Inertial particle separation in helical channels: A calibrated numerical analysis

Size-based particle separation plays a key role in many biomedical and environmental applications¹ such as the separation of tumor cells^2,3 and leukocytes⁴ from blood samples, detection of pathogenic bacteria,⁵ and cell synchronization.⁶ Several microfluidics-based approaches have been developed for this purpose including dielectrophoretic,⁷ magnetophoretic,⁸ and acoustophoretic⁹ separation. More recently, inertial microfluidic methods have been developed for size-based separation. Unlike the other approaches, inertial microfluidics relies solely on particle and fluid properties to manipulate particles within the flow to induce separation and does not incorporate external forces.^10–12 

Particles flowing through a straight rectangular microchannel will migrate to distinct equilibrium positions based on their size. These equilibrium positions are determined by balancing a shear-gradient lift force, which moves particles toward the walls, and a wall-induced lift force, which shifts particles away from the walls.^10,12,13 The shear-gradient lift force results from the parabolic nature of the fluid velocity profile, where a relatively large particle will experience a shear gradient across its diameter and migrate normal to the bulk flow. The wall-induced lift force results from an increase in pressure between the particle and the channel wall as it approaches the wall. In these channels, up to four equilibrium positions may exist for each particle size, one on each wall,¹⁴ presenting challenges for the extraction of particles into separate outlets.

To aid in constraining particle equilibrium positions, secondary flows are often used to destabilize all but one position per particle size. Secondary flows result from the centrifugal forces generated by flows in serpentine,^10,15 spiral,^6,16–21 or helical^5,22–24 channels, generating a pair of counter-rotating vortices perpendicular to the bulk flow.²⁵ The Dean number, De, characterizes the strength of the secondary flow as

where D_h is the hydraulic diameter of the channel, R is the radius of curvature of the channel, and Re is the Reynolds number of the flow. The Reynolds number Re is defined by the fluid density, ρ_f, average fluid velocity, U, and fluid viscosity, μ, as

The equilibrium position of particles in channels with these secondary flows is dictated by a balance of the Dean drag force, which pushes the particles toward the outer wall, and an inertial lift force, which moves the particles toward the inner wall. The Dean drag force can be approximated as²⁶ 

where a is the particle diameter and U_Dean is the average secondary flow velocity given by

The inertial lift force is the resultant of the shear-gradient and wall-induced lift forces described above. The following expression was derived by Ho and Leal²⁷ to describe the inertial lift force acting on particles in circular channels for Re ≪ 1,

where U_max is the maximum fluid velocity in the channel and C_L is a lift coefficient that depends on Re, the lateral position of the particle in the channel, and the velocity gradient across the particle. Asmolov²⁸ extended the analysis of this expression for particles flowing between infinite parallel plates for different values of Re, up to ∼1000, based on the mean and local velocity gradients. This approach has been adopted to predict the inertial migration of particles in rectangular microchannels.¹⁰ In addition, Matas et al.²⁹ experimentally showed a dependence of Re and velocity gradient across a particle during inertial focusing. Investigators have conducted direct numerical simulations (DNS) to determine C_L as a function of the lateral position in rectangular channels.^30–32 Although DNS can accurately predict particle focusing locations, it is computationally expensive³³ and thus impractical for the rapid assessment of device design. In inertial microfluidics, typically Re < 100 and C_L may be assumed to be constant at 0.5 to provide a qualitative estimate of the inertial lift force.¹⁰ 

The ratio of the magnitude of the inertial lift force to the Dean drag force, R_f, serves as a useful qualitative tool when designing microfluidic devices for particle separation. Comparing Eqs. (3) and (5), it is seen that F_L increases to the fourth power of the particle diameter, while F_D is proportional to it. Therefore, particles with larger diameters will tend to equilibrium positions closer to the inner wall of a curved channel relative to particles with a smaller diameter. When R_f = O(1), the particle is predicted to be inertially focused and will occupy an equilibrium position near the inner wall of the channel. Based on this phenomenon, spiral channels have been developed for size-based particle separation where up to three distinct inertially focused particle diameters were extracted through separate outlets near the inner wall.^6,26

To improve the separation efficiency of the spiral channels, Guan et al.¹⁹ increased the distance between the equilibrium particle positions by using a trapezoidal channel. Dean flow vortices that are in the center of a rectangular channel are shifted toward the outer, wider, wall in a trapezoidal channel. Instead of separating particles based on inertial migration positions, particles were separated by their focusing state. Particles with R_f ≪ 1 are dominated by the Dean drag force, and as a result, they circulate within the Dean drag vortices near the outer wall and exit through an outlet at the top of the channel, referred to as the Dean drag state. As in rectangular channels, particles with R_f = O(1) occupy an equilibrium position near the inner wall and will exit through an outlet at the bottom of the channel, referred to as the inertial migration state. By designing the channel and flow parameters such that one particle has R_f on the order of one and the other has R_f much less than one, particles can be separated into streams on opposite channel walls, simplifying particle extraction at the end of the channel and improving device efficiency.

To improve control over particle focusing, Palumbo et al.²⁴ developed the first sheath-less device capable of separation and sorting in a trapezoidal helical channel. Unlike planar, spiral channels in which the channel radius must increase with each turn, helical channels offer a constant radius throughout the length of the channel, resulting in a constant Dean drag force along the length of the channel and ultimately a smaller device footprint.

Despite the advantages of helical and trapezoidal channels for size-based particle separation, few studies have been performed to investigate the impact of channel parameters on particle focusing capability. Current analytical design guides, such as R_f, neglect the potential effects of the channel pitch and taper angle on particle focusing. To date, only the flow rate tapered channel has been studied experimentally, where particles in planar, spiral channels were shown to transition from the inertial migration state to the Dean drag state with increasing flow rates.^19,21

The present paper presents a parametric study of the effect of particle diameter, channel pitch, channel diameter, taper angle, and channel depth and width on the focusing state of the particles in a helical channel using a simplified numerical model that relies solely on local flow parameters. The results of this analysis allow the ability to optimize microfluidic channel designs by understanding the effects of individual channel parameters.

A reference channel is shown in Fig. 1(a) with a width of 160 µm, a depth of 760 µm, a taper angle of 3.5°, a helical pitch of 575 µm, and a length of 100 mm was modeled in SolidWorks 2018 (Dassault Systèmes, Vélizy-Villacoublay, France) to match the experimental conditions described in Sec. II C. Additional channels were modeled by individually varying the channel pitch, diameter, taper angle, depth, and width from the reference channel dimensions. The flow fields in the helical channels were calculated by solving Eqs. (6) and (7) for the conservation of mass and momentum, respectively, in the computational fluid dynamics (CFD) software ANSYS Fluent 19.2 (ANSYS, Inc., Canonsburg, PA, USA),³⁴ where u is the fluid velocity vector, ρ_f is the fluid density, P is the static pressure, and μ is the fluid viscosity,

A steady-state incompressible laminar flow model was used since Re did not exceed 1000 in any simulation. The density and viscosity of water were used for the fluid properties. The fluid entered the channel with a constant velocity inlet and exited the channel with a zero-pressure outlet. All models were iterated until all velocity and continuity residuals were below 10⁻⁶.

Hexahedral cells were used to mesh the fluid domain in the direction of fluid flow. A mesh independence study was conducted for the reference channel to find an optimal fluid domain cell size of 2033 µm³ (maximum cell width and depth of 10 µm), which is used for all channel variations. Figure 1(b) shows the mesh of the reference channel at the inlet face.

Since particle volume fractions are very low (<1%) in applications of size-based particle separation,^{4–6,16,17,19,21,24} particle–particle interactions were neglected and one-way particle–fluid coupling was used. It is noted that particles in a microchannel may significantly alter the local flow field,^35,36 thus affecting the inertial lift and Dean drag forces. This interaction is ignored in the present work whose aim is a qualitative assessment of the influence of helical and trapezoidal channel geometry on the force balance using a computationally inexpensive model relying solely on local flow parameters.

Particle trajectories were calculated by injecting ∼1000 neutrally buoyant particles with zero initial velocity evenly across the inlet face of the solved flow fields using the Lagrangian discrete phase model and solving for the accelerations per unit particle mass, as described in the ANSYS Fluent Theory Guide³⁴ using the following equation:

The first term on the right-hand side represents the viscous drag force, where a, C_D, and Re_rel represent the particle diameter, drag coefficient, and relative Reynolds number, respectively. Properties with the subscript p refer to the particle. C_D was defined as

where a₁, a₂, and a₃ are constants derived by Morsi and Alexander.³⁷ Re_rel was defined as

The second term on the right-hand side of Eq. (8) represents the buoyancy force, where g represents the acceleration due to gravity and is neglected for the case of neutrally buoyant particles. The third term in Eq. (8) represents a virtual mass force to capture the acceleration of the fluid surrounding the particles, where t represents time and C_vm represents a virtual mass factor that was set to the default value of 0.5. The fourth term includes the inertial lift force, F_L, described in Eq. (5).

A user-defined function (UDF) was written to incorporate this term in the CFD analysis, where the inertial lift force was calculated using cartesian coordinates. C_L was calculated as a function of the local velocity gradient and tangential velocity magnitude (non-dimensionalized by the channel hydraulic diameter and maximum fluid velocity), and a fitting parameter, ϕ. ϕ was assumed to be constant and tuned to match the experimental results of the reference channel at Re = 76. Although it is more accurate to define a fitting parameter as a function of Re²⁸ and aspect ratio (AR),³¹ the experimental validation of channels with various Re and AR in Sec. III A shows good qualitative agreement. Therefore, the effect of Re and AR is deemed negligible for the purposes of this analysis. Re = 76 was used for all cases in the present study to disregard the effect of Re on ϕ. The resulting C_L neglects the influence of the wall-interaction force. As a result, the particles in the simulation may find equilibrium positions closer to the channel walls than in reality. As particles are injected uniformly across the inlet face, some trajectories may become trapped along the wall in the absence of the wall-interaction force. Since the present study is concerned with the particle focusing state at the end of the channels, only particle trajectories exiting the outlet were considered. A detailed description of C_L and a comparison to DNS data in rectangular channels with various aspect ratios reported by Liu et al.³² are presented in the supplementary material. Good agreement was observed between C_L used in the present study and the reported DNS data in the central portions of the channel, with an expected decrease in accuracy near the walls resulting from the neglect of the wall-interaction force.

A schematic of the forces acting on the particles is shown in Fig. 1(a). The effectiveness of particle separation was determined by examining the particle focusing states at the end of the channel.

To calibrate and validate the numerical model, a particle solution of 26 µm (std. dev. 3.0 µm) and 9.9 µm polystyrene microspheres (Microgenics, Fremont, CA, USA) were injected into three helical channels with a helical pitch of 575 µm, a taper angle of 3.5°, and various widths and depths at a flow rate of 2 ml/min using a syringe pump. The first channel, referred to as the reference channel (Fig. 1), had an AR of 4.8 (160 µm width × 760 µm depth). The remaining channels had an AR of 2.8 (230 µm width × 640 µm depth) and 3.1 (150 µm width × 465 µm depth). The particles were suspended at a concentration of 10 000 particles per ml for each particle size in deionized water with Tween 20 at 0.4 v/v% (Sigma Aldrich, St. Louis, MO, USA) to prevent particle aggregation. Particle locations relative to the inner wall of the channel at one turn from the outlet were measured using wide-field fluorescence microscopy (ZEISS, Oberkochen, Germany) by taking images over a range of focal elevations with a spacing increment of 20 µm. A detailed description of the experimental procedure is provided by Palumbo et al.²⁴ 

The fitting parameter, ϕ, was used to ensure that the CFD trajectories matched the experimental particle focusing state results as described by Palumbo et al.²⁴ at the outlet of the AR = 4.8 channel at a flow rate of 2 ml/min. Figure 2 shows the spatial distribution of these experimental results and 95% of the simulated particle trajectories as a function of ϕ at the end of the 10-loop channel, where 0 represents the inner wall and 760 represents the outer wall. Note that diameters of 23 μm and 26 μm were considered when modeling the large particle trajectories to account for the 3 μm standard deviation of the nominal 26 μm particles used in the experiments. It is seen that the small particles were predicted to remain entrained in the Dean vortices over the range of ϕ evaluated so that they congregated near the outer wall of the helical channel. At low ϕ, the inertial force was not large enough to push all large particles to the inner wall of the channel and separation was incomplete. Increasing ϕ increased the magnitude of the lift force on the particles so that larger particles moved closer to the inner wall, becoming more concentrated within a narrower range of channel heights. A ϕ of 0.10 provided the best agreement with the experimental results and was thus used for the present study.

The model was validated based on the particle sorting in an AR = 5.2 helical microfluidic device reported by Palumbo et al.²⁴ The model was further validated against the fluorescence microscopy data of Palumbo et al.²⁴ for channels with aspect ratios of AR = 2.8 and AR = 3.1 at a flow rate of 2 ml/min, resulting in Re = 79 and Re = 113, respectively, using ϕ = 0.10. Figure 3 shows good agreement between the observed and predicted focusing states for each channel using the constant fitting parameter assumption in channels with various Re and AR, where heights of 0 and 1 represent the wall closest and furthest from the helix axis, respectively. In the AR = 2.8 channel, all particles were entrained in the Dean vortices, in agreement with the simulated particle trajectories. In the AR = 3.1 channel, the large particles began to migrate toward the inner wall, though no separation was observed, in agreement with the simulated particle trajectories. In the AR = 4.8 channel, the large particles focused closer to the inner wall relative to the AR = 3.1 channel, and particle separation was observed, again in agreement with the simulated particle trajectories. The discrepancy between experimental and simulated particle positions in the channel may be attributed to a lack of particle–wall and particle–fluid interactions. Since C_L is a function of Re, the accuracy of the model to predict particle positions decreased as Re changed from the tuned conditions.

A channel length of 10-loops, or 100 mm, was chosen to match the length of the microfluidic devices used in the experiments. Particles with diameters of 9.9 μm, 23 μm, and 26 μm were released from the center of each cell on the inlet face of the reference channel to determine if the channel length was adequate for larger particles to be inertially focused near the inner wall of the channel. As seen from the particle trajectories in Fig. 4, the 9.9 μm particles became entrained in the Dean vortices and occupied positions around the Dean vortices, focusing in more compact streams with each additional turn. The combined 23 μm and 26 μm particle stream migrated to the inner channel wall where the stream position oscillated from 3 to 7 turns, at which point they reached equilibrium.

The geometric parameters of the reference channel (160 µm width × 760 µm depth, a taper angle of 3.5°, a helical pitch of 575 µm, and a total length of 100 mm) were varied to study their effect on the particle focusing state. Figure 1(a) shows the parameters studied and the axial and Dean flow fields in the reference channel at Re = 76. The CFD models were also used to predict the relative effects of the particle diameter on the focusing state.

As discussed in the Introduction, the inertial lift force acting on a particle with diameter a scales with a⁴, while the drag force is directly proportional to a. By exploiting this difference in scaling, particles can be separated based on their diameter.

Figure 5 shows the predicted positions for various particle diameters in the reference channel at Re = 76. Small particles become entrained in the Dean vortices and circulate the Dean vortex cores at the top of the channel; in the Dean drag state, particles remained in the Dean drag state until a critical diameter was reached, from 19 µm to 22 µm, where particles occupied most of the channel and were unable to be separated. This will be referred to as the transition state. In the transition state, some particles remain entrained in the Dean vortices, while others begin to migrate toward the inner wall. Particles in the transition state are unable to be separated. At particle diameters of 23 µm and beyond, inertial lift forces were sufficient to push all particles to the inner wall of the channel in the inertial migration state, where they may be separated from particles in the Dean drag state. These predictions suggest that the present reference device is capable of separating particles ≤16 µm and ≥23 µm at a flow rate of 2.0 ml/min (Re = 76). Particle diameters of 9.9 µm, 21 µm, and 26 µm are considered for the remainder of the present analysis to observe the effect of each parameter on particles in each of the focusing states. Particle diameters of 16 µm and 23 µm are also included to observe effects on particles near the limits of each focusing state.

Current analytical models for inertial particle separation have been developed for planar curving channels, such as spiral channels. In these channels, secondary flows are influenced by the radius of curvature of the channel. In contrast, liquids flowing through helical channels can involve significant momentum exchange along the helix axis in the y-direction in Fig. 1(a). The additional momentum exchange results in more complex secondary flows compared to the symmetric counter-rotating vortices observed in spiral channels.¹¹ In addition to the radius of curvature, secondary flows in helical channels are influenced by the helical torsion, defined as the rate of change of a helical curve in the direction orthogonal to the radius of curvature. Figures 6(a) and 6(b) show the secondary flow vectors at Re = 76 for helical pitches of 0.20 mm and 2.08 mm, respectively, assuming the reference channel dimensions. The channel with a 0.20 mm helical pitch shown in Fig. 6(a) had a helical torsion of ∼0.01 m⁻¹, and the influence of the pitch on the secondary flows was small. Two symmetric counter-rotating vortices were present near the outer wall, and the secondary flow near the inner wall was negligible. The channel with a 2.08 mm helical pitch shown in Fig. 6(b) had a helical torsion of ∼0.12 m⁻¹, one order of magnitude larger than the channel with a pitch of 0.20 mm. The effect of the additional momentum exchange due to the increased torsion of the channel became apparent near the inner wall of the channel. Additional secondary flow vectors were observed near the inner wall of the channel in the area where the larger particles migrated. A slight asymmetry in the vortices was also observed, though the vortex cores remained in the same location for each pitch analyzed.

Locations of particles with a diameter of 9.9 µm and 26 µm are shown in Fig. 6(c), while locations of 16 µm and 23 µm particles are shown in Fig. 6(d) for the reference channels with a constant length of 100 mm and helical pitches ranging from 0.20 mm to 2.08 mm at Re = 76. The 21 µm particles are omitted as they are in the transition state over the range of pitches analyzed. The 9.9 µm and 16 µm particles remained in the Dean drag state, regardless of the pitch. Increasing the pitch resulted in a more compact range in elevation for the 9.9 µm particles, potentially resulting from the added secondary flows moving the smaller particles away from the inner wall. In contrast, the elevation range of the 16 µm particles remained constant, potentially due to them having more inertia compared to the 9.9 µm particles. Therefore, increasing the helical pitch had a minor effect on particles in the Dean drag state, focusing smaller particles into more compact streams. The average positions of the 23 µm and 26 µm particles remained relatively constant at pitches up to 0.95 mm; however, as shown in Fig. 6(d), the elevation range for the 23 µm particles increased with the increasing helical pitch, entering the transition state at a channel pitch of 0.95 mm. The result was an additional particle diameter in the transition state, where the difference in particle diameters must be greater to separate particles in channels with a large helical torsion compared to channels with a lower torsion. The 26 µm particles remained in the inertial migration state, though their average position moved further from the inner wall with the increasing pitch.

Therefore, the separation of particles with a relatively small size difference is enhanced by minimizing the helical pitch. In cases where particle diameters are far from the transition state, the effect of the increasing helical pitch is relatively insignificant, though a smaller helical pitch will result in inertial migration closest to the inner wall.

Equations (3) and (5) showed that the Dean drag force scales with the channel diameter according to D^−0.82, while the inertial lift force is unaffected, resulting in an increase in the force ratio with the increasing channel diameter. Helical channels offer greater control over this parameter compared to spiral channels as the diameter can remain constant over the length of the channel. Additionally, smaller diameters can be achieved since unlike spiral channels, the channel diameter is not required to increase by the depth of the channel with each turn.

Figure 7(a) shows the locations of particles with a diameter of 9.9 µm, 21 µm, and 26 µm for the reference channel with channel diameters ranging from 1.7 mm to 14.4 mm with a constant channel length of 100 mm at Re = 76. Figure 7(b) shows the positions of particles with a diameter of 16 µm and 23 µm in these channels. In comparison, typical planar spiral geometries have had initial diameters of ∼10 mm.^19,26 The average position of the 9.9 µm particles remained relatively consistent for all channel diameters studied; however, the range in elevations increased with the increasing channel diameter, suggesting that the particles were near the threshold of the transition state beyond a channel diameter of 10 mm. The 16 µm particles near the limit of the Dean drag state in the reference diameter of 3.24 mm entered the transition state at a channel diameter of 6.42 mm. Separation of this size would not be possible for channel diameters between 6.42 mm and 14.4 mm. The 21 µm particles that were initially in the transition state at the reference diameter were in the inertial migration state at diameters greater than 4.84 mm, allowing for separation from the 9.9 µm particle stream. As the channel diameter increased from the reference diameter, the 23 µm and 26 µm particles migrated closer to the inner wall in more compact streams. In contrast, the 23 µm and 26 µm particles occupied the transition state as the Dean drag force significantly influenced particle positions. As a result, there were no particles in the inertial migration state, and therefore, particle separation was not possible.

Increasing the diameter allowed smaller particles to inertially migrate to the inner wall. It should also be noted that the range of channel diameters constituting the transition state is larger for smaller particles. For example, Fig. 7(a) shows that 21 µm particles transitioned from the Dean drag state to the inertially focused state as the channel diameter increased from 1.67 mm to 4.84 mm, whereas the 16 µm particles began to transition at a channel diameter of 6.42 mm and remained in the transition state at a channel diameter of 14.36 mm. Therefore, the effect of channel diameter may be exploited to shift a particle size from the transition state to either the Dean drag state, by decreasing the channel diameter, or the inertial migration state, by increasing the channel diameter.

Dean vortices occur in the center of a rectangular channel, as shown in Fig. 8(a), but shift to the outer wall of a tapered channel, as shown in Fig. 8(d). Guan et al.¹⁹ showed that this can improve separation efficiency by increasing the distance between particles in the inertial migration and Dean drag states.

The influence of the taper angle on particle positions was studied by varying the taper angle of the reference channel and keeping the width of the center of the channel constant. This allowed for D_h to remain constant for each iteration. Analytically, this results in the same force balance for each taper angle, suggesting that the particle position should remain unchanged for all taper angles. Taper angles up to 5.5° were considered in the present study. At this taper angle, the width of the inner wall of the channel was 40 µm. Increasing the taper angle further would make this width too small for the 26 µm particles to migrate to the inner wall. Reverse tapers up to −3° were also considered, where the inner wall was wider than the outer wall, which resulted in the Dean vortices shifting toward the inner wall, as shown in Fig. 8(b).

Figure 8(e) shows the locations of particles with a diameter 9.9 µm, 21 µm, and 26 µm in channels with the reference dimensions and taper angles ranging from −3° to 5.5° at Re = 76. Figure 8(f) shows the positions of particles with a diameter of 16 µm and 23 µm in these channels. At negative taper angles, all the particles were in the Dean drag state, circulating the Dean vortices near the inner wall, as shown in Fig. 8(b). The 9.9 µm and 16 µm particles remained in the Dean drag state for every taper angle, demonstrating the shift in the Dean vortices with changing taper angles. In the rectangular channel (taper angle: 0°), the centers of the Dean vortices were located at the midpoint between the inner and outer walls and occupied the majority of the channel, as shown in Fig. 8(a). This resulted in no separation between the 9.9 µm particles and particles in the inertial migration state. Instead, this channel would only be capable of separating particles in the inertial migration state at different elevations, as particles in the Dean drag state occupy most of the channel. Increasing the taper angle up to 2° caused the Dean vortex cores to shift toward the outer wall; however, the vortices remained large, as shown in Fig. 8(c). As a result, the 9.9 µm and 16 µm particles remained in most of the channel. Increasing the taper angle caused the vortices to become more compact, as shown in Fig. 8(d). The result was that 9.9 µm and 16 µm particles occupied increasingly compact streams near the outer wall with an increase in taper angle.

The 21 µm particles were in the Dean drag state for taper angles up to 1° and greater than 4°, in the inertial migration state at 2°, and in the transition state at 3.5°. This dependence of focusing state on the taper angle demonstrates the utility of optimizing the taper angle, which is not captured in current analytical models.

The 23 µm and 26 µm particles were in the transition state from 0° to 3.5° and 2°, respectively, migrating closer to the inner wall. Increasing the taper angle further resulted in the average particle location shifting further from the inner wall. The 23 µm and 26 µm particles entered the transition state at taper angles of 4.5° and 5.5°, respectively. As the taper angle increased, the width of the inner wall shortened, resulting in a near-triangular channel cross section. At these steep taper angles, particles were unable to reach the inner wall, raising their position in the channel. When the position was sufficiently raised, the particles began to interact with the Dean vortices and occupy most of the channel. This can be seen with the 23 µm particles in the channel with a 4.5° taper angle and above. Therefore, steep taper angles should be avoided when designing a channel for size-based particle separation. A balance must be obtained between minimizing the size of the Dean vortices by increasing the taper angle while maintaining an adequate inner wall width for larger particles to inertially migrate.

Changing the depth of a channel with a constant width will change D_h of the channel, thus affecting the inertial force balance on a particle. Analytically, Eqs. (3) and (5) showed that the Dean drag force scales with D_h^0.82 and the inertial lift force scales with D_h⁻²;^26,28 therefore, increasing the channel depth will increase the relative magnitude of the Dean drag force relative to the inertial lift force.

The reference channel with an initial depth of 760 µm was modified by maintaining a constant width at the mid-point of the channel. This allowed for deeper channels to be studied without a large reduction in the inner wall width. The channel with a depth of 360 µm had an outer wall width of 136 µm, and the channel with a depth of 960 µm had an outer wall width of 172 µm. All channels were simulated at Re = 76. The D_h of a tapered channel scales with the depth of the channel, D, width at the mid-point of the channel, W_mid, and taper angle of the channel, θ, according to

Assuming the reference dimensions of W_mid = 113.5 µm and θ = 3.5°, D_h increases asymptotically to 227 µm with increasing D, just as it does in a rectangular channel. Therefore, increasing the channel depth while maintaining a constant width at the center of the side walls results in a continuous increase in the Dean drag force relative to the inertial drag force.

Figure 9(a) shows the normalized positions of 9.9 µm, 21 µm, and 26 µm in channels with reference dimensions and various depths, where 0 represents the inner wall and 1 represents the outer wall. Figure 9(b) shows the positions for 16 µm and 23 µm particles. Decreasing the channel depth resulted in an increase in the elevation range for particles in the Dean drag state (9.9 µm and 16 µm), as the Dean vortices occupied a larger portion of the channel cross section. The 21 µm particles remained in the transition state for all the depths that were studied. The normalized height of the 23 µm and 26 µm particles remained relatively constant for each depth, though at a depth of 460 µm and below, the particles interacted with the Dean vortices and entered the transition state. Both 23 µm and 26 µm particles remained in the inertial migration state with the increasing depth, though the 23 µm particles were in the transition state at a depth of 960 µm as D_h became too large.

Although the normalized position and range of particles remained relatively unchanged for channel depths between 660 µm and 860 µm, the true distance between particles in the Dean drag and inertial migration states increases with the increasing depth, suggesting an improvement in separation efficiency. Care should be taken to ensure that the depth is not too large, causing the desired particles to move to the transition state, as shown with the 23 µm particles in Fig. 9(b).

Alternatively, maintaining a constant width at the top of the tapered channel results in Eq. (12) for D_h, where W represents the width of the outer wall. Assuming the reference channel dimensions W = 160 µm and θ = 3.5°, D_h increases with the increasing depth to a maximum at D = 518 µm and then decreases with further increases in D. As a result, channels with different depths modified in this way may have the same D_h. For example, channels with depths of 460 µm and 560 µm both have D_h = 205 µm,

Figure 10(a) shows the normalized positions of 9.9 µm, 21 µm, and 26 µm particles, and Fig. 10(b) shows the normalized positions of 16 µm and 23 µm particles in the reference channels with a constant outer wall width of 160 µm and various depths, where 0 and 1 represent the inner and outer walls, respectively. Similar to the constant channel mid-width case described above, the 9.9 µm and 16 µm particles remained in the Dean drag state for all channel depths with an increasing elevation range as the channel depth decreased.

The 21 µm particles were in the transition state for all channel depths, though their average position approached the inner wall with the increasing depth, suggesting that they were approaching the inertial migration state. The 23 µm particles, which were in the transition state for channel depths of 360 µm–460 µm and at 960 µm and in the inertial migration state for depths of 560 µm–860 µm for the constant mid-width case, were in the transition state for depths of 360 µm–660 µm and in the inertial migration state for the remaining depths. The 26 µm particles, which were in the transition state up to a channel depth of 460 µm in the constant mid-width case, were in the transition state up to a channel depth of 560 µm. The differences in particle focusing states for the same channel depth suggest that the particle location largely depends on the channel width, as discussed in Sec. III B 6.

As with the channel depth, increasing the channel width increases the hydraulic diameter, thus increasing the relative influence of the Dean drag force. To study the effect of the channel width on the particle separation, the width of the channel, and hence D_h, was varied, while all other geometric parameters were held constant at the reference values and Re = 76, as shown in Fig. 11(a). Figure 11(b) shows a similar plot, but D_h was changed by modifying the channel depth while maintaining a constant width of 113.5 µm at the center of the channel, as was done in Sec. III B 5. This facilitates an assessment of the relative effects of changes in channel depth and width. It is seen that modifying the channel width had a much more significant influence on the particle position than did changing the depth. In the case of changing the channel width, the average position of the 16 µm particles remained relatively constant. In contrast, decreasing the channel depth resulted in the average position of the 16 µm particles moving toward the inner wall, occupying a larger portion of the channel. The 21 µm particle stream also began to focus at decreasing D_h for changes in width compared to changes in depth, whereas the 21 µm particles remained in the transition state for all cases. The 23 µm particles were in the inertial migration state over a wider range of D_h values when changing the width compared to when changing the depth, with the most compact streams occurring in the smallest width cases. It should be noted that for the range of D_h studied, the channel width was varied by a total of 20 µm, whereas the depth was varied by 517 µm. Therefore, particle separation was much more sensitive to changes in width than changes in depth.

Figures 11(c) and 11(d) show the particle positions for a wider range of channel widths with the reference depth of 760 µm. Increasing the channel width to 190 µm resulted in all particles becoming entrained in the Dean vortices. As the channel width was increased, the width of the Dean vortices increased, resulting in an increase in the range in elevation for the 9.9 µm and 16 µm particles in the Dean drag state. Decreasing the width to 150 µm resulted in 21 µm particles shifting from the transition to the inertial migration state, while the 16 µm particles remained in the Dean drag state, thus allowing for 16 µm and 21 µm particles to be separated. Decreasing the width to 130 µm resulted in the 16 µm particles entering the transition state where they are unable to be separated. The position and range of the 23 µm and 26 µm particles remained relatively constant in the inertial migration state, up to channel widths of 150 µm and 164 µm, respectively. At a channel width of 130 µm, where the inner wall had a width of 37 µm, the 23 µm and 26 µm particles were shifted upward due to the influence of the side walls.

To maximize the separation distance, the channel width should be kept to a practical minimum so that the size of the Dean vortices is reduced in order to migrate the particles in the inertial migration state closer to the inner wall in a compact stream.

The present paper examined the effects of various geometric parameters on size-based inertial particle separation in a helical channel using a calibrated numerical model, which shows good agreement with the experimental data in channels with various Re and AR. Particles occupied one of the three possible states: Dean drag state, where the particles were entrained in Dean vortices; inertial migration state, where they migrated to the inner wall of the channel; and transition state, where they were present throughout the channel and thus unable to be separated.

Increasing the pitch of the channel had negligible effects on particles in the Dean drag and transition states. Particles in the inertial migration state near the threshold of the transition state shifted to the transition state with the increasing pitch as the drag force on the particles from the secondary flow near the inner wall became significant. The helical pitch should be kept to a practical minimum to reduce the effects of the secondary flows near the inner wall.

Increasing the channel diameter reduced the magnitude of the Dean drag force, resulting in smaller particles migrating to the inner wall. For smaller particle diameters, the relative increase in channel diameter becomes significant, and may not be practical, to shift particles from the Dean drag to the inertial migration state.

Changing the taper angle was shown to change the focusing state of particles. The influence of the taper angles is not captured in current analytical models and should be considered when designing a particle sorting device.

Increasing the depth of a channel with a constant width at the center of the channel had negligible effects on the average position of the particles in the channel. In contrast, increasing the channel depth with a constant outer wall width allowed particles to remain in the inertial migration state for larger channel depths, suggesting that the particle focusing state is significantly impacted by the channel width.

A decreased channel width resulted in smaller particle diameters occupying the inertial migration state. An increased channel width increased the range of the Dean drag state particle streams, and increasing the width of the channel by 30 µm from the reference channel width resulted in all particles occupying the Dean drag state.

The present computationally inexpensive model is to be used as a design guideline to provide a quick assessment of microfluidic designs. Several assumptions were made to develop a model that relies solely on local flow parameters, such as neglecting particle–fluid interaction and the dependency of the lateral position in the channel. To obtain accurate particle positions, a more computationally expensive model is suggested, such as DNS.

See the supplementary material for a detailed description of the numerical model, including comparison to DNS data and pseudocode used to implement the inertial lift force.

This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC, Grant No. RGPIN-2019-04633).

The data that support the findings of this study are available within the article and its supplementary material and from the corresponding author upon reasonable request.