Effect of Prewhirl on the Performance of Centrifugal Compressors

In comparison with axial flow compressors, centrifugal compressors have distinct relative merits such as: higher stage pressure ratio, simplicity and ruggedness of construction, shorter length, better resistance to foreign object damage, less vulnerable to blockage, handling small volume flows, better matching characteristics, and less manufacturing costs. However, the axial flow compressor has a large flow capture per unit area which is important to aircraft applications. In this study, prewhirl is introduced to alleviate the compressibility effect at the convex side of the eye to avoid the formation of shock waves and the consequent losses. However, this caused reduction of work absorbing capacity W and pressure ratioR. This problem was overcome by twisting the prewhirl vanes between the tip and the root linearly, then parabolically, where the prewhirl angleαw1 is maximum at tip αw1t and zero at rootαw1r . αw1t was varied over a range of 20◦–50◦. Computations showed that the linear variation recovered about 40% of W and R, whereas the parabolic variation of αw1t up to 40◦ recoveredW and R totally.

produced by the rotation of the impeller.Instead of arising from the conversion of kinetic energy into thermal energy in divergent passages, as in the rotor of the axial flow compressor, it arises from the change in potential energy of the fluid in the centrifugal force field of the impeller.It is therefore, less limited by the problems of boundary-layer growth and separation in adverse pressure gradient (Kerrebrock, 1992).
Due to lower mass capture per unit frontal area, centrifugal compressors are widely used in small gas turbine engines such as small turboprops, turboshafts and auxiliary power units (APUs), and for the high-pressure spools in small turbofans.
Centrifugal compressors in comparison with axial flow compressors have some relative merits, such as higher stage pressure ratio, simplicity and ruggedness of construction, suitability to handle small volume flows, shorter length, better resistance to foreign object damage (FOD), less vulnerability to blockage and the ability to operate efficiently over a wide range of mass flow between surging and choking at a particular rotational speed (alleviating the problem of matching of operating conditions with those of the associated components) and flow direction of discharge air that is convenient for the installation of an intercooler and/or heat exchanger, Kerrebrock (1992) and Cohen, Rogers, and Saravanamuttoo (1998).Centrifugal compressors are used when the size is small enough for the Reynolds number of the flow to become small, and when the blade tip clearance becomes relatively large.They will cost less to manufacture than the equivalent multistage axial compressor in almost every size, Wilson and Korakianitis (1999).
Generally, dynamic compressors convert velocity to pressure continuously.This is in comparison with the rotary compressor of the vane or screw or other type, which compresses continuously in portions separated by moving walls and boundaries, Harman (1981).
Assuming the air flow approaches the compressor intake axially, then before passing into the radial channels of the impeller, the air flow is deflected through a certain angle in the eye of
the impeller where there is a possibility for the flow to break away from the convex face, which may lead to a shock wave with consequent losses.In this case the velocity relative to the vane V 1 will be maximum at the eye tip where the vane speed U is greater.Hence, the maximum Mach number is at the eye tip M v1t given by: where T 1 is the static temperature at the inlet, and it is the lowest in the flow passage.Taking C w2 as the whirl component at impeller tip r 2 then the torque, which is equivalent to the change of angular momentum, is given by: torque = C w2 r 2 , and Power = C w2 r 2 ω = C w2 U 2 .Hence introducing the slip factor σ = C w2 /U 2 and the power input factor ψ, which takes into account friction between the casing and the air carried round by the vanes, plus disc friction or windage, results in actual power = ψσ U 2 2 .Although M v1t may be satisfactory under ground conditions (about 0.9), it becomes excessively high at altitude if installed in an aircraft gas turbine engine (more than 1.0).Assuming that T alt /T gr ∼ 3 4 , then M v1,alt /M v1,gr ∼ 1.15.Flow distortions due to aircraft maneuvers may cause a more severe change in flows and hence M, thereby they may have a significant impact on compressor stability.However, our main concern in this work is to the effect of prewhirl on the compressor performance.Hence, it becomes necessary to reduce V 1 by introducing prewhirl at the intake, which involves passing the air over curved inlet guide vanes attached to the compressor casing, before being drawn into the impeller eye.Assuming the same mass flow in both cases, axial velocity C a remains nearly the same, however V 1 is reduced and the curvature of impeller vanes at inlet is reduced, i.e. α 1 increases as shown in Figure 1.

THEORETICAL ANALYSIS
Since the air flow has an initial whirl velocity C w1 then, by introducing prewhirl using inlet guide vanes (IGV) as shown in Figure 1, the torque which equals (C w2 r 2 − C w1 r 1 ) is reduced.Assuming C w1 is constant over the eye, C w1 r 1 will increase from root to tip.Hence, the work done on each unit mass flow of air depends on the radius at which it enters the eye.Since M v1 is high only at eye tip, it is preferable to vary the prewhirl angle α w1 gradually reducing it from a maximum at the tip to zero at the root of the eye.Thereby, the prewhirl vanes get twisted, Cohen et al. (1998).
In this work, three patterns of twisting the IGV are considered.

Case 1
Constant α w1 from root to tip.Hence C w1 = C a tan α w1 = constant value along the eye.But C w1 r 1 increases from root to tip, thereby also increasing the drop in the work capacity from root to tip.For this case α w1 was taken 30 • .Hence These performance parameters are calculated for different values of α w1t , namely 20 • , 30 Case 2 α w1 increases linearly from root to tip of the eye.It starts with α w1 = 0 at root.Hence, where "a" and "b" are constants which can be found by choosing certain values for r r /r t and α w1 then, solving two equations by the boundary conditions at root and tip.Hence for r r /r t = 0.54, and α w1t = 20 • the linear variation of α w1 along the eye can be given by: tan α w1 = 0.728 r e − 0.  Alternatively, carrying out numerical integration using the trapezoidal method, the result in Equation ( 13) is obtained.This procedure is repeated for the other values of α w1t , namely 30 Case 3 α w1 increases in a parabolic relation from 0 • at root to α w1t at tip.
Take tan α w1 = ar e + b [17] Then using r e = 0.5 and α w1t = 20 • , the result is an equation similar to that produced previously namely, tan α w1 = 0.728r e − 0.364 [18] Consequently, After some mathematical manipulations and where C a = 143 m/s is taken to provide a suitable compromise between high flow per unit frontal area and low frictional losses in the intake, After some manipulations, one can obtain Alternatively, by numerical integration of the relationship between C w1 and √ tan α w1 , the result in Equation ( 22) is obtained.This procedure is repeated for the other values of α w1t , namely 30

DISCUSSION OF RESULTS
The particulars of geometry and the design point operating variables were chosen for a typical centrifugal compressor of a small aircraft engine, Cohen, Rogers, and Saravanamuttoo (1998) and Mattingly (1996).However the results are generalized for different geometries and operating variables.
These particulars are: impeller radius r 2 = 0.25 m; eye tip r t = 0.15 m; eye root radius r r = 0.075 m; rotational speed  The main performance parameters namely: M v1 , torque, work, and pressure ratio R were calculated for the basic case of no prewhirl; and the three specified cases of prewhirl over a wide range of angles of 20 • -50 • .The results are shown in Tables I-III.However, to generalize the results, they have been nondimensionalized by dividing the value of each performance parameter in these tables over its corresponding value at the basic case of no prewhirl.The results are shown in Table IV for the parabolic case for illustration, and all presented in Figures 2-4.
Figure 2 shows the variation of the normalized relative Mach.Number M v1t , with α w1t .It is clear that M v1 can be decreased to 55% of its value at no prewhirl.This makes the operating conditions very safe as far as compressibility effect is concerned even at high altitude with aircraft engines.Figure 3 shows the variation of nondimensional work with α w1t for the three cases

FIGURE 3
Variation of normalized work with prewhirl angle α w1t for the three cases.

FIGURE 4
Variation of normalized pressure ratio with prewhirl angle α w1t for the three cases.
of prewhirl.It is noticed that a sharp drop in the work absorbing capacity of the compressor takes place with constant angle of prewhirl, reduced to about half that value with linear variation of prewhirl.The parabolic variation shows nearly perfect recovery of work up to α w1t ∼40 • .Similar trends are shown in Figure 4 for the obtainable pressure ratio.

CONCLUSIONS
In spite of alleviating the compressibility effect, introducing constant prewhirl along the height of the eye in centrifugal compressors reduces the work absorbing capacity (W ) and pressure ratio (R) down to about 80%.
Reducing the prewhirl angle, from tip to root of the eye linearly improves performance and increases W and R, up to more than 94% of the nonprewhirl values.
Reduction of prewhirl angle in a parabolic relationship, restores W and R to the level of no prewhirl when using a prewhirl angle at tip up to α w1t = 40 • .This is in addition to reducing the danger of compressibility effect.

FIGURE 2
FIGURE 2 Variation of the normalized relative Mach No. (M v1t ) with the prewhirl angle at eye tip α w1t .N = 290 rps; air mass flow rate m = 9 kg/s; axial flow velocity C a = 143 m/s; slip factor σ = 0.9; power input factor ψ = 1.04; compressor isentropic efficiency η c = 0.78; ambient conditions: T 01 = 295 K and P 01 = 1.1 bars.The main performance parameters namely: M v1 , torque, work, and pressure ratio R were calculated for the basic case of no prewhirl; and the three specified cases of prewhirl over a wide range of angles of 20 • -50 • .The results are shown in Tables I-III.However, to generalize the results, they have been nondimensionalized by dividing the value of each performance parameter in these tables over its corresponding value at the basic case of no prewhirl.The results are shown in TableIVfor the parabolic case for illustration, and all presented in Figures2-4.Figure2shows the variation of the normalized relative Mach.Number M v1t , with α w1t .It is clear that M v1 can be decreased to 55% of its value at no prewhirl.This makes the operating conditions very safe as far as compressibility effect is concerned even at high altitude with aircraft engines.Figure3shows the variation of nondimensional work with α w1t for the three cases

TABLE I
Variation of performance parameters with prewhirl angle α w1 (constant from root tip) • , 40 • , 50 • .The results are shown in Table I compared with the case of no prewhirl.

TABLE II
Variation of performance parameters with prewhirl angle α w1 (linear variation from root to tip)

TABLE III
Variation of performance parameters with prewhirl angle α w1 (parabolic variation from root to tip) w1 r ) m = rt rr • , 40 • and 50 • .The results are shown in Table III compared with the case of no prewhirl.

TABLE IV
Variation of normalized performance parameters with prewhirl α w1t (parabolic variation from root to tip) radial position between eye root and tip [r /r t ] r r /r t root to tip radius ratio of the eye r m mean radius of the eye[m]