Pressure filter design: headloss, backwash fluidisation and media selection

A pressure filter is a rapid granular filter enclosed in a vessel, so that the driving head is supplied by pumping rather than by gravity. Its process design rests on depth filtration theory, the hydraulics of flow through packed and fluidised beds, and the grading of the media. Because the filter is a pressure vessel, its mechanical design in the UK also falls under the Pressure Equipment (Safety) Regulations 2016.

Technical note, RB Solutions Ltd, 22 October 2026

Depth filtration mechanisms

In a rapid granular filter, particles much smaller than the pores are removed throughout the depth of the bed rather than strained at its surface. Removal is treated as a two-step process: transport of a particle to the surface of a media grain, followed by attachment [1, 2]. Transport occurs by three mechanisms acting together: Brownian diffusion, which dominates for particles below about one micrometre; interception, in which a particle following a streamline passes within one particle radius of the grain; and gravitational sedimentation onto the upper surfaces of the grains [2]. The sum of the three gives the single-collector transport efficiency η, and attachment is described by an efficiency α, which depends on the surface chemistry and therefore on upstream coagulation.

Iwasaki described the decline of particle concentration with depth as first order [3]:

\frac{\partial C}{\partial z} = - λ C

where C is the particle concentration (mg L⁻¹ or m⁻³), z the depth into the bed (m) and λ the filter coefficient (m⁻¹).

The filter coefficient of a clean bed is linked to the single-collector efficiency by the filtration model of Yao, Habibian and O'Melia [2]:

λ = \frac{3}{2} \frac{(1 - ε)}{d_{c}} α η

where ε is the bed porosity (dimensionless), d_c the media grain (collector) diameter (m), α the attachment efficiency (dimensionless, 0 to 1) and η the single-collector transport efficiency (dimensionless).

The model shows why smaller media and better coagulation both improve removal, and why the particle size range of about one micrometre, where neither diffusion nor sedimentation is effective, is the most difficult to remove [1, 2]. It describes a clean bed only; as deposits accumulate, λ and the head loss both change, so filter run length is set empirically by either turbidity breakthrough or terminal head loss.

Filtration rate and solids loading

The filtration rate is the superficial velocity v = Q/A, conventionally expressed in m h⁻¹. It sets the clean-bed head loss, the depth of penetration of solids and the run length. One published design standard sets a maximum of 12 m h⁻¹ for iron-removal filtration unless a pilot study or a closely similar installation supports a higher rate, and states that, at 10 m h⁻¹, a solids loading up to 10 mg L⁻¹ can generally be handled by direct filtration, with upstream clarification required above that [4]. These are reference values from a non-UK utility standard; a UK design should confirm the rate by pilot or by comparison with operating plant treating similar water.

Clean-bed headloss: Carman–Kozeny and Ergun

At the low particle Reynolds numbers of water filtration, the head loss through a clean bed is predominantly viscous. The Kozeny equation, with the constant established by Carman [5, 6], is:

\frac{h_{L}}{L} = \frac{180 μ {(1 - ε)}^{2} v}{ρ g ε^{3} φ^{2} d^{2}}

where h_L is the head loss (m), L the bed depth (m), μ the dynamic viscosity (Pa s), v the superficial velocity (m s⁻¹), ρ the water density (kg m⁻³), g the standard acceleration of gravity, 9.806 65 m s⁻² [7], φ the grain sphericity (dimensionless) and d the grain diameter (m). The constant 180 equals 36 × 5, with 5 the Kozeny constant [6].

Ergun added an inertial term, so that a single equation spans viscous and inertial flow [8]:

\frac{Δ p}{L} = 150 \frac{μ {(1 - ε)}^{2} v}{ε^{3} {(φ d)}^{2}} + 1.75 \frac{ρ (1 - ε) v^{2}}{ε^{3} φ d}

where Δp is the pressure drop (Pa); h_L = Δp/(ρg). The ratio of the inertial to the viscous term is proportional to the modified Reynolds number Re′ = ρvφd/(μ(1 − ε)).

The two equations differ in the viscous constant, 180 against 150, so for viscous-dominated flow the Ergun equation predicts about 17 % less head loss than Carman–Kozeny. The difference is within the uncertainty in porosity and sphericity, to which both equations are highly sensitive: head loss varies approximately as (1 − ε)²/ε³, so an error of 0.02 in porosity near ε = 0.42 changes the predicted head loss by more than 15 %. For a graded media, d is taken as a representative mean size, and porosity is measured on the media as placed.

Headloss development and run length

As solids deposit, the porosity of the upper layers falls and the specific surface rises, so head loss increases with the mass of solids retained. Over much of a run, the increase is approximately linear in the cumulative solids mass applied per unit filter area [1]. Run length is reached at whichever comes first: the available head loss, set by the pump and vessel pressure rating minus the clean-bed and underdrain losses, or the onset of turbidity breakthrough. A well-balanced design reaches both at about the same time; terminal head loss well before breakthrough indicates that the media is too fine or the solids load too high, and breakthrough well before terminal head loss indicates the reverse.

Backwash fluidisation and bed expansion

Minimum fluidisation velocity

At incipient fluidisation, the pressure drop across the bed equals the buoyant weight of the media per unit area:

\frac{Δ p}{L} = (1 - ε_{mf}) (ρ_{p} - ρ) g

where ε_mf is the porosity at minimum fluidisation (dimensionless) and ρ_p the media grain density (kg m⁻³). This pressure drop remains approximately constant as the bed expands further.

Equating this to the Ergun equation gives the minimum fluidisation velocity v_mf. Wen and Yu correlated the result in terms of the Archimedes number without requiring ε_mf and φ explicitly [9]:

{Re}_{mf} = \sqrt{{33.7}^{2} + 0.0408 Ar} - 33.7 Ar = \frac{d^{3} ρ (ρ_{p} - ρ) g}{μ^{2}}

where Re_mf = ρ v_mf d/μ (dimensionless) and Ar is the Archimedes number (dimensionless).

Because a graded bed contains a range of sizes, the backwash rate must fluidise the coarse fraction, so v_mf is calculated for a coarse percentile such as d₉₀. Viscosity enters v_mf strongly, so the backwash rate is checked at the warmest backwash water temperature, when a given velocity produces least expansion, and the media loss risk at the coldest.

Bed expansion

Above v_mf, the porosity of the expanded bed follows the Richardson–Zaki relation, and the expanded depth follows from conservation of media volume [10]:

\frac{v}{v_{t}} = {ε_{e}}^{n} L_{e} = L \frac{1 - ε}{1 - ε_{e}}

where v is the backwash superficial velocity (m s⁻¹), v_t the terminal settling velocity of a single grain (m s⁻¹), ε_e the expanded-bed porosity, n an exponent that depends on the grain Reynolds number at terminal velocity [10], and L_e the expanded depth (m).

A published standard requires at least 20 % bed expansion for effective cleaning, with 25 % as a suitable design target, and requires the backwash velocity to be selected from the media expansion curve at the most onerous water temperature [4]. Freeboard above the expanded bed in a pressure vessel must accommodate this expansion without media carry-over to the backwash outlet.

Media selection: effective size and uniformity coefficient

Media grading is specified by the effective size, ES = d₁₀, the sieve size passing 10 % by mass, and the uniformity coefficient, UC = d₆₀/d₁₀ [1]. A low UC gives a bed whose layers stratify less on backwashing and whose pores are more uniform. In dual-media filters, a coarse, low-density anthracite layer over a fine, dense sand layer allows solids to penetrate deeper into the bed, increasing storage; the grain sizes and densities are matched so that the two layers re-stratify in the correct order after backwash.

Table 1. Reference media specification for granular filters
Media	Effective size d₁₀	Uniformity coefficient	Other requirements	Source
Quartz filter sand	0.6 mm	< 1.3	specific gravity > 2.5, acid solubility < 5 %	[4]
Anthracite filter coal	1.2 to 1.3 mm	< 1.5	specific gravity > 1.4, acid solubility < 5 %	[4]
Dual media arrangement	500 mm of 1.1 mm coal over 500 mm of 0.6 mm sand			[4]

For drinking water in the UK, filter media are specified to BS EN 12904 (silica sand and gravel) and BS EN 12909 (anthracite), with test methods in BS EN 12902 [11], and must satisfy regulation 31 of the Water Supply (Water Quality) Regulations 2016 for products in contact with water for supply [12].

Worked example: clean-bed headloss and backwash velocity

Stated inputs. Single-medium quartz sand bed, depth L = 0.8 m; representative mean grain size d = 0.75 mm; coarse fraction d₉₀ = 0.9 mm; sphericity φ = 0.85; fixed-bed porosity ε = 0.42; grain density ρ_p = 2650 kg m⁻³ [1]; filtration rate 10 m h⁻¹ (2.778 × 10⁻³ m s⁻¹); water at 10 °C, μ = 1.306 × 10⁻³ Pa s and ρ = 999.70 kg m⁻³ [13, 14]. Sphericity and porosity are stated inputs to be confirmed on the supplied media.

Clean-bed headloss

Route 1, Carman–Kozeny in grain-diameter form: h_L = 180 × 1.306 × 10⁻³ × 0.58² × 2.778 × 10⁻³ × 0.8/(999.70 × 9.806 65 × 0.42³ × 0.85² × (0.75 × 10⁻³)²) = 0.595 m.

Route 2, specific-surface form: S_v = 6/(φd) = 9412 m⁻¹; h_L = 5μ(1 − ε)²S_v²vL/(ρgε³) = 0.595 m. The routes agree. The Ergun equation gives 0.496 m (viscous) + 0.014 m (inertial) = 0.510 m, with Re′ = 2.3, confirming that flow is viscous-dominated and that the two equations differ mainly through the 180/150 constant. A design should carry the higher value, 0.60 m, plus underdrain and nozzle losses.

Minimum fluidisation velocity of the coarse fraction

Route 1, Wen and Yu: Ar = (0.9 × 10⁻³)³ × 999.70 × 1650.3 × 9.806 65/(1.306 × 10⁻³)² = 6915; Re_mf = (33.7² + 0.0408 × 6915)^1/2 − 33.7 = 3.95; v_mf = Re_mfμ/(ρd) = 5.74 × 10⁻³ m s⁻¹ = 20.7 m h⁻¹.

Route 2, Ergun solved at incipient fluidisation with ε_mf = 0.42 and φ = 0.85: the quadratic in v_mf gives 20.8 m h⁻¹. The routes agree within 1 %. The backwash rate must exceed this value by enough to reach the 20 to 25 % expansion target [4], which is read from the media expansion curve or calculated with the Richardson–Zaki relation; and it must be rechecked at the summer temperature, since v_mf rises as viscosity falls.

Consistency check

The head loss across the fluidised bed is L(1 − ε)(ρ_p − ρ)/ρ = 0.8 × 0.58 × 1650.3/999.70 = 0.77 m. The clean-bed head loss at the 10 m h⁻¹ filtration rate (0.51 to 0.60 m) is lower, as it must be for a bed that is not fluidised at that velocity.

Vessel design under the Pressure Equipment (Safety) Regulations 2016

A pressure filter vessel placed on the market in Great Britain is within the scope of the Pressure Equipment (Safety) Regulations 2016 where its maximum allowable pressure PS exceeds 0.5 bar [15]. Classification depends on the fluid group, the state of the fluid, PS and volume V. The conformity assessment tables are in Schedule 1B; water is a group 2 fluid [15].

For a vessel containing a group 2 liquid whose vapour pressure at the maximum allowable temperature does not exceed 0.5 bar above atmospheric, Table 4 applies, and a category is assigned only where PS exceeds 10 bar and PS·V exceeds 10 000 bar L, or PS exceeds 1000 bar [15]. A water-only filter vessel below those limits is designed and manufactured in accordance with sound engineering practice and does not carry the conformity marking for a category.
Where the vessel contains a gas under pressure, for example from air scour or an air cushion above the water, Table 2 for group 2 gases applies, and a category is assigned once V exceeds 1 L and PS·V exceeds 50 bar L [15]. Most air-scoured pressure filters therefore fall in a category, with corresponding conformity assessment modules.

Mechanical design is normally carried out to BS EN 13445, the harmonised and designated standard for unfired pressure vessels, which gives a presumption of conformity with the essential safety requirements [16]. In service, a system that contains a relevant fluid, which includes gas under pressure above 0.5 bar, is subject to the Pressure Systems Safety Regulations 2000, including a written scheme of examination [17]. The process designer's inputs to these obligations are the design pressure and temperature, including pump shut-off head and any backwash or air scour pressure, the fluid classification, and the internals that determine volume and freeboard.

Design sequence

A pressure filter design proceeds as follows: establish the influent solids load and the treated-water target; select filtration rate and media from pilot data or comparable plant; calculate clean-bed head loss by two methods and add underdrain losses; estimate run length from available head loss and solids storage; size backwash for fluidisation of the coarse fraction and target expansion at the governing temperatures; set the vessel freeboard; and define the design pressure, fluid classification and conformity route under the Regulations.

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.
Yao, K.M., Habibian, M.T. and O'Melia, C.R. (1971) Water and waste water filtration: concepts and applications. Environmental Science & Technology, 5(11), 1105–1112.
Iwasaki, T. (1937) Some notes on sand filtration. Journal AWWA, 29(10), 1591–1602.
Water Corporation (2025) Design Standard DS 114: Granular Media Filtration, version 2, revision 4. Perth: Water Corporation.
Kozeny, J. (1927) Über kapillare Leitung des Wassers im Boden. Sitzungsberichte der Akademie der Wissenschaften in Wien, Abteilung IIa, 136, 271–306.
Carman, P.C. (1937) Fluid flow through granular beds. Transactions of the Institution of Chemical Engineers, 15, 150–166.
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.
Ergun, S. (1952) Fluid flow through packed columns. Chemical Engineering Progress, 48(2), 89–94.
Wen, C.Y. and Yu, Y.H. (1966) A generalized method for predicting the minimum fluidization velocity. AIChE Journal, 12(3), 610–612.
Richardson, J.F. and Zaki, W.N. (1954) Sedimentation and fluidisation: Part I. Transactions of the Institution of Chemical Engineers, 32, 35–53.
BS EN 12904, Silica sand and silica gravel; BS EN 12909, Anthracite; BS EN 12902, Inorganic supporting and filtering materials: methods of test. Products used for treatment of water intended for human consumption. London: BSI.
The Water Supply (Water Quality) Regulations 2016, SI 2016/614, regulation 31. legislation.gov.uk.
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.
The Pressure Equipment (Safety) Regulations 2016, SI 2016/1105, Schedule 1B, paragraph 4, Tables 2 and 4. legislation.gov.uk.
BS EN 13445, Unfired pressure vessels. London: BSI.
The Pressure Systems Safety Regulations 2000, SI 2000/128. legislation.gov.uk.

Related technical notes

RB Solutions provides process design for pressure filtration: media selection, headloss and run-length estimates, backwash and air scour sizing, and the design data required for vessel classification and mechanical design.

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