Testing the Expansion Curve

The usual manner of testing the expansion curve of a diagram is to compare it with a curve representing Mariotte’s law for the expansion of a perfect gas.

A theoretic expansion curve that will accord with Mariotte’s law may be constructed on the diagram by the following method:

The diagram, as drawn by the indicator, will have the atmospheric line upon it, and from this as a basis we may mark in the line of no pressure or line of perfect vacuum.

To do this we draw, beneath the atmospheric line, a line as far beneath it as will represent the atmospheric line, on the same scale as the spring used, in the indicator, to draw the diagram.

Suppose, for example, that a 30 lb. spring was used, and assuming the atmospheric pressure to be 15 lbs. per inch, then the line of no pressure would be drawn half an inch below the atmospheric line, because 15 lbs. pull on the spring would cause it to distend half an inch.

The clearance line must then be drawn in, according to directions that have already been given.

The next thing to do is to divide the length of the diagram into any convenient number of equal parts, by vertical lines parallel to, and beginning at, the clearance line, as shown in Fig. 3369. These lines are numbered as shown, ten of them being used because that is a convenient number, but any other number would do.

We next decide at which part of the diagram its expansion curve and the test curve shall touch, and in this example we have chosen that it shall be at line 10.

We have now to find what pressure the length of line 10 represents on the scale of the indicator spring, which in this case we will suppose to be 25 lbs., the line measuring ⁄30 of an inch, and a 30 lb. spring having been used to draw the diagram. Next multiply the pressure (25 lbs.) by the number of the line (10), and divide the product (250) by the number of each of the other lines in succession, and the quotient will be the pressures to be represented by the lines.

For example, for line 9 we have that 250 divided by 9 gives 27.7, hence line 9 must be long or high enough to represent a pressure of 27.7 lbs. above a perfect vacuum, or in this case 27.7⁄30 of an inch.

For line 8 we have that 250 divided by 8 gives 31.25 lbs., hence line 8 must be high enough to represent a pressure of 31.25 lbs. above a perfect vacuum.

The atmospheric line is, in this case, of no other service than to form a guide wherefrom to mark in the line of no pressure, or of perfect vacuum.

Now take the case of line 5, and 250 divided by 5 gives 50, hence the height of line 5 must represent a pressure above vacuum of 50 lbs.

Having carried this out for all the lines from line 10 to line 1, we draw in the true expansion curve, which will touch the tops of all the lines.

Another method of drawing this curve is shown in Fig. 3370. Having drawn the clearance line b c, and vacuum line d c, as before and chosen where the curves shall touch (as at a), then draw from a a perpendicular a a.

Draw line a b, parallel to the vacuum line, and at any convenient height above or near the top of the diagram.

From a draw a c, and from a draw a b parallel to d c, then from its intersection with a c, erect the perpendicular b c, locating on a b, the theoretical point (c) of cut-off.

From a number of points on a b (which may be located without regard to equally spacing them), such as e, f, g and h, draw lines to c, and also drop perpendicular lines, as e e, f f, g g, h h.

From the intersection of e c with b c, draw a horizontal line to e. From the intersection of f c with b c, draw a horizontal line, and so on; and where these horizontals cut the verticals (as at e, f, g, h) are points in the curve, which begins at c, and passes through e, f, g, h, to a.

But this curve does not correctly represent the expansion of steam. It would do so if the steam remained or was maintained at a uniform temperature; hence it is called the isothermal curve, or curve of same temperature. But in fact steam and all other elastic fluids fall in temperature during their expansion, and rise during compression, and this change of temperature slightly affects the pressure.

A curve in which the combined effects of volume and resulting temperatures is represented is called the adiabatic curve, or curve of no transmission; since, if no heat is transmitted to or from the fluid during change of volume, its sensible temperature will change according to a fixed ratio, which will be the same for the same fluid in all cases.

A sufficiently close approximation to the adiabatic curve to enable the non-professional engineer to form an idea of the difference between the two may be produced by the following process:

Taking a similar diagram to that used for the foregoing illustrations, as in Fig. 3371. Fix on a point a near the terminal, where the total pressure is 25 pounds. As before, this point is chosen in order that the two curves may coincide there.

Any other point might have been chosen for the point of coincidence; but a point in that vicinity is generally chosen, so that the result will show the amount of power that should be obtained from the existing terminal. This point is 3.3 inches from the clearance line, and the volume of 25 pounds 996, that is, steam of that pressure has 996 times the bulk of water.

Now if we divide the distance of a from the clearance line by 996, and multiply the quotient by each of the volumes of the other pressures indicated by similar lines, the products will be the respective lengths of the lines measured from the clearance line; the desired curve passing through their other ends. Thus, the quotient of the first or 25 lb. pressure line divided by 996 is .003313; this, multiplied by 726, the volume of 35 lbs. pressure, gives 2.4, the length of the 35 lb. pressure line; and so on for all the rest.

The application of either of the above curves will show that some diagrams are much more accurate than others, even though taken from engines of the same design and quality of workmanship.

As a general rule, those from large engines will be more correct than from small ones, and those from high more correct than from low speeds, and in either case efficiently covering the steam pipes and jacketing the cylinder, to prevent condensation, will improve the diagram.

The character of the imperfection in the expansion curve, shown by the application of a test curve, is generally too high a terminal pressure for the point of cut off, the first part of the curve being generally the most correct, and nearly all the inaccuracy appearing in the last half.

The usual explanation of this is, that the steam admitted during the live steam period condenses because of having to heat the cylinder, and that this water of condensation re-evaporates during the latter part of the stroke when this water of condensation is at a higher temperature than the expanded steam, and thus increases the pressure.

A leaky admission valve may generally, however, be looked for (or else wet steam), if the expansion curve rises much during its lower half.