An Inclined Manometer Is Used To Measure The Pressure Drop Between Two Taps
11. The threefluid manometer illustrated in Fig. 4P11 is used to measure a very small pressure difference (P1 — P2). The crosssectional area of each of the reservoirs is A, and that of the manometer legs is a. The three fluids have densities pa, pb, and pc, and the difference in elevation of the interfaces in the reservoir is x. Derive the equation that relates the manometer reading h to the pressure difference P1 — P2. How would the relation be simplified if A ^ a?
12. A tank that is vented to the atmosphere contains a liquid with a density of 0.9 g/cm3. A dip tube inserted into the top of the tank extends to a point 1 ft from the bottom of the tank. Air is bubbled slowly through the dip tube, and the air pressure in the tube is measured with a mercury (SG = 13.6) manometer. One leg of the manometer is connected to the air line feeding the dip tube, and the other leg is open to the atmosphere. If the manometer reading is 5 in., what is the depth of the liquid in the tank?
13. An inclined manometer is used to measure the pressure drop between two taps on a pipe carrying water, as shown in Fig. 4P13. The manometer fluid is an oil with SG = 0.92, and the manometer reading (L) is 8 in. The manometer reservoir is 4 in. in diameter, the tubing is 1in. in diameter, and the manometer tube is inclined at an angle of 30° to the horizontal. The pipe is inclined at 20° to the horizontal, and the pressure taps are 40 in. apart.
(a) What is the pressure difference between the two pipe taps that would be indicated by the difference in readings of two pressure gages attached to the taps, in (1) psi, (2) Pa, and (3) in. H2O?
(b) Which way is the water flowing?
(c) What would the manometer reading be if the valve were closed?
14. The pressure gradient required to force water through a straight horizontal 4 in. ID tube at a rate of 2 gpm is 1.2 psi/ft. Consider this same tubing coiled in an expanding helix with a vertical axis. Water enters the bottom of the coil and flows upward at a rate of 2 gpm. A mercury manometer is connected between two pressure taps on the coil, one near the bottom where the coil radius is 6 in., and the other near the top where the coil radius is 12 in. The taps are 2 ft apart in the vertical direction, and there is a total of 5 ft of tubing between the two taps. Determine the manometer reading, in cm.
15. It is possible to achieve a weightless condition for a limited time in an airplane by flying in a circular arc above the earth (like a rainbow). If the plane flies at 650 mph, what should the radius of the flight path be (in miles) to achieve weightlessness?
16. Water is flowing in a horizontal pipe bend at a velocity of 10 ft/s. The radius of curvature of the inside of the bend is 4 in., and the pipe ID is 2 in. A mercury manometer is connected to taps located radically opposite each other on the inside and outside of the bend. Assuming that the water velocity is uniform over the pipe cross section, what would be the manometer reading in centimeters? What would it be if the water velocity were 5 ft/s? Convert the manometer reading to equivalent pressure difference in psi and Pa.
17. Calculate the atmospheric pressure at an elevation of 3000 m, assuming (a) air is incompressible, at a temperature of 59°F; (b) air is isothermal at 59°F and an ideal gas; (c) the pressure distribution follows the model of the standard atmosphere, with a temperature of 59°F at the surface of the earth.
18. One pound mass of air (MW = 29) at sea level and 70°F is contained in a balloon, which is then carried to an elevation of 10,000 ft in the atmosphere. If the balloon offers no resistance to expansion of the gas, what is its volume at this elevation?
19. A gas well contains hydrocarbon gases with an average molecular weight of 24, which can be assumed to be an ideal gas with a specific heat ratio of 1.3. The pressure and temperature at the top of the well are 250 psig and 70°F, respectively. The gas is being produced at a slow rate, so conditions in the well can be considered to be isentropic.
(a) What are the pressure and temperature at a depth of 10,000 ft?
(b) What would the pressure be at this depth if the gas were isothermal?
(c) What would the pressure be at this depth if the gas were incompressible?
20. The adiabatic atmosphere obeys the equation
P/pk = constant where k is a constant and p is density. If the temperature decreases 0.3°C for every 100 ft increase in altitude, what is the value of k? [Note: Air is an ideal gas; g = 32.2 ft/s2; R = 1544 ft lbf/(°R lbmol)].
21. Using the actual dimensions of commercial steel pipe from Appendix F, plot the pipe wall thickness versus the pipe diameter for both Schedule 40 and Schedule 80 pipe, and fit the plot with a straight line by linear regression analysis. Rearrange your equation for the line in a form consistent with the given equation for the schedule number as a function of wall thickness and diameter:
and use the results of the regression to calculate values corresponding to the 1750 and 200 in this equation. Do this using (for D) (a) the nominal pipe diameter and (b) the outside pipe diameter. Explain any discrepancies or differences in the numerical values determined from the data fit compared to those in the equation.
22. The ''yield stress'' for carbon steel is 35,000 psi, and the ''working stress'' is onehalf of this value. What schedule number would you recommend for a pipe carrying ethylene at a pressure of 2500 psi if the pipeline design calls for a pipe of 2 in. ID? Give the dimensions of the pipe that you would recommend. What would be a safe maximum pressure to recommend for this pipe?
23. Consider a 90° elbow in a 2 in. pipe (all of which is in the horizontal plane). A pipe tap is drilled through the wall of the elbow on the inside curve of the elbow, and another through the outer wall of the elbow directly across from the inside tap. The radius of curvature of the inside of the bend is 2 in., and that of the outside of the bend is 4 in. The pipe is carrying water, and a manometer containing an immiscible oil with SG of 0.90 is connected across the two taps on the elbow. If the reading of the manometer is 7 in., what is the average velocity of the water in the pipe, assuming that the flow is uniform across the pipe inside the elbow?
24. A pipe carrying water is inclined at an angle of 45° to the horizontal. A manometer containing a fluid with an SG of 1.2 is attached to taps on the pipe, which are 1 ft apart. If the liquid interface in the manometer leg that is attached to the lower tap is 3 in. below the interface in the other leg, what is the pressure gradient in the pipe (Ap/l), in units of (a) psi/ft and (b) Pa/m? Which direction is the water flowing?
25. A tank contains a liquid of unknown density (see the Fig. 4P25). Two dip tubes are inserted into the tank, each to a different level in the tank, through which air is bubbled very slowly through the liquid. A manometer is used to measure the difference in pressure between the two dip tubes. If the difference in level of the ends of the dip tubes (H) is 1 ft, and the manometer reads 1.5 ft (h) with water as the manometer fluid, what is the density of the liquid in the tank?
26. The tank shown in the Fig. 4P26 has a partition that separates two immiscible liquids. Most of the tank contains water, and oil is floating above the water on the right of the partition. The height of the water in the standpipe (h) is 10 cm, and the interface between the oil and water is 20 cm below the top of the tank and 25 cm above the bottom of the tank. If the specific gravity of the oil is 0.82, what is the height of the oil in the standpipe (H)?
27. A manometer that is open to the atmosphere contains water, with a layer of oil floating on the water in one leg (see Fig. 4P27). If the level of the water in the left leg is 1 cm above the center of the leg, the interface between the water and oil is 1 cm below the center in the right leg, and the oil layer on the right extends 2 cm above the center, what is the density of the oil?
Figure 4P26
28. An open cylindrical drum, with a diameter of 2 ft and a length of 4 ft, is turned upside down in the atmosphere and then submerged in a liquid so that it floats partially submerged upside down, with air trapped inside. If the drum weighs 150 lbf, and it floats with 1 ft extending above the surface of the liquid, what is the density of the liquid? How much additional weight must be added to the drum to make it sink to the point where it floats just level with the liquid?
29. A solid spherical particle with a radius of 1mm and a density of 1.3g/cm3 is immersed in water in a centrifuge. If the particle is 10 cm from the axis of the centrifuge, which is rotating at a rate of 100 rpm, what direction will the particle be traveling relative to a horizontal plane?
30. A manometer with mercury as the manometer fluid is attached to the wall of a closed tank containing water (see Fig. 4P30). The entire system is rotating about the axis of the tank at N rpm. The radius of the tank is r1, the distances from the tank centerline to the manometer legs are r2 and r3 (as shown), and the manometer reading is h. If n = 30 rpm, r1 = 12 cm, r2 = 15 cm, r3 = 18 cm, and h = 2 cm, determine the gauge pressure at the wall of the tank and also at the centerline at the elevation of the pressure tap on the tank.
31. With reference to the figure for Prob. 30, the manometer contains water as the manometer fluid and is attached to a tank that is empty and open to the atmosphere. When the tank is stationary, the water level is the same in both legs of the manometer. If the entire system is rotated about the centerline of the tank at a rate of N (rpm):
(a) What happens to the water levels in the legs of the manometer?
(b) Derive an equation for the difference in elevation of the levels (h) in the legs of the manometer as a function of known quantities.
32. You want to measure the specific gravity of a liquid. To do this, you first weigh a beaker of the liquid on a scale (WLo). You then attach a string to a solid body that is heavier than the liquid, and while holding the string you immerse the solid body in the liquid and measure the weight of the beaker containing the liquid with the solid submerged (WLs). You then repeat the procedure using the same weight but with water instead of the "unknown" liquid. The corresponding weight of the water without the weight submerged is Wwo, and with the solid submerged it is Wws. Show how the specific gravity of the "unknown" liquid can be determined from these four weights, and show that the result is independent of the size, shape, or weight of the solid body used (provided, of course, that it is heavier than the liquids and is large enough that the difference in the weights can be measured precisely).
33. A vertical Utube manometer is open to the atmosphere and contains a liquid that has an SG of 0.87 and a vapor pressure of 450 mmHg at the operating temperature. The vertical tubes are 4 in. apart, and the level of the liquid in the tubes is 6 in. above the bottom of the manometer. The manometer is then rotated about a vertical axis through its centerline. Determine what the rotation rate would have to be (in rpm) for the liquid to start to boil.
34. A spherical particle with SG = 2.5 and a diameter of 2 mm is immersed in water in a cylindrical centrifuge with has a diameter of 20 cm. If the particle is initially 8 cm above the bottom of the centrifuge and 1 cm from the centerline, what is the speed of the centrifuge (in rpm) if this particle strikes the wall of the centrifuge just before it hits the bottom?
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