Investigation of the influence of supersonic rotor Byington, Melville R.

Investigation of the influence of supersonic rotor Byington, Melville R.
Calhoun: The NPS Institutional Archive
Theses and Dissertations
Thesis Collection
1963
Investigation of the influence of supersonic rotor
velocities on turbine performance and design.
Byington, Melville R.
University of Michigan
http://hdl.handle.net/10945/12435
NPS ARCHIVE
1963
BYINGTON, M.
library
U.
S.
Naval Po*»^tick»ate
Monfcerey, Co#t'>'
S<
INVESTIGATION OF THE INFLUENCE OF SUPERSONIC ROTOR
VELOCITIES ON TURBINE PERFORMANCE AND DESIGN
Lt. Melville R.
Byington, Jr.
U.S. Navy
Conducted in partial fulfillment of requirements
for the Degree of
AERONAUTICAL AND ASTRONAUTICAL ENGINEER
at the University of Michigan
August, 1963
ihy
Library
U.S. NavaJ Po^aauate
icftooi
Monterey, Catttomia
ACKNOWLEDGMENTS
The author is indebted to Professor R.
University of Michigan and to Professor M.
H.
B.
Keller of the
Vavra of the United
States Naval Postgraduate School for their patient and constructive
guidance.
Partial cost of this research was borne by the United
States Naval Postgraduate School.
11
.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS
ii
LIST OF TABLES
v
LIST OF FIGURES
vi
vii
ABSTRACT
I
.
II.
INTRODUCTION
1
ANALYSIS OF LOSSES AND FLOW PATTERNS IN SUPERSONIC
TURBINE STAGES
2
1.
2.
3-
k.
5.
6.
III.
A RATIONAL TECHNIQUE FOR DESIGN POINT SELECTION FOR
SINGLE-STAGE IMPULSE TURBINES
1
2.
3.
h.
5.
IV.
General Discussion
Development of a Method of Loss Prediction
in Supersonic Turbine Blading
Comparison of Loss Prediction Theory with
Available Test Results
Comparison of Typical Rotor Velocity
Coefficient Estimates Obtained by Different
Methods
Detailed Analysis of the Effect of Rotor
Profile Geometry on Losses
Analysis of the Effects of Rotor Incidence
On the Supersonic Flow Through Turbine
Blading
General Discussion
Static Efficiency as a Function of Velocity
Coefficients and Rotor Deflection
Limiting Stress Requirements
Combination of Performance and Stress Requirements
Procedure for Application
ILLUSTRATIVE DESIGN PROBLEM
1.
2.
3-
k.
5.
6.
Specifications and General Design Philosophy...
Design Point Selection
Detailed Design Point Calculations
Nozzle Flow Analysis and Profile Layout
Rotor Flow Analysis and Profile Layout
Discussion of Resulting Design
111
2
5
10
11
12
l8
25
26
27
29
30
33
35
~$h
k2
^5
k-9
51
TABLE OF CONTENTS CONT'D
Page
V.
VI
.
CONCLUSIONS AND DISCUSSION
53
REFERENCES
%
APPENDICES
A.
NOMENCLATURE
B.
DEVELOPMENT OF EQUATION
57
58
II-3-1
IV
62
LIST OF TABLES
Table
I-a
Page
Pressure Coefficient as Function of
Y, M.
I-b
I-c
I-d
II
7 = 1.1+0
63
Y, M.
Velocity Coefficient as Function of
7 = 1.40
6k
Y, M.
Pressure Coefficient as Function of
7 = 1.25
65
Y, M.
Velocity Coefficient as Function of
7 = 1.25
66
Comparison Between Estimated and Experimentally
Determined Loss Coefficients of Transonic and
Supersonic Turbine Blading Rows
67
LIST OF FIGURES
Figure
Page
Comparison of Theoretical Efficiency Esimates
for (Subsonic) Single Stage Impulse Turbines
With Results of Supersonic Turbine Tests
68
Profile Loss Coefficients for Conventional Section
Blades at Zero Incidence
69
3
Secondary Loss Acceleration Parameter
70
k
Pressure Coefficient as Function of M,
Y.
y - 1.U0.
71
5
Velocity Coefficient as Function of M,
Y.
7 = 1.40.
72
6
Estimate of Rotor Velocity Coefficient
Impulse Blade in Supersonic Flow
-
1
2
Typical
73
7
Cascade Loss Results of Reference 13
7^
8
Comparison of Rotor Incidence from Experiment
and Two Theories - For Turbine of Reference 2
75
Velocity Triangle Nomenclature
76
9
and F" Functions for Turbine Design Problem
10
F'
11
Constant Stress Efficiency Curves for Turbine
Design Problem
Mean Radius Conditions...
77
78
12
Design Velocity Triangle
13
Sharp-Corner Nozzle Geometry Considerations
80
l4a
Design Nozzle Profile
8l
lUb
Sharp-Corner Nozzle Cascade
8l
15
Design Rotor Profile
82
-
VI
79
-
ABSTRACT
An analysis has been conducted of the capabilities of high
single-stage impulse turbines.
pressure ratio,,
Some recent experi-
mental work has shown considerable promise for these machines.
This
work provides a tentative basis for better understanding the potentialities and peculiarities of such high performance turbines.
The present analysis indicates that well designed supersonic
rotor impulse turbines can be expected to produce static efficiencies
closely approximating those predicted by well-known (subsonic) curves
4..p.p.
-p
of
static
efficiency
J versus
•
4.
blade -speed
E
T
lsentropic
jet speed
-:
.
.
.
.
.
.
,
Apparently, there presently exists no generally applicable
method of blade row loss estimation in the case of supersonic flow.
An extension to the Ainley-Mathieson loss prediction procedures to
the supersonic case is proposed.
Experimental and estimated supersonic -rotor loss coefficients
of the present report differ little in magnitude from well-established
results for subsonic blades of similar deflections.
The effect of rotor incidence, in supersonic flow, has a
profound influence on pressure and velocity distributions throughout
the stage.
theory.
The effect is inadequately explained by a recently proposed
The present analysis proposes an alternative method for esti-
mating the effect.
A logical and straightforward procedure is developed for
selection of the "optimum" design point for a single-stage impulse
VII
turbine, given a required RPM, power output, and rotor blade root
stress.
The preliminary design of a large rocket-turbine is con-
ducted as an exercise in application of the foregoing developments.
This study was conducted as partial fulfillment of the
requirements for the Degree of Aeronautical and Astronautical Engineer
at the University of Michigan.
vm
)
I.
INTRODUCTION
In recent years., considerable interest has centered around
the development of high pressure ratio turbine stages for rocket-
Desirable character-
pump drive and other specialized applications.
istics are reliability and light weight, with high efficiency as a
desirable objective of lesser importance.
One type of machine attractive
for these applications is the single-stage impulse turbine.
This turbine
combines low axial thrust on the rotor with potential for high pressure
ratios and specific work output.
The high pressure ratios normally
result in supersonic flow through the rotor, and very few experimental
results concerning these stages are available.
The present analysis is undertaken to explore the potential-
ities of such stages with the benefit of some recent test results.
study is directed along three parallel, but distinct topics.
The
These are:
(1
analysis of flow patterns and loss distribution within the stage,
(2)
development of a scheme for selecting the design point to best
achieve specified design goals, and (3) a typical design problem
intended to illustrate the results of topics
-1-
(l
)
and (2).
II.
1.
ANALYSIS OF LOSSES AND FLOW PATTERNS IN SUPERSONIC TURBINE STAGES
General Discussion
While a wide body of theoretical and experimental data are
available (e.g. Reference l) for predicting flow patterns and blade
losses at low Mach numbers, little information is available at supersonic velocities.
As a consequence, turbines designed to operate in
these regimes frequently fail to achieve the expected performance and
pressure distributions.
Two examples are the turbines of Reference 2
and J wherein the flow patterns, pressure distributions, and efficiencies
differed considerably from estimated design values.
In both cases, the
results would indicate efficiency was lower than necessary, had the
losses been accurately predicted in advance and compensated for in
the design geometry.
The method of Reference k was utilized by the N.A.S.A. to
successfully correlate rotor blade momentum thickness and "specific
blade loss" with "blade surface diffusion parameter" for one family
of several transonic-rotor turbines.
This method does not include
the effect of total rotor deflection angle, and has not been satis-
factorily generalized.
The method predicts a grossly optimistic rotor
velocity coefficient for the turbine of Reference
tion angle is 1^0
2.
The rotor deflec-
in the latter rotor as compared to only 8j
in the
N.A.S.A. family, explaining the deviation.
While the loss prediction methods of Reference 1 are comprehensive, they are only recommended by the authors in cases wherein
-2-
3-
the flow is everywhere suhsonic.
Hence, an extension in generality
to include supersonic rotor flow is essential to provide for accurate
flow channel layout and a more precise estimate of performance.
Figure 1 presents the performance of the two supersonic
rotor turbines of Eeference 2 and
transonic family of References
5
J,
together with that of the N.A.S.A.
through 10.
For comparison, there
are the theoretical curves from References 11 and 12 and an unpublished
The theoretical curves presumably
curve from the Aerojet-General Corp.
are intended to apply to subsonic stages only as no mention is made of
Mach number levels.
Slight deviations between the curves are explained
by minor differences in the geometry and loss assumptions used in
deriving them.
They may be assumed to fairly approximate the range
of efficiencies to be expected for single stage impulse turbines
having negligible Mach number effects.
Comparison of these curves with the experimental results of
the high performance turbines shows agreement which is enlightening,
if not surprising.
It seems that no appreciable deterioration in
performance may be attributed to Mach effects.
The mediocre effic-
iencies of the turbines of References 2 and J are evidently due to very
high leaving losses, while the N.A.S.A. family has low leaving losses.
The turbine of Reference 2 shows performance that is quite
encouraging when compared to theoretical values
.
It is no surprise
that this is the most recent and probably the most carefully designed
of the three.
This turbine is designed for high (l40°) rotor deflec-
tion and maximum static efficiency.
On the other hand, the N.A.S.A.
h-
family of turbojet research turbines are designed for high total
efficiencies, which average about 85$The relatively inferior performance of the turbine of
Reference 3 appears to result from causes other than the high Mach
number level, although insufficient data are reported to provide for
a precise analysis of losses.
The performance evidently is degraded
by extremely severe flow separation from the rotor suction surface.
This explanation is substantiated by the almost negligible change;
in flow characteristics which resulted from a major modification of
the rotor profile.
The modification consisted of reducing the blade
thickness about
by removing metal from the suction surface.
hO^o
The
negligible resulting flow change indicates the flow was separated in
both cases, and demonstrates the ineffectiveness of profile modifications downstream of the separation point.
Since the profile
violates the criteria of Reference 13, as discussed in
able separation is to be expected.
consider-
In addition to the separation
problem, the design apparently suffers
large tip clearance and "lap".
II-5.,
somewhat from relatively
"Lap" refers to a sudden annulus
enlargement immediately downstream of the nozzle exit.
Returning to considerations of the turbine of Reference
Figure 1 also illustrates the margin of improvement
percentage points in static efficiency
-
-
2,
about two
obtained by increasing
rotor flare enough to compensate for actual rotor losses.
This
flare increase allowed design impulse conditions to be closely
approached, and illustrates the effect of accurate loss estimates
upon geometry and performance.
Comparison of theoretical performance
-5-
with the best performance of this turbine strongly indicates that
any losses associated with supersonic effects must indeed be
This deduction diametrically conflicts with standard
negligible.
works (e.g. Section
I+.5
of Reference l)
which indicate a rapid
increase in losses as the rotor flow begins to exceed local sonic
velocity.
Apparently a phenomenon exists which is similar to the
"sound barrier" drag increase of aircraft wings.
Evidently, when
the flow just exceeds local sonic velocity, strong and near-normal
shocks exist which cause separation.
Once the flow is fully super-
sonic, the flow field appears to stabilize to the extent that separa-
tion losses need be no more severe than in subsonic rotors of similar
geometry.
The design criteria for minimizing rotor flow separation
are considered in II-5-
For the present, it seems reasonable to assume
that the combined profile, secondary, and tip losses of the supersonic
rotor far outweigh any losses credited to supersonic effects.
2.
Development of a Method for Loss Prediction in Supersonic Turbine
Blading
In view of the foregoing examination of Mach number effects,
the loss prediction methods of Reference 1 bear re -examination.
The
basic Ainley-Mathieson loss parameter is
Y =
(Total Inlet Pressure) - (Total Exit Pressure)
(Total Exit Pressure) - (Static Exit Pressure)
The total pressures are referred to relative and absolute velocities in
the rotor and stator cases respectively.
total loss coefficient for a blade row,
Following Reference 1, the
Y^,,
is the sum of profile losses,
Y-nt
secondary losses, Y^-,, and tip losses, Yv
.
Profile losses for
stator or rotor blading of arbitrary reaction are empirical functions
of solidity, deflection angle, and thickness ratio.
Profile losses
are easily determined for blades of arbitrary reaction using the
methods of Reference 1 with Figures 2a and 2b, which are reproductions
of Figures
ks.
In the present case attention is
and 4b of Reference 1.
directed only to the special cases of nozzles and impulse rotor blades.
In this case the value Yp is determined from Figures 2a or 2b, as
T
appropriate, for standard thickness — = .20.
The profile loss estimate
for the nozzle or impulse rotor blade in question is then simply
The secondary loss equation is developed from relations given
in Reference 1.
From Equation
From Appendix
there is Cx = 2(s/c)
I,
(k)
of Reference 1, there is
(tanP x
- tan|3
y )cosPm
(3)
and
(C
v
n
)
D'sec
= Y
sec
cos 5 P /cos 2 f3
(-)
v
m
c
7
(4)
where the vector mean angle,
Pm = tan"
Here
|3
1
[| (tan P x
is the gas angle -relative to
as appropriate
-
-
tan
|3
y )]
(5)
moving rotor or stationary nozzle
and x and y refer to inlet and exit conditions
'
7-
Combining 2,
respectively.
Tsec "
3,
tan P x-tan P
^f
"
and h, there results
y
(6.
)
The coefficient X is an empirically derived coefficient which accounts
for the effect of acceleration on secondary losses.
Figure 17 of
Reference 1 correlates \ as a function of hub-to-tip radius ratio
and (Ay/ Ax )
p
.
Since these data were presumably gather for low sub-
sonic velocities, it is assumed (P^./Ax )
gas velocity.
2
2
=
(W
/w
Based on the foregoing assumption,
)
where W is relative
,
(W
appropriate acceleration parameter independent of M.
x
2
/W
)
should be an
y
The assumed
variation of X with acceleration and hub-to-tip ratio is shown as
Figure
3.
It should be noted that experimental points of the original
figure presented considerable scatter, whereas Figure 3 represents a
best fit of the data.
The tip clearance loss equation is developed from
^
Cjf
.
i 2t/i
2
h
/
of Reference 1, page 13.
(
7
)
c
Equation
(7)
is attributed to Carter,
was developed on a theoretical basis for small deflections.
and
However,
Reference 1 points out satisfactory agreement between (7) and tip
effects at large deflections, hence the extension seems justified
in the absence of a better theory.
Similar to (k) above, there is
also
KV
*r fc)
^B.
COS^
y
(8)
Upon combining (3),
_
2(
^
(l),
and (8), there results
^=o^
n
For single stage stators, B
(y
K
. e gt)
)
s
f
(tan
cos Pm
h
=
and (9) simplifies to
!if%
(10)
cos Bm
Similarly, for impulse rotors with B
= R
r
= 3,
there results
8(^)sin 2 R
=
(YK ) B
(9)
•>
(11)
The secondary loss relation (6) may also be simplified.
In the case of impulse rotors
(Y
=
)
(6)
becomes
2
l6x sin R
(12)
sec R
In the case of nozzles , Figure 3 indicates that for any practical
stator, X s = .OO55.
(Y
sec
)
s
=
Accordingly,,
M-oo55)x
sln
(6)
h
becomes
=
cosPm
.022
^LhL
(13)
cosR m
The stator coefficients, calculated in II-3, tend to be
excessively optimistic if (13) is used.
Furthermore, the theoretical
development based on Reference 1 and leading to (13) has so far made
no allowance for effects of blade height on the magnitude of stator
secondary losses.
However, the data of Figure 15b of Reference 1, as
well as Reference 9 a ^d earlier tests, confirms that stator secondary
losses are inversely proportional to blade height, while solidity,
aspect ratio, and chord have negligible effect on stator losses.
In
order to resolve the apparent cont rad.it ion,
(13 ) has been modified
by the coefficient 6/h, chosen empirically based on test results
presented in II-3-
Hence, the best estimate of stator secondary
losses is felt to be
The total losses of a blading row are determined by addition
of the profile, secondary, and tip losses calculated above.
Y
Y
"
T
+ Y
P
sec
+ Y
That is
fW)
K
The Ainley-Mathieson loss parameter may be converted to a pressure
coefficient or velocity coefficient to simplify subsequent calculations
.
One may write
Y
=
(note P" becomes P'
C
Px
—
"
P
"
-
P
P"/P
x
"
L. =
-
'
P
1
y
y
-
"
1
-
I
(16)
"
P /P
y
y
in the case of stationary blading).
p"V'
/P
x"
1
=
1
+Y [i
-
(1
+
Z^^
Then
^_
(1T)
-7/r-i
)
]
The pressure coefficient is seen to be a function of the relative exit
Mach number, M
,
as well as the loss coefficient and gas properties.
The conversion to a velocity coefficient is made using
-10-
which is developed in Appendix III of Reference lk.
Solving for
\|r,
there results
t
= [i + -
—1-5
(i
-
c/nr
2
(is)
Equations (17) and (18) were programmed on a 7090 computer
and the results tabulated in Table
I
for 7 = 1.^-0 and 7 = 1.25-
The
results are also plotted for 7 = 1.U0 and appear as Figures k and
3.
5-
Comparison of Loss Prediction Theory With Available Test Results
The loss prediction theory of II-2 was compared with test
results for 7 rotors and
5
stators of experimental turbines.
These
represent the currently available results that have been published
with sufficient information to provide a comparison between predicted
and experimental loss coefficients for blading rows.
The comparisons
covered a moderately wide spectrum of Y, NL, degree of reaction,
deflection angle, solidity, aspect ratio and other parameters.
While
the volume of test data is quite limited, agreement between theory
and experiment is reasonably good.
The average deviation between
predicted and experimentally determined coefficients is about one
percentage point.
The precision of both predicted and experimental
coefficients is probably no better than plus or minus one point, for
reasons discussed below.
The comparative results are shown in Table II.
It was necessary to apply certain assumptions and approxi-
mations in compiling the data of Table II.
with reasons therefore, are:
The more important of these,
)
-11-
(a)
The N.A.S.A. rotors of Reference
5
through 9
were assumed to have tip clearance Ah/h = .02
or Ah = .042", since clearance was not specified.
The errors should be small, as Y < t Ym in all
K
cases.
(b)
Geometric and velocity characteristics are deter-
mined for mean radius conditions, deemed representative of the stage.
This assumes that three-
dimensional effects are minor.
(c
The effects of Reynolds Number variation are
neglected.
Reference 1 assumes (Re)
= 2 x 10
,
while the experimental turbines operated at (Re)
only 2 to 3 times this value.
(d)
The rotor pressure coefficients were not quoted
explicitly in References
5
through
9-
The co-
efficients were deduced from
(c
P)H =
K
?k 7
~
P
2
(0
p)s
[(l-r,
T)
T^rj-i
z,i-rz=
+1,0^7 ]*r
which is developed in Appendix
is exact,
p°.
o
B.
While this relation
some interpolation was required to estimate
and Pu'/P
'
^ rom ^he published turbine maps.
stator coefficients were deduced from Reference
k.
(i)
The
9-
Comparison of Typical Rotor Velocity Coefficient Estimates Obtained
by Different Methods
Figure 6 displays the estimated variation of rotor velocity
coefficient, as a function of deflection, for typical impulse blading.
-12-
The subsonic reference curve is taken from Reference 12, and repre-
sents a mean obtained from a large body of experimental results.
The two supersonic curves were obtained following the method of
Tl~2, for a typical geometry.
interesting features.
The comparison illustrates several
The magnitude of the rotor coefficient estimate
differs relatively little between the subsonic and supersonic curves.
This is not too surprising in view of the comparisons shown in II-l.
However, the shape of the supersonic curves at high deflections appears
somewhat suspicious, as subsonic experience would anticipate a more
rapid deterioration of velocity coefficient.
Since there is insuffic-
ient test data to confirm or deny the suspicions, one would probably
be well advised to use the more conservative estimate, or perhaps
a mean.
One also notes the increase of
\|/t,
with M, for fixed losses,
Y.
This trend is also seen in Figure 5.
5.
Detailed Analysis of the Effect of Rotor Profile Geometry on Losses
Reference 13 reports extensive two-dimensional cascade tests
of rotor profiles.
These tests compared conventional constant-
curvature steam turbine profiles to several experimental profiles
designed to minimize flow separation in supersonic rotor channels.
The inlet Mach number was 1.9 and deflection angle was 1^0
.
Schlieren photographs and total pressure measurements indicated
that the resulting "shockless" profiles essentially eliminated
shock-induced separation from the suction surface.
The over-all
velocity coefficient correspondingly increased from .929 to .952.
-13-
Reference 2 compared the conventional and shockless profiles
in a test turbine, which also operated with several combinations of
rotor "lap" and flare.
Exact comparison between cascade and turbine
tests is difficult due to the limited number of configurations reported.
Furthermore, the rotor inlet Mach number was about 1.6 instead of the
design value of 1-9-
The reduced Mach number resulted from off design
nozzle pressure ratio, the reasons for which are discussed in Section
II-6.
The maximum rotor velocity coefficient achieved in the turbine
was .872.
This peak performance was obtained using zero lap, but
sufficient flare to approximately compensate for actual losses and
approach the design exit flow conditions.
In another test, the
shockless profile increased total head efficiency about 2-l/2 percentage points, under otherwise identical conditions, while static efficiency
changed little if any.
These results will be discussed in more detail
presently.
The shortcomings of the conventional, constant -curvature
passages are thoroughly explored on two-dimensional theoretical grounds
in Reference 13.
Essentially, it is demonstrated that constant-curva-
ture passages cannot transform a uniform incident flow into a uniform
curved flow.
Rather, from considerations of a characteristics net, it
is evident that the pressure on either surface undergoes a continuous
zig-zag pattern of abrupt increases and decreases throughout the length
of the curved channel.
The pressure rises are expected to react with
the thickening boundary layer and produce separation from the suction
surface.
Cascade tests demonstrated the predicted separation for the
conventional profiles.
-14-
A second and complementary fault of the conventional profiles
is the likelihood of excessive channel width caused by too large a
spacing between blades.
In supersonic rotor channels, the wider the
channel, the larger becomes the uncompensated convex surface acceler-
ation before the compression waves become incident.
Hence, an
excessively wide spacing produces an increasingly severe pattern
of pressure fluctuations.
Schlieren photographs also showed a
definite tendency toward focusing of the compression "fan" if the
spacing were too wide.
and flow breakdown.
The abrupt pressure rise triggered separation
It should be noted that supersonic rotor profiles
are determined from channel flow considerations, and solidity is not
an appropriate design variable.
The "shockless" profiles of Reference 13 were designed to
eliminate the foregoing objections.
was mentioned above.
Their success in cascade tests
An example design is discussed and constructed
in Section IV-5 and Figure 15.
Essential features of the design scheme
are:
(a)
Dual curvature, having an entry or transition
curvature half as great as the central or mid-
chord curvature.
The transition curvature
extends to a position corresponding to intersections
of the first characteristics originating from the
opposite surface.
Throughout the central region a
uniform free vortex flow is theoretically established
in planes parallel to the machine axis.
A similar
transition at the channel exit reconverts the flow
-15-
to uniform parallel exit conditions.
(b)
A sufficiently small channel width to reduce to
acceptable levels the initial, uncompensated
acceleration along the convex surface.
The
maximum recommended width is that prescribed
by a transition curvature having
(JX_)
>
cos
-out tran
(c)
(sln
-l
1
wz
)
(!)
lx
A channel width sufficiently small to preclude
deceleration to sonic velocity along the concave
surface.
This requirement is generally less
stringent than (b), unless the incident Mach
number is near unity.
A comparison of cascade tests of the two profiles is seen
in Figure 7> where the data were collected from Figures 7 and 19 of
Reference 13B,
Aside from the obvious general superiority of profile
a closer analysis yields an insight into the nature of the compar-
ative losses.
The general spanwise variation of losses indicates
the effects of secondary flows, wherein the end wall boundary layers
are "centrifuged" toward mid-span by secondary vortices
I3.7 of Reference 12 for complete discussion).
(see Art.
A fair estimate of
purely profile losses is probably obtained from the curve peaks,
near root (and tip).
The greatly increased losses in the central
span region point out the predominance of secondary losses
—
in
qualitative agreement with the estimates of Table II for the rotor
-16-
of Reference 2.
The spanwise uniform superiority of profile B may-
be explained by a sharp decrease in profile losses with secondary
losses essentially the same in both cases.
Hence one concludes
that the improvement shown by profile B is due to a reduction in
profile losses caused by virtual elimination of shock-induced
suction surface separation.
The ratio of losses between the profiles, in cascade, is
(
ya/ y b)
Cascade
=
"^
•
- ^-It
obtained from Figure
5
using the reported
37
velocity coefficients.
From Equation III-2-2, static efficiency is
directly proportional to
(l + tp)
parameters assumed fixed.
for impulse turbines with all other
Thus, the relative improvement in
r\
due to
profile B, if duplicated in the turbine, would be a factor of only
i-±-^2
1#012>
__
1 + .929
The reported data of Reference 2 allows at least a rough
comparison of losses obtained using profiles A and B in the test
turbine.
Total efficiencies were .610 and .635 respectively, with
only the rotor profiles changed.
From Equation II-J-1,
(Cp)g/(Cp)^ =
1.06 for stator velocity coefficient and total pressure ratio fixed.
Also, using Figure k for My * i.k, together with an estimated Y * .8k
A
from Table II, one may show (Y /Y_)_
A
improvement from, say Y
about .013 increase in
.
~ 1.18.
Using Figure 5, the
,8k to Y = .dh/l.lB = .71 is equivalent to
\|r
R
.
Because
tj
is proportional to
(l + \|0,
it is understandable that the static efficiency improvement, due to
the improved rotor profile, was indeed negligible in the test results.
Several important observations may be made from the results
of the cascade and turbine tests of the two profiles.
It is obvious
-17-
that the design features of the shockless profile B prove beneficial
in all cases.
The relative improvement of profile B is considerably
less in the turbine, however, under this single set of test conditions.
This result, together with a velocity coefficient reduction from
tc
ca(i e
= -952 to i
= .872,
R
(a)
may be at least partially explained by:
the likelihood of moderate flow separation in the
turbine rotor, while virtually none occurred in
cascade tests of profile B
(b)
the degrading effect of tip losses, not present
in cascade, which also are proportionally more
detrimental to rotor profile
(c)
off -design operation of the rotor, which
operated at
(d)
B.
=1.6 instead
M,,
of 1-9-
other three-dimensional effects, such as radial
pressure gradients, which would be expected to
degrade the performance of moving blades.
In addition to the effects of profile changes, Reference 2
also illustrates another important facet of rotor losses and the
necessity for their accurate estimation during design.
series, an improvement from
T]
=
.50 to .52 and
\|/
R = .Qkk to .872 was
accomplished by a slight increase in the rotor flare.
used the "shockless" profile.
on an estimated
\|/
R
=
.
In one test
Both tests
In the first case the flare was based
91 and, due to the magnitude of the actual
losses, the exit swirl angle
(3r
was constrained by continuity to be
considerably less than its design value.
The increased flare approx-
imated actual losses and produced the increased exit swirl and efficiency.
-18-
It is evident that an accurate estimate of the actual velocity-
coefficient may be considerably more important than the questionable ability to increase it a few percentage points.
6.
Analysis of the Effects of Rotor Incidence on the Supersonic Flow
Through Turbine Blading
On the basis of experiment, Reference Ik concludes that rotor
blading with non-zero edge thickness, in supersonic flow having a subsonic axial component, is constrained by continuity to operate at a
unique incidence relative to the oncoming gas flow.
This incidence
determines the gas direction relative to the rotor.
Likewise, the
absolute discharge angle of the stator, as well as effective stator
exit area, is determined uniquely and is virtually independent of the
stator geometry.
The rotor incidence effect, if uncompensated, can
drastically reduce the stator discharge area, pressure ratio, and
Mach number.
The stator pressure ratio of the test turbine of
Reference 2 was reduced to between
l/'2
and 2/3 of design value, for
example, due to existence of rotor incidence of k.5
It is evident
•
care must be taken to estimate the rotor incidence and thus determine
the required blading geometry to minimize losses and obtain the
desired pressure and velocity distributions.
References 2 and l4 set forth a theory for determining the
unique incidence,
i
s Pg -
P^
This theory will be stated briefly,
and its validity examined, in view of the importance of the incidence.
The incidence is postulated to be that for which the expan-
sion, corresponding to a Prandtl-Meyer turn of
(3
2
~
equal to the geometrically available expansion ratio
Pv
(s
^s
exac "tly
cos £*-t)/scos
f3
,
-19-
There is assumed to be a slender wedge leading edge with attached
shock, and negligible total pressure loss accompanying the expansion.
Hence, the problem as postulated resolves into simultaneous solution
of the continuity and Pr and tl -Meyer relations.
The Prandtl -Meyer relation in this case simply v +
(3
=
constant or
v
(% 5
= v (Mw
)
2
)
+
02
-P5) = v
(%)
+ ±
CO
A convenient form of the one -dimensional, steady continuity equation
may be developed form
p
VA =
(
)(M
—
RT
n/tRT) A
constant
=
(2)
Hence,
PMA
constant
(3)
n/T
But
T =
T"
-I
ETWk
rTTETMj
v
and
ana
P =
r
.
:
Zzi
(1
+
v~
2
2
'
O
^_
777-1
Since T" is constant through a constant diameter rotor,
P"MwA
(1
7 ~ L ML 2)7+1/2(7-1)
1
=
constant
(+)
2
Examining stations (see Figure 9) 2 and
3,
just upstream
and downstream of the leading edge, one writes
A5
=
A2
V
M^
1 +
P,"
M„
1
3
+
2^
My,, 2
Z^M„ 2 2
7+1/2(7-1)
(5)
.
.
-20-
From the blading geometry, there is also
A
cos
TA2
-
(3,
t/s
^
\
=
COS Po
Using C p = P,"/ P p" and
5
coml:> i nin g
^
cP
(5)
and (6), there results
<*>
^J)
(rT
Equations (l) and (7) may he solved for two unknowns.
Typically,
one knows
My-
2f
and Po.
P*, and t/s and wishes to find M^
2
3
The incidence is thus determined for a specified incident Mach
number and rotor blade profile.
Reference lk illustrates the theory for an example rotor
having P,
= 70
and
1VU
=1.9.
Reference 2 compared estimated and
experimentally deduced incidence, with fair agreement, for
and M^
= 1.6l.
See Figure 8.
P-z
= 70°
Isentropic conditions are assumed
throughout
Upon close scrutiny, however, the above theory may be
shown to become absurd as the rotor blade angle, p,, is reduced
somewhat.
The anomaly is easily demonstrated by a simple numerical
example
Suppose the (relative) incident flow occurs at M^
=1.6.
The corresponding Prandtl-Meyer angle for 7 = l.k is Vp = lk.9°-
Further suppose a fixed incidence of any reasonable magnitude, say
3
.
Now using Equations (l) and (7), calculate the allowable edge
thickness t/s, for various P*.
C
p
= 1.0.
Assume isentropic conditions,
.
-21-
By (l), v,(M
Mr,,
=
I.7O3.
Equation
=17-9° which yields
)
(7)
a corresponding
becomes
3
t/s = cos p,
-
cos
(P, + 3
= cos P
For P, =
70°.,
t/s = .027.
-
5
Ox 1.
1 + .2(1.703)
2
)
1 + .2(1.60) 2
703
1.075 cos
5
+ 3
(8)
)
However, upon reducing P* it is seen that
t/s decreases monotonically until t/s =
for P, = 52°
•
The same
conclusions are reached regardless of the selected numerical values.
In particular,
P
if C
5
p
it is evident that breakdown occurs at even larger
< 1.0 or M»
> 1.6.
The physical explanation of the breakdown of the theory is
that, at sufficiently small blade angles, the turned and accelerated
Prandtl -Meyer flow (assumed) fails to encounter the requisite geometric
expansion necessary to satisfy continuity.
In addition to the fore-
going arguments, it is also evident that the isentropic, irrotational
Prandtl -Meyer flow assumptions cannot hold in an actual turbine rotor.
In article 9-10 of Reference 12, Vavra has pointed out the inherent
danger in applying two dimensional analysis to such three dimensional
flows
Aside from the merits of the foregoing theory, rotor incidence
remains a physical reality with very important implications.
Hence, a
more satisfactory explanation of its nature must be sought out.
One must agree the continuity relation (7) is valid.
Having
decided the Prandtl -Meyer relation is not applicable in this case,
there remains one more unknown than available equations.
-22-
Expanding the Mach number
terra of
(7)
in a binomial
series, it may be shown that
A+^M^ y^-i)
-s
where
e
=
.?
Substituting in
=
t/s
^
2
_
i} + s(e2) +
- 1.
r
(
j),
cos
there results
-
I
5
From
_ i + €
(9)
y+1
2
+ e( ~2~^2 - !)
cos P
2
(9)
CP
it is evident that for moderate incident Mach
numbers, the magnitude of the incidence is relatively insensitive to
small differences between
Equation
and M^
M^-
was compared with the experimentally deduced
(9)
incidence of Reference
For design speed conditions, one is able
2.
Assuming
to deduce Cp/l + 1.9e)= •84.
.Qk
<
C
p < 1.00
.
<
;
e
< .10
e
>
0, one then concludes that
.
As stated earlier, a second valid equation in the unknowns
is apparently lacking.
Since it seems probable
interest, one might assume
t
-
„
« cos p^
-
=
e
cos
(3
e
«
1 in cases of
and rewrite Equation (9) as
Q
/,^\
(10)
l£
Cp
The theoretical incidence of Equation (10 ) and of Reference
lk was compared with the experimental incidence of Reference
comparison is shown in Figure
8.
2.
There is fair agreement, at the
The
•
-23-
design point, between the theoretical estimates for isentropic
conditions.
Furthermore, Equation (10) is seen to agree with
experiment for C p = .84 at design speed and
.
90 at zero speed.
This trend seems reasonable and agrees with the expectations of
Higher relative losses at design speed
the authors of Reference 2.
are attributed to the reduced Mach number with correspondingly
stronger, and possibly detached, shocks.
By examining the design speed case further, one may deduce
the apparent proportion of total rotor loss, occurring at the leading
edge, which results in the 4.5° observed incidence.
losses were reported as
sponds to (YrjOp = 1.12.
\|/
=
.872.
Over-all rotor
Using Figure 5, this loss corre-
Similarly, from Figure k the apparent lead-
ing edge loss, Cp = .84, corresponds to
(Yttp)-d =
-25-
Hence, the
apparent ratio of leading edge to total rotor loss is .25/1.12 = 2/9As noted above, this apparent ratio represents an upper limit of
actual leading edge losses and, in effect, incorporates the effect
of any flow acceleration, e, which may occur at the rotor entrance.
From Equation (10), for a given blade (or gas) angle and incidence,
leading edge losses would be expected to increase with t/s.
The
foregoing considerations are utilized to estimate incidence in the
design problem of IV- 5
The result of a single experiment hardly constitutes a
reliable design rule.
However, there is provided a reasonably con-
sistent account of the interaction between gas angles, blade angles,
and rotor geometry.
As mentioned previously, care must be taken to
-2k-
accurately estimate incidence and preclude serious rotor-stator
mismatching with resulting Losses.
One test cited in Reference 2
clearly indicates the criticality of the problem.
having negligible edge thickness
-were run.
Rotor blades
During the course of
the test, the nozzle pressure ratio decreased from l6/l to
I'^j'L,
corresponding to the accumulation of only a thin carbon deposit
on the rotor blades.
Nevertheless, the buildup caused the
incidence to evidently increase some 2
observed effect.
in order to produce the
III.
A RATIONAL TECHNIQUE FOR DESIGN POINT
SELECTION FOR SINGLE-STAGE
IMPULSE TURBINES
1.
General Discussion
Given a design goal in terms of RPM and power required,
expansion ratio, working fluid, and thermodynamic inlet properties,
one is faced with the task of selecting a design point to satisfy
the requirements.
Literally an infinite number of design points, in
terms of blade speed and gas angles, can meet the requirements.
Hence,
the designer must choose the most desirable combination of high
efficiency, compact size and weight, and tolerable stress limits.
An obvious approach is a trial and error process.
This results in
laborious calculations of debatable final success in determining
the "best" design point.
A single-stage turbine, especially one with a high expansion
ratio, will tend to have sizeable
is determined
leaving losses.
Hence, efficiency
primarily by blade to isentropic jet ratio as seen in
Figure 1, while the blade speed is limited by material stress considerations.
Experience and the mission requirements generally
dictate desirable blade materials and allowable stresses.
Thus one
might be well advised to restrict the infinite number of possible
design points to only those which satisfy some specified blade root
stress.
One could then proceed to decide upon a desirable size,
weight, and efficiency combination to meet design requirements.
The following method is one way of logically deciding upon
the "best" design point.
It requires neither machine calculations
-25-
)
The technique is applicable to single-
nor infinite human patience.
stage impulse turbines, totally independent of Mach number levels.
The loss coefficients which enter may be estimated by any means what-
soever, and are not related necessarily to loss prediction methods of
II-2.
The nomenclature of the development is defined in Appendix A.
The method is applied to a design problem in IV.
2.
Static Efficiency as a Function of Velocity Coefficients and Rotor
Deflection
In terms of Figure 9> Euler's Turbine Equation may be
expressed as
=
T)
2
JL
C
u
(W
2
sin
f3
+ W 4 sin PjJ
2
(l
o
For symmetrical impulse blading of the present case,
P
Equation
(l
)
2
W^
and
= P^ = p
=
\|r
R
W
2
may be rewritten,
r\
=
lO
R
2 -iL (1 +
y
C 2
o
" X1 P,
Wo
K2
"2 sin
or
2
By definition,
\|r
= Vp /C
T]
=
,
2_
£2
C
C
o
(1
+ *
)
sin P
(2)
o
and (2) may be expressed
2^ s
2
(1
+
\|r
R
)
g
sin p
(3)
£sin P
(4)
From the geometry of Figure 9>
=
<Jl
-
S
2 cos2p -
-26-
-27-
Hence (3) may
tj
is a
\|r-o
expressed
"be
2
=
2\|z
o
s
(1
O
+
sin p
known function of both
pendent of
£
and
(3
but weakly dependent on
,
S^cos^P
-
8(s/l
-
r\
£
in general.
,
(3
Ssin 0)
-
through CU.
\|/
(5)
is inde-
Equation
(5)
allows calculation of efficiency for selected design point coordinates
6
,
P.
While Equation
(5)
might be considered to "organize" the
search for a suitable design point, it places no restriction on the
blade stresses.
Incidentally, Equation (5) also illustrates the relative
importance of
\|f
fl
as compared to tp>
point improvement in
improvement in
\Jr
For example, a one percentage
should be roughly equivalent to a four point
\|f
R , since
»
\|/
«
\|/
1.
Limiting Stress Requirement
3.
For a selected blading material of a specified taper ratio,
including constant cross section blading as a special case, one may
show that blade root stress is proportional to the product of radial
If bending stresses are neglected,
acceleration and blade height.
a
~ h ¥i
R
(6)
By continuity, the mass flow rate is,
o
¥
=
density x axial velocity x annular area,
or
o
W The exit density,
the blade exit.
p]^
p.
,
cos P Rh
(7)
depends on pressure and temperature at
The pressure may be assumed independent of the vari-
ables and equal to exhaust pressure, provided the axial component of
-28-
The temperature will vary somewhat with
velocity is subsonic.
T]
=
T)
(6jP)-
If ° ne neglects this variation, Equation (7) becomes
W ~ W^ cos P Rh
(8)
1
For fixed power output, W
.
Since
Wi,
t
T]
(8)
= t-oW„. Equation
X\
d
becomes
i ~
^ R W 2 cos p Rh
(9)
Substituting,
1
h ~
^R
W 2^ cos P
into (6), there results
U 2
a
~
^
i
n+B W 2 cos P
(
10 )
One notes that U/R, the angular velocity, is fixed in the
present case.
independent of
Further, letting W* = % V^ ~
8
S
,
by assuming V,
and/6*, Equation (10) becomes
T|i|r
R
cos P
or
o"H^r S cos P
= B
=
constant
(ll)
The parameter B has stress units and may be viewed as an invariant
property of the particular design specifications and working fluid.
Experience indicates that B is indeed virtually constant over quite
-29-
a sizeable range of
$
as illustrated in IV.
P variation,
,
If one now selects a specific working stress
objective, there results
D =
—
=
R
as a design
cos P
5
T)\|f
a
a'
(12)
Thus, for a specified stress, the parameter D is a unique constant.
Furthermore, any design point satisfying (12) automatically satisfies
the imposed stress requirement,
c~.
Combination of Performance and Stress Requirements
k.
By combining the results of Equation (5) and (12)
2^ s
2
+ fo)
n sin
(l
S
(
sll
^cos^P
-
J2
Ssin P) =
-
.
Vr> 6
COS P
(13)
or
S
(
cos 2 P
-F
n/1
Ssin P)
-
I"
2t s
Ol
T
+ V R ) sin P cos P
R'
(ik)
For convenience, one may define parameters
2^ s 2
F"
=
2
i
\|r
{Jl
R (l + R
\|/
-
)
^cos^P
GAa)
sin P cos P
-
Ssin p)
(lite)
Note that F" is a universal function of
Since
\)r„
=
i|r
R ( S>P) and
D is a known constant
(lh)
.
\|/
$
and P and could be tabulated.
is assumed independent of P, while
(for a specified
3"),
one may evaluate Equation
There remains an unlimited number of admissible
which are known to satisfy the imposed stress limit.
the selection of the most attractive
design point decision.
S
}
S
,
P coordinates
The problem is
P that represents a final
.
-30-
As an aid in this decision one might examine some variable
additional relationships
For example, from (13),
=
,(S,p)
\|r
R
6
D
cos p
(15)
The rotor relative Mach number variation may be examined by noting,
W2
*»
2
6
n//RT
n//RT
2
Furthermore, if Vp = ^ S C
V2
is
2
independent of P, as previously assumed,
it follows that To is likewise constant.
\^2
Hence one notes,
~ S
(16)
Variation in radius is simply determined from
U
~
R
~
9
(17)
Reference 15 gives a useful empirical estimate for single-stage rocketturbine weight as
m,lb.
=
70jt
(R,ft.)
2
=
1-53 (R,in. ) 2
(18)
Finally, since a ~ Rh ~ 9h, for constant RPM, blade height variation
may be deduced from
—
=
constant
(19)
a
5
.
Procedure for Application
Given power and RPM requirements, together with available
expansion ratio and turbine inlet conditions for a specified working
-31-
fluid, one may proceed in an orderly manner to select the "best"
design point, as follows:
(A)
Based on experience and operational objectives,
one makes a tentative selection of blade material
and one or more values of allowable centrifugal
blade root stress.
(B)
A trial design point is selected in terms of an
arbitrary velocity triangle.
As a minimum, one
evaluates, for the trial design point:
W
a
B
P
%
%
h
e
*os
S
A single calculation of this type should suffice
provided the calculated stress is within, say + 50
percent of the selected values,
cf.
Equation (lII-4-ll).
(C)
Calculate the invariant B
(d)
Calculate D corresponding to selected a
-
-
Equation
(lIlA-12).
(E)
Plot F' against
(3
(for a reasonable regime of P) -
Equation III-4-l4a.
lated in step
(b).
Consider
Consider t
\|/
R
constant as calcu-
s
=
^(P.M^
)
~ V (P),
R
using constant M^. as calculated in step (b).
The
-32-
curves F' theN represent constant stress lines.
See Figure 10 for an example.
(F)
Plot F" against $, for selected S = constant
II-4-lVb.
-
Equation
Intersections of the constant S and constant
a curves determine unlimited compatible
(
S
,
a)
3,
design point possibilities.
(G)
For any desired
,
$, a)
,
calculate:
-
Equation (III-4-15)
¥^
-
Equation (III-4-16)
9
-
Equation (lII-lf-4)
R
-
Equation (III-4-17)
U
-
Equation (lII-U-17)
m
-
Equation (III-4-18)
h
-
Equation (lII-U-19)
T)
(H)
( £,
As an additional aid in visualizing the variations,
plot
t\
against
(3
The addition of
for a = constant.
£ = constant and 9 = constant points provides for ready-
visualization of the variation in
t\,
h along the constant stress curve.
(i)
From the plot of
(H)_,
Mw
,
U, m,
9, R,
and
See Figure 11.
one may then select the most
attractive combination of features to fix the design
point
(£>,
3,
a).
Estimate
t\,
for the selected design point.
the estimates above.
Mw
}
9, R,
U, m,
and h
Repeat (b) to verify
One notes that the approximations
introduced into the technique are now discarded.
A
second iteration of (C) through (i) would appear unnecessary
unless (i) indicates intolerable lack of agreement between
the rough and smooth design point estimates.
IV.
1.
Specifications
and.
ILLUSTRATIVE DESIGN PROBLEM
General Design Philosophy
An example of a preliminary design is presented to illustrate
the application of previously discussed methods of loss prediction
and design point selection.
The specifications and gas properties
arbitrarily chosen are those which might correspond to turbodrive
requirements of a large liquid propellant rocket using reasonably
energetic chemical propellants.
Assumed output requirements are:
28,800 SHP at 15,000 RPM.
Assumed gas properties are:
molecular weight,
On
=
12.0
specific heat ratio, y
=
1.25
constant pressure
heat
specific
*
=
c
lb/mole
.827 Btu/lb-°R
P
total inlet pressure P' =
^50 psia
o
exhaust static presPl =
sure
30 psia
gas constant,
R
=
128.8 ft -lb/lb- R
total inlet temperature,
T'
=
2200°R
The turbine is to be of the single-stage, impulse type to
minimize axial thrust on the rotor bearings.
A reasonable compromise
between static efficiency, size and weight, and simplicity is to be
achieved.
-33-
-34-
2.
Design Point Selection
The design point selection method described in Section III
A single, arbitrary trial design point serves
is to be followed.
as a basis for selection of the final design point.
The available isentropic energy is,
^is
V
- cp
n
(^
-
)r " 1/7]
-
827 x 220 ° [1
^
-
o
y20] -
76 °
tst
The isentropic jet velocity,
C
=
o
N/2gJ£H
B
^IS
=
n/5
x 10 4 x 760
(^)
7 " 1/7 1]
6l60 ft/sec
=
The isentropic Mach number,
=
(M_) jq
J JL
[
=
JQ [15'^
- 1]
= 2.40
The subsequent steps correspond to those of Section III-5.
STEP A
Suppose consideration be given to blade root stresses of
Most suitable blading materials
30,000, 37,500, and 45,000 psia.
are known to have densities of about O.3O lb/in^.
STEP B
For the first trial arbitrarily select a p
From Figure 1, the expected efficiency is
.
60 <
blade speed is, U = .3 x 6l60 = 1840 ft/sec.
_
IBM
2it
x 60
x 15000
x
_
lUa „
T)
= 65
<
,
.67-
U/C
= .3.
The mean
The mean blade radius is
2
-55-
(l)
Determination of Stator Velocity Coefficient,
\|/
g
—
The mass flow rate is
550
o
SHP x
=
fj%
t]AH
28800 x .707
.64 x 760
^
is
The annulus area,
o
A2
2rt Rh
=
2
o
£
=
H
-
=~
2 ax,
\|/
RT 2
s
C
Q
cosa
2
WRT 2
li
2
^s P2 C o cosa2 2irR
For an estimated
T
2
= T
'
-
«
\|r
——
2 AH
\|r
s
°
.95.
=
2200
-
.95
cP
2
I^2_
=
1570°R
.827
and
lb
42—
x
1
h
2
_
a
ft ~ lb
o
00 1^^x1370
R
128.8
,__
.95 x 30 ±°
x 6l60
in^
2-
=
=1.12"
cos 65
x
2rt
x 14.1 in
sec
approximate annulus height.
The nozzle secondary losses are calculated from Equation (II-2-14),
fv
x
\
\
sec/s
_
ill
,.
i
2
sln2a P
cos a
m
where
Q^
1
tan" 1 (| tan C^) = tan"
=
(^J2l)
Thus
sec
s
_13
sin 2 65°
1.12
cos 47°
m
lk
= 47
°
9
-
56-
Stator profile losses are estimated from Figure 2a to be
Total stator losses,
(Y
=
T )S
(Y
sec
)
+ (Y
s
)
»
q
(2)
=
Vax
2
P„
'p
=
-
=
C
^s o
.
=
6° 00 COS 65 °
tan"
UQli
1
Vp sin
(-^—Tj
v
OLo
-
U
2
ax, 2
w
2
Vax 2
cos P
2
=
=
^390
cos 54.7°
S
=
w 2 /v2 =
4390/6000
=
.752
9
=
u/v2
18^8/6000
=
.308
T
T
=
2 ^ISl
-
«
=
2200
-
M
vP
g
=
—
=
-—
C
=
1328
827
= 2.29
72.0 n/1328
-J/gRTp
%
.
6000
=
—
.97^
Cp
Revised
„
5^0
=
2535
2 ^^
=
?
^
ciiic
1 ^55
tan
=
W2
=
=
it59£
1.675
2620
72.0 n/t 2
Determination of rotor velocity coefficient,
\|/-n
Rotor secondary losses are, using Equation (II-2-12), and
estimating
(
X = .03
Y sec)R
"
from Figure
-
W
sin
%
3,
=
-^
sin
2
5^7°
06.
See Figure 9
-
_-]
,
)
v
=
.
97^«
ft/sec
-97^ x 6l60 = 6000
V cos a
2
2
=
-.
(3)
=
\|r
Determination of some velocity triangle components
V2
=
)
From Table I-d,
.20.
for M2 » 2.2, the velocity coefficient estimate is
(Y
-
Og
Tip losses, assuming Ati/h = .03, from Equation (lI-2-ll)
-
—
-37-
(YT
8-0
=
)
K R
2
—
sin P
d
h
= 8.0 x .03 sin
Profile losses are estimated from Figure 2b,
Estimated total rotor losses are
=
(Ym)-o
\|f
54-7° - -16
(Yp) R ~ .12
-60.
With an estimated rotor exit Mach number M^
velocity coefficient from Table I-d is
2
s 1.6, the estimated
R = .904.
The rotor coefficient compares with an estimated
from the subsonic reference curve of Figure 6 for
(4)
\|;
R
= 54-7°
|3
= .898
•
Completion of velocity triangle and performance calculations
=
%W2
P4 -
P2 =
W4
V
„ =
V^
=
^
.
ax, 2
work output,
=
cos
Static efficiency,
2290
—
=
p,
4
4
^Vj^
V=
337°
54-7°
W,
cos
=
'9° k x 459 °
+
(Wi,
-1/W^
HjJ-)
^
=
gj
cos p u - U)
- cos
T)
(—
2680
-1/2290 v
)
.
„ _
3^3
UW2 (l+^ R )sinp 2 = i848x 4?90 x 1.904 sin 54-7 o
gJ
2.5 x 10 4
17
=
;
-^—
AHIS
=
—
504
W
=
=
50l+
Btu /^-
.663
760
This estimate appears reasonable compared with Figure 1.
(5)
Temperatures, annulus (rotor blade) heights, and rotor blade stress
Since total relative temperature is constant through the rotor
of constant mean radius,
—
.
- 3 8-
V
w2
To
2
=
*
Tk +
=
*
2gJC p
W 42
2gJC p
=
T>"
Since
w,
VV
-
^
W2 2
+
T
(1
V>
-
M„
=
4
_\
i
\f =
Check work output,
c
p AT'
W*
720
2680
,
^39°
,^oO +
= 1528
2\
/-,
ig%
2
1413°R
=
T,
T
=
r
k
uTlfx^
(1
-**
1.45
n/T,,
2
n
o
(2200 - 1589) = 505 Btu/lb.
= .827
This constitutes a check for consistency of the velocity and temperature
distributions
The mass flow,
550 QWP
bH^
p
W
77S
—
=
=
V
2 q 8oo
—J70T
=
!
40.4 lb/sec
504
o
The annulus, or rotor leading edge, height is ho
W R To
=
2jtRP 2 vax,2
ho
d
40.4 x 128.8 x 1328
-
1.024"
=
2* x 14.1 x 30 x 2535
o _
h
WRT 4
4
2 * RP V
4 ax,4
= h.
"
2 BL
T
I-
2%
=
1-024 x
^
1528
x
-JL
= 1.207"
"**
The blade root stress is calculated assuming a blade of height
h.
and density
.
30 lb/ in*.
The centrifugal stresses only are considered,
as bending stresses are expected to represent a small fraction of total
root stress.
>
-39-
-A
The root force, F - ma = (pA
c
lb ~ sec
)
F
F/A
=
a
£i-
g
12
^
h,
- p
C
ft/sec 2 )
wuw n 2
/2it 15000
(**
)
60
R
z±
^t
*_
g
12
(3
QQ0Q *)*
60
and R = 14.1"
1.207"
h^ =
for
h l+
p ac
p
=
2
co
'12
ft
g
(5L.
=
1910
— psi
h],R
a = 32600 psi
,
Summary of step (b) results
(6)
=
.663
\|/
=
.904
W
=
kO.k lb/sec
/&
=
54-7°
a
=
32600 psi
1^
=
1.675
R
=
14.1"
M^
=
1.45
h
=
1.207"
9
=
.308
.97^
s
=
.732
T)
^s =
STEP C
=
B
0fT}t-R
R
S cos P a
8250 psi
,
the invariant
STEP D
h )
B [1)
-7TT
1
a^
D^
8250
B
=
=
=
=
)
-275
30000
D^
.220
=
.I83
STEP E
Constant stress lines represented by F'
^
x
>
=
p-
a|f
are plotted in Figure 10.
\|f^
R
=
\Ir_
R
(S
x
.
'
Mwl
)
;
For this calculation,
= 11^(3,
1.^-5)'
t
v
'
R
is assumed.
i]r
=
s
^ R (l+ ^ R )sinPcosp
.97^- =
Constant, and
The total rotor losses are
-kO~
esimated following step (B-3) to be
<Vr
=
<
Y
A
+ Y
sec
The velocity coefficient,
\|/
+
(
Vr
Steps
+
'
12
M^.
=1.45.
A simple working
= tr,(P) constructed from Table I-d is a convenience in
ty
(E)
^
72
=
R , is then easily calculated, as in Step
(B-3) as a function of variable B with
plot of
=
through (h).
STEP F
£t constant are plotted in Figure
Curves of F" {&) for
The range
.
10.
52 < 6 < .76 represents practical limits of interest and
increments of .02 are satisfactory.
STEPS G AND H
From intersections of
S= constant and
Figure 10, Equation (III-4-15) was solved for
a = constant curves of
Plots of T}(B)_
r\.
a - const,
are shown for the selected stresses in Figure 11.
While not expected
to be quantitatively precise, Figure 11 condenses the design variables
into an easily digested presentation.
only on 0, £
,
B,
Since
T),
M^
,
R, U,
hi
depend
the selection of a design point to fulfill a
specified mission is considerably simplified.
In the present case, one can write,
(Eq.
(Eq.
II-4-16),
III-4-4),
Mv
9
~
1.675
-J-
2
.
=
©
(
S,P),
732
=
2.29
S
plotted for convenience in the
regime of interest.
(Eq.
III-4-17),
R
» lil
—%s
= 45.8 9
.
'
-41-
(Eq.
U «
III-U-17),
——
1848
6000 9
=
.308
(Eq.
HI-4-18),
m &
(Eq.
III-4-19),
h.
(U5.8 9)
1.53
«
——
"k
2
=
3220 9
2
lb
,
.
x 1.207 = l.lU x 10" 5
i
32600
o
STEP I
A working stress must be selected.
The blade temperatures
may be estimated, for a (turbulent) recovery factor of about
to
9j
.
be,
2
.9 --%^ 2gJC
—
p
< T +
blade - 2
T x,n
2
=
+ .9
x t-^22
1328
J
'
4.13 x 10 4
=
.
i747°
— R
1
= 1287°F.
J
!
From Figure 11, one notes that efficiencies are reasonably good for
with relatively little gain to be achieved at higher stresses.
= 3750O,
ij
Experience would indicate a - 37500 to be a logical choice for a
rocket turbine application of short running time.
Ample safety factor
should be available for a blade material such as Inconel 703C a "t this
temperature
In Figure 11 for c = 37500, the regime,
.
60 < S < .68 appears
A rough idea of the trade-off between fuel consumption
most practical.
and turbine weight may be quickly made if desired.
paring points at
$-,=
For example, com-
g„ = .62 shows the latter turbine would
.66 and
weigh approximately An = 3220 (9^
-
o|) = 3220 (.k2
2
-
-37
2
= 127 lb. more
)
The corresponding reduction in fuel consumption would be
o
aw
o
o
=
¥ dP
-
w-i
L
=
—
530/778 SET
ah is
1
— —
,1
-
{
^
n2
v
;
=
.707 * 28800
76o
.
{
—1 —
.761+
-
1
-;
.794
1.34 lb/sec
.
=
,
42-
The choice of
S=
.62 would appear advantageous only if the burning
time exceeds
An
=
-o
ZW
—
127L-
n
95, sec,
=
l.jk
provided the objective is simply minimizing the sum of turbine weight
and total fuel consumed.
In the light of foregoing considerations a choice was made,
&= .6k,
selecting:
=
(3
with the assurance a « 37500.
60
From
Figure 11 and Step H, the additional estimates are immediately available
i)
W
~
-395
~
.78
~
3h
%~
lb/sec
1A6
~
18.1"
h^ ~
1.08"
R
U
~
2370 ft/sec
m
~
500 lb.
The design point selection technique has allowed a logical
and systematic choice of operating design point.
It appears that a
considerable improvement over the original arbitrary choice has been
However, detailed design calculations must follow to verify
achieved.
the choice and remove the inherent approximations.
3„
Detailed Design Point Calculations
The absolute gas angle required is
0t
2
= cos" 1
(
S cos P
2
)
= cos- 1 (.6k cos 60°) = 71-35°
9
-43-
a2
9 = 1.0 sin
and
S cos P 2 = sin 71-55° - -64 sin 60° =
-
Recalculation of stator velocity coefficient,
(a)
In the manner of Step (B-l),
*
2200
-
.96
-^
2
392
s
must be changed to incorp-
i|f
orate the revised Q^ and nozzle height.
T?
\|f
.
For an estimated
* .96,
ty
1355°R
=
827
3U x 128.8 x 1355
d
annulus height. For am = tan
(Y
v
-1
c
)
*s
Total stator losses,
224
L
-tot's
(Y +n+
=
.28.
For M
.06
(Y p )
P's
cos 55. P
92
approximate
.92".,
x 18.1
2ir
(| tan 71-35°) = 55-9°
sin 2 71.35
,13
sec
=
(.96 x 6l60 cos 71.35°)
30
a
2.2, Table I-d gives
is = -965.
It is evident that both (Cp) s and
unknown, value of Mo
it is assumed
\|r
fl
-
estimated to be 2.2.
depend on the, as yet
\|/_
In subsequent calculations
= .965 to determine the actual Mo.
The value of Cp
will then be revised, where necessary, to coincide with the newly
calculated Mo.
Determination of some velocity triangle components
(b)
V
=
.965 x 6l60 = 3950 ft/sec
vax 2 = 595 ° GOS 71-35
w2
=
sv2
U
= ^s
T
= 2200 -
u
=
.64 x 5950 = 3810
2330
378
6l60
=
%,
72.0
3810
2640
n/13^5 = 2640
1.44
d
= 5950 x .392 = 2330
.965
2
76 ° = 13l+5°R
.827
V2
d
o9
a
2
= 1900
= 5950
2640
__
2 25
.
—
2
)
-kk-
Estimate
(c
^
—
W2
^R
-**
^-p-
=
M, Tl
—91
•
»
72.0 Vtl.
x 38IO
"
_
sin 2 60° = 06
.48
-W 81x^60°
.36
W)r==
(Y_„) p
(Yp)
w
R
From Table I-d, for M^
k
An estimated
=
\|/
.
4
y t^R =
(
—>
=
~ 1.28,
=
\|/
R
*R
2
.03 sin 60° = .18
-^6
.871
6.
Completion of velocity triangle and performance calculations
(d)
Wr^" = .871 x 38IO = 3320 ft/sec
V^ =
^1655^ + (3320 cos 60°
V
,
SIX
2330
-
2
a
=
V
\|r
SIX
x\
^ M-
- cos"
1
,
—
= .871 x 1900 = —
1655
_
^ £i
1655
-
18.
(i^rr)
C
'1740'
= 1740
R = 6o x 2330
2jt x 15000
mean radius,
Work output,
4.
i
i=
T7
Vv
W
=
x 12 =
8"
17.
—
!
x I.871 sin 60°
x 381O
~
=
JT^
2.5 x 10 4
2330
—
—
575
m^'W
^
_.
,_.
~^
°
-707 x 28800
w =
s
"
__,.
575 Btu/lb.
=
gA
,_
,
,
lb/sec
The calculated efficiency appears reasonable but perhaps
slightly optimistic, compared with Figure 1.
(e)
)
88l is obtained for comparison from the subsonic
reference curve of Figure
tt
N
\|/p
(Y k )
R = 8 x
and estimated
.12 ,
Tj,
72.0^ 1^13
Determination of rotor velocity coefficient,
)
,
^
« 1^1 3 (Step B-5
.
with
= 1.28,
.
,
Temperatures and total head efficiency
T
^ T2+ 2gt a-0=^+
V=T,
4
h
+
j£2gJC
p
^^
2
= 1^29
y +
llk °
k
4.13 x 10^
,
=
1502°R
x. 2^1=1429^
-45-
"\J =
m
C
p
.827 (2200
=
W4
2
_
^
AT'
72.0
^
72.0 nTt^
4.
P^'
=
^
=
P^ (1 +
W
_
=
aF~
IS
^
3320
_
>y
1^29
72.0
n/
1429
2
30
1#2 2
+ .125 x .64
(1
575
.827 x 2200 [1 -
.
—
_
72.0
^ % F±
Btu/lb
1502) = 577
-
(2§Z5"poy
2 5,0
)
= 38.5 psia
^
o iq
=
1+50
Nozzle Flow Analysis and Profile Layout
A detailed analysis of the stator losses and required geometric
layout is necessary.
Pertinent assumptions are:
(a)
flow within the nozzles (stations 0-1
(b)
all stator losses occur in and downstream of the exit plane,
between stations 1 and
beyond station
2.
by the coefficient
)
is
considered isentropic,
and a uniform flow is established
2,
The losses between 1 and 2 are represented
\|/
=
.965?
(c)
continuity is satisfied between 1 and 2,
(d)
tangential momentum is conserved between 1 and
(e)
the flow is adiabatic, Tp' = T,'.
2.
The continuity equation is developed analogous to that
expressed by Equation (II-6-7) except using total temperatures and
pressures.
cos a,
The resulting relation is
=
r
t
(-)\
cP
+l r
M2
_£
Ml
r
1 +
-
125
{
1 + .125
^-5
mA
1—)
MQ 2
COS
CH
-46-
From previous calculations, a
g
=
71.35
M
,
=
The pressure
2.25.
coefficient,
C
P
=
^L
P
= 30
(1
+ .125 x 2.25
2 )^°
=
7T5
^0
o'
Hence, there is
cos °1
t
(j) s
=
+ .125 M-,^)
(1
+
-
06l 3
Tangential momentum conservation gives
M2
N/fgRT
sin a,
1
Tt
= M, >/fgR
sin a, = Vp sin
V-,
__.._
„,
sin
a,
tl
1/
(!)
Mi
_
=
„ rr-3r
Mg \/?gT
Oi^
T
1/
1/
'
2
sinCU
or
sin
%=^
1
\
/
V
^
! +
Ml 2
sin a
2
Ml
-V1+.125 x 2.25*
Zli M 2
2
1 +
^l + .125Mf sln
2.25
=
2
sin a,
I.67O
=
+ .125 M-l2
n/1
±-
(2)
M1
For a nozzle with no trailing edge thickness, simultaneous
solution of
(l)
and (2) gives
M1
=
2.150,
Q^ = 77.2°
.
For structural integrity a nozzle having 10$ "blockage," or
t/S cos
Qip
=
.10,
is selected.
,
cos a,
=
n
.0682
Equation
(1
(l)
then becomes
2 )^'
+.125
'— Mi
—J
M
5
±
(3)
l
Solution of
(2)
and (3) gives
Q^ = 75-3°^
\
=
2.178
for
the required nozzle exit conditions consistent with the desired (velocity
triangle of Figure 12) discharge flow.
The blade angle turns out to be
about the largest practically acceptable value.
If desired, the magnitude
^o
.
-hi-
of
could be decreased by providing a slight annular expansion
d-^
between stator and rotor.
The type of nozzle profile is probably not critical.
A
"sharp-corner" nozzle of the type discussed in Reference l6 was chosen
somewhat arbitrarily.
The sharp-corner nozzle, in general, provides
for the shortest possible nozzle length capable of a uniform discharge
Its properties become much more favorable in cases having larger
flow.
nozzle pressure ratios and lower discharge angles than the present
example
Rather than using a full characteristics network, the approximate nozzle contour required was obtained by calculating the required
throat angle, over-all area ratio, and the approximate orientation of
the limiting characteristics.
—
simply
where
v-,,
v-,
The angle of turn at the throat is
the Pr and tl -Meyer property angle corre-
= v-,(Mt)
sponding to the exit.
—
and
(y,
)
fE£ tan" 1
=
v
===1 7*9
p i78'
M
Z^. (MX 2
^
>
-
tan"
-
1)
the required angle.
Reference 17, problem 4.20),
Pq'
a
~i
(
=
A
t
y'
For M
1
=
(
PT~
^
7+1/27
j
[(
7-1
7
^)
Pi
" ±/7
W
(
-i]
2
/
7+1/2(7-1)
21 78,
Pq7 pq
=
(1
+ -125 x 2.178
P 5-0
)
=
^F^l
The required area
ratio is obtained from one -dimensional flow theory.
(e.g.
1
10.22
The relation is,
—
-48-
and
Al
(I0.22)-
=
90
UJo /m
oo*20
^8
(l0.22^
A+.
t
_
tf-5
2<20
9
1
The schematic nozzle layout is shown in Figure 13.
The limiting characteristic of the throat fan, ab, stands
at approximately
19»6
+ ^ =
fi
-
+ sin"
17.9
[—
1
-]
=-17. 9
+ sin
-1
(y_7g^)
The limiting reflected characteristic, be, stands at approximately
•
u = sin"
(
£)
= 27- 4
Thus the minimum nozzle length is
.
2.I70
wt
2° f
+
:o
tan 19. 6°
tan 27.4
j-
-
(2.8 + 4.25) v t = 7.05 w t
However, due to the required truncation of the nozzle at —
- d-^
= ik.'J
wt
= 8.38 w +
Hence,
t
tan 14.7°
minimum throat to discharge dimension, parallel to nozzle axis, is
^
the dimension bg must be no less than
eb + bg = 2.8 + 8.4
w.|-
11.2 w^.
=
.
The resulting nozzle profile layout
is shown in Figure l4a.
Since the edge thickness, t, is l/lO the total width,
l/9 x 2.20 w t = .244 w t
Total width is w = 2.444. w t
'The spacing
2.444 w+
,<w —
between blades is s =
w z+
— =
9«o5
The number of
^G = cos q;^
cos „^
75-3
blades Z = EjtR _ 2it x 17.8 = 11.59
For a selected throat diment =
.
.
—
.
s
sion
w-j-
.
S
9*
= .290",
65
W-^
W-j-
Z s = 40 blades.
The nozzle cascade is illustrated
in Figure l4b.
The height of the nozzles
(and annulus), may be more pre-
cisely calculated,
o _
WRTp
h.
2
PC VQV
c.
ax,
=
2*R
35.4 x 128.8 x 1345
30 x 1900 x 2* x 17.8
B
^ng!" = h
\
d.
The total throat area is
Z s hw
t
= 40 x
.
961 x .290 = 11.14 in 2
,
-49-
With the nozzle dimensions fully determined, the calculated
stator coefficient of Ill(a) should he checked.
For a thickness ratio
t/c = .21 and solidity c/s = 1.U+, Figure 2a and Equation II-2-1 gives
Pi
(Y p ) q
=
x .05 = .053-
-
—
Si2
The secondary losses are
(Y„ Q
J
C
'?• ^
= .272, where the blade angle rather than the gas
cos 62. 3°
.961
angle is employed to provide the most conservative estimate.
Total
-1±2_
losses
(Y-j-
-j.)
s
.33 exceed the estimated .28 of previous calculations,
=
Hence, from Table I-d, the estimated
.965 may be about .004 too
=
\|/
optimistic, but recalculation of IV- 3 seems unnecessary.
5-
Rotor Flow Analysis and Profile Layout
A leading edge thickness ratio, t/s, must be selected first
so that incidence may be determined.
expected solidity c/s
a;
2.5, and a desired edge thickness of about
.020", there is t/s = t/c
c/s » .020 x 2.5 = .050.
•
The incidence, Pg
(H-6-10), cos
(EU
J
=
— +
S
For a chord of about 1", an
-
£^, may be determined from Equation
—Lp—§
„
From considerations discussed in
.
II-6, it is assumed that about one-quarter of the rotor losses occur
around the leading edge
-
For
-
Table I-c,
Hence
(Y I^) R =
(Cp)^
p, = cos
-1
between stations 2 and
(Y
T )R
=
Y = .165)
= C p (M = l.kk,
(.050 +
^~-
=
COS
)
=
52.8°
.165
=
3.
there results from
-90
and the incidence is
7. 2°.
.90
The assumed pressure loss has increased the incidence by about 3-7°
from the isentropic case.
It is of interest that, were the blade
angle, £,, equal to 60°, the relative incident gas angle would be
about 66.1° and the nozzle pressure ratio would be greatly reduced
from the desired 15 to 1 value.
-50-
The rotor profile was laid out following the scheme
of
Reference 13 as summarized in Section II-5 of the present report.
As discussed in Section II-6, the rotor entrance Mach number, M^
assumed equal to M
w
= l.kk.
,
is
may exceed this value slightly, as
It
suggested by the discussion of II-6.
By Equation II-5-1,
—
— tran
r
+ r.„ n
>
(^) tran
>
(
)
2_v
^tran
cos
=
\±
(sin"
^
5
(^n" 1 jjL)
^os
r ° ut
=
-
f
cos
"
out
"
T±
<
S
r out + r ±
2(1
=
.720
-
-720)
^
± +
)
~
2Q
, p.
'3 2b
Theoretically a channel width-to-radius ratio of .326 would
just insure tangency of Mach waves with the convex surface, while allow-
ing an excessively large Prandtl -Meyer acceleration on this surface.
Hence, the authors recommend at least a 20 to 30^ reduction in
'
,W
Accordingly, a design value,
V
arbitrary construction scale, let
(
r o)tran = 1.11,
(r.
)
+
ran
in the central section,
(r
=
=
(xr-)-t- T , QY1
)
M tran =1.0, w tran
)
=
^M
tran
For an
w cen
=
.22,
Since the curvature is to be doubled
«89«
M cen
(r
-22 was selected.
(=r-),
=
-50,
(r out
)
cen
=
.61,
(r^) cen = -39-
The profile is constructed, to arbitrary scale of 1.10"
channel width, in Figure 15.
leading edge consists of a 10
The transition curvature is 15° and the
wedge.
This is slightly less than the
maximum angle consistent with an attached shock.
The edges could prob-
ably be rounded at a slight sacrifice in performance.
From construction,
-51-
=2.88 and
the solidity, c/s
the thickness ratio t/c = .l60.
For a
selected chord of l", thickness at the shoulder chamber is approximately
.020" and the scale of Figure 15 is 5.74/1.
The number of blades required is calculated from
Z
.
2»R
E
J*R_
=
__
2* x 17.8
S
S
c
, 2g blades
=====:
_
X
c
'
The rotor losses may be recalculated more precisely for com-
parison with (Ym) = .66 obtained in IV-3.
R
Figure 2b and Equation (lI-2-l) gives
For t/c = .16 and s/c
^—
(YtJt, = 1
v K
=
H
^
R (Y
M
.6k,
=
= 1.22)
From IV-k,
h 2 = h* =
\|/
.
,
\j/
R
B
=
=
cjti^^
R
The
.868
From Table I-d,
,
From Equation (III-2-)), for
rj
=
.7^8.
961", and
.
J_
96liii29x
=
1.172"
13^5
1910 Rh^ = 1910 x 17-8 x 1.172 = 398OO psi
S cos p = 398OO x
Total weight, mi=3220 \ 2
6.
.64.
the final efficiency estimate is
_,
^ 1-=
h,=h
P
^
2T
2 %
From IV-2, a
=
=
compared to the previous estimate of .871.
the final estimates of
.35,
.20
revised best estimate of rotor loss is (Ym) R
\|r
x .13 = .104.
=
=
.7I+8
x .868 x .6h cos 60° = 827O
3220 (-392)
2
=
^95 lb.
Discussion of Resulting Design
The major numerical results are compiled for comparison with
estimates of IV-2.
52-
.7^8
T)
B
8270
6o°
psi
O
W
35-4 Lb/sec.
P
a
39800 psi
%
=
1.1+4
R
17.8"
%k
=
1.22
:
*s
=
.961
9
*R
=
.868
s
=
7I+8
agrees well with that
The calculated efficiency of
indicated in Figure 1 for U/C Q
=
.378.
.
•392
.64
Hence, a confident prediction
of efficiencies well above .7 offers encouragement for improvement
over the previously demonstrated performances of supersonic turbines.
The Design Point Selection Technique, leading to Figure 11,
proves to be of great assistance in presenting a thumbnail sketch of
available design choices.
The slight variation from the predicted
values of IV-2 was obviously due to the reductions in both
from the approximations assumed.
\|/
R
and
\|r
s
Of particular interest is the observa-
tion that the "invariant" B changed only .24$ from the value based on
the original arbitrary design point.
Perhaps the most questionable area of the problem was that
of estimating rotor incidence.
The theoretical basis is certainly
open to speculation, as discussed in II-6.
.
V.
CONCLUSIONS AND DISCUSSION
This investigation indicates that well designed supersonic -
rotor single-stage impulse turbines are capable of rather impressive
performance.
These machines appear to be an attractive compromise
between simplicity and high efficiency.
Utilizing recent test results,
an effort is made to better understand the losses and flow peculiarities
of supersonic stages.
Many questions are encountered which cannot be
decisively resolved without more tests of the quality previously reported
by the Aeronautical Research Council of Great Britain.
Analysis of supersonic turbine tests indicate that no appreciable losses need be attributed solely to Mach number effects, pro-
vided certain design criteria are followed.
The major rotor blade
design objective is the prevention of shock-induced flow separation.
Profiles have been recently developed by the Aeronautical Research
Council which virtually eliminate separation.
In supersonic flow these
blades have much lower profile losses than the conventional constant-
curvature blades commonly used in subsonic rotors.
Carefully designed
profiles yield considerably greater improvements in total efficiency
than in static efficiency.
Properly chosen rotor blade flare, accurately
reflecting actual losses, has a considerable effect on static efficiency.
Since Mach number level need have no direct effect on blade
losses, an extension of the Ainley-Mathieson loss prediction procedures
to the supersonic case is proposed.
Some straightforward modifications
to the basic procedure are adopted.
The extension to supersonic flow
appears tentatively justified by the meager amount of available test
results
-53-
.
-5U-
The existence of a unique rotor incidence, in supersonic
flow through blades of finite edge thickness, profoundly influences
pressure and velocity distributions throughout the stage.
The
incidence results from continuity requirements which demand a turning,
and probably an increase in velocity, at the rotor entrance.
The
explanation of the phenomenon presented in Reference Ik and based on
Prandtl -Meyer corner flow is shown to be unsatisfactory.
An alterative
explanation is developed based on the assumption of negligible velocity
increase.
More experimental work is required to yield a clear under-
standing of this important phenomenon.
A technique is developed to assist the designer in selecting
the "optimum" design point for a single-stage impulse turbine having
specified RPM, power, and blade root stress.
This method is independ-
ent of Mach number level and the method of estimating blade losses.
It greatly simplifies the decision by promptly resolving design point
selection criteria into a readily digested form.
The preliminary design of a large rocket-turbine is developed
as a numerical illustration of the foregoing procedures.
Blade speeds,
but not necessarily stresses, are typically very high in order to
reduce leaving losses.
The results indicate static efficiencies
exceeding .7 may reasonably be anticipated from well designed turbines
of this type
.
VI.
REFERENCES
1.
Ainley, D. G., and Mathieson, G. C. R.
An Examination of the
Flow and Pressure Losses in Blade Rows of Axial-Flow Turbines.
A.R.C.
R. and M. 2891, 1955-
2.
Johnston,, I. H. , and Dransfield, D. C.
The Test Performance
of Highly Loaded Turbine Stages Designed for High Pressure
Ratio.
A.R.C. R. and M. 3242, 1962.
3.
Moffitt, T. P.
Design and Experimental Investigation of a
Single-Stage Turbine With a Rotor Entering Relative Mach Number
NACA RM E58F20a, 1958.
of 2.
4.
Stewart, W. L. , Whitney, W. J., and Miser, J. W.
Use of Effective
Momentum Thickness in Describing Turbine Rotor-Blade Losses.
NACA RM E56B29, 1956.
5.
Stewart, W. L. , Wong, R. Y. , and Evans, D. G.
Design and Experimental Investigation of Transonic Turbine With Slight Negative
Reaction Across Rotor Hub.
NACA RM E53L29a, 1954.
6.
Investigation
Wong, R. Y., Monroe, D. E., and Wintucky, W. T.
of Effect of Increased Diffusion of Rotor-Blade Suction-Surface
Velocity on Performance of Transonic Turbine.
NACA RM E54F03,
1954.
7.
Whitney, W. J., Monroe, D. E., and Wong, R. Y.
Investigation of
Transonic Turbine Designed for Zero Diffusion of Suction-Surface
Velocity.
NACA RM E54F23, 1954.
8.
Investigation of
Whitney, W. J., Wong, R. Y. , and Monroe, D. E.
a Transonic Turbine Designed for a Maximum Rotor-Blade SuctionNACA RM E54G27, 1954.
Surface Relative Mach Number of 1.57-
9.
Effect of Stator and Rotor
Wong, R. Y., and Monroe, D. E.
NASA MEMO
Aspect Ratio on Transonic -Turbine Performance.
2-11-59E, 1959-
10.
Effect of a
Miser, J. W. , Stewart, W. L. , and Wong, R. Y.
Reduction in Stator Solidity on Performance of a Transonic
Turbine.
NACA RM E55L09a, 1956.
11.
Analytical Investigation of Multistage -Turbine
Stewart, W. L.
Efficiency Characteristics in Terms of Work and Speed Requirements.
NACA RM E57K22b, 1958.
12.
Aero-Thermodynamics and Flow in Turbomachines
Vavra, M. H.
John Wiley and Sons, New York, i960.
-55-
-56-
13.
Stratford, B. S., and Sansome, G. E.
Theory and Tunnel Tests
of Rotor Blades for Supersonic Turbines.
A.R.C. R. and M.
3275, 1962.
Ik.
Stratford, B. S., and Sansome,
sonic Turbine Nozzles.
A.R.C.
G.
R.
The Performance of Superand M. 3273, 1962.
E.
15.
Stewart, ¥. L. , Evans, D. G., Whitney, W. J.
A Method for Determining Design Characteristics for Rocket Turbodrive Applications.
NACA RM E57K25a, 1958.
16.
Edelman, G. M.
The Design, Development, and Testing of TwoDimensional Sharp -Cornered Supersonic Nozzles. Rep. No. 22,
M.I.T., May 1, 19J+8.
17.
Shapiro, A. M.
The Dynamics and Thermodynamics of Compressible
Fluid Flow, Vol. 1, The Ronald Press Company, New York, 1953«
APPENDICES
-57-
APPENDIX A
NOMENCLATURE
A
-
area
B
-
function defined by Equation Ill-ij—11
c
-
chord
-
constant pressure specific heat
Cp
-
pressure coefficient; ratio of actual to isentropic total
pressure
C
-
isentropic jet velocity corresponding to stage pressure
ratio
D
-
function defined by Equation III-A-12
F
-
force
F'
-
function defined by Equation III-U-l4a
F"
-
function defined by Equation III-4-l4b
g
-
gravitational constant
H
-
enthalpy
h
-
blade or annul us height
c
XT
Ah/h-
rotor tip clearance ratio
i
-
rotor
J
-
mechanical equivalent of heat, 778 Btu/lb.
M
-
Mach number
m
-
mas s
N
-
RPM
P
-
pressure
R
-
mean blade radius
R
-
gas constant
incidence angle
-58-
-59-
s
-
blade spacing
T
-
temperature
t
-
blade edge thickness
U
-
mean blade speed
V
-
absolute gas velocity
W
-
gas velocity measured relative to moving rotor
-
mass flow rate
-
specific work
w
-
channel width
Y
=
total inlet pressure - total exit pressure
total exit pressure - static exit pressure
o
W
\fj
^
Ainley _ Mathieson
loss parameter.
Z
-
number of blades
GREEK LETTERS
a
absolute gas angle
-
relative gas angle
/
-
ratio of specific heats
s
«
w 2/v
2
Mw
-
e
Mw 2
5
Mw
2
o
=
u/v2
X
-
secondary loss acceleration parameter of Figure
-
wall angle
-
velocity coefficient; ratio of actual to isentropic
velocity
r\
-
efficiency (static)
a
-
rotor blade centrifugal root stress
p
-
density
\J/
J.
-6o-
t
-
maximum blade thickness
v
-
Pr and tl -Meyer angle
sin"
the Mach angle
angular velocity
-
co
(tt)j
SUPERSCRIPTS AND SUBSCRIPTS
total gas properties referenced to absolute gas velocity
11
-
total relative gas properties, refernoed to relative gas
velocity
-
stator inlet
-
stator exit plane
o
-
downstream of stator, after mixing
^
-
just after rotor leading edge
-
downstream of rotor, after mixing
-
profile A of Section II-5
q
1
k
A
«y '
B
axial component
profile B of Section II-5
cross section
c
central or mid-chord rotor channel
cen
inner or convex rotor profile surface
l
-rg
-
isentropic
yr
-
rotor tip loss
rotor leading edge
LE
J4
m
out
-
mean rotor profile curvature
vector mean angle
outer or concave rotor profile surface
-61-
p
-
profile loss
d
-
rotor
o
-
stator
secondary
J loss
sec
m
-j-
-
total loss, or total head efficiency
-
nozzle throat
tran
tt
-
rotor entrance transition channel
-
corresponding to relative velocity
-
upstream, or inlet, position
downstream, or exit, position
APPENDIX B
DEVELOPMENT OF EQUATION II-3-1
By reference to a T-S diagram one may easily show (e.g.
Appendix B of Ref. h) that
(c ? }
T.
With
Ti
'
H
)
V/V
R ~
'/T
TC
ioo
p4'/ p o'
_
Ps'/Po'^'/To')
777
"1
"
(C
ma y now ^ e expressed in terms of
'
T]m
777 " 1
and Ph'/p
V
Vis
o'
[1
'
)
(V/V)
"
and
"
'
1
Since
\
'
•
the temperature corresponding to an isentropic pressure
decrease top^', there is T^'jg = T Q (P^'/Pq
T
(V/V)
p)s
5
T °'
-
V
V
-
Vis
V V 1711177
^I^V/V)
-
=
]"
there is
7 "1 ?
V/V
= (1
-
nT
)
+ n T (pi+'/Po')
and finally,
(0'P^R
P)r
V/V
(c
p )s
Ki-V
+ti T (V/P
-62-
,
)
(7"1)/
^
7/7_1
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-63-
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j
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Figure 1.
'
C
Comparison of Theoretical Efficiency Estimates for
(Subsonic ) Single Stage Impulse Turbines with Results
of Supersonic Turbine Tests.
-
-6 9 -
NOZZLE BLADES (£ x =0
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0.12
£
0.10
VALUES OF GAS
OUTLET ANGLE
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Figure 2.
Profile Loss Coefficients for Conventional
t/c = .20;
Section Blades at Zero Incidence,
Re = 2 x 10 5 ; M < 0.6. (From Ref l)
.
-70-
0.035
0030
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0.020
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0.010
0.005
0.20
0.10
0.30
(Wx /Wy
0.40
p
[l+UD/O.D.)]
Figure
3.
Secondary Loss Acceleration Parameter.
050
0.60
-71-
0.90
0.80
0.70
0.60
050
0.40
PRESSURE COEFFICIENT, Cp
Figure k.
Pressure Coefficient as Function of M, Y, 7 = l.k.
0.30
-72-
Y
0.6
1.0
098
096
0.94
0.92
030
088
086
VELOCITY COEFFICIENT, f
Figure 5.
Velocity Coefficient as Function of M, Y, 7 = l.k.
0.84
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1.00
t/c
c/s
0.98
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2.5
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0.84
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SUBSC )NIC
CU RVE
0.82
F
tEF.^ \
\
\
(REF. 12)
0.80
25
30
35
40
45
fi,
£'
Figure 6.
50
55
60
65
70
DEGREES
Estimate of Rotor Velocity Coefficient Typical Impulse Blade in Supersonic Flow.
75
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0.98
0.96
H
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0.94
LlI
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8
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0.88
0.86
0.84
0.20
0.30
0.40
FRACTION OF SPAN
Figure 7.
Cascade Loss Results of Reference 13.
0.50
0.60
-75-
0.2
0.4
BLADE SPEED
Figure
8.
0.6
/
0.8
DESIGN SPEED
Comparison of Rotor Incidence From Experiment
and Two Theories - For Turbine of Reference 2.
-76-
STATIONS
A
_vf3
/
_V2
>V
_
jr-s^i
02
Figure 9.
=
^4
Velocity Tri&ngle Nomenclature.
:
-77-
5=. 68
0.175
•
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a-
=
30000
psi
RA
O
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^
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0.145
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0.140
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46
48
50
52
54
/3
Figure 10.
F'
56
58
60
62
64
66
-DEG.
and F" Functions for Turbine Design Problem.
68
-78-
0.56
44
46
48
50
52
54
fyFigure 11.
56
58
60
62
64
DEG.
Constant Stress Efficiency Curves for Turbine
Design Problem.
66
68
-79
U=2330
Figure 12.
Design Velocity Triangle
Mean Radius Conditions.
-
-8o-
AXIAL
DIRECTION
THROAT
n
igure 13.
Sharp-Corner Nozzle Geometry Considerations
(Not to Scale)
-8l-
SCALE
Figure ika.
Design Nozzle Profile.
Figure lVb.
Sharp-Corner Nozzle Cascade
••
1.72/1
-82-
SCALE-- 5.74/1
T/C
=.160
c /s - 2.88
Figure 15.
Design Rotor Profile.
thesB954
Investigation of the influence of supers
3 2768 002 08876 7
DUDLEY KNOX LIBRARY
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