Modeling of the Transitional Pressure Drop of Fibrous Filter Media Loaded with Oil-coated Particles

Received: January 1, 2019
Revised: April 29, 2019
Accepted: June 1, 2019

Download Citation: RIS|BibTeX|https://doi.org/10.4209/aaqr.2019.01.0002  

Cite this article:

Hsiao, T.C. and Chen, D.R. (2019). Modeling of the Transitional Pressure Drop of Fibrous Filter Media Loaded with Oil-coated Particles. Aerosol Air Qual. Res. 19: 1625-1635. https://doi.org/10.4209/aaqr.2019.01.0002

HIGHLIGHTS

• Pressure drop of fibrous filter media loaded with oil-coated particles was studied.
• A semi-empirical model was proposed to describe the filter pressure drop curve.
• The measured data set for glass-fiber and cellulose media was used in the modeling.
• The model is limited to the cases with coating oils of low surface tension.

Keywords: Fibrous filter; Aerosol filtration; Loading behavior; Oil-coated particle.

INTRODUCTION

Filtration of aerosol particles is a dynamic process (Bao et al., 2015; Choi et al., 2017; Sachinidou et al., 2017). The filter loading process is typically recorded as the filter pressure drop in the function of time (when the size distribution of sampled particles is known) or particle mass/volume collected by filter media of unit area. As particles are trapped in filter media or collected on the media surface, the particle collection efficiency of filter media often continually improves and the pressure drop across the media monotonically increases (Park et al., 2012). While the increase of particle collection efficiency of filtration media is a benefit, the increase in the filter pressure drop is undesirable. It is because the increase of filter pressure drop often results either in the reduction of filtration velocity or the increase in the load of air movers (consequently reducing the lifetime of air movers).

Various models have been proposed to estimate the time evolution of filter pressure drop when loaded with particles (Kanaoka and Hiragi, 1990; Endo et al., 1998; Thomas et al., 1999; Leung and Hung, 2008; Saleh and Tafreshi, 2014).

The modelling to predict the time evolution of filter pressure drop when continuously loaded with particles, especially in the depth and transitional filtration phases, remains a challenging task. To understand the transitional-loading behavior of filter media, the effect of various factors (e.g., filter medium, physical properties of particles, filtration face velocity, and relative humidity) have been extensively studied. A majority of previous studies have been focused on the loading of filters with only solid particles (Japuntich et al., 1994; Veerapaneni and Wiesner, 1997; Endo et al., 1998; Chen et al., 2001; Thomas et al., 2001; Song et al., 2006; Müller et al., 2010; Joubert et al., 2011; Saleh and Vahedi Tafreshi, 2015) and few on the filtration of liquid particles (Liew and Conder, 1985; Agranovski and Braddock, 1998a, b; Zhang et al., 2017). For liquid particle loading, the droplet migration on fibers in filter media is a very complex process and it is affected by a variety of factors, such as the operational flow condition and the physical property of loaded liquid (liquid viscosity and surface tension) (Chang et al., 2016).

In addition to pure solid and liquid particles, particles with a mixed phase (in the form of solid particles coated or mixed with liquids) often exist in the real world. These multi-phased particles are typically called “greasy oils.” Examples of such greasy particles or oil-coated particles are those emitted from internal combustion engines or generated in vehicle crankcases, and those produced during grinding and milling operations (with metal working fluid) (Hsiao and Chen, 2015; Wei et al., 2017). Due to their physical property difference, the pressure drop evolution curve of a fibrous filter media loaded with oil-coated particles is very different from that of a medium loaded with either pure solid or pure liquid particles. As a result, the existing pressure drop models for filter media loaded with pure solid or liquid particles cannot be directly applied to describe the filter pressure drop evolution when the filter media is loaded with oil-coated particles. One possible method to holistically modeling the loading behavior of a fibrous filter medium loaded with oil-coated particles is to integrate and modify the pressure drop models for filter media loaded with pure liquid or solid particles. The objective of this study is thus to develop a model for predicting the pressure drop evolution of fibrous filter media when loaded with oil-coated particles.

PRESSURE DROP OF AN AEROSOL FILTER

Prior to the presentation of our modeling work on the pressure drop of filter media loaded with oil-coated particles, the pressure drop models of clean filter media and loaded filters are briefly reviewed.

Clean Filter

The basic concept to estimate the clean filter pressure drop is on the force balance. It is assumed that the overall pressure drop results from the summation of drag forces acting on all the fibers in filter media of a unit area. At a low Reynolds number (i.e., ρUd_f/µ << 1), the drag force on a single fiber can be approximated by the product of the flow velocity, viscosity, and drag coefficient (f). The pressure drop of a clean filter, ΔP₀, can then be formulated as:

where F_D is the fiber drag force per unit of fiber length; L_f is the total fiber length per unit area of filter media (which can be derived from the packing density, α_f, and thickness of the filter, Z, when a single fiber diameter, d_f, is given).

The accuracy of the above filter pressure drop modeling is highly dependent on the f. Four general expressions of f for estimating the pressure drop across a clean filter are summarized in Table 1.

The expressions for f proposed by Happel (1959), Kuwabara (1959), and Fuchs and Stechkina (1963) were all derived theoretically by solving the flow field in a system of parallel and circular cylinders at a low Reynolds number with different sets of boundary conditions. Happel assumed that the normal velocity and tangential stress at the cylindrical surface are 0, whereas Kuwabara assumed that the normal velocity and vorticity are 0. However, experimental results generally indicate that the accuracy of these expressions in predicting filter pressure drops is less favorable than the empirical expression proposed by Davies (1953). The discrepancy between theoretical prediction and experimental measurements is possibly due to the random orientation of fiber cylinders and the flow interference among them in experiments (Yamada et al., 2011; Zhou et al., 2017). When Davies’ f expression is chosen and combined with Eq. (2), the pressure drop of a clean filter can be expressed as:

The above equations are semi-empirical and do not take into consideration the fiber size variation in filter matrix. A thorough survey by Jackson and James (1986) indicated that the prediction begins to deviate from the experimental measurements when the packing density is less than 0.001.

Filter Loaded with Solid Particles

Kanaoka and Hiragi (1990) developed an early theoretical model of the pressure drop across a dust-loaded filter, ΔP_l. They also employed the concept of summing the total drag forces acting on the dust-loaded fibers, F_l, which was evaluated in the manner of Newton’s resistance law. The diameter of a dust-loaded fiber (d_fm) was chosen as a representative parameter of dendritic structure.

The drag coefficient of dust-loaded fibers per unit filtration area (C_dm) and the diameter of a dust-loaded fiber (d_fm) are two critical parameters in this model. They were correlated with the filtration condition and collection mechanism along with the dimensionless accumulated particle volume, V_ac, defined as the ratio of loaded particle mass to particle density and filter packing density per unit filter volume. Kanaoka and Hiragi (1990) further classified the rate of increase of d_fm into three stages: no growth at low V_ac, rapid growth at intermediate V_ac, and dampened growth at high V_ac. This model has been claimed as applicable to predict the pressure drop of a dust-loaded filter under any filtration conditions. However, the value of d_fm and C_dm are difficult to be estimated theoretically. They were given by empirical fitting with the accumulated volume of captured particles and cannot be determined without performing experiments.

Instead of evaluating the drag forces of particle dendrites formed by deposited particles, Bergman et al. (1978) considered the particle dendrites as newly formed fibers and modified the Davies equation to include an additional pressure drop due to these newly formed fibers. In addition, to correct the interference between the dendrites and filter fibers, they increased the fiber and dendrite volume fraction by the factors (L_f + L_p)/L_f and (L_f + L_p)/L_p respectively:

where L_p can be evaluated as follows:

By combining with Eq. (6), the expression for the pressure drop across a loaded filter can be derived:

The general criticism of the model of Bergman et al. (1978) is about the assumption of a uniform distribution of deposited particles on the filter matrix. Hence, Thomas et al. (1999, 2001) proposed to divide the entire filter into multiple layers and to evaluate the layers’ collection efficiency and pressure drops at every time step based on the information obtained from prior layers. Although such a method is ultimately closer to realistic conditions, the calculation is cumbersome and the result can depend on the number of layers, which is arbitrarily determined.

Filter Loaded with Liquid Particles

For a fibrous filter loaded with liquid particles, only a few models have been established. These models are primarily applicable to filters under steady-state saturation conditions. Liew and Conder (1985) performed various tests on filters with mean fiber diameters of 4, 8, 12, and 22 µm and developed an empirical equation to predict the filter pressure drop at the final steady-state stage, ΔP_s:

where γ_LV is the liquid surface tension and θ is the contact angle between a deposited droplet and a fiber.

Raynor and Leith (2000) provided another empirical expression for ΔP_s, which is correlated with the steady-state saturation ratio, S_e, and the steady-state packing density, α_e:

They also constructed an empirical expression for S_e against the dimensionless numbers (Ca, Bo, Dr) using the commercial statistical software SAA/STAT:

where Bo is the bond number (ρgd_f²/γ_LV × 10⁵), Ca is the capillary number (µU⁄γ_LV × 10⁵), and Dr is the nondimensional drainage rate.

Both models are purely empirical and only applicable for certain operational flow ranges. For example, the model of Liew and Conder (1985) requires the filter packing density to be larger than 0.02, whereas that of Raynor and Leith (2000) is only feasible for a filter thickness of less than 0.88 cm. Moreover, these two models only predict the steady-state pressure drop of a liquid-loaded filter.

Frising et al. (2005) attempted to establish a pressure drop model for different filtration and loading stages based on the Davies equation. To imitate the gradual clogging of a filter, they characterized the entire loading process into four stages and divided the filter media into n_p layers with thickness dZ (i.e., Z/n_p). In the first stage, the loaded liquid particles are assumed to perfectly coat the filter fibers and form liquid films on or around individual fibers. Therefore, the fiber diameter, d_f, and packing density, α_f, are replaced with the “coated” fiber diameter, d_f,w, and new packing density (α_f + α_l), which includes the loaded droplets.

The second stage is defined as the formation of the liquid bridge and liquid film at the fiber intersection, and the diameter of a “coated” fiber remains constant in this stage. Because the air flow is greatly influenced by the presence of liquid bridge and films, the air velocity must be modified. The pressure drop equation for the second stage is given as:

where α_tube is the maximal packing density resulted by the liquid coating on the filter fibers. However, α_tube cannot be determined either theoretically or experimentally. In the work of Frising et al. (2005), the above value was determined empirically to optimize the prediction of the proposed model. In the third stage, the liquid packing density reaches the maximum, and the liquid migration between filter layers begins. The pressure drop is considered constant at this stage. Once the liquid begins to drain out of the filter, the loading process enters the fourth stage, in which:

where α_film is the maximal liquid packing density and can be estimated by weighing the test filter before and after experiments. Frising et al. (2005) reported reasonable agreement between their experimental results and the predictions of their model and stated that the model requires only two parameters—α_tube and α_film. However, Mullins and Kasper (2006) argued that the assumption of perfect liquid coating of filter fibers does not always hold. For liquids with high surface tension, deposited droplets create liquid beads, rather than forming a film, on the fibers in filter media (Brown, 1993).

MODELING THE PRESSURE DROP OF FILTERS LOADED WITH OIL-COATED PARTICLES

In our previous experimental study (Hsiao and Chen, 2015), the co-solvent method was applied for generating oil-coated particles. A master solution of oil-coated particles was prepared by mixing two parent solutions, coating oil dissolved in 2-propanol and potassium chloride (KCl) dissolved in DI water, using a volume ratio of 1:1. Four coating oils (DEHS, light mineral oil, castor oil, and glycerol) were selected and tested. Accordingly, the loading curve of a filter loaded with oil-coated particles changes from that of a filter loaded with pure solid particles to that of a filter loaded with pure liquid droplets as the thickness of the liquid coating increases. When the liquid volumetric percentage is less than 50%, the pressure drops of glass-fiber filters and cellulose filters are primarily caused by the solid fraction of the loaded oil-coated particles. Moreover, the loading curve for a low-surface-tension liquid becomes independent of the liquid’s viscosity because the total loaded volume is multiplied by the solid volumetric percentage. In other words, it was found that all the curves are more closely positioned for both the glass fiber and the cellulose filter media when the pressure drop evolution curves were re-plotted using the solid core particle volume per unit filter area as the abscissa (Hsiao and Chen, 2015). Thus, the model for a filter loaded with oil-coated particles having a liquid volumetric percentage of less than 50% can be established based on the modified Bergman method. Because the impaction is the major filtration mechanism in filter media used in our previous testing, the front layer was assumed to collect all particles, and the rear layer was assumed to remain as a clean filter. Thus, the total pressure drop, ΔP, of a filter medium is then the linear sum of the pressure drops across these two layers, and Eqs. (3) and (8) were used to estimate the values.

The thickness of the front layer (Z_fL) is the critical parameter for the model’s predictions, and was estimated from scanning electron microscopy (SEM) images of the loaded filters. The depths of the front layers used for the glass-fiber filter and cellulose filter were 190 µm and 55 µm, respectively. The other filter characteristics required for the model are listed in Table 2, and α_p was calculated based on the loaded particle volume.

As shown in Fig. 1, the curves predicted by the modified Bergman model demonstrated reasonable agreement with the experimental results for both the glass-fiber filter and cellulose filter up to the pressure drop ratio (defined as the ratio of the loaded filter pressure drop to the clean filter pressure drop) of 4. By recalculating the loaded particle volume, the model can be extended to predict the loading behavior for oil-coated particles, which have a coating with a maximum of 50% liquid by volume.

Fig. 1. Experimental data and model prediction for (a) the glass-fiber filter loaded with 2.5-μm solid KCl particles and (b) the cellulose filter loaded with 2.5-μm solid KCl particles.

For the loading case with the liquid volumetric percentage higher than 50%, the transitional behavior is much more dynamic and strongly dependent on the liquid’s properties (both liquid viscosity and surface tension) as well as the filter medium’s characteristics (i.e., either absorptive or non-absorptive). No theoretical/empirical model existed to describe the transitional behavior in the cases. We thus proposed the power-law equation with two parameters, i.e., the exponent, n, and critical volume, V_cr, to fit the experimental data. In the proposed equation, n controls the curve growing slope, and V_cr controls the curve horizontal scale:

where ΔP₀ is the initial pressure drop of filter, and V is loaded particle volume per unit filtration area.

A similar approach was employed by Hermans and Bredée (1936) and Gonsalves (1950) in hydrosol filtration and was later applied to aerosol filtration by Emi et al. (1982). Although a satisfactory result was reported by Emi et al. (1982). for collection efficiency data, its correlation with pressure drop was not evident. In the present study, however, the loading curves for different oil-coated particles are well fitted by Eq. (17) for at least up to 4 times the initial filter pressure drop, shown in Figs. 2(a) and 2(b).

Fig. 2. Experimental data and fitting results for (a) the glass-fiber filter loaded with 2.5-μm 50% DEHS-coated particles and (b) the glass-fiber filter loaded with 2.5-μm 100% DEHS particles.

To establish an empirical model for calculating the filter pressure drop when loaded with oil-coated particles, the data reported in Hsiao and Chen (2015) were used. The detailed information about different oil-coated particles and the corresponding loading behaviors can be found there. In this study, the data for loading with glycerol-coated particles were not included in this fitting. It is because the surface tension and viscosity of glycerol are very different from those of other coating liquids. Moreover, the effects of surface tension and viscosity on filter pressure drop were difficult to qualitatively differentiate based on the data previously collected. Therefore, the parameters, V_cr and n, were only correlated to the viscosity of coating liquids in this work.

To illustrate the fitting result, both V_cr and n parameters are plotted as the fraction of solid core in the overall particle diameter (X), instead of the volumetric percentage of coating liquid. The solid-core diameter fractions in oil-coated particles corresponding to 0%, 20%, 50%, 88%, and 100% of liquid-volumetric percentage were 1.0, 0.93, 0.79, 0.49, and 0. As shown in Figs. 3 and 4 (for glass-fiber and cellulose filter media, respectively), the values of V_cr and n became invariant when the solid core diameter fraction was greater than 0.79 (for coating liquids with similar surface tension). The effect of liquid viscosity on the fitted V_cr and n parameters was negligible. However, below the above critical diameter fraction, the V_cr and/or n values started to vary. The observed variation was then correlated with the viscosity of coating liquids.

Fig. 3. Critical volume (V_cr) of different coating liquids vs. the core diameter fraction for the glass-fiber filter.

Fig. 4. Exponent (n) of different coating liquids vs. the core diameter fraction for the glass-fiber filter.

For the glass-fiber filter, as the solid core diameter fraction approached 0, the value of V_cr for different coating liquids approached to the same value (Fig. 3). It implied that the surface tension effect dominated the viscosity effect when loading pure liquid droplets (mist) on a glass-fiber filter medium. Differently, the variation in the power n for different coating liquids was minor for glass-fiber filter media (Fig. 4). The parameter of n was approximately estimated by a polynomial of the third order:

where X is the diameter fraction of solid core in oil-coated particles.

For cellulose filter media, the variations of V_cr and n values as a function of solid-core diameter fraction were rather complex (Figs. 5 and 6). It is because not only could coating liquid flow over the fiber surfaces of cellulose media, but it could also be absorbed by fibers (Hsiao and Chen, 2015). Moreover, the viscosity of coating liquid completely affected the relative position of the loading curves in a range of solid core diameter fractions. To include the viscosity effect in the filter pressure drop model, Table Curve 3D™ was used to establish the relationship among V_cr, n, liquid viscosity, and solid-core diameter fraction of test particles in the cases with cellulose media. Note that the same procedure was also applied to the parameter V_cr in the cases with glass-fiber filter media.

Fig. 5. Critical volume (V_cr) of different coating liquids vs. the core diameter fraction for the cellulose filter.

Fig. 6. Exponent (n) of different coating liquids vs. the core diameter fraction for the cellulose filter.

In the analysis, we normalized the values of V_cr for different liquids when filter media was loaded with pure liquid particles, and normalized the viscosity of coating liquids, µ_liq, by the water viscosity, µ_w. As shown in Figs. 7–9, all three sets of data can be fitted by a polynomial equation, Eq. (19). The parameters included in Eq. (19) are listed in Table 3. In general, the values of V_cr and n can be obtained from Eqs. (18) and (19) for glass-fiber and cellulose filter media loaded with particles coated with oil liquids.

Fig. 7. Fitting result for V_cr with the glass-fiber filter.

Fig. 8. Fitting result for V_cr with the cellulose filter.

Fig. 9. Fitting result for n with the cellulose filter.

By Eqs. (17)–(19), the pressure drop evolution curve of glass-fiber and cellulose filter media can be calculated when loaded with oil-coated particles having more than 50% percentage in liquid (i.e., if the solid -core diameter fraction is less than 0.79):

where A, B, C, D, E, F, G, H, I, and J are fitting constants. 

CONCLUSIONS

The oil-coated particles (i.e., greasy particles) are generally found in machining factories and co-emitted by fuel combustion sources and required to be filtered. When filter media is loaded with such particles, the buildup rate of the filter pressure drop decreases, and the transitional point of the loading pressure drop curve shifts to the heavily loaded mass regime as the oil volumetric percentage of the coated particles rises. In this work, we developed an empirical model to describe the pressure drop curves (specifically, the filter pressure drop as a function of loaded particle volume per filter media unit area) for filter media loaded with particles coated with oil layers of various thicknesses.

To fit our experimental observations of distinct loading curve characteristics above and below a liquid volumetric percentage of 50%, the model consisted of two parts. When the particles possessed a liquid volumetric percentage of less than 50%, the Bergman model for solid-particle loading was modified to predict a filter pressure drop that increased up to 4 times of the initial pressure drop. In this part of the modeling, a loaded filter was assumed to have two layers: one layer to collect all of the oil-coated particles and the other to remain as a clean medium. The thickness of the first layer can be measured from SEM images. The total pressure drop of the loaded filter media was then assumed to be the summation of the layer pressure drops.

When the particles possessed a liquid volumetric percentage of greater than 50%, a power-law equation with two parameters, viz., the exponent (n) and the critical volume (V_cr), was introduced to fit the experimental data. The correlations of the above parameters with the solid-core diameter fraction of particles and the viscosity of the coating oil were established. The overall pressure drop of filter media loaded with oil-coated particles can thus be estimated. However, because the effects of surface tension were not included in the experimental data on which this model is based, the proposed equations (Eqs. (17)–(19)) are applicable only when the coating oil has low surface tension (< 35 mN m⁻¹).

NOMENCLATURE

ρ                Fluid density
U                Fluid velocity
d_f               Fiber diameter
d_fm          Diameter of a dust-loaded fiber
d_f,w           Coated fiber diameter
µ               Fluid viscosity
µ_liq          Viscosity of coating liquids
µ_w             Water viscosity
γ_LV          Liquid surface tension
f                Drag coefficient
F_D             Fiber drag force per unit of fiber length
F_l            Total drag forces acting on the dust-loaded fibers
L_f              Total fiber length per unit area of filter media
L_p           Total fiber length of newly formed dendrite per unit area of filter media
Z            Thickness of the filter
Z_fL            Thickness of the front (loaded) layer
α_f              Filter packing density
α_p             Packing density due to newly formed particle dendrites
α_e             Packing density at steady-state saturation condition (Raynor and Leith, 2000)
α_l              Packing density due to perfectly coated liquid on filter fibers
α_tube        Maximal value due to the liquid coating in the filter
α_film        Maximal liquid packing density
ΔP          Total pressure drop
ΔP₀           Pressure drop of a clean filter
ΔP_l          Pressure drop across a dust-loaded filter
ΔP₀ʹ          Pressure drop of a clean filter layer
ΔP_lʹ         Pressure drop across a dust-loaded filter layer
ΔP_s         Filter pressure drop at the final steady-state stage
C_dm        Drag coefficient of dust-loaded fibers per unit filtration area
V_ac           Dimensionless accumulated particle volume
V_cr           Critical volume
V_cr,liq       V_cr for filter media loaded with pure liquid
θ            Contact angle between a deposited droplet and a fiber
S_e            Steady-state saturation ratio
Ca            Capillary number (dimensionless)
Bo            Bond number (dimensionless)
Dr          Drainage rate (dimensionless)
X              Diameter fraction of solid core in oil-coated particles

ACKNOWLEDGEMENT

This research was financially supported by National Taiwan University from Excellence Research Program - Core Consortiums (NTUCCP-107L891310) within the framework of the Higher Education Sprout Project by the Ministry of Education (MOE) in Taiwan and partially supported by the Center for Filtration Research at the University of Minnesota. This manuscript was edited by Wallace Academic Editing.