Page 90 - Industrial Ventilation Design Guidebook

4.1 FLUID FLOW 55

5
When Re > 10 , the following equation, derived by means of the logarith-
mic velocity distribution by Prandtl and the empirical research results of Ni-
kuradse, is valid:

Surface Roughness
In the previous section it was assumed that the surface of the flow duct
was smooth. In reality duct surfaces are rough to varying degrees, which
has an effect on the magnitude of friction. Thus Eqs. (4.47) and (4.49) rep-
resent the lowest possible levels of /; in other words, the effect of roughness
is zero.
To allow for the effect of roughness one can use the results of empirical
tests in ducts that have been artificially roughened with particles glued on the
surface. This approach allows roughness levels to be determined as a function
of the particle diameter k. The following friction factor equation has been de-
rived for large Reynolds numbers:

N I
This is an ultimate case, when the friction factor is no longer a function of the
Reynolds number and is a function of roughness; the pressure loss is now
Ap ~ w/% where w is the fluid velocity in the duct. The surface roughness of
typical manufactured ductworks varies between the values of a theoretically
fully smooth duct and an artificially roughened one. Accordingly the pressure
loss varies between Ap ~ w ' - w~ and £ = /"(Re, roughness).
With most forms of duct, the roughness given by the following Colebrook
and White equation can be used (Eq. (4.51)). This equation has been deter-
mined by calculating an equivalent roughness, corresponding to the sand par-
ticle tests results and taking into account that with large Reynolds numbers
the friction factor's dependency on the Re value is minimal.

This equation represents the change-over section between a smooth tube and a
fully developed rough flow.
In practice the friction factors are calculated either by integration of Eq,
(4.51) or by reference to a Moody chart. This is based on Eq. (4.51) by using
equivalent roughness values representing the sand particle roughness (see
Table 4.3).
Figure 4.4 shows the Moody chart for tubes when k — 0.03 mm, which is
the case for steel tubes. Friction factors for other values of k can be attained
by using the following ratio:

and determining the corresponding diameter from the Moody chart, which is
derived from this equation.