El cuarto piso: la modernización-en-la-dependencia

The traditional explanation for why Nikuradse‟s measurements differ so much from the Moody diagram has always been that the Moody diagram is about commercially available pipes, while Nikuradse investigated a rather artificial sort of pipe with sand grains glued to their surface. The Moody-diagram, it is argued, is valid for typical surfaces. This shows that two assumptions are built into the Moody diagram: That such a thing as a typical surface actually exists, and that at the time when Colebrook &

Whites‟ equation was developed, the instruments to measure surface roughness in the relevant way were available. As shown by Thomas (1982), both of these assumptions are at best questionable. One reality the differences between the two diagrams make clear, however, is that roughness does not only affect the friction factor at a particular Reynolds number, it affects the shape of the curves, too.

To get a feeling for what having several roughness sizes at the same time in one pipeline leads to, let us first consider this very simple example: Two pipes of identical length and diameter, but different roughness, are coupled in series. For a particular Re, the first pipe is going to have friction described by f1, while the second‟s friction factor is f2. The average friction factor for the two pipes together is obviously going to be (f1 + f2)/2. If the first pipe was of the Nikuradse-type and described by the 1/120-curve and the second by the 1/1014-curve in figure 2.3.1, the average curve would obviously lie somewhere in-between, and we realize it would have a more smoothed out minima than each individual curve.

Now suppose we had a nearly similar situation with two distinct roughness values, but instead of having them in two separate pipes, they both occur super-imposed on each other in one pipe. If they affect the flow independently, an assumption which admittedly is not completely accurate, we would end up with the same overall friction

factor as for the two separate pipes described above. We can extend this theory further by combining far more different roughness values than just two. To do so, let us start by creating some simplified curve fits to Nikuradse‟s diagram.

By comparing the local minima in the partly turbulent zone of Nikuradse‟s measurements, figure 2.3.1, we see they become less and less prominent as the relative roughness decreases. By reading the minima carefully for all the curves and plotting them in a logarithmic diagram it can be seen that they lie in a nearly straight line described by:

( ) (2.6.1)

The location of the line describing where the flow becomes fully turbulent can be formulated as:

( )

(2.6.2)

We may use these results together with equations 2.5.1 and 2.5.2 for smooth and rough pipes to extrapolate Nikuradse‟s diagram. Figure 2.6.1 has been created that way.

The results show that for very high Re the partly turbulent zone becomes narrower and the friction factor is more constant than for lower Re. For high Re, the curves are in fact very similar to the diagram one may create by using AGA‟s recommendations, which are discussed in the next chapter. Figure 2.6.1 is based on extrapolating quite far beyond the area covered by Nikuradse‟s measurements, and it deals with surfaces of a kind not frequently encountered in real pipes. But it has one very distinct advantage which the classical Moody diagram does not have: It deals with only one relative roughness at a time. That gives each curve a clearer definition than those obtained by measuring on commercially available pipes, where an undefined combination of different roughness sizes exists simultaneously.

Keeping these limitations in mind, it is interesting to see how different sand grain sizes could have combined to create a friction factor curve if the grain size had a Gaussian distribution. For instance, consider a surface where the sand grain size has a relative

Figure 2.6.1. Extended Nikuradse diagram.

mean µs = 3∙10^-6 m and variance σ² = 1∙10^-12 m². From statistics, we know the probability density function for a Gaussian distribution is:

( )

√ * ( )

+ (2.6.3)

By splitting the sand grains into 100 different sizes spread from µs - 3σ to µs + 3σ, and by weighing each grain size according to equation 2.6.3, we may use the extended Nikuradse diagram in figure 2.6.1 to compute the average weighed friction factor for

that grain size. In figure 1.5.2, the curve for that has been drawn as the lowermost thick, blue line, curve 2.

Figure 2.6.2. Theoretical pipes with Gaussian surface roughness distributions show on the extended Nikuradse diagram.

We can clearly see how this strategy leads to a smoothened curve without a dip in the partly turbulent zone, and with a less abrupt transfer to smooth pipe behavior, just like in Moody‟s diagram. Even though the theoretical experiment shown here assumes the different sand grain sizes can be combined linearly, something which is a very rough simplification, it does lend weight to the argument that pipes with a combination of different roughness sizes have smoother transitions between „smooth‟ and „partly turbulent‟ flow than do pipes with only one imperfection size.

We also realize that the larger the spread in the imperfection‟s sizes, the smoother the curves. That is something to keep in mind when investigating real pipes, for instance when comparing coated and un-coated pipes: Comparisons cannot be done for only one Reynolds number, since very different surfaces must be expected to lead to very

Pipes can also have non-Gaussian surfaces. Some irregularities may be inherent in the materials, others come from distinct steps in the manufacturing process. Each of these sources of irregularities may lead to their own surface characteristics, and the final results can be some sort of weighed sum of the different roughness values. For that reason, more than one dominant roughness size may exist in one pipeline. Suppose, for instance, that a pipe is described by three distinct relative roughness values: ks1/d = 6∙10

-3, ks2/d = 6∙10^-4, ks3/d = 6∙10^-5. Suppose also that the first roughness contributes 25% of the total roughness, the second 50%, and the last 25%. By taking the friction factor from Nikuradse‟s diagram for each relative roughness and plotting the weighted average for

many different Reynolds number, curve 1 can be drawn.

Again, the tendency is that the curves begin to look more like the ones in the traditional Moody diagram. The agreement would be even better if a wider distribution of roughness values were used.

Note that no roughness distribution could be expected to create Moody-like curves for Reynolds numbers for 2,300 < Re < 4,000. In that area, Nikuradse‟s measurements show that all different curves merge, and transition between laminar and turbulent flow is relatively smooth. That contradicts how this area is displayed in the Moody diagram.

The discrepancy was also recognized by Moody, who pointed out that his diagram is very inaccurate in this area. In fact, no convincing measurements seem to offer any support to the way the Moody diagram presents friction factors for 2,300 < Re < 20,000, particularly not for relative roughness values ks/d > 0.02.

We have seen that different surface roughness distributions can lead to different friction factors. Some of Nikuradse‟s results have been confirmed thoroughly by numerous measurements. Lots of measurements have shown that the friction factor curves do become horizontal for high Reynolds numbers for all normal, commercial metal pipes.

We also know that in the horizontal part, corresponding to the „fully turbulent‟ area of the Nikuradse charts, the friction factor is determined by the relative roughness, making it sufficient to measure it for only one pipe diameter (corresponding to one relative roughness). But how is the curve‟s shape affected by the pipe diameter in the

When accurate friction calculations are required, we need more information about the surface than what can be compressed into an

'average' equvivalent sand grain roughness k_s.

„partly turbulent‟ area? A commercial pipe may for instance be manufactured from a rolled steel plate. If that plate is used to manufacture 3 different pipes with different diameters, the surface would be the same for all three. But the relative roughness would not. That is what is illustrated in figure 2.6.3, where the surface is assumed to have a Gaussian distribution of peaks and valleys, similar to the one described for curve 2 in figure 2.6.2. Setting µs = 3∙10^-5 m and variance σ² = 1∙10^-8 m², the curves are plotted for diameters 10, 1 and 0.1 m, leading to average relative roughness values of 9.25∙10^-4, 9.25∙10^-5 and 9.25∙10^-6.

Figure 2.6.3. The same surface plotted for 3 different relative roughness values.

Figure 2.6.3 shows that this leads to relatively similar shapes. The surface structure uniformity factor us, which will be defined later, does in fact fit all three curves if we set us = 3. If we change µ and σ², the shape changes, which shows a pipe‟s friction properties cannot be described by ks alone.

Put another way: No amount of effort can make it possible to compress the entire truth about a surface into a single, equivalent sand grain roughness. In the so-called „partly

ks/d = 9.25∙10^-4

ks/d = 9.25∙10^-5

ks/d = 9.25∙10^-6

turbulent‟ zone, we also need some way of describing the friction curve‟s surface-texture dependent shape.