### Introduction

Forced air cooling remains essential for dissipating the heat generated by high-power-density devices in modern electronics [1]. Axial fans are widely used in information technology, telecommunications, automotive, and aerospace applications for air cooling. The aerodynamic performance (P-Q) curve of fans is essential for designing and optimizing air cooling systems. However, fan performance modeling can be challenging and is dependent on numerous factors that must be accurately accounted for. The lumped fan (LF) model is a well-established practice for modeling electronic enclosures with forced convection cooling, but it simplifies fan geometries as a plane, making it difficult to predict swirl velocity profile and fan performance precisely, leading to some margin of error in thermal-flow designs [2]. Although higher fidelity three-dimensional computational fluid dynamics (CFD) methods, such as the steady/unsteady Reynolds-averaged Navier-Stokes approach (U/RANS), and large eddy simulation (LES), have been proven more reliable in modeling fan performance, the high computing resource demand limits their use in rapid industrial design and optimization cycles [3].

This study aims to develop and validate an analytical model for the fast and accurate prediction of the aerodynamic performance (P-Q) curve of axial cooling fans, accounting for actual fan geometry and the tip clearance effect. This prediction tool is anticipated to be highly appropriate for prompt design, analysis, and optimization of electronics cooling fans and enclosures, thereby facilitating future co-design of thermal-flow and noise.

### Analytical Model

An analytical model for predicting the aerodynamic performance of fans was previously introduced by the authors in [4], [5]. This quasi-3D method was developed based on the lifting line theory combined with the blade element theory (BET) and the viscous vortex panel solver XFOIL [6]. The method is parametric and utilises the actual blade geometry while accounting for tip loss. It assumes minimal radial flow, allowing the fan blades to be con- sidered as a series of 2D radial elements. The shroud improves fan performance by suppressing tip vortices, and the method treats this suppression as an increase in sectional lift coefficients. The key equations of the analytical model are presented in this section, and it was implemented in MATLAB.

*Figure 1* illustrates the velocity triangles of a typical fan element. In practical applications, downstream stationary objects such as streamlined stators/guide vanes or non-streamlined struts are often present. This study models the downstream structure as an ideal stator with an imaginary shape that can fully recover the pressure loss due to the rotational velocity component, which is equal to ρv_{u}^{2}/2.

The thrust* dT* and torque *dM* for a radial element *dr* can be expressed as:

where C* _{l}*, C

*are the lift coefficient and the drag coefficient, respectively.*

_{d}*r, Ω, B*and

*c*denote the blade radius, rotational speed, number of blades and chord length, respectively.

*á*is the tangential induction velocity factor (u

_{2}= Ωr(1 –

*á*)), which can be expressed as:

where *w _{1}* is the total relative velocity at the blade leading edge.

The root and tip loss for a finite-span blade were accounted for by assuming an elliptical lift distribution along the span. Based on the lifting line theory, the induced angle of attack αi can be determined as:

The downwash resulting from the tip vortices exerts a diminishing effect on the lift coefficient slope of an airfoil with an infinite span. Therefore, the finite airfoil lift coefficient C_{l} is expressed as:

where C_{l,∞} is the lift coefficient of a 2D airfoil with infinite span, obtained by XFOIL [6]. m is the slope of C_{l,∞} with respect to α_{2}. Lift and drag coefficients are calculated by polynomial interpolation from a pre-computed table, based on *α<sub2*. To ensure numerical stability, all aerodynamic data were computed using XFOIL in the range of -20o to +20o, extended to ±180o with the Viterna method [7]. XFOIL uses the e^{n} method to predict laminar-to-turbulent flow transition [6]. The parameter N_{crit}=5 was utilized to represent relatively high turbulence inflows.

By substituting *Equation 4* with *Equation 5*, and solving iteratively, the effective angle of attack α for each blade element can be determined. Then the forces and velocities can be obtained by solving *Equation (1) – (3).*

The static to total pressure can be determined as:

where, *T* is the total thrust of the rotor. *A* is the fan area.

This method considers the impact of tip clearance on fan performance using the effective aspect ratio *AR _{eff}*, which incorporates the correction factor

*M*with the geometric aspect ratio

_{AR}*AR*. A shroud can mitigate tip vortices and improve blade performance, but its effectiveness decreases with increasing tip clearance. To account for this, the tip clearance effect is modeled as an increase in

*M*

_{AR, }which can be expressed as:

*Equation 7* is an empirical equation calibrated based on several CFD studies and experiments. *T _{R}* is the tip clearance ratio which is defined as T

*, and it has a valid range from 0% to 10%. t and Rt denote the tip-gap width and the blade tip radius.*

_{R}=t/R_{t}The theoretical fan performance based on Euler’s work equation can be used for comparison with the developed method [8]. Assuming flow alignment with the geometrical camber line and ideal stator recovery of downstream pressure loss, the ideal static to total pressure can be expressed as [8]:

where v_{u}=u–vcotφ2. φ2 can be approximated as the angle between the tangent of the camber line at the trailing edge and rotor plane.

### Experimental Setup

To experimentally validate the proposed fan model, a wind tunnel was developed based on ANSI/AMCA 210-16 [9], as shown

in *Figure 2*. A modification was made to the setup by using a standardized orifice plate, in accordance with ISO 5167-2:2003 [10], to measure airflow rates. This method provides high accuracy and is easier to implement compared to the Pitot tube method used in the original AMCA standard. A honeycomb flow straightener is used to rectify non-axial airflow, and a throttling device at the end of the tunnel enables control of the fan backpressure. Due to the presence of system resistance inside the tunnel, the static pressure around the fan outlet cannot reach the free delivery condition (zero backpressure). In this case, an auxiliary fan could be added as an exhaust system. The total volumetric flow rate, Q, and the static-to-total pressure rise across the test fan, P, can be obtained following the ANSI/AMCA standardized procedures [9].

### Case Study

An 80 mm diameter cooling fan with 5 blades and a blade tip clearance ratio of 2.5% was selected to validate the aerodynamic model. The fan, which operates at 9000 RPM, is used in a commercial multiple hard disk drive enclosure and has been previously studied in [4], [5]. The blade tip radius and hub radius are 37.35 mm and 18.7 mm, respectively, while the tip Mach number is approximately 0.1 and the averaged chord-based tip Reynolds number is around 60,000.

The blade was divided into sections and twist and chord distributions were extracted from a CAD file (see *Figure 3*). Airfoil profile coordinates were non-dimensionalized to calculate aerodynamic coefficients. In cases where CAD files are unavailable, 3D scanning can be used to extract blade information. If none of these options is available, airfoil databases such as [11] may be used to find a similar airfoil based on maximum camber and thickness measurements. For the test case, the mid-span airfoil profile had a maxi- mum thickness of 6.7% at 23.6% chord and a maximum camber of 2.2% at 49.0% chord. This information aided in identifying similar airfoils in [11], specifically AG26 and AG17, which were selected to

assess the model’s sensitivity to airfoil coordinates.

To determine the optimal aspect ratio correction factor *M _{AR}*,

*Equation 7*was employed with a tip clearance ratio of

*T*= 2.5%, resulting in a value of 4.6. The fan performance curve (P-Q) was calculated using the analytical model outlined in Section II.

_{R}*Figure 4* shows the predicted P-Q curves with the optimal correction factor (MAR = 4.6) using the three similar airfoil profiles, compared to the reference data provided by the manufacturer, experimental data. Reasonable agreement is observed between the predicted results using the real fan airfoil profile and the experimental data, except for some discrepancies near and after the stalling dip, as shown in *Figure 4*. The normalized root-mean-square error (NRMSE) is 5.0% in the nominal working region, while it increases to 13.0% in the stall region. The flow becomes unstable and separates in the stall region, increasing the radial flow component, causing the axial fan to behave like a mixed- flow fan [12] and invalidating the zero-radial flow assumption. Although this XFOIL [6] based model can partially predict near-stall and post-stall fan performance, it is less accurate in the stall region due to the absence of radial flow consideration. Moreover, *Figure 4* demonstrates that using similar airfoil profiles (AG26 and AG17) in this method can provide comparable predictions except for the stalling dip region, which appropriately relaxes the minimum requirement of this model for airfoil profile data.

*Figure 5* depicts the impact of effective aspect ratio correction factor on predicted P-Q curves for different M_{AR} values (1 ≤ M_{AR},

≤ ∞) in comparison to the reference, experimental, and ideal curves. Generally, an increasing M_{AR} value leads to higher predicted fan performance. Specifically, M_{AR} = 1 indicates a fan with finite span and large tip clearance, while M_{AR} → ∞ approximates an ideal fan curve for a fan with infinite span and zero tip clearance. The proposed aspect ratio correction factor, MAR, demonstrates promising analytical capabilities in accounting for finite

span and tip clearance effects in predicting fan curves.

The proposed aerodynamic model can be seamlessly integrated with the aeroacoustic model proposed in [4], [5] to predict far-field noise levels. Figure 6 shows the noise prediction results in comparison to experimental results. The model presents reasonable predictions that capture the fundamental trends and exhibit a good match in the high-frequency range above 2 kHz. However, the combined model is still under development, and additional outcomes will be published in the future.

The combined model takes approximately 10 minutes on an Intel Xeon E5-2630v3 CPU utilizing 8 cores (1.3 core-hours), which is more than three orders of magnitude faster than the high-fidelity method (9,600 core-hours) in [3].

### Conclusions

In this study, an analytical model was presented to analyze the aerodynamic performance of electronics cooling fans in a computationally efficient and accurate manner. The model accounts for the actual fan geometry and the impact of tip clearance. This model was validated against experimental tests conducted on a standardized test facility, yielding a prediction error within 5% in the fan’s nominal working conditions. Overall, the proposed analytical model can reasonably predict fan performance with an ultra-low computational cost. This model could be beneficial in accelerating and optimizing the thermal-flow design of fan-cooled electronics systems, while also offering valuable insight into future noise control.

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