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ratio of inertial forces to viscous forces. 
Dv P - inertial forces I (1.4.1-1) viscous forces ReynoldsNumber = Re = I-L - 
In pipe flow if the value of the Reynolds number is less than about 2100, the 
flow is laminar, and as the value of the Reynolds number increases beyond 2100 
the flow undergoes a transition to turbulent. This transitional value is different 
for different geometrical configurations, for example, flow over a flat plate. 
l 2 Japan, S. 0. M. E., Ed. (1988). Visualized flow: Fluid motion in basic and 
engineering situations revealed by flow visuulization. Thermodynamics and Fluid 
Mechanics. New York, NY, Pergamon, p. 25. 
Chapter I : Essentials 21 
1.4.2 Newtonian fluids 
When two "layers" of fluid flow past each other as sketched in Figure 1.4.2- 
1, where the upper layer is moving faster than the lower layer, the deformation 
produces a shear stress tyx proportional to velocity gradient. (The profile in 
Figure 1.4.2-1 is shown as linear for ease of illustration.) The faster-moving 
layer exerts a force which tends to accelerate the slower-moving layer, and the 
slower-moving layer exerts a force which tends to decelerate the faster-moving 
layer, much as if the layers rubbed against one another like the successive cards 
in a deck which is being spread (even though this is not what actually happens, 
it is a convenient model for visualization purposes). 
This differs from a solid, in which the shear stress can exist simply because 
of relative displacement of adjacent layers without relative motion of the 
adjacent layers. 
The shear stress here is normal to the y direction. Different fluids exhibit 
different resistance to shear. 
"X 
Figure 1.4.2-1 Shear between layers of fluid 
A Newtonian fluid follows Newton's law of v i s~os i ty .~~ Newton's law 
of viscosity relates the shear stress to the velocity gradient as 
l 3 All fluids that do not behave according to this equation are called non-Newtonian 
fluids. Since the latter is an enormously larger class of fluids, this is much like 
classifying mammals into platypuses and non-platypuses; however, it is traditional. 
A complete definition of a Newtonian fluid comprises 
(footnote continued on following page) 
22 Chapter 1: Essentials 
(1.4.2- 1) 
where p is the coefficient of viscosity, v, is the x-component of the velocity 
vector, y is the y-coordinate, and zyx can be regarded as either (both quantities 
have the same units) 
a) the flux of x-momentum in the y-direction, or 
b) the shear stress in the x-direction exerted on a surface of 
constant y by the fluid in the region of lesserI4 y 
Units of shear stress are the same as units of momentum flux. 
TYY = - 2 p $ + 7 ( P - k ) ( V * V ) a v 2 
7, = -2p-$t&c)(v.v) a v 2 
(\u201d. \u201d.) 
- C L dX+x 2, = 2, = 
where k is the vlsc-. The one-dimensional form shown will suffice for 
our purposes. For further details, see Bird, R. B., W. E. Stewart, et al. (1960). 
Transport Phenomena. New York, Wiley; Longwell, P. A. (1966). Mechanics of 
Fluid Flow, McGraw-Hill; Prager, W. (1961). Introduction to Mechanics of Continua. 
Boston, MA, Ginn and Company; Bird, R. B., R. C. Armstrong, et al. (1977). 
Dynamics of Polymeric Liquids: Volume 1 Fluid Mechanics. New York, NY, John 
Wiley and Sons; and Bird, R. B., R. C. Armstrong, et al. (1987). Dynamics of 
Polymeric Liquids: Volume 1 Fluid Mechanics. New York, NY, Wiley. 
l4 Note the actual, not the absolute sense is implied; i.e., y = -10 is less than y 
= -5. 
Chapter I : Essentials 23 
Equation (1.4.2- 1) is often written without the minus sign? We choose to 
include the minus sign herein because the shear stress can be interpreted (and, in 
the case of a fluid, perhaps more logically be interpreted) as a momentum flux.16 
Momentum "runs down the velocity gradient" as a consequence of the second law 
of thermodynamics much as water runs downhill; i.e., flows from regions of 
higher velocity to regions of lower velocity. It therefore seems more logical to 
associate a negative sign with zyx in the above illustrated case. Since the 
Newtonian viscosity is by convention a positive number, and the velocity 
gradient is shown as positive, hence the minus sign to ensure the negative 
direction for the momentum flux zyx. 
l5 For example, see McCabe, W. L., J. C. Smith, et al. (1993). Unit Operations of 
Chemical Engineering, McGraw-Hill, p. 46; Bennett, C. 0. and J. E. Myers (1974). 
Momentum, Heat, and Mars Transfer. New York, NY, McGraw-Hill, p. 18; Fox, R. W. 
and A. T. McDonald (1992). Introduction to Fluid Mechanics. New York, NY, John 
Wiley and Sons, Inc., p. 27; Welty, J. R., C. E. Wicks, et al. (1969). Fundamentals 
of Momentum, Heat, and Mass Transfer, New York, NY, Wiley, p. 92; Ahmed, N. 
(1987). Fluid Mechanics, Engineering Press, p. 15; Potter, M. C. and D. C. Wiggert 
(1991). Mechanics of Fluids. Englewood Cliffs, NJ, Prentice Hall, p. 14; Li, W.-S. 
and S.-H. Lam (1964). Principles of Fluid Mechanics. Reading, MA, Addison-Wesley, 
p. 3; Roberson, J. A. and C. T. Crowe (1993). Engineering Fluid Mechanics. Boston, 
MA, Houghton Mifflin Company, p. 16; Kay, J. M. and R. M. Nedderman (1985). 
Fluid Mechanics and Transfer Processes. Cambridge, England, Cambridge University 
Press, p. 10. 
l 6 This follows the convention used in Bird, R. B., W. E. Stewart, et al. (1960). 
Transport Phenomena. New York, Wiley, p. 5; Geankoplis, C. J. (1993). Transport 
Processes und Unit Operations. Englewood Cliffs, NJ, Prentice Hall, p. 43. 
24 Chapter I : Essentials 
Ay I 
PLANES OF 
CONSTANT Y 
Figure 1.4.2-2 Momentum transfer between layers of fluid 
Figure 1.4.2-3 shows the various cases, Quadrant 1 being the example 
above. 
Y A M. MOM NTUMFLUX 
MOMENTU FLUX )qb 
VX "X 
Figure 1.4.2-3 Sign convention for momentum flux between 
layers of fluid 
A similar interpretation can be made for regarding zyx as a shear stress. 
Chapter I : Essentials 25 
"X vX 
Figure 1.4.2-4 Sign convention for shear stress on surface layers 
of fluid 
Note that 
in Quadrant 1 the fluid in the region of lesser y has lower 
velocity in the positive x-direction than the fluid above it; 
therefore, the shear stress exerted is to the left (negative) 
in Quadrant 2 the fluid in the region of lesser y has lower 
velocity in the negative x-direction than the fluid above it; 
therefore, the shear stress exerted is to the right (positive) 
in Quadrant 3 the fluid in the region of lesser y has greater 
velocity in the negative x-direction than the fluid above it; 
therefore, the shear stress exerted is to the left (negative) 
in Quadrant 4 the fluid in the region of lesser y has greater 
velocity in the positive x-direction than the fluid above it; 
therefore, the shear stress exerted is to the right (positive) 
all consistent with (b) above. By Newton's first law the shear stresses on the 
corresponding opposite (upper) faces of the planes of constant y are the negative 
of the forces shown in Figure 1.4.2-4. 
26 Chapter I : Essentials 
The cases are summarized in Table 1.4.2- 1. 
Table 1.4.2-1 Summary of sign convention for stress/momentum 
flux tensor 
We frequently refer to the action of viscosity as being "fluid friction." In 
fact, viscosity is not exhibited in this manner. Rather than friction from layers 
of fluid rubbing against one another (much as consecutive cards in a deck of cards 
sliding past each other), momentum is transferred in a fluid by molecules in a 
slower-moving layer of fluid migrating to a faster-moving region of fluid and 
vice versa, as shown schematically in Figure 1.4.2-2, where some molecules 
from the A layer are shown to have migrated into the B layer, displacing in turn 
molecules from this layer. 
b c c 
I t 
X 
Figure 1.4.2-5 Migration of momentum by molecular motion 
In