible.
Mr. Wayne Gallant and the staff of Section SI of the Naval Air
Engineering Center in Philadelphia, Pennsylvania. These men kindly
provided the background material, supporting information on the M-21
absorber, test data, and the original computer simulation of the
analysis. Their full cooperation was most beneficial.
11
I. INTRODUCTION
Over the years, the speed, size, and weight of carrier based air-
craft have steadily increased, thus demanding an ever larger energy
absorption capacity from the shipboard arresting gear. Up to now,
design improvements to meet these increasing demands have been limited
to mere enlargement of the basic linear hydraulic energy absorber,
which has been of the piston type. Modern aircraft, however, have
loaded this system to its upper limit, and make any further development
along these lines impractical due to weight and space limitations. The
need exists for a new concept which will minimize space and weight
demands while providing the necessary increase in energy absorption
capacity.
The solution proposed involves the use of a turbo-type rotary
hydraulic energy absorber. In essence the fluid is put into motion by
a suitably designed rotor and is retarded by a stator and by the
resistance of the flow passage. The power input to the rotor is
absorbed by the hydraulic resistance of the system, and is eventually
dissipated as heat.
The energy absorber is directly coupled to a cable storage drum.
The drum profile is programmed to match drum rotational velocity to that
required for optimum retarder performance. A device of this type was
introduced in 1968, and is known as the M-21 energy absorber. The
initial design gives satisfactory performance, but is of rather limited
capacity, thirty million ft. lbs. Therefore, a program has been under-
taken to redesign the absorber to increase the capacity to fifty-five
million ft. lbs.
12
A development program was initiated by the Naval Air Engineering
Center (NAEC) (Ref. 1). A full scale test of the programmed drum
coupled with the M-21 absorber was conducted to determine the
feasibility of the overall concept and to obtain performance data
needed in the redesign program. A schematic diagram of the proposed
system is shown in Figure 1
.
A theoretical approximation of the flow conditions in a machine
of this type was derived by Professor Michael H. Vavra (Ref, 2), To
facilitate investigation of the effects of various design parameters
NAEC formulated a computer program. From the computer analysis, a
workable combination of design parameters was chosen such that torque
and energy requirements were satisfied at the design point, the con-
dition at which the absorber is expected to perform most often.
Professor Vavra has also been commissioned to develop ana test a
flow model of the new design. This model will be useful for refining
losses, checking for flow separation, and obtaining information that
is not available from the mean streamline analysis.
In this paper, a theoretical simulation and performance analysis
of the energy absorber in all modes of operation is presented in terms
of dimensionless parameters that facilitate comparisons of various
designs and interpretation of model tests. The retraction problem
is also studied, and preliminary estimates of the power needed to
retract the programmed drum are given,
A throttling system is incorporated into the design to aid in
arresting aircraft of various sizes. The discussion and analysis
presented here are concerned with the condition of open throttle only.
13
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14 _
II. DESIGN VALUES AND CONSIDERATIONS
The main features of the energy absorber design that is analyzed
here are summarized in Figures 2 and 5» and Table I,
The customary approach in the design of turbomachinery is first to
optimize conditions with respect to a single specified design point.
The resulting performance over a broad range of off-design conditions
is then investigated to determine overall acceptability. The condition
at which the absorber is expected to operate most often is naturally
chosen as the design point. Specifically, a design torque of
340»000 ft. lbs. is required at a rotational speed of 76 rad./sec.
The maximum rotational speed the cable will impart to the drum, and
consequently to the rotor, is about 8? rad./sec. This limit is fixed
by the allowable landing speed of the aircraft, and by drum design.
The mechanical power imparted to the rotor is continuously
converted to thermal energy of the fluid. The temperature rise of the
fluid is significant and imposes a constraint on the design. The
temperature limit is the maximum allowable temperature the fluid may
attain without causing cavitation. Obviously, heat must be removed to
maintain equilibrium conditions. For this reason, provisions are made
to bleed off a portion of the circulating fluid continuously and to
replace it by fresh cool fluid. During a rapid series of arrests, the
cooling problem is especially critical.
The fluid used in the absorber is an ethyl glycol mixture. This
fluid is acceptable because it remains usable over a wide range of
temperatures.
15
UPPER PASSAGE
ROTOR
LOWERPASSAGE
*±\ _£
R4 R-
SCHEMATIC DIAGRAM OF ENERGY ABSORBER
FIGURE 2
16
Se^r ion A -A of ^i»urp 2 Through Rotor
Section B-B of Fieure 2 Throueh Stator
Figure 3 DIAGRAM OF ADSORBER BLADING
17
Table I PARAMETERS OP THE A3SOKBER
R = 13,61 inches1
R = 16,25 inches
R, = 16.07 inches3
R = 14.93 inches
b^ = 2,02 inchesn.
b = 2.02 inches
r = 4« 18 incheso
r. = 2.02 inches1
(L . + L )~ =U0
- 10.26 inches
(L, . + L, )
= 10.26 inches
Number of blades for rotor and stator: 72
eiB=57°
P2B
= 53°
«3B=
57°
«4B=53°
^2 " - 87
^3= - 9°
*B4 " - 87
"VlAX = 83 rad./sec.
Fluid properties:
p = 2.095 slugs/ft.2
cf
= .016
v = 3,76 x 10"5 ft,2/sec c = .7
18
A detailed discussion of the effects of various parameters and loss
coefficients is undertaken in the analysis section. A few words,
however, are in order here. The radius of curvature r. of the inner1
contour of both the upper and lower passage must not be smaller than
the blade height. This insures that the flow does not separate, and
it also increases absorption capacity. Estimates of losses in this
machine are obtained through various empirical turbomachinery cor-
relations as is shown in the analysis. Enthalpy losses in the passages
are significant. Much of this loss is due to the bending of the
passage with friction responsible for the remainder. However, a
change in the coefficient of friction c„ which expresses the direct
wall friction losses in the passages has little influence on overall
performance. A value of c„ = 0.016 was chosen since it gives best
agreement with M-21 test data.
The blade shapes first optimized by Professor Vavra from a
performance standpoint were found by NAEC to be unacceptable on a
stress basis. Blade thickness, spacing and angles were accordingly
revised by NAEC, and a blade of sufficient strength was found; its
characteristics are listed in Table 1,
The reliability of this analysis can only be judged experimentally
by correlation with test data. The theory correlates very well with
actual test results on the M-21 design. Prom this it appears that the
theoretical analysis is reasonable and that the new design should be
workable, if not optimum.
Naval carrier operations often require a series of rapid and
repeated arrests. The drum must be retracted thus reversing the
19
rotational direction of the rotor. This process is investigated to
obtain preliminary estimates of the power required to retract the
absorber.
20
III. ANALYSIS OF THE PROBLEM
A. MODES OF OPERATION
The energy absorber is capable of operation in four modes. These
modes are summarized in Table II.
Table II MODES OF OPERATION
Flow Direction Shaft Rotation
Normal Reverse
Normal Mode 1 Mode 2
Reverse Mode 3 Mode 4
Mode 1 corresponds to design point conditions. In this mode, the
fluid passes successively through stations 1, 2, 3, 4 ^^ back to 1,
in that order. Note that the flow leaving the rotor at station 2 is
moving radially outward. See Figures 2 and 3« Fo^ fluid circulation
in this sense, conditions at stations 2 and 4 are particularly sig-
nificant. At the rotor exit, station 2, the direction of the flow
with respect to the rotor, is assumed to remain constant at all flow
rates. At stator exit, station 4» the direction of the flow with
respect to the stator is likewise assumed to remain constant. While
this assumption of fixed relative and absolute flow directions is not
exact, it represents an excellent approximation when the blades are
closely spaced as in the present design.
In Mode 2, the sense of the fluid circulation is exactly as
described above. However, the direction of shaft rotation is reversed.
21
Naturally, stations 2 and 4 are still governing in the sense that flow
directions at these stations are known.
In Mode 3» the shaft rotation is the same as for Mode 1, but the
direction of fluid circulation is reversed. Consequently, the fluid
passes successively through stations 2, 1, 4» 3 and back to 1, in
that order. Note that the rotor exit is now at station 1 , and the
direction of flow at this station is radially inward. In this case,
stations 1 and 3» are controlling. The relative flow direction at
rotor exit, station 1, and the absolute flow direction at stator exit,
station 3> are now assumed to remain fixed.
In Mode 4» the sense of fluid circulation is the same as in
Mode 3> but the direction of shaft rotation is reversed. Of course,
stations 1 and 3 still govern flow direction as in the previous case.
The same net of equations describe Modes 'I and 2, but a separate
set of equations is needed to describe the reverse flow Modes, 3 and 4<
Note that only two sets of equations are needed for the four Modes,
and each set is governed by the flow direction relative to the blade
exits. Two separate computer programs are needed since one program
will be used to solve each set of equations. Program Normal finds the
solution for Modes 1 and 2, and program Reverse finds the solution for
Modes 3 and 4» Both programs will be discussed in the appendix. All
quantities and results discussed throughout the paper are obtained
from these programs.
The velocity diagrams for each mode are presented in Figures 4> 5»
6, and 7* All angles and velocities are drawn to scale.
The thermodynamic processes in the absorber are shown on the
enthalpy-entropy diagrams, Figures 8 and J. The process for normal
22
^u2 H J— Vf, fc U2—*
Scale: 0.5 inch = 100 ft/sec
-V,'U3
H
Figure 4 Velocity Diagrams for Mode 1 Operation at2
CU
I4AX 2 , . „.6^e
- - 4.8 x 10
23
'UJLWu^
-wUt—4*-vu , —
I
Scale: 0.5 inch- 25ft/sec
*-vUl|—
1
Figure 5 Velocity Diagrams for Mode 2 Operation at
'
p 2
^e v= 4 '8x 10
24
Scale: 0.5 inch- 100 ft/sec
uv-
Figure 6 Velocity Diagrams for Mode 3 Operation at
25
-4%d
Scale: 0.5 inchr 25 ft/sec
Figure 7 Velocity Diagrams for Mode 4 Operation at
2
"W 2 , _ ._6
^e = —7 = 4.8x10
26
</>
>-Q_
oen}—
Ixl
J-H AdnVHlN3.1V101
27
(/>
>-CL
Otrh-
<VJ
X1 H AdlVHJLNB TVJLOI
3 o
28
circulation, Modes 1 or 2, is shown in Figure 8, while that for
reverse circulation, Modes 3 or 4, is shown in Figure 9« The
direction of shaft rotation affects the exact quantitative enthalpy
values. The graphs, however, are merely qualitative.
A "total head" H (per unit mass) is used which differs from the
standard total enthalpy H~. This is discussed in greater depth in the
analysis section, but a few words are needed at this point to clarify
the discussion.
The fluid is considered incompressible which allows the thermal
and mechanical effects to be uncoupled. A "total head" is
conveniently defined as
P 2
p p 2
Henceforth H is referred to simply as head. The head and total
enthalpy are linked by the simple relation
H = Hip - u
where u represents internal energy per unit mass.
B. DIMENSIONAL ANALYSIS
Results are presented in terms, of appropriate dimensionless
quantities. The use of non-dimensional terms simplifies the cor-
relation of data, the comparison of various designs, and the inter-
pretation of model tests.
The three quantities chosen as consistent dimensional reference
parameters are the radius of the rotor outlet R? , the fluid density P,
and the shaft rotational speed u) or m , whichever is appropriate.
29
Using these parameters, one defines the effective Reynolds number
for this design to be
h '
wR2
2
ve
where v is the kinematic viscosity. And the dimensionless total head is
"h=
2^ 2<u R
2
In the discussion of results, it is convenient to define the flow
coefficient $ in the following specific way, namely,
V,IV
1
COR,
where the algebraic signs of uj and $ are always positive for normal
shaft rotation and negative for reversed shaft rotation. This sign
convention for shaft rotation is maintained regardless of the direction
of the fluid circulation, hence the use of the absolute magnitude |VM9 |
in the definition.
In the actual computer programs NORMAL and REVERSE, however, some-
what different sign conventions are more convenient. All equations are
written in such a way that the three quantities V _ ojR and § are always
positive. For this reason the four modes are computed as separate cases.
For comparison of various modes and designs the dimensionless torque,
power and rotational speed coefficients are respectively
2T> 5
" MAXR2 P
30
<?
BM R2 P
n =M
MAX
The detailed analysis considered later also usee various
dimensionless quantities for convenience in the derivations. The
parameters used to facilitate the non-dimensionalization in that case
are not necessarily the same as introduced in the present section, and
the separate cases should not he confused.
C. ENERGY BALANCE. AT EQUILIBRIUM
In general the rotor adds energy to the fluid. In traveling
through the passages and returning to its initial state the fluid
dissipates this mechanical energy through various losses. Under
equilibrium these losses exactly cancel the original energy input.
It turns out that the energy input of the rotor can be expressed
in terms of the flow coefficient $ in the form
A HIN
= A + B $
where A and B are true constants. The losses can be expressed as
^LOSS " A ' + B ' - +C ' ^
where A , B , and C may be considered constant to a first approximation.
Actually these quantities are weak functions of §, a fact which must
be considered in the exact solution. The energy relations which fix
the equilibrium state are shown in Figures 10 and 11. Figure 10
represents the two modes which correspond to normal flow direction.
51
ftj
ra
o•H-P•H
OO<D
Si-p
uO C\J
•H
•H—
I
•H
no
<D
O
o
CO
CM
ua
EH
— O
-Pc3
Pio•H
tj
aofH
•HO
o
o
32
ro
o
o
II S39NVHD A9U3N3
o•H-p•H
sooa>
pH
I
•H
•H
•H
uo
0)
o
pq
sa)
EH
O
00
CMcv«
te3
pni
o•HP
oU•HO<D
uCD
!>
<D
P=J
«HO
Figure 11 shows the two modes for reverse flow. These results are
drawn to scale, except that the parabolic lines are only approximate.
The final equilibrium points, where the losses balance the energy
input, however, and the relative distribution of losses at these
points are in correct scale.
Initially, it was thought that only two of the above four modes
would prove to be physically possible, one corresponding to normal
shaft rotation, the other corresponding to retraction. Surprisingly
enough, the present analysis has demonstrated that operation in all
four of the modes is theoretically possible. This point is clearly
indicated in Figures 10 and 11.
D. MODEL TESTING
Professor Vavra ha.s proposed that a model testing program be under-
taken. The model will not be a true dynamic model, but will be a purely
static device that uses air as the working fluid. Construction of the
model will constrain the air flow to the normal direction only. However,
the scale model should be useful in checking for flow separation
through the passages and refining the loss coefficients for the
analysis of the absorber. The effects of the tangential component of
velocity will also be investigated.
The discussion of the above static model is not within the scope
of this thesis. However, a hypothetical model is considered which is
a true dynamic model of the actual full scale device. The size of the
hypothetical model is taken as identical to that of the static model.
The model is scaled to O.569 of the size of the actual absorber. In
both cases the fluid is air. When expressed in dimensionless terms,
34
the performance of the hypothetical model is "basically the same as that
of the full scale machine, except for Reynolds number effects. The
full scale machine operates at a maximum Reynolds number, as defined
above, of 4»88 x 10 while the maximum Reynolds number of the model is
53.70 x 10 . The results for this model are presented along with the
results for the real absorber.
E. ASSUMPTIONS AND LIMITATIONS OP THE ANALYSIS
A one-dimensional mean streamline analysis is obviously only a
crude approximation of the actual performance of any turbomachine
,
but is a starting point necessary in most development programs. The
validity of the assumptions used are very important to the reliability
of the analysis. The primary assumptions used in this report are
summarized below.
The fluid is treated as incompressible. This assumption is
excellent as long as the provisions to maintian the fluid at a
temperature lower than the cavitation temperature are adequate. This
conveniently uncouples mechanical and thermal effects and allows a
form of "total head" H to be defined which is independent of
internal energy.
If the velocity distribution across every section were turly
uniform, the meridional component V at. the mean streamline could be
computed simply from a continuity relation of the form
v =-2-
where Q is the volumetric flow rate and A^ is the meridional cross-
section area. The actual velocity distribution, however, is non-
35
uniform because of the effect of streamline curvature, "boundary layers
and so on. To allow for these effects, approximately, the actual
meridional velocity V^ is assumed to be governed by a relation of the
form
where, the so-called blockage factor K~ is treated as a constant. The
blockage factors were extrapolated from M-21 test data. Their reliability
in the reverse flow modes is doubtful, however, since all test data
were obtained only from Mode 1 operation. An extension of these data
to all modes would be very useful.
The present analysis treats operation of the absorber as a steady
state phenomenon. Strictly speaking, the operation is actually
unsteady. However, the arrest and retraction processes require a
sufficient number of drum revolutions that the steady state assumption
provides a fair approximation.
A major assumption is that the relative flow direction at the rotor
outlet and the absolute flow direction at the stator outlet are
constant, For the case of normal flow this is an excellent ap-
proximation. However, the validity is reduced for the reverse flow
modes. In Modes 3 sxid 4> the flow encounters a sharp leading edge,
and exits past a blunt trailing edge. The implication is that the
flow is not as well guided as in the normal flow case. Thus the
actual turning angle of the flow is undoubtedly less than predicted
by the theoretical analysis. The power absorbed in both reverse flow
modes would consequently be less than predicted.
36
Losses, in general, are approximated by various empirical cor-
relations and are reasonable as shown by test results. They are
discussed in depth in the detailed analysis section.
Incidence losses, however, pose a problem. The incidence loss
approximations introduced in the analysis section are reasonable as
long as the incidence angles remain small. This is obviously not
the case for the retraction modes as shown in Figures 5 Stiid. 7» The
approximations break down physically at these severe incidence angles.
It was thought earlier that two of the four theoretical modes
might show themselves to be physically impossible. For example, a
large incidence loss might conceivably preclude the attainment of the
necessary energy balance. This has not proved to be the case, however.
It appears from the present analysis that operation in all four modes
remains possible physically. It is true, of course, that the predicted
performance for the two retraction modes, Modes 2 and 4> will be
inaccurate because of these large incidence effects, but this in itself
does not eliminate the existance of these modes.
At extreme angles of incidence, a distinct possibility of stalling
exists. In normal operation, this limits the attainable power absorption.
In a retraction mode, however, stalling would not be totally undesirable.
If the rotor stalls, the energy absorbed by the device is greatly
reduced, and the power necessary to retract the rotor is minimal.
All of the foregoing assumtions are thought to be reasonable.
Refinements of various points would be useful, especially for large
incidence angles and blunt trailing edges.
37
IV."
RESULTS AND DISCUSSION
The discussion of the problem up to this point has already included
a number of significant results reached in the analysis as shown in
Figures 4 through 1 1
.
Additional detailed results for all modes of the absorber are
presented in Figures 12, 13, 14, and in Table III. The results for
Mode 1 of the dynamic model are also included.
The main results of this investigation are summarized concisely in
Figure 12. This is a logarithmic plot of the dimensionless power II
as a function of the dimensionless shaft speed uu/od . Performance' MAX
of the proposed absorber, of the hypothetical model, and of the
existing M-21 machine are compared on this graph. Note that the
theoretically predicted performance of the M-21 . version as shown by
the continuous line is in excellent agreement with the actual measured
performance as shown by the data points.
The actual horsepower is plotted as a function of absolute shaft
speed for all four modes of the proposed absorber in Figure 13^ The
power absorbed during .arrest and the power expended during retraction
are presented for a range of possible operating speeds.
The maximum dimensionless power is also shown as a function of
Reynolds number R^ in Figure 14» This graph, although based on only
e
three computed points, does correctly portray, approximately, how
scaling affects power absorption. It would be a simple matter to
calculate additional points on this curve from program NORMAL.
38
10.
1.0
t=J
UJ
<J 0.1U.U.UJoooruj
oQ-
0.0
.001
— Proposed Absorber
-- Model—-M-2I
M"2I Test D
Mode
*MX
Figure 12 Non-dimensional Povrer vs. Non-dimensionalRotational Speed
39
ROTATIONAL SPEED (R.P.M.)
Figure 13 Absolute Power vs. Shaft Speed for the FourOperating Modes of the Proposed Energy Absorbe:
40
c
UJ
oU_U-UJ
ouenUJ
oQ.
6-
x< 4
Q.
2.-
Mini
G Computed Poinls
J I I I I I
4 5 6 78 10 5 6 7 8
REYNOLDS NUMBERw„„ R^
Figure 14 Maximum Power Coefficient vs. Maximum ReynoldsNumber for the 3 Designs Considered
41
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44
As mentioned earlier, the actual existence of four stable modes is
a surprising result. The original expectation, based largely on
intuition rather than analysis, was that only one mode would be possible
for each direction of shaft rotation. But the absorber is an unusual
machine which tends to confound intuitions based on experience with more
conventional devices. Consider for example the fact that losses are
necessary for the machine to absorb the mechanical energy input. The
smaller the loss coefficients, the higher the final equilibrium flow
rate will be, and the greater the energy absorption. Thus we are led
to the seeming paradox that the lower the loss coefficients the higher
the energy lost!
The power curves of Figures 12 and 13 are interesting and
significant in that all curves have nearly constant and nearly equal
slopes. The power can be expressed in the form
v= nfox k
<S)
^MAX
where the slope of the curve is roughly T)~3 for all cases.
The horsepower curves of Figure 13 indicate that Mode 1, the
design point mode, absorbs the greatest amount of energy. Mode 3>
however, is a very close second. Refinement of the analysis should
lower the predicted absorption of'Mode 3 somewhat. Nevertheless, the
favorable comparison of this mode to the design mode is interesting.
Figure 13 also shows that the power needed to retract the rotor is
reasonably small for Modes 2 and 4» especially in comparison to the
power absorbed during arrest.
The question of which mode the absorber will actually operate in
under various conditions of use is interesting, but by no means
45
completely clear at this time. It is thought possibly that if the
absorber be started from rest with shaft rotation in the normal
direction, Mode 1 should occur. If retraction follows immediately,
the retraction mode will probably be Mode 2. This is based on the
fact that during arrest the flow will build up a large momentum which
will tend to persist in the same sense during the subsequent retraction.
If retraction be started after the fluid motion ceased, however, the
resulting mode of operation remains uncertain.
Some method of sensing and/or controlling flow direction would be
useful to insure that Mode ? was occurring on all arrests. Mode 3
operation might not have enough absorption capacity to handle large
aircraft
.
The general question of which mode occurs is an area for further
study. Testing of the actual absorber with the idea of definitely
determining the operating modes under various conditions would be
extremely useful.
46
V. CONCLUSIONS AND RECOMMENDATIONS
The development of the problem and the exact solution of the
two sets of equations describing normal and reverse flow conditions
clearly show that four stable operating modes exist. The analysis
adequately predicts the absorption capacity of the proposed device
for Mode 1. However, the analysis is not as accurate for the
reverse flow or retraction modes.
It is recommended that the retraction problem be further studied
to refine loss coefficients and to more accurately predict the power
necessary to retract the absorber. It is also recommended that
further study be made into the question of predicting which mode
occurs under various operating conditions . This study would
encompass the unsteady phase of the problem. Actual testing of the
absorber should also be conducted with the intention of determining
which mode will occur. A method of flow control or sensing would
also be useful to guarantee proper performance in that Mode 3
absorption capacity may be insufficient in certain cases.
This turbo-type of energy absorber appears to be completely
adequate to meet the demand of carrier operations now and in the
forseeable future. The potential for future development and
enlargement is ample.
47
APPENDIX A
DETAILED ANALYSIS OF THE ENERGY ABSORBER WITH THE
FLUID CIRCULATING IN THE NORMAL DIRECTION,
MODES 1 AND 2.
A one dimensional mean streamline analysis after Vavra (Ref . 2)
will be developed with the following assumptions:
3p = constant
a = constant
Uniform flow at stations 1 to 4 except for blockage factors.
Blockage factors are independent of the flow.
The fluid is incompressible.
Steady state conditions exist.
1.0 Torque and Velocity Relations
The torque T„ exerted on the fluid by the rotor is positiveR
when acting in the direction of rotation.
TR = " [R2Vu2
" R1 V 0)
where m is the mass flow rate.
m = p A Vm
- p(2" h, y vMi
K^(2)
f^ 2" E2 V V
M2 %The quantity 2tt R b is the flow area, V .. and V
?- are the meridional
velocities at the mean streamline, and K_. and K_ are blockage factors
which are assumed to be independent of the flow.
48
By introducing a flow coefficient
V VI = J£ = -M (3)U
2ujR
2V>
all necessary velocity and thrust equations can be written in terms of
known geometric parameters and the flow coefficient. An iteration
technique is later employed in program NORMAL to calculate §. The
following velocity relations are developed now in order to simplify
the energy relationships which follow.
The sign convention used is that all peripheral velocity com-
ponents are positive when in the direction of rotation. From Figures
4 or 5 simple geometric relations are obtained which become, using
eq. J.
W2
= VM2/
C0S 32
= ^2 $/C0S 32 M
Wu2
= VM2
Tan 32
= (UR2
§ Tan P2 ^)
V l9= V, + tuR = u;R [1 + $ Tan B_] (6)
Also
u2 ~ u2 2 2L ^2
Vu2
= VM2
Tan Q2
= ^2 $ Tan a2 (7)
An expression for or in terms of the known relative exit angle can be
obtained by equating the previous two equations, or
Tan a = Tan p + 1/$ (8)
similarly at the rotor inlet
Wu1 - V
H1Tm P
1(9)
Vu1
=VM1
Tm B1+<"E
1(10)
The mass flow equation and the definition of the flow
coefficient yield
VM1 *2 R
1KB1
U1;
and
R K~Vu1 • «2 ij K^ Tan P
1+
""l < 12 >
The mass flow equations must be extended to include flow
through the stator.
£ = 2n E2
bR p V
M2K^ = 2n H, b
sp V^ K^
(13)2ti R
4bS p V
M4 VFrom the above relations, the meridional velocity equations are developed.
VM1
R2 h>2
(14)M2 "1
JL Kg
5a !?. 5* *aVM2
=R3 ^3 b
s
'
(15)
^^2^2^VM2~ R
4 V bs
'
( 16 >
1 ,1 Frictional Losses in Upper Passage
Let M„ be the frictional moment from forces along the wallsfu
of the upper passage from station 2 to 3» acting on the flow opposite
to the direction of rotation,
m R.. V , « m R- V - M
.
(17)5 u3 2 u2 fu
50
Setting Mfu = ^ mR
2Vu2 ( 18 )
eq. 17 becomes
Vu3 " \u i V
u2 (20)
From the mass flow equation
R?
K_?
b
v = $ cuR — — — (21)VM5R2R
5K^ b
sW
1,2 Frictional Losses in Lower Passage
With M„» being the frictional moment caused by forces along
the lower passage walls from station 4 to 1 acting on the flow opposite
to the direction of "rotation, the equations for the lower passage
are developed in a manner similar to the previous development.
% - h™ R4Vu4
(22)
(23)
•• (24)
(25)
From Figure 4 it can be seen that
V . = Vw , Tan a. (26)u4 M4 4
% = 1 - hR4
Vu1
=\jL
Vu4 R^
V = $ cu R —«4 2E
4 V bs
Using the previous three equations gives
1 34 s
E? L 9
b
Y „ = T1, 3 a) R -£ -££ -£ Tan 0/ (27)
51
From Figures 4 or 5» one can relate the absolute inlet angle
to absolute velocity components as
V ,
Tan or = -=i (28)M1
Using the relations developed for the inlet velocity eqs, 11 and 27,
the absolute inlet angle is
Tan ai
= \i Tssj Tm °4 (29)
Similarly the relative inlet angle is
Vu1 - » R
1Tan 3, =
1 V' VM1
coR1
= Tan a - ==
—
1 VM1
Substituting for V . yields
\ ^1 K1\ 2 ^1
The relative and absolute velocities can now be obtained from
their respective components by employing the following general equations:
V2=YM2+Vu2
(31)
W2
= VM2+ W
u2 "
(32)
W = VM/cos S (33)
52
1.3 Torque Relations
The torque as defined in eq, 1 can now be expressed in
terms of known geometric parameters and the flow coefficient. Using
eqs. 37 i 6, and 12 in eq. 1 yields
V bs
—"4JJ (54)
The non-dimensional torque coefficient iv, is now introduced asR
t
TR
TR "2n(R
2 )5 \ P K^2 '
bs
V* + * l
Tan B2- V ^ r Tan V (35)
The torque T which the stator exerts on the fluid is positive
when acting in the direction opposite to rotation
T = m (R, V , - R/V J (36)s
v3 u3 4 u4' w '
Substituting the values for the whirl components gives
». - 2TtR25(j{) p
"2
SaLV *(1+Manp2 ) " *
2
(if) K^Tan °4
Again introducing a non-dimensional torque coefficient
T
T =
5 /V2TT R/ Irr- ) p CO K_
2 VR2/ T32
(37)
b T
2 i
'
m „ R ^2- \u *
+ * LAiuTan p
2 - r k^ T- «4J
53
The general torque equation is
R fu fZ s
The torque exerted on the rotor by the fluid minus frictional moment
losses is equal to torque exerted on the fluid by the stator.
The dimensionless torque coefficient introduced earlier is
again given as
nT
=~~~T 5 ( 58 )
^ MAX V2.0 Energy Relations
The torque and velocity equations developed in the preceeding
section remain in terms of the unknown flow coefficient §. In this
section an algebraic energy equation will be formulated and solved
to yield §,
For the sake of this development, a total head term H will be
defined which differs from the standard definition of total enthalpy
H~ which is
Hj - h + \ (39)
or
H^u*-*-£ + 4 - « + ^ (40)
where h is the static enthalpy
p is the static pressure
u is the internal energy
and P is the total pressure.
54
Since the fluid is treated as incompressible, one can uncouple
the thermal effects from the mechanical effects, and define a 'total
head 1 H which does not include thermal effects. This new term is
denoted by
P 2
Obviously the enthalpies are related by
ftp - u) = H (42)
This new definition of total head H is useful in that constant H and
constant P lines are coincident. Throughout the remainder of the
analysis, useful total head H will be referred to as simply head. Note
that this total head has the units of enthalpy (per unit mass).
The process undergone in the absorber is now represented in
Figure 8, Obviously the process does not return to the original total
enthalpy EL , although it does return to the original total pressure
and head.
From Figure 8 the energy rise across the rotor is shown to be
equal to the energy losses across the stator and through the passages,
or
AEL. - m^ - AHX
- AHS
= (43)
Therefore
,
(H2
- RL) - (H2
- H3) - (H
5- H
4) - (H
4- H,) = (44)
where all the above terms are written as positive quantities.
55
The four terms of eq. kh will "be derived in terms of the flow
coefficient and the known geometric parameters as in Section 1.0. Eq.
kk is then straightforwardly solved to yield i. Program NORMAL is
employed to accomplish this task. Since various loss parameters
depend on $, an iteration technique must be used.
2.1 Conditions in Rotor
The incidence angle of the rotor blading i is defined asK
or merely the angle between the tangent to the mean camber line at the
leading edge and the relative inlet velocity. The incidence angle i
is seen to become more negative in the direction corresponding to a
greater lift coefficient for the blade.
The relative velocity ¥ can be resolved into two com"ponents
respectively tangential and normal to the mean camber line . The energy
t
associated with the tangential component W is assumed to be largely
recoverable. The energy associated with the normal component is
assumed to be totally lost.
The useful relative velocity at the rotor inlet is then
¥.' « W_ cos i_ . (1*6)
-L J. K
and the velocity component lost due to incidence is similarly
¥li
= Wl
Sin h (V7)
The basic energy equation of turbo-machinery is
"ffi- "ll - U
2 V " Ul V < U8)
56
which upon expansion becomes
2 2 2 2 2 2V-HJp -W_ V_ +u
n-w/
*T2 " % - P " o ^Using static enthalpies, regrouping, and cancelling terms in eq. ^-8
gives
2 2 2 2w - u w - u2 2 1 1
h2
-2-^-S- - hx
+ -i_A. (50 )
Substituting for the static enthalpy and rearranging yields
2 2 2 2p w
2 p w u2
-Uj.
T ~ = 7 + "2 2(U
2 - V (51)
Upon resolving W /2 into its components and regrouping further, one
obtains
/P2 + V\ A ,
Wl'
2
,
U2
2- UA /
WlAV7 ~2~v
=vy
+ -r- +—2—;-
lu2 - u
i - —) (52)
This leads naturally to the definition of the "equivalent relative
total pressure" as
and
'2 2 P
(5>0
and to the definition of the rotor blading losses as
APfYi \
Wl il /^El " P
E2\ ,__^— = [(u2
- u2
) --g-J
={ )
(55)
57
It also proves useful to express this loss in terms of a dimensionless
coefficient defined as
fef^)2 2
(56)
Recalling that
H = H - uT
(1+2)
and using the basic energy equation, eq. h8, one obtains the energy
rise across the rotor in the form
W W(H
2- Hl ) = U
2 Vu2 - UxV^ - {sin
!l
iR 4. + Y
R -§-} (57)
This equation can be conveniently non-dimensionalized by
2dividing by U . The velocity expressions developed in Section 1.0 can
not be utilized to simplify eq. 57? and the dimensionless energy rise
across the rotor (II A H) can be written as
(nAU )AH yR ' 2^ 2co Rg
H2
- E±
(sin iR ) , R
n x2
(£)Rg
4-\
§{Tan P
2 ' ^ bf \i. ™ .0 ""A} (58)
- 5
^in2i
feftfaf X2
bR
' 2
R
2cos2p,
-}
58
where the dimensionless head coefficient is defined as
nAH " "Ifa (59)
The loss coefficient Y of eqs. 56 and 58 is a function of theR
deflection A{3, where
and the aspect ration /L., defined by
This will be farther discussed in Section 3.0.
2.2 Performance of Rotor Blading
The rotor is designed for the maximum useful enthalpy rise
across the blades. If this were a frictionless process, the useful
energy rise would naturally be greater than in an actual process
involving friction. Therefore, the rotor efficiency is defined as
H " H.
\ = iT V (6l)B H
2 ^ Hx
where the total heads H , H , and H m, are shown on the H-s diagram,
Figure 8. Of course the pressure rise through the rotor is related
very simply to the corresponding energy rise according to the
relation
H = ^ (62)P
59
The numerator of eq. 6l is obtained from either eqs . 57 or
58. The denominator of eq. 6l is similarly obtained, but for the
theoretical case both the incidence and blading losses are non-existent,
It follows from eq. 58 that
H2Th " *!!(II AF y _ f 2Th 1\
u) R2
= i + nT- p2 - \z ^ ^ Tan «J
(63)
is the theoretical dimensionless energy rise across the rotor. The
efficiency can obviously be written as
\ - wm^Jl i6h >
O.
jj = m A H (65)
It is convenient to introduce a dimensionless power coefficient as
2.pUJMAX
R2
Recalling that
np - -f—
-
5(66)
VM2w R
2(3)
and
" = 2n P \ \ VM2 h2 &
"A H = -1P2 (59)a,R
2
60
the dimensionless power and head coefficients can now be related as
The power is related to the torque by the simple expression
2.3 Energy Losses in the Upper Passage
The energy loss in the upper passage, assumed as only a total
pressure loss due to friction between the rotor and stator, is expressed
as
A Hu \ - H
3"
Vj
f - ¥ (69)
or
,p„ vA /P. v,2
In a frictionless process
p v,2
H2
= (H3
)T H= ^3H -f •
. (71)
and from eq. 17 with the frictional moment M equal to zeror\ III
(V ) = v — = -^ (72)^ Vu3 ;TH V R 2 u^
J' Mu
Resolving V_ /2 into its components, one can write eq. 71 as
\ = (h3
)th- 5S + Zfl +
I^3L (73)^ 3 TH p 2
.2 ^
61
Similarly for an actual process
2 2Po VM_ V _
the energy loss can now be written using the two previous equations as
2P3TH " P
3,
( 1t } \
'2 "3" ^L^2 +(
x.l)^ (?5)
A convenient dimensionless pressure loss coefficient can now
be denoted as
• cn -^f^l . (76)
(-£)
r
where C will be shown to depend primarily on the radius ratio — ofu r.
the upper passage in Figure 2. This is similar to energy loss
coefficients developed for bends in smooth pipes.
Eq. 75 is non-dimensionalized, and using eq. 76 one obtains
2 2
(^K-f ^H-f-- 1)^- (77)
* E2 1 Ma »
R2
Using previously developed expressions for V one obtains an equation
similar in form to eq. 58 or
(RAH) a I 3v yu 2 2
cu R2
*
' ft 3 %¥ •^## - *>
62
2.4 Performance of Stator Blading
From Figure 4 "the stator incidence angle i iss
is
=°3 " °3
B (79)
i
and the useful inlet velocity V, is
V ' = V, cos i (80)3 3s v '
The total head ahead of stator blades is given by
p* v2
PH =-i + -2_ = -l2
(81 )3 p 2 p
v y
while the useful head is
p (v ') 2 p'H3 =f + -I~ = ^i (82 >
The above three equations combine to yield
(v3')
2- (v/
H3
= H3
+2
V 2* 2 2
= H5" 2 ^ sin V
(83)
Similar to rotor blading losses, stator blading losses will be defined
as
T jAlI^ji^i (84)S
| \2
(V4
2/2)
or
v.2
„ V,2 V 2
H, = H' - Y -f- - H, (sin2i ) -3-_ Y -4- (85)
A
4 " iX3 s 2 3 v "V ^ " x
s 2
63
Subtracting EL from both sides yields
V2
V2
H4
" H1
= H3
" H1
" (sin V "J- " Ys "I" (86 )
The same non-dimensionalizing technique used in Sections 2.1, and 2.3
is again employed and the energy loss in the lower passage is now
available in the general form as
,i2 - 2
'4 "1 ~3EL - EL H, - H
1
T]^u
R2
. y - 1" lo ;(sini
s)co R
2ou R
2
+ $ [(sin2is) 2 Tan PjJ + *
2 {(sinV) [(^ ^ 5|)2
+ Tan2
PJ
Y R b KL O (87)
«, 2 L.R, b K_. cos a, J J )V 4 s ^ 4 .J
The above equation is actually
H4
- H1
H;- H
1H3
- H4
2D 2 2D 2 2^ 2co R co R co R
?•
This can be further reduced to give
H4
- H1
H2- H1
H2" H3
H3" H4
2^2 2 D 2 2^2 2'2co R
2co R
2co R
2co R
2
which gives the energy losses in the lower passage in terms of the
energy changes throughout the remainder of the system. The next
step is to find the energy loss in the lower passage and solve eq.
44.
(88)
64
Again the loss coefficient Y is a function of deflection ands
the aspect ration, where
A3s
= a3
" °4 (Q^
and
3 4
2.5 Losses in Lower Passage
The development of the energy loss in the lower passage
closely follows the technique of Section 2.3.
The energy loss in the lower passage is
2 2
for a frictionless process
R.
(V ) = y _1 (q 2 )1 u1% u4 R1
V7 '
or relating theoretical to actual velocities yields
2
H4
= (Hl)Th = —7^ +— + 2liT-; (94)
while
h. - Zi + !m! + Ld! (95)1
p 2+
2
65
the pressure loss coefficient is
p= c
i ^r^ (96 )
The expressions for H. and H . are now substituted into eq. 1
,
along with eq. 6 to yield
V2
R 2
This is easily put into the dimensionless form used previously or
<"*>* - t ^)2
[°i(^)
2
+(1 - V 2
)(!r zg T- a4 )
2
] <98 >
Eq, 97 can now be substituted into eq, 88 and § can be solved
for straightforwardly. Iteration is necessary, however, since all loss
coefficients depend on $,
3,0 Loss Coefficient in Rotor
Vavra (3) gives a relation for the losses in axial turbine
bladings as
, . o.99 - ^f A P - 4f^&(99)
where AB is the flow deflection angle in degrees. The quantity Y is
the velocity coefficient defined as the ratio of the actual and
theoretical discharge velocities, which for the rotor is
W
* w2Th
and
ABR
. 32
- P1
(101)
66
The theoretical case occurs for Y_ equal zero which inK
accordance to eq. 56 gives P = P . Substituting eq. 100 into
eq» 53 yields
E1
P
E2Th
P+
2Y2
(102)
The actual rotor blading losses using eq, 102 can now be written as
W, W,
YR
=
W//2
- 1
Y'
(103)
In accordance with Horlock (Ref. 4) the loss coefficients depend
on Reynolds numbers R^ and aspect ratio A of eq. 60, Hence let YR
of eq, 103 now denoted as YR be the loss coefficient across the rotor
5blading for A_ = 3 and R_._, = 10". The latter term is deiined asu R iMR
^R =DhR
V2
(104)
where D, is the hydraulic diameter of the discharge area of the flowhR
channel between adjacent blades. IfyC/a is the rotor blade snacings atn.
the radius R_, and a„ is the opening at the discharge as shown in
Figure 3» oxi approximation of the discharge opening is given by
a^-^cos 32
(105)
then
D.4 (flow area) ^R
hR wetted perimeter "'2 (a_ + b
)
(106)
67
Substituting for a_, and rearranging yields
2bT
D.R
hR1 +
R(107)
^ cos 32
The blade spacing AL can be found with respect to the radial
extension Rp
- R. by using the blade loading criterion of Zweifel (Ref. 5)»
For axial bladings this approximation is
MR 0-425R2 ~ R
1 (Tan 32
- Tan 3^) cos2p2
Substituting this equation into the expression for the hydraulic
diameter yields
4)
(108)
DhRR2
:
(bR/R
2)(Tan3
2- TanB^cos^ (109)
1 +
0,425(1-^/1^)
Placing the above equation into the expression for the Reynolds number
and recalling that
(4)W2
ojR $
cos B
one obtains
hst
2(^) . . R2
2
(bR/R
2)(Tan3
2- TanB^) cos P
2_,
(110)
cos32+
0.425 (1 - r./r2
68
The loss coefficient is dependent on the aspect ratio and
Reynolds number through the following relations after Horlock (Ref, 4).
< - 4? <.»75 *2E) - 1 (112)
and
where the coefficient YR
holds for the actual aspect ration only at a
Reynolds number equal to 10 , By combining the above equations the
generalized loss coefficient is simply
YR -
ly2 I'
975 TAR
;1J VR^y v-h)
These relations allow the actual loss coefficient of the energy absorber
to be determined under varied conditions.
Obviously the Reynolds number is not totally specified by
geometric parameters, but depends on the flow coefficient §. In
t
iterating for §, the process must be assumed to begin at Y„ , or aR
5Reynolds number of 10 , and subsequently correct the loss coefficient
as improved values of § and K are calculated. This procedure must
also be employed for the velocity coefficient Y which is dependent on
§ through a dependence on the actual inlet angle (3... The iteration
must be continued until convergence is achieved for all values
dependent on $.
69
3»1 Loss Coefficient in Stator
This section follows closely the development in Section 3»0,
Eq. 99 is again used with
^=7^ (115)s v
4Th
and
A3s
= a3
~ aA
(116)
For a frictionless process in the stator, the total pressure
(P . ) at stator exit would equal total useful inlet pressure P ..,
or
^PT4^ThFT
;*4 'V Th ,.. n ,
p=
p=
p+
2 tmj
Substituting eq, 115 into eq, 117 yields
P p Y 2s
For an actual flow
n4
=^3 + !i! (119 )
which yields through eq. 84
s
This was as expected from Section 5*0* The loss coefficient can now be
written after eqs. 112, 115, and 11 4 as
70
and finally
^(•^ + 1fMGg)'25
.
(i 23 )
where the aspect ratio is defined by eq. 90 as
b
A =S
s IL. - R.3 4
The Reynolds number for the stator is
K v.
where for a —^ coscv,, the hydraulic diameter iss s 4
T\
2bs
Dhs
1s' s
+cos a.
4
The blade spacing according to Zweifel (Ref. 5) is
-^s 0,425
(125)
3"
4 (| Tan a - Tan a |)cos ot ^ '
and from eqs. 25 and 26
The Reynolds number can now be expressed using the above four equations
as
71
b\\ 5329 tu R,
^s =
2 Kcos a + — (| Tan a - Tan a/ | ) cos a
0.425 (r3/r
2) (1-
3 -J
(128)
A series of iterations is again needed to insure proper
convergence of all parameters.
4.0 Pressure Loss Coefficients in Upper and Lower Passages
In accordance with eq. 76 the pressure losses in the upper
passage are expressed "by
(P5)Th - P3
n %= Cu 2
and similarly for the lower passage by eq. 96 as
(129)
(piV- pi
V.
= cMl
I 2(130)
With Apb being the pressure drop in the bend, the loss
coefficients of these bends defined as
V£"2 (131)
are shown in Figure 15 as functions of the Reynolds number R^, where
similar to eqs. 104 and 124
V. D,
\ =~ n(132)
Vavra (Ref. 2) has developed the set of curves shown in Figure 15.
These curves are based on the data of Sprenger (Ref. 6). Interpolation
72
was to obtain values for ratios of r./b different than those found
in Sprenger,
The hydraulic diameter D is again
\ = 4flow area
(135)wetted perimeter
For a ring channel with flow area of 2tt R b the wetted perimeter is
2(2ttR), or
\4(2tt R)b », . N=2(2n R)
= 2b( 1 34)
Hence at the rotor inlet there is
\ = 2b* (135)h R
and at the stator inlet
Uh= 2b
s(136)
These above two values are introduced into eq_. 132, with V
equal to the corresponding value of V , that is, VM1 before the rotor
and V before the stator. The result is
VM1
2\
The Reynolds number is then used to obtain the loss coefficients § of
Figure 15*
The loss coefficients ^ of Figure 15 have been obtained from the
measured pressure drops by subratcting the pressure drops tha"c would
73
occur in a straight duct at the corresponding Reynolds number and
surface roughness.
There are several conclusions which can be drawn from Figure 15
and from additional data in Sprenger (Ref, 6), In general, the loss
coefficients for bends made up of two concentric radii increase with
decreasing i/b ratios. The curves £ versus R^ that exhibit a
minimum indicate the flow separations occur at the inner radius, and
should be avoided. Finally, Sprenger (Ref. 6) shows that two bends
with ri/b =0,5 arranged in series to form a 180° bend have a loss
coefficient of 1.665; at R^ = 10 where § is the loss coefficient
of one of the 90 bends. This data will be a first approximation in
choosing the loss coefficient for the 180 bends in the energy
absorber.
^ 1 Estimate of the Pressure Loss Coefficients in the Ub^er
and Lower Passage
The losses in the upper and lower passages will be assumed
due to two components, passage bending and friction. The losses due
to duct bending were discussed in the previous section. An arbitrary
assumption is to take these loss coefficients equal to 2,0 and 2.5
for the upper and lower passage respectively. The values of £ are
given in Figure 15 for various value of i/b. It is recommended that
ri/b be chosen between 0,5 and 1,0, Any bend with a value of ri/b
larger than 1,0 will be assumed to have the same losses as a bend with
i/b equal 1,0.
To the above losses must be added the frictional losses in a
straight duct of hydraulic diameter D, and average length L, where
L. + L
L--^-^ (139)
74
CM
CD
-P
Q)
•HO•HfH<HCD
oora
CQ
OH^
<HO
CD
-P
B•H-PW
LO!
(7>
6CO
6 25 JLN3l3!dd300 SSO'l 9NI0N38oo
75
and L. is the distance along the inner contour and L that along the1 o
outer contour in the meridional plane from the trailing edges of one
blade row to the leading edges of the other row. (See Figure 2)
In accordance with eqs . 132 and 136 this general frictional
pressure loss is
Apf
' = Xf±
f VM2
(1U0)
The friction factor X_ is a function of both Reynolds number
and surface roughness . The value of X is obtained from the curve of
Figure l6, also from Vavra (Ref. 2), for a relative surface roughness
of
£ - -\ UM)h 10
where K is the peak-to-valley roughness of the surface. Hence, for
this case
K = -^ = 200 b u - in. (lU2)r
10
If it is assumed that the rms roughness be one-half K , the surfacer
finish must be better than 100 b micro-inches
.
The applicability of this' data to the three dimensional flow
channels of the energy absorbed is obviously questionable. The losses
defined below must , at best, be considered as first estimates only.
The loss coefficient C of eq . 12Q will be taken as
Lcu
= 2>0*
+2TT V (lI+3)
R
76
1
/
/
\
I
.
1
J
/
/
/
Q/ «M
/ -
5 ob
r j
lOVd NOI lOIdd
CO
CO
to
ro
zcj a:
orLUCO
7L
CO
o _j~" o
>-CO LU
ro
C\J
%
«H
P=i
oJ
\>
r<
UO
o
Pio•H-P
•H
Fr
Hn3
o
0)
-P
6•rH
-Pra
^o
•H
OJ
77
and C. of eq. 130 is
s
where 5 is obtained from Figure 15 and \ from Figure l6 for
-2b
STM3 U „ *2 *B2
bR\
2bs
^u= —— = r E2i^ 5^ y— (l!* 5)
and
In eq. U the length L is given by
(L ). + (L )
T u 1 uoLu - 2
where (L ). and (L ) are the lengths of the inner and euteru 1 up
meridipnal contours, respectively, pf the walls pf the upper passage
between stat iens 2 and 3« Similarly in eq. 133
where (L„). and (L„) are the cpunterparts pf (L. ) and (L ) fpr thev
Si 1 i o 1 u x u o
lower passage between stations k and 1.
5.0 losses of Angular Momentum in Upper and Lpwer Passages
The frictional moments along the upper and lower passages
were introduced by eqs. 18 and 22 respectively as
M» = \ m R_ V (1U8)fu u 2 u2
and
M« h i \ \k (lt9)
78
No data is presently available for the losses of angular
momentum in passages of the type needed between the rotor and stator
of the present absorber. A very crude approximation of these moments
is obtained, however, by assuming that the shear stresses Y on the
walls are merely those acting on a flat plate with a peripheral velocity
V at R being the velocity outside the boundary layer. This over-
simplification ignores the effect of the meridional velocity components
which may cause large increases in the frictional moments.
For fully turbulent boundary layers along rough walls the
shear stress is
T - «, | \2
(150)
where c„ is assumed to be invarient with the Reynolds number of the
flow. On the element of wall area 2n Rd L of Figure 2 there acts a
frictional moment
dM„ = 2rr RdL Y Rf o
which becomes when using eq. 3
dMf
= tt p cf
(R Vu )
2d L (151)
It is assumed that the passages have been designed such that the flow
is not separated from the walls, and all stresses external to the
boundary layer are an order of magnitude smaller than stresses along
the walls, and therefore negligible. This results in the product (RV )
being constant outside of the boundary layers and equal to R V ~, the
value at the rotor discharge, for the upper passage. Integrating
eq. 151 from station 2 to station 3 yields
79
,(3) 2
MfM " I
{2)" P C
f(R
2V ° («2)
Mfu " " P c
f(R
2 V' [CVi + <Vo] ^53)
where the inner wall is of length (L ) . and the outer wall is ofu 1
length (L ) . Hence by eq. lU8u o
tt p c. R V f(L ). + (L ) 1= f 2 u2 ,. u i u_oJ
(15 ,
Substitution of the mass flow equation and velocity relations into
eq. 15*+ gives
1 + $ Tan p f(L ) + (L ) 1X = c £ ±=-^ ^ (155)
* 2 bR h?Id
from eq. 19
to
1 + $ Tan R f(L ). + (L ) 1•n = 1 - e - L u 1 u oJ (1S6)
For the lower passage a similar procedure and derivation leads
c Tan as |~(L ) + (L ) ]
and
Tan ™ F(Lj. + (L.) 1±m _* k_±_J^ ^oj
(158)
2 SU bs
cf
80
In eqs. 155 through 158, the absolute value of the flow angles
must be introduced since the loss is a function of absolute deflection.
For fully turbulent boundary layers, as in the present case,
the local skin friction coefficients c depend on the surface roughness
ration K /x, where K is the peak to average value roughness and x iss s
length along the plate.
hCx.
=- .OOU for K /x = 10"f s'
and
c^ ~ .006 for K /x = 10~ 3f s'
Since there will be flow irregularities and possible flow -separations
,
it is advisable to use a larger value for c . A c = .016 was chosen
since it gives best agreement with M-21 test data.
81
APPENDIX B
DETAILED ANALYSIS OF THE ENERGY ABSORBER WITH
THE FLUID CIRCULATING IN THE REVERSE DIRECTION, MODES 3 AND k
A separate set of equations must be derived for the reverse flow
modes. The reason, mentioned earlier, is that the relative flow exit
angles which are assumed constant are now different.
This development follows very closely to the previous analysis.
Therefore, the intermediate steps of this derivation are omitted.
1. Assumptions
a. P-i~ constant
a_ = constant
The remaining assumptions are identical to the normal flow case.
2. Analysis
The geometric relations employed are obtainable from Figures 6
t
and 7. A new flow coefficient $ is conveniently defined as
VM1
feffl^1 -M2'
The torque exerted on the fluid by the rotor when acting in the
direction of rotation is
a)
TR " m
(Rlvul " R
2V ^
where the mass flow equations are identical to those for the normal
case.
82
The frictional moments acting from stations 1 to k and stations
3 to 2 are respectively
Mfi = h m \ \l
M . = X m R_ V _fu u 3 u3
where
\ \km m E
lVul - M«
2 u2 3 u3 fu
The relative flow inlet angles are
bR *B2 m /*2\ *B2
and
ly (1 + i' Tan p1 )
*!
where
'2 rm Q **
83
(3)
(*0
Tan^2
= V bj K^ ™ ^3 " V 4 l/$'
(5)
T£m*u
=,
i^, e: (6)
^ bs
The torque is now reduced to
TE " 2 " P R
l
5
(^) Si »2
TR <7)
Tr - * + *2
[Tan Pl - jg JJ ^ Tan oj (8)
Proceeding to the rotor, one can write,
V =¥2
C°S*Z
h = P2B " P2
YR
P - PE2 El
£ W 2
2 1
and finally
^R Hl- H2 .
Sl" ^E /"g
2p 2 .A 2 " " 2 Wcu R., 1
K b
+ *'{Tan pi " Grr ^ V Tan *
3 )cos " ^k^ b
g'tau ^ "3/ ^ ^J (9)
•2 r
sin % ^lyr/W /Si bR - T \
21 _2r i- $ \—2— (id lvk—;
+ fc b- v Tan<v J
+ :—2~i^V ^3 2cos P1
where
^ " R2
- \
for this case
n A HA H2„ 2
oj R
In the lower pas;sage
.
pUTh " PU
1-£ v
2
2 MU
(10)
(11)
84
and the energy drop is
Hi
- H4
(1 - td /R
i
2D 2 2 VR.u> R 4
1
4
2 VKB4
bs V 2
Nor for the stator
i
V. = V. cos i4 4s
i =0/ — o/
s 4B 4
v =P - PT4 T3
Is p
2 VAs
=bs
R?" R
4
Ap Of. - °^s 3 4
and the energy drop across the stator is
H4
" H3 sin2 i ? /
Rr 2i" r 9 ? /
R1\ 2
co2r
2 - ~ir^ ^ m 0;) + * {t«*\n m Gr) *» *J1 4 4
2,_ sin i ,R4 ^2 K_ . b 2
*' 2_ A \ Sin
2
+_ 2
Ys VR 2 b LJ
2cos a 3 s 1332
85
In the upper passage
P2Th " P2
Cu
="~
V
( 1 4)M2
P "5-
and
H -H |2 R 2 r 2 b If 2
J^tt V (57) (cu Gg) - - ^Xr § Tan
"J } (15)
The loss coefficients are now given as
Lu
s
Cu
= 2 '° § -+ W" Xf ( 16 )
LXu =
0fTan or r-|- (17)
s 134
X„ = c„ Tan
The Reynolds numbers ot obtain § and \ graphically are
2\ ..
The expressions for Y, YR , and Y remain the same as in the previous
analysis, but the Reynolds numbers are now
86
b , § co RR \ 1
<B
^ =v
cosp.. +(bR/R
1Tan ^ - Tan 8
2BJcos ^
R2
Ir;- 1
^ ^r
*ns=
*B3
R.Tan a - Tan or_
4-B 2.
2COS O'
coso/., +,R .
. /R-7
°- 425(s7)(^- 1 -
(19)
The equation which is finally solved to obtain $ is
A IL, = A H +AH + AH,\R u s &
or
H1
- H2
- (H4
- H3) - (H
3- H
2) - (H
1- H
4) == (20)
The equation is solved by the program. An iteration technique must be
t
carried out until all parameters dependent on $ converge.
87
APPENDIX C
PROGRAM NORMAL
Program NORMAL is essentially a modification of the program
developed by NAEC.
Program NORMAL solves the equations governing normal circulation,
Modes 1 and 2, for a given design. The solution is based on an
iteration technique which is repeated until the flow coefficient
$ converges to within a specific accuracy, A starting value of §
is picked as are initial values of the various Reynolds numbers which
are also § dependent. The initial iteration step occurs with all
losses based on standard initial Reynolds numbers. This subsequent
value of * is then used to improve the estimates of the Reynolds
numbers and consequential losses. The technique is carried out
until $ converges to the proper value.
If the design parameters and angular speed are input data, the
program will compute the flow coefficient, torque, velocities, losses,
energy changes over each component, and the dimensionless torque,
power, speed, and energy. The flow chart of Table IV depicts this
process.
The method of data input is clearly indicated in the program
itself. The program is felt to be self-explanatory in that comment
cards have been extensively used. The Fortran symbols employed are
given in the list of symbols.
Subroutines have been used liberally. Each subroutine accomplishes
one individual task such as determining a loss coefficient, or
88
enthalpy change across the stator, etc. The subroutines Pig. 15 and
Fig. 16 are closest point approximations of the curves depicted in
Figures 15 and 16 of the analysis. Thses points are given by Block
data which is called by each subroutine.
89
Table IV FLOW CHART FOR COMPUTER PROGRAMS FORMAL AW RFVFRSE
>New a ndRev-iolds \Josnow nsed
.
Dat ^ T lpt t : Ocs i on si;^e
Ass i gn In i t i a ^ vn 1 JfS ofand Re vnolds numbers
to be?. i n i ten tion
N =
_JLPlMUct <*:
c* I
v = o
^v--
1
PftSZ rirST
r1 1-.? \nalysis: Calculates and Prints
Veloc i ti es
Enpr^v DifferencesLosses 3nd Loss Co~ r fieient c
Non-dimensionali zes nil pertinent quantitiesPressure DifferencesTorqu.e
Flow Coef f ic ; ent
90
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APPENDIX D
PROGRAM REVERSE
Program REVERSE solves the equations for reverse circulation,
Modes 3 and 4» The program is very similar to program NORMAL, and
the flow chart of Table IV is the same. Further discussion would be
mere repetition of Appendix C.
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REFERENCES
1. Naval Air Engineering Center, Engineering Department (Si) ReportMisc-08998, Theoretical Performance of Programmed Drum ArrestingGear , by V/illiam Clark, 1 July 1970.
'
2. Vavra, M. H. , Energy Absorber , April 1970.
3. Vavra, M. H. , Aero-Thermodynamics and Flow in Turbomachines. New.York: John Wiley and Sons, 1960.
4. Horlock, J.H., Journal of the Institute of Mechanical Science ,
Vol. 2, p. 48-75, 1960.
5. Zweifel, 0., "The Spacing of Turbomachine Bladings", Brown BoveriCompany Review , December, 1945*
6. Sprenger, H. , Schweiz Bauzeitung , 87th year, no. 15, p. 223-251, 1969-
7. Biolley, A., Schweiz Bauzeitung, Vol, 118, no. 8, p. 85-86, 1941.
129
INITIAL DISTRIBUTION LIST
No. Copies
1. Defense Documentation Center 2
Cameron StationAlexandria, Virginia 22314
2. Library, Code 0212 2
Naval Postgraduate SchoolMonterey, California 93940
3. Professor T. H. Gawain, Code 57Gn 2
Department of AeronauticsNaval Postgraduate SchoolMonterey, California 93940
4. Chairman, Code 57 1
Department of AeronauticsNaval Postgraduate SchoolMonterey, California 93940
•5, Assoc, Professor R. D. Zucker, Code 57Zu 1
Department of AeronauticsNaval Postgraduate SchoolMonterey, California 93940
6. Professor M. H. Vavra, Code 57Va 5
Department of AeronauticsNaval Postgraduate SchoolMonterey, California 93940
7. ENS Leo S. Rolek, Jr. USN 1
9808 Avenue J
Chicago, Illinois 60617
130
Security Classification
DOCUMENT CONTROL DATA -R&D'Security classification of title, body of abstract and indexing annotation mu\l be entered when the overall report is classified*
iriGinatinG activity (Corporate author)
Naval Postgraduate SchoolMonterey, California 939^0
Za. REPORT SECURITY CLASSIFICATION
Unclassified2b. GROUP
1EPOR T TITLE
Performance Analysis of a Turbo-Type Energy Absorber for an Aircraft CarrierArresting Gear
DESCRIPTIVE NOTES (Type of report and, inclusive dates)
Master's Thesis; June 1971IUTHORI5I (firsl n«m«, middle initial, last name)
Leo S. Rolek, Jr.
IEPOR T DATE
June 1971
7a. TOTAL NO. OF PAGES
132
7b. NO. OF REFS
CONTRACT OR GRANT NO.
PROJEC T NO
S«. ORIGINATOR'S REPORT NUMBERISi
9b. OTHER REPORT NOISi (Any other numbers thai may be assignedthis report)
DISTRIBUTION STATEMENT
Approved for public release; distribution unlimited
SUfl-LtMtNlAk* NOlLS 12. SPONSORING milIT Ah'i ACTi
Naval Postgraduate SchoolMonterey, California 939^0
ABSTRACT
The increasing weight and speed of carrier based aircraft are taxing the
limit of conventional piston type energy absorbers which are used for arresting
gear. Turbo-type absorbers have been proposed as an alternative. This study
investigates a turbo-type energy absorber for an aircraft arresting gear under
development by the Naval Air Engineering Center. A mathematical analysis,
computer simulation and performance prediction are given for each mode of
operation. It was initially expected that the absorber' could operate in just
two modes, forward and reverse, but the analysis shows that four distinct
modes are possible, since not only shaft -rotation but the path of fluid flow
can be reversed. Theoretical performance predictions are also compared with
test data from an existing smaller scale version of the absorber. The
agreement is excellent. An approximation of scaling effects on power
absorption is also presented. It is concluded that the turbo-type absorber
is basically adequate to meet the demands of carrier operation for the
forseeable future.
DFORMI NOV 65
I 0101 -807-681 1
1473 (PAGE 1)
151Security Classifies
Security Classification
KEY WORDS
urbo-type energy absorber
rresting gear
ROLE W T
F0RM 1^.73 (BACK)I NOV «t I S1 / <J
101 -507-69 ; I 132 Security Classification
128616
6.1 f a turbo--tVP
absorber for an , ng
craft carrier a
gear.
J 1
Thesis 128616R68945 Rolekc.l Performance analysis
of a turbo- type energy
absorber for an air-
craft carrier arrestinggear.
thesR68945
3 2768 001 98109 5DUDLEY KNOX LIBRARY5