Chapter XXIII: Appendix (1)

_On the Relation between the Temperature, Pressure, and Density_
_of Common Steam._

There is a fixed relation between the temperature and pressure of common steam, which has not yet been ascertained by theory. Various empirical formulæ have been proposed to express it, derived from tables of temperatures and corresponding pressures which have been founded on experiments and completed by interpolation.

The following formula, proposed by M. Biot, represents with great accuracy the relation between the temperature and pressure of common steam, throughout all that part of the thermometric scale to which experiments have been extended.

Let

a = 5·96131330259
log. a_{1} = 0·82340688193 − 1
log. b_{1} = −·01309734295
log. a_{2} = 0·74110951837
log. b_{2} = −·00212510583

The relation between the temperature t with reference to the centesimal thermometer, and the pressure p in millimètres of mercury at the temperature of melting ice, will then be expressed by the following formula:—

log. p = a − a_{1}b_{1}^{20 + t} − a_{2}b_{2}^{20 + t}. (1.)

Formulæ have, however, been proposed, which, though not applicable to the whole scale of temperatures, are more manageable in their practical application than the preceding.

For pressures less than an atmosphere, Southern proposed the following formula, where the pressure is intended to be expressed [Pg506] in pounds per square inch, and the temperature in reference to Fahrenheit's thermometer,—

p = 0·04948 + ((51·3 + t) / 155·7256)^{5·13} |
|. (2.)
t = 155·7256 ((p − 0·04948) − 51·3)^{1/(5·13)} |

The following formula was proposed by Tredgold, where p expresses the pressure in inches of mercury:—

p = ((100 + t) / 177)^{6}.

This was afterwards modified by Mellet, and represents with sufficient accuracy experiments from 1 to 4 atmospheres. Let p represent pounds per square inch, and t the temperature by Fahrenheit's thermometer,—

p = ((103 + t) / 201·18)^{6} |
|. (3.)
t = 201·18 p^{1/6} − 103 |

M. de Pambour has proposed the following formula, also applicable through the same limits of the scale:—

p = ((98·806 + t) / 198·562)^{6} |
|. (4.)
t = 198·562 p^{1/6} − 98·806 |

MM. Dulong and Arago have proposed the following formula for all pressures between 4 and 50 atmospheres:—

p = (0·26793 + 0·0067585 t)^{5} |
|. (5.)
t = 147·961 p^{1/5} − 39·644 |

It was about the year 1801, that Dalton, at Manchester, and Gay-Lussac, at Paris, instituted a series of experiments on gaseous bodies, which conducted them to the discovery of the law mentioned in art. (96.), p. 171. These philosophers found that all gases whatever, and all vapours raised from liquids by heat, as well as all mixtures of gases and vapours, are subject to the _same quantity of expansion_ between the temperatures of melting ice and boiling water; and by experiments subsequently made by Dulong and Petit, this uniformity of expansion has been proved to extend to all temperatures which can come under practical inquiries.

Dalton found that 1000 cubic inches of air at the temperature of melting ice dilated to 1325 cubic inches if raised to the temperature of boiling water. According to Gay-Lussac, the increased volume was 1375 cubic inches. The latter determination has been subsequently found to be the more correct one.[40]

[Pg507] It appears, therefore, that for an increase of temperature from 32° to 212°, amounting to 180°, the increase of volume is 375 parts in 1000; and since the expansion is uniform, the increase of volume for 1° will be found by dividing this by 180, which will give an increase of 208-1/3 parts in 100,000 for each degree of the common thermometer.

To reduce the expression of this important and general law to mathematical language, let v be the volume of an elastic fluid at the temperature of melting ice, and let nv be the increase which that volume would receive by being raised one degree of temperature under the same pressure. Let V be its volume at the temperature T. Then we shall have

V = v + nv (T − 32) = v (1 + n (T − 32)).

If V′ be its volume at any other temperature T′, and under the same pressure, we shall have, in like manner,

V′ = v (1 + n (T′ − 32)).

Hence we obtain

V/V′ = (1 + n (T − 32)) / (1 + n (T′ − 32)); (6.)

which expresses the relation between the volumes of the same gas or vapour under the same pressure and at any two temperatures. The co-efficient n, as explained in the text, has the same value for the same gas or vapour throughout the whole thermometric scale. But it is still more remarkable that this constant has the same value for all gases and vapours. It is a number, therefore, which must have some essential relation to the gaseous or elastic state of fluid matter, independent of the peculiar qualities of any particular gas or vapour.

The value of n, according to the experiments of Gay-Lussac, is 0·002083, or 1/480.

To reduce the law of Mariotte, explained in (97.) p. 171., to mathematical language, let V, V′ be the volumes of the same gas or vapour under different pressures P, P′, but at the same temperature. We shall then have

VP = V′P′. (7.)

If it be required to determine the relation between the volumes of the same gas or vapour, under a change of both temperature and pressure, let V be the volume at the temperature T and under the pressure P, and let V′ be the volume at the temperature T′ and under the pressure P′. Let v be the volume at the temperature T and under the pressure P′.

By formula (7.) we have

VP = vP′;

[Pg508] and by formula (6.) we have

(V′/v) = (1 + n(T′ − 32)) / (1 + n(T − 32))

Eliminating v, we shall obtain

(V/V′) = (P′/P) · (1 + n(T − 32)) / (1 + n(T′ − 32));

or,

(VP/V′P′) = (1 + n(T − 32)) / (1 + n(T′ − 32)); (8.)

which is the general relation between the volumes, pressures, and temperatures of the same gas or vapour in two different states.

To apply this general formula to the case of the vapour of water, let T′ = 212°. It is known by experiment that the corresponding value of P′, expressed in pounds per square inch, is 14·706; and that V′, expressed in cubic inches, the water evaporated being taken as a cubic inch, is 1700. If, then, we take 0·002083 as the value of n, we shall have by (8.),

VP = 1700 × 14·706 × (1 + 0·002083 (T − 32)) / (1 + 0·002083 × 180)

= 18183(1 + 0·002083 (T − 32)). (9.)

If, by means of this formula (9.), and any of the formulæ (1.), (2.), (3.), (4.), (5.), T were eliminated, we should obtain a formula between V and P, which would enable us to compute the enlargement of volume which water undergoes in passing into steam under any proposed pressure. But such a formula would not be suitable for practical computations. By the formulæ (1.) to (5.), a table of pressures and corresponding temperatures may be computed; and these being known, the formula (9.) will be sufficient for the computation of the corresponding values of V, or the enlargement of volume which water undergoes in passing into steam.

In the following table, the temperatures corresponding to pressures from 1 to 240 lbs. per square inch are given by computation from the formulæ (2.) to (5.), and the volumes of steam produced by an unit of volume of water as computed from the formula (9.).

The mechanical effect is obtained by multiplying the pressure in pounds by the expansion of a cubic inch of water in passing into steam expressed in feet, and is therefore the number of pounds which would be raised one foot by the evaporation of a cubic inch of water under the given pressure. [Pg509]

——————————————————————————————————————————————————————————
| | Volume of | Mechanical
Total pressure| | the Steam | Effect of
in Pounds | Corresponding| compared to | a Cubic Inch
per Square | Temperature. | the Volume | of Water
Inch. | | of the | evaporated
| | Water that | in Pounds
| | has | raised One
| | produced it.| Foot.
——————————————+——————————————+—————————————+—————————————
1 | 102·9 | 20868 | 1739
2 | 126·1 | 10874 | 1812
3 | 141·0 | 7437 | 1859
4 | 152·3 | 5685 | 1895
5 | 161·4 | 4617 | 1924
6 | 169·2 | 3897 | 1948
7 | 175·9 | 3376 | 1969
8 | 182·0 | 2983 | 1989
9 | 187·4 | 2674 | 2006
10 | 192·4 | 2426 | 2022
11 | 197·0 | 2221 | 2036
12 | 201·3 | 2050 | 2050
13 | 205·3 | 1904 | 2063
14 | 209·1 | 1778 | 2074
15 | 212·8 | 1669 | 2086
16 | 216·3 | 1573 | 2097
17 | 219·6 | 1488 | 2107
18 | 222·7 | 1411 | 2117
19 | 225·6 | 1343 | 2126
20 | 228·5 | 1281 | 2135
21 | 231·2 | 1225 | 2144
22 | 233·8 | 1174 | 2152
23 | 236·3 | 1127 | 2160
24 | 238·7 | 1084 | 2168
25 | 241·0 | 1044 | 2175
26 | 243·3 | 1007 | 2182
27 | 245·5 | 973 | 2189
28 | 247·6 | 941 | 2196
29 | 249·6 | 911 | 2202
30 | 251·6 | 883 | 2209
31 | 253·6 | 857 | 2215
32 | 255·5 | 833 | 2221
33 | 257·3 | 810 | 2226
34 | 259·1 | 788 | 2232
35 | 260·9 | 767 | 2238
36 | 262·6 | 748 | 2243
37 | 264·3 | 729 | 2248
38 | 265·9 | 712 | 2253
39 | 267·5 | 695 | 2259
40 | 269·1 | 679 | 2264
41 | 270·6 | 664 | 2268
42 | 272·1 | 649 | 2273
43 | 273·6 | 635 | 2278
44 | 275·0 | 622 | 2282
45 | 276·4 | 610 | 2287
46 | 277·8 | 598 | 2291
47 | 279·2 | 586 | 2296
48 | 280·5 | 575 | 2300
49 | 281·9 | 564 | 2304
50 | 283·2 | 554 | 2308
51 | 284·4 | 544 | 2312
52 | 285·7 | 534 | 2316
53 | 286·9 | 525 | 2320
54 | 288·1 | 516 | 2324
55 | 289·3 | 508 | 2327
56 | 290·5 | 500 | 2331
57 | 291·7 | 492 | 2335
58 | 292·9 | 484 | 2339
59 | 294·2 | 477 | 2343
60 | 295·6 | 470 | 2347
61 | 296·9 | 463 | 2351
62 | 298·1 | 456 | 2355
63 | 299·2 | 449 | 2359
64 | 300·3 | 443 | 2362
65 | 301·3 | 437 | 2365
66 | 302·4 | 431 | 2369
67 | 303·4 | 425 | 2372
68 | 304·4 | 419 | 2375
69 | 305·4 | 414 | 2378
70 | 306·4 | 408 | 2382
71 | 307·4 | 403 | 2385
72 | 308·4 | 398 | 2388
73 | 309·3 | 393 | 2391
74 | 310·3 | 388 | 2394
75 | 311·2 | 383 | 2397
76 | 312·2 | 379 | 2400
77 | 313·1 | 374 | 2403
78 | 314·0 | 370 | 2405
79 | 314·9 | 366 | 2408
80 | 315·8 | 362 | 2411
81 | 316·7 | 358 | 2414
82 | 317·6 | 354 | 2417
83 | 318·4 | 350 | 2419
84 | 319·3 | 346 | 2422
85 | 320·1 | 342 | 2425
86 | 321·0 | 339 | 2427
87 | 321·8 | 335 | 2430
88 | 322·6 | 332 | 2432
89 | 323·5 | 328 | 2435
90 | 324·3 | 325 | 2438
91 | 325·1 | 322 | 2440
92 | 325·9 | 319 | 2443
93 | 326·7 | 316 | 2445
94 | 327·5 | 313 | 2448
95 | 328·2 | 310 | 2450
96 | 329·0 | 307 | 2453
97 | 329·8 | 304 | 2455
98 | 330·5 | 301 | 2457
99 | 331·3 | 298 | 2460
100 | 332·0 | 295 | 2462
110 | 339·2 | 271 | 2486
120 | 345·8 | 251 | 2507
130 | 352·1 | 233 | 2527
140 | 357·9 | 218 | 2545
150 | 363·4 | 205 | 2561
160 | 368·7 | 193 | 2577
170 | 373·6 | 183 | 2593
180 | 378·4 | 174 | 2608
190 | 382·9 | 166 | 2622
200 | 387·3 | 158 | 2636
210 | 391·5 | 151 | 2650
220 | 395·5 | 145 | 2663
230 | 399·4 | 140 | 2675
240 | 403·1 | 134 | 2687
——————————————+——————————————+—————————————+—————————————

[Pg511] In the absence of any direct method of determining the general relation between the pressure and volume of common steam, empirical formulæ expressing it have been proposed by different mathematicians.

The late Professor Navier proposed the following:—Let S express the volume of steam into which an unit of volume of water is converted under the pressure P, this pressure being expressed in kilogrammes per square mètre. Then the relation between S and P will be

S = a/(b + mP),

where a = 1000, b = 0·09, and m = 0·0000484.

This formula, however, does not agree with experiment at pressures less than an atmosphere. M. de Pambour, therefore, proposes the following changes in the values of its co-efficients:—Let P express the pressure in pounds per square foot; and let

a = 10000 b = 0·4227 m = 0·00258,

and the formula will be accurate for all pressures. For pressures above two atmospheres the following values give more accuracy to the calculation:—

a = 10000 b = 1·421 m = 0·0023.

In these investigations I shall adopt the following modified formula. The symbols S and P retaining their signification, we shall have

S = a/(b + P) (10.)

where

a = 3875969 b = 164.

These values of a and b will be sufficiently accurate for practical purposes for all pressures, and may be used in reference to low-pressure engines of every form, as well as for high-pressure engines which work expansively.

When the pressure is not less than 30 pounds per square inch, the following values of a and b will be more accurate:—

a = 4347826 b = 618.

_On the Expansive Action of Steam._

The investigation of the effect of the expansion of steam which has been given in the text, is intended to convey to those who are not conversant with the principles and language of analysis, some notion of the nature of that mechanical effect to which the advantages attending the expansive principle are due. We shall now, however, explain these effects more accurately. [Pg512]

The dynamical effect produced by any mechanical agent is expressed by the product of the resistance overcome and the space through which that resistance is moved.

Let
P = the pressure of steam expressed in pounds per square foot.
S = the number of cubic feet of steam of that pressure
produced by the evaporation of a cubic foot of water.
E = the mechanical effect produced by the evaporation of a
cubic foot of water expressed in pounds raised one foot.

Then we shall have E = PS; and if W be a volume of water evaporated under the pressure P, the mechanical effect produced by it will be WPS.

By (10.) we have

SP = a − bS.

Hence, for the mechanical effect of a cubic foot of water evaporated under the pressure P we have

E = a − bS. (11.)

Let a cubic foot of water be evaporated under the pressure P′, and let it produce a volume of steam S′ of that pressure. Let this steam afterwards be allowed to expand to the increased volume S and the diminished pressure P; and let it be required to determine the mechanical effect produced during the expansion of the steam from the volume S′ to the volume S.

Let
E′ = the mechanical effect produced by the evaporation of
the water under the pressure P′ without expansion.
E″ = the mechanical effect produced during the expansion
of the steam.
E = the mechanical effect which would be produced by
the evaporation under the pressure P without expansion.
_E_ = the total mechanical effect produced by the evaporation
under the pressure P′ and subsequent expansion.

Thus we have

_E_ = E′ + E″.

Let s be any volume of the steam during the process of expansion, p the corresponding pressure, and e″ the mechanical effect produced by the expansion of the steam. We have then by (10.)

p = (a/s) − b;

∵ de″ = (ads/s) − bds.

Hence by integrating we obtain

e″ = a log. s − bs + C;

[Pg513] which, taken between the limits s = S′ and s = S, becomes

E″ = a log. S/S′ − b(S − S′). (12.)

But by (11.) we have

E′ = a − bS′,
E = a − bS;
∵ E′ − E = b(S − S′);
∵ E″ = a log. S/S′ − E′ + E;
∵ _E_ = E″ + E′ = a log. S/S′ + E. (13.)

Or,

_E_ = a (1 + log. S/S′) − bS. (14.)

Hence it appears that the mechanical effect of a cubic foot of water evaporated under the pressure P may be increased by the quantity a log. S/S′, if it be first evaporated under the greater pressure P′, and subsequently expanded to the lesser pressure P.

The logarithms in these formulæ are hyperbolic.

To apply these principles to the actual case of a double acting steam engine,

Let
L = the stroke of the piston in feet.
A = the area of the piston in square feet.
n = the number of strokes of the piston per minute.
∵ 2n AL = the number of cubic feet of space through which
the piston moves per minute.
Let cLA = the clearage, or the space between the steam valve
and the piston at each end of the stroke.
∵ The volume of steam admitted through the steam valve
at each stroke of the engine will be 2n AL(1 + c).

Let
V = the mean speed of the piston in feet per minute,
∵ 2nL = V.

The volume of steam admitted to the cylinder per minute will therefore be VA (1 + c), the part of it employed in working the piston being VA.

Let
W = the water in cubic feet admitted per minute in the form
of steam through the steam valve.
S = the number of cubic feet of steam produced by a cubic
foot of water.

[Pg514] Hence we shall have

WS = VA (1 + c);
∵ S = (VA(1 + c))/W. (15.)

Since by (10.) we have

P = a/S − b;
∵ P = [Wa/(VA(1 + c))] − b. (16.)

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

VAP = W(a/(1 + c)) − VAb; (17.)

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

VA (e + c);

and the ratio of this volume to that of the water producing it being expressed by S′, we should have

S′ = (VA(e + c))/W.

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

S = (VA (1 + c))/W.

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

P′ = [Wa / (VA(e + c))] − b. (18.)

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

E′ = P′AVe = [Wae / (e + c)] − AVbe;

and the effect E″ per minute produced by the expansion of the steam will by (12.) be [Pg515]

E″ = Wa log.[(1 + c) / (e + c)] − bVA(1 − e).

Hence the total effect per minute will be

_E_ = Wa [(e/(e + c)) + log.([1 + c]/[e + c])] − bVA. (19.)

If the engine work without expansion, e = 1;

∵ _E′_ = ( Wa/(1 + c)) − bVA, (20.)

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

_E_ − _E′_ =

Wa [(e/(e + c)) − (1/(1 + c)) + log.([1 + c]/[e + c])]; (21.)

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 [Pg516] must therefore be equal to the total resistance multiplied by the space through which that resistance is driven. Hence we shall have

[R(1 + m) + r]VA = Wa[(e/(e + c)) + log.([1 + c]/[e + c])] − VAb;

∵ RVA(1 + m) = Wa[(e/(e + c)) + log.([1 + c]/[e + c])] − VA(b + r).

For brevity, let

e′ = a[(e/(e + c)) + log.([1 + c]/[e + c])];

∵ RVA(1 + m) = We′ − VA(b + r). (22.)

By solving this for VA, we obtain

VA = We′/(R(1 + m) + b + r);

∵ RVA = We′R/(R(1 + m) + b + r). (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

U = We′R/(R(1 + m) + b + r). (24.)

Or by (22.),

U(1 + m) = We′ − VA(b + r). (25.)

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 33,000 pounds instead of one pound. Hence we shall have

H = U/33000. (26.)

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 [Pg517]

U′ = U/W. (27.)

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

D = U/F. (28.)

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

W = [U(R(1 + m) + b + r)]/Re′, (29.)

W = (1/e′)[U(1 + m) + VA(b + r)]. (30.)

By eliminating U, by (26.), we shall have

W = [33000 H(R(1 + m) + b + r)]/Re′, (31.)

W = (1/e′)[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.[41]

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 [Pg518] 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 2-1/2 lbs. per square inch more than in the condenser, and that the pressure in the latter being usually 1-1/2 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 [Pg519]

a = 3875969 b = 164 c= 0·05;

e′ = 3875969([e/(e + 0·05)] + log.[1·05/(e + 0·05)]).

In engines which work without condensation, and therefore with high pressure steam, we shall have

a = 4347826 b = 618 c = 0·05

e′ = 4347826([e/(e + 0·05)] + log.[1·05/(e + 0·05)])

To facilitate computation, the values of e′ corresponding to all values of e, from e = ·10 to e = ·90, are given in the following table:—

———————————————————————————————————————————————————————————————
|Condensing|Non-condensing|| |Condensing|Non-condensing
| Engines | Engines || | Engines | Engines
e | e′. | e′. || e | e′. | e′.
————+——————————+——————————————++————+——————————+———————————————
·10 | 10126265 | 11359029 ||·51 | 5966367 | 6692708
·11 | 9956867 | 11169008 ||·52 | 5903837 | 6622565
·12 | 9793136 | 10985344 ||·53 | 5842288 | 6553525
·13 | 9634926 | 10807875 ||·54 | 5781693 | 6485552
·14 | 9482029 | 10636364 ||·55 | 5722024 | 6418619
·15 | 9334219 | 10470560 ||·56 | 5663251 | 6352693
·16 | 9191251 | 10310186 ||·57 | 5605353 | 6287745
·17 | 9052888 | 10154978 ||·58 | 5548297 | 6223742
·18 | 8918896 | 10004675 ||·59 | 5492064 | 6160662
·19 | 8789043 | 9859014 ||·60 | 5436628 | 6098478
·20 | 8663120 | 9717760 ||·61 | 5381969 | 6037166
·21 | 8540918 | 9580682 ||·62 | 5328065 | 5976699
·22 | 8422242 | 9447559 ||·63 | 5274896 | 5917057
·23 | 8306916 | 9318193 ||·64 | 5222444 | 5858219
·24 | 8194770 | 9192396 ||·65 | 5170684 | 5800159
·25 | 8085644 | 9069984 ||·66 | 5119605 | 5742860
·26 | 7979392 | 8950796 ||·67 | 5069186 | 5686304
·27 | 7875870 | 8834674 ||·68 | 5019410 | 5630469
·28 | 7774952 | 8721468 ||·69 | 4970263 | 5575340
·29 | 7676514 | 8611048 ||·70 | 4921727 | 5520894
·30 | 7580447 | 8503284 ||·71 | 4873790 | 5467121
·31 | 7486640 | 8398056 ||·72 | 4826434 | 5414000
·32 | 7394990 | 8295250 ||·73 | 4779648 | 5361519
·33 | 7305407 | 8194760 ||·74 | 4733417 | 5309659
·34 | 7217807 | 8096496 ||·75 | 4687728 | 5258408
·35 | 7132097 | 8000352 ||·76 | 4642569 | 5207751
·36 | 7048206 | 7906249 ||·77 | 4597928 | 5157676
·37 | 6966058 | 7814100 ||·78 | 4553794 | 5108170
·38 | 6885585 | 7723832 ||·79 | 4510155 | 5059218
·39 | 6806720 | 7635365 ||·80 | 4466999 | 5010808
·40 | 6729408 | 7548642 ||·81 | 4424317 | 4962931
·41 | 6653578 | 7463580 ||·82 | 4382096 | 4915569
·42 | 6579187 | 7380132 ||·83 | 4340332 | 4868720
·43 | 6506174 | 7298230 ||·84 | 4299010 | 4822368
·44 | 6434491 | 7217822 ||·85 | 4258120 | 4776500
·45 | 6364099 | 7138858 ||·86 | 4217658 | 4731113
·46 | 6294944 | 7061285 ||·87 | 4177613 | 4686192
·47 | 6226989 | 6985058 ||·88 | 4137974 | 4641728
·48 | 6160190 | 6910126 ||·89 | 4098737 | 4597713
·49 | 6094510 | 6836450 ||·90 | 4059893 | 4554140
·50 | 6029916 | 6763992 || | |
————+——————————+——————————————++————+——————————+———————————————

[Pg520] In engines which work without expansion we have

e′ = a/(1 + c).

For condensing engines without expansion, we shall then have

e′ = 3875969/1·05 = 3691399; (33.)

and for non-condensing engines,

e′ = 4347826/1·05 = 4140787. (34.)

As the diameters of the cylinders of engines are generally expressed in inches, the corresponding areas of the pistons expressed in square feet are given in the following table, so that the values of A may be readily found:—

——————————————————————————————————————————————————————————————————————
Diam. | Area. | Diam. | Area. | Diam. | Area. | Diam. | Area.
———————+—————————+———————+—————————+———————+—————————+———————+————————
Inches.| Sq.feet.|Inches.| Sq.feet.|Inches.| Sq.feet.|Inches.| Sq.feet.
10 | 0·545 | 48 | 12·566 | 86 | 40·339 | 124 | 83·863
11 | 0·660 | 49 | 13·095 | 87 | 41·283 | 125 | 85·221
12 | 0·785 | 50 | 13·635 | 88 | 42·237 | 126 | 86·590
13 | 0·922 | 51 | 14·186 | 89 | 43·202 | 127 | 87·970
14 | 1·069 | 52 | 14·748 | 90 | 44·179 | 128 | 89·361
15 | 1·227 | 53 | 15·321 | 91 | 45·166 | 129 | 90·763
16 | 1·396 | 54 | 15·904 | 92 | 46·164 | 130 | 92·175
17 | 1·576 | 55 | 16·499 | 93 | 47·173 | 131 | 93·599
18 | 1·767 | 56 | 17·104 | 94 | 48·193 | 132 | 95·033
19 | 1·969 | 57 | 17·721 | 95 | 49·224 | 133 | 96·479
20 | 2·182 | 58 | 18·348 | 96 | 50·265 | 134 | 97·935
21 | 2·405 | 59 | 18·986 | 97 | 51·318 | 135 | 99·402
22 | 2·640 | 60 | 19·635 | 98 | 52·382 | 136 | 100·880
23 | 2·885 | 61 | 20·295 | 99 | 53·456 | 137 | 102·369
24 | 3·142 | 62 | 20·966 | 100 | 54·542 | 138 | 103·869
25 | 3·409 | 63 | 21·648 | 101 | 55·638 | 139 | 105·380
26 | 3·687 | 64 | 22·340 | 102 | 56·745 | 140 | 106·901
27 | 3·976 | 65 | 23·044 | 103 | 57·863 | 141 | 108·434
28 | 4·276 | 66 | 23·758 | 104 | 58·992 | 142 | 109·977
29 | 4·587 | 67 | 24·484 | 105 | 60·132 | 143 | 111·532
30 | 4·909 | 68 | 25·220 | 106 | 61·283 | 144 | 113·097
31 | 5·241 | 69 | 25·967 | 107 | 62·445 | 145 | 114·674
32 | 5·585 | 70 | 26·725 | 108 | 63·617 | 146 | 116·261
33 | 5·940 | 71 | 27·494 | 109 | 64·801 | 147 | 117·859
34 | 6·305 | 72 | 28·274 | 110 | 65·995 | 148 | 119·468
35 | 6·681 | 73 | 29·065 | 111 | 67·201 | 149 | 121·088
36 | 7·069 | 74 | 29·867 | 112 | 68·417 | 150 | 122·719
37 | 7·467 | 75 | 30·680 | 113 | 69·644 | 151 | 124·361
38 | 7·876 | 76 | 31·503 | 114 | 70·882 | 152 | 126·013
39 | 8·296 | 77 | 32·338 | 115 | 72·131 | 153 | 127·676
40 | 8·727 | 78 | 33·183 | 116 | 73·391 | 154 | 129·351
41 | 9·168 | 79 | 34·039 | 117 | 74·662 | 155 | 131·036
42 | 9·621 | 80 | 34·907 | 118 | 75·944 | 156 | 132·732
43 | 10·085 | 81 | 35·785 | 119 | 77·236 | 157 | 134·439
44 | 10·559 | 82 | 36·674 | 120 | 78·540 | 158 | 136·157
45 | 11·045 | 83 | 37·574 | 121 | 79·854 | 159 | 137·886
46 | 11·541 | 84 | 38·485 | 122 | 81·180 | 160 | 139·626
47 | 12·048 | 85 | 39·406 | 123 | 82·516 | 161 | 141·377
—————+—————————+———————+—————————+———————+—————————+———————+————————

[Pg521] The practical application of the preceding formulæ will be shown by the following examples.

EXAMPLES.

1. _A 36-inch cylinder with 5-1/2 feet stroke is supplied by a boiler evaporating effectively 60 cubic feet of water per hour, and the piston makes 20 strokes per minute without expansion;-- what is the power of the engine and the pressure of steam in the cylinder?_

Let it be assumed that r = 6 × 144 = 864 and m = 0·1. Since the engine is a condensing engine, we have b = 164 and e′ = 3691399. By the formulæ (25.) and (26.) we have

H = [We′ − VA(b + r)]/[33000(1 + m)];

and since by the data we have

W = 1 A = 7·069 V = 2nL = 40 × 5·5 = 220,

the formula, by these substitutions, becomes

H = (3691399 − 220 × 1028 × 7·069) / (33000 × 1·1);
∵ H = 57·6.

Since e = 1, the pressure P of steam in the cylinder, by (18.), is

P = (We′/VA) − b.

Therefore

P = (3691399/1555·18) − 164 = 2210;

which being the pressure in pounds per square foot, the pressure per square inch will be 15-1/3 lbs.

2. _To find the effective evaporation necessary to produce a power of 80 horses with the same engine. Also, find the pressure of steam in the cylinder, the speed of the piston being the same._

By the formula (32.), with the above substitutions, we have

W = (33000 × 80 × 1·1 + 220 × 7069 × 1028)/3691399 = 1·22.

The evaporating power would therefore be only increased 22 per cent., while the working power of the engine would be increased nearly 40 per cent.

The pressure P in the cylinder will be given, by (18.), as before.

P = [(1·22 × 3691399)/1555·18] − 164 = 2732;

which is equivalent to 19 lbs. per square inch. [Pg522]

3. _What must be the diameter of a cylinder to work with a power of a hundred horses, supplied by a boiler evaporating effectively 70 cubic feet of water per hour, the mean speed of the piston being 240 feet per minute, and the steam being cut off at half stroke? Also, what will be the full pressure of steam on the piston?_

Taking, as in the former examples, m = 0·1, b = 164, and r = 864, we shall have

H = 100 W = 7/6 V = 240,

and by the column for condensing engines, in table, p. 519, we have e′ = 6029916, where e = 0·50. Making these substitutions in

We′ = 33000 H (1 + m) + VA (b + r),

we shall have

(7/6) × 6029916 = 3300000 × 1·1 + 240 × 1028 × A.

Whence we find

A = 13·8;

and by the table, p. 520, the corresponding diameter of the cylinder will be 50-1/3 inches.

If P′ be the full pressure of the steam, we shall have, by (18.),

P′ = (Wa/VA(e + c)) − b.

Making in this the proper substitutions, we have

P′ = ((7/6) × 3875969) / (240 × 13·8 × 0·55) − 164 = 2318;

which being in pounds per square foot, the pressure per square inch will be 16-1/10 lbs.

FOOTNOTES:

[40] M. de Pambour states that the increased volume is 1364
cubic inches.

[41] 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.

[Pg523]

INDEX.

Air, elasticity of, 28;
May be partially expelled from a vessel by the application of
heat, 44.

America, steam navigation first established in, 487;
Circumstances which led to it, 488;
Fitch and Rumsey, their attempts to apply the single-acting
engine to the propulsion of vessels, 489;
Stevens of Hoboken commences experiments on steam navigation,
489;
Experiments of Livingstone and Fulton, 489;
Fulton's first boat, 490;
The Hudson navigated by steam, 491;
Extension and improvement of river navigation, 492;
American steamers, 494;
Difference between them and European steamers, 494;
Steamers on the Hudson, 494;
American paddle-wheels, 495;
Sea-going American steamers, 496;
Speed attained by American steamers, 497;
Lake steamers, 499;
The Mississippi and its tributaries, 499;
Steam-boats navigating it, 500;
Their structure and machinery, 500;
New Orleans Harbour, 503;
Steam tugs, 503.

Atmosphere, 38;
Weight of, 39.

Atmospheric air, mechanical properties of, 38;
Composition of, 253.

Atmospheric engine, Thomas Newcomen the reputed inventor of, 62;
Description of, as first constructed by Newcomen, 67;
The operation of considered, 69;
Not unfrequently used in preference to the modern steam
engine, 72;
Advantages which it possessed over Savery's, 73;
Considerably improved by Beighton, 75;
John Smeaton investigates this machine, 76;
Brindley obtains a patent for improvements in, 76;
Applied by Champion of Bristol to raise water, 181;
Possessed but limited power of adaptation to a varying load,
151;
Expedient to remedy this, 151;
Working-beam, cylinder, and piston applied to by Newcomen,
322.

Atmospheric pressure rendered available as a mechanic agent by
Denis Papin, 38;
Means of measuring the force of, 39;
The idea of using against a vacuum or partial vacuum to work
a piston in a cylinder, suggested by Otto Guericke, 73.

Barometer gauge, 272.

Barton's piston, 248.

Beighton, his improvement of the atmospheric engine, 75.

Black, Dr., his doctrine of latent heat, 93.

Blasco de Garay, his contrivance to propel vessels, 16;
The contrivance of, probably identical with that of Hero, 17.

Blinkensop, his locomotive engine, 337.

Blowing-box, 429.

Blowing out, Seaward's method of, 454.

Blow-off cocks, 452.

Boiler, forms of, most convenient, 255;
The waggon boiler adopted by Watt, 255;
Furnace, 256;
Method of feeding, 257;
Combustion of gas in flues, 260;
Mr. Williams's method of consuming the unburned gases which
escape from the grate, and are carried through the flues,
260;
Construction of grate and ash-pit, 261;
Magnitude of heating surface of boiler, 262;
Capacity of, must be proportioned to the quantity of water to
be evaporated, 263;
Water-space and steam-space in boiler, 263;
Proportion of water-space in the boiler, how to be regulated,
264;
Position of flues, 264;
Method of feeding, 265;
The magnitude of the feed should be equal to the quantity of
water evaporated, 265;
Different methods for indicating the level of the water in
the boiler, 266;
Level guages, 266;
Self-regulating feeder, 267;
Another method of arranging, 269;
Steam gauge, 270;
Thermometer gauge, 271;
Barometer gauge, 272;
The indicator to measure the mean efficient force of the
piston invented by Watt, 274;
The counter contrived by Watt, 278;
Safety valve, 279;
Fusible plugs used in high pressure boilers, 280;
Self-regulating damper, 281;
Self-regulating furnace invented by Brunton, 283;
Duty of a boiler, 294;
Boilers of locomotive engines, 351;
Construction of the boiler of Gurney's steam carriage, 423;
All boilers require occasional cleansing, 427;
Gurney's method of removing crust of deposited matter in
boilers, 427;
The boiler of Dr. Church's engine formed of copper, 439;
Boilers in marine engines, 449;
Effects of sea-water in, 450;
Remedies for them, 451;
Substitution of copper for iron, 460;
Expedient of coating boilers with felt, applied by Watt, 463.

Booth, Mr., his report on locomotive engines, 361.

Boulton and Watt's experiments on the horse power of engines,
288.

Branca, Giovanni, his machine for propelling a wheel by a blast
of steam, 22.

Brindley (James) obtains a patent for improvements in
atmospheric engine, 76;
Undertook to erect an engine at Newcastle-under-Lyne, 76;
Discouraged by the obstacles thrown in his way, 76.

Brougham, Lord, his sketch of Watt's character, 313;
Inscription from the pen of, on Watt's monument in Westminster
Abbey, 320.

Buffers, 404.

Cartwright's engine to use the vapour of alcohol to work the
piston, 245;
His piston, 247.

Cawley and Newcomen obtain a patent for the atmospheric engine,
64.

Champion applies atmospheric engine to raise water, 181.

Chapman, Messrs., their locomotive engine, 337.

Chlorine introduced in bleaching by Watt, 310.

Church, Dr., his steam engine, 439;
The boiler formed of copper, 439.

Coals, the virtues and powers which steam has conferred upon,
6;
The amount of labour a bushel of performs by means of the
steam engine, compared with horse power, 7;
Constituents of, 252;
Process of combustion, 252.

Coal mines, apprehensions as to the possibility of the
exhaustion of groundless, 8.

Cocks, friction on, 240.

Cocks and valves, 227.

Combustion of gas in flues, 260.

Condensation by injection, accidental discovery of, 69.

Condensation in the cylinder incompatible with a due economy
of fuel, 120.

Condensing principle, circumstance which led to Savery's
discovery of, 47.

Condensing pipe in Savery's engine, 52.

Condensing out of the cylinder, 120.

Condensing jet, 191.

Conical steam valves, 228.

Conversion of ice into water, 103;
Of water into steam, 105.

Copying press invented by Watt, 302.

Cornish system of inspection, 297.

Cornish engines, improvement of, 298;
Historical detail of the duty of, 299.

Cylinders, Wilkinson's machine for accurately boring the
insides of, 149.

D valve, 230.

Dalton and Gay-Lussac, law of, relating to the pressure of
elastic bodies, 171.

Dixon, Mr. The substitution of brass for copper tubes in
locomotive engines ascribed to him, 370.

Double clack-valve, 228.

Eccentric, 225;
Two expedients to reverse the position of, 379.

Effect of an engine, 285.

Elastic fluids. The law according to which the pressure of,
increases with their temperature, discovered by Dalton and
Gay-Lussac, 171.

Evaporation of water and other liquids, physical and
mechanical principles connected with, 97.

Expansion of common steam, effects of, 173.

Expansive action of steam, 159;
Stated by Watt in a letter to Dr. Small, 157;
Its principle explained, 158;
Mechanical effect resulting from it, 161;
Computed effect of cutting off steam at different portions
of the stroke, 162;
Involves the condition of a variation in the intensity of
the moving power, 163;
Expedients for equalising the power, 164;
The expansive principle in the engines constructed by
Boulton and Watt, limited, 165;
Its more extensive application in the Cornish engines, 165;
Methods of equalising, 174;
Description of Hornblower's engine for this purpose, 174.

Expansive principle, application of in marine engines, 466.

Farey on the steam engine, quotation from, relative to
Savery's engine, 58;
His evidence before the House of Commons, 435.

Field, construction of his split paddle, 478.

Fitch and Rumsey, their attempts to apply the single-acting
engine to the propulsion of vessels, 489.

Flues, position of, 264.

Fluids, of two kinds, 25;
Mechanical properties of, 25;
Elastic, 27;
Experimental proof that they press equally in all
directions, 41.

Fly-wheel, 205.

Four-way cock, 239;
Disadvantages of, 240.

Fuel, means of economising, in marine furnaces, 463.

Fulton and Livingstone, their experiments in steam navigation,
489.

Fulton's first boat, 490.

Furnace, self-regulating, invented by Brunton, 283.

Fusible plugs used in high-pressure boilers, 280.

Galloway, his paddle-wheel described, 476.

Gas, elasticity of, 28.

Gay-Lussac and Dalton, law of, relating to the pressure of
elastic bodies, 171.

Governor, adaptation of, 209.

Gradients, restrictions on, 411;
Disposition of, should be uniform, 415.

Great Western Railway, Dr. Lardner's experiments on, 408.

Griff, proposals to drain a colliery at, mentioned by
Desaguliers, 64.

Gurney's steam carriage, 423;
Construction of the boiler of, 423;
His method of removing crust of deposited matter in boilers,
427;
His experiments on common roads, 432.

Hall, his condensers described, 458.

Hancock, his steam carriage, 436;
In what manner it differs from that of Gurney, 437.

Harris, Dr., mentions Savery's engine in his "Lexicon
Technicum," 56.

Heat, effects of upon water, 29;
Waste of in atmospheric engine, 89;
An examination of the analogous effects produced by the
continued application of, to water in the liquid state,
102;
Radiation of, 254.

Heating by steam brought forward by Watt, 303.

"Hecla," experiments with the, 412.

Hero of Alexandria, description of his machine, 12.

High pressure engines described, 321;
One of the earliest forms of the steam engine, 322;
Obscurely described in the "Century of Inventions," 322;
Construction of the first, by Messrs. Trevethick and Vivian,
324.

Hooke exposes the fallacy of Papin's project, 64.

Horse carriages compared with steam, 435.

Horse power of steam engines, 288;
Smeaton's estimation of, 288;
Boulton and Watt's experiments on, 288.

Howard's description of his marine engine, 464.

Hudson, the, navigated by steam, 491.

Hull, Jonathan, his application of the steam engine to water
wheels, 180.

Humphrey. His marine engine described, 470.

Huskisson, Mr., death of, 329.

Hydrogen, 253.

India, steam navigation to, 483.

Indicator invented by Watt, 274.

Jeffrey, Lord; his sketch of the character of Watt, 315.

Kinneal, description of Watt's experimental engine at, 131.

Lake steamers, 499.

Lardner's, Dr., experiments on the Manchester Railway in 1832,
357;
His experiments in 1838, 406;
Experiments on the Great Western Railway, 408.

Leupold's engine, description of, 323.

Level gauges, 266.

Linen, machine for drying by steam, invented by Watt, 303.

Liverpool and Manchester railroad, effects of the introduction
of steam transport on, 329;
Want of experience in the construction of the engines, 329;
Death of Mr. Huskisson, 329;
Proceedings of the directors, 342;
Premium offered by them for the best engine, 344;
Experimental trial, 344.

Livingstone and Fulton, experiments of in steam navigation,
489.

Locomotive engine, history of, 328;
Blinkensop's engine, 337;
Chapman's engine, 337;
Walking engine, 337;
Mr. Stephenson's engine at Killingworth, 339;
Defect of, 341;
Description of the "Rocket," 345;
The "Sanspareil," 347;
The "Novelty," 349;
Superiority of the "Rocket," 350;
Subsequent improvements in the locomotive engine, 352;
Table, showing the economy of fuel gained by subdividing the
flue into tubes, 354;
Engines constructed in the form of the "Rocket" subject to
two principal defects, 354;
These defects remedied, 355;
Improved by the adoption of a more contracted blast pipe,
356;
Dr. Lardner's experiments in 1832, 357;
Adoption of brass tubes, 361;
Mr. Booth's report, 361;
Detailed description of the most improved locomotive
engines, 364;
Substitution of brass for copper tubes ascribed to Mr.
Dixon, 370;
Mr. Stephenson constructed the driving wheels without
flanges, 383;
Pressure of steam in the boiler limited by two safety-
valves, 402;
Buffers, 404;
Steam whistle, 404;
Water tank, 404;
Power of locomotive engines, 405;
Evaporation of boilers, 406;
Dr. Lardner's experiments in 1838, 406;
Resistance to railway trains, 407;
Dr. Lardner's experiments on the Great Western Railway, 408;
Restriction on gradients, 411;
Experiment with the "Hecla," 412;
Disposition of gradients should be uniform, 415;
Method of surmounting steep inclinations, 415;
Steam carriages on common roads, 419;
Difference between steam engines on railways and those used
to propel carriages on turnpike roads, 422;
Gurney's steam carriage, 423;
Construction of the boiler of, 423;
Escape of steam from the engines on the Liverpool road, 428;
Blowing-box, 429;
Separator, 430;
Difficulties in the practical working of steam carriages
upon common roads, 432;
Gurney's experiments on common roads, 432;
Prejudice against locomotive engines on common roads, 432;
Not more destructive to roads than carriages drawn by
horses, 433;
Report of the committee of the House of Commons, 433;
Weight of steam carriages, 433;
Two methods of applying locomotives upon common roads, 434;
Horse carriages compared with, 435;
Farey's evidence before the House of Commons, 435;
Risk of accident from explosion extremely slight, 435;
Hancock's steam carriage, 436;
In what manner it differs from that of Gurney, 437;
Ogle's steam carriage, 438;
His evidence before the House of Commons, 439;
Dr. Church's steam engine, 439;
The boiler of formed of copper, 439.

Lunar Society, Boulton and Watt leading members in, 302.

Marine engines, form and arrangement of, 441;
Difference between marine and land engines, 443;
Engine-room, arrangement of, 446;
Boilers in, 449;
Effects of sea-water on boilers, 450;
Remedies for them, 451;
Blow-off cocks, 452;
Indicators of saltness, 452;
Seaward's indicator, 454;
His method of blowing out, 454;
Method of Maudslay and Field to preserve freshness of water
in the boiler, 456;
Brine pumps, 457;
Tubular condensers applied by Mr. Watt, 457;
Hall's condensers, 458;
Substitution of copper for iron boilers, 461;
Process of stoking, 462;
Marine furnaces, 463;
Expedient of coating boilers with felt applied by Watt, 463;
Means of economising fuel, 463;
Description of Howard's engine, 464;
Application of the expansive principle in marine engines,
466;
Recent improvements of Messrs. Maudslay and Field, 467;
Humphrey's engine, 470;
Common paddle-wheel, 472;
Defect of, 474;
Feathering paddles, 474;
Galloway's paddle-wheel, 476;
Field's split paddle, 478;
Proportion of power to tonnage, 480;
Iron steam vessels, 482.

Mariotte's law relating to pressure, 171.