Contribution à l'étude de l'effet d'échelle dans les turbines Kaplan

Scale effect and Kaplan turbines

Chef de la Division "Station d'Essais de Turbine", E.D.F., Centre de Recherche et d'Essais de Chatou.

Résumé

When testing a model turbine, geometrical and kinematic similarity (Combes-Rateau) are assumed to be as rigorous in a viscous as in a non-viscous fluid, and head losses to always be proportional to the square of the velocity and independent of Reynolds number. This only really holds good at high Reynolds numbers and equal roughness. Head losses can in fact be considered in several different ways. Where, for instance : ΔH = λ V2/2g λ is a constant for Borda (sudden enlargement) losses, where geometrical similarity applies. It is also a function of Reynolds number for hydraulically smooth flow, and of roughness in fully turbulent flow. Modern scale-up formulae are directly based on these remarks, it being assumed for simplification that λ is a constant for n % to the total losses, and that it is proportional to Re l/4 or Re 1/5 for (100 - n) % of the losses. A point to note also is that this value n % has been decreasing steadily since the first Ackeret formula (1930), from 50 % then to 30 % for Hutton (1954), and to 20 % for Hutton-Fauconnet-Fay-Kvyatovskii (1959) and the authors's results obtained between 1960 and 1962. It is certainly no simple matter to express by a simple general formula a phenomenon affected by Reynolds number, frictional resistance, turbine design and its effect on loss distribution, mechanical turbine features and their effect on the scaling-up of loses, also turbine operating point and its effect on the relative magnitude of losses. Thus, Kaplan turbine test results cannot be used for a bulb turbine for instance, nor even for a Kaplan turbine of widely differing design and manufacture. When studying the head losses in a Kaplan turbine operating on its combination cam, that is to say at optimum guide vane setting and minimum swirl at the runner exit, the following assumptions can be made : - a) A well-designed turbine operating "on the cam" should be free from any pronounced flow separation (breakaway) in its spiral casing and intake (stay and guide vanes). Losses should only depend on Reynolds number ; b) From the guide vane outlet to the runner exit, no flow separation occurs at the blades when the guide vanes are at their best setting and when there is no cavitation ; c) Considering between the runner exit and the end of the draught-tube, it can be shwn that when there is minimum swirl in the runner discharge ("on the cam" operation), enlargement losses in the draughttube (excluding those at the outlet) amount to only about 10 % of the total draught-tube losses. In a draught elbow, flow separation would most probably affect this state of affairs. Most of the losses in the turbine can thus be assumed to be connected to Reynolds number by a relationship which is apt to vary in different parts of the hydraulic duct. On the strength of this, it has been assumed for this study that all losses should be scaled-up proportionally to Re 1/5, and that the necessary reduction in the resulting efficiency rise is due to the effect of prototype roughness on its skin friction coefficient. In order not to have different roughness effects on model and prototype, it would be necessary to define a permissible wall roughness condition which, for engineering applications, is considered to be the maximum depth of roughness not causing any more drag than a smooth wall. In the case of turbulent boundary layers, roughness has no effect and the wall is hydraulically smooth as long as all roughness particles lie entirely within the laminar sub-layer whose thickness is only a small fraction of that of the boundary layer. It has been found that the condition for this is that the "roughness Reynolds number" be smaller than 5, being calculated from friction velocity and permissible depthof roughness K. According to Schlichting, the corresponding Reynolds numbers calculated for infinite V in the flow and permissible depth of roughness K are : Re <= 100 The range of roughness values for model and prototype Kaplan turbines is given in Table 1. Such values obviously cannot be achieved and maintained in operation, and consequently, flow conditions in the prototype are always rough. The same calculation for the model turbine (dimensions and head scale 1: 10) gives K values three times as high, which are commonly experienced with a well-finished model. Hence, flow conditions in a clean, well-finished model are always turbulent and "smooth". Considering the friction coefficient diagram for the blades or any other part of a turbine : (i) The operating point for the model is at (1) ; (ii) Variation of the friction coefficient as Re 1/5 gives point (2) for the prototype ; (iii) On the prototype, the decrease in Cf due to the increase in Reynolds number is less because of roughness, giving operating point (3) for the prototype. The ratio of prototype to model losses is then as follows : δ / δ' = 1 - K+ K (R'/R) 1/5 where K = 1 - C'1/C1) / 1 - )R'/R)1/5 formula (5) This is identical to Hutton's formula, except that the influence of roughness is allowed for by coefficient K. Other consequences of the above are : - a) When operating at another point with greater discharge, the model coefficient (C'f) decreasce but the prototype coefficient (Cf) increases. The difference between points 3' and 2' is greater than between points 3 and 2, i.e. coefficient K is smaller. Efficiency increases less for point 1' than for point 1. Hence, the relationship between scale effect and discharge is as described by Hutton, though with a different explanation. b) The closer conditions in a given part of a turbine approximate hydraulically smooth flow, the better the friction coefficient. Thus, less improvement in this coefficient is to be expected in the draught-tube than in the intake for instance. This is another aspect of the distinction Hutton made between operation of various parts of a turbine. The author has tried to apply these ideas to scale-up the results the Logis Neuf power plant with the following characteristics : Type ... Vertical-shaft Kaplan. Nominal discharge... 330 cu. m/sec. Speed ... 79 r.p.m. Nominal head... 11.3 m. Output... 33,000 kW Runner diameter... 7.0 m Model diameter... 0.5 m Test head... 1.4 m Reynolds number was calculated for each part of the turbine and the following Moody K8 equivalent sand roughness) values were assumed : - Blades and guide vanes... 50 μ , Stay vanes... 100 μ (equivalent to actnal roughness of 30 μ and 60 μ respectively). Concrete spiral casing and draught-tube : 0.5 mm. Comparison between prototype Cf and model C'f values gives a K value for each part of the turbine. With the loss distribution measured on this turbine, the weighted K value for the complete turbine was found to be 0.77 at normal discharge and 0.54 at maximum discharge. In other words, roughness reduces the improvement in efficiency to 77 % of the improvement obtainable in theory with a hydraulically smooth prototype. Calculated K values and those deduced for the three Kaplan turbines tested by comparison of the model and prototype are shown on Figure 4. It is also possible to calculate K by this method, i.e. the ratio of prototype losses to model losses, for the following difrerent cases, though always considering the same kind of turbine (same design, manufacture and loss distribution) : - 1. In predicting prototype efficiency from model tests with different runner sizes under varying test heads (model always assumed to be hydraulically smooth) ; 2. In predicting the efficiency of turbines having different runner sizes (though always geometrically similar) from the same model tests (see Figs. 5 and 7). In conclusion, it seems from Figure 6 that each Hutton-type formula has its own particular validity range depending on model test characteristics, i.e. : - Hutton, Blackstone for high test heads and small runners. Fay, Kviatovsldi, Fauconnet for low Reynolds numbers.