The analysis of a turbomachine and the estimation of the resulting performance is a really tough issue because it involves complex three di-mensional, viscous, unsteady and irreversible flows occurring in both sta-tionary and rotating boundaries. On the other hand, efficiency is one of the most relevant parameter for most turbomachines and the evaluation of the irreversibilities arising in the flow plays a fundamental role because the latter can be equated to a certain amount of lost work and thus a loss in efficiency. In general terms, the increase of the entropy in the flow across a turbomachine is the consequence of several phenomena: skin friction on
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the blade surfaces, viscous and turbulent dissipative mechanisms of vor-tices and shocks, vorvor-tices that cause some parts of the fluid to move in directions other than the principal one, fluid leakage in the gap between the blade tip and the shroud, etc. As all these mechanisms reduce the useful work they are commonly referred as losses.
The direct mathematical treatment of local losses is not trivial be-cause involves the solutions of the Navier-Stokes equations with a suitable turbulence model. Moreover, for viscous layers and vortices the interest is usually focused on overall or integrated losses rather than local losses.
Thus, what is commonly done in turbomachinery engineering (at least in a preliminary design phase) is to use an ideal flow model which takes into account the effects of the irreversibilities by means of loss coefficients. In other words, loss coefficients allow to determine the real flow conditions starting from those which would be obtained in case the irreversibilities were absent (i.e., the flow was ideal).
According to Japikse and Baines [48] loss coefficients can be broadly grouped according to the methods adopted to define them in the following categories.
• Gross (or bulk overall) loss coefficients. They are fixed numbers or very simple functions of few overall fundamental performance pa-rameters and strongly depend on the experimental dataset used to determine their values. They provide reasonable estimates only in the range where data exist, so they are unfit to explore the effect of geometry modifications on the expected performance. Gross loss coefficients do not give any insights about the physics of the phe-nomenon which underneath the loss mechanism.
• Correlated coefficients. By the use of test data taken from a sub-set of similar machines, loss coefficients are correlated to key design parameters in place of overall performance parameters. The bigger
the size of the experimental dataset, the higher the coefficients reli-ability. Correlated coefficients should be applied only to families of machines with similar characteristics or, at most, they can be care-fully used to investigate the effects of limited geometry changes on the performance.
• Fundamental (or physically based) coefficients. They attempt to reproduce the actual physical processes in the flow: each basic loss-generating process within the flow field (e.g. friction, secondary flows, etc.) is accounted separately by a loss coefficient and modeled as a function of the fundamental geometric, cinematic and thermo-fluid dynamic parameters. Then, these loss coefficients are summed to-gether to get the overall loss coefficient and, in turn, the real flow conditions. However, it must be borne in mind that, whilst the partition of the global loss mechanism in more fundamental loss-generating processes reduces a complex problem to smaller propor-tions, it is artificial because loss-generating processes interact and influence each other. Compared to the previous categories these coef-ficients are the most suitable for design purposes because they should be able to predict with good accuracy the performance of turbines other than those for which they were developed; nevertheless, few empirical input (i.e., tuning constants) are still required to calibrate the loss modeling system with respect to the experimental findings and to indirectly account for possible lacks in the flow modeling.
In turbomachine engineering practice loss coefficients are usually de-fined on velocity/enthalpy or pressure basis. In the following the most relevant definitions for nozzle and rotor are collected and they are related to each other.
Figure 4.1a shows an expansion process across a nozzle from the initial state 1 to the final state 2; state 2s is the isentropic (or ideal) final state
4.3. LOSS COEFFICIENTS 115
Figure 4.1: Expansion process across a (a) nozzle and (b) rotor.
and lies on the same isobar line of state 2. The velocity loss coefficient (ϕ) compares the real discharge flow velocity to the ideal one
ϕ = c2 The enthalpy loss coefficient (ζ) is often defined as
ζ = h2− h2s
h01− h2
= h2− h2s
c22/2 (4.13)
or, more rarely, if the isentropic total-to-static enthalpy drop is consid-ered, as
ζ′ = h2− h2s
h01− h2s
= h2− h2s
c22s/2 (4.14)
Equations (4.12), (4.13) and (4.14) allow to calculate the real discharge enthalpy value (h2) which is higher than the ideal one (h2s) because the difference h2− h2s is not used to accelerate the fluid to the ideal velocity discharge value c2s.
Simple rearrangements permit to interrelate the above definitions ζ = h2− h2s
Total pressure loss coefficient (Y ) is usually preferred by English au-thors and the most diffused definition is (see, e.g., Horlock [49] for a com-prehensive list of the possible variations)
Y = p01− p02
p02− p2
(4.16) from which the total outlet pressure and, in turn, the outlet thermody-namic state can be determined. Under the hypothesis of ideal gas behav-ior it is possible to analytically relate the enthalpy loss coefficient ζ (Eq.
(4.13)) and total pressure loss coefficient (Eq. (4.16)). It results
ζ = 2 where γ and M2 denote the specific heat ratio and the outlet Mach number, respectively. Equation (4.17) for M2 ≤ 0.4 is well approximated by
ζ = Y
1 + γ2M22 (4.18)
Similarly, loss coefficients ζ′ and Y can be related by relationships analogous to Eq. (4.17) and (4.18) (see [32] for the details).
A number of turbomachine designers (e.g., Balje [50] and Rodgers [51]) prefer to use loss coefficients based on the stagnation enthalpy loss (h02− h02s, being the state 02s on the total pressure isobar line p02and s02s = s1), non-dimensionalized by the tip speed u of the rotor which follows the nozzle
∆q = h02− h02s
u2 (4.19)
Figure 4.1b shows an expansion process across a rotor from the initial state 2 to the final state 3; state 3s is the isentropic (or ideal) final state and lies on the same isobar line of state 3. Subscript R stands for relative frame of reference. The velocity loss coefficient (ψ) compares the real relative discharge flow velocity to the ideal one
ψ = w3