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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.