Cross-flow Fan Research Articles

Bionic microstructure design of cross flow fan for high aerodynamic and low noise performances

In this paper, a novel approach was introduced for cross flow fan design by incorporating bionic microstructures on the blade surface. Two types of bionic microstructures, riblet and convex hull, were constructed and studied to achieve high aerodynamic efficiency and low noise design. Numerical simulations were conducted to understand the influence of bionic microstructures on the fan’s aerodynamic and acoustic performance, and the impact of design parameters on these performances was investigated. After printing the bionic microstructure fan samples, the experimental tests were carried out. The flow field and sound field data of the fans with bionic microstructures and the original fan were compared. The results demonstrated that both riblet and convex hull microstructures effectively improved the fan’s aerodynamic performance by reducing turbulent kinetic energy on the blade surface. Additionally, the microstructures inhibited boundary layer separation on the suction surface of adjacent blades, which contributed to reducing vortex noise. However, the pressure gradient formed near the convex hull and the edge of the small-diameter riblet resulted in deterioration in the noise of the corresponding fan. By optimizing the diameter and spacing of the riblet microstructure, the best noise reduction performance was achieved for the riblet fan. The optimized fan showed a 5.3% increase in maximum air volume flow rate and a 2.4 dB(A) reduction in noise. Overall, the cross flow fan design method based on bionic microstructures has valuable application potential in improving the aerodynamic and acoustic performance of various rotating machinery.

Computational Fluid Dynamics-Based Optimisation of High-Speed and High-Performance Bearingless Cross-Flow Fan Designs

To enhance the fluid dynamic performance of bearingless cross-flow fans (CFFs), this paper presents a CFD-based optimisation of both rotor and static casing wall modifications. High-performance CFFs are essential in industrial applications such as highly specialised laser modules in the semiconductor industry. The goal for the investigated rotor modifications is to enhance the CFF’s mechanical stiffness by integrating reinforcing shafts, which is expected to increase the limiting bending resonance frequency, thereby permitting higher rotational speeds. Additionally, the effects of these rotor modifications on the fluid dynamic performance are evaluated. For the casing wall modifications, the goal is to optimise design parameters to reduce losses. Optimised bearingless CFFs benefit semiconductor manufacturing by improving the gas circulation system within the laser module. Higher CFF performance is a key enabler for enhancing laser performance, increasing the scanning speed of lithography machines, and ultimately improving chip throughput. Several numerical simulations are conducted and validated using various commissioned prototypes, each measuring 600mm in length and 60mm in outer diameter. The results reveal that integrating a central shaft increases the rotational speed by up to 42%, from 5000rpm to 7100rpm, due to enhanced CFF stiffness. However, the loss in fluid flow amounts to 61% and outweighs the gain in rotational speed. Optimising the casing walls results in a 22% increase in maximum fluid flow reaching 1800m3/h at 5000rpm. It is demonstrated that the performance of bearingless CFFs can be enhanced by modifying the geometry of the casing walls, without requiring changes to the CFF rotor or bearingless motors.

Operation Mechanism Analysis and Parameter Optimization of Bean Impurity-Separation Device for Tiger Nut Harvester

This study aimed to enhance the efficiency of tiger nut combine harvesters by reducing impurity and loss rates during processing. Scholars focused on analyzing the composition and suspension speed of the bean mixture, leading to the development of a wind-screen impurity-removal method. The wind-screen-type bean-separation device was designed with a cross-flow fan, louver screen, frame, and driving mechanism. Theoretical analysis was employed to discuss the motion characteristics and behavior of the sieve body and material, thereby revealing the screening dynamics of tiger nuts and impurities. Factors such as crank radius, crank speed, and fan speed were identified as crucial for optimizing separation performance. Initial single-factor tests helped narrow down the range of influencing factors. Subsequently, a three-factor, three-level Box–Behnken test was conducted with crank radius, crank speed, and fan speed as variables and impurity rate and loss rate as evaluation indexes. This led to the establishment of a multiple regression equation linking these factors to the evaluation indexes. Through response surface analysis and multi-objective optimization using the regression model, the optimal operational parameters for the device were determined: crank radius of 45 mm, crank speed of 497 r/min, and the fan speed of 1100 r/min. Theoretical calculations predicted an impurity rate of 2.42% and a loss rate of 0.51%. Verification tests confirmed these findings, showing an average impurity rate of 2.53% and a loss rate of 0.56%, which met the mechanized harvesting standards for tiger nuts. Overall, this study introduces a novel method and technical framework for effectively separating tiger nuts from impurities, thereby advancing the mechanization of tiger nut harvesting processes.

Numerical Simulation and Experimental Study on the Aerodynamics of Propulsive Wing for a Novel Electric Vertical Take-Off and Landing Aircraft

The electric vertical take-off and landing (eVTOL) aircraft offers the advantages of vertical take-off and landing, environmental cleanliness, and automated control, making it a crucial component of future urban air traffic. As competition intensifies, demands for aircraft performance are escalating, including forward flight speed and payload capacity. The article presents a novel eVTOL design with propulsive wings and establishes methodologies for propulsive wing unsteady numerical simulation and wind tunnel experiments, analyzing its aerodynamic characteristics and lift enhancement mechanism. The results indicate that the cross-flow fan (CFF) provides unique airflow control capabilities, enabling the propulsive wing to achieve remarkably high lift coefficients (exceeding 7.6 in experiments) and propulsion coefficients (exceeding 7.1 in experiments) at extreme angles of attack (30°~40°) and low airspeeds. On the one hand, the CFF effectively controls boundary layer flow, delaying airflow separation at high angles of attack; on the other hand, the rotation of the CFF induces two eccentric vortices, generating vortex-induced lift and propulsion. The aerodynamic performance of the propulsive wing depends on the advance ratio and angle of attack. Typically, both lift and propulsion coefficients increase with the advance ratio, while lift and drag coefficients increase with the angle of attack. The propulsive wing shows significant advantages and prospects for eVTOL aircrafts in the low flight velocity range (0–30 m/s).

Hybrid lattice Boltzmann method using Cartesian and body-fitted grids for turbomachinery aeroacoustic simulations

In this study, we developed a hybrid approach using the lattice Boltzmann method (LBM) and finite-volume lattice Boltzmann method (FVLBM) to perform efficient aeroacoustic simulations with moving boundaries. An entire domain, including the flow and acoustic fields, was computed using the standard LBM with a Cartesian grid. Local domains around the moving objects were computed using the FVLBM with body-fitted grids. These simulations were coupled using the direct forcing method to consider moving boundaries. The hybrid method was validated for several problems including turbulent flows and flow-induced sounds under low-Mach-number conditions. These validation problems covered flows with Reynolds numbers up to 2.0×105 and Mach numbers less than 0.2. In the simulations of the aeolian tone generated from stationary and rotating cylinders, the hybrid method results were consistent with those of the conventional methods. In the simulation of the turbulent flow and broadband sound of the isolated airfoil, the hybrid method was 15 times faster than the standard LBM and produced results consistent with the experimental results. Furthermore, the application of a cross-flow fan demonstrated that the hybrid method successfully simulated the flow and acoustic fields around the rotor.

<b>Evaluation Method of Passenger Thermal Comfort Considering Effects of Airflow by Cross-flow Fan in Commuter Vehicles in Summer</b>

<b>Evaluation Method of Passenger Thermal Comfort Considering Effects of Airflow by Cross-flow Fan in Commuter Vehicles in Summer</b>

Supercritical Operation of Bearingless Cross-Flow Fan

This paper presents a decoupled bearingless cross-flow fan (CFF) that operates at a supercritical speed, thereby increasing the maximum achievable rotational speed and fluid dynamic power. In magnetically levitated CFF rotors, the rotational speed and fan performance are limited by the bending resonance frequency. This is primarily defined by the low mechanical bending stiffness of the CFF blades, which are optimised for fluid dynamic performance, and the heavy rotor magnets on both rotor sides, which add significant mass but a minimal contribution to the overall rotor stiffness. This results in detrimental deformations of the CFF blades in the vicinity of the rotor bending resonance frequency; hence, the CFF is speed-limited to subcritical rotational speeds. The novel CFF rotor presented in this study features additional mechanical decoupling elements with low bending stiffness between the fan blades and the rotor magnets. Thus, the unbalance forces primarily deform the soft decoupling elements, which enables them to pass resonances without CFF blade damage and allows rotor operation in the supercritical speed region due to the self-centring effect of the rotor. The effects of the novel rotor design on the rotor dynamic behaviour are investigated by means of a mass-spring-damper model. The influence of different decoupling elements on the magnetic bearing is experimentally tested and evaluated, from which an optimised decoupled CFF rotor is derived. The final prototype enables a stable operation at 7000 rpm in the supercritical speed region. This corresponds to a rotational speed increase of 40%, resulting in a 28% higher, validated fluid flow and a 100% higher static pressure compared to the previously presented bearingless CFF without decoupling elements.

An innovative air-cooling system for efficiency improvement of retrofitted rooftop photovoltaic module using cross-flow fan

This study presents an innovative air-cooling photovoltaic (PV)system using cross-flow fan with speed regulation to optimize performance of rooftop PVsystem in tropical climates like Malaysia. Air passed through the impeller enters perpendicularly to the motor shaft, deflected by the fan blades and evacuated, allowing the fan to operate at its most efficient operating point. The airflow provided within the rear of the PV modules and the roof surface blow out the trapped hot air. Changes in the module temperature (Tcell) are detected and the fan speed are adjusted accordingly to the PWM. This method was tested for 12 hours continuously from 7:00 am on the existing PV system at German Malaysian Institute (GMI) Bangi. The highest Tcell achieved 72.88 °C and 55.75°C without and with air-cooling system with average power 210.22 W and 246.67 W per peak sun factor (PSF) respectively. There was a 17.34% increase in average power with a 13.18% in average net output power and achieved 6.68% energy efficiency using the proposed cooling system. Tcell increases more swiftly and reaches higher temperatures in the absence of a cooling system, whereas Tcell increases more slowly and at lower temperatures when a cooling system is present. The projected system's power rating was 6.48 W, which is 2.6% per PV module, and it really attained 6.32 W, which is 2.53% per PV module, while total energy consumption by the fan was 51.89 Wh per day, which is only 3.89% per PV module.

Research on Aerodynamic Test Validation and the Vector Force Control Method for an E-STOL Fan Wing Unmanned Aerial Vehicle

The concept of the Fan Wing, a novel aircraft vector-force-integrated device that combines a power unit with a fixed wing to generate distributed lift and thrust by creating a low-pressure vortex on the wing’s surface, was studied. To investigate the unique propulsion mechanism of the Fan Wing, a Fan Wing test platform was developed, and experiments were conducted in a wind tunnel. At the same time, numerical simulations were established. In order to further improve the aerodynamic efficiency of the Fan Wing and decouple the control of lift and thrust, an improved scheme for the leading-edge structure of the Fan Wing was proposed, and a numerical analysis was conducted. A Fan Wing unmanned aerial vehicle (UAV) was designed and manufactured using the Fan Wing as the source of lift and thrust for the aircraft, and flight verification was conducted. The wind tunnel tests have proven that the main factors influencing the lift and thrust of the Fan Wing are rotation speed of cross flow fan, angle of attack, and incoming flow. The numerical analysis results of slotting on the leading edge show that the lift and thrust of the Fan Wing can be improved, but also the strength and position of the low-pressure vortices can be controlled. The results of flight tests show that the distributed lift and thrust of the Fan Wing can be directly applied to aircraft without the need for additional propulsion devices. In summary, the aerodynamic characteristics of the Fan Wing can be applied to electric short takeoff and landing (E-STOL) scenarios in urban air traffic.

Numerical study on the hydrodynamic performance and wake dynamics of propulsive wing propulsors with different cross-flow fans

The propulsive wing propulsor (PWP), which means an underwater thruster equipped with a wing, a cross-flow fan (CFF), and a deflector, is capable of generating both horizontal thrust and vertical lift, thus enhancing the maneuverability of underwater vehicles and serving as a propulsion device. The hydrodynamic performance of the PWP is significantly influenced by the blade number it possesses. An unsteady numerical method based on the Reynolds-averaged Navier–Stokes equations was developed to examine the impact mechanism of blade number on the hydrodynamic performance, load fluctuation, and wake evolution of the PWP. The results indicate that as the blade number increases, the hydrodynamic forces, power, and propulsive efficiency of the PWP gradually increase. When the blade number exceeds 26, the performance of the PWP tends to stabilize. Insufficient blades can lead to turbulence in the internal flow of the CFF, intensifying interference between blade vortices, resulting in secondary peaks and frequency-domain bifurcations in hydrodynamics. With an increasing blade number, disturbances to the blade vortices decrease, enhancing the periodicity of PWP hydrodynamic fluctuations, but there may be an increase in high-frequency noise levels. The wake modes of the PWP undergo four transitions: double vortex pair mode, single vortex pair mode, single vortex pair + single vortex mode, and vortex strip mode. Disturbed blade vortices promote the transition of vortex pair shedding modes in the PWP wake, thereby causing variations in the periodicity of PWP hydrodynamics. Excessive amplitude and frequency may lead to structural fatigue damage in the PWP.

Direct aeroacoustic simulation of a cross-flow fan with Helmholtz resonator using lattice Boltzmann method

It is necessary to reduce the aerodynamic sound generated by fans since they are often used for air-conditioners, which are placed in human living areas, such as homes and offices. To reduce the aerodynamic sound, we focused on the Helmholtz resonator. To develop a resonator-type silencer for the aerodynamic sound of air-conditioners, the present study investigated the flow and acoustic fields around a cross-flow fan with a resonator and the effectiveness of the resonator for blade-passing frequency (BPF) sound using aeroacoustic simulations. As a numerical method, we used the lattice Boltzmann method (LBM), which can simultaneously simulate aerodynamically sound generation in the fan and the acoustic resonance in the resonator. Consequently, the simulation results showed that a reduction of 2.9 dB for the BPF sound can be realized by installing a resonator in the tongue part of the fan casing.

АНАЛИЗ КОНСТРУКЦИЙ ПНЕВМОСЕПАРИРУЮЩИХ СИСТЕМ

One of the main methods of separating a grain heap is pneumatic separation based on the difference in the aerodynamic resistance of the components of the grain mixture to the air flow. A classification of pneumatic separating devices is given according to the following parameters: method of air flow circulation, number of separating channels, method of using systems, direction of air flow in separating channels and chambers, method of supplying air to the pneumatic separation zone, frequency of processing, shape of pneumatic separating channels, method of feeding grain mixture into the channel , type of devices for separating light impurities, method of regulating air flow in the channel. The formulas developed by different authors for the coefficient of particle resistance to air flow are generalized. The optimal dimensions of the pneumatic channel have been established: the depth of the channel should not exceed 200 mm, and the width should not exceed 1500 mm. It has been established that the most commonly used shape of the pneumatic channel is rectangular. It is shown that various devices are used to introduce grain material into the pneumatic channel: pitched boards, support nets, corrugated rollers, spreading discs, vibration, belt, pneumatic, and combined devices. According to the direction of the air flow, separators are distinguished: vertical, inclined, horizontal. It has been established that louvered inertial dust collectors are usually used to clean the air from dust. The most modern designs of cross-flow fans are considered. Based on the analysis performed, recommendations were given to developers of aspiration systems: to ensure a margin of pneumatic channel width so that with an increase in the supply of grain material, the efficiency of air cleaning does not decrease, but at the same time the uniformity of the air flow in the pneumatic channel does not decrease.

Full-scale numerical simulation and experimental validation of the thermal environment under the heating mode in an air-conditioned room

The numerical simulation of the air flow of the room equipped with the crossflow fan air conditioners has been conducted a lot. However, the simulation of the air flow distribution under the heating mode has always been difficult to match the experimental data well, due to the fact that both the air temperature and the flow are affected by the buoyancy. In addition, very few full-scale simulations were conducted. To precisely predict the air flow and temperature distribution of the room with the crossflow fan air conditioner, a full-scale steady numerical simulation was conducted with the Computational Fluid Dynamics (CFD) package. The full-scale physical model was constructed strictly according to the real settings, including the detailed components of the air-conditioner, such as the crossflow fan and the heat exchangers. The Shear-Stress Transport (SST) k-ω turbulent model was selected to simulate the low Reynold flow near the inner surface of the room, and to simulate the high Reynold and transitional flow near the fan blades of the air-conditioner. The porous medium model was used to simulate the heat exchange fins in the air-conditioner. Furthermore, an experiment under the same condition was also conducted to verify the accuracy of the simulation. Results showed that the simulated supply air flow rate was 537.48 m3/h, producing a 4.97% deviation from the experiment result. Meanwhile, the upward movement of the supply air from the outlet of the air-conditioner was observed, leading to higher air temperatures in the upper region of the room. Furthermore, among all 210 measured temperature data points in the room, 149 data points had the deviation within ±5%, and 194 data points had the deviation within ±10%. 9 abnormal data points with greater deviation near the door were detected, which might be resulted from the outdoor cold air leakage through the crack between the door and the floor in the experiment. Conclusively, the full-scale numerical simulation in this study produced an accurate and reliable solution to predict the air temperature distribution in buildings, thus providing a potential method for the design or optimization of the air-conditioners.

Novel performance enhancement of straight blade vertical axis wind turbines through cross-flow fan integration

In this study, a novel technique for enhancing the output power of Vertical Axis Wind Turbines (VAWTs) is introduced through the integration of a cross-flow fan (CFF) for active flow control, a first-of-its-kind approach. The CFF, positioned on the airfoil's trailing edge, employs suction to regulate flow separation. A thick airfoil with a 35% thickness-to-chord ratio is utilized to accommodate the CFF. A VAWT, featuring a diameter of 1 m and a height of 0.5 m, is designed, fabricated, and subjected to experimental testing in a wind tunnel. Additionally, a standard NACA 0018 wind turbine with identical dimensions and solidity is manufactured for performance comparison purposes. A custom test rig is developed to measure turbine output, encompassing a fixation mechanism, torque measurement system, and a mechanical brake mechanism for load variation. The impact of the CFF on the wind turbine performance is investigated by measuring turbine static torque due to the fan, fan motor power consumption, turbine torque, and rotational speed across various wind speeds. Power assessments are conducted to estimate the real turbine power increase, taking into account the power consumed by the fans. A fan map is generated to determine the optimal conditions and rotational speeds for fan operation.

Comparative Evaluation of High-Speed Bearingless Cross-Flow Fan Designs for Lithography Excimer Lasers

This study conducted a comparative evaluation of two bearingless cross-flow fan designs for applications in deep ultraviolet lithography excimer lasers, where maximizing the speed and power of the fan has a direct influence on the throughput and scanning speed of these devices. Using bearingless motor technology enables a combined generation of bearing force and drive torque and leads to a compact, hermetically sealed, and conveniently maintainable drive system. With identical bearingless motors on both rotor sides, it is possible to drive the cross-flow fan symmetrically to high rotational speeds at low torsional loads. The rotor prototypes were optimised, analysed, and pushed to high-speed operation and evaluated with respect to their rotor dynamic and fluid dynamic performance using finite element methods and experimental measurements. For both prototypes, successful numerical studies were performed, where a modal analysis enabled theoretical predictions of expected resonance frequencies, and a CFD analysis visualised local flow effects and provided cross-flow fan design comparisons. A stable operation of up to 12,000 rpm and 5500 rpm was accomplished for the two elaborated designs.

Numerical Study on Fluid Dynamic Characteristics of a Cross-Flow Fan

This paper focuses on the application of a cross-flow fan (CFF) to generate propulsion for a submersible aircraft with a flying wing configuration. A numerical method is established to simulate the CFF operating both in the air and underwater. This paper then investigates the fluid dynamic characteristics of the CFF, including the velocity field, the pressure field, the cavitation distribution, the lift, and the thrust. It is concluded that proper lifts and thrusts can be obtained when the rotating speed and the angle of attack are reasonably designed. This work provides a valuable numerical methodology for studying the fluid dynamic characteristics of the CFF operating in different media and offers a technical basis for the selection of a motor system for submersible aircraft.

Intensification of thermal efficiency of a cross-flow heat exchanger under turbulent flow conditions using CuFe2O4/water nanofluid

The use of nanofluids has opened new intellectual horizons in improving the performance of heat exchangers. Therefore, in this work an experimental study was conducted for investigating the effect of CuFe2O4/water nanofluid on the heat transfer rate enhancement in a cross-flow type heat exchanger under turbulent flow conditions. Heat transfer rate were examined at different volume percentages of CuFe2O4 nanoparticle (0≤ φ ≤ 0.9% vol.) with a nominal diameter of 28 nm at three inlet fluid temperatures of 72, 80 and 88 °C, water pump rotational speeds of 2000 rpm and 3000 rpm, and cross-flow fan speeds of 1500, 2000 and 2500 rpm. According to the results, the heat transfer rate in the case of using nanofluids with a volume percentage of 0.1% was significantly higher than that of distilled water in all test conditions. The highest relative increase in the universal heat transfer coefficient (U) equal to 92.3%, compared to the use of water (φ = 0) was obtained at 0.1% vol% of nanoparticles, the lowest temperature of fluid entering the heat exchanger (72 °C), medium pump speed (2000 rpm) and low fan speed (1500 rpm). The maximum Performance Evaluation Criterion (PEC) improvement of 44.46% also occurred at this working condition.

Numerical investigation of the quasi-vortex-ring state of the propulsive wing in vertical descent

A multi-axis propulsive wing aircraft is a new type of vertical takeoff and landing aircraft. When the aircraft descends vertically, it enters a quasi-vortex-ring state (QVRS) similar to the vortex-ring state of the rotor. Based on the computational fluid dynamics method, the sliding grid technique was used to simulate and analyze the QVRS of a single propulsive wing in vertical descent. First, a computational method for the propulsive wing flow field was established, and the variation laws of the aerodynamic forces of the propulsive wing were analyzed. Then, the change in the flow field structure and wake of the propulsive wing was further studied, and the evolution mechanism of the QVRS of the propulsive wing was explained. Finally, recovery strategies for the QVRS and windmill–brake state (WBS) of the propulsive wing were proposed. The simulation results show that when the descent speed is less than the critical speed, the propulsive wing enters the QVRS. The interaction between the propulsive wing wake and reverse airflow provides an aerodynamic amplification of up to 17.3%; thus, the propulsive wing has satisfactory safety in the QVRS. When the descent speed is greater than the critical speed, the propulsive wing enters the WBS and its aerodynamic force is almost zero. The aerodynamic force of the propulsive wing can be increased by increasing the rotational speed of the cross-flow fan. Then, the descent speed of the propulsive wing slows down and the propulsive wing recovers from the dangerous state.

Development and evaluation of a fluidic facemask for airborne transmission mitigation

Recently, a fluidic facemask concept was proposed to mitigate the transmission of virus-laden aerosol and droplet infections, such as SARS-CoV-2 (COVID-19). This paper describes an experimental investigation of the first practical fluidic facemask prototype, or “Air-Screen”. It employs a small, high-aspect-ratio, crossflow fan mounted on the visor of a filter-covered cap to produce a rectangular air jet, or screen, in front of the wearer’s face. The entire assembly weighs less than 200 g. Qualitative flow visualization experiments using a mannequin clearly illustrated the Air-Screen’s ability to effectively block airborne droplets (∼100 µm) from the wearer’s face. Quantitative experiments to simulate droplets produced during sneezing or a wet cough (∼102 µm) were propelled (via a transmitter) at an average velocity of 50 m/s at 1 m from the mannequin or a target. The Air-Screen blocked 62% of all droplets with a diameter of less than 150 µm. With an Air-Screen active on the transmitter, 99% of all droplets were blocked. When both mannequin and transmitter Air-Screens were active, 99.8% of all droplets were blocked. A mathematical model, based on a weakly-advected jet in a crossflow, was employed to gain greater insight into the experimental results. This investigation highlighted the remarkable blocking effect of the Air-Screen and serves as a basis for a more detailed and comprehensive experimental evaluation.

Towards an Accurate Aerodynamic Performance Analysis Methodology of Cross-Flow Fans

Cross-flow fans (CFFs) have become increasingly popular in recent years. This is due to their use in several domains such as air conditioning and aircraft propulsion. They also show their utility in the ventilation system of hybrid electric cars. Their high efficiency and performance significantly rely on the design parameters. Up to now, there is no general approach that predicts the CFFs’ performance. This work describes a new methodology that helps deduce the performance of CFFs in turbomachinery, using both analytical modeling and experimental data. Two different loss models are detailed and compared to determine the performance–pressure curves of this type of fan. The efficiency evaluation is achieved by realizing a multidisciplinary study, computational fluid dynamics (CFD) simulations, and an optimization algorithm combined to explore the internal flow field and obtain additional information about the eccentric vortex, to finally obtain the ultimate formulation of the Eck/Laing CFF efficiency, which is validated by the experimental results with good agreement. This approach can be an efficient tool to speed up the cross-flow fans’ design cycle and to predict their global performance.