fluid_mechanics

7.7 Correlation of Experimental Data 353
density, r, fluid viscosity, m, and the velocity, V. Thus,
and application of the pi theorem yields two pi terms
ß 1 D ¢p /
and ß
rV 2 2 rVD
m
Hence,
D ¢p /
rV 2
To determine the form of the relationship, we need to vary the
Reynolds number, Re rVDm, and to measure the corresponding
values of D ¢p /rV 2 . The Reynolds number could be varied
by changing any one of the variables, r, V, D, or m, or any combination
of them. However, the simplest way to do this is to vary the
velocity, since this will allow us to use the same fluid and pipe.
Based on the data given, values for the two pi terms can be computed,
with the result:
D¢p RV 2 RVDM
0.0195
0.0175
0.0155
0.0132
0.0113
0.0101
0.00939
0.00893
¢p / f 1D, r, m, V2
f a rVD
m b
4.01 10 3
6.68 10 3
9.97 10 3
2.00 10 4
3.81 10 4
5.80 10 4
8.00 10 4
9.85 10 4
These are dimensionless groups so that their values are independent
of the system of units used so long as a consistent system is used.
For example, if the velocity is in fts, then the diameter should be in
feet, not inches or meters. Note that since the Reynolds numbers
are all greater than 2100, the flow in the pipe is turbulent 1see Section
8.1.12.
A plot of these two pi terms can now be made with the results
shown in Fig. E7.4a. The correlation appears to be quite good, and
if it was not, this would suggest that either we had large experimental
measurement errors or that we had perhaps omitted an important
variable. The curve shown in Fig. E7.4a represents the general relationship
between the pressure drop and the other factors in the range
of Reynolds numbers between 4.01 10 3 and 9.85 10 4 . Thus,
for this range of Reynolds numbers it is not necessary to repeat the
tests for other pipe sizes or other fluids provided the assumed independent
variables 1D, r, m, V2 are the only important ones.
Since the relationship between ß 1 and ß 2 is nonlinear, it is not
immediately obvious what form of empirical equation might be
used to describe the relationship. If, however, the same data are
______ D Δp
ρ V 2
______ D Δp
ρ V 2
0.022
0.020
0.018
0.016
0.014
0.012
0.010
0.008
0 20,000 40,000 60,000 80,000 100,000
_____ ρ VD
Re =
μ
(a)
4
2
10 –2 8
6
4 × 10 –3 10 3 2 4 6 8 10 4 2 4 6 8 10 5
F I G U R E E7.4
plotted on logarithmic graph paper, as is shown in Fig. E7.4b, the
data form a straight line, suggesting that a suitable equation is of
the form ß 1 Aß n 2 where A and n are empirical constants to be
determined from the data by using a suitable curve-fitting technique,
such as a nonlinear regression program. For the data given
in this example, a good fit of the data is obtained with the equation
ß 1 0.150 ß 0.25
2
(Ans)
COMMENT In 1911, H. Blasius 11883–19702, a German
fluid mechanician, established a similar empirical equation that is
used widely for predicting the pressure drop in smooth pipes in
the range 4 10 3 6 Re 6 10 5 1Ref. 162. This equation can be
expressed in the form
D ¢p /
rV 2
_____ ρVD
Re =
μ
(b)
0.1582 a rVD
m b 14
The so-called Blasius formula is based on numerous experimental
results of the type used in this example. Flow in pipes is discussed
in more detail in the next chapter, where it is shown how
pipe roughness 1which introduces another variable2 may affect the
results given in this example 1which is for smooth-walled pipes2.
For problems involving
more than
two or three pi
terms, it is often
necessary to use
a model to predict
specific characteristics.
As the number of required pi terms increases, it becomes more difficult to display the results
in a convenient graphical form and to determine a specific empirical equation that describes the
phenomenon. For problems involving three pi terms
ß 1 f1ß 2 , ß 3 2
it is still possible to show data correlations on simple graphs by plotting families of curves as illustrated
in Fig. 7.5. This is an informative and useful way of representing the data in a general