TABLE I. Properties of Steam at the Wilson Line.
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4J
0) C
a
tj|
SZ "
S
si
S"
o'o
a
C3 '5
.2 ^
.5 aj
'o
"On
QJ PH
- ^" t^
u p
P!p
2 c
Sc
3 C
C
u:
3
o c
(U O
D. *
tfl O CJ t/3
o
O ui
si
"3
'3 o
II
fif
n
cr c
W
c
t,.
pw
t'w
Hw
^ s
yV,
"tw
Ib. per
cub. ft.
cub. ft.
deg. C.
sq. in.
per Ib.
F.P.C.
deg. C.
per Ib.
o
0-9888
295-30
593-79
38-52
325-85
9098
10
1-739
173-33
598-28
49-41
191-50
8641
20
2-935
106-11
602-64
60-33
116-91
8220
3
4-764
67-339
606-85
71-19
74-218
7825
40
7-478
44-091
610-90
82-13
48-474
7476
5
n-39
29-732
6I4-75
93-00
32-666
7141
60
16-86
20-566
618-36
103-89
22-575
6830
7
24-36
I4-53I
621-70
114-80
15-957
6537
80
34-41
10-506
624-56
125-66
"537
6261
90
47-60
7-720
627-47
136-57
8-638
5998
IOO
64-64
5-778
630-01
147-48
6-353
5748
no
86-28
632-10
158-40
4-832
5508
I2O
"3-37
3-376
633-89
169-48
3-713
1-5278
Along the Wilson line the relation between pressure and volume is
- iven with considerable accuracy by the equation
0-9401 log pi + log (V 0-016) = 2-4651.
At the supersaturation limit moisture is formed and settles out in
- he form of minute droplets.
To proportion rationally the blading of a steam turbine it is lecessary to know the relationship between the pressure and the volume, or between the pressure and the total heat of the steam puring the expansion. The discovery that wet steam does not ixpand through a turbine in a condition of thermal equilibrium, vhilst affording an explanation of certain anomalies experienced in >ractice, has raised new difficulties, since we are no longer in a losilion to determine with certainty the volume of wet steam at lifferent points of the expansion. So long as the expansion is not arried beyond the supersaturation limit, or the " Wilson line," he behaviour of the steam is in accord with the equations given hove. At the supersaturation limit, however, an overdue change bruptly occurs, and it is a matter of general experience that when a umlition of unstable equilibrium is suddenly upset the subsequent ihenomena are commonly incalculable. In such cases there is fre- uently found to be a period of transition during which " repeat " xperiments fail to give consistent results. Once, however, the transi- ion is fairly effected, a new steady state is generally established. n the case of steam, this steady state appears to be obtained if the xpansion is continued considerably beyond the supersaturation mit. In this steady state, such evidence as is available goes to show iat the water of condensation which remains suspended in the m in the form of minute droplets, has a temperature approxi- lating to that of saturated steam of the same pressure, whilst the aseous portion of the steam has a temperature corresponding to iat of steam just on the point of condensing at the supersaturation mit. The dryness fraction of the exhaust steam from a turbine is icrefore given approximately by the relation
H w -t,
i here H, denotes the total heat in one Ib. of the exhaust steam, j the temperature corresponding .to saturation at the same pres- I ire while H w is the total heat of one Ib. of dry steam at the exhaust '.ressure but at the limit of supersaturation, as given in table I iX>ve for various pressures. The volume V, of the exhaust steam
- equal to yV w where V w is taken from a table similar to table_l.
, In general, engineers express exhaust pressures as so many in. of , .ercury. The standard barometric height is taken as 30 in. of mer- iry, and a vacuum of 29 in. of mercury, corresponds therefore to i absolute pressure of one in. of mercury, or 0-491 Ibs. per sq. inch. . alues of Hu,, V w , and t w for different vacua are tabulated below :
TABLE 2.
Vacuum Jin. of mercury).
tw
(C).
Hto Ib. centigrade units.
Vu,
cub. ft. per Ib.
29
28
27 26
-"37 - 0-15 + 6-98
+ 12-21
588-59 593-67 596-92 599-22
569-3 296-3 202-5 154-5
will be seen that the determination of V, depends" upon a lowledge of H e , whilst H, = Hi indicated work done. The indicated work done in the expansion of wet steam can only s matters stand to-day) be found as the result of experience with tual turbines, and our knowledge is accordingly empirical in
795
character. If we take steam expanding from the saturation line to ordinary exhaust pressures, the following rule for the effective ther- modynamic head V, engendered is in good accord with experience U, = ^M S . Where u, denotes the adiabatic heat drop, assuming the expansion from the initial to the final pressure to be effected under condition of thermal equilibrium, whilst
^=1-1070+0-02212 1(0-1638 + 0-0286)
In this expression x denotes the ratio of the initial pressure to the exhaust pressure and i\ is the hydraulic efficiency, which is taken to be the same as if the turbine were operated with steam in a super- heated condition throughout the whole range of expansion.
The coefficients in this formula for have been selected so as to make the indicated work done the same as if computed by Bau- mann's empirical rule (Inst. E. E. 1921), but the relatively small amount of work done by expansion below the saturation line is attributed to the effective thermodynamic head being less than if the expansion had been effected under conditions of thermal equi- librium. Baumann's empirical rule on the other hand assumes that the efficiency decreases by I % for every I % of moisture in the steam, the latter being assumed to expand in thermal equilibrium.
From the above expression for U, we can find H, from the gen- eral relation H e = H]. ?jU; and from this we can calculate V. as already explained.
To determine the volume of the steam at other points of the expansion it is perhaps sufficient in the present state of our knowledge to use an interpolation formula which shall give correctly the initial and final volumes at the saturation line and at the exhaust, and which shall also give correctly the work done between these two limits. No doubt there is force in the argument that as we know accurately the relation between the volume and the pressure of the steam in expanding down to the Wilson line, it would be more logical not to make use of an interpolation formula until the necessity actually arose by this line being crossed, but in the present state of pur knowledge the simpler procedure seems adequate for practical needs. It is certainly nearer the truth than the assumption hitherto adopted, that the steam expands in thermal equilibrium.
Callendar has devised a very simple and easily applied interpola- tion formula which satisfies the required conditions. It may be written as
log (H-C) = log A + u log p. The value of the constants are determined by writing
In practice it is seldom necessary to determine either A or u, whilst C is most easily obtained by writing
Hi-C =
(H.-HQ
The use of the formula will be made clearer by taking a practical example. Thus, suppose dry saturated steam at a pressure of 20 Ib. per sq. in. to be expanded down to a vacuum of 29 in. with an hydraulic efficiency of 0-7. (This efficiency is low for a modern turbine but the method is of course applicable whatever the value of ri.) Then from Callendar's steam tables it will be found that Hj = 642-82 centigrade heat units; Vi =20-08 cub. ft. per Ib., whilst
, = 125-46 units. In this case - = - = 0-4075, so that
1^ = 0-9932 and U s = 124-61 centigrade heat units. The " indicated" work done is therefore 0-7X124-61 =87-23 and hence H e = H 2 = Hi 87-23 = 555-59. The dryness fraction at exhaust is therefore
given by y =
= 0-9413, so that V e =j-V w = 0-9413 X
569'3 =535-9 cub. ft. per Ib.
Had the steam expanded in thermal equilibrium and an equal amount of work been taken out of it, its volume on exhaust would have been 594-4 cub. ft. per Ib. Hence at high vacua the volume of steam to be provided for at exhaust is some 10% less than on the old theory in which it was assumed to expand in thermal equilibrium.
The requisite data for constituting Callendar's interpolation formula are now available. Thus we haveiVi=4Oi-6; 2 V 2 = 263-I ; whilst Hi -H 2 = 87-23. Hence
i _.
Thus 0=389-8 and H 2 -C = 165-8.
We therefore get log (Hi-C)-log (H 2 -C) =0-1835 and log pi log pi = I -6099.
We divide each of these differences by 10 (say) and can then calcu- late corresponding values of log (H C) and log p by repeated sub- traction of these dividends, giving the figures tabulated in columns 2 and 3 in table 3 below. To determine corresponding values of log V we proceed in an exactly similar manner, determining the difference between log Vi and log V 2 and repeatedly adding one
