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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 (&quot;&quot; (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