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DOI: 10.1243/09544062JMES1023

643 2009 223:Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science

M Pascu, M Miclea, P Epple, A Delgado and F Durstfan blade

Analytical and numerical investigation of the optimum pressure distribution along a low-pressure axial

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643

Analytical and numerical investigation of the optimumpressure distribution along a low-pressure axial fan bladeM Pascu1∗, M Miclea1, P Epple1, A Delgado1, and F Durst2

1Institute of Fluid Mechanics LSTM, Friedrich-Alexander University, Erlangen-Nuremberg, Erlangen, Germany2Fluid Measurements and Projects FMP Technology GmbH, Erlangen, Germany

The manuscript was received on 28 January 2008 and was accepted after revision for publication on 11 July 2008.

DOI: 10.1243/09544062JMES1023

Abstract: In the field of axial flow turbomachines, the two-dimensional cascade model is oftenused experimentally or numerically to investigate fundamental flow characteristics and overallperformance of the impeller. The core of the present work is a design method for axial fan cascadesaiming to derive inversely the optimum blade shape based on the requirements of the impellerand not using any predefined aerofoil profiles.

While most design strategies based on the aerofoil theory assume constant total pressure at allstreamlines, i.e. free-vortex flow, this paper investigates the possibility of varying the total pressurealong the blade and based on that, an analytical expression of the outlet blade angle is determined.When computing the blade profile at a specified radius, critical parameters reflecting on the flowcharacteristics are observed and adjusted (i.e. sufficient lift and controlled deceleration of theflow on the contour) so that the resulting profile is derived for minimum losses.

The validation of this design strategy is given by the numerical results obtained when employedas an optimization tool for an industrial fan: 10–20 per cent absolute increase in the staticefficiency of the optimized impeller.

Keywords: non-free-vortex flow, optimum pressure distribution, design strategy, blade profile

1 INTRODUCTION

The rapid design of a blade that performs well and sat-isfies machining requirements is one of the goals indesigning fluid machinery blades, and computationalfluid dynamics (CFD) is nowadays used as a commontool to predict the performance of such devices. In thefield of turbomachinery, the design of new blades canbe achieved through direct or inverse design meth-ods. While the direct methods are frequently used topredict the performance of the impeller under designand off-design conditions by solving Reynolds aver-aged Navier–Stokes equations [1], the inverse methodscompute the geometry of the blades according todesign specifications of flow features given as inputand the necessary boundary conditions are givenempirically, according to statistical data [2].

∗Corresponding author: Institute of Fluid Mechanics LSTM,

Friedrich-Alexander University, Erlangen-Nuremberg, Cauer-

strasse 4, Erlangen, 91058, Germany.

email: [email protected]

Of high complexity in calculation are the three-dimensional inverse methods that are formulatedwith different choices of prescribed quantities. Thefirst would be the pressure distribution along theblade suction and pressure surfaces [3]. The preferredmethod, however, uses the blade pressure loading (thestatic pressure difference between the suction andpressure surfaces) and blade thickness distribution,resulting in the calculation of the mean camber line[4]. Other fully three-dimensional inverse methodscompute the mean through-flow [5] or make intensiveuse of the algorithms based on the potential theory topredict flow instabilities [6].

In the field of axial fans, the major issue is design-ing high-efficiency fans at a given flowrate and forgiven pressure-duties, and in this case, design tech-niques are typically based on engineering experi-ence and may involve much trial and error beforean acceptable design is found. The most commonapproach is making use of the aerofoil theory, which isinvariably associated with the free-vortex flow energydistribution along the radius, i.e. the blade profilesare designed for the same total pressure in span-wise

JMES1023 © IMechE 2009 Proc. IMechE Vol. 223 Part C: J. Mechanical Engineering Science



644 M Pascu, M Miclea, P Epple, A Delgado, and F Durst

direction. The work of Wallis [7] is also concerned withsuch design assumption: the considered system is ofthe free-vortex flow type and several important param-eters (e.g. lift/drag ratio) are fixed at reasonable values.Similarly, by employing the free-vortex flow, Dugaoet al. [8] achieved considerable improvement in theefficiency of the fan, with noise reduction as a secondadvantage.

However, even though this assumption is preferredby most designers, there is nothing in the aerofoiltheory to prevent any desired pressure distributionalong the blade [9] and combining an optimizationalgorithm with the arbitrary vortex flow might enablethe designer to investigate a larger range of designalternatives in an efficient manner. In this sense,the work of Sørensen et al. [10, 11] is mainly con-cerned with employing an arbitrary vortex flow modelfor fan analysis, allowing the investigation of thefan under various operating conditions. The advan-tage of employing such a design assumption in thesense of better fan performance throughout a largerrange of flows was shown. Also by employing thenon-free-vortex flow analysis at an early design stage,Albuquerque et al. [12] developed a model foundeffective in the optimization of axial flow hydraulicturbines.

In the field of axial fans, a major challenge is con-stituted by the large range of geometries availableand their fixed use, and often, the design methods inthis area are developed for a single class of applica-tion. Therefore, it seems desirable to develop a designstrategy that, for a specified fan application, deter-mines if improvements in the performance can beachieved by employing the free-vortex flow or somevariation is better suited. Hence, the motivation of thepresent work is to determine the optimum pressuredistribution along an axial fan blade, by consider-ing both free and non-free-vortex flow assumptionsand parameterizing the solution according to the fanspecifications.

The initial design considerations and the core ofthe elaborated design strategy are stated in section 2,the derivation of the optimized profiles is presentedin section 3, and in sections 4–6, the results obtainedwhen optimizing a reference impeller (i.e. an axial fanof a hub ratio of 0.5 currently used in the automotiveindustry for cooling purposes) are presented.

2 OPTIMUM BLADE PROFILE

2.1 Theoretical considerations of the flow motionin an axial impeller

The complex three-dimensional flow in axial impellers(where the flow particles enter and leave the impellerat the same radius, i.e. constant peripheral velocity)can be treated as the superposition of a number of

Fig. 1 Sketch of the two-dimensional cascade accordingto Lewis [13]

two-dimensional flows that lead to more manage-able blade design and profile selection techniques, i.e.two-dimensional cascades. If a cylindrical cut is madethrough the impeller at specified radius and the cylin-der is then developed onto a plane, a row of bladeprofiles will result [13], as depicted in Fig. 1.

The advantage of this simple approach is that theEuler equation for turbomachinery can be applied toeach cascade section independently to determine theinlet and outlet velocity triangles for that particularblade section

�pt = ρ

2

⎡⎢⎢⎢⎣(w2

1 − w22)︸︷︷︸

staticcomponent

+ (c22 − c2

1)︸︷︷︸dynamiccomponent

⎤⎥⎥⎥⎦ (1)

The first design assumption that one can make, con-firmed by both experiments and CFD calculations,is that at the design stage, axial entry of the fluidcan be assumed and this is shown in the inlet veloc-ity triangle in Fig. 2 (zero pre-swirl upstream of theimpeller and hence w1u = u). Based on this assump-tion, the expression of the inlet blade angle can bedetermined as

tan γ1 = wm

w1u(2)

Further analysis of the cascade pressure difference inequation (1) yields to

�pt = ρ

2(w2

1u − w22u + c2

2u)

= ρ

2[u2 − w2

2u + (u − w2u)2]= ρu(u − w2u) = ρuc2u (3)

2.2 Blade angle distribution

In the previous paragraph, axial entry of the fluid inthe impeller was assumed. This consideration makes

Proc. IMechE Vol. 223 Part C: J. Mechanical Engineering Science JMES1023 © IMechE 2009



Optimum pressure distribution along a low-pressure axial fan blade 645

Fig. 2 Inlet (for axial entry) and outlet velocity triangles at specified blade section

the derivation of the inlet blade angle simple

tan γ1 = wm

uQ

2Aπrn(4)

For the determination of the outlet blade angle, oneshould come back to the purpose of the design, namelyto maximize flow and minimize losses. When fur-ther deriving the expression of the static componentof the pressure difference for one profile cascade inequation (1), then

�ps = ρ

2(w2

1u + w21m − w2

2u − w22m) = ρ

2(w2

1u − w22u)

= ρ

2(u2 − w2

2u) = ρ

2

(u2 − w2

m

tan2 γ2

)

= ρ

2

(u2 − Q2

A2

1tan2 γ2

)(5)

If referring to the system characteristic then maximumflowrate can be achieved for �Ps = 0. Hence

u2 − Q2MAX

A2

1tan2 γ2

= 0

and this immediately yields to

QMAX = uA tan γ2 (6)

When analysing equation (6), it can be readilyobserved that the maximum flowrate will occur for anangle of 90◦ (when tan γ2 → ∞).

Analysing the outlet velocity triangle, one can imme-diately notice that γ2 = 90◦ implies w2u = 0 and c2u =u. Since c2u represents the swirl component of theabsolute flow at the exit of the cascade, a good designshould limit this component because it is a directmeasure of the cascade losses. Moreover, through thedeceleration of this component, a recovery of the staticpressure can be achieved. Since the hub section hasthe highest work coefficient (loading) and the pro-file should not turn more than axial here, the outlet

angle of 90◦ is best suited. For the other blade sec-tions, i.e. middle and tip, the outlet blade angle will bedetermined based on this assumption, as shown in thefollowing paragraph.

2.3 Pressure distribution along the blade and itsinfluence on the blade shape

As already mentioned in the introductory paragraph,most design methods for axial fans assume that thesame total pressure rise is applied at all sections alongthe blade, i.e. the free-vortex flow. Considering vari-ous sections of the blade, hub, and an intermediarysection, as indicated in Fig. 3(a), this can be written as

�Pt,r

�Pt,h= 1 (7)

According to Carolus [14], employing such an assump-tion introduces, from the design stage, a correct bladeloading, at the designated section. A proper bladeloading at the tip section (where the radius is larger)demands considerable smaller values of c2u (respon-sible for the swirl) than at the hub section (wherethe radius is smaller). This means that close to thehub the absolute flow, characterized by higher c2u,has to be redirected to the tip section and thus keep-ing the product c2ur constant. The desired effect ofsuch consideration is that the flow detachments willoccur earlier, and hence, the hub section will be morestressed than the tip.

This assumption is directly connected to the swirlcomponent at the exit of the cascade and it is employedwhenever aerofoil profiles are used in designing anaxial flow machine. However, there is nothing in theaerofoil theory to prevent the desired head distribu-tion along the blade radius.

By changing the pressure distribution along theblade (and thus the outlet blade angle at the cor-responding radius), designs providing higher overallperformances may be achieved with small impact on





Fig. 3 (a) Two-dimensional sketch of an axial impeller; (b) view of specified blade section in (x, y)coordinate system

the dimensions of the swirl. Hence, it is the designer’stask to investigate, for the specified design parame-ters and operating conditions, the optimum pressuredistribution that balances the efficiency and the swirlcomponent at the same time.

For this it was assumed that starting from the huband advancing to the tip, the Eulerian pressure differ-ence variation can be expressed as a function of theradius, as indicated below

�Pt,r

�Pt,h= f (r) (8)

When employing the expression of the total pressuredifference indicated by equation (3), a recurrent rela-tionship between the outlet blade angles, at differentblade sections, can be determined

f (r)2πrhn(

2πrhn − wm

tan γ2h

)

= 2πrn(

2πrn − wm

tan γ2r

)

rwm

tan γ2r= 2πn[r2 − f (r)r2

h] + f (r)rhwm

tan γ2h

1tan γ2r

= 2πnrwm

[r2 − f (r)r2h] + f (r)

rh

r1

tan γ2h

(9)

Equation (9) can be further simplified with anotherassumption made in the previous section, namely thatat the hub, the outlet blade angle is 90◦

1tan γ2r

= 2πnrwm

[r2 − f (r)r2h] (10)

It can be noticed in equation (10) that the expressionof the outlet blade (metal) angle can be parameter-ized according to the prescribed variation of the totalpressure difference in span-wise direction. The useof such formulation can be further extended if theequation can be expressed in terms of relevant geo-metrical parameters for the design process, such asthe hub ratio.

The actual application of such impellers (enginecooling fans) is an important factor for the parameteri-zation of the pressure variation, since such devices arenormally low-pressure high-flow delivering machines.Hence, the pressure should not be increased to unre-alistic values in the range of which the impeller willnever operate anyway. Also rapid variations that mightresult from employing exponential or higher orderpolynomial laws should be avoided.

Another major constraint in the parameterizationprocess of the function f (r) is the immediate impactthat this expression has on the value of the outlet bladeangle. The angle, at the specified blade radius, deliv-ered by f (r) should be a realistic value (no negativevalues) and should be incorporated in the geometry ofthe full blade, i.e. to respect the trend imposed by theneighbouring sections.

Furthermore, it was the authors’ aim to derive anexpression in which, by changing one factor, thedesired pressure variation along the blade can beachieved.

Solving all these constraints simultaneously finallyled to the following expression for the function f (r)

f (r) = x(

rtip

rhub

)x

(r − rh)1.35 + 1 (11)

The parameter to be changed during the design pro-cess is x. It can be immediately noticed that forx = 0 the classical assumption of the constant pres-sure (free-vortex flow) is employed, since f (r) = 1. Forx = 1, the pressure variation is linear with the inverseof the hub ratio, for x = 2 parabolic, and so on.

Equation (10) then becomes

1tan γ2r

= 2πnrwm

{r2 −

[x

(rtip

rhub

)x

(r − rh)1.35 + 1]

r2h

}

(12)

The expression in equation (12) allows the deter-mination of the outlet blade angle, at the specifiedradius, based on a pressure variation from hub to tip.





Essentially, this variation has a major influence onthe resulting blade shape, and as a consequence, anoptimum profile demands a proper pressure prescrip-tion. Hence, several test cases of axial fan designs willbe investigated, including the classical constant pres-sure assumption, in order to determine, for specifieddimensions and operating conditions, the optimumpressure variation in span-wise direction, which deliv-ers high efficiency and small losses due to the swirlcomponent.

2.4 Blade shape computation

In a cascade, the action of the fluid on the profile canbe considered similar to that taking place on an aero-foil in a wind tunnel, provided that the velocity of theundisturbed flow w∞ is an average of the inlet andoutlet relative velocities [15].

The ratio of the blade chord length to the blade spac-ing l/t is an important design parameter, being anindex of the solidity. The chord spacing ratio gener-ally increases from tip to hub due to aerodynamicalreasons. When attaching a coordinate system to theprofile cascade, the blade profile, i.e. blade shape,will have the aspect indicated in Fig. 3(b). Basically,the blade shape at a specified section can be read-ily computed if imputing the y coordinates and angledistribution γ (y), since

tan γ (y) = �y�x

(13)

To effectively apply any driving action on the fluid,the blade angle is increased from γ1 to γ2. The differ-ence between the two (γ2 − γ1) is a measure of theblade curvature along any given blade section. Theincrease from γ1 to γ2 can be estimated by any typeof variation, either linear or higher degree polynomial.According to Pascu and Epple [16], for axial fans aparabolic distribution of the blade angles is appropri-ate. It was shown by means of streamline analysis thatthis assumption, made at early design stage, was con-firmed by the actual flow angles, thus achieving verygood flow-blade congruency.

When considering such a distribution γ (y) = Ay2 +By + C

y = y1 → γ = γ2

y = ym → γ = γm

y = y2 → γ = γ1

(14)

The blade shape computation is now carried out withone degree of freedom given by (γm, ym). The geom-etry is iterated until the axial chord constraints arematched as well as corrected for abrupt flow param-eters. The entire procedure for computing the blade

Fig. 4 Summarized derivation of the blade shape

shape at specified radius can be summarized for betterunderstanding as presented in Fig. 4.

2.5 Further design assumptions based on theprofile theory

Considering the flow over a profile, the fluidapproaches the profile from upstream with the veloc-ity w1 at an angle γ1 and leaves the profile with thevelocity w2 at an angle γ2, as indicated in Fig. 5.

In Fig. 5, ε is defined as the drag to lift ratio. Inthe literature, ε is often referred as the ‘gliding’ angleof the profile and several optimum values are pro-posed based on extensive experimental results [17].The value of ε is an important design choice since itdenotes the impeller losses by friction. Compared tothe ideal case of the friction-less flow, when ε = 0,the drag force introduced by ε in the direction ofw∞ (Fig. 5) calls for a loss in the pressure difference.Returning again to the application of the design, it isthe design choice of the authors to initially considerε = 0, since the computed profiles should not createadditional losses in the flow around them. The nextdesign assumption is that, at all points on the com-puted profiles, flow-blade congruency is achieved, i.e.flow and computed blade angles are identical and thisimplies zero incidence and deviation angles.

Another design parameter often used in the profiletheory is the camber angle, θ . Basically, the result ofthe design solver, as indicated in Fig. 4, is the camberline, and by attributing some thickness around it, a

Fig. 5 Flow over a profile cascade





Fig. 6 Performance curves for a constant thicknessblade and an airfoil, Eckert [18]

profile is obtained. The thickness distribution is againan important design choice and one has to carefullyweigh if profiling really pays. Most design methodsfor axial fans use a variable thickness for the profile,according to the thickness distribution functions exis-tent in the literature, i.e. NACA and C4 British series[19]. However, such methods were originally proposedfor axial flow water turbines where the operatingregime requires high pressures. For axial fans, wherethe pressures are in the range of thousands of Pascals,a variable thickness along the camber is unnecessarysince, according to Eckert [18], when measuring twoimpellers, one with aerofoil profiles and the otherone with constant thickness, their performance isidentical, as shown in Fig. 6.

Hence, for the calculated profiles, constant thick-ness will be applied, thus introducing the last assump-tion at design stage, i.e. due to constant thicknessdistribution, there will be no point of maximum thick-ness, and hence, the camber angles, at all points on theprofile, will coincide with the computed blade angles.

3 CAD MODELS OF THE NEW DESIGNS

An alternative design method for axial flow impellershas been presented so far. The method is basedon computing the appropriate blade profile at eachsection of the impeller, so that optimum flow condi-tions are achieved. Essentially, the blade is computedas a succession of several cascades characterized bydifferent radius and chord length and the profilesat different radii are derived based on the variationof the pressure along the blade. Hence, calculatingthe optimum blade profile is directly dependent onthe determination of the optimum pressure variation,

according to the specified dimensions and operatingconditions of the impeller.

In order to determine the optimum pressure dis-tribution, several new designs are derived by simplyvarying the term x in equation (12):

Design I: x = 0 (corresponds to the classical assump-tion of free-vortex flow and hence constantpressure at all radii is assumed)

Design II: x = 1Design III: x = 2

During the preparation of the present work, the CADmodel of a reference impeller currently used as enginecooling ventilator in the automobile industry wasavailable, and for optimization purposes, all the newdesigns were derived for the same dimensions anddesign parameters as this impeller:

(a) hub radius, rhub = 147 mm;(b) tip diameter, rtip = 280 mm;(c) rotational speed, n = 3000 r/min;(d) design flowrate, Qdesign = 4 m3/s;(e) number of blades, z = 8.

The new designed blades were obtained by computingthe blade shape according to the prescribed pressuredistribution for several cascades: hub, tip, and twointermediary sections, namely 187 and 227 mm. Thesolidity of the reference is variable with the radius, asindicated in Fig. 7, and it is kept constant from onedesign to the other.

The computer camber lines, for each pressure dis-tribution, are presented in Fig. 8.

Obviously, for the hub section, where the outletangle is always fixed to 90◦, the aspect of the camberwill not change no matter the pressure distribution.The differences in the computed cambers appear fromthe following sections, inducing a difference betweenthe prescribed angles as well, as indicated in Fig. 9.

Constant thickness of 3.2 mm was applied to allmodels. Three impellers were generated, each having

Fig. 7 Variation of the solidity with the radius





Fig. 8 Differences in the computed camber lines for different pressure distributions

Fig. 9 Variation of the prescribed angle distributions with different pressure variations





Fig. 10 CAD model of the impeller

eight blades and corresponding to the dimensionsindicated previously. The three impellers were placedinside of pipe and the tip clearance was of ≈10 mm. Tobring the simulation results closer to the real-life oper-ating conditions, two extra components were added tothe CAD model: at the inlet of the impeller a rig sectionlong enough so that the flow entering the impellerdomain can be considered fully developed (the lengthof the rig section was chosen 3Dpipe); and at the outlet(ambient), another pipe with the length of 2Dpipe wasincluded. An exemplary depiction of the CAD modelsis shown in Fig. 10.

4 MESH GENERATION AND NUMERICALSIMULATIONS

The grid (Fig. 11) was generated in ANSYS ICEM 11.0(http://ansys.com/) using tetrahedral mesh and itssize was around 400 000 nodes for the flow domaincorresponding to one blade. The general mesh size for

the domain fan was set to 8, with a finer size aroundthe blade tips and leading and trailing edges (2), andthe suction and pressure surfaces of the blade (4). Toresolve more accurately the flow at the wall portion,prismatic layers were employed. A coarser mesh wasapplied for the flow domain around the impeller andthe pipe walls.

The numerical simulations were carried out withthe commercial code ANSYS CFX 11.0. The simu-lation type was set initially to be steady. The fluidflow was set to be viscous (air properties as an idealgas were considered). The flow was solved with theNavier–Stokes equations, assuming conservation ofmass, momentum, and energy.

Typical values for the Reynolds number, calculatedas Re = wmDtip/ν are around 800 000. One of the mainproblems in turbulence modelling is the accurateprediction of flow separation from a smooth sur-face. Standard two-equation turbulence models oftenfail to predict the onset and the amount of flowseparation under adverse pressure gradient condi-tions. To avoid this problem, the model used in thepresent computations of the air flow is the shear stresstransport model [20]. The model works by solving aturbulence–frequency-based model (k-ω) at the walland k-ε model in the bulk flow. A blending functionensures smooth transition between the two models.The interface between the different frames of refer-ence is taken to be a frozen rotor general grid interface(GGI). The frozen rotor model has the advantageof being robust, using less computer resources thanthe other frame change models, e.g. stage interfacemodel, which is not suitable for applications withtight coupling of components and/or significant wakeinteraction effects and may not accurately predict

Fig. 11 Detail of the generated mesh





loading. The frozen rotor model treats the flow fromone component to the next by changing the frame ofreference while maintaining the relative position ofthe components. This model must be used for non-axisymmetric flow domains, such as impeller/voluteor classifier/casing.

Inlet boundary with specified mass flow was appliedto the rig domain, flow direction normal to theboundary condition, and medium turbulence inten-sity (5 per cent). A rotational speed of 3.000 r/minwas applied to the fan domain. The outlet bound-ary was applied to the ambient domain by specifyinga 25 ◦C temperature (air density 1.05 kg/m3). Typi-cal Mach numbers were ∼0.25 (it is known that forMach numbers <0.3, the fluid can be considered asincompressible [21]). Several simulations were car-ried out for all models by varying the flowrate at 2,3, 4, 5, 6, and 8 m3/s. The simulations were com-pletely converged; all runs reached the convergencecriterion (residual type RMS and residual target todefault value 10−4). The convergence criterion valuecan be increased, but this means also a considerableincrease in the required computational time and notat the cost of significant differences in results. Besides,during the simulations, the most important physicalquantities, such as flowrate, pressure, and efficiency,were monitored and it was observed that all theserelevant quantities converged properly with the 10−4

residual target. All simulations were solved by paral-lel running on 16 CPUs and typical computation timerequired for full convergence is in wall clock secondsof ∼5.0E + 03.

5 NUMERICAL RESULTS

5.1 Flow analysis on the blades

Qualitatively, the flow behaviour can be tracked bymeans of velocity streamlines around the blades, thuspicturing if the prescribed profile angles at designstage are indeed achieved. Such exemplary depictionsare shown in Fig. 12 at one blade section. Apart fromno obvious differences between the three designs, thestreamlines also indicate s smooth flow that appearsto follow completely the blade contours.

Pressure plots on the blade are helpful in identifyingthe stressed spots, since according to reference [22],they are responsible for losses in the system caused bysound sources, i.e. the leading edge of the blade andareas close to tip section, due to the tip clearance.

Although the prescribed pressure distribution forthe three designs is essentially different (e.g. Design IIIhas twice the pressure at the tip section than at the hub,while Design I has a constant pressure through out theblade length), when plotting the static pressure coeffi-cient, no major differences between the three designscan be observed, as indicated in Fig. 13.

Another aspect to be analysed for the comparisonof the flow over the investigated profiles is the velocitydistribution, i.e. flow velocity in the span-wise direc-tion, as shown in Fig. 14. At this point, the authorswould like to emphasize that the results presentedin both Figs 13 and 14 correspond to the full profilecascade at the indicated radius. For the early blade sec-tions (i.e. hub and very close to the hub), the cascade

Fig. 12 Velocity streamlines around the blade for Qdesign at r = 227 mm





Fig. 13 Static pressure coefficient distribution along the blade profiles for Qdesign

Fig. 14 Span-wise velocity distributions along the blade profiles for Qdesign





effect can be clearly observed in the velocity distri-butions, since more variation curves can be identifiedfor the same design. The reason behind these differ-ences is due to the wall influence from the hub andthis basically means that, in these regions, the flowover one profile is not symmetric and obvious influ-ences from the neighbours occur. This, however, is notthe case for the upcoming sections, where hardly anydifferences can be observed, meaning that from midsections to near-tip, there is hardly any cascade influ-ence (see corresponding plots for r227 and r270). Thewall influence in the flow also explains why, for theextreme sections (hub and tip), negative velocities inspan-wise direction appear, thus indicating recircula-tions of the flow on the suction surface of the blades inthese regions. While at the hub, the velocity gradientis not too large, and close to the tip sections, strongnegative values can be observed, due to the tip clear-ance phenomenon. Such gradients can be avoided byreducing the gap between the tip and the pipe walls.

As in the case of the computed Cp, there is nomajor difference value wise in the velocity distributionbetween the three designs at the investigated section,thus confirming the similar flow pictures depicted inFig. 12.

5.2 Performance assessment of the new designs:optimum pressure distribution

A correct assessment of a turbomachine depends ondefining appropriate performance indicators. Such anindicator is the efficiency of the impeller, which forfans can be defined as

Efficiency =total (hydrodynamic) energy input to

fluid in unit time

power input to coupling of shaft

In other words, the efficiency of the impeller can bewritten as the ratio between the hydraulic power andthe shaft power

Efficiency = hydraulic powershaft power

(15)

Ideally, the two parameters should be equal so thatthe efficiency of the machine is 1. However, such adefinition becomes useless at design stage since itsvalue will never change no matter the design [23].Hence, more precise performance indicators have tobe defined.

The shaft power developed by the impeller can bewritten as the product between the angular speed andthe required torque

Pshaft = ωT

If the static pressure difference between the inletand the outlet of the impeller is measured, then the

hydraulic power is

Phydraulic = �PsQ

When the total pressure difference across the impelleris considered, then the hydraulic power becomes

Phydraulic = �PtQ

Accordingly, two efficiencies, conveniently namedtotal-to-static and total-to-total efficiencies, can be

Fig. 15 Evaluation of the performance of the threedesigns: (a) static efficiency; (b) total efficiency;(c) polytropic efficiency





defined

ηt−s = �PsQωT

ηt−t = �PtQωT

(16)

Also of high relevance for practical use is the polytropicefficiency, defined as ηpoly = ((γ − 1) ln π)/(γ ln τ)

where the calculated air ratio is γ = 1.4, π = p2/p1, andτ = T2/T1.

After processing the results of the numerical simu-lations and evaluating the performance of the threedesigns according to equation (16), the performancecharts in Fig. 15 can be plotted. It can be readilyobserved that, while at the beginning of the inves-tigated flow range and at the design point, all threedesigns reach similar performances, for higher flowsthe advantage of the third design becomes obvious,especially towards the end of the flow range whereDesign III achieves the following efficiency increaseswhen compared with Design I: 8 per cent static effi-ciency, 9 per cent in total efficiency, and 10 per cent inpolytropic (all absolute differences).

At this point, the advantage of considering at designstage the possibility of a non-free vortex flow (i.e.assuming that in span-wise direction the total pressureis not constant) becomes obvious. Employing such adesign assumption impacts directly into the extensionof the flow range under which the fan can effec-tively operate and as indicated in Figs 13 and 14, theflow around the computed profiles, both pressure andvelocity wise, is similar from one profile to another.Hence, for the investigated hub ratio, the authors sug-gest that the optimum pressure distribution along theblade should have a parabolic distribution with theratio rtip/rhub, as follows:

�Pt,r

�Pt,h= 2

(rtip

rhub

)2

(r − rh)1.35 + 1 (17)

A further increase in the pressure distribution (morethan parabolic with rtip/rhub) is not recommended,since a fourth design, corresponding to x = 2.5, wasinvestigated and decreases in all performance indica-tors, compared to x = 2, were observed.

5.3 New design versus industrial impeller

As mentioned previously, for optimization purposes,all the new designs that were investigated werederived for the corresponding dimensions and oper-ational parameters of an industrial impeller. Sinceit was found in the previous section that Design IIIachieves overall higher performances, this model willbe employed from now on for further analysis duringthe optimization process.

However, before comparing the performances of thereference and suggested design, it becomes interesting

Fig. 16 Total pressure distribution in span-wise direc-tion for the investigated models

Fig. 17 Streamlines around the blade profiles for Qdesign at r = 227 mm: (a) reference impeller;(b) Design III (x = 2)





to analyse the differences in design concepts for thetwo models.

In Fig. 16, the total pressure distributions in span-wise direction prescribed at design stage, for the refer-ence blade and the new models, developed accord-ing to the presented design strategy, are depicted.As already stated, what has been used as referenceis an axial fan which is currently manufactured forapplication in the automotive industry for enginecooling purposes. The CAD model of this fan was

Fig. 18 Performance comparison for Design III (x = 2)and the industrial impeller: (a) static efficiency;(b) total efficiency; (c) polytropic efficiency

made available by the manufacturer for optimizationpurposes. After reading out the geometry of the blade,the corresponding design parameters were deter-mined and the pressure distribution along the bladewas determined. Figure 16 highlights the main goalof the proposed design strategy: to design blade pro-files strictly according to the operational requirementsof the impeller and not to use predefined ones, whichmay have been computed for completely different flowconditions, thus ensuring that what has been pre-scribed at design stage can actually be achieved duringoperation. It remains to be seen, however, which ofthe two models (reference and Design III) actuallyperforms better under identical operating conditions,and for this, another set of numerical simulations wascarried out.

A first analysis of the numerical results is a quali-tative one, i.e. the flow aspect around the blades. Byplotting the velocity streamlines for both models atthe same radius, it was observed that, while the flowin the industrial impeller is characterized by obviousdetachments, smooth streamlines completely follow-ing the blade profiles result in the case of new design,as exemplary depicted in Fig. 17.

Furthermore, when comparing the performancecurves of the two models as plotted in Fig. 18, signifi-cant differences were observed: 7–28 per cent in staticefficiency, 2–9 per cent in total efficiency, and 4–17per cent in polytropic efficiency absolute increases forDesign III (x = 2).

6 CONCLUSIONS

An integrated optimization procedure based on adesign strategy focusing on computing the optimumblade profile at the specified section according to theoperating requirements of an axial fan impeller waspresented.

While most design methods for axial fans that relyentirely on the extensive data on aerofoil profilesassume constant total pressure at all streamlines basedon the free-vortex flow theory, the proposed strat-egy computes the blade shape after determining theoptimum pressure distribution along the blade.

An analytical investigation was carried out and aparametric expression for the total pressure distribu-tion on the blade, based on the dimensions of theimpeller, was computed.

Three new designs matching the constructive mea-sures of an industrial impeller used in the auto-mobile industry for cooling purposes were derived,each design being characterized by different pressurevariation (including the classical assumption of con-stant pressure). All three designs were numericallyinvestigated for boundary conditions reproducing theactual working conditions and it was found that forthe investigated class of impellers (hub ratio of 0.5),





the optimum pressure in the span-wise direction isdescribed by a polynomial that increases the pressurefrom hub to tip and that employing a non-free-vortexflow assumption at the design stage impacts directlyin the extension of the flow range under which the fancan effectively operate.

This conclusion was further employed as a startingpremise for the optimization process of the industrialimpeller. Flow pictures in the two impellers clearlyshow that while the flow around the reference bladeis characterized by obvious flow detachments, thestreamlines around the new design follow completelythe blade profile and the flow-blade congruency isachieved.

For a further comparison, appropriate performanceindicators were defined, i.e. the static, total, and poly-tropic efficiencies. These plots revealed the improvedperformance of the new designs with considerableincreases compared with the reference impeller. Theseresults stand for a fully optimized impeller andunderline the validity of the proposed design strat-egy. However, the optimization process is completelybased on a design method focusing on theoreti-cal aspects of the flow characteristics in an axialimpeller. Further improvements to the new designcan be achieved by employing genetic algorithms[24] or by improving the geometry of the system[25], and higher performances of the impeller canbe obtained, bringing the impeller closer to the idealmachine. It has to be kept in mind, however, thatthese later methods are more costly and require a goodstarting solution, which can be given by the presentmethod.

ACKNOWLEDGEMENTS

The authors would like to acknowledge the financialsupport of the Bavarian Science Foundation.

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APPENDIX

Notation

A area (m2)c absolute velocity (m/s)l blade chord (m)n rotational speed (r/min)P power (W)Q flowrate (m3/s)r radius (m)T torque (Nm)w relative flow velocity (m/s)wm meridional component of the relative

velocity (m/s)wu tangential (peripheral) component of the

relative velocity (m/s)

γ blade angle (◦)�P pressure difference (Pa)η efficiencyπ pressure ratio between the fan outlet and the

inlet static pressuresρ density (kg/m3)τ temperature ratio between the fan outlet and

inlet temperaturesω angular velocity (rad/s)

Subscripts

1 blade inlet2 blade outletd dynamich hubs statict tip, total




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