Fig.D-12
Laminar flow: This flow regime is governed by the Hagen-Poiseuille law

Hagen, a German experimenter, found what is basically this law, by experiments as long ago as in 1839 (of course, he could not know the concept of Reynolds number and instead of working with viscosity, he presented his results as function of temperature - upon which, of course, viscosity is dependent, Fig.D-6). Poiseuille, professor of physics on Sorbonne in Paris (at faculty of medicine) derived this relation theoretically, although originally not in a completely correct manner.
Turbulent flow: - is expected to take place if there is . No theoretical solution exists - just a handful of empirical or semi-empirical relations. If the pipes in question have smooth inner surface and Reynolds number is not large (Re Prandtl formula is valid within a larger Reynolds number range and generates more accurate results - but, being implicit, is incovenient to use.
Fig.D-13 This schematic illustration presents general shapes of pipe characteristics - the dependences between the overal difference of available specific energy between the pipe ends upon the mass flow rate. For laminar flow, as might be evaluated by inserting the Hagen-Poiseuille law, the dependence upon velocity (and, hence, flow rate) is linear. For the turbulent flow calculated by the Blasius formula, the energetic drop is proportional to 1.75th power of flow rate. The two characteristics have a common intersection point which - very roughly - corresponds to the critical value of Reynolds number. The important conclusion from this results is that not only Nature tends to increase entropy by introducing the frictional loss mechanism, but if there are available alternatives, Nature always select the one which results in fastest rate of entropy growth. Here we can see that of the two available characteristics, the one is actually followed (laminar for low Reynolds numbers, turbulent for large Reynolds numbers) which leads to larger dissipation of ordered motion energy.


Rough pipe walls: Quite often the pipes used in practical application ahve rough inner surfaces, either due to imperfections of the manufacturing process or dure to effects of corrosion. Both effects are difficult to include accurately into a calculation. Their influence upon the loss, however, is too large to be neglected. If there is large enough Reynolds number, a "fully rough" region is reached in which the dependence upon ceases to be important (Fig.D-12 above). Because of lack of any better calculation procedures, it is recommended to apply in this region the Nikuradse formula:

- which, however, was derived from experiments with artifically made roughness (glued sand grains, of constant diameter obtained by sieving) which does not represent actual behaviour of "natural" surface imperfections. For new steel pipes, the root mean square height of deviations from perfect surface is between and . Used, corroded steel pipes require inserting .

Noncircular pipe cross sections: For laminar flows it is necessary to solve the two-dimensional problem (if the flow is not fully developed, the problem is three-dimensional). For turbulent flows it is possible to apply the approximate method described in Fig.D-14 : the "hydraulic diameter", to be inserted into formulas for circular pipes, is evaluated from pipe cross sectional area and the length of the section perimeter.


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This is page Nr. D07 from textbook Vaclav TESAR : "BASIC FLUID MECHANICS"
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