= Flow of steam in. Ib. per second.
= Number of pressure stages in an ideal turbine.
= Number of pressure stages in a practicable turbine.
= Blade height of an ideal turbine, in in.
= Blade height of a practicable turbine in in.
=Mean diameter in in. of a row of blades.
= Drum diameter of a reaction turbine.
=Joules equivalent.
(j \ 2 /RPM\ 2 ) (_7^Ty moving rows only being included in the
summation.
From the standpoint of hydraulics there is a somewhat close anal-
ogy between a steam turbine and one operated by water. An
| essential feature in both cases is that the potential energy which a
, fluid possesses in virtue of its pressure is utilized to maintain a flow
, through a set of nozzles or guide vanes. In the ideal case of fric-
tionless flow the energy possessed by unit mass of the fluid is the
same whether it be at rest in the reservoir or whether it forms part
of the jet and has accordingly a kinetic energy due to its velocity.
The theoretical velocity of efflux of a gas can accordingly be deter-
mined by equating the kinetic energy to the work which the same
mass of fluid could perform were it allowed to expand, behind the
piston of an ideal engine, from the pressure of the reservoir down to
that of the receiver into which the discharge takes place. In thus
expanding behind a piston, W, the theoretical work done per Ib. of
- the fluid is given by the equation
[-(if]
where W denotes the work in foot pounds, p and pi the initial and final pressures, respectively, expressed in Ib. per sq. in., while Vo represents the original volume of the fluid in cub. ft. at pressure po and j is the index of adiabatic expansion, on the assumption that the relationship between the volume and the pressure during such an expansion can be represented by the formula i
p y V = constant.
By the principle already stated, the theoretical velocity of efflux will be obtained by writing
r s j, ^ -. ,1
(i)
M-om this expression it appears that as pi becomes smaller and smaller becomes greater and greater. When, however, the velocity )f efflux becomes equal to the velocity of sound in the escaping luid, any further reduction in pi occasions no increase in the weight lischarged from the nozzle per second. This follows because the velocity at which any impulse is transmitted through a medium is he same as that of sound in the medium. Hence, if, starting from 'in equality of pressure in reservoir and receiver, the receiver pres- sure is progressively reduced, "news " of each successive, reduction is transmitted back along the jet into the reservoir at the speed of ' ound, and as a consequence the pressure gradients there undergo a eadjustment and_the flow into the. nozzle is increased. Once, how- ' ver, the speed of issue exceeds that of sound, no " news " as to any
- urther reduction in the external pressure can reach the interior of
i he reservoir. The pressure gradients therein consequently remain Unaltered, and the weight of fluid fed to nozzle per second remains nchanged. This reasoning, which originated with Osborne Rey- .olds, applies to all cases of the efflux of fluids, although in the case f a liquid such as water it has no practical significance, as the head ecessary to generate a velocity equal to that of sound in water
- ould be many miles in height.
In the case of superheated or supersaturated steam, the speed of Dund is attained when the ratio of the lower pressure pi to the pper pressure po is equal to 0-5457. No further reduction of the >wer pressure will increase the weight of steam flowing per second, ut final velocities of efflux greatly exceeding the velocity of sound an be attained by making use of a nozzle converging first to the .iroat and then slowly diverging again. The theoretical velocities nder such conditions can be calculated from equation (i). In practice the actual velocity of efflux is less than the theoretical n account of losses due to nozzle friction. The maximum weight hich can be discharged per second from a convergent-divergent ozzle is fixed by the area of the throat. In the case of steam, for ich sq. in. of throat area the maximum weight which can be passed er second is
here p a denotes the absolute pressure of supply in Ib. per sq. in. ad Vo the corresponding specific volume of the steam in cub. ft. 3r Ib. This equation holds whether the steam is superheated or wet. In equation (i) above, the work due from one Ib. weight of steam ader pressure po is expressed in ft. Ib., but in steam turbine prac-
793
tice it is generally more conveniently expressed in heat units, and
i_
the convenience is the greater because the equation p y V= con- stant, is an inexact representation of the relationship between pres- sure and volume in the adiabatic expansion of steam. By working in heat units this difficulty is avoided.
If Ib.-centigrade heat units be adopted, the theoretical velocity of efflux is given by the relation z) = 300-2 V where u denotes the adiabatic heat drop and is conveniently measured from a Mollier chart, of which many have been published. A diagrammatic chart of this kind is reproduced in fig. 13, in which the ordinates represent
entropy, and the abscissae are total heats of steam (see 25.827).
The curves drawn on the chart represent lines of constant pressure,
constant temperature or constant wetness. The use of the chart
is best illustrated by an example. To find the velocity of efflux
from a nozzle supplied with steam at an absolute pressure of 200 Ib.
per sq. in. and at a temperature of 300 C., which is discharging into
a receiver maintained at an absolute pressure of 120 Ib. per sq. in.,
the point A is marked on the chart at a position corresponding to
the initial conditions and a straight line is drawn horizontally (i.e.
with constant entropy) to cut the i2O-lb. pressure line at B. The
length AB, as measured by the scale of total heats, represents 30-6
Ib. centigrade heat units. The theoretical velocity is therefore
3OO-2V3O-6 = l,66o ft. per second nearly.
Owing to nozzle friction the actual velocity will be less than this figure, which has accordingly to be multiplied by a coefficient, the value of which is commonly taken to be 0-95 or 0-96. With conver- gent-divergent nozzles the loss is much greater. The function of the moving wheel of an impulse turbine is to convert the kinetic energy of the jet into useful work on the shaft. The method of drawing a velocity diagram and estimating therefrom the probable efficiency of conversion is explained in the earlier article on STEAM ENGINE (25.843). With impulse steam turbines a stage efficiency of about 0-80 can be realized if the blade velocity be sufficiently high. To obtain such an efficiency the ratio of blade speed to steam speed should be about 0-47. For commercial reasons this figure is seldom obtained, but if S represents the actual ratio of blade speed to steam speed, and 81 the ratio corresponding to maximum efficiency ili, then the efficiency i\ corresponding to d can be obtained from the equation
[25 S IT
A steam impulse turbine generally consists of a series of elemen- tary turbines or stages arranged in succession on the same shaft. Suppose the first of the series has unit efficiency and expands the steam from a pressure of say 200 Ib. per sq. in. and a temperature of 300 C. to a pressure of 120 Ib. per sq. inch. Then, as shown above, in the absence of frictional losses, the state of the steam as delivered to the next elementary turbine would be represented by the point B on the chart, fig. 13, where the pressure is 120 Ib. per sq. in. and the total heat 698-2 Ib. centigrade heat units. The whole of the 30-6 units due in an adiabatic expansion from the initial conditions to a final pressure of 120 Ib. per sq. in., would in the assumed case of a perfect turbine be converted into useful work on the shaft. In practice, rfowever, only a part of this adiabatic heat drop will be usefully converted, the remainder being wasted in fric- tion and added as heat to the steam, before it is delivered to the next elementary turbine, or stage. If the efficiency of conversion is 0-7, the heat which would be added to the steam in the above example will be 0-3 X3O-6, or 9-18 Ib. centigrade units, thus making the total heat of the steam on delivery to the second-stage 698-2 + 9-18 = 707-4 nearly. This gives point C on the chart.
If it be assumed that the second stage expands the steam down to 80 Ib. per sq. in., the adiabatic heat drop will be found as before by drawing a horizontal line from C to cut the curve for 8o-lb. pres- sure at D. The length of this line as measured on the scale of total heats is 22-8 Ib. centigrade heat units. If, as before, we assume that but 0-7 of this is converted into useful work, the remainder being added to the steam as heat, the total heat of the steam as delivered to the third stage will be 707-4 0-7X22-8=691-5 heat units, giving
