Hydraulic Optimization of a Low-Pressure Hydrocyclone for Drip-Irrigation Pretreatment: Effects of Inlet Geometry, Vortex-Finder Submersion, and Underflow Diameter

2.1 Experimental design and operating conditions

A controlled laboratory experiment was conducted to determine the effects of geometric and operating parameters on pressure characteristics and water-flow redistribution in a low-pressure hydrocyclone. Clean water was used as the working liquid. The principal response variables were the flow rates and relative flow ratios at the overflow and underflow outlets.

The experimental factors summarized in Table 1 included inlet geometry, inlet pressure, inlet orientation in the vertical and horizontal planes, vortex-finder submersion, and underflow outlet diameter. Each operating condition was tested in triplicate.

Three interchangeable inlet nozzles with circular, square, and semicircular cross-sections were examined. The circular nozzle represented the conventional reference configuration. The square nozzle was selected to evaluate the effect of flat walls and corner regions on tangential-flow development, whereas the semicircular nozzle combined a flat guiding boundary with a curved outer boundary. To isolate the effect of inlet geometry, the three nozzles had equal cross-sectional areas [26]. The common inlet area corresponded to a circular inlet diameter of 36 mm and was calculated as 1,017.9 mm². On an equal-area basis, this corresponds to a nominal square side length of 31.9 mm and a nominal semicircular diameter of 50.9 mm.

The inlet pressure was varied at three levels: 0.1, 0.3, and 0.5 kgf/cm², corresponding to 9.81, 29.42, and 49.03 kPa, respectively. These pressures are equivalent to hydraulic heads of approximately 1, 3, and 5 m and therefore represent very-low-pressure and gravity-assisted operating conditions relevant to small-scale and energy-constrained drip-irrigation systems. The pressure of 9.81 kPa was used as the lower-bound operating condition, whereas 29.42 and 49.03 kPa corresponded to the pressure heads generated by water-storage tanks elevated by approximately 3 and 5 m.

The inlet inclination in the vertical plane, denoted by θ, was set at 0°, 2°, 4°, and 6°. The inlet deviation in the horizontal plane, denoted by φ, was set at 0° and 2°. The relative vortex-finder submersion was expressed as:

$\alpha_c=\frac{h_{f d}}{H_{c y l}}$            (1)

where, h_fd is the vortex-finder submersion depth, and H_cyl is the height of the cylindrical section. The tested values of α_c were 0, 0.77, and 1.00, corresponding to absolute submersion depths of 0, 84.7, and 110 mm, respectively. These levels represented no submersion, partial submersion, and submersion to the cylinder-cone junction.

Three interchangeable underflow outlets with diameters of 9, 11, and 18 mm were tested. Their respective cross-sectional areas were 63.6, 95.0, and 254.5 mm². Thus, the selected diameters provided an approximately fourfold range in underflow area and represented restricted, intermediate, and comparatively wide underflow conditions. The corresponding d_sand/D_cyl ratios were 0.049, 0.060, and 0.098.

Table 1. Geometric and operating parameters of the experimental hydrocyclone

The parameter β = 15° denotes the cone half-angle measured between the conical wall and the longitudinal axis of the hydrocyclone. The corresponding full included cone angle was therefore 30°.

2.2 Experimental hydrocyclone-pump unit

The experimental recirculation unit was assembled using a 1.5-K-6 centrifugal pump and an adjustable cylindrical-conical hydrocyclone, as shown in Figure 1.

The unit consisted of an electric motor, centrifugal pump, suction pipeline, pump discharge pipeline, regulating valve, inlet pressure gauge, flexible pressure hose, interchangeable inlet nozzle, horizontal and vertical adjustment brackets, fixing bolt, protractor, hydrocyclone chamber, adjustable overflow pipe, overflow-line pressure gauge, overflow hose, interchangeable underflow outlet, main tank, auxiliary collection tank, and drain valve.

The regulating valve installed in the pump discharge line was used to establish the prescribed inlet pressure. The inlet assembly included a flexible connection and independently adjustable vertical and horizontal brackets. This arrangement allowed the vertical inclination θ and horizontal deviation φ of the inlet nozzle to be changed without replacing the hydrocyclone body.

The overflow pipe was installed through the upper hydrocyclone cover using a threaded connection, sealing gasket, and lock nut. Its vertical position could therefore be adjusted and fixed at the prescribed submersion depth. The interchangeable underflow outlets were installed at the lower opening of the cone.

Pressure was recorded at two measurement points: immediately upstream of the hydrocyclone inlet and in the overflow line. Accordingly, the experiment measured inlet and overflow-line pressures rather than the spatial distribution of pressure inside the hydrocyclone chamber. The inlet and overflow-line pressures were recorded using hydraulic pressure gauges. Outlet-water volumes were determined using a graduated measuring vessel, the sampling duration was measured using a stopwatch, and the inlet angles were set using a protractor. The same measuring devices and collection procedure were used throughout all experimental series.

The geometric arrangement of the hydrocyclone and the positions of the adjustable structural elements are shown in Figure 2.

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2.3 Experimental procedure

The main tank was filled with clean water. The pump was switched on, and the inlet pressure was adjusted using the regulating valve. The experiments were first performed at 0.1 kgf/cm² and were subsequently repeated at 0.3 and 0.5 kgf/cm².

Before each experimental series, the required inlet geometry, vertical inclination, horizontal deviation, vortex-finder submersion, and underflow outlet diameter were set. The circular, square, and semicircular nozzles were installed consecutively. The vertical inlet inclination was varied from 0° to 6°, the horizontal deviation was varied from 0° to 2°, the vortex-finder submersion ratio was varied from 0 to 1.00, and the underflow diameter was varied from 9 to 18 mm.

Water entered the cylindrical section tangentially and formed an outer downward-rotating flow and an inner upward flow. The upward stream left the hydrocyclone through the overflow pipe, whereas the lower stream left through the underflow outlet. Because clean water was used, the two outlet streams are hereafter referred to as the overflow and underflow streams.

The overflow and underflow volumes were collected simultaneously over the same time interval. During sampling, the overflow hose was redirected from the main tank to the auxiliary collection tank. At the end of the sampling interval, the hose was returned to the main tank, and the collected overflow water was transferred to a graduated measuring vessel. The underflow volume was collected separately and determined by the same volumetric method. The common collection interval was measured using a stopwatch.

The overflow flow rate was calculated as:

$Q_{d r}=\frac{w_{d r}}{t}$          (2)

where, Q_dr is the overflow flow rate, L s^-1; W_dr is the collected overflow volume, L; and t is the sampling interval, s.

The underflow flow rate was calculated as:

$Q_{\text {sand}}=\frac{w_{\text {sand}}}{t}$        (3)

where, Q_sand is the underflow flow rate, L s^-1, and W_sand is the collected underflow volume, L.

The inlet flow rate was determined from the hydraulic balance:

$Q_{\text {input}}=Q_{d r}+Q_{\text {sand}}$          (4)

The overflow and underflow flow ratios ($R_{d r}$ and $R_{\text {sand}}$) were calculated as:

$R_{d r}=\frac{Q_{d r}}{Q_{\text {input}}}$        (5)

$R_{\text {sand}}=\frac{Q_{\text {sand}}}{Q_{\text {input}}}$        (6)

Because the two outlet streams constituted the total inlet flow:

$R_{d r}+R_{\text {sand}}=1$       (7)

The overflow ratio R_dr represented the proportion of inlet water discharged through the overflow pipe, whereas the underflow ratio R_sand represented the proportion discharged through the lower outlet. The ratios were used as complementary indicators of hydraulic flow redistribution.

The pressure difference between the inlet and overflow line was determined as:

$\Delta P=P_{i n}-P_{d r}$           (8)

where, P_in is the measured inlet pressure and P_dr is the pressure measured in the overflow line.

2.4 Dimensionless hydraulic analysis

The inlet velocity was calculated from the total inlet flow rate and the common inlet cross-sectional area:

$V_{\text {input}}=\frac{Q_{\text {input}}}{A_{\text {in}}}$           (9)

where, V_input is the mean inlet velocity, m s^-1, and A_in = 1.0179 × 10^-3 m².

The measured inlet pressure was converted to an equivalent pressure head:

$H_{\text {input}}=\frac{P_{\text {in}}}{p_\omega g}$          (10)

where, H_input is the inlet pressure head, m; $p_\omega$ is the density of water, kg m^-3; and g = 9.81 m s^-2. For the three tested pressures, H_input was approximately 1, 3, and 5 m.

A dimensionless inlet-energy parameter was calculated as:

$X=\frac{V_{\text {input}}^2}{g\left(H_{\text {input}}+T\right)}$          (11)

where, T = 0.405 m is the total internal height of the hydrocyclone. The use of pressure head in Eq. (10) ensured dimensional consistency.

The hydraulic relationships were expressed as:

$R_{\text {sand}}=f(X)$         (12)

$R_{d r}=f(X)$      (13)

The inverse relationship proposed for the underflow ratio was evaluated as an empirical regression model:

$R_{\text {sand}}=\frac{N}{X}$           (14)

where, N is a fitted, configuration-dependent coefficient. The coefficient was estimated separately for the tested inlet geometries rather than being treated as a universal hydrocyclone constant. Its experimentally observed order of magnitude was approximately 10^-3.

2.5 Statistical and regression analysis

Each operating condition was tested three times, and the arithmetic mean of the three measurements was used to calculate the hydraulic response variables and construct the flow-ratio curves. Because R_dr and R_sand are complementary variables, R_dr was used as the principal response variable in inferential comparisons, while R_sand was retained for engineering interpretation of underflow discharge.

The effects of inlet geometry were evaluated using analysis of covariance, with inlet geometry as the categorical factor and the dimensionless inlet-energy parameter X as the continuous covariate. Comparisons among the three underflow outlet diameters and among the vortex-finder submersion levels were performed using one-way analysis of variance for operating conditions with equivalent pressure and inlet orientation. Tukey’s honestly significant difference test was used for pairwise post-hoc comparisons. Statistical significance was accepted at p < 0.05.

The inverse model for R_sand and the nonlinear response curves for R_dr were fitted by least-squares regression. Model performance was evaluated using the coefficient of determination (R²), adjusted R², root mean square error (RMSE), and mean absolute error (MAE). Model residual standard deviation and bootstrap 95% confidence intervals based on 10,000 resamples were used to characterize uncertainty in the fitted relationships.

Common-X comparisons and the X values corresponding to specified overflow ratios were estimated using shape-preserving piecewise cubic Hermite interpolation (PCHIP), which preserves the monotonicity of the experimental response curves.

The approximate transition to a stable hydraulic regime was evaluated using segmented regression. The breakpoint and its 95% confidence interval were estimated separately for the inlet geometries, thereby avoiding selection of the convergence threshold solely through visual inspection of the curves.

Statistical calculations and regression fitting were performed in Python 3.13.5 using SciPy 1.17.0 and statsmodels 0.14.6.

4.1 Influence of inlet energy on hydraulic performance

The principal finding of the study was that hydraulic flow redistribution depended predominantly on the dimensionless inlet-energy parameter. In both datasets, the statistical effect of X was highly significant, whereas the overall effect of inlet geometry was not. Inlet geometry therefore influenced the location and rate of transition toward the high-overflow regime, but sufficiently high inlet energy caused all three configurations to converge toward similar terminal flow ratios.

This behavior is physically consistent with the development of rotational flow inside a hydrocyclone. Increasing inlet velocity raises tangential momentum and strengthens the outer rotating stream. The more organized vortex directs a larger proportion of the liquid toward the internal upward flow and overflow pipe, while the relative underflow fraction decreases. The progressive convergence of the three inlet geometries indicates that, after a sufficiently strong vortex has formed, the influence of the initial nozzle boundary shape becomes secondary.

Gençoğlan et al. [19] similarly found that water velocity was a major determinant of hydrocyclone performance in micro-irrigation pretreatment. Average separation efficiency increased from 71% at 1.0 m s^-1 to 84% at 1.5 m s^-1 and 90% at 2.0 m s^-1. Their reported average efficiencies for three hydrocyclone geometries were 69%, 88%, and 88%, indicating that the effects of operating velocity could be as important as, or greater than, differences among structural configurations. These findings are consistent with the present model-based result showing a significant X effect but a nonsignificant overall inlet-geometry effect.

The nearly inverse exponent obtained for the broad-range underflow response provides a quantitative explanation for the asymptotic form of Figure 3. However, the fitted coefficient N should be applied only within the tested pressure and geometric ranges. Its magnitude may change with hydrocyclone diameter, cone angle, vortex-finder diameter, underflow diameter, surface roughness, or the presence of an air core. The fitted coefficient is therefore an engineering correlation rather than a universal hydrocyclone constant.

4.2 Mechanistic interpretation of inlet geometry

The semicircular inlet did not provide a statistically higher terminal overflow ratio. Instead, it reduced the value of X required to reach the high-overflow region. This distinction is important because it identifies the principal benefit of the geometry as a reduction in the hydraulic input required to establish a favorable flow split.

The observed behavior may be attributed to the combination of a flat guiding boundary and a curved outer boundary. The flat section can direct the incoming stream tangentially along the hydrocyclone wall, while the curved section avoids the two pronounced corner regions present in a square inlet. This configuration may reduce local flow separation and produce a more coherent peripheral vortex.

The square inlet also reached the high-overflow region earlier than the circular reference inlet. However, its corner regions may create localized recirculation zones and nonuniform velocity gradients. The circular inlet avoids corners but lacks the flat guiding surface of the semicircular inlet. These competing effects explain why differences were apparent during vortex development but became small after the flow approached its terminal hydraulic regime.

The proposed explanation is based on the measured outlet-flow response. Internal tangential velocity, turbulence intensity, air-core dimensions, and recirculation zones were not directly visualized. Consequently, the mechanism should be confirmed in future studies using particle-image velocimetry or validated computational fluid dynamics.

4.3 Effects of inlet inclination and vortex-finder submersion

A moderate vertical inclination of 2°–4° produced the most favorable hydraulic response. A small downward component of inlet momentum can support the external descending vortex and promote rapid development of rotational flow. At θ = 0°, this downward component is absent. At θ = 6°, the stronger axial component may increase asymmetry and mixing between the outer descending and inner ascending streams. Therefore, the optimal condition was not the greatest tested inclination but an intermediate range that supported vortex development without excessive axial disturbance.

The comparatively weak influence of changing the horizontal deviation from 0° to 2° indicates that small departures from the nominal tangential direction do not substantially alter the hydraulic flow split under most operating conditions. The stronger response observed at θ = 6°, α_c = 0, and φ = 2° suggests an interaction among the orientation parameters rather than a uniform independent effect of horizontal deviation.

Increasing vortex-finder submersion shifted the flow toward the overflow outlet, particularly at θ = 2°–4°. A deeper vortex finder places the overflow entrance farther from the inlet disturbance and can reduce direct short-circuit flow between the inlet and overflow opening. It may also improve the organization of the internal ascending stream and reduce the sensitivity of the flow split to small variations in inlet orientation.

The present hydraulic optimum is consistent with previously published two-phase testing of an adjustable hydrocyclone, in which the greatest clarification was reported for a semicircular inlet, a vertical inclination of 2°–4°, and vortex-finder submersion to the cylinder-cone junction [26]. That earlier particle-laden experiment provides supporting evidence for the practical relevance of the configuration identified by the current clean-water hydraulic analysis; it is not presented here as a new result of the present experiment.

The recommended submersion ratio of 0.77–1.00 remains specific to the tested cylinder height, cone geometry, and overflow diameter. Excessive vortex-finder insertion in a different hydrocyclone may increase hydraulic resistance or alter the air-core structure. Therefore, the geometric ratios should be preserved when the design is transferred to another unit.

4.4 Influence of the underflow outlet diameter

The underflow outlet controls the balance between water recovery and the discharge capacity of the lower stream. The cross-sectional areas of the 9, 11, and 18 mm outlets were 63.6, 95.0, and 254.5 mm², respectively. Thus, reducing the diameter from 18 to 9 mm decreased the outlet area by 75%, producing a strong hydraulic restriction and increasing the fraction of water recovered through the overflow pipe.

The 9 mm outlet was hydraulically favorable because it produced the lowest water discharge through the underflow. Nevertheless, the smallest outlet is not necessarily optimal under all particle-laden conditions. A restricted underflow opening may increase the probability of blockage, solids accumulation within the cone, or transition from spray discharge to an unstable rope-like discharge when the water contains high concentrations of coarse particles.

Accordingly, the 9 mm outlet is recommended for low-pressure operation where high-water recovery is a priority and the sediment load is moderate. The 11 or 18 mm outlets may be more appropriate when the source water contains a high coarse-sediment concentration and greater solids-discharge capacity is required. This trade-off should be evaluated using controlled suspensions before selecting the outlet diameter for a particular water source.

4.5 Relationship between hydraulic flow distribution and particle separation

The present response variables characterize hydraulic flow distribution rather than filtration efficiency. A high overflow ratio means that a large proportion of the inlet water is recovered through the upper outlet; it does not necessarily mean that the overflow contains a lower concentration of suspended solids.

Particle separation also depends on particle diameter, density, shape, inlet concentration, residence time, turbulence, and the mass distribution of solids between the overflow and underflow streams. Gençoğlan et al. [19] reported average separation efficiencies ranging from 78% for 0.5 mm sand to 84% for 2.5 mm sand. This demonstrates that the same hydraulic configuration may have different performance for different particle-size fractions.

Mailapalli et al. [24] also showed that hydrocyclone performance changed with the concentration of suspended solids and operating duration. Their experiments included initial clean-water testing followed by testing at suspended-solid concentrations of 300, 600, 900, and 1200 mg L^-1. The resulting variation in pressure loss and removal efficiency confirms that clean-water hydraulic behavior is an essential baseline but cannot replace particle-specific validation.

The hydraulically preferred configuration identified in the present study should therefore be considered the starting operating window for subsequent solid-phase testing. Direct validation should include turbidity, total suspended solids, particle-size distribution, solids recovery through the underflow, mass-balance closure, and long-term emitter-clogging performance.

The absence of particle measurements also means that the comparison between the conventional circular inlet and the semicircular inlet represents a hydraulic before-and-after comparison. At the common value X = 0.024, the semicircular inlet increased the overflow ratio from an estimated 0.896 to 0.933. This improvement demonstrates more favorable water redistribution but should not be described as a measured reduction in particle concentration or turbidity.

4.6 Practical applicability and scalability

The tested pressures of 9.81–49.03 kPa correspond to hydraulic heads of approximately 1–5 m. The two upper pressure levels are consistent with gravity-assisted drip-irrigation systems supplied from storage tanks installed approximately 3–5 m above the irrigated area. This operating range is relevant to small farms, remote fields, and water-scarce regions where high-pressure pumping is technically or economically undesirable.

At the selected underflow ratio of 0.024, every 1000 m³ of water passing through the hydrocyclone would produce approximately 976 m³ of overflow and 24 m³ of underflow. This does not necessarily represent an irreversible loss. The underflow stream may be discharged into a sedimentation chamber, after which the clarified supernatant can be returned to the source tank. Such recirculation would preserve the solids-discharge function while reducing net water loss.

Table 6. Scenario-based seasonal underflow volumes at R_sand = 0.024

The practical scale of the underflow volume can be illustrated using published irrigation requirements for southern Kazakhstan. Reported drip-irrigation norms are 3900 m³ ha^-1 for soybeans, 4450 m³ ha^-1 for grain corn, 5450 m³ ha^-1 for vegetables, and 5500 m³ ha^-1 for sugar beet [27]. Drip-irrigated apple orchards have reported seasonal irrigation rates of 718–961 m³ ha^-1 [28] with the corresponding scenario-based seasonal underflow volumes summarized in Table 6.

These calculations illustrate the importance of underflow collection and recirculation in water-scarce regions. Even a relatively small underflow ratio may correspond to a considerable seasonal volume when the hydrocyclone serves a large irrigated area.

For large-scale systems, several hydrocyclones of the tested geometry should preferably be installed in parallel. Parallel installation maintains the validated geometric ratios, permits individual units to be isolated for inspection, and allows treatment capacity to be adjusted according to irrigation demand. Direct proportional enlargement of the hydrocyclone body without further testing could alter the vortex structure and should therefore be avoided.

The design is also suitable for modular retrofitting. The inlet nozzle, underflow outlet, and vortex-finder position can be changed without replacing the entire hydrocyclone body. The main potential economic benefits are lower required operating pressure, reduced loading of downstream disc or media filters, reduced cleaning frequency, and recovery of most of the treated water through the overflow outlet.

A complete cost-benefit assessment would require local values for fabrication, pump efficiency, measured pressure loss, electricity price, annual operating time, maintenance labor, downstream-filter replacement, and the economic value of recovered water. Therefore, the present analysis establishes technical feasibility and the principal cost drivers but does not claim a measured payback period.

Crop and soil type do not directly change hydrocyclone hydrodynamics. They influence the required system flow rate, seasonal operating duration, and downstream irrigation demand. Water-source properties-including suspended-solid concentration, particle-size distribution, particle density, organic matter, algae, and mineral precipitation-are more directly relevant to hydrocyclone selection and maintenance.

4.7 Limitations and directions for further research

The current experiments quantified clean-water flow redistribution. Turbidity reduction, total suspended solids, particle-size separation, cut size, solids recovery, and long-term emitter clogging were not directly measured. The proposed configuration is therefore hydraulically preferred but requires particle-laden validation before it can be described as optimal in terms of filtration efficiency.

The model-based statistical assessment showed that dimensionless inlet energy had a strong and significant effect, whereas the overall inlet-geometry term was not significant. The recommendation of the semicircular inlet is consequently based on its earlier transition into the high-overflow region, its lower underflow fraction at moderate and high X, and its agreement with previously published particle-laden testing, rather than on a statistically higher terminal overflow ratio.

The empirical convergence values of approximately 0.025 for the semicircular inlet and 0.030 for the square inlet should not be regarded as universal self-similarity thresholds. Their applicability is restricted to the hydrocyclone dimensions, pressure range, outlet diameters, and inlet orientations used in this study. Future replicated experiments should estimate these breakpoints together with confidence intervals.

Pressure was measured upstream of the hydrocyclone and in the overflow line. Consequently, the experimental setup did not determine the spatial location of the maximum pressure inside the chamber. The relationships between internal pressure distribution, air-core position, tangential velocity, and hydraulic stability should be investigated using additional pressure taps, flow-visualization methods, or computational fluid dynamics.

Further work should compare the circular baseline configuration and the hydraulically preferred semicircular configuration under identical pressure, flow, and particle-loading conditions. The recommended validation matrix should include the three underflow diameters, controlled particle-size fractions, several suspended-solid concentrations, and at least three independent replicates per condition. This would allow hydraulic water recovery, particle-removal efficiency, pressure loss, blockage risk, and operating cost to be optimized simultaneously.

This work is the result of collaboration among the authors; however, N.A. and P.K. contributed to all sections of the study. B.M. contributed to methodology, analysis, interpretation of results, and original project preparation. A.B. contributed to methodology and case-study analysis. A.K. contributed to methodology and interpretation of results. T.G. was responsible for writing and language editing. Y.Z. contributed to data curation, validation, and visualization. A.S. was responsible for supervision, project administration, manuscript coordination, and review and editing of the final version.