- Email: [email protected]
Prediction of the collection efficiency, the porosity, and the pressure drop across filter cakes in particulate air filtration
ARTICLE IN PRESS
Atmospheric Environment 39 (2005) 51–57 www.elsevier.com/locate/atmosenv
Prediction of the collection efficiency, the porosity, and the pressure drop across filter cakes in particulate air filtration Awni Y. Al-Otoom Chemical Engineering Department, University of Mutah, P.O.Box 78, Mutah—Karak 61710, Jordan Received 24 February 2004; accepted 15 September 2004
Abstract This study presents a new statistical model to predict the collection efficiency, cake thickness, cake porosity, and pressure drop across filter cakes during the particulate filtration of gases. This model is based on generation of a random distribution of particle sizes and particle falling locations. The model predicts the cake collection efficiency, which was found to be strongly dependent on the ratio of the mean particle size to the mean pore size of the filter medium. The average cake porosity decreases with increasing cake thickness and the pressure drop increases when the mean particle diameter decreases. r 2004 Elsevier Ltd. All rights reserved. Keywords: Filter cake; Model; Porosity; Pressure drop
1. Introduction Since the industrial revolution, the total amount of emissions has been increasing significantly. This led many governments around the world to introduce new legislations to control such emissions. The diversity of these emissions led researchers to look for practical solutions and to search for different methods for the removal of unwanted materials. Particulate emissions are one of the main types of undesirable industrial wastes. Nearly all industries have particulate emissions of some kind, which are generally removed by filtration, including cake filtration, vacuum filters, centrifugal filters, and many others. In cake filtration, the solid particles are removed from a slurry stream by exposing them to a filtration medium. A cake of the solid material is formed and its thickness increases with the time of filtration. The science of cake filtration Tel.: +962 796780123; fax: +962 65155058.
E-mail address: [email protected] (A.Y. Al-Otoom).
has been practiced for more than 70 years and many investigations have been published. (A comprehensive review is available in Shirato et al., 1987.) During cake filtration, some particles do not participate in the formation of the filter cake. They either escape from the filtration media, causing a decrease in the efficiency of the filter, or clog the septum of the filter media. Clogging of the filter media has been studied by many investigators including Notebaert et al. (1975); Granger et al. (1985); Lee (1997), and many others. Clogging, in addition to the problem of increasing the pressure drop across the filter media, causes errors in the prediction of the real cake thickness and the porosity of the filter cake. The wide distribution of both particle and filter pore sizes makes the prediction of particle escape from the filter cake difficult. Little research has been done to account for the wide range of particle distribution in the prediction of particle escape from the filter cake. The strength of the cake formed during filtration is relatively low. The cake can be easily disintegrated
1352-2310/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.atmosenv.2004.09.057
ARTICLE IN PRESS 52
A.Y. Al-Otoom / Atmospheric Environment 39 (2005) 51–57
during handling or during attempts to measure physical parameters, such as average porosity. Instead, experimental data of the pressure drop across the filter cake are adopted to calculate such parameters. Some investigators like Aguiar and Coury (1996) used cake fixation resins to obtain a strong filter cake for further analysis. The process is difficult to perform and requires extremely careful handling. Theoretical approaches are generally considered the key to predicting the cake filtration performance. The diversity and complexity of filtration media makes the evaluation of the performance of such media extremely difficult to perform experimentally. The wide range of particle size and the variability of the operating conditions of pressure and temperature add more difficulties in performing such tasks. Most of the theoretical studies present in the literature follow the Darcian-type analogy to predict the filtration progress. A recent 1-D model was developed by Chan et al. (2003) based on Darcy’s law to describe the filtration in centrifuges and to extract the filter cake parameters. A similar study was also developed earlier (SØrensen et al., 1997) which focused on the blinding of filter cake and filter media during filtration utilizing Darcian flow equations. In this work, a new approach will be utilized. A statistical model based on a random distribution of particle sizes will be presented. This model predicts the ratio of the particles that escape from the filter cake (Either clogging the filter or escaping from the filter media). In addition, this model predicts the cake thickness with filtration time, and calculates the pressure drop across this cake by predicting the porosity changes with the time of filtration.
2. The model 2.1. Model description
5. Particles have uniform spherical shapes. 6. Particles are incompressible. 7. Particle size distributions, and the particle falling location are randomly distributed. 2.3. Assigning pore location The pores centers are determined according to the total filter porosity (Ef), the length (L), width (W), and height (H) of the filter media. The distance between two successive pores in either X, Y direction is determined by these equations No. of pores in X-direction x¼
L ; S
where S is the distance between two successive pores, and will be presented in Eq. (5). No. of pores in Y-direction. y¼
W : S
2.2. Model assumptions 1. Filter media has a uniform surface porosity with an average pore diameter, pd and a constant distance between all successive pores. 2. Pores in filter media are cylindrical. 3. Particles that enter pores escape from the filter cake. 4. Particles that come into contact with other particles adhere to the surface with these particles.
(2)
Total no. of pores xy ¼
LW ; S2
(3)
but total no. of pores: Total pore volume LWHE f LW E f ¼ ¼ : Pore volume of one pore ðp=4Þp2d H ðp=4Þp2d
(4)
Equating Eqs. (3) and (4), results in the distance between two successive pore centers sffiffiffiffiffiffiffiffiffiffiffiffi p=4p2d S¼ : (5) Ef The X and Y pore coordinates are determines by X-coordinate of the pore Plxðu; pÞ ¼ Plxðu 1; pÞ þ S;
The model is designed to provide statistical analysis based on random function for formation of a wide range of particle size distributions and also uses a random function to anticipate the location of the falling particles. A computer program described by the flow chart presented in Fig. 1 is used to describe this model.
(1)
(6)
where u and p are integer array numbers generated for every pore location in X, and Y coordinates. Plx(u1,p) is the previous pore X-coordinateY-coordinate of the pore Plyðu; p 1Þ þ S;
(7)
Ply(u,p1) is the previous pore Y-coordinate Fig. 2 shows a section of filtration media with pore coordinates assigned. 2.4. Generating particles radii Particles radii, RP (i), are provided by a random number generated by the computer program and the maximum diameter of the particles
ARTICLE IN PRESS A.Y. Al-Otoom / Atmospheric Environment 39 (2005) 51–57
53
Fig. 1. Flow diagram of the computer program used for the model calculations.
(dpmax) according to Rp ðiÞ ¼
d pmax Random no:ð Þ : 2
(8)
The computer program generates random numbers between 0 and 1. It was noted that the mean particle diameter generated is equivalent to around half the value of the maximum particle diameter indicated above. A typical particle size distribution generated by this random number is presented in Fig. 3 for a
maximum particle diameter of 31 mm and a mass mean diameter of around 15 mm. The figure shows that the distribution generated by the random function resembles the normal distribution that can be found in practical situations. 2.5. Assigning particles falling location A particle falling position (x0(i), y0(i)) is also assigned randomly by a random number to provide a fraction of
ARTICLE IN PRESS A.Y. Al-Otoom / Atmospheric Environment 39 (2005) 51–57
54 0.05
0.04
y, cm
0.03
0.02
0.01
0 0
0.01
0.02
0.03
0.04
0.05
X, cm
Fig. 2. Sample of the pore allocation map in a filtration media.
25
Percent
20
Fig. 4. A sample for the deposition model for 4000 particles (top view).
15
particles deposited on the filter.
10 5
Ek ¼ 1
0 0
0.00001 0.00002 0.00003 Particle Diameter, m
0.00004
Fig. 3. A sample of particle size distribution generated by the random function for a maximum particle diameter of 31 mm and a mean particle diameter of around 15 mm.
the length and width of the filter media. x0 ðiÞ ¼ L Random no:ð Þ;
(9)
y0 ðiÞ ¼ W Random no:ð Þ:
(10)
2.6. Criteria for filter cake formation The model calculates the distances between a new falling particle and all other particles deposited on the filter media. It also finds the minimum distance among other particles, and compares this distance with the summation of the radii of the new particle and closest particle. If the minimum distance is smaller than this summation, it makes the new particle adhere to the surface of the closest particle. Depending on the X-Y coordinates of both particles and using geometrical analysis, a new height will be calculated. This new height corresponds to a new cake thickness. The model also calculates the average cake porosity (Ek) by summation of the total volume of the
Particles volume : Filtration area Cake thickness
(11)
Fig. 4 shows a part of the output from the computer program describing the top view of the deposition of filter cake for a small number of particles. If the minimum distance between the new falling particle and the closest particle are greater than the summation of their two radii, then the particle will move to the filtration media. Then the particle either escapes from the filter media or forms the first particle to deposit on its new location. The model then compares, individually, these particles in terms of X, and Y coordinates with all pore coordinates present in the filter, calculating the distance between this particle and all pore locations. It also finds the minimum distance among pore locations. If this distance is less than or equal to the particle radius, the particle will fall or escape from the filter cake. 3. Results and discussion The model developed in this study utilizes complete adhesion of particles if they fail to pass the filtration media. Adhesion of small particles is governed primarily by Van der Waals’ forces (FV). It is a function of the particle diameter and the distance between the particle and the surface (h). FV ¼
A Dp ; 6 h2
where A is Hamaker’s constant.
(12)
ARTICLE IN PRESS A.Y. Al-Otoom / Atmospheric Environment 39 (2005) 51–57
95 90
Collection efficiency, %
85 80 75 70 65 60 55
dp/pd = 0.025 dp/pd = 0.5 dp/pd = 0.75
filter cake collection efficiency, %
110
Fig. 5. The transient collection efficiency for an average filter porosity of 55%.
90 80 70 60 Ef = 0.1 EF= 0.35 Ef= 0.55 Ef= 0.75 Ef= 0.95
50 40 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Mass mean particle diameter / Mean pore diameter
1
Fig. 6. The steady-state filter cake collection efficiency.
7.E-04 dp = 1 micron
6.E-04
dp = 5 micron dp =7 micron
5.E-04 4.E-04 3.E-04 2.E-04 1.E-04
0.E+00 1.E+05 1.E+10 2.E+10 3.E+10 4.E+10 5.E+10 6.E+10 7.E+10 8.E+10 9.E+10 1.E+11
Net No. of Particles Deposited onthe Filter
Fig. 7. Filtration cake thickness as a function of the particle mean diameter and the number of particles deposited on the filter.
decreasing the filter porosity will also increase the cake collection efficiency. The cake thickness is calculated using the abovementioned model and the deposition criteria. It was found that cake thickness is strongly affected by the number of particles and the particle mass mean diameter. As illustrated in Fig. 7, the cake thickness increases with the particle size as well as the net number of particles deposited. This figure provides quantitative information about the transient progress of cake thickness, which can be helpful in determining the cleaning intervals for filtration media. Using a non-linear regression model, this relationship best fits (correlation coefficient, R2 ¼ 0:9) the following correlation: cake thickness ¼
50 1.00E+07 1.00E+08 1.00E+09 1.00E+10 1.00E+11 1.00E+12 1.00E+13 (Particle concentration* Fluid flow rate*Time) / Filtration Area, m-2.
100
30
Cake Thickness, m
The other force that may enhance the adhesion of particles is the electrostatic force. These combined forces are expected to overcome other forces affecting the particle trajectories such as Stock’s drag force and Brownian forces. It is important here to note that Brownian motion is important for submicron particles. Other forces like Thermophoresis is significant when there is a temperature gradient in filtration system. Van der Waals force, as shown in Eq. (11), linearly decreases with decreasing the particle diameter. On the other hand, other forces decrease as a stronger function of particle size such as the particle weight. Particle weight changes as the cube of the particle diameter change. The adhesion force is expected to be at least an order of magnitude higher than other forces (Kim and Lawrance, 1988) The model developed in this study predicts the collection efficiency of filtration, which strongly correlates with the ratio of the mass mean particle diameter and the mean pore diameter. During the initial stages of filtration, the collection efficiency increases with time, because the formation of the filter cake acts as a new filter media. This new cake increases the effective filtration area. It will increase the chances of collision and adhesion of the new coming particles to a fixed cake or to the filtration media. Fig. 5 demonstrates an example for the transient filtration for filter porosity of 55%. More than 80% efficiency can be achieved if the ratio of the particle mean size to the pore size is greater than 75%. As mentioned earlier, this figure describes the collection efficiency at constant filter porosity. Fig. 6 shows the steady state of cake collection efficiency. Increasing the ratio of the mass mean particle diameter to the mean pore diameter increased the steady-state collection efficiency on the filter cake. As expected,
55
2878d 0:8 p 756 þ 23100d 2p
ðparticle concentration fluid flow rate timeÞ0:033 : ðparticle concentration fluid flow rate timeÞ0:122 ð13Þ
ARTICLE IN PRESS A.Y. Al-Otoom / Atmospheric Environment 39 (2005) 51–57
56
8.E+06 Experiment( Aguiar and Coury,1996) Present Model Rudnick-Happel Ergun
1
7.E+06
0.8
6.E+06
0.6
5.E+06
∆P/(µ.u), m-1
Average Porosity
1.2
0.4 0.2
4.E+06 3.E+06 dp = 1 micron dp = 5 micron dp = 7 micron dp = 15 micron
2.E+06
0 0
0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 Cake Thickness,m
Fig. 8. Average porosity as a function of cake thickness for a mean particle size of 6.9 mm.
1.E+06 0.E+00 0.E+00 2.E-07 4.E-07 6.E-07 8.E-07 1.E-06 1.E-06 1.E-06 Mass of filter cake / ρ, m3
Predicting the porosity changes in the filter cake can provide essential parameters for the design of most filtration systems. The relationship between the average porosity and the cake thickness were calculated in this model. It was found that as the cake thickness increases, the average cake porosity decreases. This could be explained by the coverage of the initial layer with larger particles, which blocks the other particles from reaching the inner layers. Fig. 8 shows this relationship. The figure also includes experimental data obtained from Aguiar and Coury (1996) while comparing these data to Ergun’s (1952) and Rudnick–Happel’s equations. These equations are presented in Eqs. (13) and (14), respectively. The figure shows that the current statistical model fits the data better in higher cake thicknesses (4800 mm), while Rudnick–Happel’s and Ergun’s equations predict the experiment better at lower cake thicknesses. It is believed that at lower cake thickness, the particles are influenced by greater drag force resulting from pressure drop differences between the positive and negative sides of the filter. This means that particles at this level may be dragged towards the filter media with less resistance from other deposited particles. At higher thickness, this resistance becomes greater (due to the higher number of particles), and the assumption that all colliding particles adhering to the surface becomes more appropriate. DP ¼
DP ¼
150QmV ð1 Þ t; 3 Arp d 2p 150QmV Arp d 2p 2 4
(14)
3
5
3 þ 2ð1 Þ3 5
5
3 4:5ð1 Þ3 þ 4:5ð1 Þ3 2ð1 Þ2
5t; ð15Þ
Fig. 9. Pressure drop across filter cakes for diffrent means of particle size.
where e is the cake porosity, m is the fluid viscosity, V is the fluid velocity, rp is the particle density, dp is the particle mean diameter, Q is the volumetric flow rate, and A is the cross-sectional area. Once the porosity changes during filtration are predicted, estimation of the pressure drop can be easily obtained using one of the above-mentioned pressure drop equations. Fig. 9 shows the variation of the pressure drop predicted with the mean mass particles diameter and the mass of filter cake. It is clearly seen that the pressure drop across the filter cake increases with decreasing mean particle size. This is expected since smaller particle sizes tend to form lower average porosities inside the filter cake thus increasing the pressure drop across it. For model verification purposes, experimental pressure drop data with a wide range of parameters were obtained from different studies (Aguiar and Coury, 1996; Ergu¨denler et al., 1997). Table 1 summarizes the operating conditions of each of these studies, while Fig. 10 illustrates the ability of the model to predict the experimental data with good agreement between the two. The developed model can provide a good indication for the progress of the filter cake and the properties associated it. However, future development of this model is needed to focus on other important issues like the compressibility effect or the drag effect, which can be extremely important particularly in pressure filtration, This future development should also take into account the sphericity factor of the particles used in filtration. In addition, a criterion should be introduced to study clogging of filtration media, which is expected to highly depend on the internal structure of the filtration media.
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Table 1 Summary of the data used for verification
Filter type Filtration fluid Material to be filtered Mean particle size (mm) Particle density (Kg/m3) Filtration face velocity (m/s) Temperature (1C)
1st verification data (8)
2nd verification data (8)
3rd verification data (9)
Polyester felt Air Limestone 3.3 2750 0.075 25
Polyester felt Air Limestone 6.9 2750 0.075 25
Ceramic candle filter Air Char 15 384 0.003 700
25 Experiment (Aguiar, and Coury, 1996) model
∆P, Kpa
∆P, Kpa
20 15 10 5 0
0.005
0.01 0.015 Cake mass, Kg
∆P, Kpa
0
4.5 4 3.5 3 2.5 2 1.5 1 0.5 0
0.02
8 7 6 5 4 3 2 1 0
Experiment (Aguiar, and Coury, 1996) model
0
0.005 0.01 0.015 0.02 0.025 0.03 Cake mass, Kg
Experiment( Ergüdenler et.al.,1997) model
0
10
20
30
40 50 60 Time, min.
70 80
Fig. 10. A comparison of the model prediction and experimental data obtained from studies for pressure drop across filter cakes.
References Aguiar, M.L., Coury, J.R., 1996. Formation in fabric filtration of gases. Industrial & Engineering Chemistry Research 35, 3673–3679. Chan, S.H., Kiang, S., Brown, M.A., 2003. One-dimensional centrifugation model. Journal of American Chemical Engineering Institute 49, 925–938. Ergu¨denler, A., Tang, W., Brereton, C.M.H., Lim, C.J., Grace, J.R., 1997. Proceedings of the 14th International Conference on Fluidized Bed Combustion. In: Preto, F.D.S. (Ed.), vol. 2. American Society of Mechanical Engineers, New York, NY, pp. 1065–1073. Granger, J., Dodds, J., Leclerc, D., 1985. Filtration of low concentrations of latex particles on membrane filters. Filtration and Separation 22, 58–60.
Kim, S., Lawrance, C., 1988. Suspension mechanics for particle contamination control: review article. Chemical Engineering Science 43, 991–1016. Lee, D.J., 1997. Filter medium clogging during cake filtration. Journal of American Chemical Engineering Institute 43, 273–276. Notebaert, F.F., Van Haute, A.A., Wilms, D.A., 1975. Conditioning of aerobically stabilised sludge. Water Research 9, 1037–1046. Shirato, M., Murase, T., Ivitari, E., Tiller, F.M., Alciatore, A.F., 1987. Filtration in chemical process industry. In: Matteson, M.J., Orr, C. (Eds.), Filtration: Principles and Practice, second ed. Marcel Dekker, New York. Sørensen, B.L., Keiding, K., Lauitzen, S.L., 1997. A theoretical model for blinding in cake filtration. Journal of American Chemical Engineering Institute 69, 168–173.
Atmospheric Environment 39 (2005) 51–57 www.elsevier.com/locate/atmosenv
Prediction of the collection efficiency, the porosity, and the pressure drop across filter cakes in particulate air filtration Awni Y. Al-Otoom Chemical Engineering Department, University of Mutah, P.O.Box 78, Mutah—Karak 61710, Jordan Received 24 February 2004; accepted 15 September 2004
Abstract This study presents a new statistical model to predict the collection efficiency, cake thickness, cake porosity, and pressure drop across filter cakes during the particulate filtration of gases. This model is based on generation of a random distribution of particle sizes and particle falling locations. The model predicts the cake collection efficiency, which was found to be strongly dependent on the ratio of the mean particle size to the mean pore size of the filter medium. The average cake porosity decreases with increasing cake thickness and the pressure drop increases when the mean particle diameter decreases. r 2004 Elsevier Ltd. All rights reserved. Keywords: Filter cake; Model; Porosity; Pressure drop
1. Introduction Since the industrial revolution, the total amount of emissions has been increasing significantly. This led many governments around the world to introduce new legislations to control such emissions. The diversity of these emissions led researchers to look for practical solutions and to search for different methods for the removal of unwanted materials. Particulate emissions are one of the main types of undesirable industrial wastes. Nearly all industries have particulate emissions of some kind, which are generally removed by filtration, including cake filtration, vacuum filters, centrifugal filters, and many others. In cake filtration, the solid particles are removed from a slurry stream by exposing them to a filtration medium. A cake of the solid material is formed and its thickness increases with the time of filtration. The science of cake filtration Tel.: +962 796780123; fax: +962 65155058.
E-mail address: [email protected] (A.Y. Al-Otoom).
has been practiced for more than 70 years and many investigations have been published. (A comprehensive review is available in Shirato et al., 1987.) During cake filtration, some particles do not participate in the formation of the filter cake. They either escape from the filtration media, causing a decrease in the efficiency of the filter, or clog the septum of the filter media. Clogging of the filter media has been studied by many investigators including Notebaert et al. (1975); Granger et al. (1985); Lee (1997), and many others. Clogging, in addition to the problem of increasing the pressure drop across the filter media, causes errors in the prediction of the real cake thickness and the porosity of the filter cake. The wide distribution of both particle and filter pore sizes makes the prediction of particle escape from the filter cake difficult. Little research has been done to account for the wide range of particle distribution in the prediction of particle escape from the filter cake. The strength of the cake formed during filtration is relatively low. The cake can be easily disintegrated
1352-2310/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.atmosenv.2004.09.057
ARTICLE IN PRESS 52
A.Y. Al-Otoom / Atmospheric Environment 39 (2005) 51–57
during handling or during attempts to measure physical parameters, such as average porosity. Instead, experimental data of the pressure drop across the filter cake are adopted to calculate such parameters. Some investigators like Aguiar and Coury (1996) used cake fixation resins to obtain a strong filter cake for further analysis. The process is difficult to perform and requires extremely careful handling. Theoretical approaches are generally considered the key to predicting the cake filtration performance. The diversity and complexity of filtration media makes the evaluation of the performance of such media extremely difficult to perform experimentally. The wide range of particle size and the variability of the operating conditions of pressure and temperature add more difficulties in performing such tasks. Most of the theoretical studies present in the literature follow the Darcian-type analogy to predict the filtration progress. A recent 1-D model was developed by Chan et al. (2003) based on Darcy’s law to describe the filtration in centrifuges and to extract the filter cake parameters. A similar study was also developed earlier (SØrensen et al., 1997) which focused on the blinding of filter cake and filter media during filtration utilizing Darcian flow equations. In this work, a new approach will be utilized. A statistical model based on a random distribution of particle sizes will be presented. This model predicts the ratio of the particles that escape from the filter cake (Either clogging the filter or escaping from the filter media). In addition, this model predicts the cake thickness with filtration time, and calculates the pressure drop across this cake by predicting the porosity changes with the time of filtration.
2. The model 2.1. Model description
5. Particles have uniform spherical shapes. 6. Particles are incompressible. 7. Particle size distributions, and the particle falling location are randomly distributed. 2.3. Assigning pore location The pores centers are determined according to the total filter porosity (Ef), the length (L), width (W), and height (H) of the filter media. The distance between two successive pores in either X, Y direction is determined by these equations No. of pores in X-direction x¼
L ; S
where S is the distance between two successive pores, and will be presented in Eq. (5). No. of pores in Y-direction. y¼
W : S
2.2. Model assumptions 1. Filter media has a uniform surface porosity with an average pore diameter, pd and a constant distance between all successive pores. 2. Pores in filter media are cylindrical. 3. Particles that enter pores escape from the filter cake. 4. Particles that come into contact with other particles adhere to the surface with these particles.
(2)
Total no. of pores xy ¼
LW ; S2
(3)
but total no. of pores: Total pore volume LWHE f LW E f ¼ ¼ : Pore volume of one pore ðp=4Þp2d H ðp=4Þp2d
(4)
Equating Eqs. (3) and (4), results in the distance between two successive pore centers sffiffiffiffiffiffiffiffiffiffiffiffi p=4p2d S¼ : (5) Ef The X and Y pore coordinates are determines by X-coordinate of the pore Plxðu; pÞ ¼ Plxðu 1; pÞ þ S;
The model is designed to provide statistical analysis based on random function for formation of a wide range of particle size distributions and also uses a random function to anticipate the location of the falling particles. A computer program described by the flow chart presented in Fig. 1 is used to describe this model.
(1)
(6)
where u and p are integer array numbers generated for every pore location in X, and Y coordinates. Plx(u1,p) is the previous pore X-coordinateY-coordinate of the pore Plyðu; p 1Þ þ S;
(7)
Ply(u,p1) is the previous pore Y-coordinate Fig. 2 shows a section of filtration media with pore coordinates assigned. 2.4. Generating particles radii Particles radii, RP (i), are provided by a random number generated by the computer program and the maximum diameter of the particles
ARTICLE IN PRESS A.Y. Al-Otoom / Atmospheric Environment 39 (2005) 51–57
53
Fig. 1. Flow diagram of the computer program used for the model calculations.
(dpmax) according to Rp ðiÞ ¼
d pmax Random no:ð Þ : 2
(8)
The computer program generates random numbers between 0 and 1. It was noted that the mean particle diameter generated is equivalent to around half the value of the maximum particle diameter indicated above. A typical particle size distribution generated by this random number is presented in Fig. 3 for a
maximum particle diameter of 31 mm and a mass mean diameter of around 15 mm. The figure shows that the distribution generated by the random function resembles the normal distribution that can be found in practical situations. 2.5. Assigning particles falling location A particle falling position (x0(i), y0(i)) is also assigned randomly by a random number to provide a fraction of
ARTICLE IN PRESS A.Y. Al-Otoom / Atmospheric Environment 39 (2005) 51–57
54 0.05
0.04
y, cm
0.03
0.02
0.01
0 0
0.01
0.02
0.03
0.04
0.05
X, cm
Fig. 2. Sample of the pore allocation map in a filtration media.
25
Percent
20
Fig. 4. A sample for the deposition model for 4000 particles (top view).
15
particles deposited on the filter.
10 5
Ek ¼ 1
0 0
0.00001 0.00002 0.00003 Particle Diameter, m
0.00004
Fig. 3. A sample of particle size distribution generated by the random function for a maximum particle diameter of 31 mm and a mean particle diameter of around 15 mm.
the length and width of the filter media. x0 ðiÞ ¼ L Random no:ð Þ;
(9)
y0 ðiÞ ¼ W Random no:ð Þ:
(10)
2.6. Criteria for filter cake formation The model calculates the distances between a new falling particle and all other particles deposited on the filter media. It also finds the minimum distance among other particles, and compares this distance with the summation of the radii of the new particle and closest particle. If the minimum distance is smaller than this summation, it makes the new particle adhere to the surface of the closest particle. Depending on the X-Y coordinates of both particles and using geometrical analysis, a new height will be calculated. This new height corresponds to a new cake thickness. The model also calculates the average cake porosity (Ek) by summation of the total volume of the
Particles volume : Filtration area Cake thickness
(11)
Fig. 4 shows a part of the output from the computer program describing the top view of the deposition of filter cake for a small number of particles. If the minimum distance between the new falling particle and the closest particle are greater than the summation of their two radii, then the particle will move to the filtration media. Then the particle either escapes from the filter media or forms the first particle to deposit on its new location. The model then compares, individually, these particles in terms of X, and Y coordinates with all pore coordinates present in the filter, calculating the distance between this particle and all pore locations. It also finds the minimum distance among pore locations. If this distance is less than or equal to the particle radius, the particle will fall or escape from the filter cake. 3. Results and discussion The model developed in this study utilizes complete adhesion of particles if they fail to pass the filtration media. Adhesion of small particles is governed primarily by Van der Waals’ forces (FV). It is a function of the particle diameter and the distance between the particle and the surface (h). FV ¼
A Dp ; 6 h2
where A is Hamaker’s constant.
(12)
ARTICLE IN PRESS A.Y. Al-Otoom / Atmospheric Environment 39 (2005) 51–57
95 90
Collection efficiency, %
85 80 75 70 65 60 55
dp/pd = 0.025 dp/pd = 0.5 dp/pd = 0.75
filter cake collection efficiency, %
110
Fig. 5. The transient collection efficiency for an average filter porosity of 55%.
90 80 70 60 Ef = 0.1 EF= 0.35 Ef= 0.55 Ef= 0.75 Ef= 0.95
50 40 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Mass mean particle diameter / Mean pore diameter
1
Fig. 6. The steady-state filter cake collection efficiency.
7.E-04 dp = 1 micron
6.E-04
dp = 5 micron dp =7 micron
5.E-04 4.E-04 3.E-04 2.E-04 1.E-04
0.E+00 1.E+05 1.E+10 2.E+10 3.E+10 4.E+10 5.E+10 6.E+10 7.E+10 8.E+10 9.E+10 1.E+11
Net No. of Particles Deposited onthe Filter
Fig. 7. Filtration cake thickness as a function of the particle mean diameter and the number of particles deposited on the filter.
decreasing the filter porosity will also increase the cake collection efficiency. The cake thickness is calculated using the abovementioned model and the deposition criteria. It was found that cake thickness is strongly affected by the number of particles and the particle mass mean diameter. As illustrated in Fig. 7, the cake thickness increases with the particle size as well as the net number of particles deposited. This figure provides quantitative information about the transient progress of cake thickness, which can be helpful in determining the cleaning intervals for filtration media. Using a non-linear regression model, this relationship best fits (correlation coefficient, R2 ¼ 0:9) the following correlation: cake thickness ¼
50 1.00E+07 1.00E+08 1.00E+09 1.00E+10 1.00E+11 1.00E+12 1.00E+13 (Particle concentration* Fluid flow rate*Time) / Filtration Area, m-2.
100
30
Cake Thickness, m
The other force that may enhance the adhesion of particles is the electrostatic force. These combined forces are expected to overcome other forces affecting the particle trajectories such as Stock’s drag force and Brownian forces. It is important here to note that Brownian motion is important for submicron particles. Other forces like Thermophoresis is significant when there is a temperature gradient in filtration system. Van der Waals force, as shown in Eq. (11), linearly decreases with decreasing the particle diameter. On the other hand, other forces decrease as a stronger function of particle size such as the particle weight. Particle weight changes as the cube of the particle diameter change. The adhesion force is expected to be at least an order of magnitude higher than other forces (Kim and Lawrance, 1988) The model developed in this study predicts the collection efficiency of filtration, which strongly correlates with the ratio of the mass mean particle diameter and the mean pore diameter. During the initial stages of filtration, the collection efficiency increases with time, because the formation of the filter cake acts as a new filter media. This new cake increases the effective filtration area. It will increase the chances of collision and adhesion of the new coming particles to a fixed cake or to the filtration media. Fig. 5 demonstrates an example for the transient filtration for filter porosity of 55%. More than 80% efficiency can be achieved if the ratio of the particle mean size to the pore size is greater than 75%. As mentioned earlier, this figure describes the collection efficiency at constant filter porosity. Fig. 6 shows the steady state of cake collection efficiency. Increasing the ratio of the mass mean particle diameter to the mean pore diameter increased the steady-state collection efficiency on the filter cake. As expected,
55
2878d 0:8 p 756 þ 23100d 2p
ðparticle concentration fluid flow rate timeÞ0:033 : ðparticle concentration fluid flow rate timeÞ0:122 ð13Þ
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56
8.E+06 Experiment( Aguiar and Coury,1996) Present Model Rudnick-Happel Ergun
1
7.E+06
0.8
6.E+06
0.6
5.E+06
∆P/(µ.u), m-1
Average Porosity
1.2
0.4 0.2
4.E+06 3.E+06 dp = 1 micron dp = 5 micron dp = 7 micron dp = 15 micron
2.E+06
0 0
0.0002 0.0004 0.0006 0.0008 0.001 0.0012 0.0014 Cake Thickness,m
Fig. 8. Average porosity as a function of cake thickness for a mean particle size of 6.9 mm.
1.E+06 0.E+00 0.E+00 2.E-07 4.E-07 6.E-07 8.E-07 1.E-06 1.E-06 1.E-06 Mass of filter cake / ρ, m3
Predicting the porosity changes in the filter cake can provide essential parameters for the design of most filtration systems. The relationship between the average porosity and the cake thickness were calculated in this model. It was found that as the cake thickness increases, the average cake porosity decreases. This could be explained by the coverage of the initial layer with larger particles, which blocks the other particles from reaching the inner layers. Fig. 8 shows this relationship. The figure also includes experimental data obtained from Aguiar and Coury (1996) while comparing these data to Ergun’s (1952) and Rudnick–Happel’s equations. These equations are presented in Eqs. (13) and (14), respectively. The figure shows that the current statistical model fits the data better in higher cake thicknesses (4800 mm), while Rudnick–Happel’s and Ergun’s equations predict the experiment better at lower cake thicknesses. It is believed that at lower cake thickness, the particles are influenced by greater drag force resulting from pressure drop differences between the positive and negative sides of the filter. This means that particles at this level may be dragged towards the filter media with less resistance from other deposited particles. At higher thickness, this resistance becomes greater (due to the higher number of particles), and the assumption that all colliding particles adhering to the surface becomes more appropriate. DP ¼
DP ¼
150QmV ð1 Þ t; 3 Arp d 2p 150QmV Arp d 2p 2 4
(14)
3
5
3 þ 2ð1 Þ3 5
5
3 4:5ð1 Þ3 þ 4:5ð1 Þ3 2ð1 Þ2
5t; ð15Þ
Fig. 9. Pressure drop across filter cakes for diffrent means of particle size.
where e is the cake porosity, m is the fluid viscosity, V is the fluid velocity, rp is the particle density, dp is the particle mean diameter, Q is the volumetric flow rate, and A is the cross-sectional area. Once the porosity changes during filtration are predicted, estimation of the pressure drop can be easily obtained using one of the above-mentioned pressure drop equations. Fig. 9 shows the variation of the pressure drop predicted with the mean mass particles diameter and the mass of filter cake. It is clearly seen that the pressure drop across the filter cake increases with decreasing mean particle size. This is expected since smaller particle sizes tend to form lower average porosities inside the filter cake thus increasing the pressure drop across it. For model verification purposes, experimental pressure drop data with a wide range of parameters were obtained from different studies (Aguiar and Coury, 1996; Ergu¨denler et al., 1997). Table 1 summarizes the operating conditions of each of these studies, while Fig. 10 illustrates the ability of the model to predict the experimental data with good agreement between the two. The developed model can provide a good indication for the progress of the filter cake and the properties associated it. However, future development of this model is needed to focus on other important issues like the compressibility effect or the drag effect, which can be extremely important particularly in pressure filtration, This future development should also take into account the sphericity factor of the particles used in filtration. In addition, a criterion should be introduced to study clogging of filtration media, which is expected to highly depend on the internal structure of the filtration media.
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Table 1 Summary of the data used for verification
Filter type Filtration fluid Material to be filtered Mean particle size (mm) Particle density (Kg/m3) Filtration face velocity (m/s) Temperature (1C)
1st verification data (8)
2nd verification data (8)
3rd verification data (9)
Polyester felt Air Limestone 3.3 2750 0.075 25
Polyester felt Air Limestone 6.9 2750 0.075 25
Ceramic candle filter Air Char 15 384 0.003 700
25 Experiment (Aguiar, and Coury, 1996) model
∆P, Kpa
∆P, Kpa
20 15 10 5 0
0.005
0.01 0.015 Cake mass, Kg
∆P, Kpa
0
4.5 4 3.5 3 2.5 2 1.5 1 0.5 0
0.02
8 7 6 5 4 3 2 1 0
Experiment (Aguiar, and Coury, 1996) model
0
0.005 0.01 0.015 0.02 0.025 0.03 Cake mass, Kg
Experiment( Ergüdenler et.al.,1997) model
0
10
20
30
40 50 60 Time, min.
70 80
Fig. 10. A comparison of the model prediction and experimental data obtained from studies for pressure drop across filter cakes.
References Aguiar, M.L., Coury, J.R., 1996. Formation in fabric filtration of gases. Industrial & Engineering Chemistry Research 35, 3673–3679. Chan, S.H., Kiang, S., Brown, M.A., 2003. One-dimensional centrifugation model. Journal of American Chemical Engineering Institute 49, 925–938. Ergu¨denler, A., Tang, W., Brereton, C.M.H., Lim, C.J., Grace, J.R., 1997. Proceedings of the 14th International Conference on Fluidized Bed Combustion. In: Preto, F.D.S. (Ed.), vol. 2. American Society of Mechanical Engineers, New York, NY, pp. 1065–1073. Granger, J., Dodds, J., Leclerc, D., 1985. Filtration of low concentrations of latex particles on membrane filters. Filtration and Separation 22, 58–60.
Kim, S., Lawrance, C., 1988. Suspension mechanics for particle contamination control: review article. Chemical Engineering Science 43, 991–1016. Lee, D.J., 1997. Filter medium clogging during cake filtration. Journal of American Chemical Engineering Institute 43, 273–276. Notebaert, F.F., Van Haute, A.A., Wilms, D.A., 1975. Conditioning of aerobically stabilised sludge. Water Research 9, 1037–1046. Shirato, M., Murase, T., Ivitari, E., Tiller, F.M., Alciatore, A.F., 1987. Filtration in chemical process industry. In: Matteson, M.J., Orr, C. (Eds.), Filtration: Principles and Practice, second ed. Marcel Dekker, New York. Sørensen, B.L., Keiding, K., Lauitzen, S.L., 1997. A theoretical model for blinding in cake filtration. Journal of American Chemical Engineering Institute 69, 168–173.