Page:EB1922 - Volume 32.djvu/822

796

tenth of this difference until the value of log Vj is obtained. This latter procedure is based upon the general relation H 1 -H=ijU,

so that dH = ijdU= j qVdp

and thus V= TJ-T

144 dp

But on differentiating Calendar's relationship above, we get dH /H-O

n

T7

which gives us V

Since by hypothesis ij is constant we may write this as

a log V = log (H-C) - log p.

But log p is a linear function of log (H C) and therefore so also is log V. It may be noted that log V is accordingly also a linear func- tion of log p, so that this interpolation formula gives between p and V a relationship of the type pV*-= constant. But the value of the integral of Vdp is adjusted so as to bring the total work done into accord with the data. The formulas for V and H C are, in short, empirical interpolation formulas and must be regarded as such. 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)