THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS
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Copyright © 1982 by ASME
Liquid Particle Dynamics and Rate
of Evaporation in the Rotating
Field of Centrifugal Compressors
A model is presented which consists of injecting a liquid coolant into the gas or
superheated vapor of centrifugal compressors via slots in the rotating blades. The
aim of the injection is minimization of required compression work. The threedimensional analysis determines the relative velocities and trajectories of the liquid
particles, and their rate of vaporization as a function of the prevailing flow field
and inlet conditions. Inertia, viscous drag, centrifugal and Coriolis forces are all
included. The analysis rests on the assumption of low volumetric liquid/gas ratios
so that the aerodynamics of the flow is not appreciably disturbed. The computerobtained results show that for optimum conditions, and to avoid collision with the
blades, it is desirable that injection occur at the suction side of the blades and close
to the hub, that low rather than high initial particle velocities are preferred, and that
small droplet sizes are required both to avoid blade erosion and to assure the highest
rate of vaporization.
0. Pinkus
Consulting Engineer,
Mechanical Technology Incorporated,
Latham, NY
NOTIfENCLATURL
A
area
CD
D
D
W
work
viscous drag coefficient
c
specific heat
diameter of particle
g
gravtitational constant
particle diameter at point of injection h
enthalpy
E
denotes particle exit
hf
heat of vaporization
F
drag force
k
heat conductivity; ratio of specific heats
I
denotes particle impact with wall
1
height of full water column
I.
impact with ' wall
m
mass; meters
Iz
impact with z wall
p
pressure
N
revolutions for unit time
q
heat
R
radius of particle
r
radial coordinate
0
g
R
(R/Ro)
rl
start of full water column
Ro
radius of particle at point of injection r2
blade outer radius
Re
Reynolds number,
Re o
reference Reynolds Number, (DU R /.^ v ) =
Re o U R R
r
(r/r2)
(D o r 2 c./: v )
s
entropy
S
denotes particle stagnation
t
time
T
absolute temperature
u
circumferential velocity relative to blade
Tsat
U
saturation temperature
u
(u/r ^u)
particle velocity relative to blade v
radial velocity relative to blade
2
UR
particle velocity relative to vapor v
(v/r2w)
U
U/r2
w
axial velocity relative to blade
V
volume
w
(w/r2a)
z
axial coordinate
z
(z/r2)
*
angular distance between blades
2
(Pv/P)(r2/Ro)
Contributed by the Gas Turbine Division of the ASME.
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0.74 k/. gD0.^11 f
g
angular coordinate
a,
7
film heat transfer coefficient
absolute viscosity
m microns, 10 -6 m
kinematic viscosity
density
dimensionless time, t.
residence time of droplet until exit or impact
angle between z axis and streamline
angular velocity
BACKGROUND
Improvement in the performance of compression
cycles is usually achieved by discrete interstage
cooling which is a step-wise approximation to an
isothermal process. Of course, the larger the number of cooling stages, the closer the approximation
to isothermal compression. In actuality, however,
it is impractical and even unprofitable to have a
large number of stages. A more direct method of
achieving isothermal conditions would be a quasicontinuous cooling by the injection of a coolant
from the rotating blades along the path of compression. By full or partial vaporization, the
injected liquid would cool the gas, or superheated
vapor, and keep it below adiabatic compression
temperatures.
The particular attraction of such an injection
system in centrifugal compressors is that no special
equipment is needed for the delivery and compression
of the coolant. As shown in figure 1, the liquid
admitted at atmospheric conditions to the rotating
shaft is pumped by centrifugal force outward along
channels cut in the blades or the other adjacent
surfaces. Due to high speeds of rotation, the liquid
can be raised to considerable pressures so that the
dimensions of the channels and discharge orifices
can be kept extremely small.
The process of liquid injection into the compression path involves at least four areas of investigation, as follows:
a) Liquid Injection. This relates to the
centrifugally generated pressures and flow of the
liquid in terms of the radial spokes of the rotating
system and the orifice restriction at the discharge
end. A good treatment of this subject is given in
Ref. 1. However, this region is of little interest
for our purposes. Given a predetermined rate of
injection, together with slot dimensions, this flow
will determine the required pumping pressure, p
at the inlet to the compression path. o
b) Particle Damics. This relates to the
relative velocities and trajectories of the liquid
particles in the rotating field. The flow map of
these particles is a function of many variables:
the initial conditions, that is, point of injection,
particle size, velocity, etc.; the flow field of
the gas into which the droplets are injected; the
geometry of the blades; and on the angular velocity
of the rotor which imparts (via the rotating fluid)
centrifugal and Coriolis forces on the particles.
c) Rate of Vaporization. Since the aim is to
reduce the temperature by particle evaporation, the
expected rate of vaporization must be determined as
a function of particle dynamics and the state of the
gas. In turn, particle vaporization, which brings
ENLARGED SECTION
OF WATER PATH
i"fll Y't: ny
.0
about a variation in particle size, has a direct
bearing on particle dynamics.
d) Two-1'hase Flow. As mentioned in parts
b) and c), the underlying flow map has an important
bearing on particle trajectory and rate of vaporization. While this flow map, i.e., velocities,
pressures, temperatures, etc., can initially be
taken from an aerothermodynamic analysis applied
to the gas alone, the presence of droplets will
tend to alter this map. Should the volumetric
liquid/gas ratio be very high, a new map of pressures
and velocities must subsequently be obtained on the
basis of a two-phase flow system.
There are a number of referenced works deal.lug with gas-particle flows, but these contain
only elements of the problem under study here.
Perhaps the closest in subject matter and scope is
the work of Ref. 2 which is concerned with the
trajectories of entrapped solid particles in turbine cascades, the emphasis being on minimizing
collision with the stationary and moving surfaces.
The present case, of course, deals instead with
deliberate particle injection, and instead of solids
it deals with liquid particles which, due to evaporation, undergo continuous variation in size.
The particular system studied here deals with water
droplets injected into the field of superheated
steam, and it is confined to cases where the
volumetric (droplet/steam) ratios remain sufficiently low so as not to alter appreciably the aerodynamics of compression. The emphasis here is on
droplet dynamics aimed at providing guidelines for
the optimum positioning, sizing, and metering of
the required injection system.
2
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The viscous drag force on a spherical particle
is (3)
T:iii AT IONS FOR P ARTICI,P'NAM1CS
Particle Flow in a Rotating Pluid
F - 67r bRU
To account for the spatial configuration of a
centrifugal compressor, t' e equations will he
written in three dimensions of a cylindrical coordinate system. 'Ihe coordinates are fixed in the
rotating system, with z along the axis of rotation
and ('Onstn" 0 representing a mcrrdiona1 plane as
shown to Iigure 2. All velocities are relative to
u
'19s
PU2 1 „3
2
600 OF
Co the rotating blades ant tiii other bounding; surfaces.
Iioi',e 'or , as is c); or from Figure 2, the
1
12
(3)
pL'R RN, Re R
0 \ r 2 12 k r 2 )
Now the Reynolds no. is of the order 1 to 100
'l_ereas R, the particle radius, is of the order of
1 to 100 microns, yieldi zg_ for the (P IF ) ratio
radial and axial ve loo iti' are also absolute velocities, whereas the rel1t 0' circumferential velocity
u 0Ifferes from the ahoolute circumferential velocit.
ho the quantity r
values anytivwhere from i0
In this scheme t''ic three. relevant momemtum
equations ('-e
h^
,
2 to 10 6 .
P P
,
2
Fr d2`
(
: '
(la)
dt
The viscous force F can in general be represented
I = H. 2 'AUR
1' ere the coefficient C D is a function of the Reynolds
cl t
111.
d`r
pA' — =
dt2
+ pVr (w +
r
d0 2
cis
)
Re =
(l)
the relative velocity_ between the vapor and
and U
particle, is given by
V
cV d_z - I'z
dt 2
Above,l' is the viscous drag force, the last term
in E ioLion ii is the Coriol-is force and the last term
In Equation lb is the centrifugal force, both imparted
on the particle by the dri- of the rotating fluid. No
pressure forces are inclu_'d because these are small
by comparison, as can be seen from the following:
/
66 ) 2
(uv
I l
dr = dt = of r
\
2
so that the pressure Force acting on a particle is
2
r
R.nR^
(2)
2
^`^
V .Vr
/
rc^ /
VABS
VBABS
dr 2
dt)
+ (w v
dz 2 1/2
dt )
(4)
(5a)
R ) F
I r = ABU
t
(5b)
(URz/L R ) 1'
(5c)
F Normalizing all velocities by r 2 :.; all distances
hi r 2 ; and writing
R
v ) (?
) LR
r2 2 = 3C (__
t=
,rw
s
-
izrZ
m
+ (v v
The component drag forces in Equations 1 thus
become
RE
STREAMLINE
\ I ^STREANLINE
r
r dir
L 2 + U 2+U 2 1/2
Rz J
hr
L ^ii
F = (h r ; /V R )
(0 order of)
R 1 = Gp.A =
Ulll:
(lb)
we obtain the normalized form of Equations I
VrABS
i2
- ( 2 = _H
dT
R
(uv u)
2 (1 + d^
)v(6a)
11m ABS
VZABS
d2r - 0U R C D (v v - v) + r(1 + 00) 2(6b)
R
d:
d 2z
du `
i
=
3LRC R
(wV
- w)
(6c)
R
where
U 11 = [in - u) 2 + (v v - v) 2 + (,v _ w)21112
r2
Q
=
2
(ii) (__)
R
Fig. 2 Coordinate System and Velocities
0
3
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Above, IZ = (R/R o ) where R is the particle size
at point of injection, and C D is a function of the
U RD
Reynolds number (-).
All subscripts v refer to
1f
v
the superheated vapor, whereas the unsubscripted
quantities refer to the droplets. The trajectory
coordinates 0, r, z and velocities u = r
, v =
di
dr, w = dz are the unknowns to be solved for all
di dT
relative to the rotating blades. For purposes of
numerical evaluation the dependence of C D on Re was
where
0.74k
2
g 0
o fg
If the variation in T v is not very large then
is approximately constant, with only Re and °:T
varying. Tf AT varies over a wide range then certainly
hf g , and to some degree also k, would vary and Equation (11) should more properly be written as
dR
o
z
CD=
0.05 exp 6.193
% 0.398
for 0.6<Re<1500 (7)
for Re >1500
Evaporations of Liquid Particle
In all the previous equations the size of the
droplet, R, is a variable and a function of time.
The evaporation of the particle is in fact the main
objective of this injection scheme. The variation
in droplet size can be obtained from a heat balance
between the heat required for evaporation and the
rate of heat transfer occurring between the superheated vapor and droplet. The amount of heat required to induce a given rate of vaporization is
hli fg
dm
dV
(8)
= (g') h
g
fg dt
Above, it is assumed that the liquid injected
into the vapor is at a temperature corresponding to
the saturation temperature of the local pressure,
and the only heat required is the vaporization
enthalpy hf g . If the liquid is below that temperature, it would first have to be heated requiring
dca _d T
cm dt
dt
with T going from T o to l s , t', ) belnc the inlet teliserature. This quantity, however, is bound to be small
as compared with that given by Equation (3).
The heat required for vaporization has to be
transferred from the superheated vapor to the liquid
particle by conduction, or
dt = ?.A (T v -T s )
(9)
Writing A = 4R, V = 4 and equating the two
rates of heat transfer, we o^tain
dR =
dt
(T vT s )
g,,h fg
Mow the film coefficient ' can be related to
the heat conductivity between the vapor and water
droplet, via (4)
k
D = 0.37 Re
0.6
dR
= - Re 0.6 [T
1
fg
h (;
r, z).
(12)
Do -
The above differential equation for R (r) must
now be added to the set of Equations (6) for a complete solution of the problem of evaporating droplets
Method of Solution
As derived in the previous section, we have a
set of four differential equations to solve, namely
- (d 2 )
: U R C D
t2rUR
d^2
d z'UR
R
CD
- - d
d; dr
- ) - 2(1 - d ) d .
- (uv r
(13a)
13b
(v - dr ) + r (1 + dc) 2(
v
)
d
d^
^
D (w v - dz )
(13c)
RaR = -i Re0.6 [Tv - T y ]
(14)
As indicated above, U , C O3 and Re are functions
since they all depend on partic
of
le velocity. The
solution sought is of course the position of the
particle P(,, r, z), and its radius R as a function
of time, which would, via its spatial derivatives,
also provide the particle velocities u, v, and w
relative to the blade. The history of particle flow,
in particular its trajectory and rate of vaporization,
can then be mapped in detail.
Equations (13) and (14) contain as unknowns u
v , w and T . These are the characteristics of the
superheated vapor which represent input data to be
taken either from equations, tables, graphs or any
other available separate solution. In most cases the
presence of droplets should not influence the underlying flow so as to affect droplet dynamics too much,
even though they may modify to some degree the vapor
flow itself. however, should the resulting two-phase
flow modify the original flow in a substantial way,
some iteration between the particle equations and
the pure vapor solutions may be required.
Equations (13) and (14) constitute an initial
value problem and we need starting conditions in
order to proceed. These conditions are supplied by
the mode of injection, namely.
= 0
The above inserted into Equation (10) and
properly normalized, yields
R
Is
(T moving in a rotating high-speed fluid.
which is a close representation of C as given in
Ref. 3.
°
dt
0.74
^
Re-0.1496-
0.6
_
€c'D =
put in equation form, as follows
24 for Re <- 0.6
/{Re
riP.c
0.74
a_
Point of Injection o,
r , z
0
Velocity of Injection u , v , w
v
- T]
s
(11)
0
0
Particle Size
R
Relevant Constants
Re ,
0
0
= l
0
0
4
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With Re = Re o , U R R, and C as given by Equation
(7) evaluated at each point in the mesh, the solution
can then proceed in such time intervals / as to provide solutions of sufficient accuracy. The equations
were solved on a computer using a fourth order RungeKutta method. For particles in the range of 50-100um,
time intervals of L. = 0.01 gave sufficiently accurate results. However, as particle sizes of the
order of 1 pm were approached, a finer time mesh was
required in which case 3t's of the order of 0.001
were used.
PARAMETRIC SOLUTIONS
Two sets of solutions will be presented here. The
first serves as a brief introductory survey of the
basic behavior of droplet flow in a rotating system.
The second and main series will deal with an actual
compressor whose characteristics are described
in detail in Ltef. 5 along with its aerothermo-
dvnamic flow conditions. All cases deal. ith
water being injected into centrifugally compressed
superheated steam.
Preliminary Runs
In order to bring out some of the basic features
of droplet behavior, a few preliminary runs were made under the following conditions:
• u = v =w
v
v
v
• v
v
Values of the velocities, pressures, and temperature throughout the flow field are documented in
some detail in Ref. 5. In summary fashion, their
.Hain characteristics are as follows:
• The static pressure rise in the compressor
is 60%, yielding maximum superheat temperatures of the order of 38 C (50 ° F).
• Most of the compression occurs in the outer
reaches of the blade, so that in the present context the inner half of the blade is
of little interest.
1.0
PARTICLE EXIT
0.9
R
r
/
uo = 0.2
0.8
POINT OF
INJECTION
U, = 0.5
= 0
= 0.5; u =w
v
v
= 0
The first represented a case where the gas
moves integrally with the blades. The results show
that the drag imposed on the droplet tends to keep
it at a nearly constant radius, in phase with the gas.
Thus, the Coriolis and centrifugal forces seem to
play a small part as compared to the viscous drag, at
least for the 10 ',.m particle considered in this
example. The second set of conditions represents a
situation of constant gas velocity in the radial
direction only. Small particles are completely
dominated by the viscous drag; however 30 _.m
particles have sufficient inertia to remain in the
flow field until they are caught up and struck by
the upstream blade.
The rate of change in particle diameter due to
evaporation, as shown in Figure 3, seems to go
through three stages. It is high at the beginning of
injection, probably because at that point its relative
velocity is high; it then tapers off, to increase
again toward the end of the path. The latter increase is due to the decrease of particle size,
which, as will be seen later on, yields higher rates
of evaporation. Since the variation in mass is proportional to (1 - k) 3 , the above variation in slope
will be even more pronounced for the particle's
rate of change of mass.
y= 1.65 • 10
Do = 10µm
0.1
vV
3
=0.5 7 =WV= Vo =WO= 0
0.1
0.9
0.3
0.2
Fig. 3 Rate of Droplet Vaporization
A08n'
p 362°F
i
333.1°F,
i i
i
i
i
i
t
/
k I!
S = CONST
-
250°F
30 psia
213°F
—_15 psia
Centrifugal Compressor
Compressor Characteristics. The required
performance of the eentrifupal compressor under consideration here is mapped out in Figure 4. Other
relevant data are as follows:
Mass of Vapor Flow - 0.9 kg/sec (2 lb/sec)
Angular Speed - 3142 cad/sec (30,000 rpm)
Maximum Blade Radius - r = 0.14 m (0.455 ft)
Tip Speed - 437 m/sec (133 ft/sec)
Lumber of Blades - 17 or . 20°
SATURATION LINE
h
S
— COMPRESSION WITH 80 010 ADIABATIC EFFICIENCY
• • • • ADIABATIC COMPRESSION
0
10 — COMPRESSION WITH CONTINUOUS DESUPERHEAT
------- INTERCOOLING AT p = CONST.
Fig. 4 Path of Continuous Desuperheat
5
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In the outer half of the blade, the circumferential vapor velocities vanish and both
the radial and axial components reach values
equal to blade tip velocity.
Velocities are higher on the suction side
than on the pressure side; and higher near
the shroud than near the hub.
The circumferential velocities (u ) are all
negative; that is, they are all irected
toward the pressure side of the blade. They
decay rapidly as the straight radial portion
of the passage is approached, so that they
vanish at r > 0.55. The radial velocities
(v v ), on the other hand, are low in the
curved portion and approach a maximum at
exiting chances. Thus, as seen in Figure 8, by
reducing the injection velocity to nearly zero, and
placing the injection point at z o = 0.55 even th
D o = 50 pm particles succeed in exiting. For th
large particles collision with the pressure side of
the blade occurs whenever u o > 0.55
1.0
N
-
zo =0.475
D o = 50^ m
... ...uo=0.2
— u o = 0.5
0.9
about r = 0.8.
Particle Trajectory and Evaporation. Solutions
were obtained for particle injection at three
different planes: on the pressure side with injection in thedirection of rotation (e = 0); at the
suction side with injection against the rotation
(0 = 20 ° ); and from the hub surface (z = 0.55).
Several injection radii will be examined. The
advantage of a low radius is that it provides the
droplet with a longer trajectory in the channel and
thus higher evaporation; however, it also increases
the likelihood of collision with the walls, which is
to be avoided. For the higher radius the conditions
are reversed. Then, too, at any given radius the
particle can be injected closer to the hub or to the
shroud, that is at various values of z
Aside from the location of injection point, the
next parameters of importance are the injection velocity and direction. Two velocities, 20% and 50%
of tip velocity, were chosen which correspond to the
pressure differentials that the centrifugal forces
are likely to generate in the water column. Finally,
three particles sizes were examined, 2, 10 and 50
microns in diameter.
A summary of the more relevant results is contained in Figures 5 through 8. The particle dynamics
tied to the prevailing vapor velocities suggests the
following:
Injection from Pressure Side
Physically this mode of injection means that
the particle has little opportunity to move
away from the plane of injection. As a
result, small particles whose inertia are low
are, in most cases, struck by the pressure
side of the blade. Thus no results are offered for particle sizes of 2 ;gym and 10 cm.
Increasing the u velocity from 0.2 to 0.5
did not seem to make much difference. Likewise, runs in which the angle of injection
was varied from 90 ° to 45 ° , i.e., with v o =
± u 0 , did not improve the exiting chances,
and in some cases made the situation worse.
Injection from Suction Side
Injection against rotation, i.e., with the
injection plane moving away from the particle,
provides satisfactory trajectories, enabling
the particles in most cases to exit from the
chamber. Here, seen from Figure 6, lighter
particles are pre ferred, as with the D o =
50 pm particles the vapor does not succeed
in sweeping them out the channel before they
are struck by the approaching upstream blade.
The very light particles are swept out of the
channel along a path nearly parallel and
very close to the injection plane (0 0 = 20 ° ).
Here, in contrast to the situation with 6 = 0,
the lower the injection velocity the better the
0.8
0.7
0.1
10
12
14
16
18
9 , Degrees
Fig. 5 Particle Trajectories for Injection Frol 71 7
Pressure Side
Injection from the hub Wall
Here injection near the pressure side of the
blade (0 0 = 6-1/2 ° ) results in unsatisfactory
trajectories due to the stagnant flow pockets
prevailing there; injection near the suction
side (6 0 = 13 ° ) improves the chances of
particle exit. As shown in Figure 8, the
small particles tend to be lodged in stagnant pockets of vapor and remain there.
A radius of injection higher than r o = 0.5
and particle sizes larger than D o = 10 um
are required to assure particle exit,
provided its injection is near the suction
side of the blade.
Particle Vaporization
Overall levels of vaporization are low for
all cases, less than 5%. This is due primarily to the low levels of AT available,
('L 50 ° F). The smaller particles yield
higher percentage levels of vaporization.
Even though the actual mass of vaporization
per particle is a product of the percentage
Am and mo , still for a fixed rate of water
injection the number of smaller particles
present will be correspondingly higher.
Consequently, the percentages represent the
total mass of vaporization. For suction
6
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i
1.0
J o = -0.2
zo = 0.475
0.9
T
D ----m 2
i
1
i
10
50
IIiLii\
iIiiiii
e o =6.6°
0
y
•••••CIRCUMFERENTIAL FLOW (9)
0.52
0.8
W
5
.1..
12
10
8
6
4
14
16
r
10(To =0.8)
10
^
1o(ro =0.6)
0.7
1
it
20
18
q/61
9, DEGREES
i
NUMBERS REFER TO D o INµm
10
4
0.i
0.E
zo = 0.55
—RADIAL FLOW (r)
6
0.8
0
IF
Fig. 6 Trajectories for Different Particle Sizes
0
2
3
4
r
75
0.5
6
80776.1
Fig. 8 Time History of Particles for Injection
from Hub Wall
side injection the highest values are of the
order of 4%; for hub wall injection (excluding the stagnant particle data which are meaningless) the highest values are of the order of
5%.
1.a
Do =50µm
To = 0.525
-...._U
-0.5
—O n --0.2
----u0 = -0.05
.
0.1
0.1
S...
F
v
^^
0.
S. -. ..
0.1
5-.-..--
0.50
.5-.-
18
20
16
10
12
14
urn,
e, DEGREES
Fig. 7 Effect of Varying u o on Particle Trajectory
2
4
6
8
Op timum Injection Syste m
Based on the results of the parametric study
the design of the injection system for the prototype
compressor could then proceed subject to the follow-
inm considerations:
• The lower the injection point in the compression passage, the longer the droplet
residence time, and thus better vaporization.
However, low injection points raise the risk
of droplet collision with, and erosion of,
the passage walls.
• A large number and dispersion of orifices
are desirable in order to maximize the
spatial distribution of the droplet sprays
within the compression passage. But here
again, good dispersion of the droplet spray
increases the risks of wall impingement; in
addition, many orifices require a large
number of radial channels in the blade, which
the blade may not be able to accommodate.
• Higher injection velocities yield better
atomization, higher evaporation rates, ana
better chances of traversing stagnant steam
pockets; however, they also increase chances
of collision and require more input power
for centrifugal pressurization.
Based on the above, five injection ports were
selected at the locations and initial conditions as
shown in Figure 9. This arrangement yields the following flow characteristics:
7
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• As seen in Figure 10, the heaviest particles
in orifices 1 and 2 just impinge on the upper
tip of the pressure side of the blade. This
was done intentionally in order to extend the
droplet fan over the widest possible area.
For all other cases the droplets succeed in
emerging from the passage without colliding
with any of the surfaces.
• The smallest particles, D o < 2 pm, are swept
out of the channel along a path nearly
parallel and close to the injection surface.
The heaviest particles, due to their larger
inertia, penetrate the deepest in the blade
passage.
_
• For trajectories of the particle in the z
plane, i.e., in plane at the blade (the
previous plots were in the plane of rotation
Of), there is no variation in the axial excursion between the large and small particles.
1.I
0A
0.
0.'
12
SU
ORIFICES NOS. 1
#1-4=0.5
2-!° 0.525
0 0 = 0.2
NOS. DENOTE D o , µm
0.i
#1
0.'
1
2
8
6
4
10
12
14
1e
lb
zU
8, DEGREES
Fig. 10 Particle Trajectories for Orifices
No. 1 and No. 2
is of some significance from the standpoint of
erosion as the smaller particles, even where
they impinge on the blade, would do so carrying
negligible momentum and thus cause no damage.
This once again emphasizes the great importance
of obtaining small particle sizes.
Effect of Injection Angle
We shall consider the effects of varying the
injection angle, while keeping the total velocity
constant, i.e.,
U = u 2 +v +w 2 = 0.2
0
0
0
0
In discussing the results for various injection
angles we shall maintain the following convention
W
U.DZ3 U.MU
8.4
0.3
8.2
1
Pig. 9 Location of Orifices
8.1
• ^ e = 0 is the injection normal to the suction
plane or U. = - 0.2
0
em35-1
•
= 45 ° refers to a velocity vector tilted in
the re plane 45 ° toward the exit, i.e.,
toward positive r
_ - 45 ° is the same as above but with a tilt
•
• The residence time of the particles decreases
with an increase in injection radius, r, as
expected. Also, the heavier particles have a
longer residence period, as can be surmised
from the r = constant plots in Figures 10 and
11. The residence times range from about
T = 0.5 to T = 1.3 which for w = 3152 rad/sec
yields
0.16 x l0
< t < 0.37 x l0
• Near the exit from the path, the small particles, 2 pm < D < 10 pm, have a nearly zero
relative circumferential velocity (u f ti 0);
whereas the particles in the 50 pm range have
relative velocities of the order of 30f. This
•
away from the exit (- r)
z
= 45 ° refers to a velocity vector tilted in
the (8z) plane 45 ° toward the hub (+ z)
• 4 = - 45 ° is the same as above with a tilt
ztoward the shroud (- z)
For the small particles D o = 2 pm, varying the
injection angle has no effect whatsoever. Also the
vaporization rates are little affected by varying the
orientation of the velocity vector. Furthermore, as
shown in Figure 11 the effect of varying 4 6 on the
droplet trajectories are also minor. Thus the only
area of possible interest is that pertaining to the
effect of varying the angle 4 on the behavior of
the larger droplets.
8
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The effects of orienting the flow 45 ° either
towards the hub or towards the shroud are shown in
Figure 12. First, it will be noted that for either
plus or minus values of 4 the trajectories of the
particles below 10 lim are z very close to that of
normal injection. However, there are important
effects on the trajectories of the heavier particles
For ¢ = + 45 ° the result is that the larger particles collide with the wall of the hub. However,
directing the flow toward the shroud, 4 =
produces in both orifices #1 and 03 a significant excursion away from the 4 = 0 solution.
Comparing the results of Figures 11 and 12
with the results for normal injection, the conclusion is reached that orienting the water jet towards
the shroud (4 = 45 ° ) can at times be extremely useful for the following reasons:
• Whereas normal injection produces sheetlike
sprays in the rz plane, that is, particles
of various sizes follow the same trajectory,
orientation towards the shroud produces a
wide and satisfactory fanlike spray also in
the plane of the blade. This, of course, is
of importance in that such a fanlike distribution inundates the maximum space with
liquid droplets and maximizes vaporization.
• Whereas in orifice #1 normal injection led
to collision with the pressure side of the
blade for Do = 50 um, an angle of 4 z = - 45 °
eliminates that danger yielding a trajectory
in which even the heaviest particles exit
the chamber with ease.
1.0
10
10 :
0.95
ALL
0.90
DD
0.85
0.80
50
0.75
50
S HROUD
0.70
0.65
ORIFICE #3
U1 D ; = 0.1
0.60
HUB
„^„^ 0 Z = 0°
0.55
NOS REFER TO D D IN µ m
z
0.50
To give a comprehensive view of the spatial
domain that the sprays from the five orifices are
expected to cover, the trajectories from all the
orifices were superimposed on each other. Figure
13 gives the expected domain of various particle
spectrums in the plant of rotation (re); while
Figure 14 gives the expected domains in a plane
passing through the axis of rotation (zO). Finally,
Figure 15 gives the expected rates of droplet vaporization as a function of size for the various
orifices.
0.3
0.425
0.4150.50
0.45
0.525
0.55
0.575
'" in
Fig. 12 Effect of Varying 4 on Particle Trajectory,
z
Orifice 03
in - ., ,2
2 > Oo
OF
T=0.5,
50
I
Q2
ID
I'll'
I
is
:E #5
ICES
0844
¢B = 45 °
;E #2
I-
ORIFICE #3
8
Y s=0.5
O o h = 0.2
14
16
......... o° 2µm
_„„_^ O D , 10 fLmDEMARCATION LINES
09=45
.
12
18
20
:E #1
8, DEGREES
NOS. DENOTE 0 ,µm
•
10
III
0, DEGREES
IL
i.
10
If
Fig. 11 Particle Trajectory for Orifice No. 3
^^0 0 =50µm
[ffi7
Fit. 13 Domains of Droplet Sizes
20
w=
9
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'.0
Injection System
• Orifices. A large number should be used, so
that their diameter is as small as possible
while their length/diameter ratio is high. The
orifices, barring other considerations, should
be located at points of high gas velocity.
• State of the Injected Coolant. Liquids of low
viscosity and at the highest possible temperature should be used. It should have high
enthalpy of vaporization and low surface tension.
The lowest possible ratio of (liquid/gas) flows
should be aimed at.
0.95
0.90
0.85
0.80
Mode of Injection
• Injection from the pressure side nearly always
leads to collision with the injection surface
and is not recommended.
• Injection from the hub wall leads to particle
arrest due to stagnant velocity pockets and
is also not recommended.
• Injection from the suction side at r o is the
most satisfactory mode.
• Angles of injection slanted towards the shroud
are useful in widening the spray in the meridional plane and in reducing the chances of
collision for larger drops.
• High injection velocities do not change substantially the particle trajectory for large
particles; lower injection velocities are, in
fact, preferable.
• Small size particles, below 10 pm in diameter,
increase the chances of particle exit.
0.15
0.10
0.65
0.60
0.55
U.U3
0.5UU
0.415
0.450
0.425
u.oau
Evaporation and Particle Size
• Droplet size is probably the most crucial parameter in a liquid injection system. Smaller
particles have the highest rates of vaporization, whereas large drops are also more likely
to collide with walls.
• For a AT = 38 ° C (50 ° F) and D = 2 pm the maximum values obtained (for par?icles with assured
exit from the channel) were 4% of the original
mass; for AT = 380 ° C (500 ° F) evaporation of
the order of 20% to 30% of original mass can
be realized.
811183
Pig. 14 Domain of Droplet Flow in Blade Plane
ORIFICE #2
3.5
#4
ACKNO WEED GhMENTS
3.0
Researcif performed under Subcontract No. 36X24713C with Dechanical Technology Incorporated under
Union Carbide Corporation Contract W-7405-eng-26 with
the U. S. Department of Energy.
Credit is also due to iir. Andrew Tuzinkiewicz of
MTI who provided the input material on the aerodynamics and thermodynamics of the steam flow, and to
Mr. Herman Leibowitz of MI'I for his interest and
support in carrying out the program.
#1
#5
2.5
#3
0
2.0
1.5
1.
1.0
2.
0.5
3.
4.
1
2
5
10
ZU
aU
PARTICLE SIZE, µm
100
5.
Dakin, J.T., "Viscous Liquid Films in Nouradial
Rotating Tubes", GE Report No. 77CR133.
Hussein, M.F. and Tabakoff, W., "Computation and
Plotting of solid Particle Flow in Rotating
Cascades", Computer and Fluids, Vol. 2 No. 1-A,
pp. 1-55, Pergamon Press, 1974.
Schlichting, II., "Boundary Layer Theory", McGrawHill, 1955.
McAdams, W.H., "Heat Transmission", 3rd Edition,
McGraw-Hill, 1954.
Pinkus, 0., "Particle Dynamics in a Rotating HighSpeed Vapor", MTI Report No. 80TR19, March 1980.
Fig. 15 Rate of Droplet Evaporation
10
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