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of 
heat transfer and sometimes mass transfer. 
This large and very interesting area of fluid mechanics is still developing 
rapidly. For further information there are a wide range of multiple resources, 
from the applied to the theoretical. The reader might find a starting point in one 
of the references cited bel0w.1~ 
1.4.4 Compressible vs. incompressible flows 
Flows in which the density variation in the fluid can be neglected may be 
modeled as incompressible, even fluids usually thought of as compressible, such 
as gases. Only if the density variation must be considered are the flows modeled 
as compressible. 
Most liquid flows can be modeled as incompressible. (Cavitation is an 
exception.) Gas flows at low speeds can frequently be modeled as 
incompressible. For values of the Mach number (another dimensionless group, 
which represents the ratio of the local velocity to the speed of sound) less than 
0.3, the error is less than 5% in assuming incompressible behavior. For Mach 
l 8 Tadmor, 2. and C. Gogos (1979). Principles of Polymer Processing. New York, NY, 
John Wiley and Sons; Dealy, J. M. and K. F. Weissbrun (1990). Melt Rheology and its 
Role in Plastics Processing. New York, NY, Van Nostrand Reinhold; Mashelkar, R. 
A., A. S. Mujumdar, et al., Eds. (1989). Transport Phenomena in Polymeric Systems. 
Chichester, W. Sussex, England, Ellis Horwood, Ltd.; Joseph, D. D. (1990). Fluid 
Dynumics of Viscoelastic Liquids. New York, NY, Springer-Verlag; Bird, R. B., R. C. 
Armstrong, et al. (1987). Dynamics of Polymeric Liquids: Volume I Fluid Mechanics. 
New York, NY, Wiley; Bird, R. B., C. F. Curtis, et al. (1987). Dynamics of Polymeric 
Liquids: Volume 2 Kinetic Theory. New York, NY, Wiley. 
Chapter I : Essentials 33 
numbers greater than 0.3, gases normally should be modeled as compressible 
fluids. 
1.5 Averages 
We make extensive use of the concept of average, particularly in 
conjunction with area-averaged velocity, bulk or mixing-cup temperature, and 
bulk or mixing-cup concentration, all of which are attributes of flowing streams. 
All averages we will use may be subsumed under the generalized concept of 
average discussed below. 
Among other uses, averages link microscopic models to macroscopic 
models, the variables in the macroscopic models being space and/or time 
averages of the variables in the microscopic models. Time averages are used as 
models for velocity fields in turbulent flow, where the rapid local fluctuations in 
velocity are often either not mathematically tractable or not of interest. 
1.5.1 General concept of average 
In one dimension, consider a function 
Y = f(J0 (1.5.1-1) 
By the average value of y with respect to x we mean a number such that, if we 
multiply it by the range on x, we get the same result as would be obtained by 
integrating y over the same range. We write this definition as 
Yaverag (b - a) = yx (b - a) l = y f ( x ) d x ( 1.5.1 -2) 
We use the overbar to indicate an average and subscript(s) to indicate with respect 
to which variable(s) the average is taken.19 If it is clear with respect to which 
variable the average is taken, we will omit the subscript. 
This definition can be rewritten as 
l 9 There is obviously overlap between this notation and the notation sometimes used 
for partial derivatives. The context will always make the distinction clear in cases 
treated here. There are also a number of notations used for special averages, such as 
the area average and bulk quantities. Treatment of these follows. 
34 Chapter I : Essentials 
(1.5.1 -3) 
where we have omitted the subscript because there is only one independent 
variable with respect to which the average can be taken. 
This same definition can be extended to the m e where y is a function of 
many variables. We define the average of y with respect to x,, as 
( 1.5.1 -4) 
Sometimes we take the average with respect to more than one independent 
variable; to do this we integrate over the second, third, etc., variables as well. 
For example, the average of y with respect to XI, X6, and x47 is 
(1.5.1-5) 
Notation for averages does not always distinguish with respect to which 
variables they are taken. This information must be known in order to apply an 
average proprly, as must the range over which the average is taken. 
We will in this course of study frequently use three particular averages: the 
area average, the time average, and the bulk average. 
le 1.5.1-1 Time-a vera pe vs distga ce-a vergge SD& 
Suppose I drive from Detroit to Chicago (300 miles), and upon arrival am 
asked "What was your average speed?" If I answer, "60 mph," I am probably 
thinking somewhat as follows: My speed vs. time (neglecting short periods of 
acceleration and deceleration) might be approximated by the following: 
Chapter I : Essentials 35 
First 45 minutes - speed about 40 mph during time to get 
out of Detroit 
Stopped for 15-minute coffee break 
Resumed driving, drove at 70 mph (exceeding speed limit by 
5 mph) for 3.5 hours 
Passed highway accident, reduced speed to 50 mph for last 30 
minutes to Chicago city limits 
A plot of my approximate speed versus time, s = f(t), is shown in Figure 
1.5.1-1. 
Figure 1.5.1-1 Time-average speed 
ACTUAL SPEED, MPH iJ 
for travel between two points 
Here speed is the dependent variable and time the independent variable: 
s = f(t) 
The distance traveled is the integral of speed by time. 
( 1.5.1 -6) 
By declaring my average speed to be 60 mph, I mean that multiplying 60 
mph, the average speed, by the elapsed time of 5 hours will yield the total 
distance, 300 mi. In Figure 1.5.1-1 this means that the area beneath the line at 
36 Chapter I : Essentials 
the average speed is the same as the integral of the actual speed vs. time curve. 
Writing this in the form of our definition of average we have: 
S tinv amage 
(0.75 - 0) (40) + (1 - 0.75) (0) + (4.5 - 1) (70) + (5 - 4.5) (50) 
(1.5.1-7) 
(5 - 0) hr 
Note that we could, however, equally well regard speed as a function of 
distance, x, not time, t, and get another function s = g(x) as shown in Figure 
1.5.1-2. 
- 
10- 
0 
0 50 100 150 200 250 300 
DISTANCE, MILES 
Figure 1.5.1-2 Distance-average speed for travel between two 
points 
This new function tells me how fast I was going as a function of where I was 
rather than what time it was. I can equally well define an average speed based 
on this function 
Chapter I : Essentials 37 
(30 - 0) (40) + (275 - 30) (70) + (300 - 275) (50) mi [ (300 - 0) mi I = ( 1 .5.1-8) 
(mi)* 
19,600 - - hr = 65.3 
300 mi hr 
This average speed is different, since it is intended to be used differently. Instead 
of multiplying by elapsed time we multiply by elapsed distance to get the 
integral: 
(mi) coo g(x) dx = (300) mi (65.3) = 19,590 hr (1.5.1-9) 
The integral of g(x), however, does not have the intuitive appeal of the integral 
of f(t) in the numerator of the right-hand side of Equation (1.5.1 -7), which is the 
total distance traveled. 
1.5.2 Velocity averages 
There are two primary averages that are used in conjunction with fluid 
velocity, although many are possible: 
area-averaged velocities, and 
time-averaged velocities 
Area-averaged velocity 
A special average that is used frequently enough to deserve a unique notation 
is the area-averaged velocity, <v>, which is defined as a number such that, if it 
is multiplied by the basis area through which the fluid is flowing, yields the 
volumetric flow rate. 
38 Chapter 1: Essentials 
(1 52-1) 
This is clearly just a special case of our definition of general average above, 
where the average here is with respect to area?O 
We must be careful to get the signs correct when using area-averaged 
velocities or volumetric