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ton in the case of that engine was 46 lbs. per square inch; and we have also seen that, in consequence of that resistance, the total pressure of the steam, when arriving in the cylinder, was also necessarily 46 lbs. per square inch.

The mass of water evaporated is 41.87 cubic feet per hour, or 0.70 cubic feet per minute. This water is immediately transformed, in the boiler, into steam, at the effective pressure of 50 lbs. per square inch or at the total pressure of 65 lbs. per square inch.

But we know the volume of the steam] generated under a determined pressure. Tables of that volume have been formed from experiment, and one will be found below, § 11. According to these tables, the steam, generated under a total pressure of 65 lbs. per square inch, occupies 435 times the space of the water which produced it.

Thus the water transformed into steam at the total pressure of 65 lbs. per square inch, and spent each minute in the motion, formed a volume of

0,70 c. ft. x 435 304 cubic feet.

This steam, penetrating into the cylinders, is then reduced to a pressure of 46 lbs. Its temperature, however, remains the same, because the pipes that conduct it to the cylinders and the cylinders themselves are immersed in the boiler, or surrounded by the flame that comes out of the fire-place. We know that the space occupied by the steam, when its temperature remains the same, augments in an inverse ratio to the pressure. At the moment it arrives in the cylinders, that same mass of steam occupies consequently a greater space in the proportion of 65 to 46.

Thus its total volume is then

65
304X = 430 cubic feet.
46

Now, the area of the two cylinders is 190 square inches or 1.32 square foot; thus the above volume of 430 cubic feet of steam, passing through the cylinders in a mninute, must necessarily cross them with a volocity of

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which gives us, consequently, the velocity of the piston in feet per minute with the supposed load.

gine, with the above-mentioned proportions || the circumference of the wheel to twice the is able, if in a good condition and with a stroke, thus the speed of the engine is well-animated fire, to draw a load of 100 t., tender included, with a velocity of 213 miles an hour.

The same mode of calculation may serve for any other load or any other engine. Thus, in general, making again use of the letters already employed in our research of the resistance on the piston, viz.

M representing the number of tons of the load.

n the resistance of the load per ton.
F the friction of the engine without load.
its additional friction for each to of the

load.

D the diameter of the wheel. d the diameter of the cylinder. 7 the length of the stroke. And p the atmospheric pressure per unit of surface.

R=(F+ô M+nM)

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ms PD Rd2l

or, putting for R its value found above, and passing from the speed per minute to the speed per hour, in multiplying by 60,

V = 60

ms PD
(F+M+nM) D+pd3l

It must be remarked that 60 8 is equal to S, or the evaporating power per hour; that is to say, that by employing this value it is no longer necessary to multiply by 60, and the reckoning will be simplified in its application.

The formula will then be, √ =

m PSD
[F+(d+n)M]D+pd2l

This will consequently be the general expression of the velocity of the engine per hour; expression in which everything is known by measures taken on the engine, even the evaporating power S, which results from the extent of the heating surface computed as above. m, which is the volume of the steam generated under the pressure pressure of the steam in P, is found in a table like the one below (Chap. V. Art. V. § 11.)

will be the pressure of the steam per unit
of surface in the cylinder as above demon-
strated (Chap V. Art. II.)
If, besides,

Р

express the boiler;

the total

s, The effective evaporating power of the engine expressed by the number of cubic feet the boiler is able to evaporate in a minute at the pressure P,

And m the ratio of the volume of steam, at the degree of pressure P, to the volume of water,

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is the space occupied by the steam when arrived in the cylinders.

This volume of steam, crossing the cylinders in a minute, if we divide it by the speed it must necessarily have, and consearea of the cylinders, we shall have the quently the velocity it will communicate to the piston.

Now the area of the two cylinders is d2; thus the velocity per minute will be,

ms P

d2R

By means of this formula, and by measures simply taken on the engine, it will therefore be easy to determine immediately the effect we may expect from it.

In that expression, the evaporating power S being expressed in cubic feet the resulting speed will also be expressed in feet. If we wish to have it in miles, as a mile contains 2580 ft., it will be sufficient to divide by that number, and the result will be the speed of the engine in miles per hour.

We shall see further on that the produce mP is almost invariable; and consequently we learn by the inspection of this formula, that the velocity of an engine with a given lead increases with the heating surface and the diameter of the wheel, and diminishes, cylinder and the stroke of the piston augon the contrary, when the diameter of the

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To deduce from that the speed of the engine in miles per hour, we must observe that an hour contains 60 minutes, and thus that the speed per hour will be 60 times as great; a mile containing 5280 ft., the pro-| duce must be divided by that number in In order to effect that division, the area of order to have the speed expressed in miles; the cylinders ought necessarily to be exand finally the speed of the engine, according to the proportion of the stroke to the pressed in units similar to those of the volume s. The area of the cylinders must be diameter of the wheel, is 5.887 times that of the piston. then expressed in square feet and not in inches; and the same condition is consequently required also for R, P, and p. So included.

We shall consequently have 326 × 60

5280

the x5.88721.83 miles, velocity of in the calculation we must express pres

the engine per hour.

(+n) VD

F

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After the manner that the calculation has been established, it is clear that the value we shall find for M, will be the number of tons of the total load, that is to say, tender

sures in lbs. per square foot, which puts § 3. Of the Heating Surface that must be them at the same rate as if expressed in the usual manner.

Thus we see that the evaporation supposed above, must necessarily produce a Passing from this expression to the velovelocity of 213 miles per hour for the en-city of the engine, we know that it is to the gin; that is to say, that a locomotive, en-velocity of the piston in the proportion of

adopted to obtain from an Engine a determined Velocity with a given Load. The same equation may also serve to determine any one of the indeterminate quantities in the general problem of loco

motive engines. Thus, for instance, it will|tion of the velocity of the engine to that of tion will, in that case, give for the diamet show the extent of heating surface, or the the piston, the result will naturally be the of the wheelevaporating power necessary to enable an relative speed of the engine. engine to draw a known load at a fixed speed. For that, we have only to draw from the general equation the value of S. It will be,

S-V[(8+) MDn FD+pd2]. m PD

The result thus obtained will be the effective evaporating power of the engine in cubic feet of water per hour; and as we have seen (Chap. V. Art. IV. § 3) that the effective evaporating power is equal to of the heating surface expressed in square feet, we shall easily obtain the last by multiplying the result by the fractional num

D

(P-p) d2 l
(d+ n) M+F

$9. Of the effective Pressure necessary in the Boiler of an Engine, the Dimensions of which are already fired, in order that the Engine may draw a certain Maximum Load.

We also see that in the case of a maximum load, the pressure of the steam in the cylinder will be the same as in the boiler, It is understood that this method can and that its velocity will be the very veloc-only succeed within certain limits, and that ity at which the steam is generated in the the diameter of the wheel cannot be reduced boiler; results which besides are, of them- beyond certain dimensions, fixed by the selves, evident to an attentive mind, and other requisites of the business. which have already been pointed out. loads, the engine-men never urge it so as In regard to the limit of speed with small to risk an accident, by too great a velocity in the motion of the piston, or other parts of the mechanism. Only one single instance, in the experiments we shall relate Finally, if the length of the stroke, the below, will be found, in which the engines diameter of the cylinders, and that of the attained a speed of 35 miles an hour. This wheel are already fixed, we may calculate § 4. Of the Maximum Load of an En-velocity is the greatest that has been ob- what is the pressure that must be produced gine with a given Pressure. served, until the present moment, except in the boiler to enable the engine to attain We found above (§ 2) the expression of during some extremely short intervals. the maximum load required. The same the load an engine is able to draw at a giv- take care partially to shut the regulator, the quantity P considered as unknown, When the train is too light, the engine-men equation resolved in that case, in regard to en velocity; the gives

ber.

and not to animate the fire to its highest
pitch, as we shall mention hereafter.
§ 6. Of the Diameter that ought to be giv
en to the Cylinder, to render an Engine
capable of attaining a fixed Maximum

Load.

more considerable may be the load. We
must, however, add that in all cases, for
the motion to be possible, the resistance on
the piston must not be greater than the force
that is to move it. Consequently, the re-
sistance we have expressed by R must, at
most, be equal to P. This observation
fixes the limits of the possible load, with a
determined pressure. Beyond that point
the equation may continue to give results,
to determine the diameter that ought to be
but they will no longer suit the question. given to the cylinders of an engine to ren-
The limit of the load with the pressure Pder it capable of drawing a fixed load at
will thus be known by the equation R

or,

[F+(8+ n) M]

which gives

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P;

(P — p d2 l M (8+ n) D s + n This equation will give the maximum load of the engine, including the weight of the tender, subservient, however, to the conditions of adhesion explained hereafter, in Chap. VIII.

$5. Of the Velocity of the Engine corres

concluded above (§ 4) the limit of possible
The same equation from which we have
loads with a given pressure, may

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also serve

d= /D [ ( 8+n) M + F ].
(P-p)
This diameter will be expressed in feet,
according to the manner the calculation was
made.
It will be easily reduced to the
common expression in inches.

a

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This pressure will be expressed, according to the adopted measures, in pounds per square foot, but, by taking the part of it, of pounds per square inch. we may reduce it to the usual expression

The same would take place in regard to easily found ;--we shall not stop any longer any other research. These deductions are add, that the values given by those equaon this point. It is scarcely necessary to tions are only applicable to the questions, in as far as they are not in opposition to the practical rules of construction. Thus, the pressure determined above must in no case exceed the resistance of which the metal of the boiler is capable; neither must the diameter of the wheel be large enough to put nor small enough to destroy its speed, &c. the engine in danger in going off the rails, &c.

§ 10. Synoptical Table of the preceding Formula.

$7. Of the Length that ought to be given to the Stroke of the Piston of an Engine, the Cylinders of which have already a fixed Diameter, so as to enable that Engine to draw a certain Maximum Load. In the same manner, also, if the diameter In a view to facilitate practical researchof the cylinder has already been chosen ones, we shall collect here those different foraccount of some other consideration, we formulæ into a table. may, in a certain degree, produce the same effect; that is to say, render the engine able to attain the maximum load required, by adopting for the stroke of the piston a If we write this expression under the fol- suitable length. In that case the equation lowing form

ponding with the Maximum Load. Putting that value of M in the formula that gives the speed, we have the speed corresponding with the maximum load. After the necessary reductions we find— m SD V = del

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we shall perceive at first sight, that it is
exactly the speed produced by the passage
in the cylinders of the steam of the boiler, pressed, according to the adopted meas-
when that steam undergoes no reduction ures, in feet and decimals of feet; one may
of pressure. In fact,

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The signs employed having the following significations, viz. :—

M, representing the number of gross tons
of the load, tender included.
n, the resistance per ton of the load, or ac-
cording to the determination already
made, n = 8 lbs.

F, the friction of the engine without load,
taken, according to the average of the
above experiments, in case the engine is
not yet constructed; that is to say, at
15 lbs. per ton of its presumed weight.
In case the engine is already constructed,
and one wishes to obtain a very accurate
result, F must be determined by a direct
the additional friction of the engine per
experiment made on the engine itself.
ton of load, or according to the determin-
&tion hereabove, d= 1 lb. ; and, conse-
quently, (6+n) = 9 lbs.

§8. Of the Diameter that ought to be giv.
en to the Wheel of an Engine, so as to
enable it to draw a fixed Maximum Load.,
We may also obtain the same result by
reducing, in a suitable proportion, the di-
ameter of the wheel, by which the speed of
the engine will be diminished, and a great-D, the diameter of the wheel, expressed in
er power of traction given to it. The equa- feet.

d, the diameter of the cylinder, also expressed in feet and decimals of feet.

1, the length of the stroke, in feet and decimals of feet.

P, the total pressure (or atmospheric pres-m, sure included) of the steam in the boiler, expressed in pounds per square foot; that is to say, 144 times the pressure per square inch.

p, the atmospheric pressure expressed in pounds per square foot as above, that is to say, p= 2117 lbs. ; and, consequently (P-p), the effective pressure of the steam in the boiler, being expressed in the same manner, viz., in pounds per square foot.

e

lected, that the reduced heating surface We must remark that these formulæ ar
itself consists of the sum of the heating not such as are called empiric ones; tha
surface of the fire-place, more the third is to say, imaginary suppositions, corres-
part of the heating surface of the tubes.) ponding more or less exactly with experi-
being the ratio of the volume of the ence. They are, on the contrary, rigorous
steam at the total pressure P, to the vol- deductions from the most solid principles
ume of water that has produced it, accord-of mechanics; their elements have been
ing to the known tables, one of which determined by direct experiments, and their
will be found in one of the following par-results will soon be confirmed in the same
agraphs.
way.

V, finally, being the velocity of the engine In all cases, these formulæ suppose the in feet per hour, that velocity being ne- engine drawing its load on a dead level. If cessarily expressed in that manner for it be required to apply them to the case of the general harmony of the calculation; an inclined plane, it will suffice to take for but as a mile contains 5280 feet, it can M, not the nominal load of the engine, but easily be reduced to the speed in miles, its real load; that is to say, not merely the S, being the effective evaporating power of resistance of the wagons, but their resistthe engine per hour, or otherwise, accord- These different signs being thus well un-ance in ascending the inclined plane in ing to the described experiments, S being derstood, and the letters n and ♪ being re- question, as will be seen in Chap. VII. Art. the of the number of square feet in the placed by their values, 8 lbs. and 1 lb., the II. reduced heating surface. (It will be recol-formulæ above give the following table :-§ 11. Table of the Volume of the Steam

SYNOPTICAL TABLE OF THE PRACTICAL FORMULE OF LOCOMOTIVE ENGINES.

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and vice versa.

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3. Heating Surface that must be given to the boiler of an Engine, in order that it may
The equation gives the effective evaporating power per hour, from which the heating
surface may be deduced by multiplying by the fractional number 10:-
draw a known load with a fixed velocity.

6. Diameter that must be given to the cylinder of an Engine not yet constructed, in order
that, if necessary, it may draw a certain maximum load.

The diameter being expressed in feet and decimals of feet, its expression in inches will
be found in multiplying by 12:—

This stroke will be expressed in feet, and may be transformed into inches, as above:-

7. Length of stroke of the Piston that may replace the diameter of the cylinder and produce the same effect of maximum load.

8. Diameter of the wheel of an Engine, in order to render it able to draw the same maximuin load :

This speed being expressed in feet, the speed per mile will be its 32 part:—

9. Effective pressure that must be produced in the boiler of a given Engine, in order to render that Engine capable of drawing a certain maximum load.

This pressure being expressed in pounds per square foot,

4. Maximum load that an Engine is able to draw at a determined pressure.
This load is expressed in gross tons, and includes the tender :-

This lead will be expressed in gross tons, tender included :—

5. Velocity of an Engine, with its maximum load.

its T

the pressure

per inch will

( F + 9 M ) D + p d2 i

mPSD pd2l V Ꮽ Ꮩ Ꭰ

' V [(9 M+F) D+pd2l]

mPD

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(P — 'p) =

D (9 M + F):

QUESTIONS TO BE SOLVED.

1. Velocity which an Engine of known proportions will take, when working at a given pressure, and drawing a determined load.

The result being the speed in feet per hour, the speed in miles will be obtained by dividing by 5280:

V =

2. Load that a given Engine will be able to draw, with a known pressure, and at a determined velocity.

M

generated under different degrees of Pressure, necessary for the application of the Formula.

The use of the formulæ we have obtained, necessitating a knowledge of the volume of the steam at different degrees of pressure, we subjoin here a table which we have calThe incuted from 5 to 5 lbs. pressure. termediate degrees may be easily filled up; but it would be an unnecessary operation, as we shall see that the pressure in the boiler has so little influence on the speed, that we may, in our calculations, take from the table the pressure nearest to the one we require, provided we also take the volume corresponding with that approximate pressure.

The reason of the little influence the pressure has on the result is, that in proportion as the pressure augments, the volume of the steam diminishes, so that the produce mP, that the equation contains, remains constant for such values of P as are very near to each other. We shall very shortly be witnesses of the fact, which will be explained in the calculation we shall make of the velocity of the engine at different pressure.

TABLE OF THE VOLUME OF THE STEAM GENE-
RATED UNDER DIFFERENT PRESSURES.

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9

§ 12. Of the combined Proportions that ought|| 1254
to be given to the parts of an Engine, in
order that it may fulfil several conditions at
the same time.

=

=

239 t.

feet..

10
9

143 square, back

H

139 t. on a dead level. The surpine must be 43 × of resistance occasioned by the inclination Finally, this last condition will be fulof the plane is, therefore, equal to the rac-filled by giving, for instance, to the fireWe have given, above, separate from tion of 139 t. on a level. Consequently place a heating surface of 50 square feet, each other, the different practical formula the total traction on the rising nd wiljand to the tubes a surface of 280 square of locomotion; but we may also combine be 139 t. + 100 t. those formulæ with one another. To give This example indicates sufficiently the an example of this, and at the same time Thus, in this case, the load on the in-manner in which the calculation is to be a practical application of the results obclined plane will be V = 239 t. made. It would be the same with any tained hitherto, we shall suppose that it is And the load on the dead level = 100 t. other combination that might occur. Evirequired to build an engine capable of The engine being supposed to weigh dently, nothing is required but to bring drawing a certain given maximum load, tion of about 180lbs. If, besides, we suppose the different, unknown quantities, and to 12 t., with coupled wheels, will have a fric-together the different equations concerning and, at the same tine, capable of attaining it to have a wheel of 5 feet, with a stroke of express that they exist simultaneously.

a certain speed, with another load also known,

In this case we may determine the diameter of the cylinder, according to the first condition; and the heating surface of the boiler according to the second. Let ting, therefore, M' be the given maximum load, M" the second load mentioned above, and V" the velocity of the engine corresponding with that second load, we shall have simultaneously the two following, equations (See § 6 and 3.)

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This case is evidently that of a railway. on which it would be required that the average trains should have on a level a certain regular speed, and that, at the same time, the engines might ascend with those trains, and without any extra help, an acclivity occurring on a point of the road.

Let us then suppose that it is wanted to build an engine with coupled wheels, capable of drawing a train of 100 gross tons, at a speed of 20 miles an hour on a dead level; and that it is required, at the same time, that that engine be able to ascend without extra aid, and with the same load (reducing, however, its speed,) a plane inclined in the proportion of

We know that an engine working upon a level undergoes, from its load, a certain degree of resistance, which proceeds from the friction of the wagons; but in going up an inclined plane, the load presents not only that same friction of the wagons, but also a surplus of resistance proceeding from the tendency of the train to roll back towards the foot of the plane. The force that draws the train backwards, depends on the weight of the train and on the inclination of the plane. It is the gravity along the plane, and is equal to the mass that is to be moved, divided by the number that marks the inclination of the plane.

On an inclination of, the gravity of a weight of 112 t., which is the weight of the train and engine together, is in pounds. 112 x 2240 = 1254 lbs. 200

Now, 1254 lbs.. at the rate of 9 lbs. per ton (including the increase of friction in the engine,) represents the resistance of

16 in. or 1.33 ft.; and if we wish the effec-
tive pressure (P-p) in the boiler, during|
the ascent, not to exceed 60 lbs. per aquare
inch, or, in other words, 8640 lbs. per squar
foot, the first equation will give, for the di
ameter of the cylinder-

=

1 foot

AND

RTICLE VI.

TARLES OF THE PROPORTIONS
ECTS OF THE ENGINES.

A Practicable Table of the Diameter of the Cylinder and Pressure of Steam, nedrssary to enable a Locomotive Engine to draw a given Maximum Load.

5 (9 × 239 + 180) 8640 × 1.33 Thus the cylinder must have I ft or i 2 in. in diameter. We Fovest calculated, in a special case, the diameter necessary for the cylinThis value must be introduced in the der of an engine working at a given pressecond equation with the other data of the sure, so that it may draw a certain maxiproblem. Observing, moreover, that dur-mum load. In continuing the same calcuing the journey one may reduce the effec-lation through a series of different cases, tive pressure in the boiler to 50 lbs. (or 65 after the formula § 6, we form the following lbs. total pressure) per square inch, which practical table, which will show either the gives for the corresponding volume of the diameter of the cylinder when the pressure steam m =435 (see the table given in the preceding paragraph,) the second equation will give

= 42.65

is given, or the pressure in the boiler, when it is the diameter of the cylinder which is determined, or, finally, the maximum load when the two other data are fixed beforehand.

S=20×5280 (900+180) 5+21 17 × 1×1.33 435 X (65X 144) × 5 It must be understood that the engines By which we see that the effective evap-||will not be able to draw the loads marked orating power S of the engine must be 43 in the table, unless the rails are in such a cubic feet of water per hour. And, as we state as to offer a sufficient adhesion to the know, by the experiments related above, wheels; without which condition, the that the effective power is equal to of movement could not be effected, as will be the reduced heating surface, this surface" explained in Chap. VIII.

A PRACTICAL TABLE OF THE DIAMEter of the cyLINDER AND PRESSURE OF STEAM

CORRESPONDING TO GIVEN MAXIMUM LOADS.

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§2. A Practice Table of the length of use, the following table is formed, which || length of stroke of the piston and diameter Stroke of the Pis, and Diameter of will show, at first sight, either the length of the wheel. Wheel, necessary to enable an Engine to of stroke of the piston, or the diameter of draw a fixed Maximum Load at a given Pressure ar

the wheel which an engine ought to have, § 3. A Practical Table of the Area of Heat for it to draw a maximum load at a given In solving the formula § 7, in a series of pressure; or, again, the maximum loads cases adapted to the engines the most in corresponding to given dimensions for the

A PRACTICAL TABLE OF THE LENGTH OF STROKE AND DIAMETER
OF WHEEL, CORRESPONDing to giveN MAXIMUM LOADS.

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ing Surface capable of producing a given Velocity with given Loads.

In order to faciliate practical researches, we shall extend, to a certain number of the most ordinary cases, the calculation of the heating surface capable of producing predetermined effects.

The table which we are thus going to form after the formula in § 3, may serve, not only to determine the heating surface" capable of producing desired effects, but also the velocity of given loads, when the heating surface is already determined.

The table supposes the engine working at 50 lbs. effective pressure, per square inch, in the boiler. As, however, the pres sure has no perceptible influence on the velocity, as will be seen hereafter, if the engine works at a higher pressure, it will be able to attain a more considerable maximum load; but for all the loads of the table, it will, nevertheless, require the same heating surface in order to produce the same velocity. In consequence, the table may serve for any pressure, either above or be low 50 lbs. The only difference will be in the maximum loads, which, agreeable to the pressure, will be greater or smaller than those fixed in the table.

By recurring to § 10 of the preceding Article, it will be seen in what manner the area of heating-surface is to be computed.

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