Modelling laminar flow in V-shaped filters integrated with catalyst technologies for atmospheric pollutant removal
Samuel D. Tomlinson, Aliki M. Tsopelakou, Tzia Ming Onn, Steven R. H. Barrett, Adam M. Boies, and Shaun Fitzgerald 1 Department of Engineering, University of Cambridge, Cambridge, CB2 1PZ, UK 2 Centre for Climate Repair, Cambridge, CB3 0WA, UK 3 Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA
Abstract
Atmospheric pollution from particulate matter, volatile organic compounds and greenhouse gases is a critical environmental and public health issue, leading to respiratory diseases and climate change. A potential mitigation strategy involves utilising ventilation systems, which process large volumes of indoor and outdoor air and remove particulate pollutants through filtration.
However, the integration of catalytic technologies with filters in ventilation systems remains underexplored, despite their potential to simultaneously remove particulate matter and gases, as seen in flue gas treatment and automotive exhaust systems. In this study, we develop a predictive, long-wave model for V-shaped filters, with and without separators. The model, validated against experimental and numerical data, provides a framework for enhancing flow rates by increasing fibre diameter and porosity while reducing aspect ratio and filter thickness. These changes lead to increased permeability, which lowers energy requirements. However, they also reduce the pollutant removal efficiency, highlighting the trade-off between flow, filtration performance and operational costs. Leveraging the long-wave model alongside experimental results, we estimate the maximum potential removal rate and minimum cost if a billion Vshaped filters integrated with catalytic enhancements were deployed in operation. These findings highlight the feasibility of catalytic filters as a scalable, high-efficiency solution for improving air quality and mitigating atmospheric pollution. 1 Introduction
Atmospheric pollution poses a severe threat to human health, ecosystems and climate stability, making its mitigation a global priority. Particulate matter (PM), volatile organic compounds (VOCs) and nitrogen oxides (NO x ) contribute to respiratory and cardiovascular diseases, premature mortality and environmental degradation. In parallel, the accumulation of GHGs such as carbon dioxide, methane and nitrous oxide is accelerating global warming and climate instability. While strategies to mitigate outdoor pollution are essential, it is also important to manage indoor air quality, given that people now spend the majority of their time indoors. Ventilation systems play a pivotal role in indoor air quality by filtering airborne pollutants and reducing exposure to harmful pathogens. Ubiquitous in residential, commercial and industrial environments, ventilation systems process large volumes of indoor and outdoor air, predominantly removing PM and typically leaving GHGs and VOCs unaddressed. Similar air-processing systems are found in applications such as mines, combustionintake systems, cooling towers and agricultural facilities, where they help manage air quality by moving PM and gases such as CH 4 . Catalytic filters are used in flue gas treatment to remove NO x and dust, thereby reducing the operational costs associated with pollution control. A similar approach is employed in automotive exhaust systems, where catalytic diesel particulate filters capture PM while simultaneously reducing NO x emissions through selective catalytic reduction. Motivated by these successes, recent studies have proposed deploying catalytic filters that combine PM capture with catalytic conversion for GHGs or VOCs in ventilation systems.
The potential of such catalytic filtration systems, however, remains under-explored, particularly in the context of flow optimisation and filter design. Another similar integration of removal technologies was proposed during the COVID pandemic, where a number of approaches to tackle virus elimination were proposed such as high grade filters, UV-C light and antimicrobial coatings. Furthermore, filters are, or should be, replaced at regular intervals, which facilitates the integration of new technologies into existing systems.
As a result, the design life of anew product can be shorter and more readily achievable. Filtration efficiency is governed by fluid-structure interactions within a porous medium, with filter sheets classified as either fibre-based or pore-based. While fibre-based filters are easier to manufacture, they offer less control and can have a higher environmental impact than porebased alternatives, as they are more prone to clogging and may often be discarded. In contrast, pore-based filters can be cleaned and reused, reducing their environmental impact. Pleating the filter sheet increases the surface area, thereby enhancing airflow or reducing pressure drop, but may introduce velocity gradients across the filter, leading to variations in pollutant removal efficiency and clogging. Predicting the permeability of porous media is essential for optimising filter performance, with the Kozeny-Carman relation providing an empirical link between permeability, porosity and structural properties of the filter sheet. The theory of filtration relates permeability to particle collection on isolated fibres and identifies the main capture mechanisms in fibrous media as interception, diffusion and inertial impaction. As particles are filtered out, the filter sheet blocks, either through the accumulation of particles on the fibres or within the pores, which affects the permeability and hence the airflow rate through the sheet. The level of filtration can be measured in terms of the removal efficiency, which represents the percentage of particles removed larger than a certain size. Advances in filtration have combined modelling, computational simulations and experimental validation. Prior studies have resolved the velocity and pressure field in various V- and U-shaped filter geometries in laminar and turbulent flows, using reduced models and computational fluid dynamics (CFD) simulations. Pereira et al. refined these modelling approaches using a laminar linear long-wave theory within a periodic filter with dead ends, identifying designs to maximise airflow rates through the device. While CFD simulations enhance the understanding of airflow through filters, their high cost limits their use for optimisation. Experimental investigations using techniques like Hot Wire Anemometry and Particle Image Velocimetry have validated these models across various filter designs. This study focuses on laminar experiments with V-shaped filters that provide data on pressure drops and flow fields, where the effects of dust loading are negligible. A brief summary follows. Fabbro et al. explored the relationship between pressure drop and velocity through a V-shaped filter across filter geometries and efficiencies. Mrad et al. examined the streamwise velocity field upstream and downstream of the V-shaped filter, comparing their results to simulations performed for laminar and turbulent flows. Zhang et al. investigated the airflow rate of V-shaped and cylindrical filters, measuring the deposition of particles in filters with different porosities and fibre diameters. Al-Attar et al. investigated the effects of pleat density, pleat orientation and face velocity on V-shaped filters from ventilation and gas-turbine applications. Despite these advancements, uncertainties remain in filter performance. Firstly, existing models are validated on a case-by-case basis rather than being predictive across devices to allow for the optimisation of filter design. Secondly, a balance must be struck between highresolution simulations, which can be computationally expensive, and reduced-order models, which may lack precision. Thirdly, the influence of separators, which structurally support filters and impact airflow dynamics, remains unquantified. Lastly, the integration of catalytic technologies into filters will introduce modifications to the filter-sheet structure and pollutant removal efficiency, as discussed in. To address these uncertainties, we develop a predictive non-linear long-wave model for laminar flow in V-shaped filters, incorporating the effects of separators. The long-wave model incorporates relations that enable the flow rate and permeability of the filter to be determined from measurable design parameters: fibre diameter, porosity, filter length, height and thickness. We validate the long-wave model using numerical and experimental data, demonstrating its applicability across filter designs. We examine how design parameters influence the potential removal efficiency of particulate matter and different gases for V-shaped filters integrated with catalytic technologies.
Our findings suggest that catalytic filters could serve as a solution for improving indoor and outdoor air quality, with implications for public health and climate mitigation. 2 Methods
In this study, we analyse airflow through a V-shaped filter, where the geometry and fibre-sheet properties will influence the velocity and pressure field. Table 1 summarises the key flow characteristics for V-shaped filters, including length, velocity, pressure and material properties used in the analysis. The streamwise, wall-normal and spanwise directions are represented by ˆx-, ˆy- and ˆz-coordinates, where hats indicate dimensional quantities. Assuming the fluid is incompressible and Newtonian, consistent with standard models, the velocity field is given by ˆu = (ˆu(ˆx, ˆy, ˆz), ˆv(ˆx, ˆy, ˆz), ˆw(ˆx, ˆy, ˆz)) and the pressure field is ˆp = ˆp(ˆx, ˆy, ˆz). The V-shaped filter is assumed to be periodic in the normal direction and symmetric around the centre line of each period. Consequently, we focus on a single half-period of dimensions L [ˆ] in length, 2 H [ˆ] in height and W [ˆ] in width. Assuming the filter sheet thickness (t [ˆ] ) is negligible (e)
Figure 1: A pressure-driven airflow through a (a) duct containing a V-shaped filter, (b) a single half period of the V-shaped filter, (c) duct containing a separated V-shaped filter and (d) a single half period of the separated V-shaped filter. In (b, d), we position the origin of the coordinate system, (ˆx, ˆy, ˆz), at the start of the V-shaped filter, midway between the filter sheets and transverse width. Each period of the V-shaped filter has length L [ˆ], height 2 H [ˆ] and width W [ˆ] .
The location of the filter sheet is given by ˆy = ˆs(ˆx); the area below and above the filter sheet defines subdomains D [ˆ] 1 and D [ˆ] 2, respectively. (e) Photograph of a φ = 80% porosity fibre sheet and SEM image of the microstructure with fibres of around d [ˆ] = 100µm thickness, reproduced from Huang et al., licensed under CC BY 4.0. Name Symbol Unit Value
Pressure difference ∆P [ˆ] Pa 300
Half-period length Lˆ m 0.1
Half-period height Hˆ m 0.004
ˆ Half-period width Wm 0.1
ˆ Thickness t m 10 [−]
ˆ Permeability km 10 [−]
Number of half-periods N – 200
Table 1: A summary of characteristic pressure, length and material scales characteristic of Vshaped filters.
relative to the other length scales of the filter, its location, ˆy = ˆs(ˆx), is given by
sˆ(ˆx) = H [ˆ]. To account for the different flow characteristics above and below the filter sheet, we divide the half-period into two subdomains
ˆ D 1 = {xˆ ∈} × {yˆ ∈} × {zˆ ∈}, Dˆ 2 = {xˆ ∈} × {yˆ ∈ [ˆs, ˆH]} × {zˆ ∈},
as in Fig. 1(b, d). The flow enters the half period through D [ˆ] 1 at ˆx = 0 with a constant inlet pressure
pˆ = ˆp in,
and exits through D [ˆ] 2 at ˆx = L [ˆ] with a constant outlet pressure
pˆ = ˆp out . The imposed pressure drop, ∆ˆp ≡ pˆ in − pˆ out - 0, drives the airflow. To capture airflow behaviour, we solve the Navier–Stokes equations in D [ˆ] 1 and D [ˆ] 2
ˆ u xˆ + ˆv yˆ + ˆw zˆ = 0,
ˆ ˆ µ(ˆu xˆxˆ + ˆu yˆyˆ + ˆu zˆzˆ ) − p xˆ = ˆρ(ˆu · ∇ [ˆ] )ˆu,
ˆ ˆ µ(ˆv xˆxˆ + ˆv yˆyˆ + ˆv zˆzˆ ) − p yˆ = ˆρ(ˆu · ∇ [ˆ] )ˆv,
ˆ ˆ µ( ˆw xˆxˆ + ˆw yˆyˆ + ˆw zˆzˆ ) − p zˆ = ˆρ(ˆu · ∇ [ˆ] ) ˆw,
where ˆµ is the dynamic viscosity and ˆρ is the density of air. Since filters are usually enclosed within rigid housings, at the ends of the V-shaped filter in the transverse direction, we impose no-slip and no-penetration conditions at ˆz = 0, W [ˆ] :
uˆ = ˆv = ˆw = 0. We examine two common filter configurations, one with and one without separators, allowing for the comparison of flow behaviour from small-scale residential to large-scale industrial systems. The configuration affects the imposed boundary conditions and therefore the airflow behaviour.
If the V-shaped filter does not have separators, we impose symmetry conditions across each half-period at ˆy = 0, H [ˆ] :
ˆ u yˆ = ˆv = ˆw yˆ = 0. If the V-shaped filter has separators, we impose no-slip and no-penetration conditions at ˆy = 0, H [ˆ] : uˆ = ˆv = ˆw = 0,
To account for both viscous and inertial resistance effects introduced by the filter sheet, we employ the Darcy–Forchheimer model. This model is widely used in porous-media analysis, incorporating permeability (k [ˆ] ) effects via the permeance (ˆκ = k/ [ˆ] (ˆµt [ˆ] )) and pressure losses at higher flow velocities through the inertial resistance (β [ˆ] ). At ˆy = ˆs(ˆx), we have
ˆ ˆ ˆ u · ˆn = ˆκ(ˆp [−] − p [+] ) − β [ˆ][/] u · ˆn,
where ˆn ≡ is the normal vector to the filter sheet and the superscript − (+) indicates evaluation in D [ˆ] 1. For simplicity, we impose no tangential velocity at ˆy = ˆs(ˆx)
ˆ u · [ˆ] t x = ˆu · [ˆ] t z = 0,
where [ˆ] t x ≡ and [ˆ] t z ≡ are the tangent vectors to the filter sheet. The flow rate through the filter half-period is constant as mass is conserved and is given by
Hˆ
ˆ Q =
ˆ
� y=0
Wˆ
uˆ dˆydˆz. ˆ
� z=0
Therefore, assuming that the inlet pollutant concentration ˆc is uniform in D [ˆ] 1, as there are no sources or sinks other than the filter sheet, the potential pollutant removal rate is
ˆ ˆ q = ηQ [ˆ] c,
where Q [ˆ] cˆ is the flow rate of pollutant to the filter sheet and η is the removal efficiency of the pollutant-removal technology, defined as the fraction of pollutant molecules removed upon contact with the filter sheet. We aim to determine the dependence of Q [ˆ] and ˆq on the dimensional parameters of the problem, L [ˆ], H [ˆ], W [ˆ], ∆ˆp, ˆµ, ˆκ, ˆρ, β [ˆ], η and ˆc, in V-shaped filter configurations. 2.1 Non-dimensionalisation
We begin by non-dimensionalising the governing equations– using the characteristic length (L [ˆ] ), velocity ( U [ˆ] ) and pressure scales of the flow:
[x] ˆ [ˆ], y = [y][ˆ]
L ǫ
L [z] ˆ [ˆ], u = U [u] ˆ [ˆ]
[u] ˆ [ˆ], v = [v][ˆ]
U ǫ
ǫ [v][ˆ] U [ˆ], w = U [w] ˆ [ˆ]
U [w] ˆ [ˆ], p = [ǫ] [ ˆ][L][(ˆ][p] ˆ [ −] U [ˆ] [p][ˆ] [out] [)]
µˆU [ˆ],
x = [x] ˆ [ˆ]
ǫ [y][ˆ] L [ˆ], z = L [z] ˆ [ˆ]
where ǫ ≡ H/ [ˆ] L [ˆ] is the aspect ratio of the half period. The location of the filter sheet becomes
s(x) = 1 − x. The subdomains become
D 1 = {x ∈} × {y ∈} × {z ∈}, D 2 = {x ∈} × {y ∈} × {z ∈},
where W = W/ [ˆ] L [ˆ] is the non-dimensional width of the filter. At the inlet, x = 0, the pressure condition gives p = 1,
and at the outlet, x = 1, the pressure condition gives p = 0. In D 1 and D 2, the bulk equations become u x + v y + w z = 0, ǫ (u xx + u zz ) + u yy − p x = ǫ Re(u · ∇)u, ǫ (v xx + v zz ) + ǫ v yy − p y = ǫ Re(u · ∇)v, ǫ (w xx + w zz ) + w yy − p z = ǫ Re(u · ∇)w. At z = 0, W, the end conditions give u = v = w = 0. At y = 0, 1, the symmetry conditions become u y = v = w y = 0,
and the no-slip and no-penetration conditions become u = v = w = 0. At y = s(x), the normal flow condition gives u · n = κ(p [−] − p [+] ) − β[/] u · n,
where n = is the unit normal vector at the filter-sheet surface, κ = /ǫˆ [ ˆ] L is the permeance, k = κ/(t/ [ˆ] L [ˆ] ) is the permeability and β = β [ˆ] U [ˆ] is the inertial resistance of the filter sheet. Parameters κ and k quantify the filter’s resistance to flow through the filter sheet, and β accounts for non-Darcian effects due to inertia at higher velocities. The tangential flow conditions give u · t x = u · t z = 0,
where t x ≡ and t z ≡ are tangential vectors at the filter-sheet surface. The volumetric flow rate through the half period becomes
1 Q = � y=0
W
u dydz,
� z=0
and the potential removal rate of the pollutant is given by q = ηQ. In the limit ǫ ≪ 1 and ǫ Re ≪ 1, the governing equations simplify, and we proceed to solve the leading-order boundary value problem defined in– using asymptotic and numerical methods.
We evaluate Q and q through the V-shaped filter,–, and investigate its dependence on the dimensionless groups ǫ, W, κ, β and η that characterise the geometry, filter and flow. 2.2 The long-wave limit
We begin by taking the limit as the aspect ratio of the half period ǫ ≡ H/ [ˆ] L [ˆ] → 0, substituting the expansions (u, v, p) = + ǫ + ...,
into the governing equations– and taking the leading-order approximation. In this study, we consider the limit where B ≡ βǫ = O to capture the inertial effects observed in the experiments that influence the flow through the filter sheet. At the inlet, x = 0, the pressure condition gives p 0 = 1,
and at the outlet, x = 1, the pressure condition gives p 0 = 0. In D 1 and D 2, the bulk equations become u 0x + v 0y + w 0z = 0, u 0yy − p 0x = 0, p 0y = 0, w 0yy − p 0z = 0. The long-wave limit, ǫ → 0, leads to short Navier-Stokes regions, which ensure that the end conditions are satisfied. At y = 0, 1, the symmetry conditions become u 0y = v 0 = w 0y = 0,
and the no-slip and no-penetration conditions become u 0 = v 0 = w 0 = 0.
At y = s(x), the normal flow condition gives v 0 = κ][ −] [Bv] 0 [,]
and the tangential flow conditions give u 0 = w 0 = 0. The flow rate through the half period becomes
1 Q ≈ � y=0
W
u 0 dy dz,
� z=0
and the potential removal rate of pollutant remains as is given by. The transverse flow problem is not forced directly, and we expect it to relax to a solution for which p 0 = p 0 (x) and w 0 = 0. First, we consider the V-shaped filter without separators. We can solve subject to– in D 1 to give u 0 = p 0x/2, v 0 = −p 0xx y /6 +y, and in D 2 to give u 0 = p 0x − )/2, v 0 = /3 − ) + 2p 0xs x )/2. Substituting the velocity field– into gives a system of ordinary differential equations (ODEs) along y = s(x).
We can solve this system of ODEs to give the solution in D 1 and D 2 as p 0 = p 0 (x). Along y = s(x),
Ls = κ][ −] [B][] [,] Ls−1 = κ][ −] [B][] [,]
where Ls =
and at x = 1, the outlet and no-gradient condition is p [+] 0[,] p [+] 0x[.]
Second, we consider the V-shaped filter with separators. We can solve subject to– in D 1 to give u 0 = p 0x (y − s)y/2, v 0 = −p 0xx y /6 +y,
and in D 2 to give u 0 = p 0xy + s)/2, v 0 = p 0xxy /4 − sy/2 −/12) + p 0x s x/2. Substituting the velocity field– into along y = s(x) gives
Ms = κ][ −] [B][] [,] Ms−1 = κ][ −] [B][] [,]
subject to–, where Ms = Ls /4. As u 0 = u 0 (x, y) due to the long-wave approximation, the flow rate through the half period in the configuration without separators simplifies to
1 Q ≈ Wu 0 dy = −Wp 0x /3, � y=0
and in the configuration with separators simplifies to
1 Q ≈ Wu 0 dy = −Wp 0x /12. � y=0
Under the long-wave assumption, the system parameters are reduced to just W, κ, β and η; the goal in the following section is to identify the values that maximise Q and q. 2.3 Validation of numerical solution
To validate the numerical solution to–, we compare the long-wave model’s pressure field to that of for a filter without separators. Fig. 2(a) shows agreement, where the numerical solution is shown with solid lines and the results from are represented by symbols. In, the filter sheet did not extend across the entire half-period, and the model enforces no-slip and no-penetration regions above and below the inlet and outlet. Using the same parameters from, the filter sheet is located at s(x) = a + β, which terminates at s = 0.525 and s = 0.475. In Fig. 2(b), we present the pressure field obtained from the long-wave model for the same filter geometry without separators. This allows for a direct comparison with the results from. While the pressure fields for filters with and without separators are similar, variation in the flow field and flow rate arises due to differences in boundary conditions at y = 0, 1. Following this validation step, we move on to explore filter sheets that span the full height of the pleat, with s = 1 and s = 0. 2.4 Permeability and removal efficiency
Accurate permeability estimates are crucial for predicting the flow rate and optimising filter design. The Kozeny–Carman model, widely used for estimating permeability in porous media, is given by
ˆ k = [C][ ˆ][d] [φ]
[,]
where C ≈ 0.07 is a fitting coefficient determined from in Section 3. Precise removal efficiency values are also critical for estimating filter effectiveness, particularly in environments requiring high air quality standards. We model η using an exponential model similar to the single-fibre efficiency formula described in,
η = r 1 − exp − [E] ˆ � � k
��
where E ≈ 2.1 × 10 [−] m is a fitting coefficient determined from. The coefficient r is equal to unity for particulate filtration and equal to the net probability of collision with an active site and conversion for catalytic removal. In Section 3, we assume that for PM 2.5 : cˆ = 3 × 10 [−] mol/m and r = 1, for NO x : ˆc = 2 × 10 [−] mol/m and r = 0.5, and for CH 4 : cˆ = 8 × 10 [−] mol/m and r = 0.1. These values represent a realistic and optimistic average removal efficiencies across NO x and CH 4 catalysts, e.g., reports 50% removal at 250 ◦ C and 10 3 ppm for catalytic bag filters in flue gas treatment and averages around 10% removal at 300 W/m and 10 ppm across photocatalysts in the lab. NO xis easier to remove than CH 4 via catalysis due to its higher reactivity and lower energy requirements. 3 Results
3.1 Experiments
To assess the accuracy of the long-wave model, we compare its predictions with experimentally observed relationships between the pressure drop (∆ˆp) and filtration velocity (ˆv f ) in V-shaped Figure 2: (a) Comparison of the pressure field in (symbols) with the long-wave model (lines) for a V-shaped filter without separators. (b) The pressure field in the long-wave model for a V-shaped filter with separators. In (a–b), the location of the filter sheet is prescribed by s(x) = a + β, with β = 0.05, a = 0.5 and κ = 1, as illustrated in the inset. filters.
explored ∆ˆp as a function of ˆv p and ǫ. Fig. 3(d) demonstrates the long-wave model’s correspondence with their experimental data with k [ˆ] ≈ 6 × 10 [−] m, where η = 94% for d [ˆ] p ≥ 0.1 µm, φ = 0.94 and d [ˆ] = 2.1 µm. Across all V-shaped filter datasets, the long-wave model successfully captures the effects of geometry, permeability and flow resistance, as summarised in Fig. 3(e). Having demonstrated agreement between the long-wave model and the experimental data, we next show that the permeability can be predicted from measurable filter-sheet properties, eliminating the need for case-by-case fitting. Fig. 4 shows that the Kozeny–Carman model offers a robust framework for estimating permeability, effectively capturing how variations in d [ˆ] and φ influence k [ˆ] across the experiments. The Kozeny–Carman model predicts Authors Velocity (ˆv f ) Geometry (ǫ, H [ˆ] ) Reynolds (Re) Structure (φ, d [ˆ] ) Permeability (k [ˆ] ) Efficiency (η)
Fabbro et al. 0.0746 0.0386 0.0016 4 × 10 0.92 10 [†] 1 × 10 [−][†] >99 [†]
Fabbro et al. 0.0853 0.0359 0.0014 4 × 10 0.88 100 [†] 4 × 10 [−][†] 50–55 [†]
Mrad et al. [∗] 0.3315 0.4464 0.0125 1 × 10 0.92 33 2 × 10 [−][†] 50–55
Zhang et al. 0.0358 0.1092 0.0155 6 × 10 0.84 [†] 5 [†] 1 × 10 [−][†] 99.97
Theron et al. [∗] 0.449 0.3195 0.0064 2 × 10 0.92 13.8 2 × 10 [−][†] >99 [†]
Tronville et al. [∗] 1.25 0.1280 0.0032 4 × 10 0.92 [†] 1 [†] 1 × 10 [−] [†] >99 [†]
Youssef et al. 0.2 0.125 0.0025 3 × 10 0.92 [†] 1 [†] 1 × 10 [−] [†] >99 [†]
Rebai et al.
[∗] 0.65 0.078 0.004 2 × 10 0.92 [†] 15 4 × 10 [−] [†] >99 [†]
Al-Attar et al. 0.0287 0.008 0.003 2 × 10 0.94 2.1 6 × 10 [−][†] 94
Table 2: Summary of experiments on V-shaped filters without dust clogging: their average filter velocity (ˆv f, m/s), aspect ratio (ǫ, –), half-height ( H [ˆ], m), Reynolds number (Re, –), porosity (φ, –), fibre diameter ( d [ˆ], µm), permeability and efficiency (η, –). The superscript [†] implies either that these values have been assumed or they have been calculated based on available data and–. The superscript [∗] indicates those studies for which ǫ Re ≫ 1, inertia is dominant and the flow is likely turbulent. Figure 3: Comparison of the pressure drop (∆ˆp) with the filtration velocity (ˆv f ) predicted using the long-wave model (solid lines) and experiments (symbols) in (a) for ǫ = 0.065, 0.037, 0.026, (b) for ǫ = 0.45, (c) for ǫ = 0.11 and (d) for ǫ = 0.0089, 0.0083, 0.0078, 0.0074. (e) Comparison of ∆ˆp with ˆv f predicted using the long-wave model (solid lines) and (symbols). that permeability increases with both d [ˆ] and φ, consistent with trends observed in the experimental data. By collapsing the experimental trends related to fibre diameter and porosity into a single relation, the long-wave model demonstrates its broad applicability to V-shaped filter configurations, offering a unified approach to predicting filter performance. Since permeability plays a central role in filtration performance, we extend this framework to investigate how changes in the filter-sheet properties affect the removal efficiency (η) of the V-shaped filter for particles with diameter d [ˆ] p ≥ 0.3 µm in Fig. 5. The exponential model reflects the experimental trend that higher permeability leads to lower removal efficiency, as increased pore space reduces particle capture efficiency. Thus, we have established a predictive framework that links measurable V-shaped filter properties to the permeability through the Figure 4: Comparison of the normalised permeability (a) k/ [ˆ]) and (b) k/ [ˆ] d [ˆ] with the average fibre diameter ( d [ˆ] ) and porosity (φ) using the Kozeny–Carman model (solid lines) and experiments (symbols). Figure 5: Comparison of the removal efficiency η with the permeability k [ˆ] using the exponential model (solid line) and experiments (symbols). Kozeny–Carman model, the removal efficiency through the exponential model and flow rates through the long-wave model–. When investigating changes in the flow rate due to catalytic coatings in the following section, we will ensure that the V-shaped filters meet a MERV rating of at least 9–12 for particles with d [ˆ] p ≥ 1 µm, corresponding to a minimum efficiency reporting value of 85%, as depicted by the black dashed line in Fig. 5. 3.2 Flow rate
Fig. 6 illustrates how the non-dimensional permeability (k) and filter width (W ) influence the flow rate (Q) in V-shaped filters, providing insights into optimising filter performance. Separators reduce the maximum flow rate by a factor of four compared to configurations without separators, consistent with the asymptotic analysis in A. To ground the findings within exper Figure 6: The flow rate (Q) through the V-shaped filter (a) without separators and (b) with separators, for varying permeability (k) and filter widths (W ), using. Dashed lines represent asymptotes derived in A for k ≫ 1 and k ≪ 1, and vertical lines represent the nondimensional permeability measured in experiments in (green) for varying fibre diameter ( d [ˆ] ) and aspect ratio (ǫ). imental data, we highlight a permeability estimate from Table 2 (green), whereˆ k = 2 × 10 [−], d = 1 × 10 [−] m and ǫ = 0.013.
Fig. 6(a) illustrates that increasing d [ˆ], e.g., by applying a catalyst coating with an average thickness of 5µm to the fibres that make up the filter sheet (assuming the catalyst is applied before the fibres are assembled; if applied after the filter is assembled, the relationship between k and the filter’s structure will differ), leads to a higher k, which in turn increases Q from 0.003 to 0.07 for W = 2. Larger fibres create wider pores, reducing the flow resistance and allowing for more efficient airflow. However, increasing the permeability negatively impacts removal efficiency. As k increases, η decreases from 99.97 to 29%, although the total filtered airflow ηQ still increases from 0.003 to 0.02. To maintain a removal efficiency of at least 85% for the V-shaped filter from Table 2, permeability must not exceed k < [ˆ] 1 × 10 [−]
m, limiting the increase in d [ˆ] to 2.5 times, compared to the sixfold increase shown in Fig. 6(a). Fig. 6(b), examines how reducing ǫ, e.g., by doubling the V-shaped filter length relative to the period half-height, affects permeability and flow rate.
This change increases k and raises Q from 0.003 to 0.02 for W = 2. Extending the fibre sheet increases its surface area, allowing for greater airflow and higher flow rates through the filter. In this case, the increase in k due to ǫ does not significantly impact η, as indicated by. Our findings highlight the intricate balance between permeability, flow rate and removal efficiency in V-shaped filters. While increasing fibre diameter or filter length can improve airflow, careful parameter selection is necessary to maintain high filtration performance.
These insights generalise to other parameters, including φ, t [ˆ] and L [ˆ], each of which influences permeability and flow dynamics. 3.3 Flow field
In Fig. 7, we examine how k affects the streamwise velocity field (u), normal velocity field (v), pressure field (p) and streamlines (u, v). We compare the flow field without separators (panels (a) (b) (c) (d)
(e) (f) (g) (h)
(i) (j) (k) (l)
(m) (n) (o) (p)
Figure 7: The streamwise velocity (a, e), normal velocity (b, f), pressure field (c, g) and streamlines (d, h) for (a–d) k = 1 × 10 [−] and (e–h) k = 1 × 10 [−] without separators using. The streamwise velocity (i, m), normal velocity (j, n), pressure field (k, o) and streamlines (l, p) for (i–l) k = 1 × 10 [−] and (m–p) k = 1 × 10 [−] without separators using. a–h) and with separators (panels i–p), where u is either no slip or symmetric at y = 0, 1. For high permeability, shown in panels (e–h, m–p), the filter offers little resistance to the flow passing through it. As a result, the pressure fields in regions D 1 and D 2 equilibrate and vary linearly due to the total pressure drop across the filter. This pressure drop generates a streamwise velocity in D 1 and D 2, which is converted into a normal velocity through the fibre sheet. For low permeability, shown in panels (a–d, i–l), the filter strongly resists normal flow, effectively blocking fluid from passing through the fibre sheet. Consequently, the V-shaped filter half-period experiences minimal through-flow, and the pressure fields in D 1 and D 2 are approximately uniform. The pressure in D 1 is equal to the inlet pressure, while the pressure in D 2 matches the outlet pressure. Due to the small pressure gradients in these regions, the streamwise velocity in D 1 and D 2 remains low, leading to only a small normal velocity through the fibre sheet. These results highlight the role of permeability in shaping the flow structure through the V-shaped filter. Authors Geometry ( W [ˆ], N ) Flow rate ( Q [ˆ] ) Efficiency (η) Removal rate PM 2.5, NO x, CH 4 (ˆq)
Fabbro et al. 1 200 0.61 99.97 1.8 × 10 [−] 6.1 × 10 [−] 4.9 × 10 [−]
Mrad et al. 1 200 2.03 52.5 3.2 × 10 [−] 1.1 × 10 [−] 8.5 × 10 [−]
Zhang et al.
1 200 1.02 99.97 3.0 × 10 [−] 1.0 × 10 [−] 8.1 × 10 [−]
Al-Attar et al. 1 200 2.30 94 6.5 × 10 [−] 2.2 × 10 [−] 1.7 × 10 [−]
Table 3: Examples of the potential removal rates of PM and different gases using the laminar experiments on V-shaped filters: their width ( W [ˆ], m), number of half-periods (N, –), flow rate, filtration efficiency (η, –), potential removal rate of PM 2.5, NO x and CH 4 (ˆq, mol/s). 3.4 Potential pollutant removal rates
To illustrate the findings, we analyse results from the laminar flow experiments in V-shaped filters in Table 3. To estimate the airflow rate for a V-shaped filter, we scale the measured airflow rate per half-period, assuming an average filter width of 1 m and 200 repeating half-periods, consistent with typical ventilation systems. An increase in filter sheet permeability, driven by variations in fibre diameter, porosity or geometry, from the values reported by to those reported by, results in a doubling of the airflow rate through the filter from approximately 1 to 2 m /s. Higher permeability reduces flow resistance, allowing increased airflow, but decreases filtration efficiency from 99.97 to 52.2%, likely due to a shorter residence time of particles within the porous structure of the filter medium. Despite the reduction in filtration efficiency, the increased airflow enhances the potential atmospheric removal rate slightly from 3.0 × 10 [−] to 3.2 × 10 [−] mol/s for PM 2.5, 1.0 × 10 [−] to 1.1 × 10 [−] mol/s for NO x and 8.1 × 10 [−] to 8.5 × 10 [−] mol/s for CH 4, as a greater volume of air is processed per unit time. Globally, approximately a billion filters are in operation, with an estimated flow rate through these systems ranging from 0.6 to 2.3 Gm /s, based on the values provided in Table 3. If catalyst technologies were integrated into these systems, the projected atmospheric removal rate could range from 7 × 10 [−] to 2 × 10 [−] GtPM 2.5, 6 × 10 [−] to 2 × 10 [−] GtNO x, 3 × 10 [−]
to 9 × 10 [−] GtCH 4 per year at a cost of $7 × 10 to 3 × 10 per tNO x and $2 × 10 to 6 × 10 per tCH 4 removed, assuming $17.87 per m for coating the filter sheet. These findings demonstrate that optimising permeability, combined with catalytic integration, could enable filters to function as large-scale atmospheric removal devices, removing approximately 1 GtCO 2 e per year at around $100 per tCO 2 e. 4 Discussion
This study has used a long-wave theory to model laminar flow through V-shaped filters for pollutant removal. The long-wave model simplifies the airflow problem in the V-shaped filter, preserving physical details and improving computational efficiency compared to the complete problem. Our results agree with numerical simulations and experimental data, confirming the long-wave model’s applicability to diverse V-shaped filter configurations. The long-wave model captures key relationships between pressure drop, permeability, velocities and flow resistance, predicting the removal efficiency and permeability based on measurable filter properties such as fibre diameter, porosity, filter length, half-height and thickness. Our findings explain the critical trade-off in V-shaped filter design, underscoring the need for an optimised balance between flow rate, pollutant removal and energy efficiency. Increasing fibre diameter and porosity enhances airflow but reduces available surface area for particle capture, lowering removal efficiency. Conversely, increasing the aspect ratio and thickness of the filter improves pollutant capture but increases flow resistance, raising the energy demands of the ventilation system. By modifying filter surfaces with other removal technologies, these systems could capture and convert CO 2, CH 4, N 2 O, NO x and VOCs by integrating carbon capture, catalysis and air purification within existing infrastructure. These multifunctional filters would enhance both indoor and outdoor air quality, particularly in environments with stringent air quality regulations. Future work could explore adaptive filtration strategies that incorporate real-time monitoring and dynamic adjustments to pollutant concentrations, leading to self-regulating and more efficient air purification. Nevertheless, the limitations of the approach must be acknowledged. The long-wave model’s assumption of steady-state, laminar flow excludes non-linear effects such as turbulence and clogging, which may alter performance over extended operation. While regular filter maintenance mitigates some clogging, real-world conditions may introduce unpredictable accumulation patterns or high dust loads. Turbulence could enhance pollutant transport and improve filtration efficiency. Future work incorporating turbulent flow effects will complete the laminar and turbulent performance comparison in Table 2. With these extensions, the model could potentially be generalised to flue gas treatment and automotive exhaust systems. While the exponential and Kozeny–Carman relations provide reliable estimates of removal efficiency and permeability, their empirical nature may require further calibration for different types of filters (e.g., U-shaped) or operating conditions (e.g., turbulence). Experimental validation of catalytic enhancements is currently underway to confirm the removal efficiency and the longwave model’s predictions for multifunctional filters. In conclusion, this study provides a framework that unites theoretical, numerical and experimental analyses into laminar flows through V-shaped filters. The framework provides a blueprint for optimising the next generation of filters that can mitigate PM, GHGs and VOCs.
Acknowledgments
We acknowledge Grantham Foundation for supporting this research. A Asymptotic solutions
A.1 Large-permeability solution
When the permeability of the filter sheet is large, κ ≫ 1, we can simplify the system of ODEs for pressure without separators–. In Fig. 7(a–d), the filter offers little resistance to the flow so that the pressure field becomes uniform across the channel. We expand the pressure field as follows p [−] 0 [=][ p] [−] 00 [+][ p] [−] 01 [/κ][ +][ ...,] p [+] 0 [=][ p] [+] 00 [+][ p] [+] 01 [/κ][ +][ ...,] and substitute into–. At leading order, along y = s(x), we have p [−] 00 [=][ p] [+] 00 [=][ p] [,] such that p 00xx s /3 − p 00x s s x = κ][,] p 00xx /3 − p 00x s x = κ][.]
At x = 0, the inlet and no-gradient condition is p 00 = 1, p 00x = 0,
and at x = 1, the outlet and no-gradient condition is p 00 = 0, p 00x = 0. Subtracting the equations in,– reduces to p 00xx )/3 − p 00x )s x = 0, p 00 = 1, p 00 = 0,
such that the leading-order pressure distribution is
√3) p 00 = [π][ −] .
2π
Hence, the leading-order flow rate, evaluated by substituting into, is given by
√3 Q = [W] + .... 4π
We can perform the same procedure for the system of ODEs for pressure with separators to recover. Hence, the leading-order flow rate, evaluated by substituting into, is given by
√3 Q = [W] 16π [+][ ....]
– agree with the results shown in Fig. 6. A.2 Small-permeability solution
When permeability is small, κ ≪ 1, we can simplify the system of ODEs for pressure without separators. In Fig. 7(e–h), the filter blocks the flow such that the pressure field becomes constant and equal to the inlet condition in D 1 and equal to the outlet condition in D 2 . We expand the pressure field as follows p [−] 0 [=][ p] [−] 00 [+][ κp] [−] 01 [+][ ...,] p [+] 0 [=][ p] [+] 00 [+][ κp] [+] 01 [+][ ...,]
and substitute into–. At leading order, along y = s(x),
p [−] 00xx [s] [/][ −] [p] [−] 00x [s] [s] [x][,] p [+] 00xx [] [/][ −] [p] [+] 00x [] [s] [x][.] At x = 0, the inlet and no-gradient condition is p [−] 00[,] p [−] 00x[,]
and at x = 1, the outlet and no-gradient condition is p [+] 00[,] p [+] 00x[.]
The leading-order pressure distribution in D 1 and D 2 is p [−] 00[,] p [+] 00[.]
Hence, the leading-order flow rate, evaluated by substituting into, is given by
Q = O(κ),
which agrees with the numerical results shown in Fig. 6. We can perform the same procedure for the system of ODEs for pressure with separators to recover and.