CHAPTER 17 Pressure Measurement in Mowing Fluids

1. CHAPTER 17Pressure Measurement in Mowing Fluids

2. The idea of this machine is so simple and natural that the moment I conceived it I ran immediately to the river to make a first experiment with a glass tube. Henri de Pltot (1732)

3. 1. DEFINITIONS • The pressure definitions and the measurements we have discussed thus far have concerned fluids in thermodynamic states of statistical equilibrium. • Even for the very common case of a flowing fluid, however, there is a departure from the basic equilibrium condition, since a directed kinetic energy of flow is superposed on the random kinetic energy of the fluid molecules [1]. • Three additional pressure definitions are usually offered to cover the fluid-flow station..

4. Static (or stream) pressure p is the actual pressure of the fluid whether in motion or at rest. In principle, it can be sensed by a small hole drilled perpon-dicular to and flush with the flow boundaries so that it does not disturb the fluid in any way.

5. Dynamic (or velocity) pressure p, is the pressure equivalent of the directed kinetic energy of am fluid when the fluid is considered as a continuum. Total (stagnation, impart, or pilot) pressure p, is the sum of the static anti dynamic pressures. It can be sensed by a probe that is at rest with respect to the system boundaries when it locally stagnates the fluid isentropically (i.e. without losses and without heat transfer). ; (17.1) • MATHEMATICAL RELATIONS • The relation between these three pressures is

6. This relationship is based on the general energy equation for steady flow and on the first law of thermodynamics. In a fluid continuum that is in directed motion with respect to the system boundaries, the energy relation at a single point where isentropic stagnation is postulated is (17.2) 0= where the subscript P signifies a point quantity. Here it seems appropriate to discuss the relationship between specific weight w and density ρThese are related, through Newton's second law of motion [2], to the effect that

7. F=KMa W=KMg W=Kρg where g is the local value of the acceleration of gravity. It remains to define the proportionality constant K. In the SI system of units, where l N is defined as that force which imparts to a mass of 1 kg an acceleration of 1 m/s/s, it is clear that K must have the numerical value of unity. However, in USCS units, where 1 lbf is defined as that force which imparts to a mass of 1 lbm (i.e., 0.45359237 kg) an acceleration of 32.1740 ft/s/s, it is likewise clear that K must have the numerical value of 1/32.1740 = 1/ g

8. Where It follows that Now it is necessary to distinguish between the relatively incompressible liquids and the readily compressible gases. (For simplicity, only constant density liquids and perfect gases are analyzed.)

9. For a constant-density liquid, equation (17.2 ) yields. (17.3) where the subscript inc means incompressible. Equation (17.3) bears cut the familiar relation between total, static, and dynamic pressures in accord with equation (17, 1) and identifies the dynamic pressure.

10. For a perfect gas undergoing an isentropic process, (P/ρ)γis a constant, where γ is the ratio of specific heats. Equation (17.2) yields (17.4) Where the subscript comp means compressible. To put this relationship in a form comparable to equation (17.3), we introduce the Mach number, M=V/(γgcRT)1/2which accounts for the elastic nature of the gas. We then expand in a binomial series[3]:

11. (17.5) which again confirms the simple relationship given in equation (17.1). The dynamic pressure of equation (17.5) approaches that of equation (17.3) as the Mach number approaches zero, that is, as compressibility effects become less important.

12. When an effective total pressure P, at a plane is required, it becomes necessary to modify the point definitions of total pressure Ptpas given by equations (17.3) and (17.5). For the constant-density liquid, for example, the effective total pressure can be taken as the flow-weighted average total pressure. Thus from equation (17.3), (17.6)

13. where the integrals are taken over the area A of the plane. Integration yields (17.7) The last term in equation (17.7) is simply the density ρof the flowing fluid times the actual kinetic energy per pound of flowing fluid. for note. (17.8)

14. Sometimes it is more convenient to work with the uniform velocity of continuity (i.e. , the effective one-dimensional velocity V) rather than. with the point velocity Vp : (17.9) The ratio of the actual to the uniform kinetic energies of equations (17.8) and (17.9) defines kinetic energy coefficient αwhich also can be used to express the effective total pressure at a plane as

15. (17.10) Representative values of the kinetic energy coefficient are 1.02< <1.15 in a straight pipe for fully developed turbulent flaw can be given by the simplified expression a=I+2.7f

16. where f is the familiar friction factor of equation (17.24) and Table 17.1. Far constant-density fluids, effective total pressures defined in terms of flow-weighted average total pressures provide direct measurement of the head loss h between planes [5], that is, . （17.12） where the units of each term in equation (17.12) are ft-lbf/lbm, sometimes called feet of head, but this is only true if For the compressible fluids, however, no simple counterparts of equations (17.7), (17.10), and (17.12) are available.

17. For example, the difference in total pressure heads for a perfect gas flowing without heat transfer between planes gives no indication of head loss, since This is in direct contrast to the incompressible head loss of equation (17.12). Nevertheless, basic expression have been given for pt , p and p0for both compressible and constant-density fluids, and the next sections are concerned with the measurement of these moving fluid pressures.

18. Sensing Static Pressure • Static pressures are used for determining flow rates by head-type fluid meters (e.g., venturis, nozzles, orifices). They are required in velocity determinations, to establish thermodynamic state points, and also are useful for obtaining indications of flow direction. • From its definition, static pressure can be sensed in at least three ways. • Small hales can be drilled in the surface of the flow boundary in such a way that the streamlines of the flow are relatively undisturbed.

19. The familiar wall taps, first used extensively by. Bernoulli, function in this manner. Small holes can also be so located on probes that streamline curvatures, and other effects caused by the probe presence in a flowing fluid stream are self- compensating.

20. In this class belong the static tubes, such as those designed by Prandtl. Finally, small holes can be strategically located at critical points on aerodynamic bathos at which static pressures naturally occur. • The sphere, cylinder, wedge, and cone probes are examples of this type. Briefly expanded, separate accounts of the more basic static pressure sensors will serve to broaden our understanding of some of the problems involved.

21. 17.3.1 Wall Taps It is generally assumed that infinitely small square-edged holes installed normal to flow boundaries give the connect static pressure. Now it seems a relatively simple matter to drill a small bole perpendicular to a flow boundary and hence to sense the static pressure. But in reality, small holes are difficult to machine, they are exceedingly difficult to keep burr-free, and small holes are slow to respond to pressure changes (see Chapter 18). Therefore, fine usually settles for other than the infinitely small hole, and pays the price for compromise by applying a correction to the reading of a finite-diameter pressure tap.

22. The problem of correcting pressure tap readings was discussed by Darcy as early as 1857. The American engineer Mills did an exceptionally precise experiment to determine hole size effects in 1879 [5]. Many others, including Allen and Hooper [7], in t932, have continued this work. The recommended geometry for a pipe wall tap is shown in Figure 17.1. The effect of hole size is almost always evaluated experimentally with respect to an arbitrary small-diameter reference tap (see, for example, Figure 17.2).

23. Such results are then extrapolated to zero hole size to obtain absolute errors [8],[9]. Franklin and Wallace [10]. In a very ingenious experiment based on the use of diaphragm pressure transducers set flush in the walls of a wind tunnel, avoided such extrapolations to zero hole size. (They incidentally proved this procedure to be a questionable technique.) All experimenters agree, however, that pressure top errors increase with hole size. A recent summary of the literature on static tap errors is given in [11].

24. FIGURE 17.1 Recommended static-pressure wall-tap geometry

25. FIGURE 17.2 Typical experimental determination of hole size effect for (Source: After Shaw [9] )

26. Although square-edged holes yield small positive errors, radius-edged holes introduce additional positive errors, and chamfer-edged holes introduce small negative effects (Figure 17.3). These shape effects can be visualized as follows. At the upstream edge of a square tap, the fluid separates cleanly, (In effect, the fluid fails to note the removal of the constraining boundary.) Thus there are only minor deflections of the streamlines into a square tap. However, by fluid viscosity, a slight forward motion is imparted to the fluid in the hole. It is the arresting of this motion by the downstream wall of the tap that accounts for the pressure rise.

27. On the contrary, flow over a tag with a rounded edgy does not immediately separate: but instead is guided into the hole with the resulting recovery of a portion of the dynamic pressure.

28. Finally, although the flow does separate at the upstream edge of a countersunk tap, it also accelerate, at the sloping downstream edge of the tap, this latter effect induces a sucking action that detracts from the wean, pressure in the countersunk tap. Static pressure tap size error is usually expressed as a function of the wall stress τ ,since it is believed to arise because of a local disturbance of the boundary layer, that is, the wall shear stress esters the correlation since it characterizes the flaw velocity gradient (i.e., ) and this has are important effect on the wall ta performance.

29. The dimensional curves of Shaw (Figure 17.4) indicate that L/d (of Figure 17.1) should be>1.5 to avoid tap error dependence on L/d.Thus most experimenters keep L/d between 1.5 and 15. Franklin and Wallace [10] extended. Shaw's work (which was limited to )to and because of their careful experiment work, and because they avoid extrapolations to zero tap size, their results are recommended as definitive, and are given ii Figure 17.5. It is interesting to observe that for on the order of 2000, Δp/τ is about 3.75

30. FIGURE 17.3 Effect of orifice edge form on static pressure measurement. Variation in percentage of dynamic pressure. (Source: From Rayle [8])

33. FIGURF 17.5 Non-dimensional error curve for static pressure taps. Hale diameters vary from 0.02 to 0.25 in. (Source: After Franklin and Wallace f [10].)

34. Actually, dimensional analysis leads to two acceptable pairs of non-dimensional terms, depending on the grouping of variables chosen. For one grouping we have (17.14) whereΔp is the error in static pressure caused by the tap size, and is the tap Reynolds number, define in turn by where d is the tap diameter, v is the kinematic viscosity. And V* is the friction velocity, that is,

35. (17.15) From another grouping of variables we obtain (17.16) (17.17) (17.18)

36. FIGURE 17.6 Static tap error as a function of wall shear stress and tap Reynolds number. (Source After Benedict and Wyler [11])

37. Both of these sets of dimensionless variables have been used to correlate the available data on static wall tap errors. Such graphs are shown in Figures 17.6 and 17.7. The straight lines of Figure 17.7 can be represented by the empirical equations (17.19) to describe static tap error in the range , whereas at tap Reynolds greater than 385, we have (17.20)

38. FIGURE 17.7 Static tap error in terms of p* andτ* (Source: After Benedict and Wyler [11].)

39. Still another method for presenting static pressure tap error has been suggested [12]. Here ΔP is nondimensionalized by the dynamic pressure pcrather than the wall shear stressτ, and the resulting quantity is given in terms of the pipe Reynolds number RD=VD/v instead of the tap Reynolds number Thus (17.21) Now algebraically, (17.22)

40. where , in USCS units. But it is well established (e. g, [13]) that (17.23) where f is the Darcy friction, factor defined in general, for smooth pipes, by the implicit empirical Prandtl equation [14] (17.24) Solutions to this equation are given in Table 17.1.It should be mentioned at this point that all the above work on static taps was done for smooth pipes. Static taps installed in rough pipes would yield question able results, and hence this practice is not recommended. Combining equations (17.22) and (17.23), there results

41. (17.25) The Δp/τterm is, of course, the very empirical information available froth the Franklin and Wallace-type curves of Figures 17.5 and 17.6, once we can enter tine friction Reynolds number. This we can get from the pipe Reynolds number RDvia (17.26) It is now possible to plot general correction curves for static taps in smooth pipes in fully developed turbulent flows according to equation (17.21), based on equations (17.24)-(17.26). Such a correction curve has been proposed by Wyler [15] and is presented in Figure 17.8.

42. Example 1 Water at 70℉is flowing at a velocity of 10 ft/sin a 4-in smooth pipe. The density and kinematic viscosity of this water are p=62.3lbm/ft2, v=1.11×10-5ft2/s. Find the absolute error in the static measurement sensed by a 1-in square-edged tap. Solution . The pipe Reynolds number is The friction factor from equation (17.24) or Table 17.1 is

43. FIGURE 17.8 Static tap error as a function of pipe Reynolds number (Source: After WyIer [15].) The Friction Reynolds number from equation (17.26) is

44. Now we have at four ways to get ΔP/Pc 1. By Figure 17.6, Δp/τ=2.93 2. By Figure 17.7, at log , we read which by equation (17.17) leads to Δp/τ=2.91 3 .By equation (17.20), Δp/τ=2.845 Using an average value of Δp/τ=2.9 and equation (17.23), we have

45. 4. By Figure 17.8, which is the error to be expected of this finite pressure tap.

46. Example 2. Solution . From equation(17.24) or Table 17.1 Show that the static pressure error teacher about 1% of the dynamic pressure when From equation (17.26), From Figure 17.5. From equation (17.25),

47. As a check, one could obtain the tap error directly from Figure 17.8 at d/D=0.1,that isΔP/Pc=1.16%These two answers are within the accuracy of reading Δp/τ and Δp/pcfrom the figures. Another correlation sometimes useful is that of Rayle's Δp/pcversus the Mach number (Figure17.9) Note that Rayle [8] states that a 0.030-in hale with a 0.015-in-deep countersink should give nearly true static pressure. Actually, Mach number effects have not been considered by many experimenters because they are not usually important in pipe wall tap.

48. FIGURE 17.9 Wall tap errors for compressible flows. (Source: After Rayle [8].)

49. On pressure probes, however, Mach number effects could be very important. Rainbird [16] has done some important work in this area. As a final note on wall tap performance, Emmett and Wallace[17] have stated that the magnitude of errors found in rough channel tests have is approximately one-half to two-thirds the value found in smooth channel tests. They are speaking of turbulence effects of which we usually have no firm quantitative idea, but their work suggests that static tap errors can have a large uncertainty associated with them. A work on throat tap nozzles [121 does support the Emmett and Wallace contention that static taps in high-turbulence regions may show only one-half the error of the usual low-turbulence tests.

50. 17.3.2 Static Tubes • The accuracy in static pressure measurement using static tubes depends mainly on the position of the sensing holes with respect to the nose of a tube and its supporting stem. • Acceleration effects caused by the nose tend to lower the tap pressure; stagnation effects caused by the stem tend to raise the tap pressure. In a properly compensated tube, these two effects will just cancel at the plane of the pressure holes [19], [20].