these difficulties, and rendered the most objectionable harbour of the Union a safe and good seaport, perfectly easy of approach and of egress at all times; a small steam tug will take in tow several large ships, and carry them with safety and expedition to the offing, where it will dismiss them on their voyage, and take back vessels which may have arrived. APPENDIX. 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. a1 = log. a2 = 0.74110951837 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: 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 in pounds per square inch, and the temperature in reference to Fahrenheit's thermometer, p = 0.04948 + 51.3 + 5·13 (2.) p t=155.7256 5.13 The following formula was proposed by Tredgold, where p expresses the pressure in inches of mercury:— 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,— 103 J. t = 201·18 Vp - 103 (3.) M. de Pambour has proposed the following formula, also applicable through the same limits of the scale: = 138-56229906 t = 198.562vp-98.806 (4.) MM. Dulong and Arago have proposed the following formula for all pressures between 4 and 50 atmospheres: ?= (09.0793 +0-0067555 () }. t 147·961 p 39′644 (5.) 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.* * M. de Pambour states that the increased volume is 1364 cubic inches. 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 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 n v 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, 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 10. 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 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'; 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.), 1700 × 14.706 × { 1 + 0·002083 (T — 32)} VP = (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. |