Newtonian fluids rough pipe

Note that we have an additional fluid property (m and n instead of /z), but we also assume that pipe roughness has a negligible effect, so the total number of variables is the same. The corresponding dimensionless variables are / ARe pl, and n [which are related by Eq. (6-47)], and the unknown (DF = ef) appears in only one group (/). The procedure just followed for a Newtonian fluid can thus also be applied to a power law fluid if the appropriate equations are used, as follows. 

For turbulent flow of a Newtonian fluid, / decreases gradually with Re, which must be the case in view of the fact that the pressure drop varies with flow rate to a power slightly lower than 2.0. It is also found with turbulent flow that the value of / depends on the relative roughness of the pipe wall. The relative roughness is equal to eld, where e is the absolute roughness and d, the internal diameter of the pipe. Values of absolute roughness for various kinds of pipes and ducts are given in Table 2.1. 

Turbulent flow of Newtonian fluids is described in terms of the Fanning friction factor, which is correlated against the Reynolds number with the relative roughness of the pipe wall as a parameter. The same approach is adopted for non-Newtonian flow but the generalized Reynolds number is used. 

The opposite conclusion would presumably apply to dilatant fluids and for these the pressure drop in a rough pipe may perhaps exceed that for a Newtonian fluid. A similar situation might also arise for Bingham-plastic and pseudoplastic materials which exhibit elastic recovery to a high degree. 

As in the case of Newtonian fluids (R7) one would expect that the end of the stable laminar-flow region (generalized = 2100) should not be influenced by the roughness of the pipe. Until experimental data are available, this assumption is recommended although a contrary opinion has been published (W4). 

A large body of literature is available on estimating friction loss for laminar and turbulent flow of Newtonian and non-Newtonian fluids in smooth pipes. For laminar flow past solid boundaries, surface roughness has no effect (at least for certain degrees of roughness) on the friction pressure drop of either Newtonian or non-Newtonian fluids. In turbulent flow, however, die nature... 

Extensive studies (6) have been conducted to understand the effect of pipe roughness on friction loss in turbulent flow of Newtonian fluids in rough pipes. The phenomenon of turbulent flow with non-Newtonian fluids in rough pipes, however, has received very little attention (7). 

There are insnfficient data in the literatnre to provide a reliable estimate of the effect of roughness on friction loss for non-Newtonian flnids in tnrbnlent flow. However, the influence of roughness is normally neglected, since the laminar bonndary layer thickness for such fluids is typically much larger than for Newtonian fluids (i.e., the flow conditions most often fall in the hydraulically smooth range for common pipe materials). An expression by Darby et al. (1992) for / for the power law flnid, which applies to both laminar and turbulent flow, is... 

In laminar flow/ = 16/Re, whereas in turbulent flow the dependence of/ on Re is a function of the specific rheological behavior of the fluid and roughness of the walls of the inside of the drill pipe (91). A number of functional relationships between / and Re have been proposed for turbulent flow. Ignoring the effects of the roughness of the surface of the drill pipe, / can be approximately related to Re by a generalized form of the well-known Blasius equation for Newtonian fluids (90, 95)... 

Bewersdorff, H.-W. and Berman, N. S., Effect of roughness on drag reduction for commercially smooth pipes, J. Non-Newtonian Fluid Mech. 24 365 (1987). 

We also have the Fanning friction factor,/, which equals 2(j) and the Moody friction factor/, which equals 8(, just as we saw earlier when discussing frictional pressure loss in rough and smooth pipe for Newtonian fluids. 

As with Newtonian fluids, it is a pretty safe bet that flow is laminar for Re <2100, but drag-reducing additives often seem to delay the laminar-turbulent transition to higher Re s. The friction factor-Reynolds number relation remains a function of pipe roughness. 

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