TOPOLOGÍA DE RED ESTRELLA
3.3 METODOLOGÍAS PARA LA PROPUESTA .1 POBLACIÓN .1 POBLACIÓN
The geometry of the idealized stage was defined to give the required performance at the design point: that is, at the design flow rate and rotational speed. However, any turbomachine will often be operated away from its design point. The idealized analysis can also be used to give reasonable predictions of how the stage will perform for off-design operating points.
(a) Effect of Varying Flow Rate
Consider first the effect of a reduction in flow rate at fixed blade speed U (i.e. at constant RPM). The resulting velocity triangles will look as follows:
The new velocity triangles were arrived at as follows:
(i) Based on the Euler Approximation, the flow will still leave the blade rows at the metal angle. Therefore, α1, β2 (and α3) are unchanged. Recall that there must be a set of stators or inlet guide vanes ahead of the rotor to account for the inlet swirl.
(ii) From continuity, Ca is reduced and thus so is C1. In a quantitative calculation, the new value of Ca would just be obtained from Ca =Q A, where Q is the new volume flow rate, and A is the
annulus area as established at the design point.
(iii) The magnitude of W2 is also reduced, by continuity, but the direction is unchanged.
From the velocity triangles, Cw1 has decreased while Cw2 has increased. As a result, the change in swirl velocity ΔCw has increased. From the Euler equation
g HΔ E =U CΔ w
and the head rise produced by the machine ΔH=ηΔHE will be increased. Equivalently, for a compressible flow machine, Δh0, and the corresponding pressure ratio, P02 P01, will be increased. Note that this is consistent with the increase in incidence (“angle of attack”) at the leading edge of the rotor blade. As a result
of this, the blade should develop greater lift, do more work on the fluid, and thus increase the head rise. On the other hand, increasing the incidence will eventually lead to stalling of the blade. Thus, reducing the flow rate through a compressor stage will move it towards stall. Note that the incidence was also increased for the stators, bringing them closer to stall as well.
Clearly, we can use the velocity triangles and the Euler equation to predict the quantitative stage characteristic for the idealized stage. It is convenient to express the characteristic in terms of the work and flow coefficients. The flow turning is
ΔCw =Cw2 −Cw1
and from the velocity triangles (noting that Ww2 is negative for the conventional compressor velocity triangles)
Cw1=Ca tanα1 Cw2 =U +Ww2 =U +Ca tanβ2 Then
(
)
ΔCw =U +Ca tanβ2 −tanα1 and dividing by U(
)
ΔC U C U w = +1 a − 2 1 tanβ
tanα
or ψ = +1 mφThus, the ψ versus φ curve (effectively, the head rise versus flow rate characteristic) is a straight line with slope
m=tanβ2 −tanα1
For the present case, the symmetry of the velocity triangles implies that β2 = −α1and the slope is then . For α1 > 0, as is the case here, this gives a negative slope and an inverse relationship between
m= − 2tanα1
head rise and flow rate, as inferred above. Alternatively, since the characteristic passes through the design point (say, φD and ψD), we can write m D D =ψ − φ 1
and the slope of the characteristic is seen to be determined by the choice of design point (note also that in all cases ψ = 1 at φ = 0 for the ideal characteristic). Interestingly, the characteristic will be steeper for a more lightly-loaded stage (lower design work coefficient ψD) as illustrated in the plot. 1.0 0.5 0.0
φ
0.5 1.0ψ
ψD1 ψD2 ψD3 INCREASING DESIGN-POINT LOADING φD(b) Effect of Varying Blade Speed
It is also worth looking briefly at the effect of varying the blade speed at constant flow rate. Using the Euler Approximation again, it can be shown that the change in the velocity triangles will look as follows:
From the triangles, φ=Ca U>φD since U has decreased. For the work coefficient, ψ = ΔC Uw , ΔCw has clearly decreased, but so has U. However, ΔCw has decreased more rapidly than U; as can be seen, a small further decrease in U would reduce ΔCw to zero. We therefore conclude that ψ =ΔC Uw <ψD and φ
and ψ are again seen to vary inversely. In summary, any deviation from the design point will cause the a given compressor to move along the same ψ versus φ characteristic.
It is also worth noting that the reduction in rotational speed has had a very strong effect on the absolute work transfer:
g HΔ E =U CΔ w
Since ΔCw decreases directly with U (and in fact faster than U) the head rise varies approximately as
g HΔ E ≈kU2
and the head rise delivered by the stage, at a fixed value of flow rate, will change strongly with the rotational speed: for example, reducing the speed by a factor of 2 will reduce the head rise by about a factor of 4. Thus, high rotational speed is essential to obtain high pressure rise from a compressor stage. This will be illustrated further in later sections.
As seen, the Euler Approximation results in an idealized ψ versus φ characteristic for the stage that is a straight line with a negative slope.
We have already noted that some changes in operating point will result in positive values of the incidence at the leading edge of the airfoils. If this incidence becomes too large, we would expect the airfoils to stall. Also, we would expect the efficiency of the stage to be best when the rotor and stator blades are operating at the design point. We can therefore project what the actual stage characteristic is likely to be based on the idealized characteristic:
The characteristic shown applies for all rotational speeds. As noted, there is a strong effect of rotational speed on the absolute performance (say ΔH for a given Q). To emphasize this, the characteristics are often plotted in absolute terms as variations of ΔH (or ΔP0) versus Q (or
&m
) for constant values of rotational speed N. The corresponding curves are easily calculated from the non-dimensional characteristic. The resulting map will look as follows.ψ
Dψ
η
Dφ
φ
USING EULER APPROXIMATION STALL LIKELY ACTUAL MAXIMUMη H Δ N max η Q ηCONSTANT CONSTANTOn each of the constant speed lines, there will be a point that corresponds to the design point values of φ and ψ on the non-dimensional characteristic. At each of those points, the velocity triangles will be similar, as indicated in the drawing. In each case, the relative velocity vector at the rotor inlet is lined up with the metal angle and the flow comes smoothly onto the leading edge. As shown, we would therefore expect that the machine will operate at its maximum efficiency at each of those points, apart perhaps for some small effect of differing Reynolds numbers. Also, as we will see later, frictional losses vary as V2 and thus the higher flow velocities with increasing rotational speed will result in higher frictional losses. This effect will be partly offset by the fact that the Reynolds number is also increasing.
Later in the chapter, we will examine to what degree actual machines match the performance characteristics we have inferred from the velocity triangles in this section.