Fenômentos de Transporte
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Fenômentos de Transporte


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four fundamental dimensions) 
F 
P 
Cr 
D 
gc 
V 
F M L t 
1 0 0 0 
0 1 - 3 0 
0 1 -1-1 
0 0 1 0 
0 0 1-1 
-1 1 1 -2 
(5.2.1-2) 
It can be shown that this matrix is of rank 4, so in a complete set we should be 
able to obtain 6 - 4 = 2 dimensionless products. 
Carrying out the dimensional analysis yields 
228 
W1 
Chapter 5: Application of Dimensional Analysis 
[q.),i")] p v2 D2 = 0 (5.2.1-3) 
In principle, we can solve W I explicitly for (p:fb2) - to obtain a new 
function 
(*) = W2[(.?)] 
This relation may be rewritten as 
(5.2.1 -4) 
(5.2.1-5) 
where (p v2) is proportional to the kinetic energy per unit volume and D2 is 
proportional to a characteristic area. 
For internal flows in circular conduits this may be expressed a.. 
(5.2.1-6) 
where we have used the bulk velocity as the characteristic velocity, the product 
(R L) in place of the square of the characteristic length, and the conduit diameter 
as the characteristic length in the Reynolds number (for geometrically similar 
systems, defined below, these lengths remain in constant ratio to one another). 
The Fanning friction factor, f, is then defined by the relation 
F = (2xRL)($p(v)*)f [(D(v)p,l .p (5.2.1-7) 
where the first factor on the right-hand side is the lateral area of the conduit and 
the second factor on the right-hand side is the kinetic energy per unit volume. 
The function f is just the function ~2 divided by (ng,). Consistent with our 
practice throughout the text, we do not incorporate g, in the formula, because 
Chapter 5: Application of Dimensional Analysis 229 
the definition of f is the same for all systems of units, and when g, is needed it 
is self-evident by dimensional homogeneity. The friction factor is a 
dimensionless function of Reynolds number. 
A friction factor (unfortunately also designated by f') four times the 
magnitude of the Fanning friction factor is sometimes also defined in the 
literature. This friction factor is called the Darcy or BIasius friction factor. 
Two different friction factors, both designated by f, cause no real difficulty as 
long as one is consistent in which friction factor one uses and observes the factor 
of four. The biggest danger of mistake is usually in using a value from a chart or 
table which incorporates one friction factor in a formula using the other. 
For external flows, Equation (5.2.1-5) may be written as 
F = ( q q w " ( y ) ] (5.2.1-8) 
where we now have a different function because we have a different flow 
situation. 
For flow around objects, we usually convert the square of the characteristic 
length to the cross-section of the object normal to flow. The reference 
velocity usually used in the kinetic energy term and in the Reynolds number is 
the freestream velocity (the velocity far enough away from the surface that 
there is no perturbation by the object). The characteristic length, if the cross- 
sectional area is circular, is the diameter of the circle - if not, a specific 
dimension must be specified. Then 
(5.2.1 -9) 
where CD, called the drag coefficient, is a dimensionless function of 
Reynolds number (CD and ~3 are different functional forms because of the 
constants introduced). Again, we do not incorporate g, because the definition is 
valid for any consistent system of units. 
5.2.2 Shape factors 
Dimensional analysis can seldom be used to predict ways in which 
phenomena are affected by different shapes of object. Therefore, when applying 
dimensional analysis to a problem it is usually convenient to eliminate the 
230 Chapter 5: Application of Dimensional Analysis 
consideration of shape by considering bodies of similar shape, that is, bodies that 
are geometrically similar. 
D r u force on ship hull 
Tbe drag force exerted on the hull of a ship depends on the shape of the hull. 
Hulls of similar shape may be specified by a single characteristic dimension such 
as keel17 length, L, and hulls of other lengths but similar shapes have their 
dimensions in the same ratio to this characteristic length - for example, the ratio 
of beam width (width across the hull) to length will be the same for all hulls of 
similar shape. If the hulls are considered to have the same relative water line (for 
example, with respect to freeboard, the distance of the top of the hull above the 
waterline), keel length will also be proportional to the draft (displacement). 
The drag force will then depend on the keel length, L, the speed of the ship, 
v, the viscosity of the liquid, p, the density of the liquid, p, and the acceleration 
of gravity (which affects the drag produced by waves). What are the 
dimensionless groups that describe this situation? 
So 1 u t ion 
Our function is 
(5.2.2- 1) 
The rank of the dimensional matrix may be shown to be 3 for a system with 
three fundamental dimensions. We will therefore have (6 - 3) = 3 dimensionless 
groups. 
If we carry out the dimensional analysis, we obtain 
(5.2.2-2) 
(the dimensionless group F/[pv2L2] is sometimes called the pressure 
coefficient, P) . l* 
l7 The keel is the longitudinal member reaching from stem to stern to which the ribs 
are attached. 
I * Langhaar, H. L. (1951). Dimensional Analysis and Theory of Models. New York, 
NY, John Wiley and Sons, p. 21. 
Chapter 5: Application of Dimensional Analysis 231 
To run experiments, we would fearrange (5.2.2-1) as 
and would rearrange (5.2.2-2) as 
F= P = w2(Re, Fr) 
p v2 L2 
(5.2.2-3) 
(5.2.2-4) 
In each case the experiments would consist of measuring the variable on the left- 
hand side of the equation as a function of the variables on the right-hand side of 
the equation, which would be varied from experiment to experiment. 
Notice the numerous advantages of (5.2.2-4) over (5.2.2-3). 
Using (5.2.2-3), to run three levels of each variable would 
require 35 = 243 experiments. Using (5.2.2-4), to run three 
levels of each variable would require only 32 = 9 experiments. 
. To use (5.2.2-4), we could easily vary both Reynolds and 
Froude numbers by varying velocity (which is easily 
accomplished in a towing tank) and hull length (which could 
be accomplished by using models of the same shape but 
different sizes). 
To use (5.2.2-3), we would also have to find a way to vary p 
(perhaps by vaqing temperature or fluid or adding a thickening 
agent to water), p (perhaps by varying fluid; or dissolving 
something in water to vary density; or varying temperature to 
vary density, even though this is not a very big effect), and g 
(perhaps by building two laboratories, one on a very high 
mountain and one in a mine, or building the experiment in a 
centrifuge). None of these alternatives is particularly 
appealing. 
To present the results of (5.2.2-3) in a compact form is 
difficult, as is extrapolating or interpolating the results. The 
results of (5.2.2-4), however, can be concisely presented on a 
two-dimensional graph, where the approximate results of 
interpolation or extrapolation can be easily visualized, as is 
illustrated in the following figure (the data is artificial and 
must not be used for design purposes). The figure 
shows the data from nine hypothetical experiments arranged in 
232 Chapter 5: Application of Dimensional Analysis 
a factorial design (each of three levels of Re combined with 
each of three levels of Fr, and hypothetical curve fits which 
could be used for interpolation (or extrapolation, if the curves 
were to be extended beyond the range of the data - which would 
be dangerous). 
Fr = 6 
F r = 4 
Fr = 2 
Reynolds number 
vle 5.2.2-2 Deceleration o f corUpressible -fh&i 
The important variables that determine the maximum pressure, pmax, 
resulting when flow of a compressible fluid is stopped instantly (for example, by 
shutting a valve