Mean Pressure of Expanded Steam.-For calculations of engines it is generally assumed that steam expands according to Mariotte's law, the curve of the expansion line being a hyperbola. The mean pressure, measured above vacuum, is then obtained from the formula

in which Pm is the absolute mean pressure, p, the absolute initial pressure taken as uniform up to the point of cut-off, Pt the terminal pressure, and R the ratio of expansion. Ifl length of stroke to the cut-off, L = total stroke.

Pil+pil hyp log

Pm =

L

Ꮮ

1+ hyp log R

; and if R = Pm = P1 R Mean and Terminal Absolute Pressures.-Marlotte's Law.-The values in the following table are based on Mariotte's law, except those in the last column, which give the mean pressure of superheated steam, which, according to Rankine, expands in a cylinder according to the law pvt. These latter values are calculated from the formula 1 may be found by extracting the square root of P1 R four times. From the mean absolute pressures given deduct the mean back pressure (absolute) to obtain the mean effective pressure.

Pm 17

=

16R-

R

Calculation of Mean Effective Pressure, Clearance and Compression Considered.-In the above tables no account is taken

A

Area of ABCD =

L

FIG. 137.

B

actual

of clearance, which in
steam-engines modifies the ratio
of expansion and the mean pres-
sure; nor of compression and
back-pressure, which diminish
the mean effective pressure. In
the following calculation these
elements are considered.

=

L= length of stroke, l length before cut-off, x = length of compression part of stroke, c = clearance, p1 initial pressure, Pb = back pressure, Pc pressure of clearance steam at end of compression. All pressures are absoPlute, that is, measured from a perfect vacuum.

P1(l+c)(1 + hyp log

+ hyp log+c);

C = poc(1 + hyp log * +¤) = P(x+c)(1 + hyp log *+o);

D = (P1 — Pc)C = p1c − P¿(x +c).

=P1(l+c) (1+ hyp log+c)

− [po(L − x) + P¿(x + c)(1 + hyp log 2+C)+;

1+hyp -
+P2c − Pf(x+c)]

EXAMPLE.-Let L

=

area of A

L

1, l = 0.25, x = · 0.25, c = 0.1, P1 = 60 lbs., P = 2 lbs.

Area A = 60(.25+.1)(1+hyp log)

6 36.668 mean effective pressure. The actual indicator-diagram generally shows a mean pressure consider. ably less than that due to the initial pressure and the rate of expansion. The causes of loss of pressure are: 1. Friction in the stop-valves and steampipes. 2. Friction or wire-drawing of the steam during admission and cutoff, due chiefly to defective valve-gear and contracted steam-passages. 3. Liquefaction during expansion. 4. Exhausting before the engine has completed its stroke. 5. Compression due to early closure of exhaust. 6. Friction in the exhaust-ports, passages, and pipes.

Re-evaporation during expansion of the steam condensed during admis sion, and valve-leakage after cut-off, tend to elevate the expansion line of the diagram and increase the mean pressure.

If the theoretical mean pressure be calculated from the initial pressure and the rate of expansion on the supposition that the expansion curve fol

lows Mariotte's law, pv a constant, and the necessary corrections are made for clearance and compression, the expected mean pressure in practice may be found by multiplying the calculated results by the factor in the following table, according to Seaton.

Particulars of Engine.

Expansive engine, special valve-gear, or with a separate cut-off valve, cylinder jacketed..

Expansive engine having large ports, etc., and good or-
dinary valves, cylinders jacketed...

Expansive engines with the ordinary valves and gear as
in general practice, and unjacketed
Compound engines, with expansion valve to h.p. cylin
der; cylinders jacketed, and with large ports, etc..
Compound engines, with ordinary slide-valves, cylinders
jacketed, and good ports, etc..

Compound engines as in general practice in the merchant
service, with early cut-off in both cylinders, without
jackets and expansion-valves...

Fast-running engines of the type and design usually fitted in war-ships....

........

Factor.

0.94

0.9 to 0.92

0.8 to 0.85

0.9 to 0.92

0.8 to 0.85

0.7 to 0.8

0.6 to 0.8

If no correction be made for clearance and compression, and the engine is in accordance with general inodern practice, the theoretical mean pres sure may be multiplied by 0.96, and the product by the proper factor in the table, to obtain the expected mean pressure.

Given the Initial Pressure and the Average Pressure, to Find the Ratio of Expansion and the Period of Admission.

P= initial absolute pressure in lbs. per sq. in.;

paverage total pressure during stroke in lbs. per sq. in.;

length of stroke in inches;

period of admission measured from beginning of stroke; c = clearance in inches;

Ractual ratio of expansion =

L+c

To find average pressure p, taking account of clearance,

L+c
R

Given p and P, to find R and 1 (by trial and error).-There being two unknown quantities R and 1, assume one of them, viz., the period of admission 1, substitute it in equation (3) and solve for R. Substitute this value of R in the formula (1), or l = the result is greated than the assumed value of 1, then the assumed value of the period of admission is too long; if less, the assumed value is too short. Assume a new value of 1, substitute it in formula (3) as before, and continue by this method of trial and error till the required values of R and I are obtained. EXAMPLE.-P= 70, p = 42.78, L = 60′′, c = 3′′, to find L Assume l = 21 in.

c, obtained from formula (1), and find l. If

which is greater than the assumed value, 21 inches.

Now assume l = 15 inches:

Period of Admission Required for a Given Actual Ratio of Expansion:

Pressure at any other Point of the Expansion.-Let L1

up to the given point.

WORK OF STEAM IN A SINGLE CYLINDER.

To facilitate calculations of steam expanded in cylinders the table on the next page is abridged from Clark on the Steam-engine. The actual ratios of expansion, column 1, range from 1.0 to 8.0, for which the hyperbolic logarithms are given in column 2. The 3d column contains the periods of admission relative to the actual ratios of expansion, as percentages of the stroke, calculated by formula (5) above. The 4th column gives the values of the mean pressures relative to the initial pressures, the latter being taken as 1, calculated by formula (2). In the calculation of columns 3 and 4, clearance is taken into account, and its amount is assumed at 7% of the stroke. The final pressures, in the 5th column, are such as would be arrived at by the continued expansion of the whole of the steam to the end of the stroke, the initial pressure being equal to 1. They are the reciprocals of the ratios of expansion, column 1. The 6th column contains the relative total performances of equal weights of steam worked with the several actual ratios of expansion; the total performance, when steam is admitted for the whole of the stroke, without expansion, being equal to 1. They are obtained by dividing the figures in column 4 by those in column 5.

The pressures have been calculated on the supposition that the pressure of steam, during its admission into the cylinder, is uniform up to the point of cutting off, and that the expansion is continued regularly to the end of the stroke. The relative performances have been calculated without any allow. ance for the effect of compressive action.

The calculations have been made for periods of admission ranging from 100%, or the whole of the stroke, to 6.4%, or 1/16 of the stroke. And though, nominally, the expansion is 16 times in the last instance, it is actually only 8 times, as given in the first column. The great difference between the nominal and the actual ratios of expansion is caused by the clearance, which is equal to 7% of the stroke, and causes the nominal volume of steam admitted, namely, 6.4%, to be augmented to 6.47 13.4% of the stroke, or, say, double, for expansion. When the steam is cut off at 1/9, the actual expansion is only 6 times; when cut off at 1/5, the expansion is 4 times; when cut off at %, the expansion is 22 times; and to effect an actual expansion to twice the initial volume, the steam is cut off at 46% of the stroke, not at half-stroke.

Expansive Working of Steam-Actual Ratios of Expansion, with the Relative Periods of Admission, Pressures, and Performance.

Steam-pressure 100 lbs. absolute. Clearance at each end of the cylinder 7%

of the stroke.

8

.125

2.736

159,433 11.83

32.38

2.079 6.4 .342 ASSUMPTIONS OF THE TABLE.-That the initial pressure is uniform; that the expansion is complete to the end of the stroke; that the pressure in expansion varies inversely as the volume; that there is no back-pressure of exhaust or of compression, and that clearance is 7% of the stroke at each end of the cylinder. No allowance has been made for loss of steam by cyl. inder-condensation or leakage.

Volume of 1 lb. of steam of 100 lbs. pressure per sq. in., or 14,400
lbs. per sq, ft...
Product of initial pressure and volume................

..........

.........

4.33 cu. ft. .62,352 ft.-lbs.