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