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A proposed correlation equation for predicting filter coefficient under unfavorable deposition conditions
Separation and Purification Technology 65 (2009) 248–250
Contents lists available at ScienceDirect
Separation and Purification Technology journal homepage: www.elsevier.com/locate/seppur
Short communication
A proposed correlation equation for predicting filter coefficient under unfavorable deposition conditions You-Im Chang ∗ , Wei-You Cheng, Hsun-Chih Chan Department of Chemical Engineering, Tunghai University, 40704 Taichung, Taiwan
a r t i c l e
i n f o
Article history: Received 24 July 2008 Received in revised form 17 October 2008 Accepted 21 October 2008 Keywords: Filtration Particle Deposition Correlation Unfavorable condition
a b s t r a c t A proposed correlation equation is presented in the present note, based on our previous paper [Y.I. Chang, H.C. Chan, Correlation equation for predicting filter coefficient under unfavorable deposition conditions, AIChE J. 54 (2008) 1235–1253]. When comparing with available filtration experimental data for both large and small particles, we find that the present proposed correlation equation can give a better prediction on the filter coefficient than the equation presented in our previous paper. © 2008 Elsevier B.V. All rights reserved.
1. Introduction
bed filtration [1]
In our previous paper [1], a correlation equation for predicting the filter coefficient under unfavorable deposition conditions was presented. By adopting the concept of Vaidyanathan and Tien [2], we found that those available colloidal filtration data under unfavorable deposition conditions at various ionic strengths of suspension can be described well by using a filter coefficient ratio ˛, defined as the ratio of the initial filter coefficient 0 to its value in the absence of the electrostatic repulsive force 0S (i.e. when the ionic strength of the colloidal suspension is high) as ˛=
0 0S
(1)
and ˛ was found to be the functions of parameters used to describe the magnitudes of van der Waals attractive and electrostatic repulsive energies of DLVO theory [3]. When using the triangular network model by adopting the Brownian dynamic simulation method [4], as the sum of four individual deposition mechanisms, e.g. the Brownian diffusion, the DLVO interactions, the gravitational force and the interception, our correlation equation (see Eq. (2) below) was successfully obtained by regressing against a broad range of parameter values governing particle deposition in deep
∗ Corresponding author. Tel.: +886 4 23590262; fax: +886 4 23590009. E-mail address: [email protected] (Y.-I. Chang). 1383-5866/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.seppur.2008.10.031
0.969 −0.423 2.880 1.5 NE1 NE2 NLo ˛C−C = ˛DLVO + ˛D + ˛I + ˛G = 0.024NDL 1/3
−0.715 2.687 +3.176AS NR−0.081 NPe NLo −0.514 0.125 NLo + NR−0.24 NG1.11 NLo +0.222AS NR3.041 NPe
(2)
with AS =
2(1 − 5 ) 2 − 3 + 3 5 − 2 6
and
= (1 − ε)1/3
where ε is the porosity of the filter, NR is an aspect ratio, NPe is the Peclet number, NG is the gravitational number, NLo is the van der Waals number, NDL is the electric double layer number, NE1 and NE2 are the first and the second electrokinetic numbers of the DLVO theory. The definitions of these dimensionless numbers are provided in Table 1. When comparing with the experimental results obtained by Bai and Tien [5], Elimelech and O’Melia [6], Elimelech [7] and Vaidyanathan and Tien [8], we found that our correlation equation can fit well with the experimental data especially for submicroparticles whose strong Brownian motion behavior cannot be ignored. On the other hand, our correlation equation is less precise than the Bai and Tien’s equation [5] (see Eq. (3) below) for large size particles, especially when the diameter of particles is 11.4 m as used in Vaidyanathan and Tien’s experiments [8], while ˛B−T =
0 0.7031 −0.3121 3.5111 1.352 = 2.527 × 10−3 NLo NE1 NE2 NDL 0S
(3)
Y.-I. Chang et al. / Separation and Purification Technology 65 (2009) 248–250
249
Table 1 Summary of dimensionless parameters presented in the correlation equation of Eq. (2). Parameter
Definition
Physical interpretation
NA
A 12a2 U
Attraction number
NDL
˛pi
Electric double-layer force parameter
pi
NE1 NE2
api ( p2 + g2 ) 4kB T 2( p / g ) 2
[1+( p / g ) ]
First electrokinetic parameter Second electrokinetic parameter
NG
a2 ( − f )g 2 Pi Pi 9 kB T
NPe
Udg D∞
NR
dp dg
Aspect ratio
NLo
A 6kB T
London force parameter
Gravity number; ratio of the Stokes particle setting velocity to the approach velocity of the fluid Peclet number characterizing ratio of the convective transport to the diffusive transport
The parameters in the various dimensionless groups are as follows: dg is the collector diameter, dp is the particle diameter, U is the inlet fluid velocity, D∞ is the bulk diffusion coefficient (described by Stokes–Einstein equation), A is the Hamaker constant, kB is the Boltzmann constant, T is the fluid absolute temperature, api is the ith particle radius, pi is the ith particle density, f is the fluid density, g is the gravitational acceleration, is the reciprocal of the electric double layer thickness, is the dielectric constant of the fluid, p and g are the surface (zeta) potentials of particle and collector, respectively.
with NLo = and
4A 9dp2 U
,
NE1 =
( p2 + g2 ) 3dp U
,
NE2 =
2 p g p2 + g2
NDL = dp .
2. The proposed correlation equation
Fig. 2. Comparison of experimental data with the proposed correlation equations of Eqs. (3) and (4) when the Vaidyanathan and Tien’s data are excluded.
than those deviations predicted by using Eq. (3). Since Eq. (2) showed remarkable agreement with the available experimental data especially for those sub-microparticles with significant Brownian motion behavior, we re-draw Fig. 10 excluding Vaidyanathan and Tien’s large particle data, and the result is shown in Fig. 1. In this figure, it can be found that Eq. (2) always overestimates while Eq. (3) underestimates those values of filter coefficient ratio. So, in order to provide a better agreement between the experimental data and the equation’s prediction for both large and small size particles, we further take the algebraic averaged value of Eqs. (2) and (3) as follows 1 [In(˛C−C ) + In(˛B−T )] 2
Fig. 10 in our previous paper [1] illustrated experimental data mentioned above against the corresponding values of ␣C−C and ␣B−T (i.e. the diagonal line in that figure indicated agreement between experimental data and equation’s prediction), we found that the deviations of Vaidyanathan and Tien’s data [8] from the diagonal line of that figure predicted by using Eq. (2) are larger
In˛ =
Fig. 1. Comparison of experimental data with the correlation equations of Eqs. (2) and (3) when the Vaidyanathan and Tien’s data are excluded.
Fig. 3. Comparison of experimental data with the proposed correlation equations of Eqs. (3) and (4) when the Vaidyanathan and Tien’s data are included.
(4)
The prediction results without considering Vaidyanathan and Tien’s data are shown in Fig. 2, which indicates that Eq. (4) fits better than both of Eqs. (2) and (3). Even including Vaidyanathan and Tien’s data, this proposed correlation equation Eq. (4) can still
250
Y.-I. Chang et al. / Separation and Purification Technology 65 (2009) 248–250
Table 2 Summary of the averaged variance values of Eqs. (2)–(4) with reference to the diagonal lines of Figs. 1–3. Averaged variance
Fig. 10 of Ref. [1]
Fig. 1
Fig. 2
Fig. 3
Eq. (2) Eq. (3) Eq. (4)
2.866 0.083 –
0.296 0.068 –
– 0.068 0.020
– 0.083 0.082
give a better fit than those results observed in Fig. 10 of our previous paper. The averaged variance values of Eqs. (2)–(4) with reference to the diagonal lines of Figs. 1–3 are summarized in Table 2. From this table, two important conclusions can be drawn as follows. 3. Conclusions (1) Despite that Eq. (2) shows remarkable agreement for those sub-microparticles with significant Brownian motion behavior, Eq. (3) does give a better prediction on the filter coefficient ratio of those available experimental data for both large and small particles (see Fig. 1, Table 2). (2) The proposed correlation equation of Eq. (4) does give a better prediction than that of Eq. (2) and of Eq. (3) when the Vaidyanathan and Tien’s large particle data are excluded
(see Fig. 2’s column of Table 2). Even with the consideration of the Vaidyanathan and Tien’s data, the accuracy of Eq. (4) is still as good as that of Eq. (3). Acknowledgments The authors would like to express their sincere thanks to Prof. Chi Tien at Chemical Engineering Department of Syracuse University for his valuable suggestions on this work. The financial support received from the National Science Council of the Republic of China, research grant no. NSC-96-2221-E-029-0014-MY3, is greatly appreciated. References [1] Y.I. Chang, H.C. Chan, AIChE J. 54 (2008) 1235. [2] R. Vaidyanathan, C. Tien, Chem. Eng. Sci. 46 (1991) 967. [3] E.J.W. Verwey, J.Th.G. Overbeek, Theory of the Stability of Lyophobic Colloids, Elsevier, Amsterdam, 1948. [4] C. Tien, B.V. Ramarao, Granular Filtration of Aerosols and Hydrosols, 2nd edition, Elsevier, Oxford, 2007, See chapter 8.4. [5] R. Bai, C. Tien, J. Colloid Interface Sci. 218 (1999) 488. [6] M. Elimelech, C.R. O’Melia, Langmuir 6 (1990) 1153. [7] M. Elimelech, Water Res. 26 (1992) 1. [8] R. Vaidyanathan, C. Tien, Chem. Eng. Commun. 81 (1989) 123.
Contents lists available at ScienceDirect
Separation and Purification Technology journal homepage: www.elsevier.com/locate/seppur
Short communication
A proposed correlation equation for predicting filter coefficient under unfavorable deposition conditions You-Im Chang ∗ , Wei-You Cheng, Hsun-Chih Chan Department of Chemical Engineering, Tunghai University, 40704 Taichung, Taiwan
a r t i c l e
i n f o
Article history: Received 24 July 2008 Received in revised form 17 October 2008 Accepted 21 October 2008 Keywords: Filtration Particle Deposition Correlation Unfavorable condition
a b s t r a c t A proposed correlation equation is presented in the present note, based on our previous paper [Y.I. Chang, H.C. Chan, Correlation equation for predicting filter coefficient under unfavorable deposition conditions, AIChE J. 54 (2008) 1235–1253]. When comparing with available filtration experimental data for both large and small particles, we find that the present proposed correlation equation can give a better prediction on the filter coefficient than the equation presented in our previous paper. © 2008 Elsevier B.V. All rights reserved.
1. Introduction
bed filtration [1]
In our previous paper [1], a correlation equation for predicting the filter coefficient under unfavorable deposition conditions was presented. By adopting the concept of Vaidyanathan and Tien [2], we found that those available colloidal filtration data under unfavorable deposition conditions at various ionic strengths of suspension can be described well by using a filter coefficient ratio ˛, defined as the ratio of the initial filter coefficient 0 to its value in the absence of the electrostatic repulsive force 0S (i.e. when the ionic strength of the colloidal suspension is high) as ˛=
0 0S
(1)
and ˛ was found to be the functions of parameters used to describe the magnitudes of van der Waals attractive and electrostatic repulsive energies of DLVO theory [3]. When using the triangular network model by adopting the Brownian dynamic simulation method [4], as the sum of four individual deposition mechanisms, e.g. the Brownian diffusion, the DLVO interactions, the gravitational force and the interception, our correlation equation (see Eq. (2) below) was successfully obtained by regressing against a broad range of parameter values governing particle deposition in deep
∗ Corresponding author. Tel.: +886 4 23590262; fax: +886 4 23590009. E-mail address: [email protected] (Y.-I. Chang). 1383-5866/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.seppur.2008.10.031
0.969 −0.423 2.880 1.5 NE1 NE2 NLo ˛C−C = ˛DLVO + ˛D + ˛I + ˛G = 0.024NDL 1/3
−0.715 2.687 +3.176AS NR−0.081 NPe NLo −0.514 0.125 NLo + NR−0.24 NG1.11 NLo +0.222AS NR3.041 NPe
(2)
with AS =
2(1 − 5 ) 2 − 3 + 3 5 − 2 6
and
= (1 − ε)1/3
where ε is the porosity of the filter, NR is an aspect ratio, NPe is the Peclet number, NG is the gravitational number, NLo is the van der Waals number, NDL is the electric double layer number, NE1 and NE2 are the first and the second electrokinetic numbers of the DLVO theory. The definitions of these dimensionless numbers are provided in Table 1. When comparing with the experimental results obtained by Bai and Tien [5], Elimelech and O’Melia [6], Elimelech [7] and Vaidyanathan and Tien [8], we found that our correlation equation can fit well with the experimental data especially for submicroparticles whose strong Brownian motion behavior cannot be ignored. On the other hand, our correlation equation is less precise than the Bai and Tien’s equation [5] (see Eq. (3) below) for large size particles, especially when the diameter of particles is 11.4 m as used in Vaidyanathan and Tien’s experiments [8], while ˛B−T =
0 0.7031 −0.3121 3.5111 1.352 = 2.527 × 10−3 NLo NE1 NE2 NDL 0S
(3)
Y.-I. Chang et al. / Separation and Purification Technology 65 (2009) 248–250
249
Table 1 Summary of dimensionless parameters presented in the correlation equation of Eq. (2). Parameter
Definition
Physical interpretation
NA
A 12a2 U
Attraction number
NDL
˛pi
Electric double-layer force parameter
pi
NE1 NE2
api ( p2 + g2 ) 4kB T 2( p / g ) 2
[1+( p / g ) ]
First electrokinetic parameter Second electrokinetic parameter
NG
a2 ( − f )g 2 Pi Pi 9 kB T
NPe
Udg D∞
NR
dp dg
Aspect ratio
NLo
A 6kB T
London force parameter
Gravity number; ratio of the Stokes particle setting velocity to the approach velocity of the fluid Peclet number characterizing ratio of the convective transport to the diffusive transport
The parameters in the various dimensionless groups are as follows: dg is the collector diameter, dp is the particle diameter, U is the inlet fluid velocity, D∞ is the bulk diffusion coefficient (described by Stokes–Einstein equation), A is the Hamaker constant, kB is the Boltzmann constant, T is the fluid absolute temperature, api is the ith particle radius, pi is the ith particle density, f is the fluid density, g is the gravitational acceleration, is the reciprocal of the electric double layer thickness, is the dielectric constant of the fluid, p and g are the surface (zeta) potentials of particle and collector, respectively.
with NLo = and
4A 9dp2 U
,
NE1 =
( p2 + g2 ) 3dp U
,
NE2 =
2 p g p2 + g2
NDL = dp .
2. The proposed correlation equation
Fig. 2. Comparison of experimental data with the proposed correlation equations of Eqs. (3) and (4) when the Vaidyanathan and Tien’s data are excluded.
than those deviations predicted by using Eq. (3). Since Eq. (2) showed remarkable agreement with the available experimental data especially for those sub-microparticles with significant Brownian motion behavior, we re-draw Fig. 10 excluding Vaidyanathan and Tien’s large particle data, and the result is shown in Fig. 1. In this figure, it can be found that Eq. (2) always overestimates while Eq. (3) underestimates those values of filter coefficient ratio. So, in order to provide a better agreement between the experimental data and the equation’s prediction for both large and small size particles, we further take the algebraic averaged value of Eqs. (2) and (3) as follows 1 [In(˛C−C ) + In(˛B−T )] 2
Fig. 10 in our previous paper [1] illustrated experimental data mentioned above against the corresponding values of ␣C−C and ␣B−T (i.e. the diagonal line in that figure indicated agreement between experimental data and equation’s prediction), we found that the deviations of Vaidyanathan and Tien’s data [8] from the diagonal line of that figure predicted by using Eq. (2) are larger
In˛ =
Fig. 1. Comparison of experimental data with the correlation equations of Eqs. (2) and (3) when the Vaidyanathan and Tien’s data are excluded.
Fig. 3. Comparison of experimental data with the proposed correlation equations of Eqs. (3) and (4) when the Vaidyanathan and Tien’s data are included.
(4)
The prediction results without considering Vaidyanathan and Tien’s data are shown in Fig. 2, which indicates that Eq. (4) fits better than both of Eqs. (2) and (3). Even including Vaidyanathan and Tien’s data, this proposed correlation equation Eq. (4) can still
250
Y.-I. Chang et al. / Separation and Purification Technology 65 (2009) 248–250
Table 2 Summary of the averaged variance values of Eqs. (2)–(4) with reference to the diagonal lines of Figs. 1–3. Averaged variance
Fig. 10 of Ref. [1]
Fig. 1
Fig. 2
Fig. 3
Eq. (2) Eq. (3) Eq. (4)
2.866 0.083 –
0.296 0.068 –
– 0.068 0.020
– 0.083 0.082
give a better fit than those results observed in Fig. 10 of our previous paper. The averaged variance values of Eqs. (2)–(4) with reference to the diagonal lines of Figs. 1–3 are summarized in Table 2. From this table, two important conclusions can be drawn as follows. 3. Conclusions (1) Despite that Eq. (2) shows remarkable agreement for those sub-microparticles with significant Brownian motion behavior, Eq. (3) does give a better prediction on the filter coefficient ratio of those available experimental data for both large and small particles (see Fig. 1, Table 2). (2) The proposed correlation equation of Eq. (4) does give a better prediction than that of Eq. (2) and of Eq. (3) when the Vaidyanathan and Tien’s large particle data are excluded
(see Fig. 2’s column of Table 2). Even with the consideration of the Vaidyanathan and Tien’s data, the accuracy of Eq. (4) is still as good as that of Eq. (3). Acknowledgments The authors would like to express their sincere thanks to Prof. Chi Tien at Chemical Engineering Department of Syracuse University for his valuable suggestions on this work. The financial support received from the National Science Council of the Republic of China, research grant no. NSC-96-2221-E-029-0014-MY3, is greatly appreciated. References [1] Y.I. Chang, H.C. Chan, AIChE J. 54 (2008) 1235. [2] R. Vaidyanathan, C. Tien, Chem. Eng. Sci. 46 (1991) 967. [3] E.J.W. Verwey, J.Th.G. Overbeek, Theory of the Stability of Lyophobic Colloids, Elsevier, Amsterdam, 1948. [4] C. Tien, B.V. Ramarao, Granular Filtration of Aerosols and Hydrosols, 2nd edition, Elsevier, Oxford, 2007, See chapter 8.4. [5] R. Bai, C. Tien, J. Colloid Interface Sci. 218 (1999) 488. [6] M. Elimelech, C.R. O’Melia, Langmuir 6 (1990) 1153. [7] M. Elimelech, Water Res. 26 (1992) 1. [8] R. Vaidyanathan, C. Tien, Chem. Eng. Commun. 81 (1989) 123.