Laminar and Turbulent Flow
Flow phenomena can be divided into three main types:
• Laminar (streamline)
• Transitional
• Turbulent
In laminar flow there are no disturbances, and therefore all flow particles move in the same direction. Transitional flow is the flow regime that takes place during the change from streamline to turbulent flow. In the case of turbulent flow the particles move in a given flow direction, but the flow is erratic and random.
When the Reynolds number is under 2000, it is shown empirically that the flow in a smooth tube is laminar. This flow has a parabolic velocity profile, as shown in Fig. 4.3.
Now consider a cylindrical volume element in a flow stream. The radius of the element is r and its length is L. The force produced by the flow in this volume is due to the viscosity, which is
ItttLt ~ — lirrLri^1, r ~ ~^~dr
The pressure difference (drop) between the ends of the element produces a force Apirr2, and considering the force balance,
Apirr2 = 2 TrrLv^.
R ‘ ar
Simplifying this gives
Denoting w as ml at r = 0 and noting that vm = 0 at r = R, the integration gives
W = m i = 
(4.40)
T
T 
P&P
L
A p = 8r>V™L = 32rlVmL R2 D2 ‘ 
(4.44) 
Equation (4.44) is the HagenPoiseuille law, which shows that the pressure loss during laminar flow is linearly proportional to the flow velocity. The following equations show the relationship between the pressure loss and the friction factor: 
O2irrV"’ dr = 2rl 
= 
For a parabolic velocity profile the velocity expression is Vm 
^(R2r2). 
4tjL 























When Re > 105, the following equation, derived by means of the logarithmic velocity distribution by Prandtl and the empirical research results of Ni — kuradse, is valid:
= 4.01 In (Re7/) — 0.4 . (4.49)
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) represent the lowest possible levels of f; 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 derived for large Reynolds numbers:
If k ‘
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]JS — w~ and Ј = f(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 determined by calculating an equivalent roughness, corresponding to the sand particle tests results and taking into account that with large Reynolds numbers the friction factor’s dependency on the Re value is minimal.
5.4
(4.51; 
+
Jf 
Re// 3.7 Id
This equation represents the changeover 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:
D 
Case
And determining the corresponding diameter from the Moody chart, which is derived from this equation.
TABLE 4.3 Equivalent Roughness Values for Various Materials
Material k,,quiv
TOC o "15" h z Commercial or follower steel 0.046
Asphalted cast iron 0.120
Cialvanized steel 0.150
Cast iron 0.26
Wooden surface 0.189
Concrete 0.33
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