Thermodynamics

26 | ThermodynamicsF 1 = P 1 A 11 A 1P 1A 2P 22F 2 = P 2 A 2FIGURE 1–44Lifting of a large weight by a smallforce by the application of Pascal’slaw.GasFIGURE 1–45The basic manometer.INTERACTIVETUTORIALSEE TUTORIAL CH. 1, SEC. 10 ON THE DVD.h1 2to exert a proportionally greater force than that applied to the smaller. “Pascal’smachine” has been the source of many inventions that are a part of ourdaily lives such as hydraulic brakes and lifts. This is what enables us to lifta car easily by one arm, as shown in Fig. 1–44. Noting that P 1 P 2 sinceboth pistons are at the same level (the effect of small height differences isnegligible, especially at high pressures), the ratio of output force to inputforce is determined to beF 1P 1 P 2 S F 2 F 2S A 2A 1 A 2 F 1(1–22)The area ratio A 2 /A 1 is called the ideal mechanical advantage of thehydraulic lift. Using a hydraulic car jack with a piston area ratio of A 2 /A 1 10, for example, a person can lift a 1000-kg car by applying a force of just100 kgf ( 981 N).1–10 ■ THE MANOMETERWe notice from Eq. 1–18 that an elevation change of z in a fluid at restcorresponds to P/rg, which suggests that a fluid column can be used tomeasure pressure differences. A device based on this principle is called amanometer, and it is commonly used to measure small and moderate pressuredifferences. A manometer mainly consists of a glass or plastic U-tubecontaining one or more fluids such as mercury, water, alcohol, or oil. Tokeep the size of the manometer to a manageable level, heavy fluids such asmercury are used if large pressure differences are anticipated.Consider the manometer shown in Fig. 1–45 that is used to measure thepressure in the tank. Since the gravitational effects of gases are negligible, thepressure anywhere in the tank and at position 1 has the same value. Furthermore,since pressure in a fluid does not vary in the horizontal direction withina fluid, the pressure at point 2 is the same as the pressure at point 1, P 2 P 1 .The differential fluid column of height h is in static equilibrium, and it isopen to the atmosphere. Then the pressure at point 2 is determined directlyfrom Eq. 1–19 to beP 2 P atm rgh(1–23)where r is the density of the fluid in the tube. Note that the cross-sectionalarea of the tube has no effect on the differential height h, and thus the pressureexerted by the fluid. However, the diameter of the tube should be largeenough (more than a few millimeters) to ensure that the surface tensioneffect and thus the capillary rise is negligible.A 1EXAMPLE 1–6Measuring Pressure with a ManometerA manometer is used to measure the pressure in a tank. The fluid used hasa specific gravity of 0.85, and the manometer column height is 55 cm, asshown in Fig. 1–46. If the local atmospheric pressure is 96 kPa, determinethe absolute pressure within the tank.