To find the Work done in the two cylinders of compound engines-The Woolf engine. It has already been stated that the formula ( 5 ), page 828, for the work of steam expanded in one cylinder, applies also to the work of steam in the Woolf engine, when the combined actual ratio of expansion behind the pistons in the two cylinders, is given. Thus, the net total work for one stroke of the two pistons, quoting that formula, is,

w = a P [l' (1 + hyp log R”) – c],..

(25)

RULE I. To find the net work done by steam in the two cylinders of a Woolf engine, for one stroke, with a given combined actual ratio of expansion.— To the hyperbolic logarithm of the combined actual ratio of expansion behind the two pistons, add 1; multiply the sum by the period of admission to the first cylinder plus the clearance, in feet; and from the product subtract the clearance. Multiply the area of the first piston, in square inches, by the initial pressure in pounds per square inch, and by this remainder. The product is the net work in foot-pounds.

For example, let the 2d case, pages 860, 861, be calculated by this rule:- = 1 × 63 = 63 lbs., ' = 2.42 feet, c .42 foot, and R" = 6.24.

Then,

-a P

w=63 [2.42 (1+ hyp log 6.24) - .42]
=63(2.42 x 2.8310)-.42]

= 63 (6.851-.42) = 405.20 foot-pounds,

as was before calculated, allowing for small errors of approximation. The Receiver-engine.-A complete formula for the work of the receiverengine necessarily comprises three elements:-First, the expression of the gross work, including the work of the clearances; second, the deduction. for the passive work of the clearances; third, the addition for the gain of work by the reduction of the back pressure on the first piston when there is an intermediate fall of pressure. Beginning with the first case, pages 863, 864, in which there is no intermediate fall of pressure, the total initial work of the steam admitted to the first cylinder is expressed by a P l'; whence the total work with expansion is

This measures the total area of the diagram, Fig. 347, page 862, including the clearances. The work of the clearance of the first cylinder, cc'o"o, is

a Pc.

The work of the clearance of the second cylinder is the rectangle hh'o'o, which includes the section hoo" of the first clearance; and, deducting this, the remainder, which is the rectangle h'o'o", is to be added to the first clearance. To express this remainder algebraically, the volumes of the first and second clearances, oo" and oo, are in the ratio of the areas of the cylinders, or as I to r, and the volume of the difference, o"o', is as c (r− 1). The height, o'h', is the final pressure in the first cylinder, and is equal to the

initial pressure divided by R', the actual ratio of expansion in the first cylinder; or,

Therefore the work of the excess, o"o', of the second clearance is,

to be deducted from the gross work by expansion (26). Whence the equation for the net work, in the first case:-

w = a P l′ (1 + hyp log R′′) − a P c (1+",1); or

w= a P [l' (1 + hyp log R′′) − c (1 +

when there is no intermediate fall of pressure.

R'

Before reducing this formula to a rule, it may be remarked that it gives values which approximate closely to the true values, for cases in which there are intermediate falls of pressure-such cases as usually occur in practice;— and, for ordinary practical purposes, the results of the application of this formula will be sufficiently near to exactness. It was found, in fact (page 864), that the reductions of work by intermediate falls, as compared with the work done when there was no fall, were as follows:-When the pressure falls to

3/4, 2/3, the reduction of work is,

0.2,

1.0,

1⁄2 of the final pressure in the first cylinder,

4.6 per cent. of that in the first case.

The intermediate fall of pressure is rarely so much as two-thirds; and even with this fall the reduction of work, it is seen, only amounts to 1 per cent. The slightness of the reduction results from the fact, as was before explained, that though the actual ratio of expansion, with intermediate falls, is less than when there is no intermediate fall, yet the loss of work by such reduction of expansion is practically compensated by the gain of net work on the first piston by the fall of back pressure against it.

Adopting, then, the formula (27) as applicable for all cases of receiverengines arising in practice, it is required only to give the actual ratio of expansion in the first cylinder, and to multiply this ratio by the ratio of the capacities of the two cylinders, to arrive at the ratio of expansion to be employed in the formula. This is literally the actual combined ratio of expansion for the first case, without intermediate fall of pressure, as was found (page 865), represented by R" in the formula (27).

FORMULAS FOR COMPOUND ENGINES.

RULE 2. To find the net work done by steam in the two cylinders of a receiver-engine for one stroke, with a given actual ratio of expansion in the first cylinder. Multiply the actual ratio of expansion in the first cylinder by the ratio of the two cylinders, and to the hyperbolic logarithm of the compound ratio add 1; multiply the sum by the initial period of admission to the first cylinder, plus the clearance, in feet (product A). Divide the ratio of the two cylinders, minus 1, by the actual ratio of expansion in the first cylinder; add i to the quotient, and multiply the sum by the initial clearance in feet (product B). Subtract product B from product A, giving the remainder C. Multiply the area of the first cylinder, in square inches, by the total initial pressure in pounds per square inch, and by the remainder C. The product is the net work in foot-pounds for one stroke.

This rule is applicable to any of the four cases, page 865: a = 1 square inch, P = 63 lbs. per square inch, c=.42 foot, l'= 2.42 feet, R' = 2.653, R” = 7.959, and r=3. Then, on the model of the given formula (27),

×

w=63 [2.42 (1 + 2.0743) − .42 (1 +

I 3 2.653

= 63 (7.440.737) = 422.29 foot-pounds,

Or, following the wording of as was before calculated for the first case. the rule:-The combined actual ratio of expansion is 7.959, of which the hyperbolic logarithm is 2.0743; adding 1 to this, the sum, 3.0743, is multiplied by 2.42, the initial period of admission plus the clearance, and 3.0743 x 2.427.440 (product A). Again, the ratio of the cylinders is 3, and 3-12; the actual ratio of expansion in the first cylinder is 2.653, and 22.653-754. Adding 1 to this quotient, the sum is multiplied by the initial clearance .42, or 1.754 × .42.737 (product B). The difference of products A and B is (7.440.737 =) 6.703, and this, multiplied by 63 lbs., the initial pressure per square inch, and by 1, the area of the piston in square inches, gives

6.703 × 63 × 1 =: 422.29 foot-pounds,

the work of one stroke.

COMPRESSION OF STEAM IN THE CYLINDER..

The work expended in compressing such exhaust steam as is not permitted to escape during the return-stroke of the piston, and is shut into the cylinder against the retiring piston, is to be reckoned against the quantity of steam thus reclaimed. For every phase of the distribution there is a particular period of compression, by the adoption of which the resulting efficiency of the steam, for a given distribution, is raised to a maximum. The method of determining the best period of compression will be given in the author's work on The Steam Engine. The following table, No. 298, contains the best periods of compression for several periods of admission, with 7 per cent. clearance, and for several back exhaust-pressures. It is seen, by the table, that, the more expansively the steam is worked, the greater should be the period of compression-that is, the exhaust port should be closed the earlier in the course of the return-stroke; and that the greater the proportion of back-pressure to initial-pressure, the less should be the period of compression.

Table No. 298.-COMPRESSION OF STEAM IN THE CYLINDER:-
BEST PERIODS OF COMPRESSION.

PRACTICE OF EXPANSIVE WORKING OF

STEAM.

ACTUAL PERFORMANCE OF STEAM IN THE STEAM-ENGINE.

In working steam expansively, the practical performance is affected by several circumstances. There is the influence of the wire-drawing of the steam during its admission into the cylinders; of the needful opening of the exhaust passages before the end of the stroke, for the escape of the steam from the cylinder; of the back exhaust pressure on the piston, and the closing of the exhaust passage before the end of the return-stroke, with the consequent shutting in and compression by the piston of a portion of the exhausting steam. These influences have been analyzed and measured by the author. He concluded that, when the cylinders were liberally proportioned, first, the possible loss by early exhaust was of no importance, and that the early release was, on the contrary, beneficial, in facilitating a complete exhaust during the return-stroke; second, that as the loss by wire-drawing was of little or no moment, and that, as wire-drawing was, to some degree, equivalent to an earlier cut-off, it might even prove advantageous in point of economy; third, that the loss by back exhaust pressure in excess of the atmospheric resistance in non-condensing engines, in good practice, is of little or no importance. These conclusions were based upon the performance of locomotives, fitted with the link-motion, and worked with steam of 100 lbs. effective pressure per square inch in the boiler; but they are applicable to all classes of steam-engine.1

The only obstacle to the working of steam advantageously to a high degree of expansion in one cylinder, in general practice, is the condensation to which it is subjected, when it is admitted into the cylinder at the beginning of the stroke, by the less hot surfaces of the cylinder and the piston; the proportion of which is increased with the ratio of expansion, so that the economy of steam by expansive working ceases to increase when the period of admission is reduced down to a certain fraction of the stroke, and that, on the contrary, the efficiency of the steam is diminished as the period of admission is reduced below that fraction. The initial condensation here pointed out, is succeeded by the re-evaporation of a portion of the condensed steam during the later portion of the period of expansion; because, as the pressure falls, the temperature of the steam, and of the water which it contains, also falls, until it ultimately descends below the actual temperature of the cylinder, when the heat of the cylinder is absorbed by the water, and

See Railway Machinery, 1855, pp. 69-99; also a paper on 'The Expansive Working of Steam in Locomotives," in the Proceedings of the Institution of Mechanical Engineers, 1852, pp. 60-82, and 109-128.