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By which the pressure of steam in the cylinder will be known, when the effective evaporation, the diameter of the cylinder, and speed of the piston, are given.

If it be required to express the mechanical effect produced per minute by the action of steam on the piston, it is only necessary to multiply the pressure on the surface of the piston by the space per minute through which the piston moves. This will give

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which expresses the whole mechanical effect per minute in pounds raised one foot.

If the steam be worked expansively, let it be cut off after the piston has moved through a part of the stroke expressed by e.

The volume of steam of the undiminished pressure P' admitted per minute through the valve would then be

and the ratio of this

being expressed by S',

VA (e + c);

volume to that of the water producing it we should have

VA (e + c)

S' = W

The final volume into which this steam is subsequently expanded being VA (1 + c), its ratio to that of the water will be

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The pressure P', till the steam is cut off, will be

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The mechanical effect E' produced per minute by the steam of full pressure will be

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and the effect E" per minute produced by the expansion of the steam will by (12.) be

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If the engine work without expansion, e = 1;

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-

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(19.)

(20.)

as before; and the effect per minute gained by expansion will therefore be

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which therefore represents the quantity of power gained by the expansive action, with a given evaporating power.

In these formulæ the total effect of the steam is considered without reference to the nature of the resistances which it has to

overcome.

These resistances may be enumerated as follows:

1. The resistance produced by the load which the engine is required to move.

2. The resistance produced by the vapour which remains uncondensed if the engine be a condensing engine, or of the atmospheric pressure if the engine do not condense the steam. 3. The resistance of the engine and its machinery, consisting of the friction of the various moving parts, the resistances of the feed pump, the cold water pump, &c. A part of these resistances are of the same amount, whether the engine be loaded or not, and part are increased, in some proportion depending on the load.

When the engine is maintained in a state of uniform motion, the sum of all these resistances must always be equal to the whole effect produced by the steam on the piston. The power expended on the first alone is the useful effect.

Let R = the pressure per square foot of the piston surface, which balances the resistances produced by the load.

mR = the pressure per square foot, which balances that part of the friction of the engine which is proportional to the load.

r = the pressure per square foot, which balances the sum of all those resistances that are not proportional to the load.

The total resistance, therefore, being R + mR + r, which, when the mean motion of the piston is uniform, must be equal to the mean pressure on the piston. The total mechanical effect

must therefore be equal to the total resistance multiplied by the space through which that resistance is driven. Hence we shall have

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RVA (1+m) = We' - VA (b + r).

By solving this for VA, we obtain

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(22.)

(23.)

This quantity RVA, being the product of the resistance RA, of the load reduced to the surface of the piston, multiplied by the space through which the piston is moved, will be equal to the load itself multiplied by the space through which it is moved. This being, in fact, the useful effect of the engine, let it be expressed by U, and we shall have

We'R

U =

R (1+m) + b + r;

Or by (22.),

(24.)

(25.)

U (1+m) We'-VA (b + r).

The value of the useful effect obtained from these formulæ will be expressed in pounds, raised one foot per minute, W being the effective evaporation in cubic feet per minute, A the area of the piston in square feet, and V the space per minute through which it is moved, in feet.

Since a resistance amounting to 33,000 pounds moved through one foot per minute is called one-horse power, it is evident that the horse power H of the engine is nothing more than the useful effect per minute referred to a larger unit of weight or resistance; that is to 35,000 pounds instead of one pound. Hence we shall have

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Since the useful effect expressed in (24.) and (25.) is that due to a number of cubic feet of water, expressed by W, we shall obtain the effect due to one cubic foot of water, by dividing U by W. If, therefore, U' be the effect produced by the effective evaporation of a cubic foot of water, we shall have

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If the quantity of fuel consumed per minute be expressed by F, the effect produced by the unit of fuel, called the DUTY of the engine, will, for like reason, be

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If the fuel be expressed in hundredweights of coal, then D will express the number of pounds' weight raised one foot by a hundredweight of coal.

By solving (24.) and (25.) for W, we obtain

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W =

1

{ 33000 H (1 + m) + VA (b + r)}. (32.)

The evaporation necessary per horse power per minute will be found by putting H = 1 in these formulæ.*

It will be observed that the quantities A and V, the area of the cylinder and the speed of the piston, enter all these formulæ as factors of the same product. Other things, therefore, being the same, the speed of the piston will be always inversely as the area of the cylinder. In fact, VA is the volume of steam per minute employed in working the piston, and if the piston be increased or diminished in magnitude, its speed must be inversely

• Formulæ equivalent to some of the preceding are given, with numerous others, by M. de Pambour, in his Theory of the Steam Engine. These mathematical details contain nothing new in principle, being merely the application of the known principles of general mechanics to this particular machine. M. de Pambour objects against the methods of calculating the practical effects of steam engines generally adopted by engineers in this country. Their estimates of the loss of power by friction, imperfect condensation, and other causes, are, as I have stated in this volume, vague, and can be regarded at best as very rough approximations; but, subject to the restrictions under which their methods of calculation are always applied, they are by no means so defective as M. de Pambour supposes. He proves what he considers to be their inaccuracy, by applying them in cases in which they are never intended to be applied by English engineers. Those who desire to reduce to general algebraical formulæ the effects of the different kinds of steam engines will, however, find the volume of M. de Pambour of considerable use.

varied by the necessity of being still moved through the same number of cubic feet by. the same volume of steam.

It has been already stated in the text, that no satisfactory experiments have yet been made, by which the numerical value of the quantity r can be exactly known. In engines of different magnitudes and powers, this resistance bears very different proportions to the whole power of the machine. In general, however, the larger and more powerful the engine, the less that proportion will be.

That part of this resistance which arises from the reaction of the uncondensed vapour on the piston is very variable, owing to the more or less perfect action of the condensing apparatus, the velocity of the piston, and the magnitude and form of the steam passages. M. de Pambour states, that, by experiments made with indicators, the mean amount of this resistance in the cylinder is 24 lbs. per square inch more than in the condenser, and that the pressure in the latter being usually 14 lb. per square inch, the mean amount of the pressure of the condensed vapour in the cylinder is about 4 lbs. per square inch. Engineers, however, generally consider this estimate to be above the truth in well-constructed engines, when in good working order.

In condensing low pressure engines of forty horse power and upwards, working with an average load, it is generally considered that the resistance produced by the friction of the machine and the force necessary to work the pumps may be taken at about 2 lbs. per square inch of piston surface.

Thus the whole resistance represented by r in the preceding formulæ, as applied to the larger class of low pressure engines, may be considered as being under 6 lbs. per square inch, or 864 lbs. per square foot, of the piston. It is necessary, however, to repeat, that this estimate must be regarded as a very rough approximation; and as representing the mean value of a quantity subject to great variation, not only in one engine compared with another, but even in the same engine compared with itself at different times and in different states.

In the same class of engines, the magnitude of the clearage is generally about a twentieth part of the capacity of the cylinder, so that c = 0.05.

That part of the resistance which is proportional to the load, and on which the value of m in the preceding formulæ depends, is still more variable, and depends so much on the form, magnitude, and the arrangement of its parts, that no general rule can be given for its value. It must, in fact, be determined in every particular case. In the practical application of the preceding formulæ in condensing engines we shall have

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