Raabe J. Hydro power - the design, use, and function of hydromechanical, hydraulic, and electrical еquipment
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The distance
It,
is always positive and depends on the ground rnainly as a lightning
protection. The length
hi
of the insulator depends mainly on the maximum voltage
Ir,,,,
between
its
ends, on climatic conditions (air moistness, rainfall, rime) and
on
the air
and is much larger than the spark-over distance under the conditions above
rnentioncd. The order of magnitude is very roughly given as (Fig. 4.7.1.2 a)
For
DC
transmission, U,,, equals the nominal voltage of the insulator against carth. For
3-phase
AC
transmission, U,,,,, equals the crest star voltage. Since the norginal voltage
then is defined by the effective delta voltage U,,, U,,,, becomes
U,,,,
=
J2/3
U,,.
Example: 3-phase
AC
transmission by Uej
=
735 kV, e.g. Churchill Falls-Montreal.
Then
Iti(nl)
=
7,2m! Naturally the latter is a very rough approximation.
Differentiating the relation (4.7-7) between sagging
yo
and admissible stress
o,,,
after
A,,
=
o,,/(ge), and keeping
Q
and the distance
'a'
of adjacent towers constant,
gives the following relation between stress increment
da,,
and increment of sagging
dy,
Hence
a ncgative
sagging
of the cable, caused by
its
shrinkage under low
temperatures
in
winter
creates
additional stress. Further loads
in
winter
by snow, ice and rime also have
to
be
accounted
for,
when
deciding
the
distance
and
height
of
the
towers. Moreover
the
fixing of
the
distance
'a'
has
to
account for
a
given
length
of
a
cable in consequence of
its
manufacturing method. This
finite
length
of
ally
conductor
requires
"dead end towers" in certain distances from
each
other. Also
additional
strcsscs
have
to be accounted for in
the
case of a failure of
the
cable, or
a
sudden removal
of
its
ice load,
see
[4.44]
and
[4.45].,
The cost of the towers follo\vs from the voltage Uej and the distance
'a'
of adjacent towers:
For
Uel
=
20, 110, 220,
380
kV
cost
=
7'7;
32,5;
50; 60
103US$/km
at
a
=
140; 300;
350;
400 m.
4.7.5.
The
optimum
distance
of
adjacent
towers
From (4.7-8)
it
is recognized for the general case of suspension by an inclinded insulator
(Fig. 4.7.1.1)
h,,
=
a2/(8
A,,)
+
b
+
hi cos
o!
+
h,,
(4.7
-
10)
where
a
is the angle the insulator makes with the vertical (Fig. 4.7.1.2 b). This angle
shortens on the one hand the height of the tower, and on the other hand it
prevents lateral
motions of the line under wind force. Simplifying, a design parameter
5
given by
lhi
=
hi
cos
o!
+
h,
+
b
may be introduced, taking into account that
b
is porportional to
the insulator length and so also generally
11,.
According to (4.7-9) terms may also be reduced to the voltage by h,cosa
+
h,
+
h
-
-
k,
U,,,,, where
k,
is a design-linked parameter and
U,,,
the crest voltage in
V.
The first
cost
of
a tower may be expressed by the relation
.
K3P
=
k5
jln,',,,
(4.7- 1 1)
where
k,
and
n
are empirical parameters,
n
is in the range of
n
=
2
to
3.
For a very long
line
of
length
L,
the tower number
i,
now assumed as a continuously varying value follows
from
i
=
Llo.
Thc
resulting cost of a line is
K,
,,,
=
i
K,,.
Inserting horc
i
=
Llcr,
K3,
from (4.7-- 11) and
It,,,,,,
from (4.7-
lo),
the resulting cost
as
function of thc distance
'a'
becomes
K3
r6-s
=
'5
L
Urnox
a
-
l/n
+
a2
-
I/)#
/(8 Aod)ln
-
(4.7-
12)
Under the assumption that the
(11
-
1)st power of the square bracket does not clisappear
and putting tlK,,,,Jdu
=
0,
a distance
crop
is obtained, which mit~imizcs the first cost of
the transmission line as
It
is seen from this that the optimum distance between adjacent towers increases with the
square root of
U,,,,
A,,
.
This result does not consider separately the cost of the insulators.
A somewhat more refined expression may be obtained with
Ki as the first cost of
insulators per tower for the resulting first cost of the line (always without cable)
K;,,
=
K,
Lln
+
K,,,,.
From this by putting dK;,,,/tla
=
0,
an improved version for the
optimum distance
a,,
is found
This relation
can
be solved by trial and error, starting from (4.7-13) Example:
U,,,
=
lo6 V;
./lo,
=
3320 m (aluminum, steel),
k,
=
36
-
loF6
m/V;
n
=
2,35. Hence
from (4.7
-
13):
a,,
=
505 m.
4.7.6.
Future
developments
There is an increasinz tendency to use high voltage direct current (HVDC) transmission,
i:l the
cae
of distances of about 700 km and more, between power station and the next
consumer centre (4.36
to
4.391.
Contrary to the common three phase
AC
transmission with its three cables, DC transmis-
sion
needs only two cables. The cost of a high voltage transmission line depe~~ds first on
the
cod of
he
towers resulting rrom their area and height. Both factors are influenced
favoi~rably by shortening the number and length of insulators. Each additional 100
kV
in
effective 3-phase AC delta voltage
UeJ
needs roughly
1
m
more in insulator length.
Obviously the latter results
from a crest voltage. Since this is 2'12 times the effective
voltage,
30%
of the insulator length (usually corresponding to an additional height of
tower) is saved by
DC
transmission of the same effective voltage of one pole to earth.
Moreover in
DC
transmission, one conductor may be charged positively to earth and the
other of the same voltage negatively
to earth. Thus for a given insulator length, the
effective voltage in
DC
transmission is
2
2'12 times that of a single phase
AC
transmis-
sion.
At an admissible relative power loss
hi
along the line, the losses
d
P,,
APAF
of the
DC
and
AC
line, respectively, and the powers
P,,,
PAC
transmitted along these I~nes, respec-
tively, are related by
f
11;
=
A
PAC/
PAC
=
A
PDc/
P,,
.
(4.7-
16)
Expressing the loss in the
AC
System by 3
lzf
A,
R,
in the
DC
System by
2
If,
R
(both,
systems having the same cross sectional areas of their cables and hence the same
ohmi2
H
of thenl), and introducing the power transmitted in the
AC
systcrn
ris
3
I,,
Url
,,,
and
in the
DC
Systcm as
2
I,,
U,,, tilc above relation is reduccd to
where
I,JAc
is the effective current in each phase of the
AC
line,
I,,
the current in the
DC
line, Uef the effective star voltage in each pllase of the
AC
line and
U,,
the volta_re
of each pole to tlie earth in the
DC
line. Applying the same length of insulators in both
the systems, the voltages are related by
Putting this in (4.7-17) gives
Hence the ratio of the power transmitted in both the lines
Thus the
DC
line transmits
33
%
more power than the
AC
line. If the conductors of both
the lines have the same length and the cables of both the
lincs have the samc cross
sectional area (as expressed by
R
=
const), the required mass of material for conductors
and insulators of the
DC
line is
66%
of that used for the
AC
line.
In favour of the
AC
line, the
DC
line requires the additional cost of the conysrter siati~ns
at the ends of the line, also the cost of the measures required to eliminate
the
hig1,c.r
harmonics in the
DC
line after conversion
of
the
DC
into
AC.
A
further reduction of cob!
by
DC
results from the more uniform distribution of the current over the
crnss
secticn
of the conductor and from the absence of any loss due to skin effect.
The nccessily to transport greater amounts of electric energy and simultaneously, the ever increasing
difficulties
of obtaining the rights-of-way for the transmission line, make
it
necessary, to in1,estifate
new possibilities of elcctric power
transmission.
In the following only methods are considered, wl;ich
do not convert the electric energy into other forms of energy, e.g. chernical energy, cspcciallj
hydrogen by electrolysis. As for the latter, see the respective chapter
of
Sirneon's book on hydro
power
[1.55
a].
Thc change over to even highzr ratcd voltages does not ideally solve this problem. as this ~cquires
larger cross sectional areas and land areas for the line (see Fig.
4.7.2).
In the following the transmis-
sion by electric means only will be considered.
Transmission by
compressed-gas-inzula~ed
cables wou1.l overcome these diflicu!ties, since at a ivc~
voltase,
4
to
6
times rnorc energy can be transported than with an overhead linc and the correspond-
ing land requirements are only one tenth
[4.40].
Also the preservation of landscape is helped by underground cables. Single phase gas-insulated
cables
nlith a full sheath current have
a
surge impedance of approximately
69
R
[4:10],
frorii which
e.g.
4500
hlW
eurgc impedance load can be ottained at
550
kV.
The
surge impedance load without
sbcath current would be
60%
and for
3
phases
SO%
of the above mentioned values.
One pcssibility of technical realization is t11e so called "wcllmantel" as a sheath for conventional
cables. It
combincs stability with good bending characteristics. Cables with an outer diamcter
up
to
250
mm can be wound on reels up to a length of
200
m. and they still can be transported by available
Qrriers. The "wellmantel" has been proved as
a
constr~~ction elt-me~t for thermally insulttted tubes.
Different expansions of the tubes need
no balancing elen~~,nts. Gue to the corrugation there are only
small
longitudinhl forces, which can be compensated easily.
Thus the therrnal insulation of cryogenic and
superconductive
cables is possible. Such cables will be
applied
in future to eliminate the losscs or to reduce the cross sectional arcn of
a
cable for a certain
Fig.
4.7.2.
rransrni:sion tower cear Quebec of high voltage
735
AC
transmission line fronl Churchill
1-311s
t~ Q~lebec (;..bout
1203
km) for the continuous transmission of
10000 PVIW;
ir.sul,?tor length
about
7
n-I. Strong insulators, required, for the crossing of the St. Lawrence River. (Photograph
CGilrtCSy professor Netsch, Lava1 University, Quebec).
lozd. The cooling may be effccted by pressurized gas flow in the interspace betwecn cable and sheath.
l'he
distant:
between the inner cocductor and sheath is retained by adequate spacers
[4.41].
Other a\ailab!e cable types for 3-phase hish voltage transmissiun are low pressure oil-filled cables
of
the single conductor type. and oil pressure cables in steel pipes. Upper l!mits of the tral~smission
voltzgc for cables with multilayer insulation are given by paper-impregnated or plastics-
:
imprcg!i;~tt.d components, and for cables with homogeneous insulation arc given by extruded
plastics. Influencing variables
are
the cable type, cable design. layout arrangcment, soil condition,
f
and the 1d:id hctor. Possibilities of increasing the current carrying capacity are, larger cross sections,
i
layi:~~
in
s?ecial back-filling materials, parallel systems, and forced cooling by
oil
or water. Future
j
possiblc varia~~ts for anderground bulk power tralismission can be espected from SF,-insulated
pipc
'
line cables. cryogenic. and supcr-conducting cablzs
14.421.
One of the most immediate ecollolnic objections to underground trai~smission is the obvi
di!ficulty of inspection. Non-destructive test procedures for underground power cables nzre
scribed and proven by
lioriner
et
a1
[4.43].
5.
Survey of basic hydrodynamics, with special reference
to hydro
power
The project and design of hydro power has to be based on the principles of
hydrodynamics. Starting from the early contributions of
D.
Berno~rilli
and
L.
Elllet.
[5.1],
this must include first thc kinematics of ideal fluid flow due to mass conservation
(continuity), rotation and irrotational flow in stationary or rotating frames of reference
and the components of
strain rate tensor. Velocity triangles and relative ii~hirl must be
considered.
The dynamics of fluids follows from these, when applying the equation of
motion and the
energy
theorem derived from this for stationary and rotating frames of reference. The
usual lack of
dctailed information about the flow field also requires the introduction
of
the momentum theorem. The mcaning of unsteady flow in relation to energy conversion
has to be discussed.
The
dynamics of real fluids in t~lrbo machinery has to start from liquid properties such
as compressibility
arid
viscosity together with turbulence-linked eddy viscosity. The rela-
tion between stress and strain rate yields the Navier-Stokes' equations.
Consideration
of
the boundary laycr concept and fluid behaviour related to a flat plat::
and a pipe
leads to the laminar and turbulent flow regimes. Turbulence and its con-
nected apparent shear stresses have to be considered as the sources of dissipative energy
loss. For ducts the influence of wall roughness and wall vibration on dissipation has to
be accounted for.
The flow in stationary and rotating ducts requires a knowledge of stall and secondary
flow as loss mechanisms. The significance of turbulence, wall curvature and duct rotation
in
the generation of secondary flow have to be understood. The rotor of reaction
machines needs a consideration of leakage and disk friction losses as
a
function of the
gcometry of the component parts.
Windage loss and secondary flow in curved buckets are required for the understanding
of the performance of impulse turbines.
A
knowledge of different flow regimes in diffuser channel may show remedies for lowering
loss in this element which is essential for the high efficiency
of
low head turbines and
impeller pumps.
The concepts of circulation in connection with Joukovsky's theorem and simple
aerofoil
lheory may increase the understanding of flow and loss mechanism in axial turbo-
machines. The appreciation of relative whirl, the link between the velocity triangle and
circulation and slip factor, may Cdcilitate the understanding of effects in semi axial and
radial machinery.
A
general survey and detailed trcatment of certain problems is found
in
the references
[5.2
to
5.81.
5.2.
The
kincnratics
cf
an
itlcal
flow
5.2.1.
M~ISS
conservation
(Continuity)
In steady flow the mass flow through an arbitrary cross sectional area
1.i
of a stream tube
has to
bc
constant. Whcn
11
is very smail, density
e
and fluid velocity
c,
nonnal to
A,
can
be
assumed as constant over
A.
Hence (see Fig. 5.2.la)
ril
=
~c/l
=
constant. For an
incompressible ideal
liquid
this is simplified to a constant volume
flow
along the tube
Q
r
V=
cA
=
constant.
(5.2
-
1)
For an unsteady flow the stream tbbe does not exist. Here mass conservation for the
general case of a compressible fluid can be formulated at a certain instant and volume
element assuming the fluid as a continuum.
"
;
r/*5i~
y=
dQ
4
@
=
const.
I
As
Fig.
5.2.1.
Illustrations of
kinerna:ic facts:
a)
stream
f,
EIf$
ties
J)
tubc; stream
due
b)
cor~tinuity;
to
fucction rotation
'P
c) veloci- in (curl); axi-
symmetric
flow;
e) axisym-
metric potential flow, and
graphical method;
f)
velocity
triangle of turbine.
The velocity components in the
x,
y
and z-direction at a point with Cartesian coordinates
X, y,
z
are
c,,
c,,
c,.
An elementary parallel epiped with
dx,
dy,
ciz as its edge lengths is
considered. The mass
flow through the cross sectional area
'dydz'
at the point
x
is
c,
q
dy
dz.
At the point
x
+
dx
it
has increased by [a(~cJiax] dx
dy
dz. Similar increments
of mass
flow occur
in
the other two directions (Fig.
5.2.1
b).
In
the time interval
dr
during which the surplus of mass flow hence obtained [a(c,p)/dx
+
?(e,.~)/dy
+
?(C,Q)/~Z]
dxdy
dz
crosses the surface of the volume element, the rate
of
incx-ease of density
t?p/2~
causes an increment of mass within the ~olume
clxdydz:
(dgldt)
Jtd-~dydz.
Mass conservation requires
a
compensation of both the terms
last
mentioned. Thus
div(ec)
+
i?~/at
=
0.
Hence the one dimensional flow in
a
pipe requires
q(dc/dx)
it
c(dg/dx)
+
d@/dt
=
0.
Obvio
?~/3x
=
(i-~/2r)/(dsi?r).
In
the
unsteady flow of
a
real compressible fluid the density rate
i3~
172
mainlv from
a
density
wve
will1 density gradient
dplds.
which
p;isses
rc'I~ti\e
to the
fluid
in the x-directio~~
with
the
~p~cd of sound '(1'(nlbo callcd
celerity).
Thus
(7~1'21
=
-
'J.
C'onsccll~cnti!
collti~luity reads now
~dc13.x
+
(S~ldl)
(I
-
c,'(i)
=
0.
In
a
hydl-;iliiic machinc
anti
its compol:cnt>.
the
ratio
c/a
is
usually
much
less
than
unity.
~hcrefore in general the term c/a can be neglected. Thereby continuity becomes in thc one
dime~lsional case approximately, ~(aclax)
+
doldt
=
0, or in the general case
div
c
=
-
(ae/~)/at.
(5.2
-
3)
The relative increase in density
?Q/Q
depends on the pressure increment
Sp
and the bulk
E,,
(which is assumed here to be c.onstant property of the liquid) giving
Hence the simplified continuity equation for a slightly compressible liquid reads
dive
=
-
(1/EJ dplat. (5.2
-
5)
5.2.2.
Potential
flow
and
the
curl
(vorticity)
In
the case of
a
potential flow, the velocity vector c is a gradient of a scalar function
Q,
called the velocity potential or hence brief the potential, according to
Inserting this
into the continuity (5.2-5) of an ideal inconlpressiblz flow
with
EL+
K
(from 5.2-4), the so called potential cquation results
Since the sequence of partial derivatives of
Q,
can be changed, it results from
(5.2-6)
that
Obviously (Fig.
5.2.1 c)
a
mean angular velocity
0
of the fluid about
:i
certain
axis
is
defined by the arithmetic mean value of angular velocities, that two adjacent liquid line
elements have, both normal to each other and in a plane normal
to
the axis. Hence the
components of
o
are
ox
=
(a~,/a,~
-
acYlaz)/2;
oy
=
(ac,laz
-
ac,/a~)/2 etc. (5.2
-
9)
The normal convention is, to omit the factor
'/2
by introducing the rot c or from now on
(after
C.
Maxweil)
the curl
c
as twice the angular velocity as a scalc of vorticity. Thus
curl
c
-
rot
c
=
20,
(5.2-
10)
For example a rotating solid body has a constant curl, as has a forced vor!e.u. Compr-iring
(5.2-8), (5.2-lo), it is seen, that a
flow
with
a
velocity potential, a 'potential flow' has a
vanishing curl
c.
Nevertheless there exists also an example of
a
potential flow, where the flow circulates
around an axis. This kind of circulating flow usually exists in the absence of viscous
effects, where at a distance
r
from its axis the whirl velocity coinponent c, (here equal to
the resultant velocity) follows the rule of constant moment of momentum along the
circular streamlines about this axis
According to
(5.2-61,
its
potential obeys tile
law
c.,,
=
(l/r)
c?cl,/?rp.
with
;Icp
i1s
tile
ii2crcmcnt of the pcriphcral angle (azi~nuth). 1iitcgr:ition yields
tilt.
potcnti,ll
(P
=
const
cp
-I-/(.u,
r),
(5.2
-
12)
at
which
x
and rare thc coordinates ~lormal to
the
pcripherill velocity
c,,.
l'ho
last re!ation
and the linearity of
the
potential equation
(5.2-7)
lead to the conclusioii that
any
free
vortex about an x-axis can be superimposed by a flow within the mcridian of this axis
and with potential
/(x,
r).
In
most of thc hydro turbiiics. the moment of momcntum oiflow ilpstrcam
of
the runner is ncnrly
constant. This follows from the intention, to cxtract from any strcam tubc the snmc availnblc spccrfic
energy, namely head by means of the theorem of momcnt of niornentitm irndcr the as:utnption cf
a whirl-free outlet
(c,,
=
0)
which usually occurs at least at the bep.
In
such
a turbomachine the rotary flow, given by the free vortex
c,r
=
const can
be
scperin?poscd
on an axial symmetric potential flow represented by a
potential
flow in a meridian.
5.2.3.
The
potential
flow
in
the
meridian,
stream
function
In
mzny cases
of
steady !low the flow energy pcr unit mass
(Y=
p/~
+
c2/2
+
ylt
(Cup.
5.3.1))
is
constant
a.ld the same
in
the who!e Cow
field
(except the boundary layers). This casc
may
bc
consid-
ered as natural, since no reason exists
v/hy the energies of adjacent strcamtubcs, usi~aily originating
from the same potential energy in a huge reservoir, ditTer from cacli other. Thc cqtl~titj~i
01
motioll
(5.3-
I)
shows, that in this case of grad
Y
=
0
and
dc/dr
=
0
the flow is also a potclltial flow with
curl
c
.=
0.
Now
such an axially symmetric potential flow, e.g.? upstream of the rotor of
a
turbine is
considered. It is represented
by
a
potential flow in the meridian (Fig.
5.2.1
d).
The velocity
c
has the components
c,
and
c,
in the axial and radial directions.
A
potential
flow
requires
after
(5.2
-
8)
We
assign to eacli streamline in the meridian a streamfunction
Y,
as
a
scalar due to
a
ceriain
flow
Q,
that passes betveen this streamline and the axis. Hence and fgr formal
co!ivenience:
Y
=
(2;(3n).
Any streamline in the meridian is described by
Y(r,
.uj
=
const.
Nest two neighbouring streamlines
Y
and
Y
+
dY
are considered. The flow between
these streamlines is
dQ
=
2ndY.
Consider the flow past the area described by the rotation of the distance
dr
between these
srrearnlines. Here this flow passes through an annular section of area
2nrdr
with velocity
'
c,.
Thus continuity gives (Fig.
5.2.1
d)
1
Next consider a movement
ax
to the adjacent streamline. Now the
flow
passes with
c,
through the cylinder
Znrax.
Hcnce
c,
=
-
(llr) dYji3x.
(5.2
-
15)
Inserting
(5.2
-
14)
and
(5.2-
15)
in
(5.2-
13)
gives
d'
Y/3x2
-
(:/I-)
aul/i?r
+
d2
'Y/ar2
=
0.
(5.2
-
16)
One solutio~~, cited
by
Lamb
[5.5]
has
the
fonn
174
Y
=
r2(xn-
-
{(n
-
1)
(11
-
2)/[2(2n
-
I)])
(r2
+
x2)
xn-'
+
{(n
-
I)
(n
-
2)
(n
-
3)
(11
-
4)/[2 ;4(2n
-
1)
(211
-
31)
in which
n
is a positive integer. Another solution given by the products of harmonic
functions in
x
with Bessel or Neumann functions in
r
was cSled by
J.
Ranhe
[4.11].
Owing to the linearity of the differential equa!ion
(5.2-
16)
due to the streamfiinction.
these solutions can be superimposed after multiplication by constants, so as to
br:
to the given boundary conditions
Y
=
const. Graphical method: Let
A
n
be thz
tlormal distance between adjacent streamlines, between whicl~ a discharge
AQ
flows. At
a sclected point
(Fig.
5.2.1
e),
distance
r
from the axis, the flow through the truncated
of area
2nr
An
normal to the velocity
c
becomes
A
Q
=
2x1-c
Ail.
When the
betwecn adjacent equipotcntinl lines normal to the streamlines, has the
difference
A@
along a distance
As
in the strcamwise direction: the velocity is
c
=
d
@/A
s.
A
combination of the last two relations gives, assuming
AQ
=
constant and
A@
=
constant, the following condition for the design of the grid formed by streamlines and
equipoten tial lines
A
s/An
=
const
r.
(5.2-
18)
Within
a
cylindrical distributor of a fluid machine, the flow is nearly normal to the axis. Therefore
along the equipotcntial lines in this
cornponcnt
r
=
const. Thus
As
=
const
An.
An
is obtained
by
dividing the given span
I),
of the distributor vane into just as Inany equal sections as elementary
turbines are required (usually
3
to
5).
At the draft tube inlct of thc turbine the distance
As
of two adjaccnt equipotential lines can
be
assiimcd as nearly constant. Therefore
now
the relation
r
Arl
=
conslafit holds good. An exper-
imental
;ncthod is sho\vn in
[5.9].
5.2.4.
Velocity triangle
and
relative eddy
A
stationary obscrver sees the absolute flow with its absolute vclocity
c
from a stationary
or absolute frame of reference. (Strictly speaking, according to the equation of absolute
motion
used, this frame should be an inertial frame with a rectilinear and uniform
motion.)
In
a rotor turning with local peripheral velocity
'u'
about its axis, an observer at station
r
rotating with the rotor at angular velocity
o,
observes the relative flow with its relative
velocity
tv.
From the theorem of the triangle of velocities (Fig.
5.2.1
f)
Assunle the absolute flow as a potenrial flow. Hence curl
c
=
0.
Therefore rrom
(5.2--20)
curl
lo
=
-
curl
(o
x
r).
From the well-known identity curl
(o
x
r)
=
2
o,
it follows
This curl is called the "relative whirl" or "relative eddy". That means
in
an
absolute
Potential flow, an observer, rotating in the rotor at
its angular velocity, sees always a
rotational flow with a curl parallel to the axis of rotation, in the negative sense of the real
rotation.
Assunic
thc flow
tt~rough
tlie rotor is on axial synlnlctric stre:lrn surL~ccs
(Fig.
5.2.2a).
0b:~iou~ly thc
governing
vortici~y due to the rclativc wllit-l
ill
such
a
strcanl face is
~ricnted normal
to
tf~c
I;~tfer
and
is giken
by
thc colnponcllt
-
?(I)
cos
11,
11
being the
angle, thc streanl.;urklcc makes wit11
a
plnnc nornlal to the
axis.
J-Ietlce (curl
w),
=
-
2w cos
jl
where
,I
indicates a
direction
normal
to
thc strean1 surface.
For the computation of the relative flow field in such an axial sylnnletric s&re;~msurface
thc cur1
,P
may be expressed in terms
of
the velocity components
w,
and w, in the
meridional and peripheral directions (Fig.
5.2.2a)
by
virtue of
Stokcs'
theorem, after
which (curl
w),
=
lim
4
w
tls/A
[5.3]. Thus
A
-0
where
s
now is the meridional direction and
cp
the azimuth. Inserting here (curl
IV),
=
-
2
0
cos
jl
yields
dw,/ds
+
cos
p
w,/r
+
81v,~,/(r
dcp)
=
-
2w cos p.
(5.2
--
22)
-.
1.1g.
5.2.2.
Illustration
of
further
kinematic facts. a) Relative eddy in axisymmetric relative
flow.
b)
Velocity i~crements due
to
rotation.
Assuming axial symmetric flow laminae
of
varying thickness b and linking the relative
straimline
to
a strenmfunction
Y
=
const (along the streamline), continuity enables
a
.
reduction of the velocity components to Y analogous to (5.2-14) and (5.2-15)
rv,
=
(llb)
Z\Y/as.
(5.2- 24)
Inserting these
rcla tions in (5.2-22) results in
a2
Ylas" (COS
ji/r.-
(lib)
ab/d~)
a
Y/as
+
(l/r2)
d2
'Y/dp2
=
-
20b
cos
p.
(5.2
-25)
Any graphical or analytical solution has to start from preliminary givcn relations p(s),
r(s)
and
b(s). Moreover it r.eeds the vane traces of the suction and pressure faces given
as
Y',(s,
rp)
=
0
and
Yb(s,
rpj
=
AQ/z,
where
A
Q
is the
flow
due to the flow lamina considered
and
z
the vane number of the rotor.
The author has shown
an
analytical solution
[5.10]
and has presentecl the conversion of the
differential equation into a difference equation
and
thus into a system
of
inhomc~gcncous linear equa-
176