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# Fenômentos de Transporte

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We cannot apply our equations across the boundaries between phases because the equations were derived on the assumption of a continuum, and at phase boundaries there is a discontinuity in properties. 284 6 A l Manometers Chapter 6: Momentum Transfer in Fluids A practical application of static fluids occurs in a device cake a manometer. Manometers are U-tubes (tubes bent into the shape of the letter U) partmlly filled with a liquid immiscible with and differing in density from the fluid in which the pressure is to be determined. The pressure difference between the points at the inlet of each of the legs (the vertical parts of the U) is inferred by measuring the difference in heights of the liquid interfaces in the legs and applying the equations for a static fluid. Examde 6.1.1-1 Pressure difference usinn a manometer A manometer is used to determine the pressure difference between two vessels as shown in Figure 6.1.1-1. Figure 6.1.1-1 Measurement of pressure difference with manometer The vessel on the left and the left leg of the manometer above point 2 are tilled with fluid A; the manometer contains fluid B between points 2 and 3; the right leg of the manometer above point 3 and the tank on the right contain fluid C. Find the difference in pressure between points 1 and 4. Solution We cannot apply our equations for a static fluid directly between points 1 and 4 because these equations were derived for a continuum, and the path between 1 and 4, if taken through the fluids, has abrupt discontinuities in properties at the two fluid interfaces. (If not taken through the fluids - that is, through the walls of the manometer - the discontinuities would require the introduction of solid mechanical properties.) We can, however, by writing the Chapter 6: Momentum Transfer in Fluids 285 pressure difference as the sum of differences each in a single fluid, apply the equations to these single fluids. By adding and subtracting pz and p3 we have (PI -PO) = (PI - P2) + (Pz - P3) + (P3 - P4) (6.1.1-1) Applying Equation (6.1- 14) to the various pressure drops in Figure (6.1.1 - 1) (6.1.1 -2) Example 6.1 .I-2 Pressure difference between tanks The two tanks in Figure 6.1.1-2 both contain water (fluids A and C). The manometer fluid B has a specific gravity of 1.1. Find the pressure difference (p1 - P4). Figure 6.1.1-2 Pressure difference between tanks Solution Substituting in Equation (6.1 .l-1) from the previous example 286 Chapter 6: Momentum Transfer in Fluids - (62.4) 9 (32.2) % (3.0 ft - 2.5 ft) + S 1 - (1.1) (62.4) 9 (32.2) 5 (2.5 ft - 2.0 ft) + S 1 = 2 8 ( 3 ) (6.1 .l-3) Examble 6.1.1 -3 Differential manometer A differential manometer employs two fluids which differ very little in density. It normally also has reservoirs of a large cross-section to permit large changes in position of the interface level between the manometer liquids without appreciable change in interface levels between the manometer liquids and the fluid whose pressure is to be determined. A typical differential manometer is shown in Figure 6.1.1-3. - J -27 A C A 0 1 Z O " 0 Figure 6.1.1-3 Differential manometer Two fluids, A and B, are employed to measure the difference in pressure of a gas C at points 5 and 6 as shown, Develop an equation for the difference in pressure between point 5 and point 6. Chapter 6: Momentum Transfer in Fluids 287 So 1 u t ion The pressure difference is obtained in terms of the difference between 2 and 3 by writing Since the density of a the gas is so low compared to the densities of the liquids, we neglect the pressure difference (the static head) of gas from 5 to 1 and from 6 to 4. Applying Equation (6.1.1-2) but since z4 = z1 (PS - PS) g ( P A +B) (z2 - z3) = (6.1.1-7) which shows that, for a given pressure difference, as the densities of A and B approach each other, the difference in height between points 2 and 3 grows larger - that is, the manometer becomes more sen~itive.~ 3Another way to achieve more sensitivity in a manometer in an economical fashion is to incline one leg at an angle, viz. so that v a small vertical displacement will induce a proportionately larger displacement of the interface along the right leg of the tube. Such arrangements are 288 Chapter 6: Momentum Transfer in Fluids 6.2 Description of Flow Fields We have modeled fluids as continua even though they are made up of discrete molecules. It is, however, convenient to talk of fluid \u201cparticles,\u201d by which we mean aggregates of fluid that remain in proximity to one another. Such a fluid \u201cparticle,\u201d if we could somehow paint it a different color to distinguish it from the surrounding fluid, would trace a path line if we took successive multiple exposures with a fixed camera of this particle as it passed through space. If we think of instead continuously marking each fluid particle that passes through a given point in space - experimentally, for example, by supplying a tracer such as an injected dye or a stream of small bubbles - and taking an single photograph showing the instantaneous position of all particles so marked, the line established is called a streakline. A streamline is yet another imaginary line in a fluid, this time established by moving in the direction of the instantaneous local fluid velocity to get from one point to the next on the line. A streamline is thereby everywhere in the direction of the velocity vector at a particular instant in time. By definition, then, there can be no flow across a streamline because there is no component of the velocity normal to the streamline. If, to establish yet another imaginary line in the fluid, we mark a line of adjacent fluid particles at a given instant and observe the evolution of such a line with time, we obtain a timeline which may give us insights into the behavior of the fluid. For example, we could think of marking a line in an initially static fluid and then observing the behavior of this line as the fluid is set into motion. In steady flow fluids move along streamlines, since the velocity at any point in the fluid is not changing with time; therefore, pathlines and streamlines are coincident. Since all fluid particles passing through a given point follow the same path in steady flow, streaklines also coincide with timelines and pathlines. In unsteady flow the streamline shifts as the direction of the velocity changes, and a fluid particle can shift from one streamline to another so the pathline is no longer necessarily a streamline. Similarly, successive particles of fluid passing through the same point no longer necessarily follow the same path, so sometimes called draft tubes, probably because of their original application in measuring stack draft pressures for power plants. Chapter 6: Momentum Transfer in Fluids 289 streaklines do not necessarily coincide with streamlines and pathlines in unsteady flow.4 A stream tube is a imaginary tube whose surfaces are composed of streamlines. Since there is no flow across streamlines (because the component of the velocity normal to the streamline is zero), there is no flow through the walls of a stream tube. The stream function Y is a quantity defined by (6.2-1) Notice that defining the stream function in this way satisfies the continuity equation identically - for example, in two dimensions one has because the order of differentiation is immaterial if the partial derivatives are continuous, as they are in cases of interest here. Consider an element of differential length, dr, lying along a streamline; i.e., tangent to the streamline. The condition that the velocity vector be collinear to the tangent to the streamline is expressed by