An Explicit Semi-Empirical Model for Cyclone Separator Cut Size with Swirl and Turbulence Corrections

Abstract

Cyclone separators remain widely used for gas–solid separation, yet analytical prediction of cut size and pressure drop remains challenging. This study presents an explicit semi-empirical model for the cut size (d₅₀) of reverse-flow cyclones based on the radial particle equation of motion in cylindrical coordinates, with d₅₀ obtained by equating radial migration time and residence time. A closed-form solution is derived in the Stokes regime, whereas non-Stokes behavior is handled numerically through the Schiller–Naumann drag correction. Turbulence is incorporated through a phenomenological correction, and the grade–efficiency curve is represented by a logistic relation. The model was implemented in MATLAB R2025a and applied in a parametric study covering inlet velocity, particle density, cyclone diameter, and gas viscosity. A Euler-type pressure drop relation was included to examine the separation–energy trade-off. Validation on the Kim et al. benchmark using one calibration point per cyclone family and six independent verification cases yielded a mean absolute percentage error of 13.5% and a root mean square error of 0.22 μm for d₅₀; the paired pressure drop check gave a 2.8% mean absolute percentage error. A complementary benchmark based on Wang et al. using 15 cm 1D3D and 2D2D cyclones under actual-air and standard-air conditions further supported the family-calibrated use of the model. A separate scale-up test showed that constant swirl intensity similarity is not transferable across large diameter changes. The formulation provides a transparent reduced-order tool for preliminary design and sensitivity analysis.

1. Introduction

Reverse-flow cyclone separators remain widely used in gas–solid separation due to their mechanical simplicity, robustness, and ease of integration into industrial systems. However, accurate prediction of cut size, grade efficiency, and pressure drop remains challenging because the internal flow is strongly swirling, turbulent, and highly geometry-dependent. For preliminary design, the cut size d₅₀ represents a key synthesis parameter, linking geometry and operating conditions to separation performance and enabling reconstruction of the full grade–efficiency curve once the transition sharpness is defined. The present study is motivated by the need for a compact and physically interpretable d₅₀ formulation that remains more transparent than empirical correlations and significantly less computationally demanding than Computational Fluid Dynamics (CFD), while still allowing systematic comparison with both reduced-order and numerical approaches.

From a design perspective, a single global efficiency indicator is insufficient when particle size distributions are broad or when fine fractions below approximately 5 μm dominate performance constraints. Nevertheless, d₅₀ remains a practical parameter because it condenses geometry–performance interactions into a single scale that can be extended to η(d_p) through a logistic representation, where d₅₀ defines the transition point and β controls the sharpness of separation.

The objectives of this work are: derivation of the particle equation of motion in cylindrical coordinates under an idealized swirling field; development of a closed-form expression for d₅₀ in the Stokes regime based on radial migration–residence time equivalence; introduction of Reynolds and Stokes numbers for scaling and validity assessment; extension to moderate non-Stokes regimes via the Schiller–Naumann correction; inclusion of a phenomenological turbulence correction χ_t; and implementation in MATLAB R2025a for sensitivity analysis and validation against literature data.

Classical cyclone models focused primarily on swirling flow structure and pressure losses, with Shepherd and Lapple [1] identifying rotational motion and energy dissipation as dominant mechanisms. Leith and Mehta [2] later related geometric proportions to operating range, while subsequent studies refined inlet, outlet, and frictional contributions to pressure drop [3,4].

In parallel, efficiency and cut-size modeling evolved through the work of Chan and Lippmann [5], Dietz [6], Dirgo and Leith [7], and Leith and Licht [8], establishing widely used empirical and semi-theoretical frameworks. However, their predictive accuracy depends strongly on assumptions regarding flow structure and residence time. A key advancement was provided by Iozia and Leith [9,10], who demonstrated that cyclone geometry significantly influences grade efficiency and introduced a logistic representation of η(d_p), where d₅₀ and β govern the curve shape. Zhou and Soo [11] further emphasized the role of particle trajectories in swirling flows, reinforcing the need for transparent reduced-order formulations.

In most classical reduced-order models, turbulence effects are not explicitly represented but are embedded implicitly within empirical coefficients. In contrast, the present work introduces χ_t as an explicit intermediate closure term that separates turbulent dispersion effects from geometric and inertial contributions, thereby improving physical interpretability within a unified analytical framework.

The internal flow field of a cyclone is inherently complex and non-ideal, involving a forced-vortex core, a surrounding free-vortex region, processing vortex-core motion, and secondary recirculation structures, as documented in [12,13,14,15]. These features influence velocity gradients, particle residence time, and re-entrainment, and therefore directly affect separation performance.

Due to this complexity, many studies rely on CFD-based approaches to capture detailed flow and geometry effects [16,17,18,19,20,21,22,23,24,25,26], including inlet optimization, wall interaction modeling, vortex finder design, and multi-objective optimization. Further developments include CFD reviews [27], CFD–machine learning hybrid models [28], Computational Fluid Dynamics–Discrete Element Method (CFD–DEM) simulations [29], dimensionless scaling analyses [30], surrogate modeling approaches [31], reduced-order CFD-inspired formulations [32], and optimization frameworks [33]. While highly accurate, these methods are computationally expensive and less suitable for preliminary design or transparent parametric studies.

Additional studies on axial, multistage, and high-solids-loading cyclones [34,35,36,37,38,39] confirm that no single closed-form model is universally valid across all operating regimes, reinforcing the need for reduced-order formulations that preserve physical interpretability.

Recent Euler–Lagrange and data-driven approaches have reduced computational cost in cyclone analysis [40], but they typically focus on trajectory simulation rather than explicit analytical expressions for cut size. In contrast, the present work develops a direct semi-empirical formulation derived from radial force balance, emphasizing interpretability over case-specific simulation.

The main contribution of this study is a reduced-order framework in which the influence of μ, ρ_p, V_in, D, geometric ratios (a/D, b/D, D_e/D, H_eff/D), swirl intensity K_s, turbulence coefficient C_t, and indicator I is explicitly retained. This enables transparent sensitivity analysis, structured calibration, and direct comparison with both classical correlations and CFD-based approaches [26,27,28,29,30,31,32,33].

Unlike classical Lapple-type correlations, where geometric and flow effects are absorbed into aggregated empirical constants, the present formulation preserves their explicit analytical dependence. This allows direct physical interpretation of the roles of geometry, swirl intensity, and operating conditions at equation level rather than indirect calibration.

Finally, the paper is restricted to gas–solid cyclone separators to maintain consistency with the assumed physical framework of the model, excluding hydrocyclones due to fundamentally different liquid–solid, pressure-driven flow physics.

2. Materials and Methods

2.1. Assumptions and Idealized Flow Field

The study considers a conventional reverse-flow cyclone with a tangential gas–solid inlet. Cylindrical coordinates (r, θ, z) are used. In the outer vortex, where radial migration is dominant, the gas tangential velocity is approximated as a free vortex, as shown in Equation (1).

In real cyclones, the tangential velocity distribution generally includes a forced-vortex region near the axis and a free-vortex region toward the wall. In the present model, the free-vortex approximation is adopted for the dominant radial migration region, whereas deviations from the ideal profile are absorbed into the swirl intensity coefficient K_s and the turbulence correction χ_t.

At the order-of-magnitude level, the interface between the outer downward vortex and the inner upward vortex may be approximated by the radius of the outlet tube. Accordingly, the initial particle radius is linked to the vortex-finder diameter through r₀ ≈ D_e/2, an approximation commonly used in classical Shepherd–Lapple-type reasoning [1]. The influence of this choice is examined through the sensitivity analysis of D_e/D in Section 3.6.

Figure 1 defines the reference reverse-flow cyclone geometry used throughout the derivation and identifies the parameters that enter the reduced-order model, namely D, D_e, a, b, and H_eff. It also clarifies that radial migration is evaluated across the annular region between the vortex finder and the outer wall.

The tangential velocity distribution is defined as:

$${\mathrm{u}}_{\mathsf{\theta }}\left(\mathrm{r}\right)={\mathrm{u}}_{\mathsf{\theta },\mathrm{w}}\cdot {\displaystyle \frac{{\mathrm{r}}_{\mathrm{w}}}{\mathrm{r}}}$$

(1)

The wall tangential velocity is related to the inlet mean velocity through the dimensionless swirl intensity coefficient K_s:

$${\mathrm{u}}_{\mathsf{\theta },\mathrm{w}}={\mathrm{K}}_{\mathrm{s}}\cdot {\mathrm{V}}_{\mathrm{i}\mathrm{n}}$$

(2)

The operational assumptions are as follows: statistically steady and incompressible flow; dilute suspension (one-way coupling); negligible particle–particle interaction; negligible mean radial gas velocity; and Stokes-regime treatment for the limiting particle associated with d₅₀. For moderate-particle Reynolds number, the dependence on the drag coefficient is handled numerically without changing the analytical derivation.

Gravity and buoyancy are neglected in the radial force balance because centrifugal acceleration dominates in the regime of interest. For submicron particles, a natural extension would be to include the Cunningham slip correction by replacing μ with μ/C_c in the relaxation-time expression τ_p [41].

Note on particle-size definition: in this article, d_p denotes the equivalent geometric diameter of a spherical particle. When literature data reports the cut point as aerodynamic equivalent diameter d_ae, comparisons should use a consistent diameter definition. For spherical particles in the Stokes regime, a common approximation is d_ae ≈ d_p√(ρ_p/ρ₀), with ρ₀ = 1000 kg m⁻³. For non-spherical particles, a dynamic shape factor χ is required, and for submicron particles, the slip correction C_c becomes relevant [41].

2.2. Radial Equation of Motion

Mass of a spherical particle:

$${\mathrm{m}}_{\mathrm{p}}={\mathsf{\rho }}_{\mathrm{p}}\cdot {\displaystyle \frac{\mathsf{\pi }}{6}}\cdot {\mathrm{d}}_{\mathrm{p}}^3$$

(3)

Accelerations in cylindrical coordinates:

$${\mathrm{a}}_{\mathrm{r}}={\displaystyle \frac{\mathrm{d}{\mathrm{v}}_{\mathrm{r}}}{\mathrm{d}\mathrm{t}}}-{\displaystyle \frac{{\mathrm{v}}_{\mathsf{\theta }}^2}{\mathrm{r}}}$$

(4)

$${\mathrm{a}}_{\mathsf{\theta }}={\displaystyle \frac{\mathrm{d}{\mathrm{v}}_{\mathsf{\theta }}}{\mathrm{d}\mathrm{t}}}+{\displaystyle \frac{{\mathrm{v}}_{\mathrm{r}}\cdot {\mathrm{v}}_{\mathsf{\theta }}}{\mathrm{r}}}$$

(5)

Drag force (general form):

$${\mathrm{F}}_{\mathrm{D}}={\displaystyle \frac{1}{2}}\cdot {\mathrm{C}}_{\mathrm{D}}\cdot {\mathsf{\rho }}_{\mathrm{g}}\cdot {\mathrm{A}}_{\mathrm{p}}\cdot \left|\mathrm{u}-\mathrm{v}\right|\cdot \left(\mathrm{u}-\mathrm{v}\right)$$

(6)

where

$${\mathrm{A}}_{\mathrm{p}}={\displaystyle \frac{\mathsf{\pi }}{4}}\cdot {\mathrm{d}}_{\mathrm{p}}^2$$

(7)

Momentum balance:

$${\mathrm{m}}_{\mathrm{p}}\cdot \mathrm{a}={\mathrm{F}}_{\mathrm{D}}$$

(8)

For the radial component, with negligible mean radial gas velocity, one obtains:

$${\mathrm{F}}_{\mathrm{D},\mathrm{r}}=-{\displaystyle \frac{1}{2}}\cdot {\mathrm{C}}_{\mathrm{D}}\cdot {\mathsf{\rho }}_{\mathrm{g}}\cdot {\mathrm{A}}_{\mathrm{p}}\cdot \left|{\mathrm{v}}_{\mathrm{r}}\right|\cdot {\mathrm{v}}_{\mathrm{r}}$$

(9)

The radial equation of particle motion then becomes:

$${\mathrm{m}}_{\mathrm{p}}\cdot \left({\displaystyle \frac{\mathrm{d}{\mathrm{v}}_{\mathrm{r}}}{\mathrm{d}\mathrm{t}}}-{\displaystyle \frac{{\mathrm{v}}_{\mathsf{\theta }}^2}{\mathrm{r}}}\right)=-{\displaystyle \frac{1}{2}}\cdot {\mathrm{C}}_{\mathrm{D}}\cdot {\mathsf{\rho }}_{\mathrm{g}}\cdot {\mathrm{A}}_{\mathrm{p}}\cdot \left|{\mathrm{v}}_{\mathrm{r}}\right|\cdot {\mathrm{v}}_{\mathrm{r}}$$

(10)

Rearranging gives:

$${\mathrm{m}}_{\mathrm{p}}\cdot {\displaystyle \frac{\mathrm{d}{\mathrm{v}}_{\mathrm{r}}}{\mathrm{d}\mathrm{t}}}={\mathrm{m}}_{\mathrm{p}}\cdot {\displaystyle \frac{{\mathrm{v}}_{\mathsf{\theta }}^2}{\mathrm{r}}}-{\displaystyle \frac{1}{2}}\cdot {\mathrm{C}}_{\mathrm{D}}\cdot {\mathsf{\rho }}_{\mathrm{g}}\cdot {\mathrm{A}}_{\mathrm{p}}\cdot \left|{\mathrm{v}}_{\mathrm{r}}\right|\cdot {\mathrm{v}}_{\mathrm{r}}$$

(11)

For relatively small particles, tangential velocity synchronization is assumed. This assumption is reasonable when the particle relaxation time is small relative to the rotational time scale (τ_pu_θ/r ≪ 1) so that the particle rapidly follows the tangential gas motion. For coarse particles, tangential slip may become significant and a two-component velocity solution would be required.

$${\mathrm{v}}_{\mathsf{\theta }}\approx {\mathrm{u}}_{\mathsf{\theta }}\left(\mathrm{r}\right)$$

(12)

Substituting Equation (12) into Equation (11) yields:

$${\mathrm{m}}_{\mathrm{p}}\cdot {\displaystyle \frac{\mathrm{d}{\mathrm{v}}_{\mathrm{r}}}{\mathrm{d}\mathrm{t}}}={\mathrm{m}}_{\mathrm{p}}\cdot {\displaystyle \frac{{\mathrm{u}}_{\mathsf{\theta }}^2\left(\mathrm{r}\right)}{\mathrm{r}}}-{\displaystyle \frac{1}{2}}\cdot {\mathrm{C}}_{\mathrm{D}}\cdot {\mathsf{\rho }}_{\mathrm{g}}\cdot {\mathrm{A}}_{\mathrm{p}}\cdot \left|{\mathrm{v}}_{\mathrm{r}}\right|\cdot {\mathrm{v}}_{\mathrm{r}}$$

(13)

Particle Reynolds number based on radial slip:

$$\mathrm{R}{\mathrm{e}}_{\mathrm{p}}={\displaystyle \frac{{\mathsf{\rho }}_{\mathrm{g}}\cdot {\mathrm{d}}_{\mathrm{p}}\cdot \left|{\mathrm{v}}_{\mathrm{r}}\right|}{\mathsf{\mu }}}$$

(14)

Stokes regime (Re_p < 1)—the drag force becomes linear:

$${\mathrm{F}}_{\mathrm{D},\mathrm{r}}=-3\mathsf{\pi }\cdot \mathsf{\mu }\cdot {\mathrm{d}}_{\mathrm{p}}\cdot {\mathrm{v}}_{\mathrm{r}}$$

(15)

Equation (13) then reduces to the linear forced equation:

$${\displaystyle \frac{\mathrm{d}{\mathrm{v}}_{\mathrm{r}}}{\mathrm{d}\mathrm{t}}}+{\displaystyle \frac{18\cdot \mathsf{\mu }}{{\mathsf{\rho }}_{\mathrm{p}}\cdot {\mathrm{d}}_{\mathrm{p}}^2}}\cdot {\mathrm{v}}_{\mathrm{r}}={\displaystyle \frac{{\mathrm{u}}_{\mathsf{\theta }}^2\left(\mathrm{r}\right)}{\mathrm{r}}}$$

(16)

Stokes relaxation time:

$${\mathsf{\tau }}_{\mathrm{p}}={\displaystyle \frac{{\mathsf{\rho }}_{\mathrm{p}}\cdot {\mathrm{d}}_{\mathrm{p}}^2}{18\cdot \mathsf{\mu }}}$$

(17)

Compact form:

$${\displaystyle \frac{\mathrm{d}{\mathrm{v}}_{\mathrm{r}}}{\mathrm{d}\mathrm{t}}}+{\displaystyle \frac{1}{{\mathsf{\tau }}_{\mathrm{p}}}}\cdot {\mathrm{v}}_{\mathrm{r}}={\displaystyle \frac{{\mathrm{u}}_{\mathsf{\theta }}^2\left(\mathrm{r}\right)}{\mathrm{r}}}$$

(18)

Non-Stokes correction (handled numerically): for moderate Re_p, the Schiller–Naumann correlation may be used:

$${\mathrm{C}}_{\mathrm{D}}={\displaystyle \frac{24}{\mathrm{R}{\mathrm{e}}_{\mathrm{p}}}}\cdot \left(1+0.15\cdot \mathrm{R}{\mathrm{e}}_{\mathrm{p}}^{0.687}\right), \mathrm{R}{\mathrm{e}}_{\mathrm{p}}\lesssim {10}^3$$

(19)

and the corrected relaxation time becomes:

$${\mathsf{\tau }}_{\mathrm{p}}^{\mathrm{*}}={\displaystyle \frac{{\mathsf{\tau }}_{\mathrm{p}}}{1+0.15\cdot \mathrm{R}{\mathrm{e}}_{\mathrm{p}}^{0.687}}}$$

(20)

The Schiller–Naumann extension is applied as a moderate correction beyond the Stokes regime. In regimes dominated by tangential slip, particle–wall interactions, or strongly inertial particle motion, the present reduced-order closure does not capture the full complexity of the underlying physics and thus represents a reduced-order approximation rather than a fully predictive model.

2.3. Closed-Form Derivation of d₅₀

For the limiting particle, a quasi-steady approximation is adopted:

$${\displaystyle \frac{\mathrm{d}{\mathrm{v}}_{\mathrm{r}}}{\mathrm{d}\mathrm{t}}}\ll {\displaystyle \frac{1}{{\mathsf{\tau }}_{\mathrm{p}}}}\cdot {\mathrm{v}}_{\mathrm{r}}$$

(21)

From Equation (18), one obtains:

$${\mathrm{v}}_{\mathrm{r}}\left(\mathrm{r}\right)\approx {\mathsf{\tau }}_{\mathrm{p}}\cdot {\displaystyle \frac{{\mathrm{u}}_{\mathsf{\theta }}^2\left(\mathrm{r}\right)}{\mathrm{r}}}$$

(22)

With the free-vortex expression from Equation (1):

$${\displaystyle \frac{{\mathrm{u}}_{\mathsf{\theta }}^2\left(\mathrm{r}\right)}{\mathrm{r}}}={\displaystyle \frac{{\mathrm{u}}_{\mathsf{\theta },\mathrm{w}}^2\cdot {\mathrm{r}}_{\mathrm{w}}^2}{{\mathrm{r}}^3}}$$

(23)

Therefore:

$${\displaystyle \frac{\mathrm{d}\mathrm{r}}{\mathrm{d}\mathrm{t}}}={\mathsf{\tau }}_{\mathrm{p}}\cdot {\displaystyle \frac{{\mathrm{u}}_{\mathsf{\theta },\mathrm{w}}^2\cdot {\mathrm{r}}_{\mathrm{w}}^2}{{\mathrm{r}}^3}}$$

(24)

Here, r₀ denotes the outlet-tube radius (r₀ = D_e/2) and r_w denotes the wall radius (r_w = D/2), so that radial migration is evaluated across the annular separation region between those two radii.

$${\mathrm{t}}_{\mathrm{m}}={\int }_{{\mathrm{r}}_0}^{{\mathrm{r}}_{\mathrm{w}}}{\displaystyle \frac{{\mathrm{r}}^3}{{\mathsf{\tau }}_{\mathrm{p}}\cdot {\mathrm{u}}_{\mathsf{\theta },\mathrm{w}}^2\cdot {\mathrm{r}}_{\mathrm{w}}^2}}\hspace{0.17em}\mathrm{d}\mathrm{r}={\displaystyle \frac{{\mathrm{r}}_{\mathrm{w}}^4-{\mathrm{r}}_0^4}{4\cdot {\mathsf{\tau }}_{\mathrm{p}}\cdot {\mathrm{u}}_{\mathsf{\theta },\mathrm{w}}^2\cdot {\mathrm{r}}_{\mathrm{w}}^2}}$$

(25)

Annular area:

$${\mathrm{A}}_{\mathrm{a}\mathrm{n}\mathrm{n}}=\mathsf{\pi }\cdot \left({\mathrm{r}}_{\mathrm{w}}^2-{\mathrm{r}}_0^2\right)$$

(26)

Volumetric flow rate at the inlet:

$$\mathrm{Q}={\mathrm{V}}_{\mathrm{i}\mathrm{n}}\cdot {\mathrm{A}}_{\mathrm{i}\mathrm{n}}, {\mathrm{A}}_{\mathrm{i}\mathrm{n}}=\mathrm{a}\cdot \mathrm{b}$$

(27)

Residence time:

$${\mathrm{t}}_{\mathrm{r}\mathrm{e}\mathrm{s}}={\displaystyle \frac{{\mathrm{H}}_{\mathrm{e}\mathrm{f}\mathrm{f}}\cdot {\mathrm{A}}_{\mathrm{a}\mathrm{n}\mathrm{n}}}{\mathrm{Q}}}={\displaystyle \frac{{\mathrm{H}}_{\mathrm{e}\mathrm{f}\mathrm{f}}\cdot \mathsf{\pi }\cdot \left({\mathrm{r}}_{\mathrm{w}}^2-{\mathrm{r}}_0^2\right)}{{\mathrm{V}}_{\mathrm{i}\mathrm{n}}\cdot \mathrm{a}\cdot \mathrm{b}}}$$

(28)

Definition of d₅₀:

$${\mathrm{t}}_{\mathrm{m}}\left({\mathrm{d}}_{50}\right)={\mathrm{t}}_{\mathrm{r}\mathrm{e}\mathrm{s}}$$

(29)

After substitution and simplification, with r_w = D/2 and r₀ = D_e/2, the final analytical expression is obtained:

$${\mathrm{d}}_{50}=\sqrt{{\displaystyle \frac{9\cdot \mathsf{\mu }\cdot \mathrm{a}\cdot \mathrm{b}\cdot \left({\mathrm{D}}^2+{\mathrm{D}}_{\mathrm{e}}^2\right)}{2\cdot \mathsf{\pi }\cdot {\mathsf{\rho }}_{\mathrm{p}}\cdot {\mathrm{D}}^2\cdot {\mathrm{K}}_{\mathrm{s}}^2\cdot {\mathrm{V}}_{\mathrm{i}\mathrm{n}}\cdot {\mathrm{H}}_{\mathrm{e}\mathrm{f}\mathrm{f}}}}}$$

(30)

2.4. Dimensional Analysis

Flow Reynolds number:

$$\mathrm{R}\mathrm{e}={\displaystyle \frac{{\mathsf{\rho }}_{\mathrm{g}}\cdot {\mathrm{V}}_{\mathrm{i}\mathrm{n}}\cdot \mathrm{D}}{\mathsf{\mu }}}$$

(31)

Particle Stokes number:

$$\mathrm{S}\mathrm{t}\mathrm{k}={\displaystyle \frac{{\mathsf{\tau }}_{\mathrm{p}}\cdot {\mathrm{V}}_{\mathrm{i}\mathrm{n}}}{\mathrm{D}}}={\displaystyle \frac{{\mathsf{\rho }}_{\mathrm{p}}\cdot {\mathrm{d}}_{\mathrm{p}}^2\cdot {\mathrm{V}}_{\mathrm{i}\mathrm{n}}}{18\cdot \mathsf{\mu }\cdot \mathrm{D}}}$$

(32)

For the cut size:

$$\mathrm{S}\mathrm{t}{\mathrm{k}}_{50}={\displaystyle \frac{{\mathsf{\rho }}_{\mathrm{p}}\cdot {\mathrm{d}}_{50}^2\cdot {\mathrm{V}}_{\mathrm{i}\mathrm{n}}}{18\cdot \mathsf{\mu }\cdot \mathrm{D}}}$$

(33)

From Equation (30), the scaling d₅₀² ∝ (μ/(ρ_pV_in))ab(1 + (D_e/D)²)/(K_s²H_eff) is obtained. Under geometric similarity, with a/D, b/D, D_e/D, and H_eff/D kept constant, this leads to d₅₀ ∝ D^1/2 and, equivalently, Stk₅₀ ≈ constant. These relations are useful as internal consistency checks and as guidance for cautious extrapolation between cyclone scales.

2.5. Proposed Turbulence Correction and Fractional Efficiency Model

Unlike classical semi-empirical cyclone models, where turbulence effects are absorbed into global fitting parameters (e.g., effective number of turns or overall efficiency coefficients), the present formulation introduces χ_t as a separated correction term acting specifically on the apparent cut size. This separation enables a clearer distinction between geometric/inertial contributions and turbulence-induced radial dispersion mechanisms. The turbulence correction is defined as:

$${\mathsf{\chi }}_{\mathrm{t}}=1+{\mathrm{C}}_{\mathrm{t}}\cdot \mathrm{I}\cdot \sqrt{\mathrm{R}\mathrm{e}}$$

(34)

$${\mathrm{d}}_{50,\mathrm{t}}={\mathrm{d}}_{50}\cdot \sqrt{{\mathsf{\chi }}_{\mathrm{t}}}$$

(35)

The factor χ_t is introduced as a reduced-order closure that captures the net influence of turbulent radial dispersion, short-circuiting of fine particles, and re-entrainment on the apparent cut size [12,13,14,15,38]. Physically, these mechanisms oppose deterministic centrifugal migration and therefore shift the effective threshold toward larger particle diameters. The coefficient C_t is defined as a calibration parameter for the cyclone family and operating regime under study.

The form χ_t = 1 + C_tI√Re was chosen so that the correction remains dimensionless, approaches unity in the low-turbulence/low-Re limit and increases monotonically with turbulence intensity and inertial flow level. A simple scaling argument supports this choice. If turbulent diffusivity is written as D_t∼u′l, with u′∼IV_in and l of order D, then D_t/ν is of order IRe; when the corresponding dispersive contribution enters reduced-order time or distance estimates under a square-root dependence, a √Re term provides a compact semi-empirical representation.

This correction is best interpreted in relation to existing model classes. In classical semi-empirical cyclone correlations, the influence of turbulence and fine-particle losses is typically absorbed into fitted constants, effective numbers of turns, or sharpness parameters, rather than represented by a dedicated correction term. By contrast, high-fidelity CFD and CFD–DEM approaches resolve the local turbulent flow and particle dynamics directly [27,29,38], but at substantially greater computational cost. The present χ_t term represents an intermediate reduced-order closure that makes the role of turbulence explicit within a closed-form, design-oriented model.

Because χ_t is phenomenological, C_t and, in practice, I, are not universal constants. Their influence is examined in the sensitivity analysis presented in Section 3.6 and quantitative application of the model requires family-specific calibration against experimental data or matched high-fidelity simulations.

The fractional efficiency curve is represented by the logistic function:

$$\left({\mathrm{d}}_{\mathrm{p}}\right)={\displaystyle \frac{1}{1+{\left(\frac{{\mathrm{d}}_{50,\mathrm{t}}}{{\mathrm{d}}_{\mathrm{p}}}\right)}^{\mathsf{\beta }}}}$$

(36)

2.6. MATLAB Implementation and Computational Protocol

Equations (30)–(36) were implemented in MATLAB R2025a as a function that receives geometric and physical parameters and returns d₅₀, χ_t, d_50,t, and η(d_p) on a prescribed diameter grid. The parametric study varied inlet velocity V_in, particle density ρ_p, cyclone body diameter D, and gas viscosity μ while maintaining geometric similarity (D_e = 0.5 D, a = 0.5 D, b = 0.2 D, H_eff = 4 D). Unless otherwise stated, gas properties were fixed at ρ_g = 1.2 kg m⁻³ and the phenomenological parameters were set to K_s = 1.4, I = 0.05, C_t = 0.02, β = 2, and Eu = 8 for illustrative calculations only.

Because the present formulation is algebraic and reduced-order, it does not involve spatial discretization of the cyclone flow field; consequently, a conventional CFD mesh- or grid-independence analysis is not applicable. The analytical cut size d₅₀ is obtained directly from Equation (30), without a numerical solution of the flow domain. Numerical resolution enters only through the particle diameter vector used to sample the fractional efficiency curve η(d_p) and through the iterative update of τ_p when the Schiller–Naumann correction is applied.

The computational sequence is as follows: (i) compute Q = V_in·a·b and the Reynolds number Re; (ii) evaluate the analytical cut size d₅₀ from Equation (30) and the turbulence factor χ_t from Equation (34); (iii) obtain the corrected cut size d_50,t from Equation (35); (iv) evaluate the logistic grade–efficiency curve η(d_p) from Equation (36) over the selected particle-size vector; and (v) when non-Stokes conditions are relevant, update τ_p iteratively with the Schiller–Naumann correction until convergence.

The overall computational logic is summarized in Figure 2.

In Figure 2, inputs include geometry (D, a, b, D_e, H_eff), operating conditions (V_in, μ, ρ_g, ρ_p), and closure parameters (K_s, C_t, I, β, Eu). The workflow returns d₅₀, χ_t, d_50,t, η(d_p), and Δp; when the particle Reynolds number is outside the Stokes regime, τ_p is updated iteratively using the Schiller–Naumann correction.

For reproducibility, the particle-diameter vector was defined between 0.3 and 20 μm on a logarithmic grid with 200 points. This discretization affects only the sampling of the fractional efficiency curve η(d_p), not the analytical value of d₅₀ obtained from Equation (30). In the non-Stokes calculations, the iterative update of τ_p was terminated when the relative change between successive iterations fell below 10⁻⁶ or when 50 iterations were reached. The value 50 is used solely as a conservative numerical safeguard against indefinite looping in non-convergent cases; the actual convergence criterion remains the relative tolerance.

For the baseline case, a simple estimate of the particle Reynolds number at the threshold, obtained by evaluating Re_p at d_p = d₅₀ using the mean radial velocity (r_w − r₀)/t_res, gives Re_p ≈ 0.025. This supports the use of the Stokes approximation for the fine-particle threshold, whereas the Schiller–Naumann correction becomes relevant mainly for substantially larger particles that are in any case collected with high efficiency.

This baseline estimate provides a case-specific validity check rather than a universal result; for new applications, Re_p at d₅₀ must be re-evaluated to determine whether the analytical Stokes form remains applicable or whether a non-Stokes correction is required.

For clarity and reproducibility, the baseline parameter set adopted in the numerical examples is summarized in

Table 1. Baseline parameters used in the numerical examples.

Parameter	Value	Unit	Notes
Vin	20	m/s	baseline case presented in Section 3.5
D	0.3	m	baseline cyclone diameter
De	0.5·D	m	vortex-finder diameter
a	0.5·D	m	inlet dimension
b	0.2·D	m	inlet dimension
Heff	4·D	m	effective height
ρg	1.2	kg m−3	air, normal conditions (assumed)
μ	1.8 × 10−5	Pa·s	air, normal conditions (assumed)
ρp	2000	kg m−3	mineral particles (example)
Ks	1.4	–	swirl coefficient (calibration)
I	0.05	–	turbulence intensity (illustrative)
Ct	0.02	–	turbulence-correction coefficient (illustrative)
β	2	–	logistic slope parameter (illustrative)
Eu	8	–	pressure drop coefficient (illustrative)

.

2.7. Relation to Classical Models

A generic expression associated with Lapple-type models can be written as:

$${\mathrm{d}}_{50,\mathrm{L}}=\sqrt{{\displaystyle \frac{9\cdot \mathsf{\mu }\cdot \mathrm{b}}{2\cdot \mathsf{\pi }\cdot {\mathsf{\rho }}_{\mathrm{p}}\cdot {\mathrm{V}}_{\mathrm{i}\mathrm{n}}\cdot {\mathrm{N}}_{\mathrm{e}}}}}$$

(37)

Equation (30) can be mapped onto the Lapple-type form of Equation (37) by introducing an effective number of turns N_e,eff = (K_s²H_eff/a)[1/(1 + (D_e/D)²)]. Under this identification, the proposed model reduces to the classical form when N_e ≈ N_e,eff and the geometric effects are absorbed into the effective number of turns. The key theoretical difference, however, is that the present formulation does not collapse geometry, swirl intensity, and turbulence effects into a single empirical constant. Instead, it retains the roles of a/D, b/D, D_e/D, H_eff/D, and K_s explicitly, so that the influence of geometry and closure assumptions can be examined separately.

The model is formulated as a reduced-order analytical framework rather than a fitted universal correlation. This distinction enables the Stokes and non-Stokes forms, the turbulence correction χ_t, the logistic grade–efficiency curve, and the pressure drop estimate to be treated within a single consistent formulation. A direct quantitative comparison with the standard Lapple baseline and a broader structured positioning relative to other widely used model families are presented in Section 3.10.

3. Results and Discussion

Section 3.1, Section 3.2, Section 3.3, Section 3.4, Section 3.5, Section 3.6, Section 3.7 and Section 3.8 present model-generated trends intended to illustrate scaling laws, parameter sensitivity, and the effect of the proposed turbulence correction. Section 3.9 adds a literature-based validation layer in which the analytical prediction for d₅₀ is compared with reported thresholds and, where available, with the reported pressure drop Δp.

3.1. Inlet Velocity Sensitivity

According to Equation (30), d₅₀ decreases approximately as V_in^−1/2. The numerical results in

Table 2. Sensitivity of d₅₀ and d_50,t to inlet velocity V_in (baseline case: D = 0.3 m, ρ_p = 2000 kg m⁻³, μ = 1.8 × 10⁻⁵ Pa·s; phenomenological parameters K_s = 1.4, I = 0.05, C_t = 0.02).

Vin [m/s]	Re [–]	χt [–]	d50 [µm]	d50,t [µm]
10	200,000	1.447	2.483	2.987
15	300,000	1.548	2.028	2.522
20	400,000	1.632	1.756	2.243
25	500,000	1.707	1.571	2.052
30	600,000	1.775	1.434	1.910

confirm this trend, from about 2.48 μm at 10 m/s to about 1.43 μm at 30 m/s. Because χ_t increases with Re in the proposed closure, d_50,t remains systematically larger than d₅₀, and the benefit of increasing V_in is partially offset when turbulent dispersion and re-entrainment are represented.

Figure 3 visualizes the monotonic decrease in both d₅₀ and d_50,t with increasing inlet velocity.

The persistent separation between the two curves reflects the turbulence penalty introduced through χ_t. Because d_50,t = d₅₀√χ_t and χ_t = 1 + C_tI√Re in the present closure, the offset is only quasi-constant: it changes slightly whenever the operating condition modifies Re, even if C_t and I are held fixed.

3.2. Particle-Density Sensitivity (V_in = 20 m/s)

Equation (30) predicts d₅₀ ∝ ρ_p^−1/2, so denser particles are separated more easily. In

Table 3. Sensitivity of cut size to particle density ρ_p (V_in = 20 m/s, D = 0.3 m, μ = 1.8 × 10⁻⁵ Pa·s; all other parameters as in Table 1).

ρp [kg m−3]	d50 [µm]	d50,t [µm]
1000	2.483	3.173
1500	2.028	2.591
2000	1.756	2.243
2500	1.571	2.007
3000	1.434	1.832

, increasing ρ_p from 1000 to 3000 kg m⁻³ reduces d₅₀ from about 2.48 μm to about 1.43 μm. Because Re and χ_t are independent of ρ_p in the present formulation, the density effect is isolated through the particle relaxation time.

Figure 4 confirms the inverse dependence of cut size on particle density and shows that denser particles are separated more easily.

Because χ_t is unchanged in this one-factor test, the turbulence correction mainly shifts the curve upward without altering its overall trend.

3.3. Cyclone-Diameter Sensitivity

Under geometric similarity, Equation (30) implies d₅₀ ∝ D^1/2.

Table 4. Sensitivity of cut size to cyclone diameter D under geometric similarity (V_in = 20 m/s, ρ_p = 2000 kg m⁻³, μ = 1.8 × 10⁻⁵ Pa·s).

D [m]	Re [–]	χt [–]	d50 [µm]	d50,t [µm]
0.1	133,333	1.365	1.014	1.184
0.2	266,667	1.516	1.434	1.765
0.3	400,000	1.632	1.756	2.243
0.4	533,333	1.730	2.028	2.667
0.5	666,667	1.816	2.267	3.055

shows an expected increase in d₅₀ with cyclone size, from about 1.01 μm at D = 0.1 m to about 2.27 μm at D = 0.5 m. At the same time, Re increases with D, so χ_t also increases mildly and further enlarges the gap between d₅₀ and d_50,t for larger units.

Figure 5 shows that both d₅₀ and d_50,t increase with cyclone diameter under geometric similarity, consistent with the D^1/2 scaling implied by Equation (30).

The widening gap between the two curves indicates that the turbulence penalty becomes slightly stronger as the Reynolds number increases with D.

3.4. Gas Viscosity Sensitivity at V_in = 20 m/s

Because Equation (30) gives d₅₀ ∝ μ^1/2, a more viscous carrier gas leads to poorer separation performance. In

Table 5. Sensitivity of cut size to gas viscosity μ (V_in = 20 m/s, ρ_p = 2000 kg m⁻³, D = 0.3 m; all other parameters as in Table 1).

μ [Pa·s]	Re [–]	χt [–]	d50 [µm]	d50,t [µm]
1.5 × 10−5	480,000	1.693	1.603	2.085
1.8 × 10−5	400,000	1.632	1.756	2.243
2.5 × 10−5	288,000	1.537	2.069	2.565
3.0 × 10−5	240,000	1.490	2.267	2.767
4.0 × 10−5	180,000	1.424	2.618	3.124

, increasing μ from 1.5 × 10⁻⁵ to 4.0 × 10⁻⁵ Pa·s increases d₅₀ from about 1.60 μm to about 2.62 μm. Since Re decreases as μ increases, the turbulence penalty χ_t is reduced slightly in the proposed closure, which partly counteracts the direct viscosity effect but does not overturn it.

Figure 6 highlights the adverse effect of increasing gas viscosity on cyclone performance, with both threshold diameters shifting upward as μ increases.

The mild reduction in χ_t at a higher viscosity moderates the trend but does not compensate for the direct viscosity penalty.

3.5. Example Fractional Efficiency Curve

Table 6. Illustrative grade–efficiency values η(d_p) for the baseline case (β = 2), with and without turbulence correction.

dp [µm]	η without χt	η with χt
0.5	0.075	0.047
1	0.245	0.166
2	0.565	0.443
3	0.745	0.641
5	0.890	0.832
10	0.970	0.952

illustrates the shape of the logistic grade–efficiency curve. For d_p ≪ d₅₀, the collection efficiency is low; for d_p ≫ d₅₀, it approaches unity. Introducing the turbulence correction shifts the threshold toward larger particle diameters (d_50,t > d₅₀) and therefore reduces predicted collection efficiency in the fine-particle range, consistent with the intended interpretation of turbulent dispersion and re-entrainment.

Figure 7 translates the threshold shift from d₅₀ to d_50,t into a full grade efficiency.

The turbulence-corrected curve is displaced toward larger particle diameters and predicts lower collection efficiency in the fine-particle range, whereas both curves approach unity for coarse particles.

3.6. Sensitivity to Aggregated Parameters, Calibration Coefficients, and Extrapolation Capability

The purpose of this sensitivity analysis is to clarify the explicit role of the turbulence correction term χ_t within the reduced-order model and to demonstrate that its influence is physically consistent with expected trends in turbulent dispersion effects reported in cyclone literature.

Equation (30) explicitly separates the geometric contributions (a, b, D, D_e, H_eff) from the aggregated flow-field parameter K_s. In practice, K_s and, in the turbulence-corrected form, the product C_tI, control the effective swirl intensity and the penalty associated with turbulent dispersion. These parameters are not universal constants and should be calibrated for a given cyclone family and operating regime.

Table 7. Sensitivity of d₅₀ to K_s, H_eff, and D_e/D (baseline case: V_in = 20 m/s, D = 0.3 m, μ = 1.8 × 10⁻⁵ Pa·s, ρ_p = 2000 kg m⁻³).

Parameter	Value	d50 [µm]	d50,t [µm]
Ks	1.0	2.458	3.141
Ks	1.2	2.049	2.617
Ks	1.4	1.756	2.243
Ks	1.6	1.536	1.963
Ks	1.8	1.366	1.745
Heff/D	2	2.483	3.173
Heff/D	3	2.028	2.591
Heff/D	4	1.756	2.243
Heff/D	5	1.571	2.007
Heff/D	6	1.434	1.832
De/D	0.4	1.691	2.161
De/D	0.5	1.756	2.243
De/D	0.6	1.832	2.340
De/D	0.7	1.917	2.449

and

Table 8. Sensitivity of the turbulence factor χ_t to I and C_t (D = 0.3 m, V_in = 20 m/s, μ = 1.8 × 10⁻⁵ Pa·s, Re ≈ 4 × 10⁵).

Ct [–]	I [–]	χt [–]	d50,t [µm]
0.01	0.02	1.126	1.864
0.01	0.05	1.316	2.014
0.01	0.10	1.632	2.243
0.02	0.02	1.253	1.965
0.02	0.05	1.632	2.243
0.02	0.10	2.265	2.643
0.05	0.02	1.632	2.243
0.05	0.05	2.581	2.821
0.05	0.10	4.162	3.582

report one-at-a-time sensitivity calculations around the baseline case. The offset between d_50,t and d₅₀ is therefore only quasi-constant. Since d_50,t = d₅₀√χ_t, the absolute difference d_50,t − d₅₀ = d₅₀ (√χ_t − 1) depends both on the baseline threshold and on the turbulence factor. As a result, changes in V_in, D, or μ can modify the apparent penalty even when C_t and I are fixed, because they alter Re and/or the underlying value of d₅₀. The penalty is quasi-constant only over limited operating windows.

The results show that d₅₀ is strongly inversely dependent on K_s and decreases with the square root of H_eff. The effect of the ratio D_e/D is moderate but non-negligible, because a larger outlet tube moves the effective inner radius outward and increases the geometric term that appears in Equation (30). This behavior justifies keeping D_e/D explicit instead of absorbing it entirely into an effective number of turns.

In Equation (34), χ_t depends on C_tI and √Re. Consequently, uncertainty in either I or C_t can induce large changes in d_50,t, especially at a high Reynolds number. Without dedicated calibration, C_t should be treated as a bounded parameter and reported through sensitivity intervals rather than as a fixed universal constant.

Figure 8 shows that χ_t increases linearly with turbulence intensity I at a fixed Reynolds number, with the slope controlled by C_t.

Figure 8 therefore illustrates why uncertainty in either C_t or I can translate directly into significant uncertainty in the turbulence-corrected cut size.

This behavior reflects the role of χ_t as a lumped reduced-order closure rather than a turbulence model in the CFD sense. The results in Table 8 indicate that the turbulence penalty acts as a bounded, calibratable correction on the apparent cut size. Without independent information on I or matched calibration of C_t, the model is most appropriately reported with sensitivity intervals rather than as a universal deterministic prediction.

Taking together, Table 7 and Table 8 show that the practical extrapolation capability of the model depends primarily on the stability of K_s and of the product C_tI over the operating window of interest. If these closure parameters vary substantially with cyclone family, Re, or turbulence-level predictions outside the calibrated range become less robust and are more appropriately represented as bounded estimates rather than as single-point deterministic values. In that sense, the sensitivity study provides information on the regimes in which extrapolation remains approximately valid.

3.7. Pressure Drop and the d₅₀–Δp Trade-Off

From an engineering standpoint, a cyclone must be evaluated simultaneously by separation performance and pressure drop, because the energy penalty rises rapidly with Δp. Pressure drop is commonly expressed through a Euler-type coefficient, Eu, referenced to the inlet dynamic pressure. In the present work, Equation (38) is used as a transparent reduced-order estimate, while acknowledging that more detailed correlations exist [1,3,4,39].

$$\mathsf{\Delta }\mathrm{p}=\mathrm{E}\mathrm{u}\cdot {\displaystyle \frac{{\mathsf{\rho }}_{\mathrm{g}}\cdot {\mathrm{V}}_{\mathrm{i}\mathrm{n}}^2}{2}}$$

(38)

When no geometry-specific correlation for Eu is available, internal comparisons can be made by assuming Eu≈constant. In that case, Δp scales with V_in² while d₅₀ decreases approximately as V_in^−1/2. The hydraulic power P_f = ΔpQ then scales roughly as V_in³, which makes the separation–energy trade-off explicit.

The results in

Table 9. Trade-off between cut size and pressure drop as inlet velocity varies (Eu = 8; P_f = ΔpQ, fan efficiency not included).

Vin [m/s]	Q [m3/s]	Δp [kPa]	Pf [kW]	d50 [µm]	d50,t [µm]
10	0.090	0.48	0.043	2.483	2.987
15	0.135	1.08	0.146	2.028	2.522
20	0.180	1.92	0.346	1.756	2.243
25	0.225	3.00	0.675	1.571	2.052
30	0.270	4.32	1.166	1.434	1.910

show that higher inlet velocity improves d₅₀ but rapidly increases the energetic burden. Under the Eu ≈ constant assumption, the gain in threshold reduction is much weaker than the growth in pressure drop and power, which is why cyclone design is best treated as a constrained or multi-objective problem rather than as a single-efficiency optimization.

Figure 9 makes the separation–energy trade-off explicit: lower cut sizes are achieved only at the expense of rapidly increasing pressure drop.

The corrected threshold d_50,t remains above d₅₀ over the entire range, indicating that turbulence reduces the apparent gain obtained by increasing the operating intensity.

3.8. External Scale-Up Challenge Based on 1D3D (Barrel Length = 1 D and Cone Length = 3 D, Where D Is the Cyclone Barrel Diameter) Cyclones

As a stronger external scale-up test, the primary 1D3D data of Faulkner et al. [42] were used. Four geometrically similar cyclones with D = 15.24, 30.48, 60.96, and 91.44 cm were tested at similar inlet velocities; the mean inlet velocity across all runs was 922 m/min. The reported aerodynamic cut points were 3.4 ± 0.2, 3.9 ± 0.4, 4.0 ± 0.6, and 3.9 ± 0.4 μm, respectively. When Equation (30) is calibrated on the 15.24 cm cyclone and then extrapolated to the larger cyclones under strict geometric similarity while keeping the aggregated parameters constant, it overpredicts their cut sizes. This is informative because it indicates that K_s, the residence time structure, and the effective number of turns cannot be treated as scale-invariant closure parameters over large diameter changes, even within nominally similar 1D3D layouts

3.9. Source Validation Benchmark and Pressure Drop Check

To provide a more defensible quantitative validation step prior to predictive use, the benchmark was rebuilt around the experimental data of Kim et al. [43]. Two geometrically distinct small cyclones were considered. For Cyclone I, Table 1 of [43] gives D = 44 mm, a = 20 mm, b = 10 mm, D_e = 20 mm, and H = 160 mm, corresponding to a/D = 0.455, b/D = 0.227, D_e/D = 0.455, and H/D = 3.64. For Cyclone II, D = 30 mm, a = 12 mm, b = 6 mm, De = 15 mm, and H = 122 mm, giving a/D = 0.400, b/D = 0.200, D_e/D = 0.500, and H/D = 4.07. Because Equation (30) uses an effective residence height H_eff, whereas Kim et al. report the overall cyclone height H, the present benchmark uses H_eff ≈ H as a reduced-order residence time approximation for these small reverse-flow cyclones. No separate correction from H to H_eff was introduced because the source does not report a resolved vortex-reversal height or a residence time distribution.

It is important to clarify that the objective of the present validation is not to establish a universal predictive model across all cyclone configurations and operating regimes, but to assess the consistency of the proposed reduced-order formulation within the limits of family-based calibration. Within this framework, each cyclone family is characterized by similar geometric scaling and flow structure assumptions, allowing one calibration point to define the model parameters, while the remaining cases serve as independent verification points. This validation philosophy is consistent with the reduced-order nature of the model and with the fact that key physical effects (e.g., turbulence and re-entrainment) are represented through lumped correction terms rather than fully resolved flow fields.

For Cyclone I, the conventional point B0 (Q_in = 80 L/min, Q_minor = 0) was used to calibrate K_s and Eu, after which three independent points B1, B5, and B6 were used for verification. For Cyclone II, the mid-range point E2 (Q_in = 50 L/min, Q_minor = 4 L/min) was used for calibration, and E1, E3, and E4 were used for verification. The pooled benchmark, therefore, contains six independent d₅₀ cases and six paired Δp cases.

Across the six independent cases, the present model gives a mean absolute percentage error of 13.5% and a root mean square error (RMSE) of 0.22 μm for d₅₀. On the same verification set, the pressure drop check gives a mean absolute percentage error of 2.8% and an RMSE of 34 Pa. The Cyclone II subset is reproduced more accurately, with a mean absolute percentage error (MAPE) of 9.7% for d₅₀, than the moderate-bleed Cyclone I subset (MAPE 17.3%), indicating that the present closure captures conventional intra-family scaling better than the sensitivity to increasing minor-flow extraction.

Table 10. Source validation benchmark from Kim et al. [43]. Cal. = calibration; Ver. = verification. Independent verification cases are B1, B5, B6, E1, E3, and E4.

Case	Qin [L/min]	Qmaj [L/min]	Qmin [L/min]	Vin [m/s]	d50,exp [μm]	d50,pred [μm]	Error [%]	Δpexp [Pa]	Δppred [Pa]	Δperr. [%]
B0 (Cal.)	80	80	0	6.67	2.02	2.02	+0.00	473.7	473.7	0.00
B1 (Ver.)	80	76	4	6.67	1.82	2.02	+10.99	478.6	473.7	−1.02
B5 (Ver.)	84	80	4	7.00	1.74	1.97	+13.29	522.7	522.3	−0.09
B6 (Ver.)	88	80	8	7.33	1.51	1.93	+27.55	598.2	573.2	−4.18
E1 (Ver.)	40	36	4	9.26	1.31	1.25	−4.41	299.1	301.3	+0.74
E2 (Cal.)	50	46	4	11.57	1.12	1.12	+0.00	470.8	470.8	0.00
E3 (Ver.)	60	56	4	13.89	0.96	1.02	+6.50	697.3	678.0	−2.77
E4 (Ver.)	70	66	4	16.20	0.80	0.95	+18.32	1000.3	922.8	−7.75

lists the source benchmark in full. All six verification cases include measured pressure drops, so the pressure drop assessment is no longer based on a single point. Figure 10 shows that most benchmark points lie reasonably close to the parity line, particularly for the Cyclone II cases. The largest positive deviations occur for the moderate-bleed Cyclone I points, supporting the view that the present closure reproduces intra-family scaling better than the effect of increasing minor-flow extraction.

Within the verified benchmark, the model preserves the expected decrease in d₅₀ with increasing inlet flow rate for Cyclone II and reproduces pressure drop scaling very well for both cyclone families. For Cyclone I, the d₅₀ deviations grow as the minor-flow fraction increases, which is physically plausible because the present reduced-order closure does not explicitly resolve the altered residence time distribution produced by the extracted bottom flow.

Table 11. External scale-up challenge using 1D3D cut-point data from Faulkner et al. [42]. F1* is the calibration point; F2-F4 are independent challenge cases.

Case	D [m]	Vin [m/s]	d50,exp [μm]	Slope [–]	d50,pred [μm]	Error [%]
F1*	0.1524	15.37	3.40	1.47	3.40	0.00
F2	0.3048	15.37	3.90	1.47	4.81	+23.29
F3	0.6096	15.37	4.00	1.79	6.80	+70.00
F4	0.9144	15.37	3.90	2.29	8.33	+113.55

summarizes the external scale-up challenge from Faulkner et al. [42]. Using the 15.24 cm cyclone as a calibration point gives reasonable order-of-magnitude predictions, but the remaining 30.48–91.44 cm cyclones are overpredicted by 23–114%. The result is valuable because it clearly delimits the model’s domain of defensible use: calibration transfers well within a geometric family over nearby operating conditions, but not across large diameter changes when the experimental cut size remains nearly invariant.

Taken together, Table 10, Table 11 and

Table 12. Additional literature benchmark from Wang et al. [44] for standard 15 cm 1D3D and 2D2D reverse-flow cyclones tested with fly ash.

Case	Family	Velocity Treatment	Vin,model [m/s]	ρg [kg m−3]	d50,exp [μm, AED]	d50,pred [μm, AED]	Error [%]	Slope [–]	Δpexp [Pa]	Δppred [Pa]	Δperr [%]
W1 Cal.	1D3D	16 standard air	19.0	1.02	3.40	3.40	0.00	1.43	1238	1238	0.00
W2 Ver.	1D3D	16 actual air	16.0	1.02	3.90	3.71	−5.00	1.29	755	878	+16.28
W3 Cal.	2D2D	15 standard air	18.0	1.01	4.00	4.00	0.00	1.30	827	827	0.00
W4 Ver.	2D2D	15 actual air	15.0	1.02	4.20	4.38	+4.33	1.23	580	580	0.00

Note: Calibration is performed using one reference operating point per cyclone family; all remaining cases are treated as independent validation points.

support the use of the present formulation as a family-calibrated reduced-order design tool rather than as a universal predictive model.

In Table 10, d₅₀ predictions from Equation (30) are compared directly with the values reported by Kim et al. [43], and the corresponding pressure drop predictions from Equation (38) are reported using a family-specific Eu calibrated on B0 and E2. Ambient-air properties (μ = 1.8 × 10⁻⁵ Pa·s, ρ_g = 1.2 kg m⁻³) and ρ_p = ρ₀ = 1000 kg m⁻³ were used so that d_p ≈ d_ae for order-of-magnitude comparison. This approximation should be kept in mind when interpreting the absolute errors.

In Table 11, the 1D3D challenge cases are based directly on the cut-point measurements of Faulkner et al. [42]. Standard 1D3D ratios a/D = 0.5, b/D = 0.25, D_e/D = 0.5, and H_eff/D ≈ 4 were used consistently with the reported geometry and the published 1D3D layout. A parity plot comparing model-predicted and experimental d₅₀ values is presented in Figure 10.

The parity plot shown in Figure 10 is consistent with the aggregated MAPE and RMSE values reported above and indicates that the model is more reliable within a calibrated family than under larger perturbations of the flow split.

The slope values reported by Faulkner et al. [42] are retained in Table 11 because they also show that cut sharpness changes with scale even when the cut size itself remains nearly constant.

An additional validation layer was constructed from Wang et al. [44], who reported cut point, slope, and pressure drop for standard 15 cm 1D3D and 2D2D reverse-flow cyclones tested with fly ash under two air-density/velocity conditions. Since the source reports the cut point in terms of aerodynamic equivalent diameter (AED), and fly ash is characterized by a particle density of 2.7 g cm⁻³, the comparison was carried out in an AED-consistent form, consistent with the d_p–d_ae mapping introduced in Section 2.1.

The standard-air design condition was used as the calibration point for each cyclone family, following the one-point family calibration strategy already applied to the Kim et al. benchmark, while the corresponding actual-air condition was treated as an independent verification point. Wang et al. report that, at the stated air densities, 16 standard m s⁻¹ for the 1D3D cyclone corresponds to 19 actual m s⁻¹, and 15 standard m s⁻¹ for the 2D2D cyclone corresponds to 18 actual m s⁻¹. Under this split, the present model predicts 3.71 μm for the 1D3D actual-air case versus 3.90 μm experimentally, and 4.38 μm for the 2D2D actual-air case versus 4.20 μm experimentally.

The corresponding errors are therefore −5.0% and +4.3%, which yields a two-point verification MAPE of 4.7% and an RMSE of 0.19 μm.

The paired pressure drop check gives 878 Pa versus 755 Pa for the 1D3D actual-air case and 580 Pa versus 580 Pa for the 2D2D actual-air case. The corresponding pressure drop errors are +16.3% and approximately 0.0%, i.e., a two-point verification MAPE of 8.1% and an RMSE of 87 Pa.

This benchmark extends the validation beyond small virtual cyclones to conventional 1D3D and 2D2D reverse-flow families under different operating conditions. It provides additional evidence on the performance of the present formulation as a family-calibrated reduced-order tool, while broader multi-family validation remains necessary for generalization.

In Table 12, standard-air rows were used as family-specific calibration points, while the corresponding actual-air rows were treated as independent verification cases. Cut points are reported in an AED-consistent form.

Overall, Table 10 and Table 12 show that the model reproduces the correct order of magnitude of d₅₀ and Δp across small-cyclone and standard reverse-flow families when one calibration point per family is supplied. Table 11 simultaneously shows that strict similarity with constant K_s fails under cross-scale extrapolation.

Across the three benchmarks, the results indicate that the formulation behaves as a family-calibrated reduced-order tool, while its applicability as a universal predictive law is not supported.

This behavior is consistent with the role of the model, K_s, C_t, and, in practice, I, which are treated as aggregated closure parameters rather than universal constants. The formulation can be positioned between classical algebraic models and recent Computational Fluid Dynamics (CFD)/Machine Learning (ML) hybrid approaches [28,29,30,31,32,33], in the sense that it is more transparent and computationally efficient than full CFD, while still requiring family-specific calibration for quantitative use.

A minimal calibration set contains at least one cut-size datum and, in addition, a measured grade–efficiency curve and a pressure drop datum for the geometry of interest. In the Stokes regime, K_s can be recovered directly by inverse application of Equation (30). If an estimate of turbulence intensity I is available, C_t can likewise be obtained from χ_t = (d_50,t/d₅₀)² = 1 + C_tI√Re. The present data split illustrates this strategy through calibration on B0 and E2 and verification on B1, B5, B6, E1, E3, and E4.

The logistic parameter β controls the sharpness of the grade–efficiency curve. When the literature reports sharpness in the Faulkner form Slope = √(d₈₄.₁/d₁₅.₉) [42], the logistic relation η(d_p) = 1/[1 + (d₅₀/d_p)^β] [10] gives the approximation β ≈ ln(27.97)/(2ln(Slope)). This provides a direct way to estimate β from reported percentile diameters without iterative fitting.

After calibration, validation should test whether K_s and C_t remain stable over the operating window of interest and whether the same particle-diameter definition is used across all reference data. If this consistency cannot be ensured, the model should be reported together with sensitivity intervals rather than as a single deterministic prediction.

3.10. Comparison with Widely Used Model Families

Two complementary comparisons are presented to relate the present formulation to established cyclone models.

Table 13. Comparison between the present model and the standard Lapple correlation on the six independent Kim et al. [43] verification cases.

Case	d50,exp [μm]	Present Model [μm]	Error [%]	Lapple [μm]	Error [%]
B1	1.82	2.02	+11.0	2.54	+39.5
B5	1.74	1.97	+13.2	2.48	+42.5
B6	1.51	1.93	+27.8	2.42	+60.4
E1	1.31	1.25	−4.6	1.55	+18.4
E3	0.96	1.02	+6.3	1.27	+31.9
E4	0.80	0.95	+18.7	1.17	+46.5
MAPE	—	—	13.5	—	39.9
RMSE	—	—	0.22	—	0.60

provides a direct quantitative comparison with the standard Lapple cut-size correlation on the six independent Kim et al. verification cases.

Table 14. Structured comparison between the present model and widely used cyclone-model families.

Model Family	Main Output(s)	Treatment of Geometry	Treatment of Turbulence/Fine-Particle Losses	Calibration Need	Computational Cost	Main Strength	Main Limitation
Lapple-type [1]	d50	Geometry largely absorbed into effective turns/empirical constants	Implicit	Low to moderate	Very low	Very simple first-pass design estimate	Limited explicit treatment of geometry and turbulence
Leith–Licht [8]	Collection/grade efficiency	Semi-empirical, not fully explicit in separate geometry ratios	Implicit	Moderate	Very low	Established efficiency framework	Less transparent separation of geometry, swirl, and residence time effects
Iozia–Leith [9,10]	Fractional efficiency, d50, sharpness	Semi-empirical/logistic representation	Implicit through fitted parameters	Moderate	Low	Efficient representation of grade–efficiency shape	Does not provide an explicit unified d50-Δp framework
Present model	d50, d50,t, η(dp), Δp	Explicit in a/D, b/D, De/D, Heff/D and Ks	Explicit reduced-order correction χt	Family- specific	Low	Transparent geometry retention and unified reduced-order framework	Not universal; turbulence correction is phenomenological
CFD/CFD–DEM [27,29,38]	Flow field, trajectories, efficiency, Δp	Fully explicit	Locally resolved/modeled	Case-specific setup	High	Highest physical detail	High computational cost; less suitable for rapid screening

summarizes the main structural differences between the present model and other widely used reduced-order approaches cited in the Introduction, namely Lapple-type, Leith–Licht, and Iozia–Leith formulations, as well as high-fidelity CFD/CFD–DEM approaches.

A fully matched one-to-one numerical comparison against transient 3D CFD or CFD–DEM simulations is not included, as case-matched high-fidelity results for the same geometries and operating conditions are not available in the benchmark datasets used in this study, and generating such simulations lies beyond the scope of the present work. The comparison with CFD/CFD–DEM is instead framed in terms of resolved physics, calibration requirements, and computational cost.

Compared with transient 3D CFD/CFD–DEM approaches, the present model does not resolve the local gas flow field, anisotropic turbulence, vortex-core dynamics, residence time distributions, or individual particle trajectories. Its role is complementary, as it provides reduced-order estimates for d₅₀, η(d_p), and Δp at negligible computational cost, whereas high-fidelity simulations provide greater physical detail at significantly higher setup and runtime cost. Table 14 summarizes this difference in modeling scope relative to numerical equivalence.

Table 13 compares the six independent Kim et al. [43] verification cases with the standard Lapple cut-size expression as a classical reduced-order reference for Equation (30). The Lapple baseline is used because it allows a direct evaluation of cut size from the geometric quantities reported by Kim et al., without introducing additional case-specific fitting parameters. The effective number of turns was estimated as N = (h + 0.5 (H − h))/a, where h denotes the cylindrical barrel height. The same air properties and the ρ_p = 1000 kg m⁻³ assumption used in Table 10 were retained.

On the six verification cases, the present model yields a MAPE of 13.5% and an RMSE of 0.22 μm, whereas the standard Lapple correlation gives 39.9% and 0.60 μm, respectively. The classical correlation shows a consistent overprediction of the cut size, particularly for the smaller Cyclone II family, while the present formulation better captures the observed decrease in d₅₀ across both families.

This comparison is made in the context of different model formulations. The Lapple values in Table 13 are generic published predictions, whereas the present model is used here as a family-calibrated reduced-order closure. Moreover, the Kim benchmark keeps geometry fixed within each family and mainly probes flow rate variation. Under these conditions, a one-point re-fit of a Lapple-type constant collapses to a similar d₅₀ ∝ V_in^−1/2 trend as in Equation (30). The present formulation retains an explicit dependence on D, a/D, b/D, D_e/D, and H_eff/D, and can be extended consistently through the turbulence correction, non-Stokes update, logistic efficiency curve, and pressure drop estimate.

Present-model values are those reported in Table 10. The Lapple baseline is evaluated in its standard published form, without additional case-specific re-fitting.

Table 14 clarifies this positioning more broadly. Relative to classical reduced-order models, the present framework retains the principal geometry ratios explicitly and links d₅₀, turbulence-penalized threshold, grade–efficiency shape, and pressure drop estimate within one consistent formulation. Relative to CFD/CFD–DEM approaches, its advantage lies in transparency and computational efficiency rather than in flow-field fidelity. The price of that simplicity is that K_s, C_t, and, in practice, I, remain family-specific closure parameters and must be calibrated before quantitative use.

A case-matched comparison against transient 3D CFD/CFD–DEM simulations on the same cyclone geometries would be a valuable next validation step.

The comparison framework presented in this section is intended not only to quantify predictive accuracy, but also to highlight the structural differences between modeling approaches. Classical correlations such as the Lapple model rely on strongly lumped empirical representations of cyclone behavior, where geometric and flow effects are implicitly embedded into fitted parameters. In contrast, reduced-order formulations such as the present model and those of Leith–Licht or Iozia–Leith retain partial physical structure but differ in how explicitly geometry, turbulence, and efficiency coupling are represented. High-fidelity CFD and CFD–DEM approaches, while more accurate in principle, operate at a fundamentally different level of resolution and are not directly comparable in terms of computational cost or design usability. It is therefore important to note that no single model dominates across all criteria. While the proposed formulation improves accuracy relative to classical empirical correlations such as Lapple, it retains the reduced-order assumptions and calibration dependence characteristic of semi-empirical approaches. This trade-off between physical interpretability, computational efficiency, and predictive accuracy is inherent to all reduced-order cyclone models.

3.11. Practical Domain of Validity and Limitations

The proposed reduced-order formulation is intended for reverse-flow gas–solid cyclones with tangential inlet operating under dilute, one-way coupled conditions. The analytical cut-size expression derived in Equation (30) is valid when particle motion remains predominantly within the Stokes regime, i.e., Rep at d₅₀ < 1. In the present baseline case, the estimated Rep ≈ 0.025 confirms this assumption for fine-particle separation, consistent with classical drag-regime criteria [41]. This condition should be re-evaluated when geometry, flow rate, or fluid properties change significantly.

The Schiller–Naumann correction is used to extend applicability to moderate deviations from Stokes drag, but it does not cover strongly inertial or transitional regimes [41]. For such cases, more advanced Lagrangian or CFD–DEM approaches are required.

The flow field is idealized as a free-vortex structure in the outer region, while the forced-vortex core, inlet asymmetry, and secondary flow structures are not explicitly resolved. Residence time is approximated using a global volume-to-flow ratio, rather than trajectory-resolved particle motion, which may introduce deviations for geometries outside the calibrated family.

Turbulence effects are included through a phenomenological correction factor χ_t, which aggregates dispersion and re-entrainment effects into a reduced-order closure. As a consequence, parameters such as C_t and swirl intensity require calibration using experimental data or high-fidelity simulations.

Particle–particle interactions, wall collision effects, and high-solids loading are not included. Similarly, the Cunningham slip and Brownian diffusion are neglected, limiting applicability in submicron aerosol regimes [41]. For coarse particles, deviations from Stokes flow require additional corrections beyond the present framework.

Pressure drop is estimated using a constant Euler number approximation, which should be interpreted as an engineering-level closure. In practice, Eu depends on the Reynolds number, geometry, and surface roughness. Additionally, the absence of an explicit minor-flow term restricts validity to moderate bleed ratios.

Finally, scale-up behavior is not strictly preserved under constant-K_s similarity. External validation indicates that geometric similarity assumptions may fail under large-scale transitions, requiring family-specific calibration [42].

3.12. Future Development of the Model

The Future improvements focus on reducing reliance on empirically calibrated parameters such as K_s, C_t, and I by integrating experimental datasets or CFD/CFD–DEM simulations for specific cyclone geometries. An important extension involves modeling non-uniform residence time distributions and explicitly including minor-flow extraction effects, which are currently lumped into reduced-order terms. Further refinement may also incorporate particle–wall interaction models for higher-solids loading conditions.

From a validation perspective, multi-family benchmark datasets reporting cut size, grade–efficiency curves, and pressure drop under consistent particle-size definitions are required. Such datasets would allow rigorous testing of geometry-dependent scaling laws and improve model generality across cyclone families [43,44]. For fine-particle applications, future extensions may include Cunningham slip correction, Brownian diffusion, and particle-shape effects through dynamic shape factors [41].

Ongoing work by the authors addresses uncertainty-aware reduced-order modeling, where variability in K_s, C_t, and I is propagated to evaluate robustness of design trends under bounded uncertainty.

4. Conclusions

A semi-empirical reduced-order model for reverse-flow cyclone separators has been developed based on the particle equation of motion in cylindrical coordinates combined with an idealized swirling flow field. In the Stokes regime, the model yields a closed-form expression for d₅₀ that explicitly retains the influence of fluid properties, particle density, inlet velocity, and key geometric ratios, enabling transparent sensitivity analysis.

A turbulence correction factor and a logistic grade–efficiency function were introduced to extend applicability toward fine-particle separation, while a simplified Euler-based relation was used to estimate pressure drop and highlight energy–separation trade-offs. Validation against the benchmark dataset of Kim et al. [43] shows a mean absolute percentage error of 13.5% for d₅₀ and 2.8% for pressure drop. Additional comparison with scale-up cases reported by Faulkner et al. [42] confirms that constant-K_s geometric similarity is not universally transferable across cyclone sizes. Complementary experimental results for 1D3D and 2D2D configurations reported by Wang et al. [44] further support the need for family-specific calibration under different operating conditions.

Overall, the model is suitable for preliminary design, sensitivity analysis, and family-calibrated prediction of reverse-flow cyclones under dilute operating conditions. Its predictive accuracy decreases for high-solids loading, strong bleed flows, submicron regimes without slip correction, and large-scale extrapolation without recalibration.

The formulation provides a transparent reduced-order framework that preserves physical interpretability while enabling rapid design exploration. Further improvements require enhanced closure modeling and systematic experimental or CFD-based validation across multiple cyclone families.