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Original paper Modeling the ﬁltration process with a ﬂat-type fabric ﬁlter NING MAO 1 , YOSHIO OTANI 1,∗ , YUPING YAO 1 and CHIKAO KANAOKA 2 1 Graduate

School of Natural Science and Technology, Kanazawa University, Kakuma-machi, Kanazawa 920-1192, Japan 2 Ishikawa National College of Technology, Kitachujo, Tsubata, Kahoku-gun, Ishikawa 929-0392, Japan Received 7 July 2005; accepted 8 August 2005 Abstract—The fabric ﬁltration process is divided into three stages, i.e. depth ﬁltration, transition ﬁltration and surface ﬁltration. A new model is proposed to describe the three stages under both virgin and regeneration conditions. The concept of an interface dust layer is introduced to predict the transition from the depth ﬁltration stage to the transition ﬁltration stage. The transition ﬁltration stage due to the non-uniform accumulation of residual dust on a ﬁlter is characterized by two parameters — the surface cleaning fraction, f , and the residual dust load on the uncleaned surface, Wu0 . It is found that f decreases and Wu0 increases with the number of ﬁltration cycles. This result suggests that both the fraction of uncleaned surface and the thickness of dust cake on the uncleaned surface increase with the number of ﬁltration cycles. The present model successfully describes the change in pressure drop during the ﬁltration cycles using the parameters. Keywords: Depth ﬁltration; transition ﬁltration; surface ﬁltration; fabric ﬁlter; pressure drop.

NOMENCLATURE

A, B, d C c0 df f K m N p ∗ To

coefﬁcients in (12) dust concentration (kg/m3 ) inlet dust concentration (kg/m3 ) ﬁber diameter (m) surface cleaning fraction (—) speciﬁc resistance of accumulated dust on ﬁlter (m/kg) dust retention in unit volume of ﬁlter (kg/m3 ) number of ﬁltration cycles (—) ﬁlter pressure drop (Pa)

whom correspondence should be addressed. E-mail: [email protected]

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pulse jet cleaning pressure (kPa) resistance of ﬁlter (l/m) ﬁltration time (s) ﬁltration velocity (m/s) dust load per unit surface area of ﬁlter (kg/m2 ) ﬁlter depth (m)

Greek α η λ μ ρp τ

ﬁlter packing density (—) single ﬁber collection efﬁciency (—) collection efﬁciency raising factor (—) gas viscosity (Pa s) particle density (kg/m3 ) duration of depth ﬁltration period (s) duration of transition and surface ﬁltration periods (s)

Subscript 0 c d i in m max r s u τ

initial cleaned surface depth ﬁltration stage number of ﬁltration cycle inside ﬁlter dust loaded condition maximum residual transition and surface ﬁltration stages uncleaned surface at the end of depth ﬁltration

1. INTRODUCTION

There is a growing tendency to employ bag ﬁlters in various industrial processes because of their high particle collection efﬁciency at a low pressure drop. Since bag ﬁlters are operated in cycles of dust accumulation and cleaning, it is necessary to accurately predict the ﬁltration performance during the cycles, especially the evolution of pressure drop under a wide range of operational conditions.

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Incomplete ﬁlter cleaning or patchy cleaning has a marked effect on the ﬁltration performance of fabric ﬁlters where the dust cake is detached unevenly from the ﬁlter surface during the periodical ﬁlter cleaning process. The causes of patchy cleaning are various. Dust removed from a ﬁlter during cleaning with pulse jets may redeposit on the ﬁlter when the ﬁltration is resumed [1, 2]. The variation in detachment force caused by pulse jets leads to non-uniform ﬁlter cleaning [3]. There are areas of the bag where the surface dust cake is completely removed and other areas where the dust cake is undisturbed. At the bottom of a bag, the inertia forces by pulse jets alone are not capable of removing the adhering dust layers and other mechanisms such as reverse air ﬂow play a role in dust removing [4, 5]. A number of efforts have been made to model the ﬁltration process with patchy cleaning. Although the overall fraction of cleaned areas on the ﬁlter surface may be measured or estimated, it is virtually impossible to predict the exact locations of the cleaned and uncleaned sites. Duo et al. proposed a probabilistic model based on the assumption that the cleaned sites are distributed randomly over the ﬁlter surface [6, 7]. Dittler et al. developed a two-dimensional quasi-stationary ﬂow model, in which the surface of completely or patchily cleaned ﬁlter medium was divided into small sections to predict the pressure drops as a function of regeneration efﬁciencies and regeneration patterns [8]. These models may reﬂect the actual cleaning pattern of the ﬁlter surface, but they seem too complicated for practical use. Kanaoka and Yao experimentally studied the ﬁltration phenomenon using a nonwoven fabric ﬁlter with pulse jet cleaning. They employed the type 1 ﬁlter testing rig speciﬁed in VDI/DIN 3926 to measure the development of the pressure drop during the ﬁltration cycles and showed that the evolution of pressure drop of a ﬂat-type fabric ﬁlter could be divided into three stages in a ﬁltration cycle, where the increasing rate of pressure drop gradually increases in the ﬁrst stage and then decreases in the second stage, and ﬁnally it is almost constant in the third stage, as shown in Fig. 1 [9]. Since most of the particle penetration from a bag ﬁlter occurs just after the ﬁlter cleaning in each ﬁltration cycle, the modeling of the ﬁrst and second stages in a ﬁltration cycle is especially important for preventing particle penetration through bag ﬁlters as well as for the prediction of pressure drop evolution. The objective of present work is to propose a new ﬁltration model, which can express the pressure drop development during the repeated ﬁltration cycles, taking into account the patchy cleaning.

2. MODEL DESCRIPTION OF THE FABRIC FILTRATION PROCESS

Kanaoka and Yao explained the mechanisms of the three stages in a ﬁltration cycle as follows: Stage 1 = depth ﬁltration; Stage 2 = the self-equalizing mechanism for parallel path ﬂows through unevenly distributed dust cake or the change in density of accumulated dust; Stage 3 = surface ﬁltration with uniformly

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Figure 1. Increasing pressure drop rate of ﬁlter A.

distributed dust cake [9]. The present model assumes that the second stage of the transition stage occurs due to uneven dust cake distribution on the ﬁlter surface after cleaning. Figure 2 illustrates the three stages for fabric ﬁlters under the virgin condition (the ﬁrst cycle) and the regeneration condition (the ith cycle). For a virgin ﬁlter, dust particles penetrate into ﬁlter and are collected inside the ﬁlter. As ﬁltration proceeds, the dust load in the ﬁlter reaches a maximum and dust cake starts to build up on the ﬁlter surface. When the pressure drop reaches a prescribed value, the ﬁlter is cleaned. After ﬁlter cleaning, a part of the residual dust cake remains on the ﬁlter surface. However, on a part of the ﬁlter surface, not only is the dust cake completely removed, but also part of the dust inside the ﬁlter is also cleaned. Therefore, depth ﬁltration takes place on the cleaned surface just after the ﬁlter cleaning until the dust load reaches the maximum retention in the ﬁlter. Since the ﬁltration velocity through the cleaned surface is much higher than that through the uncleaned surface, we assume that there is air ﬂow only through the cleaned surface. Once the dust begins to accumulate on the cleaned surface, which is the end of the depth ﬁltration stage, the second stage of transition ﬁltration stage begins. During the transition stage, the surface ﬁltration on the uncleaned surface also plays a role for dust collection. The local ﬁltration velocity varies over the ﬁlter surface because of the non-uniformity in the dust cake distribution. This leads to nonlinear pressure drop development of the ﬁlter. As time passes, the non-uniformity

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Figure 2. Model description of the ﬁrst cycle and the ith cycle.

will gradually disappear because of the self-equalizing mechanism for parallel path ﬂows. The difference in local ﬁltration velocity will then diminish and this will lead to uniform deposition of ﬁlter cake, which is the beginning of the surface ﬁltration stage. Moreover, the present model assumes that depth ﬁltration takes place without surface ﬁltration or, vice versa, surface ﬁltration occurs without penetration of particles into the ﬁlter.

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Figure 2. (Continued).

2.1. Stage 1: depth ﬁltration stage 2.1.1. Mathematical description of depth ﬁltration. Since ﬁbrous ﬁlters and fabric ﬁlters are similar in structure, one may apply the depth ﬁltration theory for ﬁbrous air ﬁlters to fabric ﬁlters. In fabric ﬁltration, depth ﬁltration takes place for virgin ﬁlters and also for the regenerated ﬁlter where the dust inside the ﬁlter is partially removed by ﬁlter cleaning. The depth ﬁltration stage of fabric ﬁlters is expressed by the following set of partial differential equations: 1 4 α ∂C ηm C, = ∂x π 1 − αm df 1 ∂m ∂C =− , u ∂t ∂x

−

(1) (2)

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where C, u, α and df are, respectively, the dust concentration, ﬁltration velocity, ﬁlter packing density and ﬁber diameter. The subscripts ‘m’ stands for the condition with dust load. αm and ηm are, respectively, the ﬁlter packing density and the single ﬁber collection efﬁciency under the dust-loaded condition. Substituting (2) into (1), the increasing rate of dust load is expressed by the following differential equation: 4 α 1 ∂m ηm Cu. = ∂t π 1 − αm df

(3)

The single-ﬁber collection efﬁciency with dust load can be expressed by the following equation [10]: ηm = η0 (1 + λm),

(4)

where λ is the collection efﬁciency raising factor and η0 is the single-ﬁber collection efﬁciency without dust load. The packing density of dust-loaded ﬁber is a function of the dust load, m, and the particle density, ρp , and is given by: m αm = α + . (5) ρp Substitution of (4) and (5) into (3) yields: 4Cuρp αη0 1 + λm ∂m = . ∂t π df ρp − ρp α − m

(6)

Equation (6) can be used to predict the dust load, m, at an arbitrary ﬁlter depth, x, and time, t. 2.1.2. Transition from the depth ﬁltration stage to the transition ﬁltration stage. In order to ﬁnd the transition from the depth ﬁltration stage to the transition ﬁltration stage in each ﬁltration cycle, we introduce a new concept of the interface dust layer on the ﬁlter surface. The interface dust layer is a hypothesized surface layer at x = 0. When the deposited dust fully occupies the voids of this layer or when the dust load reaches the maximum in this layer, the dust particles would not penetrate into the ﬁlter and start accumulating on the ﬁlter surface. Using the interface dust layer, we can predict the transition from depth ﬁltration to transition ﬁltration without solving (6) for the whole ﬁlter depth. Now, we apply (6) to the interface dust layer. We assume that the interface dust layer consists of the uncleaned surface and cleaned surface with the surface cleaning fraction, f , and that the ﬂow passes through only the cleaned surface at the ﬁltration velocity, u/fi−1 , during the ith ﬁltration cycle. Supposing that the depth ﬁltration ends at τi for the ith ﬁltration cycle as the dust accumulation increases from mi to the maximum, mmax , the integration of (6) gives: mmax τi ρp − ρp α − m 4c0 uρp αη0 dt, i = 1, 2, . . . , n, (7) dm = 1 + λm π df fi−1 mi 0

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where c0 is the dust concentration at the ﬁlter inlet. The solution for (7) is: ρp − ρp α π df 1 1 τi = fi−1 + 2 ln(1 + λmmax ) − mmax λ λ λ 4c0 uρp αη0 ρp − ρp α π df 1 1 − fi−1 . + 2 ln(1 + λmi ) − mi λ λ λ 4c0 uρp αη0 Since m1 = 0 and f0 = 1, we have the following equation for i = 1: ρp − ρp α π df 1 1 = τ1 . + 2 ln(1 + λmmax ) − mmax λ λ λ 4c0 uρp αη0 Substitution of (9) into (8) gives: ρp − ρp α π df 1 1 fi−1 . + 2 ln(1 + λmi ) − mi τi = τ1 fi−1 − λ λ λ 4c0 uρp αη0

(8)

(9)

(10)

One may assume that the dust retention per unit volume in the interface dust layer after ﬁlter cleaning, mi , is proportional to the total residual dust load, Wr , and that the coefﬁcient of proportionality is invariant for two consecutive ﬁltration cycles. Now, for i > 2, τi is expressed as: ρp − ρp α Wr,i Wr,i 1 1 − mi−1 + 2 ln 1 + λmi−1 τi = τ1 fi−1 − λ λ Wr,i−1 λ Wr,i−1 π df fi−1 . (11) × 4c0 uρp αη0 m2 can be determined from the experimental result of τ2 using (10). Wr,i is obtained experimentally by weighing the ﬁlter after cleaning. Since (11) is a recurrence formulae, τi for i > 2 is successively determined from mi−1 , Wr,i−1 and Wr,i . 2.1.3. Pressure drop development during the depth ﬁltration stage. To predict the pressure drop development during the depth ﬁltration stage, an empirical equation is used [11]: pd = (A + BW d )u,

(12)

where A, B and d are the empirical constants, and W is the dust load per unit ﬁlter area given by W = c0 ut for t τi . 2.2. Transition ﬁltration stage and surface ﬁltration stage 2.2.1. Stage 2: transition ﬁltration stage. The transition ﬁltration, which is an unstable ﬁltration process with uneven ﬁltration velocity distribution, is caused by a non-uniformity in the ﬁlter surface. When the non-uniformity is caused by such surface treatment as calendar, the transition ﬁltration process could occur for a

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virgin ﬁlter during the ﬁrst cycle. For the regenerated ﬁlter, the transition ﬁltration is caused by the non-uniform residual dust distribution after ﬁlter cleaning. In order to express the non-uniformity of residual dust cake distribution after ﬁlter cleaning in a simpler manner, the present model introduces two parameters, i.e. the surface cleaning fraction, f , which is the ratio of cleaned surface area to the total surface area of a ﬁlter, and the residual dust load on the uncleaned surface after cleaning, Wu0 . Here, ‘cleaned’ implies that the accumulated dust cake on the ﬁlter is completely removed and yet the dust inside the ﬁlter is also partially cleaned up. The residual dust distributes homogenously on the uncleaned surface. Both Wu0 and f change with the number of ﬁltration cycles. On the cleaned surface, depth ﬁltration takes place before the dust accumulation reaches the maximum and then surface ﬁltration until the cake resistance becomes the same as that through the uncleaned ﬁlter surface. 2.2.2. Stage 3: surface ﬁltration stage. The surface ﬁltration process is a stable ﬁltration process with a constant ﬁltration velocity all over the ﬁlter surface. The pressure drop increases linearly with the ﬁltration time. 2.2.3. Pressure drop development during the transition and surface ﬁltration stages. In order to predict the pressure drop development during the transition and surface ﬁltration stages, we need to express the resistances of the cleaned and the uncleaned surfaces, Rc and Ru , as a function of f , Wu0 and t. Since the cleaned and uncleaned surfaces give the resistance to air ﬂow in parallel, the following holds: ps = uc Rc μ = uu Ru μ = uRμ,

(13)

where R and u are the average resistance and the average ﬁltration velocity, respectively. u is related to the ﬁltration velocities through the cleaned and uncleaned surfaces, uc and uu , by: u = uc fi−1 + uu (1 − fi−1 ).

(14)

Combining (13) and (14): fi−1 1 − fi−1 1 + = . Rc Ru R

(15)

Rc and Ru vary with ﬁltration time, and are given by (16) and (17) respectively: t uc dt, (16) Rc = Rcτ,i + Kc0 τi t Ru = Ruτ,i + Kc0 uu dt, (17) τi

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Rcτ,i and Ruτ,i are the resistances of the cleaned and uncleaned surfaces at the start of transition ﬁltration in the ith ﬁltration cycle, K is the speciﬁc resistance, and c0 the inlet dust concentration. Differentiation of (16) and (17) with respect to t gives: dRc = Kc0 uc , dt dRu Ru = = Kc0 uu . dt Eliminating R in (13) by (15): Rc =

uc =

u uRu 1 = . fi−1 1 − fi−1 Rc (1 − fi−1 )Rc + fi−1 Ru + Rc Ru

(18) (19)

(20)

Substitution of (20) into (18) yields: Rc =

Kc0 uRu . (1 − fi−1 )Rc + fi−1 Ru

(21)

Now we want to eliminate Ru in (21) in order to ﬁnd the solution for Rc . From (14), we have: uu =

u fi−1 − uc . 1 − fi−1 1 − fi−1

(22)

Substituting (22) into (17) followed by the integration: Ru = Ruτ,i +

Kc0 u fi−1 (t − τi ) − (Rc − Rcτ,i ), 1 − fi−1 1 − fi−1

(23)

Ruτ,i is the sum of resistance due to the dust load on the uncleaned surface at the end of depth ﬁltration, Wuτ,i is equal to the residual dust load on the uncleaned surface, Wu0,i−1 , and the resistance of the cleaned surface at the end of the depth ﬁltration in the ith cycle (see Fig. 2). It is expressed as: Ruτ,i = Rcτ,i + KWu0,i−1 .

(24)

Substituting (24) to (23): Ru = Rcτ,i + KWu0,i−1 +

Kc0 u fi−1 (t − τi ) − (Rc − Rcτ,i ). 1 − fi−1 1 − fi−1

(25)

Substituting (25) into (21), the following equation is obtained: Kc0 u fi−1 (t − τi ) − (Rc − Rcτ,i ) Kc0 u Rcτ,i + KWu0,i−1 + 1 − fi−1 1 − fi−1 . Rc = Kc0 u fi−1 (1 − fi−1 )Rc + fi−1 Rcτ,i + KWu0,i−1 + (t − τi ) − (Rc − Rcτ,i ) 1 − fi−1 1 − fi−1

(26)

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Equation (26) is numerically integrated to obtain Rc with the initial condition: fi−1 pτ,i Rc t=τ = Rcτ,i = . μu

(27)

Finally, eliminating uc in (13) using (26) and (18), we have: Kc0 u fi−1 (t − τi ) − (Rc − Rcτ,i ) μuRc Rcτ,i + KWu0,i−1 + 1 − fi−1 1 − fi−1 . ps = Kc0 u fi−1 (1 − fi−1 )Rc + fi−1 Rcτ,i + KWu0,i−1 + (t − τi ) − (Rc − Rcτ,i ) 1 − fi−1 1 − fi−1

(28) Equation (28) gives ps in the ith cycle as a function of fi−1 , Wu0,i−1 and t. In order to obtain the surface cleaning fraction, fi−1 , and the residual dust load on the uncleaned surface after cleaning, Wu0,i−1 , two points on the experimental curve of ps at t (t = τi and t = τi +ψi ) are substituted into (28), where ψi is the duration of transition and surface ﬁltration periods for the ith cycle. At t = τi , (28) is reduced to (29) with the assumption of Ruτ,i Rcτ,i : pτ,i = μ

u fi−1

(29)

Rcτ,i .

At t = τi + ψi , (28) becomes: pmax = μu(Rcτ,i + K(1 − fi−1 )Wu0,i−1 + Kc0 uψi ).

(30)

If we assume that (29) holds for a short time period, t, at the beginning of transition ﬁltration, then we have: u u Rcτ,i + Kc0 t , (31) pτ +t,i = μ fi−1 fi−1 where the second term in the brackets is the increase in resistance due to the particles accumulated on the cleaned surface. Subtracting (29) from (31), and letting t → 0, we have: μKc0 u2 pτ +t,i − pτ,i dp = . (32) lim = 2 t→0 t dt τ,i fi−1 Since [ dp ] = [ dp ] , fi−1 can be calculated by: dt τ,i dt max,i fi−1 = μKc0 u

2

dp dt

1/2 . max,i

(33)

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Now we have fi−1 , Wu0,i−1 can be calculated by (30):

Wu0,i−1

pmax − fi−1 pτ,i − Kc0 uψi μu = . K(1 − fi−1 )

(34)

2.3. Filter cleaning Filter cleaning is carried out when the ﬁlter pressure reaches a prescribed value. The pressure drop after ﬁlter cleaning is referred to as the residual pressure drop, which is a function of the residual dust load on a ﬁlter. Although the residual dust load consists of particles both in a ﬁlter and on the ﬁlter surface due to patchy cleaning, we assume that the residual pressure drop is determined only by the residual dust in the ﬁlter since most of the air ﬂow passes through the cleaned surface of ﬁlter. Since the dust accumulated in a ﬁlter during the depth ﬁltration stage decreases exponentially with ﬁlter depth, we assumed that the residual dust in the ﬁlter exists only in a very thin layer of thickness, L, in the vicinity of the ﬁlter surface (see Fig. 3). As shown in Fig. 3, the air ﬂow passes through the thin layer of the cleaned surface at a velocity of u/fi and then spreads over the ﬁlter surface giving the air ﬂow velocity of u because of high ﬂow resistance of ﬁlter. Equation (12) of the pressure drop of the ﬁlter with dust load consists of two terms, i.e. the ﬁrst term of the ﬁlter media contribution and the second term of the accumulated particle contribution. Consequently, one may assume that the contribution of ﬁlter media may be neglected in the thin layer with densely packed particles, whereas there is no contribution of accumulated particles in the deeper layer of the ﬁlter. Therefore, the residual pressure drop in the ith cycle may be expressed by: d pr,i = Au + BWin,i u/fi ,

Figure 3. Calculation of residual pressure drop.

(35)

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where Win,i is the residual dust in the ﬁlter, which is the difference between the total residual dust and the residual dust on the ﬁlter surface: Win,i = Wr,i − Wu0,i (1 − fi ).

(36)

3. EXPERIMENTAL SETUP

The ﬁltration–cleaning cycle is performed repeatedly at a given ﬁltration velocity using the experimental setup shown in Fig. 4. Test dust of ﬂy ash fed by a screw dust feeder is dispersed by an ejector. The dust-laden gas is introduced into the top of the vertical duct and the dust in the gas is collected on the testing ﬁlter. The ﬁlter pressure is recorded by a pressure transducer (Valcom; VPRN-A4). When the ﬁlter pressure reaches a prescribed value (1000 Pa in the present work), ﬁlter cleaning is carried out by compressed air of 500 kPa-gauge through a solenoid valve connected

Figure 4. Schematic diagram of the experimental setup.

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Table 1. Test conditions Fabric ﬁlter material structure dimension Dust particle material median diameter density dust concentration Testing conditions ﬁltration velocity pulse jet pressure ﬁlter pressure before cleaning

(—) (—) (mm)

PPS felt square (300 × 300)

(—) (μm) (kg/m3 ) (g/m3 )

ﬂy ash (JIS Z8901 no. 10) 4.84 2300 5

(m/s) (kPa-gauge) (Pa)

0.033 500 1000

to a 2.5-l compressed air reservoir. The downstream dust concentration is measured by a photometer (laboratory made) as well as sampling the test particles with HEPA ﬁlters. Table 1 shows the testing conditions.

4. RESULTS AND DISCUSSIONS

4.1. Particle penetration after ﬁlter cleaning The relationship between dust penetration (or downstream dust concentration) and the pressure drop is shown in Fig. 5 for the initial nine ﬁltration cycles. Figure 5 shows that most of the particle penetration from bag ﬁlters occurs in the virgin ﬁlter and the regenerated ﬁlter immediately after the cleaning. The ﬁlter pressure increases non-linearly during this period. From the viewpoint of particle penetration, it is important to predict the pressure drop development in the initial stages of each ﬁltration cycle. 4.2. Surface cleaning fraction and residual dust load on the uncleaned surface after cleaning The experimental results of pressure drop development for the ﬁrst, ﬁfth and 30th ﬁltration cycles are shown in Fig. 6, and the increasing rates of pressure drop calculated from Fig. 6 are plotted in Fig. 7. It is seen from Fig. 7 that the peak of pressure drop increasing rate is higher for a larger number of ﬁltration cycle and that the transition stage starts at 47 and 29 s, respectively, for the ﬁfth and 30th ﬁltration cycle. The surface ﬁltration ends at 1005 and 821 s, respectively, for the ﬁfth and 30th ﬁltration cycle. fi and Wu0,i are calculated using (33) and (34), and are plotted in Figs 8 and 9, respectively. In Fig. 8, f1 is 0.56 for the ﬁrst ﬁltration cycle and gradually decreases with the number of ﬁltration cycles, indicating that the fraction of uncleaned surface

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Figure 5. Relationship between dust penetration and pressure drop.

Figure 6. Time dependency of ﬁlter pressure drop.

is about a half after the ﬁrst cycle and then increases with the ﬁltration cycles. In Fig. 9, Wu0,i increases with the number of ﬁltration cycles, suggesting that the layer of dust cake on the uncleaned surface becomes thicker with the number of ﬁltration cycles.

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Figure 7. Time dependency of increasing pressure drop rate.

Figure 8. Relationship between surface cleaning fraction and number of ﬁltration cycles.

4.3. Residual pressure drop The pressure drop increases with the dust load and the ﬁlter is cleaned when the pressure drop reaches 1000 Pa in the present work. Immediately after the ﬁlter cleaning, the ﬁlter was removed and weighed to obtain the total residual dust load of the ﬁlter. The relationship between the residual pressure drop and the total residual

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Figure 9. Relationship between residual dust load on the uncleaned surface and number of ﬁltration cycles.

Figure 10. Constitution of residual dust load.

dust load is plotted by solid diamond symbols in Fig. 10. In Fig. 10, the residual dust on ﬁlter surface, which is calculated from fi (see Fig. 8) and Wu0,i (see Fig. 9), is also plotted by hollow triangle symbols together with the residual dust in the ﬁlter, which is calculated as the difference between the measured total residual dust load and the calculated residual dust on ﬁlter surface. Figure 10 shows that there is a slight increase in the residual dust in the ﬁlter, but a signiﬁcant increase in the

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Figure 11. Relationship between residual pressure drop and number of ﬁltration cycles.

residual dust on the ﬁlter surface. This is probably because the residual dust in the ﬁlter greatly changes the sticking property of the ﬁlter to the accumulated dust cake on the ﬁlter. The residual pressure drop predicted by (35) is shown in Fig. 11 together with the experimental data. Although the predicted residual pressure drop is slightly higher than the experimental data, the predicted curve well expresses the tendency of the experimental data. 4.4. Filtration cycle time The ﬁltration cycle time for each cycle is the sum of the duration of the depth ﬁltration period, τi , and the duration of the transition and surface periods, ψi . Using (11) with fi and Wr,i given in Figs 8 and 10, respectively, τi is calculated and plotted in Fig. 12 for i > 2 and the ﬁltration cycle time is shown in Fig. 13. In Fig. 12, the duration of depth ﬁltration abruptly decreases from 191 to 79 s for the ﬁrst and second ﬁltration cycles, and then gradually decreases to 29 s for the 30th ﬁltration cycle. Similar to Fig. 12, Fig. 13 shows that the ﬁltration cycle time decreases rapidly for the ﬁrst two ﬁltration cycles from 1163 to 1088 s, and then declines gradually to 821 s for the 30th ﬁltration cycle. From Figs 12 and 13, it is clear that the initial rapid decrease in ﬁltration cycle time is mainly due to the sharp shortening of the depth ﬁltration period and the later gradual decrease in the ﬁltration cycle time is due to the reduction in the transition and surface ﬁltration periods.

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Figure 12. Duration of depth ﬁltration period.

Figure 13. Relationship between ﬁltration cycle time and number of ﬁltration cycles.

5. CONCLUSIONS

A new model is proposed for the prediction of the fabric ﬁltration process, dividing a ﬁltration cycle into three stages, i.e. the depth ﬁltration stage, the transition ﬁltration stage and the surface ﬁltration stage. The major conclusions obtained in the present work are as follows:

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(i) A new concept of a hypothetical interface dust layer is useful for predicting the transition from the depth ﬁltration stage to the transition ﬁltration stage. (ii) The transition ﬁltration is attributed to the non-uniform residual dust distribution due to patchy cleaning. The non-uniformity can be represented by two parameters — the surface cleaning fraction, f , and the residual dust load on the uncleaned surface after cleaning, Wu0 . (iii) The surface cleaning fraction, f , decreases with the number of ﬁltration cycles, indicating that the fraction of uncleaned surface is increasing with the number of ﬁltration cycles. (iv) The residual dust load on the uncleaned surface after cleaning, Wu0 , increases with the number of ﬁltration cycles, suggesting that the layer of dust cake on the uncleaned surface becomes thicker with the number of ﬁltration cycles. (v) The initial rapid decrease in the ﬁltration cycle time is due to shortening of the depth ﬁltration period, and the later gradual decrease is due to reduction in the transition and surface ﬁltration periods. (vi) The model not only successfully describes the time evolution of the ﬁlter pressure drop during ﬁltration cycles, but also quantitative evaluation of ﬁlter cleaning performance becomes possible.

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