They are not absolutely consistent with each other but the dis-
crepancy is small enough to be negligible in practice.
TABLE 3.
Sec-
log
1 A.
t \7
, H-C
H-C
tion.
(H-C)
log/)
log V
log ^
n
.
A
2-40310
1-30100
30280
2-55800
361-4
o
20-08
B
2-38475
1-14001
44543
2-53965
346-5
H-9
27-89
C
2-36640
0-97902
58806
2-52130
332-1
29-3
38-74
D
2-34805
0-81803
73069
2-50295
318-4
43-o
53-79
E
2-32970
0-65704
87332
2-48460
305-2
56-2
74-70
F
2-3II35
(>.(<)< .OS
2-01595
2-46625
292-6
68-8
103-8
G
2-29300
0-33506
2-15858
2-44790
280-5
80-9
144-1
H
2-27465
0-I7407
2-30121
2-42955
268-9
92-5
200- 1
I
2-25630
O-OI3O8
2-44384
2-41120
257-8
103-6
277-8
J
2-23795
T-85209
2-58647
2-39285
247-1
II4-3
385-9
K
2-21960
T-69IIO
2-72910
2-3745
_Y,I,-K
124-6
52VQ
To avoid an accumulation of errors and to facilitate checking,
the values of the intermediate logarithms in the above table are
tabulated to five figures but only four of these are significant. When
the additions and subtractions are accurately carried out the values
in the last line of the table must be the values at the exhaust with
which the calculation was started. The convenience of this check is
so great that it is advisable (even at the expense of the slight inac-
curacies involved) to use this type of interpolation formula even
in the case of steam superheated throughout its expansion, although
in this case exact relationships between the different functions can
be stated.
Knowing U, the general characteristics of a turbine intended to
operate with a given hydraulic efficiency can be very readily deter-
mined.
Thus if we define K as
where d denotes the mean diameter in in. of a moving row of blades,
and the summation includes the moving rows only; the efficiency
jr
of the turbine is a function of y, as will be readily understood from
the obvious consideration that K is proportional to the mean square
of the blade speed, whilst U is proportional to the mean square of
the steam speed. If the hydraulic efficiency be plotted against
!
ij the resultant curve is an ellipse, but this ellipse is not symmetri-
cal about the axis along which
this ellipse is
is measured. The equation to
T U Kj - *U Z,
where iji denotes the maximum value of i\, and y-p is the corre-
jr
spending value of VT-
i^
The relation between ij and -p, as determined by the collation
of actual test figures is given in figures 15 and 16. In both cases the
expansion is assumed to be continuous in character instead of being
effected in finite steps, a circumstance which slightly lowers the
apparent hydraulic efficiency of the impulse machine, but the error
is small and moreover cancels out when the curve is used for pur-
poses of design.
When the steam is initially superheated the value of U to be use
in the formula is given by U = U I +U, where U 1 represents th
thermodynamic head expended down to the saturation line an
UI=^M, as explained above.
07
Of
or
04
03
01
Oi
FIC
3. \5
.
-
,
.' ' "=
=
1
/
^
^
/
s Indicated Hydraulic Efficiency of Impulse Turbines
1
/
II
Effective Thermodynamic Head in Ib Cent
Z fUjfioz 31 / where d /s the mean dm:c
the Blade Path in Inches
'Into
I/
K
Tf
7
/
V
i/ues <
r
FIC.I6
^,
-
,
/
X
<
|
"5
/
/
Im
Tui
Heated Hydraulic Efficiency of Reaction
"bines (not carrectedfor Tip Leakage)
U- Effective Thermodynamic Head F.PC.
t-x&fHtff
/
/
/
/
I
/
/
/
Val
jes
fl
{
M 200M0400MOM0700U0900 1000 IPOO I2OQ 1800 MOO 1900 1600 POO MOO 1000 '8000
Suppose that an impulse turbine which is to operate with dil
saturated steam supplied at a pressure of 20 Ib. absolute an
hausted at a vacuum of 29 in. mercury is to run at a speed of 1,51
revs, per minute, the mean diameter of all the blade rows bein
44 J in. whilst the designed hydraulic efficiency is 0-7. Then fni
Tf
fig. 15 it will be seen that -JQ =436. Hence as from table 3 tl
total thermodynamic head is 124-6, the value of K must be 124-6 :
436 = 54.330-
But if v be the number of stages
whence f = i2, so that a turbine of 12 stages with wheels of 44! i
mean diameter will give the required efficiency. If v does not tut
out to be an even number, it can be made so by suitably adjustir
the value of d. Intermediate values of v are directly proportion
to the corresponding values of U and a series of such values calci
lated with an ordinary lo-in. slide rule, which is amply accurate fi
the purpose, are as follows:
Section
U .
Section
U .
v
A
o
o
1-3010
G
80-9
7-79
Q-33.SI
B
14-9
1-435
1-1400
H
92-5
8-91
C
29-3
2-82
0-9790
I
103-6
9-98
0-0131
D
43-o
4-14
0-8180
J
"4-3
II-OI
-0-1479
E
56-2
5-41
0-6570
K
124-6
12-0
0-3089
The values of v are fractional, but they are used merely for cur
plotting, the values of the different functions corresponding
integral values of v being read from the curves. Thus in fig. 17 log
has been plotted against v and it should be noted that the curve,
by no means represented by a straight line. Since v is proportion)
TURBINES, STEAM
797
to K it follows that if in any turbine log p when plotted against K
gives a straight line, that turbine, whether of the impulse or reac-
tion type, cannot be designed to operate with uniform efficiency.
In the diagram fig. 17 the values of log p represent the pressure of
the steam after discharge from the preceding stage, stage No. I
being thus conceived as being preceded by an imaginary stage No. o.
A corresponding plot of the volume would, however, give not the
volume at discharge from the guide blades, but this volume as in-
creased by the heat generated in the passage of the steam through
the moving buckets. All stages being similar, the effective thermo-
dynamic head at each stage is the same. But the apparent thermo-
dynamic head, obtained by dividing the total thermodynamic
head U by the number of stages, is somewhat greater than the adia-
batic heat drop at each stage.
1-4
,K
20 In*.
10
IS
17
IS
IS
14
13
12
II
10
9
8
7
a
3
4
3
2
Fl C. 17
I
/
A
/
12
II
1-0
09
0-8
07
M
0-5
04
0-3
02
01
OH
01
*2
0-3
A
-04
\
/
\
/
\
^7
j
\
c
\
i
\
to
3
\
f
i
t
X
ii
1
|
\
E
i
s:
\
x
~ii
<
&
|
\
?
H
t
<H
x
t
1 \
a
H
\
Va/u
IS Of
'V
/
y
\
1
z
3
4
5
>
*/r
7
8
9 1
V
12
S
^
\
i
"*
^
D
\
;=
- 'B
K
According to what has been stated above, the velocity of dis-
iarge from the_guide blades of a stage is commonly taken as
=300-2 Xo-gsVtt where u is the adiabatic heat drop. The weight
' discharged per second per sq. ft. of guide blade area is
... v__ 300-2XQ-95VM
~ V* ~ \<t>
lere V$ represents the volume of the steam after an adiabatic
pansion between the pressure above and below the stage. Instead
calculating these values it is more convenient to utilize the known
lues of U and V and to correct the above formula by using an
propriate coefficient $. As there are 12 stages in the present
se we get = ' = 10-38 =q, and the above equation may
Before be written
W =
^300-2X0-95^10-38
- ~
i interpolation formula for / which is applicable for the ordinary
ige of turbine efficiencies and for convergent guide blades is
, 1+0-13 (i i))Vx i where x denotes the ratio of the pressure
we and below the stage. The coefficient / is readily evaluated
the ordinary slide rule with quite sufficient accuracy.
In the case under consideration we note from the curve fig. 17,
it when v = i , log p = I 197 so that x = I -27 and / is therefore I -025.
The area available for flow through a row of guide blades is
-^ where h' denotes the blade height in in., and a is
..
" effective " angle of discharge, allowing if necessary for the
fact that the blades are of finite thickness. Hence if iv be the
weight of steam flowing through the turbine per second
_
6223 /' d sin aV q
Taking sin = 0-30, 5=44!, 3 = 10-38 and w = 10-3 Ib. per second,
this expression reduces to h' = -O3732V. Values of h' thus calcu-
lated for the values of V given in table 3 are plotted in fig. 17 and
from the curve thus obtained we read off the theoretical blade
heights at the different stages. These are :
Stage No. .
Theor. blade height
in in. .
I
0-94
2
1-18
3
i-5i
4
1-95
5
2-48
6 '
3-24
Stage No. .
Theor. blade height
in in. .
7
4'3
8
fcfis
9
7-69
10
10-48
ii
14-40
12
20-0
In practice the nearest even dimensions will be substituted for the
calculated heights. The calculated heights for the last three stages
are inconveniently long, but they can all be reduced to say 9 in. by
suitably increasing the effective angles of discharge. Some builders
moreover increase the pressure drop at the exhaust end, and would
accordingly combine stages II and 12 into one. These expedients
decrease the efficiency but are cheaper than the alternative of con-
structing the low-pressure end on the double flow principle.
The high-pressure end of a turbine can be proportioned in a man-
ner exactly similar to that described, but as the steam there is com-
monly superheated, the problem is correspondingly simplified and
need not therefore be discussed here. It is, however, usually neces-
sary to construct some of the high pressure stages as partial admis-
sion stages and it is also a common practice to have a large pressure
drop at the first stage with the object (at some sacrifice of efficiency)
of making a large initial reduction in the temperature and pressure
of the steam, so that the high pressures and temperatures are con-
fined to the nozzle-boxes of the first stage. To the same end a veloc-
ity compounded wheel is frequently used in the first stage. The
general theory of these wheels is described in Prof. Evving's article
(see 25.844), but it may be observed that in practice it has been found
necessary to adopt empirical methods of designing such wheels.
If designed as pure impulse wheels operated with a fluid which is
" freely deviated " the results are very disappointing. One rule
which has been used is to assume that only 85 % of the total heat
drop of the stage is utilized in the nozzles, and of the residue that
5 % is utilized in each of the three sets of blading. The wheel there-
fore works to some extent as a reaction turbine.
A -et
ts
+
*
V*
- :
:r
s - ' -
-K f*.**r\
..til J
k- / -
FIG.I8
h- -77i
Speaking generally, the principle of " free deviation " as embodied
in some water wheel designs is inadmissible in. steam turbine prac-
tice, in which the moving blades should be just sufficiently long to
avoid " spilling " of the steam delivered to them from the guide
blades. As to the exact form of the moving blades, this does not
appear to be of primary importance within reasonable limits, as,
79 8
TURBINES, STEAM
although the practice of different makers varies considerably, all
impulse turbines exhibit much the same efficiency under corre-
sponding conditions. Typical Rateau blading is illustrated in fig. 18.
The discharge angle is commonly about 30 save at the last row
of blading where it is increased to 35.
As regards nozzle and guide blade efficiencies, generally, reliable
experimental data are still lacking. It has been assumed that the
efficiency of convergent guide blades is a maximum when the speed
of efflux is equal to the velocity of sound, and though this is not
improbable from a priori considerations no conclusive evidence in
support of the view has yet been forthcoming, and turbines which
attempt to embody this theory have not shown the slightest
superiority over competing designs. A great drawback to high
steam speeds is the liability to excessive wear of the blading, and in
this respect reaction blading has a great advantage over impulse
blading in addition to the higher inherent efficiency of the former.
This higher inherent efficiency depends upon the fact that the
overall efficiency of a steam turbine depends upon its stage effi-
ciency, a stage being defined as the section of a turbine comprised
between two successive heat drops. In the case of impulse turbines
for each successive heat drop, frictional losses are experienced in
two elements, namely, the nozzles or guide blades and the moving
buckets, whereas in a reaction turbine at each heat drop there is
loss in one row of blading only.
The Design of Reaction Turbines. The proportioning of a com-
pound reaction turbine is a somewhat intricate problem, and as a
preliminary it will be convenient to discuss the flow of steam through
a series of openings or stages. At each of these a certain thermo-
dynamic head q is expended, and this is not, in general, the same
for each stage. If however U denote the total thermodynamic head
expended in forcing the steam through n stages we have
dU ,
Now Laplace's theorem in the calculus of finite differences may be
written
2,2 = q dn
+ (Ag - A 2o ) - (A'g -
If we neglect the terms comprising the differences we get
so that
d U
dq
-"
NowrfU = - Vdp whilst if (as it is frequently permissible to
assume) the velocity of flow at each stage is proportional to V q
we may write
where F denotes some coefficient, w is the weight of steam flowing
per second, V its specific volume, whilst fl denotes the area through
the stage. Making this substitution for q we get
144 r dp _ dn , dq
~~~
whence
"GO
here ^ is the mean value of ^ when plotted against n and I is a
factor depending on the coefficient of discharge. Substituting for
q, the above expression reduces to
V
o- JL ir^.
Q.
In the case of an ordinary dummy Q is constant, and the law of
expansion is expressed in this case by pV = constant. Whence if
the coefficient of discharge be unity we get, on making the proper
substitutions
HI = 68 J2. |^o
Vo
n+ loge x
Here x denotes the ratio of the initial pressure to the final pressure.
The logarithmic term becomes of great importance when n is small
and renders the formula reliable under very extreme conditions.
Suppose it is desired to replace n openings in which the area is
varied in direct proportion to the volume of the steam, by n open-
ings all of equal area, the weight of steam passed per second, and
the total pressure difference remaining constant. If we neglec
the small change such a substitution will make in the value of J
and assume that the velocity of discharge at each stage is still pro
portioned to V q we get
fin
oge "a/7\
n \ff)'
-(4).
Use will be made of this formula in proportioning the blading of ;
reaction turbine.
Let it be required to proportion the blading for a double flow reac
tion machine, the conditions being similar to those assumed for tli,
impulse turbine discussed above, save that the total discharge wil
be assumed to be 27 Ib. of steam per second, that is to say, 13-.
Ib. each way, whilst the speed is to be 2,400 revs, per minute. Tb
hydraulic efficiency will be taken as 0-7, as before, so that the quan
tities already tabulated in table 3 can be used without modification
If it were practicable to construct a reaction turbine with all it
blade rows of the same mean diameter, the problem would be a
simple as that of the impulse machine, and we shall, in the firs
instance, compute the blade heights for such an ideal turbine am
from the figures thus obtained we shall deduce the blade height:
required for the practical machine.
In this ideal turbine the blade heights are varied so that the ratii
of blade speed to steam speed is everywhere constant and from thi
perfect uniformity of conditions it follows that q (the thermodynanm
head expended at any stage) is also constant and proportional
Since the blade speed is also proportional to its mean diameter
we may write
where /3 is a coefficient. From this it follows that
2
0'=
where K is defined as above. Hence
u
If the hydraulic efficiency be decided on, the value of jf can Ix
obtained from the curve plotted in fig. 16.
144 ">V ,- .- wV
Again since v=~ =. we may write V q = G = where G it
rh d sin a ha
some constant. But V g is, as already shown, equal to
and equating these two expressions we get
R.P.M. -, /TT
h'd
IOOO
loooG wV
, , * (**
~ R.P.M. 2* \-jj-
The value of G must be determined experimentally, and from care-
ful tests it appears that for normal Parsons blades h' may be written
as,
_
-
616 !Y .. pC
R.P.M. f \TJ
It may be added, however, that the value of the coefficient is not
quite independent of the efficiency, and whilst the value 616 is appro-
priate to an efficiency of 0-7 it increases to 678 for an efficiency of
Q- O/
For a reaction turbine having an hydraulic efficiency of 0-7 it
will be seen from the efficiency curve that -^ has the value 600,
and if d be taken as 49 in. we get for the total number of rows
(fixed and moving) corresponding to the expenditure of a therni"
dynamic head U, the expression
_2K t IOOO \ t 1200 U / IOOO \ = y
~ 5* x VR.P.M J d*- VR.P.M./
Taking the values of U from table 3 the corresponding valm
t are entered in the fifth column of table 4. Taking the steam pass
as 13-5 Ib. per second each way, we get for h' the expression
From this the values of h' given in the sixth column of table 4
have been deduced.
TURBINES, STEAM
799
19
18
17
16
IS
14
13
Fl<
f
1
-V,
.t
7 8 9 10 H
. In fig. 19 the blade heights corresponding to sections H, I, J
!nd K have been plotted, and from this graph we find that in the
leal turbine, if we have a stage at i> = lo-8l then the blade heights
t stages 9-81 and 10-81 will be as follows:
Lv 8-81 9-81 10-81
h 9-20 in. 13-14 in. 18-94 in.
TABLE 4.
Sec-
tion
log/)
U
V
V
h'
h' (dY-
h
dn h
dv h'
n
A
1-3010
o
20-08
o
710
1704
1-045
1-477
o
B
1-1400
14-9
27-89
1-306
986
2367
i-43i
452
1-89
C
0-9790
29-3
38-74
2-542
1-37
3287
1-940
416
3-69
D
0-8180
43-o
53-79
3-73
1-90
4565
2-60
3/0
5-34
E
0-6570
56-2
74-70
4-88
2-64
6339
3-47
3"
6-89
F
0-4961
68-8
103-8
5-97
3-67
8792
4-58
250
8-28
G
0-3351
80-9
144-1
7-02
5-09
12230
5-97
172
9-56
H
0-1741
92-5
2OO-I
8-03
7-07
16980
7-68
1-087
10-68
I
0-0131
103-6
277-8
8-97
9-82
23570
9-79
996
11-70
J
1-8521
II4-3
385-9
9-93
13-64
32740
12-40
910
12-60
; K
1-6911
124-6
535-9
10-81
1 8 -94
45470
As the first step to the design of a practical turbine the blades
L j/=9-8i and ? = lo-8l must be replaced by two blades of equal
ight, say h, which must be such that these two blades will pass the
; me weight of steam per second as the blades they replace. As a
' st approximation, the required height is equal to the height given
fig. 19 corresponding to v =
9-81 + 10-81^
= 10-31. This height
15-7 inches. This approximation with blades so long in propor-
>n to the drum diameter is not a very good one, although when
e blades are not excessively long this simple rute gives quite good
isults. To determine a more accurate value of h we make use of
uation (4) which in this case may be written as
18-94
9^0) +4 (7 3 J ^) + (18-94)
LI
iere the factor on the right is the mean value for the value of
as deduced from Cotes' rule for the mean value of a function defined
by three equidistant coordinates, and which is exact for any curve
which can be adequately defined Jpy 4 ordinates.
From this expression we get (h) 2 =216-2, whence ^=14-7, show-
ing that the provisional value obtained from the diagram was about
7 % too long. It is only at the L.P. end of a turbine, however, where
the blades are long and where the pressure drop per blade is high,
that the error attains any such magnitude.
If we use semi-wing blades for these two rows, the height will be
two-thirds of the figure given, or 9-8 inches. Let it be taken at gf in.,
so that the drum diameter is 49 9j = 39-25 in., and to this diameter
the blading of the ideal turbine must be reduced by means of an
appropriate " transfer " formula.
If h denote the height of the blades after transfer to a drum of
diameter D and h' the height of the blades, of the ideal turbine as
already calculated, all of which have the same mean diameter d.
Then we must have
and = -
Here n denotes the number of blade rows in the practical turbine
corresponding to v rows of blades in the ideal turbine.
Values of h(d) 2 are tabulated in column 7 of table 4 and from these
values the corresponding values of h are readily deduced by means
*.
B
FIG.
20
-~~,
^
<
&
~^
-k
s
x,
X
F
X
8
\,
s
i
\
\
\,
\
Values 'ifv
\
1284567690
of a slide rule. This is done by assuming a provisional value of h.
Calling this provisional value a a better value of h is got by writing
A still closer value is then obtained by repeating the process. At the
end of each operation the value of r-, is also found, and is entered
7 J
in the adjoining column. These values of j-,= -j- have been plotted
in fig. 20 and from them the value of n corresponding to any stated
/
i
9 .
B
7
9
4
>
a
/
1
j
FIG. 21
'
/
'
1
/
/
&
/
/
/
U
/
Theoretical B/ode Height sfln$
/
/
X F
''
U
/
/
X
t
>
A Of/
/
|O^D
g
0*
/
S
fl
n'
'
/
*
t
'..
"'
c
B
7
1 ...
293 294
295 ...
459