Lamella clarifier design: Hazen theory, Yao criterion and plate hydraulics

A lamella clarifier places a stack of inclined plates in the settling zone, so that each particle needs to fall only the short distance between adjacent plates. This note derives the sizing basis from Hazen's overflow rate theory and Yao's analysis of inclined settlers, and sets out the hydraulic checks that decide whether the theoretical plate area is achieved in practice.

Technical note, RB Solutions Ltd, 22 October 2026

Hazen surface overflow theory

Hazen showed that, for discrete particles in an ideal horizontal-flow basin, removal depends on the plan area of the basin and not on its depth or detention time [1]. Camp formalised the ideal basin: uniform horizontal velocity, uniform particle distribution at the inlet, and particles removed once they reach the floor [2]. A particle with settling velocity v_s is completely removed if v_s is at least the surface overflow rate:

v_{0} = \frac{Q}{A}

where v₀ is the surface overflow rate or critical settling velocity (m h⁻¹), Q the flow rate (m³ h⁻¹) and A the plan area of the settling zone (m²).

Particles with v_s < v₀ are removed in the proportion v_s/v₀. For a suspension with a distribution of settling velocities, the overall fraction removed is [2]:

F = (1 - x_{0}) + \frac{1}{v_{0}} \int_{0}^{x_{0}} v_{s} d x

where F is the fraction of particles removed (dimensionless), x the cumulative fraction of particles with settling velocity less than v_s, and x₀ the fraction with settling velocity less than v₀.

The result applies to discrete (type 1) settling. Flocculent suspensions continue to aggregate during settling, so their removal also depends on depth and time; in that case the design settling velocity is established from settling-column tests on the actual flocculated water [3].

Discrete settling and Stokes' law

A small sphere settling at terminal velocity in the laminar regime obeys Stokes' law [4]:

v_{s} = \frac{g (ρ_{p} - ρ_{w}) d^{2}}{18 μ}

where v_s is the terminal settling velocity (m s⁻¹), g the standard acceleration of gravity, 9.806 65 m s⁻² [5], ρ_p and ρ_w the particle and water densities (kg m⁻³), d the particle diameter (m) and μ the dynamic viscosity of water (Pa s).

Stokes' law assumes creeping flow around the particle. Its domain of validity is set by the particle Reynolds number:

{Re}_{p} = \frac{ρ_{w} v_{s} d}{μ}

where Re_p is dimensionless. Stokes' law is applied where Re_p is less than about 1 [3, 6]; at higher Re_p a transitional drag correlation is required, and Stokes' law overestimates the settling velocity.

Two consequences matter for design. First, v_s varies as d², so flocculation that doubles floc diameter quadruples settling velocity, other things equal. Second, v_s varies inversely with viscosity, so settling velocity at 10 °C is about 23 % lower than at 20 °C (μ = 1.306 and 1.002 mPa s respectively [7]). The design case is the minimum water temperature.

Floc density is not constant; it falls as floc size increases because flocs are porous fractal aggregates with a large water content. Stokes' law calculations with a solid-particle density therefore overestimate floc settling velocity, which is a further reason to establish v_s by test for coagulated waters [3].

Inclined plate settling: the Yao criterion

In an inclined channel, a particle moves along the channel with the fluid while settling vertically. Yao analysed laminar flow between inclined surfaces and obtained the condition for capture of the critical particle [8]:

\frac{v_{s}}{v_{0}} (\sin θ + L \cos θ) = S_{c} L = \frac{l}{d}

where v_s is the critical settling velocity (m h⁻¹), v₀ the mean fluid velocity in the channel (m h⁻¹), θ the angle of the plates from horizontal, l the length of the plate along the flow (m), d the perpendicular spacing between plates (m), L the relative length (dimensionless) and S_c a geometry constant: 1 for parallel plates, 4/3 for circular tubes and 11/8 for square conduits [8, 11].

For parallel plates (S_c = 1), with N channels of width w, v₀ = Q/(N w d). Substituting and rearranging gives the projected-area form:

Q = v_{s} N w (l \cos θ + d \sin θ) = v_{s} A_{p}

where A_p is the total projected horizontal area of the plate stack (m²). The term l cos θ is the horizontal projection of each plate and d sin θ the small additional projection of the channel opening.

The projected-area form shows that an inclined plate clarifier behaves as a Hazen basin whose effective area is the sum of the projected plate areas. This is the physical basis of the compactness of a lamella clarifier: a plate stack can provide several times its own footprint as effective settling area.

Effective area in practice

The theoretical projected area is not fully effective. Flow maldistribution between channels, entry and exit disturbance at the plate ends, and sludge sliding counter to the flow all reduce performance. One North American standard limits the loading to the projected area of 80 % of the plates [10]; this is cited as reference practice rather than a UK requirement. An efficiency factor of this kind should be stated explicitly in any design, not absorbed silently into the design settling velocity.

Plate angle and spacing

The plate angle is a compromise between projected area, which increases as θ decreases, and sludge removal, which requires the deposited solids to slide down the plate. For counter-current settlers, in which the liquid flows upward and sludge slides down, an angle of about 55 to 60° is used to secure sludge removal [9]; for tube settlers an angle of about 60° is recommended because sludge does not slide reliably at shallower angles [11]. Co-current settlers, in which liquid and sludge both move downward, can operate at 30 to 40° [9].

Published plate geometries span a plate height of 1 to 3 m and a perpendicular gap of 3.4 to 8 cm [9]. Smaller spacing increases L and therefore the projected area per unit volume, but raises the channel velocity for a given flow, increases the risk of bridging by sludge, and makes the stack more sensitive to flow maldistribution. Spacing is therefore selected with the sludge character in mind: dense mineral solids tolerate a narrow gap, while light, voluminous hydroxide floc needs a wider one.

Laminar flow and stability checks

Yao's analysis assumes laminar flow in each channel. The channel Reynolds number is:

Re = \frac{v_{0} D_{h}}{ν} D_{h} = \frac{4 w d}{2 (w + d)}

where Re is dimensionless, v₀ the mean channel velocity (m s⁻¹), D_h the hydraulic diameter of the channel (m) and ν the kinematic viscosity (m² s⁻¹).

Stability against density currents and short-circuiting is assessed with a Froude number based on the hydraulic radius:

Fr = \frac{v_{0}^{2}}{g R_{h}} R_{h} = \frac{D_{h}}{4}

where Fr is dimensionless and R_h the hydraulic radius (m).

The published criteria are Re < 2000 for laminar flow and Fr > 10⁻⁵ for stable flow [9]. The two criteria pull in opposite directions: a low velocity favours laminar flow but weakens the inertia that resists density currents caused by temperature or concentration differences. Plate channels normally achieve both with a wide margin, as the worked example shows; the Froude check is more often critical in the open zones of the tank below and above the plate stack.

Table 1. Published design criteria for inclined plate settlers
Parameter	Value	Source
Geometry constant S_c, parallel plates / circular tubes / square conduits	1 / 4/3 / 11/8	[8, 11]
Plate angle, counter-current	about 55 to 60°	[9]
Plate angle, co-current	30 to 40°	[9]
Plate height; perpendicular gap	1 to 3 m; 3.4 to 8 cm	[9]
Channel Reynolds number	< 2000	[9]
Froude number	> 10⁻⁵	[9]
Sludge hopper wall slope (vertical:horizontal)	1.2:1 to 2:1	[11]

Worked example: plate area, channel velocity and flow checks

Stated inputs. Flow Q = 100 m³ h⁻¹; design settling velocity of the floc v_s = 0.5 m h⁻¹ (to be established by settling test for a real water); counter-current parallel plates at θ = 60°, plate length l = 1.5 m, width w = 1.0 m, perpendicular spacing d = 0.05 m; water at 10 °C, ν = 1.306 × 10⁻⁶ m² s⁻¹ [7, 12]. No plate efficiency factor is applied, so that the two routes below can be compared directly.

Number of channels

Projected area per channel: w(l cos θ + d sin θ) = 1.0 × (1.5 × 0.5 + 0.05 × 0.866) = 0.793 m². Required A_p = Q/v_s = 100/0.5 = 200 m², so N = 200/0.793 = 252.1, rounded up to 253 channels, giving A_p = 200.7 m².

Second route: Yao criterion directly

Channel velocity v₀ = Q/(N w d) = 100/(253 × 1.0 × 0.05) = 7.91 m h⁻¹ (2.20 × 10⁻³ m s⁻¹). L = 1.5/0.05 = 30. Critical settling velocity v_s = S_c v₀/(sin θ + L cos θ) = 7.91/(0.866 + 15.0) = 0.498 m h⁻¹, which is no greater than the design value of 0.5 m h⁻¹. The routes agree.

Hydraulic checks

D_h = 4 × 1.0 × 0.05/(2 × 1.05) = 0.095 2 m. Re = 2.20 × 10⁻³ × 0.095 2/1.306 × 10⁻⁶ = 160, far below 2000, so the flow is laminar. R_h = 0.023 8 m and Fr = (2.20 × 10⁻³)²/(9.806 65 × 0.023 8) = 2.1 × 10⁻⁵, above the 10⁻⁵ criterion. Both criteria in Table 1 are met. If an efficiency factor of 0.8 were applied to the projected area [10], N would rise to 316 channels, v₀ would fall to 6.33 m h⁻¹ and Fr to 1.3 × 10⁻⁵, still above the limit.

Stokes' law cross-check for a mineral particle

For a quartz particle of d = 20 µm and ρ_p = 2650 kg m⁻³ [3] at 10 °C: v_s = 9.806 65 × (2650 − 999.70) × (20 × 10⁻⁶)²/(18 × 1.306 × 10⁻³) = 2.75 × 10⁻⁴ m s⁻¹ = 0.99 m h⁻¹. Re_p = 2.75 × 10⁻⁴ × 20 × 10⁻⁶/1.306 × 10⁻⁶ = 4.2 × 10⁻³, well inside the Stokes regime. Such a particle would be fully captured at the design loading; the 0.5 m h⁻¹ design value relates to light coagulated floc, not to mineral grit.

Inlet distribution and outlet collection

Yao's result assumes equal flow to every channel. Maldistribution increases the velocity in some channels above the design v₀, and the critical settling velocity rises in proportion. Flow distribution to the plate stack is achieved by forcing the flow through a controlled head loss, such as a perforated baffle or orifice manifold, whose head loss is large compared with the velocity head variations along the distribution channel [3]. The orifice discharge relation governs this design:

q = C_{d} a_{o} \sqrt{2 g Δ h}

where q is the flow through one orifice (m³ s⁻¹), C_d the discharge coefficient (dimensionless), a_o the orifice area (m²) and Δh the head loss across the orifice (m). Since q varies with Δh^1/2, a variation of a fraction δ in Δh produces a variation of about δ/2 in q.

The distribution velocity must also be low enough not to break the floc formed upstream; the velocity gradient in the transfer zone should not exceed that of the last flocculation stage, as discussed in the coagulation and flocculation note. Clarified water is collected uniformly over the top of the stack by submerged orifice pipes or weir launders so that the upward velocity is uniform across the plates.

Sludge hopper and sludge withdrawal

Solids slide from the plates into a hopper whose wall slope must exceed the angle at which the sludge will remain static. Published guidance gives hopper slopes of 1.2:1 to 2:1 vertical to horizontal, corresponding to about 50 to 63° from horizontal [11]. Withdrawal should be frequent enough that the sludge blanket never rises to the underside of the plates, which would reduce the effective settling area and re-entrain solids. The hopper is sized from the solids mass balance: the mass of solids removed per day divided by the concentration achieved in the hopper, both of which follow from the coagulant dose and raw water solids calculated in the coagulation design.

Design sequence

A lamella clarifier design proceeds as follows: establish the design settling velocity at minimum temperature by test; choose counter-current or co-current flow and the plate angle; select spacing to suit the sludge; calculate projected area with an explicit efficiency factor; check channel Re and Fr; design the inlet distribution and outlet collection for uniform flow; and size the hopper and withdrawal system from the solids mass balance. In the UK, the discharge from a clarifier serving an industrial process or a construction site is controlled by an environmental permit under the Environmental Permitting (England and Wales) Regulations 2016 [13], and the design effluent quality must be referred to the permit conditions.

References

Hazen, A. (1904) On sedimentation. Transactions of the American Society of Civil Engineers, 53, 45–88.
Camp, T.R. (1946) Sedimentation and the design of settling tanks. Transactions of the American Society of Civil Engineers, 111, 895–936.
Crittenden, J.C., Trussell, R.R., Hand, D.W., Howe, K.J. and Tchobanoglous, G. (2012) MWH's Water Treatment: Principles and Design, 3rd ed. Hoboken: Wiley.
Stokes, G.G. (1851) On the effect of the internal friction of fluids on the motion of pendulums. Transactions of the Cambridge Philosophical Society, 9(2), 8–106.
Tiesinga, E., Mohr, P.J., Newell, D.B. and Taylor, B.N. (2021) CODATA recommended values of the fundamental physical constants: 2018. Reviews of Modern Physics, 93, 025010.
Metcalf & Eddy, AECOM (2014) Wastewater Engineering: Treatment and Resource Recovery, 5th ed. New York: McGraw-Hill.
Huber, M.L. et al. (2009) New international formulation for the viscosity of H₂O. Journal of Physical and Chemical Reference Data, 38(2), 101–125.
Yao, K.M. (1970) Theoretical study of high-rate sedimentation. Journal of the Water Pollution Control Federation, 42(2), 218–228.
Delft University of Technology (2007) Drinking Water Treatment: Sedimentation, course module CT4471, pp. 65–66. Delft: TU Delft OpenCourseWare.
Great Lakes–Upper Mississippi River Board (2012) Recommended Standards for Water Works, clause 4.2.6. Albany: Health Research Inc.
Central Public Health and Environmental Engineering Organisation (1999) Manual on Water Supply and Treatment, 3rd ed., sections 7.5.5 and 7.5.12. New Delhi: Ministry of Urban Development, Government of India.
Wagner, W. and Pruß, A. (2002) The IAPWS formulation 1995 for the thermodynamic properties of ordinary water substance for general and scientific use. Journal of Physical and Chemical Reference Data, 31(2), 387–535.
The Environmental Permitting (England and Wales) Regulations 2016, SI 2016/1154. legislation.gov.uk.

Related technical notes

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