Fenômentos de Transporte
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corresponding points experience forces in a constant ratio (which may or may 
not have the same value as the ratio for lengths). Two bodies are thermally 
Chapter 5: Application of Dimensional Analysis 26.5 
similar if at geometrically corresponding points the ratio of the temperatures 
is constant. 
Points in the model and prototype that correspond are called homologous 
points. Phenomena occurring in the model and prototype2* at homologous 
points are said to occur at homologous times, even though these events may 
not occur at the same real time. An equivalent definition of similarity is that two 
systems are similar if they behave in the Same manner at homologous points and 
times. 
The classical principle similarity from Buckingham n-theorem is 
(5.5-1) 
Two systems in which all relevant physical quantities are in a constant ratio will 
satisfy Equation (5.5-1). The z\u2019s - dimensionless groups - are the criteria for 
geometric, dynamic, thermal, etc., similarity. 
If we arbitrarily restrict the f of Equation (5.5-1) to a power function (a 
product of powers) then 
II;, = c (x2)x2 (n3)x3 * * . (n\u201c-JXn-r (5.5-2) 
where C is called the shape factor since it has been found to depend primarily on 
shape, and XI, x2 ... yn - r are empirical exponents. In this relationship, which is 
called the extended principle (really, it is a restricted principle) of similarity, 
the shape factor depends on geometry; therefore, this equation is limited to 
systems that have the same shape. 
The shape factor can be eliminated if two systems of similar geometric form 
are compared 
(5.5-3) 
28 One dictionary definition of prococype is \u201cthe original on which a thing is 
modeled.\u201d This is a little ambiguous for common engineering usage, because w e often 
use data from our model to design an object, in which case the object is modeled on 
the model as rhe original. We usually mean the laboratory or mathematical 
representation when we say \u201cmodel,\u201d and the real world object under investigation or 
to be created when we say \u201cprototype.\u201d 
266 Chapter 5: Application of Dimensional Anulysis 
where the prime distinguishes between the systems. Use of the analysis in this 
manner is called extrapolation since the values of the exponents x2, x3, etc., 
are extrapolated from one system to the other. 
It is possible to eliminate the effect of the exponents by comparing 
geometrically similar systems which have the same values of the corresponding 
dimensionless groups on the right-hand side. This makes all ratios on the right- 
hand side equal to unity, and therefore the right-hand side is equal to unity 
independent of the values of the exponents, thus 
(5.5-4) 
In general, when two physical systems are similar, knowledge about one system 
provides knowledge about the other. Equation (5.5-4) is called the principle 
of corresponding states. 
Often it is not possible to match the values for every pair of corresponding 
groups as required above. In such cases we must run our experimental models so 
as to minimize the unmatched effects. If it is not possible to do this, we get a 
scale effect. For this reason, when running models, one has to be careful that 
values of the groups are appropriate to both the model and the prototype For 
example, if in testing a model of a breakwater in the laboratory, one uses a very 
small scale (shallow depth), in the data obtained the surface tension of the water 
may have a large effect on the behavior; however, at the scale of an actual 
breakwater the surface tension of the water would usually have no perceptible 
effect. 
We can define scale factors as the ratio of a parameter in the model, 
Xmdel, to that in the prototype, Xproto. 
(5.5-5) 
le 5.5-1 Drap on immetsed body 
Consider scaling the drag on a body immersed in the stream of 
incompressible fluid. The following equation describes the flow, where 9 (. .) 
is functional notation. 
Chapter 5: Application of Dimensional Analysis 267 
(5.5-6) 
If we wish to measure the same force in the model as in the prototype, and wish 
to use the Same fluid in each, determine the parameters within which we must 
construct the model. 
Solution 
For (Pmdcl to be the same as (Ppoto, the Reynolds numbers of the model and 
prototype must be equal. 
In terms of the scale factors 
= 1 
If the same fluid is used in model and prototype 
K, = 1 
K, = 1 
Then 
2 1 KD K v K p - KD K v ( l ) 
KP - (1) 
K,K, = 1 
(5.5-7) 
(5.5-8) 
(5.5-9) 
(5.5- 10) 
so the product of D and v needs to correspond in model and prototype. Then 
268 Chapter 5: Application of Dimensional Analysis 
F 
= 1 
F- - 1 K 
Kt Kk 
which implies 
(5.5-1 1) 
(5.5-12) 
(5.5- 13) 
(5.5- 14) 
(5.5- 15) 
Therefore, so long as the same fluid is used, KF will remain equal to one for the 
same values of D and v in the model and prototype; that is, the force in the 
model and that in the prototype will be the same. 
Dle 5.5-2 Scale effech 
We wish to estimate the pressure drop per unit length for laminar flow of 
ambient atmospheric air at 0.1 ft/s in a long duct. Because of space limitations, 
the duct must be shaped like a right triangle with sides of 3,4, and 5 feet. We do 
not have a blower with sufficient capacity to test a short section of the full-size 
duct, so it has been proposed that we build a small-scale version of the duct, and 
Chapter 5: Application of Dimensional Analysis 269 
further, that we use water as the fluid medium in order to achieve larger and 
therefore more accurately measurable pressure drops in the model? 
Discuss how to design and build the laboratory model. 
Solution 
Since there is no reason to create more scaling problems than we already 
have, we will make the laboratory model geometrically similar to the full-size 
duct. We will therefore be able to characterize it by a single length dimension, 
which we will designate as D. 
In the laminar region, we can expect the friction factor, f, which is a vehicle 
(to be discussed in detail in Chapter 6) for predicting pressure drops in internal 
flows, to be a function of only the Reynolds number, the same dimensionless 
group as used in the previous example. 
friction factor = f = f (Re) = f -F ("" (5.5- 16) 
In the laminar region the friction factor is not a function of relative roughness of 
the surface; therefore, we are free to choose the most easily fabricated material to 
build the laboratory model. 
The pressure drop is related to the friction factor for this case as 
(5.5-17) 
where C is a constant. 
If we choose to make the friction factors of the model and prototype the 
same, the scaling law follows as in the immediately preceding example 
= 1 
~ ~ ~ 
29 This problem can be solved quite nicely by numerical methods if the necessary 
computing equipment is available. We ignore this alternative for pedagogical reasons 
in order to obtain a simple illustration. 
270 Chapter 5: Application of Dimensional Analysis 
(5.5-18) 
This time, however, we have prescribed the scale factors for the fluids used. 
Approximate values at room temperature are 
1 CP = 5 . 5 6 ~ 103 = K, 
1.8 x 104 cP 
Therefore, to keep the Same friction factor will require 
KDK, = 7.2 
Examining the scaling for pressure drop 
However, we intend to make Kf = 1.0, so 
(1) (772) K2, 
KD K(?) = 
K(?) = ( 7 7 2 ) z 
(5.5- 19) 
(5.5-20) 
(5 521) 
(5.5-22) 
(5 5 2 3 ) 
(5.5-24) 
(5 5 2 5 ) 
Chapter 5: Application of Dimensional Analysis 271 
We now have considerable flexibility in choosing the size of our model and the 
velocity at which we pump the water. 
For example, suppose that we decide it will be easy to fabricate a model duct 
with hypotenuse of 2 inches. This will set the scale factor