Pressure drop model for nanostructured deposits

Separation and Purification Technology 138 (2014) 144–152

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Pressure drop model for nanostructured deposits D. Thomas a,⇑, F.X. Ouf b, F. Gensdarmes b, S. Bourrous a,b, L. Bouilloux b a b

Université de Lorraine, Laboratoire Réactions et Génie des Procédés (LRGP), UMR 7274, F-54000 Nancy, France Institut de Radioprotection et de Sûreté Nucléaire (IRSN), PSN-RES/SCA/LPMA, PSN-RES/SCA/LECEV, BP 68 Gif-Sur-Yvette, 91192, France

a r t i c l e

i n f o

Article history: Received 19 June 2014 Received in revised form 19 September 2014 Accepted 26 September 2014 Available online 27 October 2014 Keywords: Pressure drop Nanoparticle Aggregate Agglomerate Cake

a b s t r a c t This study presents a new pressure drop model developed for cakes composed of nanostructured particles. The cake structure is understood as a tangle of chains composed by juxtaposed primary particles with (aggregates) or without (agglomerates) a partial overlap. Since cake porosity is one of the main parameters determining aeraulic resistance, an experiment protocol based on the changes in deposit thickness as a function of the cake mass per surface area has been developed to accurately determine this parameter. To this end, the pressure drop and the porosity of the cakes created by the filtration of carbon nanoparticles aggregates and agglomerates on PTFE membrane were measured. The aggregate and agglomerate count median mobility diameters range from 91 nm to 170 nm and from 48 nm to 62 nm, respectively. The associated Peclet numbers range from 0.19 to 53 for filtration velocities of 0.01, 0.05 and 0.09 m/s. Initial experimental results indicate that the porosity of the cakes ranges from 0.94 to 0.984 in correlation with the Peclet number of the aggregates or agglomerates. The agreement between experimental results and the pressure drop model is fairly good. Of the experimental values, 95% are within plus or minus 25% of the theoretical value. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction The manufacture of nanoparticles is increasing and opens up possibilities for new applications and economical developments. According to Roco et al. [1], the worldwide nanomaterials marketplace in 2020 will represent a $3 trillion market associated with nearly 6 million workers. Nevertheless, these economical concerns should not overshadow the social impact of nanoparticles, and research must be led on the toxicity of such products. Due to this social concern, the containment of airborne nanoparticles and hazardous particulate matter during production processes is essential in order to reduce worker exposure as much as possible and protect the environment. Fibrous media are a widely used solution, and the question of penetration of nanoparticles through the media [2–4] and clogging [5,6] of filters is still being investigated. On the other hand, the Fukushima event reminded our society of the critical subject of nuclear installation containment. In most cases this containment is achieved using a ventilation system and High Efficiency Particulate Air (HEPA) filter to create subatmospheric pressure in the facility. In the case of fire, soot particles emitted could rapidly clog the HEPA filters on the ventilation ducts and, as a consequence, modify the ventilation conditions ⇑ Corresponding author. E-mail address: [email protected] (D. Thomas). http://dx.doi.org/10.1016/j.seppur.2014.09.032 1383-5866/Ó 2014 Elsevier B.V. All rights reserved.

inside the installation. Among other things, the specific morphology of soot [7] creates challenges when describing the behaviour of HEPA filters in case of fire. Recent research conducted by IRSN [8,9] provides for the description of the complex clogging behaviour of HEPA filters in fire conditions according to an empirical model. Nevertheless, such approach is limited and most of the previous studies have focused on specific fire conditions or the filtration of micronic particles in ambient temperature and pressure conditions. To our knowledge, studies investigating the pressure drop of nanoparticle cakes are limited [10,5,11] and the phenomenological description of the clogging phenomena for nanoparticle aggregates has been poorly investigated. Several correlations used to estimate the pressure drop of the cake can be found in the literature. They can be divided in two groups: the capillary model and the particulate model. The most popular correlation based on the capillary model is the Kozeny– Carman equation in Stokes regime. In this approach, the porous medium is considered to be an assembly of capillaries of specific size and geometry through which fluid flows. The particulate model is based on flow around particles. Mauret and Renaud [12] and more recently Puncochar and Drahos [13] have determined the applicability range of these models. In the case of fibre beds, Mauret and Renaud [12] show that the capillary approach is less suitable for porosities greater than 0.75 and for Reynolds numbers below 100. Since the porosity of nanostructured deposits is very

D. Thomas et al. / Separation and Purification Technology 138 (2014) 144–152

List of variables Cc slip correction factor (–) Co overlap parameter (–) Co; p mean 2D projected overlap coefficient (–) D aggregate or agglomerate diffusion coefficient (m2 s1) d distance between the centres of two particles in contact (m) dagg aggregated or agglomerated particle size (m) dp fibre or particle diameter (m) dpG count median diameter (m) dvg geometric mean size of the volume equivalent diameter (m) Fc correction factor (–) FT drag force per unit length of fibre acting on fibres (N m1) L total length of fibres per deposit surface area (m1) Mc deposited mass of nanoparticles (kg) mS cake mass per surface area (kg m2) N number of primary particles per cubic meter of cake (–)

high, ranging from 90% to 98% [14,5,11], the approach based on capillary model is not relevant. The particulate model developed by Endo et al. [15] is currently widely used to determine the pressure drop of a nanostructured deposit. In this approach, the pressure drop across a particle layer is assumed to be equal to the fluid drag acting on all individual particles. For a particle size distribution following a log normal distribution and in a Stokes regime, the authors obtained the following expression:

DP ¼ 18

gUf

v ðaÞ

j 

Cc ð1  aÞ2 d2 exp 4ln2 rg vg



ms

qp

Pe Uf Vpp Z

peclet number of aggregates/agglomerates (–) gas velocity (m s1) volume of the primary particle (m3) nanostructured deposit or filter thickness (m) packing density of the nanostructured deposit (a = 1  e) (–) pressure drop of a nanostructured deposit (Pa) porosity of the nanostructured deposit (e = 1  a) (–) empirical constants (11 = 1.1 ± 0.1 and 12 = 0.2 ± 0.02 for Df  1.78, [19]) dynamic shape factor of the particles (–) mean free path of gas (m) gas viscosity (Pa s) particle density (kg m3) geometric standard deviation of particle size distribution (–) void function (–) filtration surface area (m2)

a DP

e 11, 12 j k

g qp rg m(a) X

with (aggregates) or without (agglomerates) partial overlapping (Fig. 1). It therefore makes more sense to use the drag force acting on the chain of particles rather on particles. Sakano et al. [16] defined the drag force per unit length of fibre acting on fibres using the Davies equation [17]:

  16pa0:5 1 þ 56a3 FT ¼ gUf Cc

   2k cdp a þ b exp  dp 2k

Cc ¼ 1 þ

ð3Þ

where a = 1.165, b = 0.483, c = 0.997 [18] and k, the mean free path (in air at 20 °C and atmospheric pressure k = 66.4 nm). The pressure drop of a fibrous filter is equal to:

DP ¼ F T L

ð4Þ

where L is the total length of fibres per deposit surface area. To calculate L, we have to take into account the structure of the deposit. For a fibrous filter characterized by a fibre diameter (dp), a thickness (Z) and a packing density (a), L is equal to:

L ¼ LF ¼

4a

pd2p

Z

2. New pressure drop model Endo et al. [15] determined the pressure drop of a particle deposit from the sum of the drag forces acting on all the particles forming the cake. However, the nanostructured deposit can be understood as a tangle of chains composed by juxtaposed particles

ð2Þ

where a is the packing density (a = 1  e), g the air viscosity, Uf the air velocity and Cc the Cunningham coefficient defined as follows:

ð1Þ

where g is the gas viscosity, a the packing density (a = 1  Porosity (e)), dvg the geometric mean size of the volume equivalent diameter, rg the geometric standard deviation of particle size distribution, Cc the slip correction factor, Uf the gas velocity, qp the particle density, m(a) the void function, j the dynamic shape factor of the particles and mS the cake mass per surface area. It should be noted that the void function makes it possible to take the effect of neighbouring particles into account. Kim et al. [5] and more recently Liu et al. [11] have shown that the Endo’s model is applicable for soot agglomerate deposits since it takes into account the size distribution of the spherical primary particles (j = 1) and not the size distribution of the agglomerates. Note that the authors have used different void functions without justifying their choice. Moreover, Kim et al. [5] used the void function defined as m(a) = 10 (1  e)/e although, according Endo, it is only applicable in the porosity range from 0.3 to 0.6. However, Endo’s model does not take into account the partial overlapping of particles making up the cake although the SEM images provided by the authors seem to prove its existence. The goal of this work is to investigate and evaluate the porosity of the cake layer formed by aggregates or agglomerates of nanoparticles and to develop a predictive pressure drop model taking into account the overlap between primary particles observed for aggregates.

145

Fig. 1. Correction factor (Fc) versus overlap coefficient (Co).

ð5Þ

146

D. Thomas et al. / Separation and Purification Technology 138 (2014) 144–152

For a deposit formed by chains of juxtaposed particles, the total length per deposit surface area (L) is given by:

L ¼ LC ¼

6a

pd2p

Z

ð6Þ

For a deposit formed by chains of particles with partial overlapping of particles (Fig. 1), L is equal to (see Appendix A for details):

4a

ð1  CoÞ L ¼ LC ¼ 2 Z h  i 2 pdp 3  Co2 1  Co3

ð8Þ

where d represents the distance between the centres of two particles in contact, while dp is the diameter of the particles (see Fig. 1). If Co = 0, the primary particles are in point contact (juxtaposed particle), while if Co = 1, the particles are completely merged. In order to link the different expressions for L we propose the introduction of a correction factor Fc defined by the ratio L/LF. This parameter takes into account the difference of the total length of chains compared to the total length of fibres for the same packing density and the same particle or fibre diameter. In the case of deposit formed by juxtaposed particles, Fc is equal to 3/2. For chains with partial overlapping, we obtain the following expression: 2 3

ð1  CoÞ  i  Co2 1  Co 3

ð9Þ

Fig. 1 shows the changes in the correction factor as compared to the overlap coefficient. Note that Fc = 3/2 for Co = 0 (i.e. particles in point contact), and Fc tends toward 1 (i.e. the chains of particles can be considered as fibre) when Co tends toward 1. The pressure drop of a nanostructured deposit is equal to:

DP ¼ F T LF Fc or DP ¼

64a

ð10Þ  0:5

1 þ 56a

 3

ag Uf Z Fc

2

Ccdp

ð11Þ

The packing density can be expressed as a function of cake mass per surface area (mS) as follows:

a¼

mS Z qp

ð12Þ

where Z is the deposit thickness and qp, the material density. Combining Eqs. (9), (11) and (12) we obtain:

DP ¼

  64a0:5 1 þ 56a3 2 Ccdp

qP

h

2 3

ð1  CoÞ  i gmS Uf  Co2 1  Co 3

ð14Þ

3. Materials and method

ðdp  dÞ dp

Fc ¼ h

h i 2 dp ¼ dpG exp 1:5ln rg

ð7Þ

with the overlap parameter Co defined by Brasil et al. [19] as follows:

Co ¼

where dp is the diameter of average mass, i.e. the particle diameter whose mass multiplied by the total number of particles gives the total mass. If the particle size distribution follows a log normal function, the diameter of average mass can be calculated by the Hatch Coate equation [20] from the count median diameter (dpG) and the geometric standard deviation (rg):

ð13Þ

The experimental setup (see Fig. 2) includes two different sources of carbon nanoparticle aggregates: a ‘‘spark discharge’’ generator (PALAS GFG 1000) with carbon electrodes and a combustion aerosol generator using propane as fuel (miniCAST 5201). A filter holder is used at the outlet of these aerosol generators. The filtration flow-rate is controlled by a mass flow-rate controller (MFC Brooks 5850) and the pressure drop is measured with a differential pressure transducer (Wöhler DC2000 Pro). An agglomeration chamber was also used for the miniCAST in order to increase the size of the soot particles without any change in their chemical properties and primary particle diameter. Depending on aerosol source and experimental configuration, the number concentration of particles and time duration of deposit ranged from 4  105 part/ cm3 to 9  106 part/cm3 and 5 min to 26 h, respectively (experimental conditions are reported in appendix Tables C.1 and C.2). The size and morphology of generated particles were investigated using a scanning mobility particle sizer (TSI 3936) and transmission electron microscope (TEM). Particles produced by these generators are structured as agglomerates (PALAS GFG 1000) or aggregates (miniCAST), with their fractal dimensions calculated according to the approach described in Ouf et al. [21]. By definition, an agglomerate is composed of primary particles bonded together by weak, Van der Waals type forces and exhibiting a low degree of overlap (Co  0). Conversely, an aggregate is composed of strongly bonded primary particles (covalent type bonds) with a high degree of overlap. The overlap coefficient Co, previously introduced, was then computed on TEM (JEOL 100CXII for miniCAST soot particles), and HR-TEM (JEOL JEM 2011 for GFG carbon nanoparticles) micrographs according to Brasil et al. [19]. Nevertheless, for GFG 1000 and in agreement with Wentzel et al. [22], we were not able to identify a significant overlap between monomers, and the overlap coefficient Co was assumed to be 0. At the same time as the Co computation, the count median diameter of the monomers and the fractal dimension of the aggregates/agglomerates were measured using Image-J and a Matlab software [21]. According to the work of Brasil et al. [19], the projected overlap coefficient Co; p may be converted to the real, three-dimensional, overlap coefficient Co, taking into account the bias induced by an additional monomer superposition due to the 2D projection:

Fig. 2. Experimental setup.

147

D. Thomas et al. / Separation and Purification Technology 138 (2014) 144–152 Table 1 Properties of nanoparticles produced in this study. Aerosol generator

a b c

GFG 1000

miniCAST

Configuration

Frequency: 500 u.a.

Frequency: 999 u.a

No chamber

Filtration velocity Uf (m/s) Count median diameter of the primary particle (nm) Geometric standard deviation rg pp Diameter of average mass of the primary particle (nm) Count mode aggregate or agglomerate diameter (nm) Geometric standard deviation rg ag Overlap coefficient (Co) Fractal dimension Particle density (kg/m3) Organic fraction (%)

0.01

0.01

0.09

0.01–0.05–0.09

54 1.6

62 1.6

91 1.6

0.09 6.7 1.20 7

48 1.5

49 1.5 0a 1.8 2000a <5c

Agglomeration chamber 0.09 23.3 1.15 24 170 1.7 0.12 ± 0.05 1.7 1745b 7

From Wentzel et al. [22]. Mean value from Park et al. [24] for diesel and Ouf et al. [7] for acetylene. From Su et al. [25] for GFG 1000 and thermo-optical measurement for miniCAST.

Co ¼ f1  Co; p  f2

ð15Þ

where Co; p is the mean 2D projected overlap coefficient, f1 and f2 are proposed empirical constants (f1 = 1.1 ± 0.1 and f2 = 0.2 ± 0.02 for Df  1.78, [19]). It should be noted that the knowledge of the fractal dimension is of major interest, since the Brasil et al. [19] approach is limited to particles with a fractal dimension close to 1.7–1.8. Depending on the carbon-to-oxygen ratio, particles produced by the miniCAST could be mainly composed of carbon in elemental form (elemental carbon EC) or in organic form (OC for organic carbon). The EC/OC ratio of the soot generated by the miniCAST was characterized using a thermo-optical method (Sunset Lab, Improve protocol), and the organic carbon content in the present case is lower than 7%. Table 1 presents the characteristics of particles used in this study and indicated a good agreement between the count median diameter measured in this study for carbon primary particles generated by PALAS GFG 1000 and the results reported by Wentzel et al. [22] and Charvet et al. [23]. Finally, PTFE membranes (Millipore FSLW04700 with 3.0 lm pore size) were used as filtration media since the penetration of particles inside this media is limited and facilitates the specific study of cake pressure drop and microstructure. One of the main parameters determining the aeraulic resistance of a cake composed of particles is the packing density (a) or the porosity (e = 1  a). The porosity represents the ratio between the volume of void in the cake and the total volume of the cake (i.e. volume occupied by the material and the void inside the cake). Using changes in cake thickness Z as a function of the cake mass per surface area (mS = Mc/X), it is possible to determine the mean porosity of the deposit from Eq. (16), assuming that there is no cake compression:

dZ 1 ¼ dmS qp  ð1  eÞ

investigated surface. For this item, a specific membrane holder, comprising a porous medium and a pump, was developed and used to measure the roughness of a blank PTFE membrane and to keep samples as horizontal as possible during the analysis. Even for this type of membrane, which could be assumed to be a reference flat and simple surface, a mean roughness of 9.2 lm was measured. In our study this will represent the lowest cake thickness that we could characterize. The measurement of the cake thickness is based on the difference of the mean thickness of the clogged membrane in the cake zone and the mean thickness of the blank membrane zone. In order to reduce uncertainty due to the fluctuation of cake thickness on the membrane surface, the deposit thickness is the mean of at least 6 measurements performed at four different locations of the membrane (two orthogonally opposite pairs). Standard deviations have been calculated for each mean value, and Tables C.1 and C.2 of the appendix present these values. In general, the standard deviation is always lower than 11 lm, and the mean standard deviation is 20% of the mean thickness. Fig. 3 presents the changes in cake thickness Z as a function of the deposited mass per surface area mS for each filtration velocity and nanoparticle sources. A linear regression is then applied to the experimental results. It should be noted that the reported regressions are not forced to equal 0 since the membrane used is not a perfectly smooth surface. Consequently, the reported ordinates are in agreement with the roughness of the membrane (9.2 lm) considering its associated standard deviation (6.8 lm for a confidence interval of 95%).

ð16Þ

using Mc the cake mass, Z the thickness, X the filtering area and qp (kg/m3) the particles density. The deposited mass of nanoparticles (Mc, kg) was measured using a weighing cell (Mettler Toledo AE 240) with a resolution of 0.01 mg. The filtration surface area (X, m2) was manually measured with an uncertainty of 5%. Finally, the most challenging measurement remains the determination of the cake thickness (Z). Recently, Kim et al. [5] demonstrated that an optical measurement of this parameter is possible even for cakes composed of nanoparticles. As a consequence we have used a focusvariation surface metrology system (InfiniteFocus ALICONA). This type of commercial device has a very good resolution in terms of height measurement (close to 1 lm); nevertheless, this good resolution could be limited by the roughness and the planarity of the

Fig. 3. Changes in cake thickness as a function of the deposited mass per surface area.

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D. Thomas et al. / Separation and Purification Technology 138 (2014) 144–152

Table 2 Experimental cake porosity results. Aerosol generator

miniCAST

GFG 1000

Configuration

No chamber

Agglomeration chambers

Frequency: 500 u.a.

Frequency: 999 u.a.

Filtration velocity Uf (m/s) Mean cake porosity Uncertainty Aggregate or agglomerate count median diameter (nm) Pe

0.01 0.973 0.002

0.09 0.946 0.020 170 52.9

0.01 0.980 0.007 48 0.2

0.01 0.984 0.020 54 0.3

1.1

0.05 0.950 0.003 91 5.7

0.940 0.002 10.0

Furthermore, the rather good agreement reported between the fit and the experimental data shows that cake porosity does not change as a function of mass. Therefore no compression of the cake during its formation is noted. The slope of the linear regression and its associated uncertainty were calculated taking into account the uncertainty associated with each experimental pair (mS, Z). Further details are available in Ouf and Sillon [26]. 4. Results and discussion 4.1. Cake porosity Table 2 shows the experimental cake porosities obtained with miniCAST and Palas GFG 1000 generators (experimental results are also available in Appendix C). These values are close to the porosities of cakes composed of nanoparticle aggregates and reported in the literature. According to Kim et al. [5], the porosity of flame-generated particle loading is equal to 0.95. More recently, Liu et al. [11] found porosity values equal to 0.953–0.961 for cake formed during the filtration of diesel soot particles. Beyond this good agreement, our experimental results also confirmed the decrease of average porosity as filtration velocity increases, as was already observed by Liu et al. [11] (e = 0.961, 0.955 and 0.953 for 5, 10 and 20 cm/s, respectively). According to Liu et al. [11], the Pe number would be a good predictor of porosity for nanoparticle aggregate cakes. Mädler et al. [10] also reported a relationship between Pe and porosity for nanoparticle agglomerate films. They explained this trend as follows: for small Peclet numbers, diffusion is dominant; therefore, an interception of the nanoparticle agglomerate with the cake takes place close to the deposit surface, forming cake with a porosity higher than 0.97. For large Peclet numbers, the aggregate penetrates more effectively into the cake because of its higher velocity,

Fig. 4. Deposit porosity versus Peclet number (present study, [5,11,27]).

0.09 0.957 0.005 49 1.8

0.09 0.966 0.020 62 3.4

which tends to form cake with porosities close to 0.94. Fig. 4 shows the changes in the cake porosity as a function of Pe number, which is defined as follows:

Pe ¼

Ufdagg D

ð17Þ

where Uf is the filtration face velocity, dagg, the aggregated or agglomerated particle size and D, the particle diffusion coefficient (calculated with dagg according to the Stokes–Einstein equation). The experimental values obtained by Liu et al. [11] and Kim et al. [5], with diesel particles (dagg = 63.8 nm) and soot agglomerates (dagg = 120 nm), respectively, are displayed on this figure. A good agreement can be observed with our results. Furthermore, for predictive reasons and in keeping with Mädler et al. [10], the continuous line proposed in Fig. 4 is the best-fit line of fitted values given by the following equation:

¼

1 þ 0:44Pe 1:019 þ 0:46Pe

ð18Þ

4.2. Pressure drop evolution Figs. 5 and 6 show the experimental and theoretical changes in pressure drop as a function of cake mass per unit area. For the theoretical calculation, parameters used in the model are summarized in Tables 1 and 2. For both nanoparticle sources, the agreement between the experimental results and our model is good. Considering porosities reported in Table 2, the model is a fairly good fit of the results even for filtration velocities ranging from 0.01 m/s to 0.09 m/s. In addition to the good agreement between the new model and the experimental results, in Fig. 7 we present a comparison between the pressure drop, normalized by the filtration velocity, computed using the Endo model associated with the void function proposed by Liu et al. [11] as a function of experimental results.

Fig. 5. DP/Uf versus ms, experimental and predicted results for GFG.

D. Thomas et al. / Separation and Purification Technology 138 (2014) 144–152

Fig. 6. DP/Uf versus ms, experimental and predicted results for miniCAST.

149

[11]. Since the overlapping coefficient is difficult to estimate from the SEM images published by the authors, we used a Co of 0.15 in keeping with Wentzel et al. [22]. The porosity reported by Liu et al. [11] is 0.955, and the mass median diameter of the primary particle is equal to 44.9 nm (with geometric standard deviation = 1.26). According to the Hatch–Coate equations [20], the diameter of average mass is equal to 41.4 nm. The 95% confidence intervals, presented in Figs. 7 and 8, corresponds to the interval around the perfect agreement (Model = Experiment) in which 95% of experimental results are located. For the Endo and Liu model, this interval represents a variation of ±36% around the perfect agreement and in most of the cases Endo’s model underestimates the real pressure drop. On the other hand, our model has a 95% confidence interval corresponding to a variation of ±25% around the perfect agreement. This variation is fully explained by the contribution of the uncertainties associated with the model’s parameters (dp, Fc, a, Uf, mS) to the overall uncertainty of the calculated pressure drop. This uncertainty has been estimated (see Appendix B for details) for all of our experimental results as equal to a mean value of 27%, a value close to the 25% of variation associated with the 95% confidence interval. A fairly good agreement is reported between our model and experimental results. Moreover, for Liu et al. [11] and in contrast with the Endo and Liu model, no assumption has been made concerning the impact of organic fraction on the density of diesel soot particles, and we used a density of 1745 kg/m3 for these particles in our model. The experiments and pressure drop model suggest that cake porosity is determined by the agglomerate or aggregate diameters, whereas the pressure drop of the nanostructured deposit is linked to particle primary diameters and overlap coefficient. It can be assumed that cake porosity is the result of two porosities: the inter-aggregate (or agglomerate) porosity and the intra-aggregate (or agglomerate) porosity. In order to dissociate these two contributions to cake porosity, further experiments will be necessary.

5. Conclusion

Fig. 7. DP/Uf theoretical versus DP/Uf experimental (Endo and Liu model).

Fig. 8. DP/Uf theoretical versus DP/Uf experimental (our model).

Fig. 8 also presents the same comparison but according to our new model. In order to extend the validation of the pressure drop model, we also test it with the experimental measurements of Liu et al.

Loading tests using carbon nanoparticle aggregates and agglomerates were performed on a PTFE membrane. Particular care was taken to characterize the cake porosity and the nanostructured particles. The aggregate and agglomerate count median mobility diameters ranged, respectively, from 91 nm to 170 nm and from 48 nm to 62 nm. These nanostructured particles were composed by primary diameters with a diameter of average mass equal to 7 nm for agglomerates and 24 nm for aggregates. The porosity values of the nanostructured deposit formed by carbon nanoparticle aggregates and agglomerates determined in this study (0.94– 0.984) confirm those reported in literature. All porosity values (this study and literature) are strongly correlated with the Peclet number of the aggregates or agglomerates in the range 0.19–53 for filtration velocities of 0.01, 0.05 and 0.09 m/s. Finally, an empirical relationship has been proposed between porosity and Pe, which can be used to predict this parameter with a reasonable agreement. A new pressure drop model has been developed for a nanostructured deposit. In this model the nanostructured deposit can be considered as a tangle of chains composed by juxtaposed particles (agglomerate) or with partial overlapping (aggregates). For the nanostructured particles studied in this paper or published, the agreement between experimental results and the theoretical model is fairly good. 95% of experimental values are within the range of plus or minus 25% of the theoretical value. This study has also made it clear that cake porosity is determined by the agglomerate or aggregate diameters, whereas the pressure drop of the nanostructured deposit is linked to particle primary diameter and overlap coefficient.

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D. Thomas et al. / Separation and Purification Technology 138 (2014) 144–152

With the overlap parameter (Co) defined by [19] as follows:

Acknowledgement This work is a part of the LIMA joint research program (The Interactions Media-Aerosol Laboratory) between the Institut de Radioprotection et de Sûreté Nucléaire (IRSN) and the Reactions and Chemical Engineering Laboratory (LRGP) of the French National Centre for Scientific Research (CNRS). Special thanks to the graduate students who worked on this subject: Phuangphet VIBHATAVATA (2009), Wenxin SUN (2010), Ying YU (2011), Ahmed KACEM (2012). Appendix A

Co ¼ ðdp  dÞ=dp

ðA6Þ

From Eqs. (A4), (A5), (A6), Eq. (A3) becomes

LC ¼

6aZ

ðA7Þ

pd2p

for juxtaposed particles

LC ¼

4aZð1  CoÞ h  i pd2p 23  Co2 1  Co3

ðA8Þ

for particles with a partial overlapping.

The total length of the chains per deposit surface area is given by

LC ¼ NZd

ðA1Þ

or

a ¼ N  V pp

ðA2Þ

where N is the number of primary particles per cubic meter of cake; Vpp, the volume of the primary particle; d, the distance between the centres of the particles (d = dp for juxtaposed particles) and Z, the cake thickness. From Eqs. (A1) and (A2), we obtain

LC ¼

aZd

V pp ¼

p

6

pd3p 2 4

  Co  Co2 1  3 3

ðB1Þ

where

ðB3Þ

ðA4Þ

  @ DP 64a0:5 1 þ 56a3 ¼ gmS Uf 2 @Fc Cudp qP

ðA5Þ

  h i   cd  2dp þ 2ak þ 2b exp  2kp k  cdp 64a0:5 1 þ 56a3 @ DP ¼ FcgmS Uf  2 @dp qP dp2 Cc

For particles with partial overlapping, Vpp is given by

V pp ¼

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2  2  2  2  2 @ DP @ DP @ DP @ DP @ DP rDP ¼ r2a þ r2dp þ r2Fc þ r2mS þ r2Uf @a @dp @Fc @mS @Uf

ðB2Þ

For particles without partial overlapping, Vpp is given by 3 dp

The uncertainty associated with the pressure drop calculated according to this model was estimated according to the following relationship:

  @ DP 32a0:5 1 þ 392a3 ¼ FcgmS Uf 2 @a Cudp qP

ðA3Þ

V pp

Appendix B

ðB4Þ

Table C.1 Experimental data obtained for the MiniCAST burner. Source

Frequency (u.a)

Uf (m/s)

Duration (min)

ms (g/m2)

DP (Pa)

Mean thickness Z (lm) (standard deviation (lm))

CAST

Without agglomeration chamber

0.01

30 60 90 120 150 – 300 – 31 60 90 120 180 240 5 10 10 20 25 30 50 90 90 90 180 60 120 180 300 – –

0.37 0.72 1.08 1.76 1.95 1.85 3.23 1.55 1.07 1.61 2.08 5.38 6.76 9.13 0.52 1.14 0.94 1.66 2.87 3.24 5.57 9.50 6.36 5.97 16.75 0.48 1.04 1.69 2.97 1.60 3.56

109 205 303 512 558 525 985 540 2068 3290 4253 8874 13,213 18,193 1621 3658 2918 5766 9248 10,756 19,160 33,049 22,938 22,340 65,019 1666 3584 5734 9854 5031 13,295

20 ± 5 27 ± 5 41 ± 8 56 ± 11 57 ± 10 49 ± 4 82 ± 6 54 ± 7 24 ± 6 36 ± 6 46 ± 11 83 ± 7 93 ± 16 116 ± 8 18 ± 5 28 ± 6 23 ± 8 37 ± 7 49 ± 7 51 ± 7 73 ± 6 111 ± 12 84 ± 16 77 ± 7 173 ± 7 13 ± 7 26 ± 5 30 ± 9 41 ± 8 45 ± 8 50 ± 8

0.05

0.09

With ageing chambers

0.09

D. Thomas et al. / Separation and Purification Technology 138 (2014) 144–152

151

Table C.2 Experimental data obtained for the GFG-1000 generator. Source

Frequency (u.a.)

Uf (m/s)

Duration (min)

ms (g/m2)

DP (Pa)

Mean thickness Z (lm) (standard deviation (lm))

GFG-1000 1 bar argon 1.5 bar air

500

0.01

30 60 405 930 1200 1560 6 10 22 54 75 95 105 120 145 195 30 120 253 540 930 5 13 30 45 65 90 120

0.07 0.10 4.99 9.67 1.28 1.56 0.10 0.19 0.26 0.62 0.84 1.04 1.16 1.35 1.53 1.91 0.10 0.29 0.57 0.98 1.46 1.97 0.10 0.26 0.62 0.97 1.26 1.76 2.40

41 70 331 646 1007 1261 664 1084 2399 5390 6871 8848 9989 11,162 12,618 16,762 74 263 552 649 940 1133 547 2618 5197 7690 11,558 16,081 21,887

7±2 15 ± 8 20 ± 4 43 ± 6 33 ± 5 43 ± 6 13 ± 5 17 ± 4 15 ± 7 22 ± 5 26 ± 5 34 ± 6 35 ± 7 39 ± 7 29 ± 3 37 ± 5 15 ± 4 20 ± 5 24 ± 4 37 ± 7 54 ± 10 76 ± 7 8±4 14 ± 7 15 ± 10 20 ± 4 27 ± 11 30 ± 6 46 ± 10

0.09

999

0.01

0.09

  @ DP 64a0:5 1 þ 56a3 ¼ FcgUf 2 @mS Cudp qP

ðB5Þ

  @ DP 64a0:5 1 þ 56a3 ¼ FcgmS 2 @Uf Cudp qP

ðB6Þ

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