Évolution de la méthode thermodynamique de détermination du rendement des machines hydrauliques

Development of the thermodynamic method

Ingénieurs à Electricité de France, Division Technique Générale.

Résumé

The principle of efficiency evaluation of a hydraulic machine by this method was described in La Houille Blanche (No. 4-5) by Wilm and Gampmas in 1954, in a form in which basic reasoning lines were taken from thermodynamics, especially so as to explain internal energy in terms of such measurable values as water pressure, temperature and physical properties. Enthalpy and entropy were both considered too unfamiliar to hydraulicians to be used at the time. The method has come into much more widespread use during the last twelve years, however, and heat engineers are becoming interested in it as a potential means of determing feed pump and boiler efficiency. This has made it necessary to produce a rather more general form of description of the method, and especially to feature the considered effects in the enthalpy/ entropy diagram. The expression for elementary enthalpy variation (1) integrated between the extreme points representing the machine inlet and outlet represents energy per unit mass given up (turbine) or received (pump). This integration brings forth the measurable quantities, i.e. pressure and temperature. Observing that the representative inlet and outlet points e and 8 can be replaced by points 1 and 2 on the same constant-enthalpy line, expression (5) is obtained. In practice, constant-enthalpy expansions at the inlet or outlet (performed on a small flow considered as a representative sample) are adjusted to simplify the measurements or to make them more accurate, depending on circumstances or the equipment available. Several alternatives are thus available ; in one, the water drawn off undergoes slight expansion, and both pressure and temperature diferences are measured ; in another, temperature equilibrium is achieved between inlet and outlet by making the water drawn off undergo carefully controlled partial expansion, and instead of measuring temperature difference, the point at which the temperatures are equal is determined ; in yet another alternative, the water drawn off can be expanded to outlet pressure, so that only the temperature difference measurement remains. In a theoretically perfect machine without losses, the difference in enthalpy between inlet and outlet is the reference value to which the previously calculated specific energy must be related in order to obtain the efficiency. This is calculated by integrating expression (5) from the truc inlet to a fictitious outlet on the same isentropic line, which yields expression (7). In practice, this integration can be calculated along the e isothermal without appreciable error. The practical expression for turbine efficiency is (11), in which energy is made to appear in terms of unit weight (head), and into which the velocity and gravity terms neglected in the initial reasoning are introduced. Use is also made of precalculated non-dimensional coefficients (1 - α1) and (1 - βc), which represent the enthalpy difference between the extreme pressures, respectively along the isothermal and isentropic lines passing through the entry point (e or 1) and with respect to a reference enthalpy difference (P8 - pc) u0, in which u0 is a reference density. The physical characteristics of water, i.e. compressibility, thermal expansion and specific heat come into the enthalpy difference calculations. The values willm and Campmas originally considered from old sources appear to be unchallenged to this day. They have received indirect confirmation by comparisons between the thermodynamic and more conventional methods, especially by the results of temperature measurement bridge calibrations by isenthalpic expansion. The calibration coefficient for these bridges remains constant from 2 °C to 20 °C, which can be considered to be a very favourable indication as to the reliability of the characteristics of the water used. Emphasis is laid on the homogeneity required for the data regarding the various physical characteristics of water. This homogeneity stems from the laws of thermodynamics ; it must in practice be complied with between coefficient α and specific heat values, in order that the physical characteristics of the water all have the same weight in the various alternative versions of the method. Until the last few years, the "partial expansion" variant was the one most frequently applied. The reason for this preference was that with this method, the temperature difference term can be dropped from the expression for efficiency, and there had also been some doubt as to the possibility of measuring temperature differences sufficiently accurately. People have now become accustomed to reasoning along thermodynamics lines, however, and platinum temperature measurement bridges can be considered sufficiently reliable for differential measurement not just to eletect equal temperatures. This reliability has been confirmed by the results of numerous calibrations. This enables the most general form of the method-without expansion-to be applied, which does not need any heat input to the equipment (such as certain recently proeluced "incorporated" pressure reducers) and thus makes for more rapid, reliable measurement. Because of its general character, this variant is not affected by any of the limitations applicable to the "partial expansion" method, for instance when used on pumps. When compared, results of both above variants are not found to differ to any significant degree. Some general features of data obtained by the thermodynamic method are discussed. Eighteen comparisons between this and more conventional methods generally show differences mostly not exceeding ± 1 %. Four of these results are particularly instructive ; they refer to comparisons with laboratory methods, and the differences are found to be 0.2 %. The closeness of the results is shown up by a set of 74 "point" efficiency measurements on the same turbine over a period of three years. Standard deviation for the scatter in the points with respect to the mean efficiency curve is equivalent to a practical temperature difference of 1/1,000 °C. Electricité de France have been systematically using the thermodynamic method since its development in 1954 in acceptance-testing hydraulic machines, and in investigating operating problems, for example the effect of wear on machine efficiency. The number of complete acceptance tests done by the thermodynamic method has been 700 out of a total 1,100. Progress in differential temperature instrument design should enable the variant without expansion to be used as current practice and slightly more accurate efficiency data obtained than has been possible so far.