on the theoretical link between design parameters and performance in cross flow fans a numerical and...

Computers & Fluids 34 (2005) 49–66www.elsevier.com/locate/compfluid

On the theoretical link between design parametersand performance in cross-flow fans: a numerical

and experimental study

Andrea Toffolo *

Department of Mechanical Engineering, University of Padova, Via Venezia, 1–35131 Padova, Italy

Received 23 June 2003; received in revised form 26 January 2004; accepted 26 April 2004

Available online 20 July 2004

Abstract

Cross-flow fan performance strictly depends on the complex configuration of the non-axisymmetricalflow field within the machine. The flow field, in turn, is deeply influenced by the design parameters of both

casing and impeller geometry. In this paper, the relationship between the design parameters of the geo-

metrical configuration and fan performance is discussed in a theoretical perspective, analyzing the features

of the corresponding flow fields. These are reconstructed by a numerical study on cross-flow fan operation

carried out for a representative set of configurations at different throttling conditions. Time-accurate

solutions for a two-dimensional viscous and incompressible model of the fan using a sliding mesh technique

are calculated with a commercial CFD code. The numerical results are validated with experimental data

obtained from tests on performance and from local measurements of the flow field.� 2004 Elsevier Ltd. All rights reserved.

1. Introduction

A well founded theory on cross-flow fan operation does not still exist because of the compli-cated flow field structure within the machine (Fig. 1). A double passage of the air across the bladerow and the formation of an eccentric vortex inside the impeller, due to the motion of the blades,make it difficult to analyze the aerodynamics and the loss mechanisms of this category of fans.

* Tel.: +39-49-827-6747; fax: +39-49-827-6785.

E-mail address: [email protected] (A. Toffolo).

0045-7930/$ - see front matter � 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compfluid.2004.04.002

mail to: [email protected]

Nomenclature

c absolute velocity (m/s)D diameter (m)e specific energy (J/kg)f normalized stream functionh height (m)L axial length (m)p pressure (Pa)Q volumetric flow rate (m3/s)R radial coordinate (m)ReD ¼ u2D2q=l Reynolds numbers thickness (m)u peripheral speed (m/s)a log spiral angle (deg)b angle of relative velocity (deg)g efficiencyl dynamic viscosity (kg/ms)q density (kg/m3)r stream functionU ¼ Q=ðLD2u2Þ flow coefficientW ¼ Dp=ð0:5qu22Þ pressure coefficientf angular coordinate (deg)

Subscripts and superscripts1 internal2 externalI first blade passageII second blade passageb bladed dischargeR rear walls statict totalu tangential componentV vortex wall

50 A. Toffolo / Computers & Fluids 34 (2005) 49–66

The geometry of both casing and impeller exerts a significant and complex influence on thesephenomena, and fan performance is, therefore, hardly predictable from a given set of designparameters values.

A systematic experimental investigation of fan performance has been recently presented in [1]on the basis of the main design parameters identified in [2]. This is, however, an empirical

Fig. 1. The most significant parameters affecting fan performance and efficiency.

A. Toffolo / Computers & Fluids 34 (2005) 49–66 51

reconstruction of the relationship between design parameters and fan performance, which is notbased on theoretical considerations drawn from the analysis of the corresponding flow field withinthe machine.

On the other hand, several analytic studies in the literature have tried to evaluate fan perfor-mance starting from the characteristics of the flow field pattern. Eck [3] and Coester [4] proposedto describe the streamlines within the impeller by a potential flow produced by a number ofvorticity sources lying on the inner periphery of the blade row. Tramposch [5] then used thismodel to calculate the energy transfer across the impeller by assuming ideal conditions of inci-dence and discharge on the outer blade edges. Ilberg and Sadeh [6] used a combined vortex (aforced vortex in the core and free vortex potential flow outside) to reproduce the velocity fieldinside the impeller. Their model, however, requires the knowledge of data retrieved from exper-imental measurements or flow visualization, and, therefore, cannot be used for predictive pur-poses. Ikegami and Murata [7] calculated the theoretical performance of a simplified cross-flowfan configuration (two straight casing walls separating suction and discharge zones) as a functionof the eccentricity and the angular position of two vorticity sources, one inside and the otheroutside the impeller. An actuator model was suggested in the experimental studies by Yamafuji[8]. The streamlines are concentric to the vortex center and have a constant velocity which de-pends on the radius of the streamline and the eccentricity of the vortex. Fan performance is thencalculated for different eccentricities and angular positions of the vortex and for different impellergeometries, defined by the ratio of the internal and external diameters and the external bladeangle.

Although these models are able to capture some performance trends that are experimen-tally verified, they all fail in predicting accurately the characteristic curves. The main causesare the lack of reliable models to represent the loss mechanisms and the over-simplifyingassumptions that do not consider some complex but fundamental features of the flow fieldpattern.

This paper aims at explaining how a specific set of design parameters corresponds to a givenperformance and efficiency in a theoretical perspective based on a detailed analysis of the flowfield. The latter is reconstructed by means of CFD simulations, the results of which are validatedwith the experimental data presented in [1,9].


2. The numerical analyses

2.1. The simulation program

The aim of the numerical analyses is to simulate a set of representative cross-flow fan config-urations, obtained using different values of the design parameters that exert the most significantinfluence on fan performance and efficiency. These parameters derive from the parameterizationsuggested in [2] and have been identified in a systematic experimental study [1]. They are the anglea of the logarithmic spiral that controls the radial width of the rear wall, the height hd and thethickness sV of the vortex wall (Fig. 1).

The choice of the configurations to be simulated is driven by the indications obtained in [1] aswell. Three values are considered for the rear wall radial width, corresponding to different shapesof the fan characteristic curve. In fact, small, intermediate and large radial widths result inascending, nearly flat and descending Wt–U curves, respectively. The considered rear walls areshown in Fig. 2. The small radial width rear wall RE is Eck’s patented rear wall and is made up oftwo circular arcs, one of which centered on impeller axis. The intermediate (R2r) and large (R3r)radial width rear walls are two log spiral rear wall, having a ¼ 17:2� and 23.6�, respectively.Vortex wall geometry can be subdivided in two main categories according to thickness. The flatthin ðsv=D2 ¼ 0:13Þ vortex wall used in [1] is combined with all the three rear walls. Only the twopositions in which high performance and efficiencies were achieved in [1] are considered and areindicated as H1 ðhd=D2 ¼ 0:185Þ and H2 ðhd=D2 ¼ 0:316Þ in Fig. 2. In the other category, twothick vortex walls are matched with the only small radial width rear wall RE, because of the highefficiencies obtained from this combination of design parameters. The two vortex walls, shown inFig. 2, are the modular vortex wall S3H1 already used in [1], having sv=D2 ¼ 0:39 andhd=D2 ¼ 0:185, and Eck’s patented vortex wall VE. All the selected casing shapes are then com-bined to the impeller for which the validity of the similarity laws has been verified in [10], having24 circular arc blades and the external and internal blade angles equal to 38� and 70�, respectively.The impeller has the external diameter D2 equal to 152.4 mm, the axial length L equal to228.6 mm, and rotates at 1000 rpm.

Each of the resulting fan configurations is simulated at the flow rates corresponding to flowcoefficients equal to 0.2, 0.4, 0.6, 0.8 and 1.0, provided that these values are lower than the flowcoefficient at free blowing. The same fan configurations, at the same flow conditions, were also

Fig. 2. The casing shapes selected for the numerical simulations.


selected in [9] for an experimental program to determine the local pressures and velocities of theflow field inside the impeller.

2.2. The CFD model

The numerical simulations are performed using the commercial general-purpose CFD codeFLUENT by Fluent Inc. A two-dimensional computational grid in the plane perpendicular tomachine axis is developed to model the flow domain to be analyzed. The axial components of thevelocity vectors are not considered because of the ratio between the axial length and the externaldiameter of the impeller is equal to 1.5. In fact, according to the literature [11], three-dimensionaleffects can be neglected if the ratio L=D2 is inside the range between 0.5 and 4.

In spite of the high computational effort required for the calculation of an unsteady solution,the motion of the impeller is modeled using a ‘‘sliding mesh’’ technique, in which part of thecomputational grid is actually rotated in a time-dependent simulation. On the other hand, the‘‘frozen rotor’’ or the ‘‘mixing plane’’ techniques would not be effective in this application, be-cause of the high non-uniformity of the flow conditions around the impeller.

The grid is subdivided into two zones: the first is the moving zone that includes the impellerblades, whereas the second is stationary and is bounded by machine walls (Fig. 3). The inlet ofthe computational domain is placed two diameters upstream the impeller, that is where the flowis supposed to be undisturbed by the local perturbations induced by impeller motion. Onthe discharge side, part of the facility in which the fans were tested is included in the compu-tational domain in order to simulate accurately the highly non-uniform flow that is formed inthe duct between impeller exit and the discharge section. Thus, domain outlet is placed afterthe plenum chamber, in the throat of the Venturi nozzle which is used to measure the volumetricflow rate (Fig. 3). This choice makes available a section in which the flow conditions are asuniform as possible, in order to have the imposed boundary conditions as close as possible toreality.

The grid of both the rotating and the stationary zone is made up of quadrilateral cells and isunstructured to allow a thickening of the nodes in the regions in which fan geometry causes largegradients of flow quantities. The region adjacent to the interface between the two zones, a cir-cumference concentric to impeller axis, is discretized using regular quadrilateral cells, the numberof which determines the overall size of the grid. The number of intervals in which the interface is

Fig. 3. Boundaries of the computational domain and detail of the blade-to-blade grid.

R2r - H1

0.0

1.0

2.0

3.0

0.0 0.5 1.0 Φ

Ψt

experimental240 cells360 cells480 cells600 cells

R2r - H1

0

10

20

30

0.0 0.5 1.0Φ

ηt %experimental240 cells360 cells480 cells600 cells

Fig. 4. Grid refinement test according to the number of cells along grid interface.


subdivided was chosen as a trade-off between the computational effort required and the accuracyof the numerical solution. In Fig. 4, fan performance and efficiency predicted for the R2r-H1configuration using four different levels of grid refinement, having 240, 360, 480 and 600 cellsalong the interface (from 10 to 25 cells across a blade passage), are compared to the experimentalresults collected in [1]. It appears that the predictions obtained with 480 cells are grid independentand sufficiently accurate (see also the complete validation of the experimental results in Section2.3). The resulting mesh contains approximately 25,000 cells in the rotating zone and from 60,000to 90,000 in the stationary zone, depending on the geometrical fan configuration to be simulated.A detail of the blade-to-blade grid is provided in Fig. 3.

Air is considered as an incompressible fluid, the pressure rise generated by the fan being in therange of decades of Pascals. Viscous effects are taken into account by means of the renormal-ization group (RNG) k–e turbulence model, as Fluent User’s Manual [12] recommends whenflows characterized by strong streamline curvature and relatively low Reynolds numbers(ReD ¼ 80; 000 in this case) are considered. Wall functions are used to model the flow in the near-wall region and the values of yþ in the unseparated regions are kept in the range between 30 and60, as suggested in [12]. The portion of the mesh containing the impeller is rotated by steps of0.000125 s, which makes the grid shift by 0.75�. The impeller, therefore, performs a completerotation in 480 steps, corresponding to 0.06 s.

At domain inlet a boundary condition of constant total pressure, equal to ambient pressure, isimposed, whereas at domain outlet a constant value of the velocity is imposed in the throat sectionof the Venturi nozzle by fixing the volumetric flow rate through the fan. This choice of boundaryconditions is preferable to the imposition of total pressure at domain inlet and static pressure atdomain outlet. In fact, fan configurations having unstable or flat Ws–U curves, for which therelationship between volumetric flow rate and discharge static pressure is not biunivocal, can besimulated without ambiguities.

The simulation of a given fan configuration at given flow coefficients starts from a quiescentflow condition and proceeds until a quasi-steady state is reached, that is when the fluctuations ofthe main flow quantities become periodic. This takes from 4 to 10 impeller rotations depending onthe configuration and the flow rate, corresponding to 20–50 h of CPU time on a four-processorCompaq AlphaServer ES40 (clock frequency 667 MHz). Fan performance and efficiency are thencalculated from the time and space average of some quantities characterizing the flow field. Thestatic pressure rise across the fan ðDpsÞ is evaluated by averaging in both time and space the local


static pressure on the fan discharge section. The total pressure rise across the fan ðDptÞ is cal-culated by adding to Dps the dynamic pressure corresponding to the mean velocity in the dischargesection, obtained by dividing the volumetric flow rate by the area of the discharge section. It isworth noting that the same procedure for the experimental measure of these quantities is followedby the UNI 10531 [13] standard on test methods and acceptance conditions for industrial fans(equivalent to ISO 5801 [14]). Finally, the torque on the blade row, which is required to calculatethe total efficiency, is evaluated by averaging in time the integral of the torque (pressure andviscous stresses) acting on the blades with respect to impeller axis.

2.3. Validation of the results

An example of the results obtained with the CFD model of the cross-flow fan described in theprevious section is given in Fig. 5, where the contours of the total pressure field with velocityvectors superimposed are plotted for three sample configurations. To validate the capabilities ofthe model, the results of the numerical simulations are compared to the experimental data aboutfan performance and efficiency collected in [1] and to those about the flow field within the impellercollected in [9]. In fact, very different flow field patterns within the same geometric configurationmay result in identical values of performance and/or efficiency. These comparisons are presentedin Fig. 6 for the characteristic curves Wt–U and gt–U of all the considered configurations, andin Fig. 7 for the local flow quantities of sample configurations on two circumferences havingR ¼ 20 mm and R ¼ 40 mm in a plane perpendicular to impeller axis.

Fig. 5. Contour plots of the total pressure fields for three sample configurations (with velocity vectors superimposed).

R3r - H1

0.0

1.0

2.0

3.0

0.0 0.5 1.00

20

40

60

R2r - H1

0.0

1.0

2.0

3.0

0.0 0.5 1.00

20

40

60

RE - H1

0.0

1.0

2.0

3.0

0.0 0.5 1.00

20

40

60

RE - S3H1

0.0

1.0

2.0

3.0

0.0 0.5 1.0

ηt%

Φ Φ

Ψt Ψt

0

20

40

60

R3r - H2

0.0

1.0

2.0

3.0

0.0 0.5 1.00

20

40

60

R2r - H2

0.0

1.0

2.0

3.0

0.0 0.5 1.00

20

40

60

RE - H2

0.0

1.0

2.0

3.0

0.0 0.5 1.00

20

40

60

RE - VE

0.0

1.0

2.0

3.0

0.0 0.5 1.0

ΦΦ

Φ

Φ Φ

Φ

ηt%Ψt Ψt

ηt%

ηt%

ηt%

ηt%

ηt%Ψt Ψt

ηt%Ψt Ψt

0

20

40

60

Fig. 6. Fan performance (circles) and efficiency (triangles) according to experimental data (white symbols) and

numerical simulations (black symbols).


RE-VE =0.2

-6

-4

-2

0

2

4

-180 -120 -60 0 60 120 180

Ψt

R=40mm (num)R=40mm (exp)R=20mm (num)R=20mm (exp)

R3r-H2 =0.2

-20

-16

-12

-8

-4

0

4

-180 -120 -60 0 60 120 180

sΨ


RE-S3H1 =0.6

0

1

2

3

4

-180 -120 -60 0 60 120 180

v/u2


R2r-H1 =1.0

-60

-30

0

30

60

90

120

-180 -120 -60 0 60 120 180

ζ

ζ ζ

ζ

yaw


Φ

Φ Φ

Φ

Fig. 7. Examples of validation of characteristic flow quantities.


The numerical simulations apparently reproduce the real performance and flow fields with asatisfactory accuracy. The largest differences between experimental data and numerical results aredue to two main causes:

• The CFD model is two-dimensional and is not able to simulate the axial components of thevelocity vectors due to three-dimensional effects. These components become comparable tothose perpendicular to machine axis as flow rate decreases ðU < 0:4Þ and the space betweenthe impeller and the rear wall increases (i.e., larger rear wall radial widths).

• The number of cells of the model has been limited due to the high computational effort re-quired by the sliding mesh technique for the calculation of the unsteady solutions. As a con-sequence, the interaction between recirculating flow and blade row is reproduced with loweraccuracy as vortex eccentricity and strength increase, that is as flow rate ðU > 0:8Þ and therear wall radial width increase, due to the highly distorted streamlines and high energy lossesin the vortex core.

3. The analysis of the results

3.1. Definition of a normalized stream function

The quantities of the flow field on the external circumference of the impeller are not analyzedhere according to the angular coordinate, because of the non-uniform distribution of the radial

Fig. 8. Definition of the normalized stream function f .


velocity that would make it difficult to formulate considerations on mean values. Instead, a nor-malized stream function f is defined as illustrated in Fig. 8 and is used for an easier comparisonamong different configurations or at different flow coefficients. First, a stream function r is definedas usual for the time-averaged velocity field calculated with the numerical simulations. Then, thezero of the normalized stream function f is assigned to the streamline that brushes the vortex walland envelopes the recirculating flow ðr ¼ rVÞ, while the value 1 is assigned to the streamline thatbrushes the rear wall ðr ¼ rRÞ. The value of f for a generic streamline r is defined as:

f ðrÞ ¼ ðr � rVÞ=ðrR � rVÞ ð1Þ

Therefore, f is equal to a number between 0 and 1 for a streamline of the throughflow, to anegative number for a streamline of the recirculating flow and to a number greater than 1 for astreamline that belonging to the portion of the flow that returns to suction.

R2r - H1

-4

-3

-2

-1

0

1

2

3

4

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2

normalized stream function

t

Φ = 0.4

flow

reci

rcul

atin

g

thro

ughf

low

retu

rnin

g to

suc

tion

Ψ

Fig. 9. The local total pressure coefficient along the external circumference of the impeller.

R2r - H1

-2

-1

0

1

2

3

4

-4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2


cu2/u2

Φ = 0.4

thro

ughf

low

retu

rnin

g to

suc

tion

reci

rcul

atin

g fl

ow

Fig. 10. The ratio between the tangential velocity and the peripheral speed along the external circumference of the

impeller.

R2r - H1

0

10

2030

40

50

60

70

80

90

100

110

120

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1


β2

Φ= 0.4

β2b

Fig. 11. The relative velocity angle along the external circumference of the impeller.


The values of some characteristic quantities of the flow field along the external circum-ference of the impeller are plotted for a sample configuration and for a single flow coefficientin Figs. 9–11, to illustrate the theoretical basis of the analysis that is developed in the fol-lowing sections. In these figures, two ordinates correspond to a single abscissa, since eachstreamline crosses twice the external circumference of the impeller (see the example inFig. 8 for a generic streamline r). In all the figures, the lower ordinate represents the valuethat the plotted quantity assumes when the streamline enters the impeller, while the upperordinate represents the value that the plotted quantity assumes when the streamline leaves theimpeller.


Fig. 9 shows the values of the local total pressure coefficient ðWtÞ along the external circum-ference of the impeller. At impeller inlet the values of the streamlines coming from suction ðf > 0Þare null, even if the perturbing effects of the blade motion appear. On the other hand, the averageof the values of the throughflow streamlines ð0 < f < 1Þ at impeller exit is the discharge meantotal pressure of the throughflow. Therefore, this value represents the total pressure rise across theimpeller and, apart from the viscous losses in the duct between impeller exit and the dischargesection, across the fan itself. The impeller makes the total pressure of the streamlines returning tosuction ðf > 1Þ rise as well, but this transferred energy is completely lost. Finally, the totalpressure of a streamline of the recirculating flow at impeller exit is not varied, apart from theviscous losses, when it re-enters the impeller. In fact, the two ordinates corresponding to the sameabscissa are practically superimposed for f < 0.

Fig. 10 shows the values of the ratio between the tangential component of the absolute velocityðcu2Þ and the peripheral speed u2 along the external circumference of the impeller. The areabounded by these values is the integral of the variation of cu2 between inlet and outlet for all thestreamlines flowing through the impeller. This area is, therefore, a measure of the mechanicalenergy transferred from the impeller to the flow. Actually, the exact expression of the specificenergy transferred to a single streamline is:

e ¼ cIIu2u2 � cIIu1u1 þ cIu1u1 � cIu2u2 ð2Þ
but the contributions of the tangential components on the inner circumference of the impeller canbe neglected since they almost annul each other. The energy fractions that are transferred to therecirculating flow, the throughflow and the flow returning to suction can be evaluated by dividingthe area of the diagram along the abscissae f ¼ 0 and f ¼ 1. The total efficiency can then beestimated by comparing the total pressure rise across the fan to the overall mechanical energytransferred by the impeller, whereas the hydraulic efficiency can be determined by comparing thetotal pressure rise across the fan to the fraction of the mechanical energy that is transferred by theimpeller to the only throughflow. The volumetric efficiency can be evaluated, as usual, as the ratiobetween the throughflow and the overall flow rate, but an interesting alternative formulation is toconsider the ratio between the energy fraction associated with the throughflow and the overallenergy transferred by the impeller. In the latter case the total efficiency can be expressed as theproduct of the hydraulic and the volumetric efficiencies.
Fig. 11 shows the values of the relative angle of the velocity vector ðb2Þ along the externalcircumference of the impeller for the only throughflow streamlines. The incidence conditions canbe estimated in this diagram by comparing the effective relative angles to the blade external angleb2b. Stall is likely to occur for the streamlines having the relative angles b2 much lower than b2b.

Figs. 12–14 show the values of Wt, cu2=u2 and b2, respectively, for all the selected configurationsat all the considered flow coefficients and are discussed in the following.

3.2. Total efficiency

For a generic configuration, the total efficiency at low flow rates is poor due to the significantvolumetric losses. In fact, the ratio between the throughflow and the overall flow rate through theimpeller is low because of both the fractions of the flow recirculating around the vortex core andreturning to suction (Figs. 12 and 13). At intermediate flow rates, the flow coefficient being around

Fig. 12. The local total pressure coefficient for the selected configurations at all flow coefficients.


Fig. 13. The ratio cu2=u2 for the selected configurations at all flow coefficients.


Fig. 14. The relative velocity angle b2 for the selected configurations at all flow coefficients.



0.6, these fractions are noticeably reduced while, on the other hand, the variation of the tangentialcomponents of the velocity across the impeller is increased for all the streamlines (Fig. 13). Asubstantial equilibrium, therefore, occurs between the reduction of the volumetric losses, whichtends to increase the total efficiency, and the hydraulic efficiency drop due to the increase of vortexeccentricity and strength. This equilibrium results in a maximum in the gt–U curve (Fig. 6).Finally, when the flow coefficient approaches 1.0, the further reduction of the volumetric lossesdoes not counterbalance the further decrease of the hydraulic efficiency due to both the increase inthe variation of the tangential velocity components and the worse incidence conditions that makestall arise in a large portion of the suction arc (Fig. 14).

The highest total efficiencies, which are obtained using thick vortex walls matched with smallradial width rear walls, can be justified considering the reduced extension of the suction arc andthe larger space occupied by the thick vortex wall itself. These geometrical features result in adrastic improvement of the volumetric efficiency (Figs. 12 and 13). In fact, at low flow rates thetotal efficiencies obtained with the configurations RE-S3H1 and RE-VE are at least twice thoseobtained with any other configuration having the thin vortex wall (Fig. 6). Moreover, thehydraulic efficiency is enhanced as well, since the vortex is less eccentric and the distribution of therelative velocity angle at impeller inlet is more uniform (Fig. 14), preventing stall phenomena atintermediate flow rates.

Conversely, in the configurations with the thin vortex wall, the wider extension of the suction arcresults in an increased portion of the flow returning to suction and the small vortex wall thicknesscauses a stronger recirculation which involves a flow rate that is nearly equal to the throughput onefor U ¼ 0:6 (Figs. 12 and 13). The volumetric losses are, therefore, the main reason for the poorertotal efficiencies obtained with these configurations. Moreover, incidence losses play a major role inmaking the total efficiency decrease at the highest flow rates. Although the angles of the relativevelocity increase with the flow coefficient in the portion of the suction arc adjacent to the vortexwall, thus improving incidence conditions, a larger and larger fraction of the throughflow ap-proaches the blade row at higher incidence angles in the portion of the suction arc that is closer tothe rear wall for U P 0:8 (Fig. 14). The combinations of the small radial width rear wall with thethin vortex wall (configurations RE-H1 and RE-H2) are particularly unfavorable because of thelittle space that is available to the vortex and the consequent low values of both volumetric andhydraulic efficiencies. A better total efficiency is obtained, in fact, with the highest positions of thevortex wall (Fig. 6). When intermediate and large radial widths of the rear wall are considered, bothlarger radial width rear walls and higher positions of the thin vortex wall tend to exert a negativeinfluence on the total efficiency because they allow the recirculating flow to expand inside themachine. This expansion results in an increased rate of the recirculating flow and in an increase ofthe energy transferred to it by the impeller (Fig. 13). These negative effects are overweighted by thereduction of the flow rate returning to suction and the associated energy. On the other hand, thedecrease of the relative angles of the velocity at impeller inlet result in worse incidence conditionsfor both larger radial width rear walls and higher positions of the vortex wall (Fig. 14).

3.3. The shape of the characteristic curve

The variation of the tangential component of the velocity increases with the flow coefficient forall the configurations, as expected from the simplified one-dimensional theory on cross-flow fan


operation formulated by Eck [3]. However, different shapes of the Wt–U characteristic curves areexperimentally and numerically obtained, as already mentioned in Section 2.1 (see also [1]),depending on the rear wall radial width.

In the configurations having a small radial width rear wall, the vortex is forced to be not verystrong and eccentric at low flow rates. In such a flow field pattern the tangential components ofthe velocity at impeller exit are small (Fig. 13). Therefore, the total pressure coefficient cannot behigh, in spite of the high hydraulic efficiency due to the low eccentricity and strength of the vortex.As flow rate increases, the vortex gets stronger and moves towards more eccentric positions. Thisresults in an increasing variation of the tangential components of the velocity, which produces anascending characteristic curve at least until the hydraulic efficiency remains high (Figs. 6 and 13).

For intermediate radial widths of the rear wall, the vortex is already strong and eccentric en-ough at low flow rates to form a flow field in which the variation of the tangential components ofthe velocity are high, generating high values of Wt (Fig. 13). As flow rate increases, the charac-teristic curve is almost constant because of the substantial equilibrium between the effects of astronger and more eccentric vortex (which tend to increase the total pressure coefficient) and thehydraulic efficiency drop due to the more complex interaction between the recirculating flow andthe blade row.

Finally, for large rear wall radial widths, the negative effects of the diminished hydraulic effi-ciency, which also depends on the worse incidence conditions with the arising of stall phenomena,prevail over the further increase of the variation of the tangential components of the velocity dueto the highest vortex strengths and eccentricities, and result in a descending trend of the char-acteristic curve (Figs. 6 and 13).

Higher positions of the thin vortex wall tend to make the characteristic curve more stablebecause they generate stronger and more eccentric vortices. Moreover, the entire flow field followsthe clock-wise movement of the vortex. This results in a larger number of streamlines having lowenergy, because of the unfavorable incidence conditions, that are included in the throughflow.These streamlines would otherwise return to suction for lower positions of the vortex wall.

3.4. The maximum flow rate

The values of the maximum flow rate obtained for different combinations of the most signifi-cant design parameters can be justified considering that larger discharge sections, which are ob-tained for larger radial width rear walls and/or higher positions of the vortex wall, result indecreasing mean velocities at fan outlet. The curve of the static pressure coefficient, therefore,increases, tending to approach the curve of the total pressure coefficient, and this makes the flowrate at free blowing increase. However, these modifications of the geometric configuration lead toa lower energy transfer as well. Consequently, an optimal combination of the design parametersexists for which the fan flow rate at free blowing has a maximum value.

4. Conclusions

The present study on cross-flow fan operation confirms the tendencies that were high-lighted in the experimental investigations by the same group of authors [1,9,10]. The influence


of the most significant design parameters on performance and efficiency, which was simplyoutlined in [1], and on the flow field pattern, which was examined in [9] inside the impel-ler, has been analyzed here in details and the flow fields determined by representative com-binations of the design parameters have been theoretically linked to fan performance andefficiency through a streamline by streamline discussion. The ‘‘circle’’ of relationships existingamong:

• design parameters (the radial width of rear wall is the most important, followed by the positionand the thickness of the vortex wall);

• flow field pattern (mainly characterized by the strength and the position of the vortex coreinside the impeller);

• performance (total pressure coefficient, total, volumetric and hydraulic efficiency, maximumflow coefficient);

is now closed. This is the basic step towards the formulation of a general theory on cross-flow fanoperation.

Acknowledgements

The author wishes to thank Prof. Andrea Lazzaretto and Prof. Antonio Dario Martegani forthe helpful discussions and their useful suggestions.

References

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[12] Fluent v5.4 User’s Manual. Fluent Inc., Lebanon, NH.

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[14] ISO 5801, Industrial fans––Performance testing using standardized airways, 1993.

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