Coagulation and flocculation design: chemistry, velocity gradient and G·t

Coagulation destabilises colloidal and dissolved material; flocculation grows the destabilised particles into flocs that settle, float or filter. This note sets out the hydrolysis chemistry of aluminium and iron(III) coagulants, the alkalinity they consume, the velocity gradient method for rapid mixing and flocculation, and the jar test that fixes the design dose.

Technical note, RB Solutions Ltd, 22 October 2026

Destabilisation mechanisms in coagulation

Particles in natural and industrial waters are commonly in the colloidal range and carry a negative surface charge, so electrostatic repulsion between diffuse double layers keeps them in suspension. Four destabilisation mechanisms are recognised [1, 2]: compression of the double layer by an increase in ionic strength; adsorption of oppositely charged hydrolysis species, leading to charge neutralisation; enmeshment in a precipitating metal hydroxide, termed sweep flocculation; and inter-particle bridging by long-chain polymers.

With aluminium and iron(III) salts, the two mechanisms of practical importance are charge neutralisation and sweep flocculation. The coagulation diagram for alum, plotting dose against pH, separates a charge-neutralisation zone, in which positively charged hydrolysis species adsorb on particle surfaces and restabilisation can occur at overdose, from a sweep zone at higher dose, in which amorphous Al(OH)₃ precipitates and enmeshes particles [3]. The diagram also explains why the optimum mixing regime differs: charge neutralisation depends on rapid dispersion of transient hydrolysis species before they precipitate, whereas sweep flocculation is comparatively insensitive to the intensity of the initial mix [3].

Hydrolysis of aluminium and ferric coagulants

A trivalent metal ion added to water hydrolyses through a series of monomeric and polymeric hydroxo complexes and, above its solubility limit, precipitates as the amorphous hydroxide [2]. Each mole of metal hydroxide formed releases three moles of hydrogen ion, which react with the bicarbonate alkalinity of the water. For aluminium sulfate tetradecahydrate the overall reaction is:

{Al}_{2} {({SO}_{4})}_{3} \cdot 14 H_{2} O + 6 {HCO}_{3}^{-} \to 2 Al {(OH)}_{3} + 3 {SO}_{4}^{2 -} + 6 {CO}_{2} + 14 H_{2} O

where the molar mass of Al₂(SO₄)₃·14H₂O is 594.4 g mol⁻¹ and of Al(OH)₃ 78.00 g mol⁻¹, from standard atomic weights [11]. Six equivalents of alkalinity are consumed per mole of alum.

For iron(III) chloride the corresponding reaction is:

{FeCl}_{3} + 3 {HCO}_{3}^{-} \to Fe {(OH)}_{3} + 3 {CO}_{2} + 3 {Cl}^{-}

where the molar mass of FeCl₃ is 162.2 g mol⁻¹ and of Fe(OH)₃ 106.87 g mol⁻¹ [11]; alkalinity is expressed as CaCO₃ with an equivalent mass of 50.04 g eq⁻¹.

The alkalinity consumed per unit dose, expressed as CaCO₃, follows directly:

A_{c} = \frac{n_{e} \cdot 50.04}{M_{c}} \cdot D_{c}

where A_c is alkalinity consumed (mg L⁻¹ as CaCO₃), n_e the equivalents of H⁺ released per mole of coagulant product (6 for alum, 3 for FeCl₃), M_c the molar mass of the product as dosed (g mol⁻¹) and D_c the dose (mg L⁻¹).

Table 1. Stoichiometric alkalinity consumption and hydroxide solids per unit coagulant dose
Coagulant (as dosed, dry basis)	Alkalinity consumed, mg CaCO₃ per mg	Hydroxide formed, mg per mg
Al₂(SO₄)₃·14H₂O	0.505	0.262 as Al(OH)₃
FeCl₃	0.926	0.659 as Fe(OH)₃

The values in Table 1 are stoichiometric for complete precipitation as the trihydroxide and are calculated from the reactions above. Commercial products are supplied as solutions of stated strength, so doses must be converted to the dry-basis product or to the metal before these ratios are applied. Where raw-water alkalinity is insufficient, the pH falls below the coagulation window and supplementary alkalinity (lime, sodium hydroxide or sodium carbonate) is required; the carbonate equilibria that govern the resulting pH are treated in a separate note on pH correction.

pH window and residual metal

Aluminium hydroxide has a narrower pH range of low solubility than iron(III) hydroxide, which is one reason iron salts are preferred for coagulation over a wider pH range [6, 7]. Adsorptive coagulation with iron salts is reported to be most effective at pH 6 to 8, and with aluminium salts over a narrower range near pH 7 [7]. For drinking water in England and Wales, residual aluminium is controlled by the prescribed concentration in the Water Supply (Water Quality) Regulations 2016, and coagulant products are subject to regulation 31 of those Regulations, which governs products and substances in contact with water for supply [9]. Product quality is specified in BS EN 878 (aluminium sulfate), BS EN 888 (iron(III) chloride) and BS EN 890 (iron(III) sulfate) [10].

Velocity gradient G: the Camp and Stein formulation

The rate of orthokinetic (shear-induced) collision between particles depends on the local velocity gradient. Camp and Stein generalised the laminar shear result to turbulent mixing by relating a root-mean-square velocity gradient to the power dissipated per unit volume [4]:

G = \sqrt{\frac{P}{μ V}} = \sqrt{\frac{ε}{ν}}

where G is the mean velocity gradient (s⁻¹), P the power dissipated in the fluid (W), μ the dynamic viscosity (Pa s), V the volume of the mixed zone (m³), ε = P/(ρV) the specific energy dissipation rate (W kg⁻¹), ρ the density (kg m⁻³) and ν = μ/ρ the kinematic viscosity (m² s⁻¹).

G is a volume-averaged quantity. Local dissipation near an impeller can exceed the mean by an order of magnitude, so G does not describe floc break-up near the impeller tip. It remains the accepted basis for scaling and comparing mixing duties, provided its limits are recognised [1].

The orthokinetic collision frequency function for two particles of diameters d_i and d_j in uniform laminar shear, after Smoluchowski [12], is:

β_{ij} = \frac{1}{6} G {(d_{i} + d_{j})}^{3}

where β_ij is the collision frequency function (m³ s⁻¹) and d_i, d_j the particle diameters (m). The collision rate per unit volume is β_ij·n_i·n_j, with n the number concentrations (m⁻³).

Because the rate is proportional to G and to the cube of collision diameter, the product of G and the hydraulic residence time t, the dimensionless G·t, is a measure of the number of collision opportunities. Camp introduced G·t as a design criterion for flocculation basins [5].

Rapid mixing design

Mechanical rapid mixers

For a baffled tank in the fully turbulent regime, impeller power is given by the power-number relation [13]:

P = N_{P} ρ N^{3} D^{5}

where N_P is the dimensionless power number of the impeller (constant for a given geometry in fully turbulent flow), N the rotational speed (s⁻¹) and D the impeller diameter (m). G then follows from the Camp and Stein relation.

Design guidance for rapid mixing varies between sources. A Dutch university module recommends a rapid-mix G of at least 1500 s⁻¹ [7]; an Indian government manual gives a detention of 30 to 60 s and G of at least 300 s⁻¹ for mechanical flash mixers, and a working range of 300 to 5000 s⁻¹ [6]. UK guidance requires coagulant to be dosed at a point of high turbulence [8]. For charge-neutralisation coagulation the instantaneous dispersion of the coagulant matters more than the residence time [3].

Static and hydraulic mixers

In a static (in-line) mixer or hydraulic jump, the power dissipated is the product of the flow and the frictional head loss across the element:

P = ρ g Q h \Rightarrow G = \sqrt{\frac{g h}{ν t}}

where g is the standard acceleration of gravity, 9.806 65 m s⁻² [14], Q the flow rate (m³ s⁻¹), h the head loss across the mixer (m) and t = V/Q the residence time in the mixing element (s).

Flocculation kinetics and flocculator design

Flocculation is a balance between aggregation and break-up. Aggregation increases with G·t, while floc break-up increases with G, so flocculators are normally divided into compartments in series with tapered G, falling from the first to the last compartment [6, 7]. Series compartments also reduce short-circuiting and bring the residence time distribution closer to plug flow.

For paddle flocculators the power dissipated is estimated from the drag on the paddle blades:

P = \frac{1}{2} C_{D} A_{p} ρ v_{r}^{3}

where C_D is the drag coefficient of the blade (dimensionless; 0.8 to 1.9 depending on blade aspect ratio [6]), A_p the projected blade area normal to motion (m²) and v_r the velocity of the blade relative to the water (m s⁻¹).

Floc integrity limits the paddle tip speed; a tip velocity above about 1 m s⁻¹ is reported to break flocs [7]. The velocity gradient at any point between the last flocculation stage and the clarifier should not exceed that of the last stage, which governs the design of transfer channels, ports and inlet distribution [6].

Table 2. Published design ranges for rapid mixing and flocculation
Parameter	Range	Source
Rapid mix G, recommended minimum	1500 s⁻¹	[7]
Mechanical flash mixer detention	30 to 60 s, G ≥ 300 s⁻¹	[6]
Flocculation G	20 to 75 s⁻¹	[6]
Flocculation G·t, aluminium coagulant	2 × 10⁴ to 6 × 10⁴	[6]
Flocculation G·t, iron coagulant	1 × 10⁵ to 1.5 × 10⁵	[6]
Flocculation detention	10 to 30 min	[6]
Paddle tip velocity for floc break-up	above about 1 m s⁻¹	[7]

These ranges are reference practice from published manuals; they are not UK statutory requirements. The design G and G·t for a particular water should be confirmed by jar testing at the design temperature.

Effect of temperature

At a fixed power input, G varies as μ^−1/2. Equivalently, the power needed to hold a design G is proportional to viscosity. The dynamic viscosity of water is 1.306 mPa s at 10 °C and 1.002 mPa s at 20 °C [15], so a flocculator sized at 20 °C needs about 30 % more power to deliver the same G in winter. Cold water also slows hydrolysis and precipitation kinetics, which is why winter jar tests often show a higher optimum dose or longer flocculation time [6].

Worked example: alum demand, flocculator power and static mixer G

Stated inputs. Flow Q = 0.25 m³ s⁻¹ (900 m³ h⁻¹); alum dose 30 mg L⁻¹ as Al₂(SO₄)₃·14H₂O; design temperature 10 °C, μ = 1.306 × 10⁻³ Pa s and ρ = 999.70 kg m⁻³ [15, 16]; flocculation residence time 20 min at G = 50 s⁻¹; static mixer head loss 0.5 m with an internal volume of 0.3 m³.

Alkalinity consumption and solids

Route 1, by mass ratio: A_c = 30 × 6 × 50.04 / 594.4 = 15.15 mg L⁻¹ as CaCO₃. Route 2, by moles: 30 / 594.4 = 0.050 47 mmol L⁻¹ alum, giving 0.100 9 mmol L⁻¹ Al; at 3 eq per Al this is 0.302 8 meq L⁻¹, and 0.302 8 × 50.04 = 15.15 mg L⁻¹ as CaCO₃. The routes agree. Aluminium hydroxide formed is 30 × 2 × 78.00 / 594.4 = 7.87 mg L⁻¹, or 170 kg d⁻¹ at the design flow, before the contribution of the raw-water suspended solids.

Flocculator volume and power

V = Q·t = 0.25 × 1200 = 300 m³. P = G²μV = 50² × 1.306 × 10⁻³ × 300 = 980 W of power delivered to the water. Second route: ε = P/(ρV) = 980 / (999.70 × 300) = 3.27 × 10⁻³ W kg⁻¹; ν = 1.306 × 10⁻⁶ m² s⁻¹; G = (ε/ν)^1/2 = 50.0 s⁻¹, recovering the design value. G·t = 50 × 1200 = 6.0 × 10⁴, at the upper end of the published range for aluminium coagulant in Table 2. The figure of 980 W is shaft power to the water; drive and gearbox losses are additional.

Static mixer

Route 1: P = ρgQh = 999.70 × 9.806 65 × 0.25 × 0.5 = 1225 W; t = 0.3 / 0.25 = 1.2 s; G = (P/(μV))^1/2 = (1225 / (1.306 × 10⁻³ × 0.3))^1/2 = 1770 s⁻¹. Route 2: G = (gh/(νt))^1/2 = (9.806 65 × 0.5 / (1.306 × 10⁻⁶ × 1.2))^1/2 = 1770 s⁻¹. The routes agree, and the result exceeds the 1500 s⁻¹ minimum in Table 2 [7].

Jar testing

The jar test is the basis for the choice of coagulant, dose, pH and mixing conditions [8]. A published procedure uses 1 L samples, rapid mixing at 200 rpm for 1 min after simultaneous dosing of all jars, slow mixing at 20 rpm for 15 min, and 30 min settlement, followed by measurement of colour, turbidity and pH in the decanted water [17]. A first pass at dose steps of about 10 mg L⁻¹ is followed by a second at 2 to 3 mg L⁻¹ around the optimum [17].

Three refinements make jar results transferable to design. First, the paddle speed should be converted to G using the power characteristic of the jar apparatus, so that the test reproduces the design G and G·t rather than an arbitrary speed. Second, tests should be run at the design minimum temperature, since viscosity and hydrolysis rate both change. Third, the jar measures chemistry and floc formation but not the residence time distribution of the full-scale tanks; full-scale mixing conditions cannot be reproduced exactly in the laboratory [6], so the design should allow adjustable dose, pH set point and flocculator speed.

Design sequence

A defensible coagulation and flocculation design proceeds in order: characterise the raw water over its seasonal range (turbidity, colour or organic carbon, alkalinity, pH, temperature); select the coagulant and pH window by jar test; calculate alkalinity demand and confirm whether supplementary alkalinity is required; size rapid mixing for dispersion at the dosing point; size flocculation for G and G·t with tapered compartments; check transfer velocities to the clarifier; and estimate the hydroxide sludge load for the solids-handling stage.

References

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.
Stumm, W. and Morgan, J.J. (1996) Aquatic Chemistry: Chemical Equilibria and Rates in Natural Waters, 3rd ed. New York: Wiley.
Amirtharajah, A. and Mills, K.M. (1982) Rapid-mix design for mechanisms of alum coagulation. Journal AWWA, 74(4), 210–216.
Camp, T.R. and Stein, P.C. (1943) Velocity gradients and internal work in fluid motion. Journal of the Boston Society of Civil Engineers, 30(4), 219–237.
Camp, T.R. (1955) Flocculation and flocculation basins. Transactions of the American Society of Civil Engineers, 120, 1–16.
Central Public Health and Environmental Engineering Organisation (1999) Manual on Water Supply and Treatment, 3rd ed. New Delhi: Ministry of Urban Development, Government of India, section 7.4.
Delft University of Technology (2007) Drinking Water Treatment: Coagulation and Flocculation, course module CT4471. Delft: TU Delft OpenCourseWare.
Drinking Water Inspectorate (2015) Manual on Treatment for Small Water Supply Systems, section 5.3. London: DWI.
The Water Supply (Water Quality) Regulations 2016, SI 2016/614, regulation 31 and Schedule 1. legislation.gov.uk.
BS EN 878, Aluminium sulfate; BS EN 888, Iron(III) chloride; BS EN 890, Iron(III) sulfate. Chemicals used for treatment of water intended for human consumption. London: BSI.
Prohaska, T. et al. (2022) Standard atomic weights of the elements 2021 (IUPAC Technical Report). Pure and Applied Chemistry, 94(5), 573–600.
Smoluchowski, M. von (1917) Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen. Zeitschrift für physikalische Chemie, 92, 129–168.
Rushton, J.H., Costich, E.W. and Everett, H.J. (1950) Power characteristics of mixing impellers. Chemical Engineering Progress, 46, 395–404 and 467–476.
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.
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.
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.
Environmental Protection Agency, Ireland (2002) Water Treatment Manual: Coagulation, Flocculation and Clarification, jar test appendix. Wexford: EPA.

Related technical notes

RB Solutions provides process design for coagulation, rapid mixing and flocculation: coagulant selection, alkalinity balance, G and G·t sizing, and the specification of jar testing for new and existing plant.

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