Performance factors for filtration of air using cellulosic fiber-based media: A review

Owens, L. P., and Hubbe, M. A. (2023). "Performance factors for filtration of air using cellulosic fiber-based media: A review," BioResources 18(1), 2440-2519.

Abstract

The filtration of air has attracted increasing attention during recent waves of viral infection. This review considers published literature regarding the usage of cellulose-based materials in air filtration devices, including face masks. Theoretical aspects are reviewed, leading to models that can be used to predict the relationship between structural features of air filter media and the collection efficiency for different particle size classes of airborne particulates. Collection of particles can be understood in terms of an interception mechanism, which is especially important for particles smaller than about 300 nm, and a set of deterministic mechanisms, which become important for larger particles. The effective usage of cellulosic material in air filtration requires the application of technologies including pulp refining and chemical treatments with such additives as wet-strength agents and hydrophobic sizing agents. By utilization of high levels of refining, in combination with freeze drying and related approaches, there are opportunities to achieve high levels of interception of fine particles. A bulky layer incorporating nanofibrillated cellulose can be used in combination with a coarser ply to achieve needed strength in a filter medium. Results of recent research show a wide range of development opportunities for diverse air filter devices containing cellulose.

Download PDF

Full Article

Performance Factors for Filtration of Air Using Cellulosic Fiber-based Media: A Review

Lindsay P. Owens ^a,b and Martin A. Hubbe ^a,*

DOI: 10.15376/biores.18.1.Owens

Keywords: Permeability; Capture efficiency; Dust; Droplets; Face masks; Interception; Triboelectricity; Nanocellulose; Hydrophobic

Contact information: a: North Carolina State University, Dept. of Forest Biomaterials, Campus Box 8005, Raleigh, NC 27695-8005, USA; b: D. S. Smith, 2366 Interstate RD, Riceboro, GA 31323;

* Corresponding author: hubbe@ncsu.edu

INTRODUCTION

Dangers resulting from airborne particulates and aerosols have captured urgent attention during the COVID-19 pandemic. Aerosol particles that contain viruses can travel through the air, spreading diseases (Tang et al. 2006; Nazarenko 2020). According to the US Centers for Disease Control and Prevention, the wearing of a facemask, i.e. air filtration, is one of the most effective and immediate ways to limit the spread of viral infections (CDC 2021). These recommendations are supported by published research (Cowling et al. 2010; Bin-Reza et al. 2012; Bunyan et al. 2013; Nazarenko 2020). The use of such masks has become commonplace in general society, as it has been for many years within hospital critical care settings (Arnold 1938). In addition, hazardous particulates and aerosol droplets can be removed by air filtration systems that are built into building ventilation equipment (Clausen 2004; Hyttinen et al. 2011; Brincat et al. 2016; Liu et al. 2017; Brochot et al. 2019; Nazarenko 2020).

The present article mainly focuses on the current and potential role of cellulosic fiber materials employed in air filtration. A recent review article highlighted the usage of cellulose as a component in facemasks (Garcia et al. 2021). Although cellulose is already very widely used as a filtering medium, it faces stiff competition from other materials, especially in higher-end applications. The present review considers various pros and cons of cellulosic fiber materials specifically in air filtration equipment. Evidence is considered from diverse studies, ranging from cotton fabric facemasks to hospital ventilation systems. Aspects of the size, structure, and chemical nature of cellulosic materials are considered with respect to what is known about the mechanisms of filtration in different air environments of interest. Because nanocellulose structures can be prepared with a very high surface area and very small pores, related research will be a particular focus of this review.

A recurrent theme of this review article will be the role of moisture relative to the effectiveness of air filters. As will be discussed, various theoretical approaches have been developed based on assumed completely dry filter media and air-borne particles. Related work has dealt with similar systems in which the solids are completely immersed in aqueous solution. But real issues encountered during evaluation of air filtration systems can span a wide range of intermediate conditions. For example, it is well known that the user’s breath can dampen a facemask from the inside, while rain can dampen it from the outside. Thus, although this review’s main focus is air filtration by cellulose-containing media, some issues related to damp or wet filter media have been intentionally included within its scope.

Desired Attributes for Filter Media

The usage of any material as filter media needs to be justified based on evidence or theories related to its performance in that role (Brown 1993). Capture efficiency, the ratio of filtered particles to particles in the incoming air, is a top priority (Abdolghader et al. 2018). The second requirement is that the resistance to flow needs to be as low as practical, depending on the application (Belkin 1997; Morgan-Hughes et al. 2001; Abdolghader et al. 2018). This is especially important in the case of facemasks, since a high resistance of the filter media promotes a greater leakage of air around the edges of the mask. An inverse relationship between capture efficiency and resistance to flow has been observed in many cases (Soo et al. 2016; Chien et al. 2018; Dziubak and Dziubak 2020). Figure 1 illustrates three kinds of substances that may need to be resisted by air filtration media.

Fig. 1. Key attributes often wanted in media for air filtration, with a focus on facemasks. The horizontal dotted line represents a hypothetical deterministic, size-based filter structure that plays a role analogous to that of a sieve.

Filter media need to be tolerant of a range of moisture contents of the incoming air, including the likelihood of aerosol droplets in some cases. They need to have adequate capacity for collected particulates and not be prone to early blockage during their use. In addition, some applications can benefit from antibacterial or antiviral activities.

In common with all manufactured products, there is a preference of filter media made mostly from natural, renewable materials, using mainly eco-friendly processing conditions (Heydarifard et al. 2016). Cellulosic materials are often well-suited to such goals, and in addition they often have a suitably low price. Cellulose fibers, as well as various nanocellulose products, can be obtained from photosynthetically renewable plant materials. The technologies needed to separate the fibrous material, including the optional removal of lignin, are well established (Fardim and Tikka 2011). Depending on such factors as later chemical modifications, the cellulosic material is generally biodegradable (de Almeida et al. 2020; Li et al. 2021; Hu et al. 2022). The key questions to be considered in this article relate to how well cellulosic materials, of various types, can meet the other critical objectives and how such performance can be improved. Thus, it is important to consider the extent to which cellulose-based filter media can either match or approach the performance of filter products that have become established in the market.

Challenges for Cellulose-containing Facemasks and Filter Media

Their interactions with water represent some of the greatest differences between cellulosic materials compared to other commonly used materials for filter media. It has been found that cellulosic filtration media may be susceptible to a rise in resistance to flow after exposure to droplets of saline solution (Turnbull et al. 2005). The capillary forces present during ordinary drying of paper will tend to draw adjacent cellulosic surfaces into molecularly-tight contact, thus greatly decreasing the surface area available for filtration (Stone and Scallan 1966; Page 1993). Such an effect is illustrated schematically in Fig. 2. As will be discussed later, it can be a great advantage to dry cellulosic media by specialized methods such a freeze-drying in order to maintain a high specific surface area for filtration (Mao et al. 2008; Nemoto et al. 2015; Yoon et al. 2016; Lu et al. 2018; Ma et al. 2018; Wang et al. 2018). It is logical to expect that such surface area may become irrecoverably lost if the material becomes wetted and redried in the course of its usage. This type of effect will be especially important in cases where the cellulosic material has been subjected to high levels of shearing while in a wet condition, leading to a microfibrillated structure. Even ordinary cellulosic filter papers tend to have a relatively high pressure drop compared to some of the competing media, at similar levels of capture (Chien et al. 2018).

Fig. 2. Concept of how intermittent wetting and then drying may lead to densification and loss of void volume of cellulose-based filter media

Finally, there can be a concern that the hydrophilic, eco-friendly surfaces of cellulosic materials may present a favorable breeding ground for microbes. Human breath is known to be rich in microbes, including viruses (Milton et al. 2013). There are two parts to this concern. On the one hand, one can ask whether the hydrophilic nature of cellulose means that sufficient water to allow microbial viability will be typically present in the media during its usage. For example, cellulose-based insulation in homes can be subject to infestation by bacterial mold (Godish and Godish 2006). A study by Maus et al. (2001) showed that mold growth in non-cellulosic filter media was mainly dependent on the presence of nutrients and high relative humidity (> 98%). In particular, the required nutrients often can be imported by collected dust. Tests can be carried out to monitor the populations of bacteria over the course of using cellulose-based filtration media for air (Bibeau et al. 2000). However, there appears to be a need for research focusing specifically on whether or not the presence of cellulose within filter media affects the development of mold. On the other hand, research results suggest that the chemical and other attributes of cellulose make it suitable for various antimicrobial treatments. Table 1 lists selected studies that have focused on the development of antibacterial or antifungal treatments of filter media containing cellulose-based materials.

Table 1. Topic Areas of Some Articles Focusing on Development of Antibacterial or Antifungal Properties of Cellulose-based Air Filter Media

AIR FILTRATION: THEORETICAL BACKGROUND

Overview

Air filtration by fibrous filter media has been widely studied, with both experimental and theoretical models thoroughly developed and documented. This section covers the theoretical foundations of air filtration by fibrous filter media, examining the mechanisms of filtration such as interception, filtration by size, and rebound or retention. Air permeability of the fibrous filter media is also modeled and discussed, while investigating the trade-off between capture efficiency and permeability. Respiratory mask leakage issues due to improper mask fitting are explored, as well as the relationship between mask leakage issues and permeability, with the goal of achieving optimal breathability and capture efficiency for the wearer. Filter capacity issues related to filter porosity, filter surface area, cake filtration, and filtration mechanism are discussed. Antimicrobial applications to air filtration are also covered.

In the discussions that follow, it will be shown in various cases that filtration efficiency, as well as the variations in pressure loss across media, can be affected by various parameters, such as particle size, filter media size, surface areas, pore dimensions, relative humidity, and electrical charge effects, among others. It is important to bear in mind that there are often interactive effects among two or more such parameters. Thus, there may be seeming discrepancies between the results of nominally similar studies, and there is an ongoing need to consider many aspects of the conditions under which tests are carried out.

Interception

Flow field

Filtration models can aid in better understanding the physical principles governing the performance of fibrous filters. Mathematical expressions, both theoretical and empirical, can be used in an effort to predict filter performance as a function of structure of other attributes. The most helpful models are multi-fiber models involving a flow field of aerosol particles passing around a filter fiber, while also considering the effect of neighboring filter fibers in this flow field (Lee and Liu 1980). Much of the work on modeling these flow fields amongst an array of fibers is based on the Kuwabara-Happel flow field model (Happel 1959; Kuwabara 1959). Such models are relevant to cellulose-based materials, due to their generally fibrous nature. Factors affecting the likelihood of interception of an individual particle onto a filament of filter medium can be divided into the general categories of stochastic and deterministic mechanisms (van de Ven 1989). The stochastic mechanism, to be described below, involves random diffusion of the particle. Deterministic mechanisms include stream-line-based interception, momentum-based deviations of particle paths from streamlines, and effects due to electrical charges, among others.

Diffusional interception involving Brownian motion

When considering very small particles, especially those of about 200 nm diameter or less, one can expect that if they were to exactly follow the streamlines of flow of air through the filter media, then most of them would not directly impinge onto any surfaces. Because air cannot pass through solids objects, the streamlines all will pass around any filament of filter medium in their way. Depending on the typical size of pore spaces within a filter medium, there may be a low probability that a very small particle would get stuck in an opening too small to allow its passage. Rather, in such cases the main mechanism of impaction (possibly leading to retention by the filter) will be diffusion. In other words, Brownian motion can be expected to have a dominant effect on the collection of such particles (Alonso and Alguacil 2001; Gustafsson et al. 2018). The general concept is illustrated in Fig. 3. Due to thermal energy, which can express itself through collisions among air molecules and particles, the momentary paths followed by such particles will be chaotic. The chaotic motions associated with diffusion will be superimposed upon the predicted motions based on the streamlines. Each entity will have an average kinetic energy of 3/2 kT, representing motion in the three dimensions of space (Hirchfelder et al. 1954), where k is the Boltzmann constant and T is absolute temperature. The velocity of diffusion increases with decreasing particle size. Thus, the importance of Brownian motion in bringing about impacts of particles onto filaments of the filter media will become increasingly important with decreasing particle size. Note that these predictions generally assume that colloidal-sized particles, which are thus affected by Brownian motion, are being collected on much larger filter media (such as fibrous filter media) in the absence of moisture.

Fig. 3. General concept of a collection mechanism of very small particles that depends on their random (Brownian) movements due to their thermal energy, which is expressed by random collisions against gas molecules

Interception by the diffusion-based mechanism just described will generally increase with increasing surface area of the filter media. In the case of cellulose-based media, two aspects are critically important. Mechanical refining, hydrodynamic shearing, or micro-grinding are methods to increase the accessible surface area of cellulosic materials in the wet state (Lavoine et al. 2012; Gharehkhani et al. 2015). However, there can be a major loss in accessible surface area when cellulosic material is dried, as will be discussed later. Thus, different capture efficiencies can be expected for cellulosic filter media, in the capture of fiber small particles from air, depending on how the filter media has been prepared.

Streamline interception

Especially when the diameters of the filaments in the filter media are of similar size or smaller than those of the particles, there will be a considerable chance of direct interception, even without any need to consider effects of Brownian motion or any forces of attraction (van de Ven 1989). This type of interception is classed as deterministic, since the outcome can be predicted based on streamlines of flow. In cases where a laminar model of flow is justified, such paths can be modeled (Ayaz and Pedley 1999).

Inertial deviations from streamlines of flow

Interception of a particle via inertial impaction occurs when a particle’s inertia causes it to stray from the original gas streamline and meet with a fiber surface (Abdolghader et al. 2018). This mechanism is illustrated in Fig. 4.

Fig. 4. Depiction of a deviation of particle motion from a streamline of flue due to the inertia of that particle, thus leading to a collision with a collector surface

The inertial filtration mechanism depends largely on the mass of the particles. The greater the particle size, the greater the inertia and the greater the inertial deposition will be. Particles with larger face velocities and densities also exhibit higher inertia.

Electrical field-induced

Filtration efficiency can be significantly improved by introduction of electrostatic forces (Abdolghader et al. 2018). Electrical fields can be especially useful in improving the filtration efficiency of particles that are of the wrong size to efficiently be captured by other mechanisms. This range is typically about 0.15 to 0.5 µm (see Fig. 8, with the label “Mixed capture mechanism”). However, for very small particles (smaller than 20 nm), electrical fields may decrease filtration efficiency (Zhu et al. 2017; Givehchi et al. 2015). Electrical field-induced filtration efficiency is governed by factors such as particle charge density, fiber surface charge density, the chemistry of the particles and fibers, and the intensity of the applied electric field (Mostofi et al. 2010).

An electrical field can be implemented either by applying a charge to the particles or by applying a charge to the filter medium (Thakur et al. 2013). The latter is referred to as an electret system, and an example of such a system is shown in Fig. 5. Note that such charge attraction can render the collection process much less dependent on the streamlines of flow or inertial effects. Although cellulosic materials generally are not good conductors of electricity, that does not appear to be an impediment to their usage in electret systems (Li et al. 2020).

Fig. 5. Rudimentary illustration of an electret system that uses electrostatic attractive forces to enhance collection efficiency

In general, a Coulombic force is created when the charges on the particle and the fiber surface are of opposite sign. It has been stated that a polarization force is generated when the fiber surface is charged and the particle is neutral, while an image force is generated when the particle is charged and the fiber surface is neutral (Abdolghader et al. 2018). The word polarization is appropriate in cases where the charge distribution within a suitably small particle is capable of redistribution; in other words, sufficient electrical conductivity is a requirement. The term image force refers to a related redistribution of charge within a conductive filter surface that initially has a net neutral charge before encountering the charged particle (Muscat and Newns 1977). Such an image force has been shown to improve filtration efficiency, even for singly charged small particles (Alonso et al. 2007). The cited work predicts that the image force will become increasingly significant as particle size increases. In addition, the capture efficiency has been to shown to rise with increasing (opposite) electrical charge of the particles to be captured (Fjeld and Owens 1988).

Particle size is an important factor in the consideration of electrical field-induced filtration efficiency. Filtration efficiency under applied electrostatic forces has been shown to increase with increasing particle size (Thomas et al. 2013). The cited tests were carried out with steel or polymer fiber mesh filters having various diameters in the range of 25 to 200 μm. Larger particles have the ability to obtain more units of net charge (Hogan et al. 2009). Since the probability of a particle smaller than 20 nm taking on two or more net charges of either polarity is virtually zero, there are only three different conditions of charge that particles smaller than 20 nm can exhibit: neutral, singly positive, and singly negative (Hoppel and Frick 1986). Particle size is also important to consider when evaluating the charge adopted by a particle when passed through a bipolar charger. Particles smaller than 20 nm obtain different charges than larger particles when passed through the same bipolar charger and thus exhibit different filtration efficiencies. For these reasons, electrical fields applied to particles smaller than 20 nm might decrease filtration efficiency in some cases, while for larger particles, electrical field-induced filtration efficiency is reliably increased (Marlow and Brock 1975; Givehchi et al. 2015). The tests by Givehchi et al. (2015), which focused on effects due to capillary forces, involved stainless steel mesh screens with diameters of 25 μm and an air flow velocity of about 0.07 m/s.

Filtration by Size

Fiber size

Filtration of air through cellulosic fiber-based media depends partly on fiber size. Fiber diameter influences both the quality of filtration and the useful life of the filter. Fiber diameter impacts filtration characteristics such as particle penetration, most penetrating particle size (MPPS), pressure drop, and filter clogging (Chattopadhyay et al. 2016). The cited authors employed a commercial glass-fiber filter with a flow rate of 2.5 L/cm²⋅min, which corresponds to about 0.04 m/s. The general rule is that filtration efficiency is higher for fibers with smaller diameters. Thus, it has been found that filter media composed of finer fibers exhibit a higher filtration efficiency for nanoparticles (Abdolghader et al. 2018). As discussed in more detail later in this article, fibrous elements in cellulose-based filter media often can be described as either “fibers” or as “nanocellulose”. The diameter of a typical fiber is often in the range of 15 to 50 μm, whereas the diameter of typical nanocellulose products is often in the range of about 10 to 100 nm.

The term nanofiber has been used to describe fibers with diameters lower than 0.5 μm. Nanofibers decrease particle penetration and the MPPS of a filter, while increasing pressure drop (Kim et al. 2008; Chattopadhyay et al. 2016; Tang et al. 2017). High pressure drop associated with nanofibers can be attributed to nanofibers’ high surface area-to-volume ratio. This characteristic renders nanofibers a poor choice for use in homogenous filter media; however, adding nanofibers as a low-density layer on top of microfibers within a composite filter achieves the benefits granted by the nanofibers while mitigating the issue of high pressure drop (Podgorski et al. 2006; Kim et al. 2008). Figure 6 presents the concept of a bulky, high surface area layer with high fractional pore volume, such that there can be a high collection efficiency of very small (Brownian-dominated) particles, while not contributing a large barrier to flow.

Fig. 6. Concept of a layer within a filter device that is designed to have a high efficiency of collection of particles by a stochastic Brownian diffusion mechanism while minimizing the resistance to air flow through the layer

Pore sizes in fibrous mat

Filtration of air through cellulosic fiber-based media depends partly on the pore sizes in the fibrous mat. Manipulation of pore size is therefore an important tool to improve the filtration efficiency of a filter medium. Pore size and structure of the cellulosic fiber mat significantly influences filtration efficiency due to the diffusion, interception, and sieving mechanisms (Ma et al. 2018). Filters with smaller pore sizes generally exhibit higher filtration efficiencies, but they also have higher pressure drops due to the smaller pore sizes (Zikova et al. 2015; Soo et al. 2016).

Another kind of deterministic capture, which is applicable to the largest dimensions of airborne particles, can be called sieving or screening. Figure 7 illustrates how this deterministic mechanism can affect particle capture efficiencies, which are expected to be a function of pore sizes.

Fig. 7. Illustration a rudimentary screen-type capture, based on the sizes of particles relative to the size of passages between filaments in the filter medium

Combined effects of particle size

Based on the mechanisms already described, filtration efficiency of fibrous filters will depend in various ways upon particle size of the aerosol particles being filtered. The dominant mechanism of filtration changes depending on particle size. At the limit of very small sizes, the diffusional mechanism based on Brownian motion dominates. As particle size increases, the streamline interception and inertial mechanisms gradually become dominant. Therefore, a phenomenon exists within the intermediate particle size region in which more than one of these filtration mechanisms are operating but none is dominating. It is usually within this intermediate region that particle penetration is at a maximum and the filtration efficiency is at a minimum. Figure 8 illustrates this behavior (Lee and Liu 1980). These data were based on experiments carried out at relatively low air velocity with high-efficiency particulate air filters (HEPA).

The phenomenon of minimum filtration efficiency at a certain particle size is well established; however, the minimum filtration efficiency and the particle size at which it occurs are known to vary with the flow velocity and filter type. For fibrous filters at relatively low flow velocity, the minimum filtration efficiency generally occurs with particle diameter around 0.3 µm (Lee and Liu 1980). At higher filtration velocities, however, the most penetrating particle size may become significantly smaller in diameter than 0.3 µm (Liu and Lee 1976).

Fig. 8. Typical size-dependency of particle collection efficiency on filter media due to a transition from primarily stochastic (Brownian diffusion) capture to deterministic (direct impingement, momentum effects, etc.) at greater particle size. Figure redrawn based on original by Liu and Lee (1976)

Rebound or Retention

Viscoelastic properties

When a particle strikes a fiber surface within a fibrous filter under dry conditions, the initial kinetic energy of the particle is converted either to elastic deformation or the energy is lost as heat in the course of plastic deformation. In the event that all of the initial kinetic energy is consumed, the particle rests and adheres to the fiber surface. However, if the energy stored as elastic deformation exceeds the adhesion energy, then the particle will rebound from the fiber surface (Wang and Kasper 1991; Givehchi and Tan 2014). Real particles can be expected to have visco-elastic behavior, and depending on details of that behavior, different portions of the energy of impact will be irreversibly absorbed so that it no longer can contribute to the possible rebound.

A particle’s adhesion to a fiber surface is partly dependent upon the particle’s impact velocity. The relationship between elastic and viscous effects within a real material often can be influenced by the rate in which a process takes place. As a familiar example, very old panes of glass in windows have been predicted to be infinitesimally thicker at their bases (Gulbiten et al. 2018), which is consistent with a gradual process of viscous flow. But the same glass will shatter in response to a sudden impact. The fact that rebounding has been observed can be attributed to the fact that particle collection on a dry surface generally occurs during a very short time period.

When the impact velocity is less than the critical velocity, then the particles adhere to the fiber surface, whereas when the impact velocity is higher than the critical velocity, particles rebound from fiber surfaces. The particle has a greater probability of bouncing from a filter surface with increasing hardness of the contact bodies, with increasing particle size, and with increasing particle velocity (Hinds 1999). Figure 9 contrasts such bouncing with a situation in which the collector or particle is covered with a liquid, which can dissipate kinetic energy.

Fig. 9. Illustration of a viscous layer in determining whether an impinging particle will bounce from a collector surface or come to rest upon it

Particle shape vs. rebounding effects

With respect to particle shape, spherical particles that impinge on a collector surface may either slide or roll (Hubbe 1985; Barquins 1992). The area of contact for a spherical particle may remain nearly constant at any point in the particle’s course of movement along a smooth surface. This behavior contrasts with that of cubic particles, which will either slide or tumble. As illustrated in Fig. 10, when a cubic particle tumbles along a fiber surface, the area of contact between particle and fiber changes significantly as a function of time. One can expect that the most frequent collisions will involve contact with a corner, or perhaps an edge of the cubic particle, thus involving relatively low amounts of attractive energy. By contrast, once a cubic particle comes to rest, facewise, on a flat surface, one can expect a large force of attraction; thus one can expect a correspondingly large frictional force that would resist subsequent sliding along the surface. This phenomenon leads to a greater initial probability of particle bounce in the case of cubic particles. But once the cubic particle has come to rest, presumably with flat surfaces in close contact, then it can be expected to be highly resistant to a rolling motion.

Fig. 10. Illustration of the contrasting ways in which a cubic particle would be expected to interact with a collector surface during an initial collision brought about by flow of particle-laden air through the device

Boskovic et al. (2005) found that, for particles between 50 and 300 nm, cubic particles experienced a lower filtration efficiency than spherical particles of the same electrical mobility diameter. This phenomenon is explained by how the different shapes physically interact with fiber surfaces (Fig. 10). The tumbling that the cubic particles exhibit can significantly alter the area of interaction between fiber surface and particle, and by this means the probability of the particle detaching from the fiber surface (particle bounce) is predicted to be high. Therefore, when all other parameters affecting filtration efficiency remain constant, the particle kinetic energy can be attributed to the difference in filtration efficiency of particles with various shapes. The higher kinetic energy of a more massive particle is demonstrated to lead to the increase in the bounce probability of a particle (Dahneke 1971; Boskovic et al. 2005).

Short-range attractions

Short range attractions influence the rebound or retention of a particle on a fiber surface. In particular, the London dispersion component of van der Waals forces acting between solids in an air medium will contribute an attractive component of force, regardless of the types of material. The force-distance relationship predicted for van der Waals attraction is illustrated in Fig. 11.

Fig. 11. Van der Waals (London dispersion component) energy as a function of distance between two solid objects

Multiple theories have been developed to calculate the adhesion energy between a particle and a surface based on elastic impaction. The most widely recognized elastic adhesion energy models are the Bradley-Hamaker (BH), Johnson-Kendall-Roberts (JKR), and Derjaguin-Muller-Toporov (DMT) models (Hertz 1882; Bradley 1932; Derjaguin et al. 1975; Johnson et al. 1971).

The Hertz elastic adhesion energy model fails to consider these short range attractions and is restricted only to small amounts of linear elasticity and deformation (Hertz 1882). Therefore, the Hertz model significantly underestimates the contact radius between particles and fiber surfaces. The BH elastic adhesion energy model does take into account van der Waals forces between two contact bodies, but it fails to consider the adhesion force from the impaction (Bradley 1932). The JKR model involves a development of the Hertz model to consider the influence of adhesion energy and contact pressure within the contact area (Johnson et al. 1971). Because the BH model neglects to consider specific adhesion energy between contact bodies, which plays a significant role in nanoparticle adhesion, the BH model is not useful for calculating the adhesion efficiency of nanoparticles. The JKR model, which does take this specific adhesion energy into consideration, is useful for calculating the adhesion efficiency of nanoparticles (Givehchi and Tan 2015). The DMT model includes the effect of van der Waals forces between the contact bodies (Derjaguin et al. 1975). The major flaw in the DMT model is that it fails to consider deformations outside the contact area (Maugis 2000). The JKR model is most applicable for soft materials, large contact radii, compliant spheres, and high adhesion energy, while the DMT model is most applicable for hard materials, small contact radii, and low adhesion energies (Maugis 2000).

Capillary Forces

Because either humidity or liquid water is likely to be present in typical situations of air filtration, significant effects of capillary forces can be expected. The capillary force effect can influence filtration performance of particles. The previously discussed adhesion energy models did not take into consideration the impact of humidity on particle adhesion energy to filter media; in these models, air was assumed to be dry. Realistically, however, ambient air usually contains moisture, so air filtration often would occur under humid conditions. These issues are especially relevant to cellulose-based filter media due to the abundant hydrophilic –OH groups at their surfaces.

Humidity in the air causes a very small meniscus to be formed in the contact area between the particle and the filter media surface (Orr et al. 1975; Chen and Lin 2008; Chen and Soh 2008). This meniscus expands until the condensation rate and evaporation rate reach equilibrium with the ambient air (Pakarinen et al. 2005). As a consequence, there is a capillary force that increases the adhesion force between particle and filter surface (Ahmadi et al. 2007; Zhang and Ahmadi 2007). The radius of curvature of the meniscus at equilibrium was first predicted by Kelvin (see Mitropoulos 2008).

For hydrophilic materials, the capillary force can be calculated as a function of the surface tension of water, as shown in Eq. 1,

F_c = 4πγR_p(sin α sin (α+ θ) + cos θ) (1)

where F_c is the capillary force, γ represents the surface tension of water (0.0735 N/m at standard temperature and pressure conditions), R_p represents the particle radius, θ represents the wetting angle, and α represents the angle between the planes perpendicular to the meniscus and filter surface. Since θ and α are usually very small, this equation can be shortened to,

F_c = 4πγR_p(2)

This equation is the standard used for spherical particles larger than about 1 µm. The physical situation is illustrated in Fig. 12. The equation may not be applicable to nanoparticles, because the capillary force for nanoparticles is partly governed by relative humidity (Pakarinen et al. 2005). Another reason that this equation may not be applicable to nanoparticles is that the size of the nanoparticles influences the surface tension force, which lessens the capillary force for such tiny particles (Pakarinen et al. 2005).

Fig. 12. Geometries for calculation of the capillary force based on the interfacial tension and the perimeter of water-air interfaces, which contribute to holding solid items together

The capillary force between nanoparticles and filter surface depends on relative humidity, particle size, and surface tension. Equations not considering capillary effects are appropriate mainly for totally dry systems or for complete immersion in liquid. To include the effect of relative humidity on capillary force, an additional term β is incorporated, which represents the ratio of the capillary force calculated using the previous equation to the actual capillary force at a particular value of relative humidity (Pakarinen et al. 2005). The equation for capillary force for nanoparticles to include the capillary force effect is,

F_c = 2βπγd_p (3)

in which β is dependent upon size and can be determined using data presented by Pakarinen et al. (2005). For another size and relative humidity not demonstrated in their experimental data, the capillary force can be determined by extrapolation.

Estimate of maximum negative pressure

Equations 1 through 3 were derived under an assumption that the geometry of contact between a particle and a filter surface can be well represented by a sphere interacting with a planar surface. The pressure within such a meniscus can be obtained from the Young-Laplace equation,

(4)

where γ is the interfacial tension, θ is the contact angle of water with the surface (drawn through the water), R₁ is the smaller radius defining the meniscus, and R₂ is the larger radius defining the meniscus. The geometrical situation is sketched in Fig. 13. The greatest negative pressure can be expected when the water has a zero degree angle with the surfaces (perfect wetting); hence, the cosine term can be set equal to one. As either the meniscus continues to advance or evaporation of the water continues, the value of R₁ becomes much less than R₂, making it possible to simplify Eq. 4 as:

(5)

At the limit where the two adjacent surfaces have come very close together, the value of ΔP is predicted to become infinitely negative. Though the validity of using the equation may become questionable at that point, what is observed in practice is that the two adjacent surfaces tend to jump into molecular contact (Campbell 1959). This mechanism helps to explain why, during the process of papermaking, it is possible to achieve high levels of relative bonded area, with the formation of hydrogen bonds directly between the two surfaces (Campbell 1959; Page 1993). Another practical consequence of such forces is that flat plates of glass can become impossible to separate if they are placed in contact while droplets of water are present.

Fig. 13. Simplified view of a meniscus formed between a pair of flat surfaces what are envisioned as perfectly flat, featureless, and parallel, giving rise to a very strong negative pressure at the limit of close approach of the surfaces, based on the Young-Laplace equation

In theory, one might expect that capillary condensation would give rise to an additional component of adhesion, thus decreasing the tendency of particles to bounce away from a dry collector surface following an impingement. However, a finite time period (often measured in seconds) is generally required for the condensation to take place and for the attractive capillary force to develop (Bocquet et al. 1998). Since the elapsed time for rebounding of a particle will be a very small fraction of a second, there may be insufficient time for capillary condensation to have a significant effect on whether or not the particle rebounds.

Oil-coated fibers

Oil that is coated on fibers has the effect of minimizing the magnitude of particle motion along the fiber following initial particle collision with the fiber surface. This approach has been shown to be effective for air intake filters for vehicle engines (Maddineni et al. 2017, 2020). The level of particle motion along a filament of the filter medium will be suppressed following initial collision, which makes extensive sliding or bouncing less likely. Oil coating on fiber surfaces also raises the adhesion energy, the deformation, and the dissipative energy (Hinds 1999). The increased adhesion energy can be understood based on the simplified Young-Laplace equation (Eq. 5), where the oil is in this case playing the role of the fluid.

Boskovic et al. (2007) carried out an investigation in which a 3 mm-thick polypropylene medium was coated with mineral oil. The filter medium had a packing density of 0.184, and the fiber diameter averaged 12.9 µm with a standard deviation of 1.4. The particles used in this experiment were cubic MgO particles and spherical polystyrene latex particles with a diameter between 50 and 300 nm, based on electrophoretic mobility testing. Filtration efficiencies were found for two face velocities, 0.1 and 0.2 m/s. This experiment demonstrated no substantial difference in the filtration efficiencies between the spherical and cubic particles of the same electrical mobility diameter. The oil coating absorbed the particles’ kinetic energy, minimizing the particle motion along the fiber following collision, and thereby reducing the probability of particle bounce. These results show that oil coating on filter fiber surfaces minimizes the effect of particle shape on filtration efficiency.

Chemical affinities

A filter demonstrates varying chemical affinities for the particulate matter or aerosols being filtered, depending on the chemical composition of the materials composing the filter. Figure 14 illustrates the concept of inter-diffusion among polymer segments at a surface, which will depend on a high level of similarity between the materials. Such similarity can be quantified based on solubility principles and involves London dispersion forces, the polar component of interaction, and hydrogen bonding ability (Hansen 2007). In principle, relatively high levels of adhesion will result in cases where the mutual solubility is high enough to allow macromolecules inter-diffusion to take place at an interface. A limitation of this mechanism of adhesion is that it requires relatively high mobility of polymer segments, i.e. a softened or melted condition. In addition, the passage of time is required for such inter-diffusion to take place. In the case of cellulosic filter media, especially under moist conditions of collection, one can expect strong molecular associations to form, based on solubility principles, with such materials as starches and proteins, due to chemical affinities and shared hydrophilic properties.

Fig. 14. The concept of inter-diffusion of polymer segments when the materials of two objects coming into contact have high similarity of such factors as dispersion interactions and polarity

Another class of chemical-based affinity is associated with triboelectric charges. Filters for medical masks and respirators are typically composed of mats of nonwoven fibrous materials, such as polypropylene, wool felt, and fiberglass paper. In electrostatic filters, resins have been implemented along with natural wool fibers to help sustain an electrostatic charge (Institute of Medicine 2006; Das and Waychal 2016). The charged character of the dry wool, i.e. its triboelectricity, can be attributed to the amine groups present within the protein that makes up the fiber (Shin et al. 2017). It has been shown that filters of natural fiber or cotton fabric exhibit less capability of sustaining a static charge compared to polyester woven fabrics due to their higher proclivity to water absorption (Konda et al. 2020a,b,c). Also, the addition of protein to cellulose-based filter media has been shown to improve collection efficiency (Liu et al. 2017; Souzandeh et al. 2017; Sun et al. 2022). Recently, much research has been directed toward more specialized materials to optimize the balance between filtration efficiency and pressure drop. These emerging materials include polymer nanofibrous membranes, carbon nanotubes, porous metal-organic frameworks, nanowire networks, silk, inorganic oxide fibrous films, chitosan, and cellulose (Liu et al. 2020; Ma et al. 2018; Chattopadhyay et al. 2016).

Permeability Models

Governing equations

A series of models have been developed to enable estimation of pressure drop both in the initial usage of a clean filter and progressively as particles build up within (as plugging) or on (as a cake) the filter media (Tcharkhtchi et al. 2021). A model can be regarded as a simplified version of reality that nevertheless may be able to suggest relationships between controlled parameters and observed parameters. Most of the expressions that have been developed to represent filter pressure drop are based on the cell models presented by Kuwabara (1959), Happel (1959), or the semi-empirical Davies equation (1953). These theoretical approaches, however, are typically only applicable to clean filters or filters that have reached equilibrium. The earlier Davies (1953) equation, which can be used to determine the pressure drop for a clean (dry) filter is,

(6)

in which ΔP₀represents the pressure drop, u₀represents the gas velocity at the filter surface, µ_grepresents the gas viscosity, Z is the filter thickness, d_fis the fiber diameter, α is the filter solidity or packing density, and the expression inside the brackets is an empirical correction in consideration of non-perpendicular fibers.

Davies modified his original approach to pressure drop to yield adequate results for early filtration stages (1973). In this expression, Davies (1973) substituted the terms for fiber diameter, d_f, and fiber packing density, α, with terms representing wet fiber diameter (d_fwet) and wet fiber packing density (α_wet). This modification yields the expression,

(7)

and u₀represents the gas velocity at the filter surface, m_liqrepresents the mass of the collected liquid, Ω represents the filtration surface area, and ρ_lrepresents the liquid density. The shortcoming of Davies’s modified model is that it necessitates the perfect, uniform wetting of the fibers and the uniform distribution of liquid through the filter (Frising et al. 2005). Therefore, it is only applicable for the early stages of filtration. Thus, when considering cellulosic filters, it is likely that attributes of the filter media will tend to become less important with the passage of time during the filtration process.

Once a filter is no longer clean but has reached a “pseudo”-steady-state equilibrium, the expression developed by Liew and Conder (1985) can be used:

(8)

In Eq. 8, ΔP_s represents the “pseudo”-steady state pressure drop, ΔP₀ represents the pressure drop for a clean filter, Z represents the filter thickness, U₀ represents the filtration velocity, d_f is the average fiber diameter, σ_LV is the liquid surface tension, θ_C is the contact angle, and µ_l is the liquid viscosity.

As illustrated in Fig. 15, the buildup of particles within and on filter media has the potential to profoundly affect both the collection efficiency and the pressure drop evaluated at a set flow rate through the device. Particles that accumulate within the filter media will eventually plug up the channels of flow. Particles that accumulate on the surface of the filter device can form a cake.

Fig. 15. Two ways in which the accumulation of particles can be effected to increase the pressure drop as particle-laden air passes through a porous filter

To account for the different stages of filtration (how factors of filtration change with filtration time), theoretical models have been developed that divide the filter into layers rather than viewing the filter as a whole. The multi-layer model proposed by Frising et al. (2005) includes different expressions for each of the four filtration stages proposed by Contal et al. (2004). In the model presented by Frising et al. (2005), the expression for filter penetration is,

(9)

in which dZ represents the filter layer thickness. This model adds terms to the formula to account for the dynamic state of packing density as the filter progressively becomes clogged with more liquid. The pressure drop expressions for this multi-layer model by Frising et al. (2005) are derived from the ‘wet’ pressure drop equation presented by Davies (1973). The expression proposed by Frising et al. (2005) for the first stage of filtration is,

(10)

in which α_lrepresents the liquid packing density and d_fwetrepresents wet fiber diameter. The expressions for the subsequent filtration stages are the same as this expression but with factors added in consideration of the increase in velocity through the filter that happens as it clogs. These added factors also account for the change in packing density in the filter that happens as it clogs with liquid. A source of error in this model is the theoretical assumption of a “liquid tube” model of film flow, in which Frising et al. (2005) theorizes a “liquid tube” forming around a fiber at the start of filtration. Mullins and Kasper (2006) demonstrated that the “liquid tube” theory is unsupported; they found that a continuous liquid film as indicated by the “liquid tube” model cannot exist in the absence of droplets, because a film in contact with an individual filter fiber will be fragmented by Plateau-Rayleigh instability (Plateau 1873). Another shortcoming of this model is the inability to predict when the transition to the next filtration stage or “layer” in the model occurs.

Blockage & the trade-off between capture and permeability

Blockage is important to consider when evaluating filtration efficiency and permeability of the filtration material. Over time, filtration efficiency increases due to clogging of the fibrous filtration media, and permeability decreases. As particle loading increases, the filtration medium’s properties change and parameters influencing filtration efficiency and permeability become more complex with time (Hubbe et al. 2009; Mahdavi et al. 2015). Such an effect is illustrated in Fig. 16.