Properties of Steam and Theory of the Steam Engine
55. We have now to consider the action of heat-engines which the working substance is water and water-vapour steam. The properties of steam are most conveniently stated referring in the first instance to what happens when steam is formed under water constant pressure. This is substantially the process which occurs in the boiler of a steam-engine when the engine is at work. To fix the ideas we may suppose that the vessel in which steam is to be formed is a long upright cylinder fitted with a piston which may be loaded so that it exerts a constant pressure on the fluid below. Let there be, to begin with, at the foot of the cylinder a quantity of water (which for convenience of numerical statement we shall take as 1 _), at any temperature tv; and let the piston press on the surface of the water with a force of P _ per square foot. Let heat now be applied to the bottom of the cylinder. As it enters the water it will produce the following effects in three stages:
(1) The temperature of the water rises until a certain temperature t is reached, at which steam begins to be formed. The value of t depends on the particular pressure P which the piston exerts. Until the temperature t is reached there is nothing but water below the piston.
(2) Steam is formed, more that being taken in. The piston (which is supposed to exert a constant pressure) rises. No further increase of temperature occurs during this stage, which continuous until all the water is converted into steam. During this stage the steam which is formed is said to be saturated. The volume which the piston encloses at the end of this stage, the volume, namely, of 1 _ of saturated steam at pressure P (and temperature t), will be denoted by V in cubic feet.
(3) If after all the water is converted into steam more heat be allowed to enter, the volume will increase and the temperature will rise. The steam is then said to be superheated.
56. The difference between saturated and superheated steam may be expressed by saying that if water (at the temperature of the steam) be mixed with steam some of the water will be evaporated if the steam is superheated, but none if the steam is saturated. Any vapour in contact with its liquid and in thermal equilibrium is necessarily saturated. When saturated its properties differ considerably, as a rule, from those of a perfect gas, but when superheated they approach those of a perfect gas move and more closely the farther the process of superheating is carried, that is to say, the more the temperature is raise above t, the temperature of saturation corresponding to the given pressure P.
57. The temperature t at which steam is formed depends on the value of P. Their relation was determined with great care by Regnault, in a series of classical experiments on which our knowledge of the properties of steam chiefly depends.1 The pressure of saturated steam rises with the temperature at a rate which increase rapidly in the upper regions of the scale. This will be apparent from the first and second columns of Table II., given on next page, which is complied from Rankines reduction of Regnaults results. The first column gives the temperature on the Fahr. scale; the second gives the corresponding pressure in pounds per square inch. Rankine has also expressed the relation of temperature and pressure in saturated by the following formula (which is applicable with other constants to other vapours1):
Logp = 6&Mac250;1007 2732 396945 ,
where p is the pressure in pounds per square inch, and r is the absolute temperature in Fahr. degrees. For most purposes, however, it is more convenient to find the pressure corresponding to a given temperature, of the temperature corresponding to a given pressure, from the table by interpolation.
58. The same table shows the volume V, in cubic feet, occupied by 1 _ of saturated steam at each pressure. This is a quantity the direct experimental measurement of which is of very great difficulty. It may, however, be calculated, from a knowledge of other properties of steam, by a process which will be described later (§ 75). The values of V given in the table were determined by Rankine by means of this process; they agree fairly well with such direct observations of the density of steam as have been hitherto made. 1 The relation of P to V may be approximately expressed by the formula2
PV17 = constant = 68500 (nearly),
when P is stated in _ per sq. ft. and V in cub. ft. per _.
TABLE II.Properties of Saturated Steam.
59. We have next to consider the supply of heat. During the first stage, until the temperature rises from its initial value tv to t, the temperature a which steam begins to form under the given pressure, heat is required only to warm the water. Since the specific heat of water is nearly constant, the amount of heat taken in during the first stage is approximately t-tv thermal units or J(t-tv) foot-pounds, J being Joules equivalent (§ 23), and this expression for it will generally serve with sufficient accuracy in practical calculations. More exactly, however, the heat taken in is somewhat greater than this, for Renaults experiments show that the specific heat of water increases slightly as the temperature rises. In stating the amount of heat required for this first stage, tv must be taken as a known temperature; for convenience in numerical statement the temperature 32&Mac251; F. is usually chosen as an arbitrary starting-point from which the reception of heat is to be reckoned. We shall employ the symbol h to designate the heat required to raise 1_ of water from 32&Mac251; F. to the temperature t at which steam begins to form. The value of h in thermal units is given, approximately, by the equation
h = t 32.
More exact values, which take account of the variation in the specific heat of water, will be found in the last column of Table II. During the first stage, sensibly all the heat supplied goes to increase the stock of internal energy which the fluid possesses, the amount of external work which is done by the expansion of ht fluid being negligible.
60. The heat taken in during the second stage is what is called the talent latent heat of steam, and is denoted by L. Of it a part is spent in doing external work,namely, P multiplied by the excess of the volume of the steam over the volume of the water,and the remainder is the difference of internal energy between 1 _ of steam at t and 1 _ of water at t. The volume of 1 _ of water, at such temperature as are usual in steam-engines, is nearly 0&Mac250;017 cubic feet. We may therefore write the external work (in foot-pounds) done during the production of 1 _ of steam under constant pressure P,
External work = P(V - 0&Mac250;017).
61. Adding together the heat taken in during the first and second stages we have a quantity designated by H and called the total heat of 1 _ of saturated steam:
H = h + L.
Reganaults values of H are very accurately expressed (in thermal units) by the formula
H = 1082 + 0&Mac250;305t.
They are given in the fourth column of Table II. A similar formula gives approximate values of L, exact enough for use in practical calculations,
L = 1114 - 0&Mac250;7t.
The total heat of formation of 1 _ of steam, when formed under constant pressure from water at any temperature tv, is of course H h0, where hv corresponds to tv.
62. Of the whole latent heat of steam, L, the part P(V - 0&Mac250;017) is, as has been said above, spent in doing external work. The remainder (in foot-pounds)
JL P(V - 0&Mac250;017)
is the change of internal energy which the substance undergoes during evaporation. This quantity, for which it is convenient to have a separate symbol, will be denoted by p in thermal units, or Jp in foot-pounds. In dealing with the heat required to produce steam we adopted the state of water 32&Mac251; F. as an arbitrary starting-point fro which to reckon the reception of heat. In the same way it is convenient to use this arbitrary starting-point in reckoning what may be called the internal energy of the substance, which is the excess of the heat taken in over the external work done by the substance during its reception of heat. Thus the internal energy I of 1 _ of saturated steam at pressure P is equal to the total heat H, less that part of the total heat which is spent in doing external work, or (in foot-pounds)
JI = JH P(V - 0&Mac250;017),
Or I = L + h - P(V - 0&Mac250;017)/J = h + p.
The notion of internal energy is useful in calculating the heat taken in or rejected by steam during any stage of its expansion or compression in an engine. When a working substance passes from one condition to another, its gain of loss of heat is determined by the equation.
Heat taken in = increase of internal energy + external work. Any of the terms of this equation may be negative; the term is negative when work is done, not by, but upon the substance.
63. The same equation gives the means of finding the amount of heat required to form steam under any assigned conditions, in place of the condition assumed at the beginning of this chapter, where the formation of steam under constant pressure was considered. Whatever be the condition as to pressure under which the process of formation is carried on, the total heat required is the sum of the internal energy of the steam when formed and the work done by the substance during the process. Thus in general
Heat of formation = I + J -1 =&Mac186;PdV,
the limits of integration being the final volume of the steam and the original volume of the water. When steam is formed in a closed vessel of constant volume no external work is done; the heat of formation is then equal to the internal energy, and is less than the total heat of formation (H) of steam, at a constant pressure equal to the pressure reached in the vessel, by the quantity P(V - 0&Mac250;017).
64. In calculations which relate to the action of steam in engines we have generally to deal, not with dry saturated steam, but with wet steam, or steam which either carries in suspension, or is otherwise mixed with, a greater of less proportion of water. In every such mixture the steam and water have the same temperature, and the steam is saturated. The dryness of wet steam is measured by the proportion q of dry steam in each pound of the mixed substance. When that is known it is easy to determine the other physical constants: thus
Latent heat of 1 _ of wet steam = qL;
Total heat of 1 _ of wet steam = h + qL;
Volume of 1 _ of wet steam = qV + (1 q)0&Mac250;017
= qV very nearly,
unless the steam is so wet as to consist mainly of water;
Internal energy of 1 _ of wet steam = h + qp.
65. Steam is superheated when its temperature is raised, in any manner, above the temperature correspo00nding to saturation at the actual pressure. When much superheated, steam behaves like a perfect gas, and may be called "steam gas." It then follows the equation
PV = 85&Mac250;5r,
and the specific heat at constant pressure, Kp is 371 foot-pounds or 0&Mac250;48 thermal unit. At very low temperatures steam approximates closely to the condition of a perfect gas when very slightly super-heated, and even when saturated; at high temperatures a much greater amount of superheating is necessary to bring about an approach to the perfectly gaseous state. The total heat required for the production of superheated steam under any constant pressure, when the superheating is sufficient to bring the steam to the state of steam gas, may therefore be reckoned by taking the total heat of saturated steam at a low temperature and adding to it the product of Kp into the excess of temperature above that. Thus Rankine, treating saturated steam at 32&Mac251; F. as a gas, gives the formula
H = 1092 + 0&Mac250;48(t 32)
to express the heat of formation (under any constant pressure) of superheated steam, at any temperature t which is so much above the temperature of saturation corresponding to the actual pressure that the steam may be treated as a perfect gas. Calculated from its chemical composition, the density of steam gas should be 0&Mac250;622 times that of air at the same pressure and temperature. The value of y of Kp/Kv for steam gas is 1&Mac250;3. These formulas, dealing as they do with steam which is so highly superheated as to be perfectly gaseous, fail to apply to high-pressure steam that is heated but little above its temperature of saturation. The relation of pressure to volume and temperature in the region which less between the saturated and the perfectly gaseous states has been experimented on by Hirn.1 Formulas which are applicable with more or less accuracy to steam in either the saturated or superheated condition have been devised by Hirn, Zeuner,2 Ritter,3 and others.
66. The expansion of volume which occurs during the conversion of water into steam under constant pressurethe second stage of the process described in § 55is isothermal. From what has been already said it is obvious that steam, or any other saturated vapour, can be expanded or compressed isothermally only when wet and that evaporation (in the one case) or condensation (in the other) must accompany the process. Isothermal lines for a working substance which consists of a liquid and its vapour are straight lines of uniform pressure.
67. The form of adiabatic lines for substances of the same class depends not only on the particular fluid, but also on the proportion of liquid to vapour in the mixture. In the case of steam, it has been shown by Rankine and Clausius that if steam initially dry be allowed to expand adiabatically it becomes wet, and if initially wet (unless very wet4) it becomes wetter. A part of the steam is condensed by the process of adiabatic expansion, at first in the form of minute, particles suspended throughout the mass. The temperature and pressure fall; and, as that part of the substance which remains uncondensed is saturated, the relation of pressure to temperature throughout the expansion is that which holds for saturated steam. The following formula, proved by Rankine5 and Clausius6 (see § 75), serves to calculate the extent to which condensation takes place during adiabatic expansion, and so allows the relation of pressure to volume to be determined.
Before expansion, let the initial dryness of the steam be q1 and its absolute temperature r1. Then, if it expand adiabatically until its temperature to r, its dryness after expansion is
q = r _q1L1 + log_ r1) .
L r1 r
L1 and L are the latent heats (in thermal units) of 1 _ of steam before and after expansion respectively. When the steam is dry to begin with, q1 = 1.
This formula is easily applied to the construction of the adiabatic curve when the initial pressure and the pressure after expansion are given, the corresponding values r and L being found from the table. It is less convenient if the data are the initial pressure and the initial and final volumes, or the initial pressure and the ratio of expansion r. An approximate formula more appropriate in that case is
Pvn = constant, or P/P1 = (v/v1)n = rn.
Here v and v1 denote the volume of 1 _ if the mixture of steam and water before and after expansion respectively, and are to be distinguished from V and V1, which we have already used to denote the volume of 1 _ dry saturated steam at pressure P and P1. The index n has a value which depends on the degree of initial dryness q1.
According to Zeuner,7 n = 1&Mac250;035 + 0&Mac250;1q1, so that for
q1 = 1 0&Mac250;95 0&Mac250;85 0&Mac250;9 0&Mac250;75 0&Mac250;7
n = 1&Mac250;035 1&Mac250;130 1&Mac250;125 1&Mac250;120 1&Mac250;115 1&Mac250;105.
Rankine gave for this index the value 10/9, which is too small if the steam be initially dry. He determined it by examining the expansion of indicator diagrams taken from working steam-engines; but, as we shall see later the expansion of steam in an actual engine is by no means adiabatic, on account of the transfer of heat which goes on between the working fluid and the metal of the cylinder and piston. When it is desired to draw an adiabatic curve for steam, that value of n must be chosen which refers to the degree of dryness at the beginning of the expansion.
68. We are now in a position to study the action of a heat-engine employing steam as the working substance. To simplify the first consideration as far as possible, let it be supposed that we have, as before, a long cylinder composed of non-conducting material except at the base, and fitted with a non-conducting piston; also a source of heat A at some temperature r1; a receiver of heat, of heat, or as we may now call it, a condenser C, at a lower temperature r2; and a non-conducting over B (as in § 40). Then we can perform Carnots cycle of operations as follows. To fix the ideas, suppose that there is 1 _ of water in the cylinder to begin with, at the temperature r1:
(1) Apply A, and allow the piston to rise. The water will take in heat and be converted into steam, expanding isothermally at constant P1. This part of operation is shown by the line ab in fig. 14.
(2) Remove A and apply B. Allow the expansion to continue adiabatically (bc), with falling pressure, until the temperature falls to r2. The pressure will then be P2, corresponding (in Table II.) to r2.
(3) Remove B, apply C, and compress. Steam is condensed by rejecting heat to C. The action is isothermal, and the pressure remains P2. Let this be continued until a certain point d is reached, after which adiabatic compression will complete the cycle.
(4) Remove C and apply b. Continue the compression, which is now adiabatic. If the point d has been rightly chosen, this will complete the cycle by restoring the working fluid to the state of water at temperature r1.
The indicator diagram for the cycle is given in fig. 14, as calculated by the help of the equations in § 67 and of Table II. for a particular example, in which p1 = 90 _ per square inch (r1 = 781), expansion is continued down to the pressure of the atmosphere, 14&Mac250;7 _ per square inch (r2 = 673). Since the process is reversible, and since heat is taken in only at r1 and rejected only at r2, the efficiency is (r1-r2)/r2. The heat taken in per _ of the fluid is L1, and the work done is L1(r1-r2)/r1, a result which may be used to check the calculation of the diagram.
69. If the action here described could be realized in practice, we should have a thermodynamically perfect steam-engine using saturated steam. The fraction of the heat supplied to it which such an engine would convert into work would depend simply on the temperature, and therefore on the pressure, at which the steam was produced and condensed. The temperature of condensation is limited by the consideration that there must be an abundant supply of some substance to absorb the rejected heat; water is actually used for this purpose, so that r2 has for its lower limit the temperature of the available water-supply.
To the higher temperature r1 and pressure P1 no limit can be set except such as is brought about in practice by the mechanical difficulties, with regard to strength and to lubrication, which attend the use of high-pressure steam. By a very special construction of engine and boiler Mr. Perkins has been able to use steam with a pressure as high as 500 _ per square inch; with engines of the usual construction the value ranges form 190 _ downwards.
In the temperature of condensation be taken as 60&Mac251; F., as a lower limit, the efficiency of a perfect steam-engine, using saturated steam, would depend on the value of P1, the absolute pressure of production of the steam, as follows:
For perfect steam-engine, with condensation at 60&Mac251; F., P1 in _ per square inch being 40 80 120 160 200
Highest ideal efficiency = &Mac250284 &Mac250;326 &Mac250;350 &Mac250;368 &Mac250;381
But it must be supposed that these values of the efficiency are actually attained, or are even attainable. Many causes conspire to prevent steam-engines form being thermodynamically perfect, and some of the causes of imperfection cannot be removed. These numbers will serve, however, as a standard of comparison in judging of the performance of actual engines, and as setting forth the advantage of high-pressure steam form the thermodynamic point of view.
70. As a contrast to the ideally perfect steam-engine of § 68 we may next consider a cyclic action such as occurred in the early engines of Newcomen or Leupold, when steam was used non-expansively,or rather, such an action as would have occurred in engines of this type had the cylinder been a perfect non-conductor of heat. Let the cycle of operations be this:
(1) Apply A and evaporate the water as before P1. Heat taken in = L1.
(2) Remove A and apply C. This at once condenses a part of the steam, and reduces the pressure to P2.
(3) Compress at P2, in contact with C, till condensation is complete, and water at r1, is left.
(4) Remove B and apply A. This heats the water again to r1, and completes the cycle. Heat taken in = h1 h2.
The indicator diagram for this series of operations is shown in fig. 15,
Here the action is not reversible. The calculate the efficiency, we have
Work done (P1 P2)(V1 - 0&Mac250;017) .
Heat taken in J(L1 + h1 h2)
The values of this will be found to range from 0&Mac250;067 to 0&Mac250;072 for the values of P1 which are stated in § 69, when the temperature of condensation is 60&Mac251; F.
71. In the ideal engine represented in fig. 14 the functions of boiler, cylinder, and condenser are combined in a single vessel; but after what has been said in chap. II. It is scarcely necessary to remark that, provided the working substance passes through the same cycle of operations, it is indifferent whether these are performed in several vessels or in one. To approach a little more closely the conditions that hold in practice, we may think of the engine which performs the cycle of § 70 as consisting of a boiler A (fig. 16) kept at r1 a non-conducting cylinder and piston B, a surface condenser C kept at r2, and a feed-pump D which restore the condensed water to the boiler. Then for every pound of steam supplied and used non-expansively as in § 70, we have work done on the piston = (p1 p2) V1; but an amount of work has to be expended in driving the feed-pump (P1 P2) 0&Mac250;017. Deducting this, the net work done per _ of steam it the same as before and the heat taken in is also the same. An indicator diagram taken from the cylinder would give the area efgh (fig. 17), where oe = P1, ef = V1, oh = P2; and indicator diagram taken from the pump would give the negative area hjie, where ie is the volume of the feed-water, or 0&Mac250;017 cub. ft. The difference, namely, the shaded area, is the diagram of the complete cycle gone through by each pound of the working substance. In experimental measurements of the work done in steam-engines, only the action which occurs within the cylinder is shown on the indicator diagram. From this work spent on the feed-pump is to be subtracted in any accurate determination of the thermodynamic efficiency. If the feed-water is at any temperature r0, other than that of the condenser as assumed in § 70, it is clear that the heat taken in is H1-h0 instead of H1 h2.
72. We have now to inquire how nearly, with the engine of the fig. 16 (that is to say, with an engine in which the boiler and condenser are separate from the cylinder). We can approach the reversible cycle of § 68. The first stage of that cycle corresponds to the admission of steam from the boiler into the cylinder. Then the point known as the point of cut-off is reached, at which admission ceases, and the steam already in the cylinder is allowed to expand, exerting a diminishing pressure on the piston. This is the second stage, of the stage of expansion. The process of expansion may be carried on until the pressure falls to that of the condenser, in which case the expansion is said to be complete. At the end of the expansion release takes place, that is to say, communication is opened with the condenser. Then the return stroke begins, and a period term the exhaust occurs, that is to say steam passes out of the cylinder, into the condenser, where it is condensed at pressure P2, which is felt as the back pressure opposing the return of the piston. So far, all has been essentially reversible, and identical with the corresponding parts of the Carnots cycle. But we cannot complete the cycle as Carnots cycle was completed. The existence of a separate condenser makes the fourth stage, that of adiabatic compression, impracticable, in the best we can do is to continue the exhaust until the condensation is complete, and then return the condensed water to the boiler by means of the feed-pump.
It is true that we may, and in actual practice do, stop the exhaust before the return stroke is complete, and compress that portion of the steam which remains below the piston, but this does not materially affect the thermodynamic efficiency; it is done partly for mechanical reasons, and partly to avoid loss of power through clearance (see chap. VI.). In the present instance it is supposed that there is no clearance, in which case this compression is out of the question. The indicator diagram given by a cylinder in which steam goes through the action described above is shown to scale in fig. 18 for a particular example, in which it is supposed that 1 cubic foot of dry saturated steam is admitted at an absolute pressure of 90 _ per square inch, and is expanded twelve times, or down to a pressure of 5&Mac250;4 _ per square inch, at which pressure it is discharged to the condenser. As we have assumed the cylinder to be non-conducting, and the steam to be initially dry, the expansion follows the law PV1&Mac250;135 = constant. The advantage of expansion is obvious, that part of the diagram which lies under the curve being so much clear gain.
73. To calculate the efficiency, we have
Work done per _ during admission = P1V1 ;
" " during expansion to volume rV1 = P1V1 P2rV1 ,
(by § 36), = (P1V1 P2rV1)/0&Mac250;135 ;
Work spent during return stroke = P2rV1 ;
,, ,, on the fee-pump = (P1 P2)0&Mac250;017 ;
Heat taken in = H1 h0.
74. These expressions refer to complete expansion. When the expansion is incomplete, as it generally is, the expression given above for the work done during expansion still supplies if we take P2 to be the pressure at the end of expansion, while the work spent on the steam during the back-stroke is PbrV1 and that spent on the feed-pump is (P1 Pb)0&Mac250;017, Pb being the back pressure. Incomplete expansion is illustrated by the dotted line in fig. 18.
It is easy, by the aid of §§ 64 and 67 to extend these calculations to cases where the steam, of being initially dry, is supposed to have any assigned degree of wetness. The efficiency which is calculated in this way, which for the present purpose may be called the theoretical efficiency corresponding to the assumed conditions of working, is always much less than the ideal efficiency of a perfect engine, since the cycle we are now dealing with is not reversible. But even this theoretical efficiency, short as it falls of the ideal of a perfect engine, is far greater than can be realized in practice when the same boiler and condenser temperatures are used, and the same ratio of expansion. The reasons for this will be briefly considered in the next chapter; at present the fact is mentioned to guard the reader form supposing that the results which the above formulas give apply to actual engines.
75. The results of § 68 have been turned to account by Rankine and Clausius for the purpose of deducting the density of steam from other properties which admit of more exact direct measurement. Let the perfect steam-engine there described work through a very small interval of temperature _r between two temperature r and r _r. The efficiency is _r/r, and the work done (in foot-lbs.) is JL_r/r. The indicator diagram is now reduced to a long narrow strip, whose length is V - 0&Mac250;017 and its breadth _P, the difference in pressure between steam at temperature r and r _r. Hence the work done is also _P(V - 0&Mac250;017), and therefore
V - 0&Mac250;017 = JL . _r .
Here _r or (in the limit dr, is the rate of increase of temperature with increase of
temperature with increase of pressure in saturated steam at the particular temperature r. It may be found roughly from Table II., p.48 4, or more exactly by differentiating the equation given in § 57. L is also known, and hence the value of V corresponding to any assigned temperature may be calculated with a degree of accuracy which it would be difficult to reach in direct experiment. The volumes given in the Table are determined in this way.1
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