## Difficultés du calcul des pertes de charge linéaires dans les conduites forcées

### Difficulties in the calculation of linear penstok head losses

Chef du Département Recherches B.V.S. - Chargé de cours à l'Ecole Nationale Supérieure des Mines de Saint-Etienne.

Numerous researchers have followed, in the wake of Prandtl's and von Karman's memorable theories [1 and 2] and the experimental work of their collaborator Nikuradse [3] with a considerable sum of experimental adjustment ; reference to the most important of them is made in the article and its bibliography. Stated simply Nikuradse's 'harp' of 1933 has become Moodys's graph of 1944 (Figs. 1 and 5). Section II of the article briefly reviews the theory on which this graph is based and gives the essential formulae for each of the various states of flow determined from the graph, as follows : - (i) Hydraulically smooth turbulent flow : Blasius's formula (4) and Prandtl-Nikuradse's formula (5), applicable for the following Reynolds Numbers (ref. [9]): - R < R' = 23/ε/D (ii) Transition turbulent flow, Colebrooks formula (9) [ref-4] or Altschoul's formula (10) [ref. 9]. (iii) Hydraulically rough turbulent flow, Prandtl-Nikuradse's formula (8) applicable for the following Reynolds numbers [ref. 9]:- R < R'' = 560/ε/D Absolute roughness is defined in terms of duct material and the type and method of application of its lining (see Fig. 2) [ref. 8]. More recent experience (discussed in Section III) has brought many points of disagreement with the above theories to light, at least for pipes in industrial service with very heterogeneous roughness as regards the shape and height of its projections. To begin with, it has been shown that the hydraulically smooth flow condition expressed by the Prandtl-Nikuradse formula (5) is in fact very far from being the absolute left-hand limit for experimental points. On the contrary, very numerous experiments [5, 8, 20, 21 and 21 bis] have shown the experimental curves to lie well to the left of the Prandtl-Nikuradse curve (5) and, in fact to tend towards Blasius's curve (Fig. 4). Blasius formula (4) thus seems to be more representative of hydraulically smooth conditions than the logarithmic formula (5). It is important to try to fill the gap between the old and new left-hand limits, but this work still remains to be done at the time of writing. A new theory in closer agreement with experience with big industrial pipes and tunnels would be welcome at the present stage. Section IV considers changes in absolute roughness ε' with pipe age. Physical and chemical properties such as pipe and lining material and method of lining application, also hydraulic parameters (V, D, p), and-above all- physical and chemical water properties are apt to play a very considerable part in increasing roughness (Figs. 6 and 7). An attempt has been made [ref. 30] to classify water in five groups and to state the roughness increase factor ε for each (see formula 20, Table land Fig. 8). Section V discusses the effect of hooping a pipe (Fig. 9) on its loss of head. The author begins by giving an indication of the maximum deformation produced, which is found to amount to very little indeed, and then shows by calculation that there is no relationship between hooping and loss of head. Experimental results for twenty-six pipes are shown in Figure 10, including twelve pipes hooped over varying lengths. These data confirm the theoretical calculations.

*© Société Hydrotechnique de France, 1966*