LECTURE XII. QUESTIONS.

1. State Boyle's law, and describe an experiment to show that the pres sure of a gas varies inversely as the space it occupies.

2. Referring to Regnault's table in this Lecture, draw a curve of volumes and corresponding pressures from 10 to 50 lbs. absolute, taking I inch to represent the volume of 1 lb. of steam, and also I lb. of pressure per square inch.

3. Steam is admitted into a cylinder at atmospheric pressure, and is cut off at half stroke. Divide the stroke into 10 equal parts, and, supposing that the pressure at the beginning of each of these portions remains uniform until the piston reaches the next in order, find the pressure at each point as well as the mean pressure by Watt's method. Ans. 83.

4. A certain quantity of steam at 40 lbs. pressure absolute is admitted into a cylinder, and then expands to 4 times its original bulk. What is its final pressure? Explain what is meant by steam of 40 lbs. absolute. Ans. 10 lbs. absolute.

5. The piston of an engine moves through 12 inches under a pressure of 30 lbs. absolute; the steam is then cut off and allowed to expand. What will be its pressure when the piston has moved through 18 inches from the beginning of its stroke? Ans. 20 lbs. absolute.

6. Steam is admitted into a cylinder at atmospheric pressure, and is cut off at stroke. Find the pressure when the piston has reached 17 of its full stroke. Ans. 4'4 lbs. absolute.

7. Steam of 40 lbs. absolute pressure is admitted into a cylinder. Find the final pressure in each of the following cases:-1st, when it is cut off at stroke; 2nd, when at stroke; 3rd, when at stroke; 4th, when at

stroke. Ans. 1st-5 lbs. ; 2nd = 10; 3rd=20; 4th=30.

8. Steam is admitted into the cylinder of an engine at 9 lbs. above the atmosphere (taken at 15 lbs. per square inch), and is expanded down to 7 lbs. below the atmosphere. Find the pressure of the steam at halfstroke, and the point of cut-off. (S. & A. Exam. 1890.) Ans. I lb. above the atmosphere, or 16 lbs. absolute; stroke.

9. When is steam said to be be saturated? Distinguish between saturated steam and superheated steam. When air is compressed without change of temperature its pressure is increased according to Boyle's law; state what happens when saturated steam is compressed in like manner without change of temperature? (S. & A. Exam. 1891.)

10. What do you understand by the expansion of air or gas according to Boyle's law? Assuming that steam expands in this way, find the pressure of steam on its admission into a steam cylinder under the following conditions:-Length of stroke of piston is 5 feet, steam is cut off after the piston has described 2 feet, and expands down to 3 lbs. below the atmospheric pressure (taken at 15 lbs.) (S. and A. Exam. 1891.) Ans. 30 lbs. absolute.

=

11. The pressure of steam is 30 lbs. above the atmosphere, and the cutoff takes place when the piston has moved 5 inches. The mean resistance of the load 18 lbs., and the steam is supposed to expand according to Boyle's law, how much farther will the piston have moved when the actual pressure of the steam just balances the resistance? (S. and A. Exam., 1893.) Ans. It may be asked, what is meant by "the mean resistance of the load being 18 lbs ?" Does it mean 18 lbs. above atmospheric pressure, or 18 lbs. absolute? Again, what is meant by the expression, "actual

pressure of the steam just balances the resistance?" The pressure will be actual, no matter what its value may be. Taking the question to read thus: "The mean resistance of the load corresponds to a pressure of 18 lbs. above atmospheric pressure, and the steam is supposed to expand according to Boyle's law, how much farther will the piston have moved when the pressure of the steam just balances the resistance?" Ans. 182 inches.

12. Assuming, as was done by Watt, that the actual expansion curve of steam is the same as that of air when expanding at a constant temperature, set out an approximate expansion curve when steam of a volume 10 cubic feet, and pressure 65 lbs. per square inch above the atmosphere, is expanded to a volume of 40 cubic feet. (S. and A. Exam., 1894.)

LECTURE XIII.

CONTENTS.—Finding the Mean Pressure from a Theoretical Diagram of Work.

Finding the Mean Pressure.-Let us try another example of Watt's diagram of work. Suppose we have an engine using steam of 100 lbs. pressure absolute per square inch, and cutting off at of the stroke. We find the curve of expansion by Boyle's law and the mean pressure precisely as we did in the last Lecture, with this difference, that we shall add the last pressure ordinate in order to get a nearer approximation to the true mean absolute pressure. In this example, as in the last, we shall assume that there is no back pressure whatever; or, in other words, that a perfect vacuum continues throughout the whole of the exhausting stroke. This, of course, does not take place in actual practice, as we shall learn further on, but our chief object at present is to impress upon junior students the simplest methods of finding the mean pressure and the area of the diagram of work.

There are several rules for obtaining approximately the mean pressure from a diagram of work such as we are now discussing. The plan most commonly adopted by engineers (as we shall see later on) in finding the mean pressure from actual indicator diagrams is, to measure by a suitable scale or rule the length of each of the ten ordinates, taken at the centre of each of the ten spaces into which the diagram is divided, add them together, and divide by their number. For instance, applying this rule to the following figure, we should measure the length of the vertical lines midway between the points o and 1, 1 and 2, 2 and 3, . . . . 9 and 10, add these ten pressure ordinates together, and divide the sum by 10, to get the mean pressure; and doing so (or calculating these pressures by pv = constant), we find them to be respectively, 100, 100, 100, 71'43, 55'5, 45°45, 38°46, 33'3, 29'41, and 26.31 lbs., giving a mean of 599 lbs., or slightly greater than that found by the calculations on the opposite page.

Economy Due to Cutting Off Early.-EXAMPLE I.—We also see the economy or gain effected by "cutting off" the steam early from the cylinder and doing work during the rest of the stroke by the mere expansive force of the steam; for, the mean

This is

pressure per square inch on the piston is p=59°7 lbs. greater than half the initial pressure, 100 lbs., although the steam was cut off from the cylinder when the piston had only moved through of its stroke. Suppose that we consider the stroke of the piston to be, L= 5 feet, then,

The work done in one stroke upon every square inch of the piston's area is equal to the mean forward pressure in lbs. per square inch multiplied by the length of the stroke in feet.

Work done per sq. in.

298.5 ft.-lbs.

If we assume the area of the piston to be 100 square inches, then,

The whole work done in one stroke = Work done upon every square inch multiplied by the area of piston in square inches.

Had steam of the full initial pressure (P = 100lbs.) been continued throughout the whole stroke, then,

Р × L X

A

= - Whole work done in one stroke.

100 (lbs.) × 5 (ft.) x 100 (sq. ins.) =

50,000 ft.-lbs.

But 50,000 (ft.-lbs.): 29,850 (ft.-lbs.) :: 100: x.

x = 597 per cent.

In other words, by cutting off at stroke and letting the steam then act expansively, we get more than half the work that we would have got from 4 times the weight of steam of the same initial pressure if used non-expansively.

What percentage gain per pound of steam used does this amount to?

For a certain weight of steam (viz., the weight of steam that fills the cylinder at 100 lbs. pressure) we get 50,000 ft.-lbs. of work.

For of this weight of steam used expansively we get 29,850 ft.-lbs. of work.

Therefore, for the same weight as before we would get29,850 × 4, or 119,400 ft.-lbs. of work. Consequently, for every 1 lb. of steam used non-expansively we have to use

50,000 (ft.-lbs.): 119,400 (ft.-lbs.): 1 (lb. steam): y.

y = 2 2°39 lbs.

Or, it would take 2.39 lbs. of steam used non-expansively at 100 lbs. pressure to do the same amount of work as 1 lb. used expansively and cut off at stroke with the same initial pressure; i.e., I: 2'39: 100% : %.

.. (239%-100%) 139 per cent. gain by the adoption of cutting off at stroke and taking advantage of the expansive properties of steam.

Advantages arising from Condensing Steam.-The advantages arising from condensing the steam during exhaust are also rendered apparent from this example. For had the exhaust taken place freely against the atmospheric pressure (instead of under