Page 197 - Bird R.B. Transport phenomena

§6.2 Friction Factors for Flow in Tubes 181

and appropriate initial conditions. The uniform inlet velocity profile in Eq. 6.2-5 is accu-
rate except very near the wall, for a well-designed nozzle and upstream system. If Eqs.
3.7-8 and 9 could be solved with these boundary and initial conditions to get vand &, the
solutions would necessarily be of the form
v = v(f,0,z,bRe) (6.2-7)
Ф = &(r, 0, z, Ь Re) (6.2-8)
That is, the functional dependence of v and Ф must, in general, include all the dimen-
sionless variables and the one dimensionless group appearing in the differential equa-
tions. No additional dimensionless groups enter via the preceding boundary conditions.
As a consequence, dvj dr must likewise depend on r,d,z, t, and Re. When dvj dr is eval-
uated at r = \ and then integrated over z and в in Eq. 6.2-3, the result depends only on t,
Re, and L/D (the latter appearing in the upper limit in the integration over z). Therefore
we are led to the conclusion that/(f) = /(Re, L/D, t), which, when time averaged, becomes
/ = /(Re,L/D) (6.2-9)
when the time average is performed over an interval long enough to include any long-
time turbulent disturbances. The measured friction factor then depends only on the
Reynolds number and the length-to-diameter ratio.
The dependence of /on L/D arises from the development of the time-average veloc-
ity distribution from its flat entry shape toward more rounded profiles at downstream z
values. This development occurs within an entrance region, of length L = 0.03D Re for
c
laminar flow or L ~ 60D for turbulent flow, beyond which the shape of the velocity dis-
e
tribution is "fully developed." In the transportation of fluids, the entrance length is usu-
ally a small fraction of the total; then Eq. 6.2-9 reduces to the long-tube form
/ = /(Re) (6.2-10)
and/can be evaluated experimentally from Eq. 6.1-4, which was written for fully devel-
oped flow at the inlet and outlet.
Equations 6.2-9 and 10 are useful results, since they provide a guide for the system-
atic presentation of data on flow rate versus pressure difference for laminar and turbu-
lent flow in circular tubes. For long tubes we need only a single curve of / plotted versus
the single combination D(v )p/JJL. Think how much simpler this is than plotting pressure
z
drop versus the flow rate for separate values of D, L, p, and /л, which is what the uniniti-
ated might do.
There is much experimental information for pressure drop versus flow rate in tubes,
and hence/can be calculated from the experimental data by Eq. 6.1-4. Then/can be plot-
ted versus Re for smooth tubes to obtain the solid curves shown in Fig. 6.2-2. These solid
curves describe the laminar and turbulent behavior for fluids flowing in long, smooth, cir-
cular tubes.
Note that the laminar curve on the friction factor chart is merely a plot of the
Hagen-Poiseuille equation in Eq. 2.3-21. This can be seen by substituting the expression
for (2P — tyj) from Eq. 2.3-21 into Eq. 6.1-4 and using the relation w = P(V )TTR ; 2 this gives
0 Z
, 16 [Re < 2100 stable 1 r* ? тп
7 = K }
Re [Re > 2100 usually unstable]
in which Re = D(V )P//JL; this is exactly the laminar line in Fig. 6.2-2.
Z
Analogous turbulent curves have been constructed by using experimental data. Some
analytical curve-fit expressions are also available. For example, Eq. 5.1-6 can be put into
the form

2.1 X 10 3 < Re < 10 5 (6.2-12)
Re i/4