Thermodynamics

28 | ThermodynamicsA relation for the pressure difference P 1 P 2 can be obtained by startingat point 1 with P 1 , moving along the tube by adding or subtracting the rghterms until we reach point 2, and setting the result equal to P 2 :P 1 r 1 g 1a h2 r 2 gh r 1 ga P 2(1–24)Note that we jumped from point A horizontally to point B and ignored thepart underneath since the pressure at both points is the same. Simplifying,P 1 P 2 1r 2 r 1 2gh(1–25)Note that the distance a has no effect on the result, but must be included inthe analysis. Also, when the fluid flowing in the pipe is a gas, then r 1 r 2and the relation in Eq. 1–25 simplifies to P 1 P 2 r 2 gh.OilEXAMPLE 1–7Measuring Pressure with a Multifluid ManometerAIR1WATER2The water in a tank is pressurized by air, and the pressure is measured by amultifluid manometer as shown in Fig. 1–49. The tank is located on amountain at an altitude of 1400 m where the atmospheric pressure is 85.6kPa. Determine the air pressure in the tank if h 1 0.1 m, h 2 0.2 m, andh 3 0.35 m. Take the densities of water, oil, and mercury to be 1000kg/m 3 , 850 kg/m 3 , and 13,600 kg/m 3 , respectively.h 1h 2h 3MercuryFIGURE 1–49Schematic for Example 1–7. (Drawingnot to scale.)Solution The pressure in a pressurized water tank is measured by a multifluidmanometer. The air pressure in the tank is to be determined.Assumption The air pressure in the tank is uniform (i.e., its variation withelevation is negligible due to its low density), and thus we can determine thepressure at the air–water interface.Properties The densities of water, oil, and mercury are given to be 1000kg/m 3 , 850 kg/m 3 , and 13,600 kg/m 3 , respectively.Analysis Starting with the pressure at point 1 at the air–water interface,moving along the tube by adding or subtracting the rgh terms until we reachpoint 2, and setting the result equal to P atm since the tube is open to theatmosphere givesSolving for P 1 and substituting, 1850 kg>m 3 1 N210.2 m24a1 kg # ba 1 kPam>s21000 N>m b 2 130 kPaP 1 r water gh 1 r oil gh 2 r mercury gh 3 P atmP 1 P atm r water gh 1 r oil gh 2 r mercury gh 3 P atm g 1r mercury h 3 r water h 1 r oil h 2 2 85.6 kPa 19.81 m>s 2 23113,600 kg>m 3 210.35m2 1000 kg>m 3 210.1 m2Discussion Note that jumping horizontally from one tube to the next andrealizing that pressure remains the same in the same fluid simplifies theanalysis considerably. Also note that mercury is a toxic fluid, and mercurymanometers and thermometers are being replaced by ones with safer fluidsbecause of the risk of exposure to mercury vapor during an accident.