- Email: [email protected]
New filterability and compressibility test cell design for nuclear products
Nuclear Engineering and Design 265 (2013) 288–293
Contents lists available at ScienceDirect
Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes
New filterability and compressibility test cell design for nuclear products J.P. Féraud a , D. Bourcier a,∗ , D. Ode a , F. Puel b,c a b c
CEA Marcoule, DTEC/SGCS/LGCI, BP 17171, 30207 Bagnols-sur-Cèze, France Université Lyon 1, Villeurbanne, France CNRS, UMR5007, Laboratoire d‘Automatique et de Génie des Procédés (LAGEP), CPE-Lyon, 43 bd du 11 Novembre 1918, 69100 Villeurbanne, France
h i g h l i g h t s • • • • •
Test easily usable without tools in a glove box. The test minimizes the slurry volume necessary for this type of study. The test characterizes the flow resistance in a porous medium in formation. The test is performed at four pressure levels to determine the compressibility. The technical design ensures reproducible flow resistance measurements.
a r t i c l e
i n f o
Article history: Received 4 April 2013 Received in revised form 10 June 2013 Accepted 10 June 2013
a b s t r a c t Filterability and compressibility tests are often carried out at laboratory scale to obtain data required to scale up solid/liquid separation processes. Current technologies, applied with a constant pressure drop, enable specific resistance and cake formation rate measurement in accordance with a modified Darcy’s law. The new test cell design described in this paper is easily usable without tools in a glove box and minimizes the slurry volume necessary for this type of study. This is an advantage for investigating toxic and hazardous products such as radioactive materials. Uranium oxalate precipitate slurries were used to test and validate this new cell. In order to reduce the test cell volume, a statistical approach was applied on 8 results obtained with cylindrical test cells of 1.8 cm and 3 cm in diameter. Wall effects can therefore be ignored despite the small filtration cell diameter, allowing tests to be performed with only about one-tenth of the slurry volume of a standard commercial cell. The significant reduction in the size of this experimental device does not alter the consistency of filtration data which may be used in the design of industrial equipment. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Nuclear fuel reprocessing with actinide oxalates requires a filtration step between precipitation and calcining. In this case, liquid/solid separation should be satisfactory with pure uranium or plutonium oxalates. A new uranium/plutonium coprecipitate now being investigated is composed of needle-shaped aggregates (ArabChapelet et al., 2007; Arab-Chapelet et al., 2008). This material exhibits poor filterability because of the particle shape and agglomerate structure. After coprecipitation (Charton et al., 2012) (Borda et al., 2011), the filtration step leaves a porous cake.
∗ Corresponding author at: CEA Marcoule, 30207 Bagnols-sur-Cèze, France. Tel.: +33 466 791 496. E-mail address: [email protected] (D. Bourcier). 0029-5493/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nucengdes.2013.06.036
Evaluating the mechanical behavior of the cake during solid/liquid separation is compulsory for orienting technological options toward devices suitable for the solid product (Tarleton and Wakeman, 2007a,b). The evaluation consists in measuring the cake flow resistance and compressibility at small scale. The flow resistance is one of the main important parameters in solid/liquid separation. It defines the difficulty of the operation, and can be measured using the filtration device presented here. Porosity and compressibility result in a friable cake, and the residual liquid captured in it cannot be easily eliminated by draining because of the creation of preferential pathways. To improve the dryness of the final product, studies have been done on depth filtration theories and on cake filtration, in order to better understand and master the filtration operation. In this study, the valuable product is the solid phase of the suspension. The technological selection criterion should ultimately
J.P. Féraud et al. / Nuclear Engineering and Design 265 (2013) 288–293
The Kozeny–Carman model can determine the value of this resistance based on experimental data (Léger, 2008). It considers the cake as incompressible, consisting of spherical particles. The value of the resistance comes from an identification of Darcy’s law in filtration:
Nomenclature d dp De kB kDL kH l n dp, P PT PG pl ps pwall Rs t T U U0 V w ˛ ˛0 ε εs ϕ
ф s g ˝
capillary diameter (m) particle diameter (m) equivalent sphere diameter (m) brinkmann’s permeability (m2 ) Happel’s corrective parameter (m2 ) Happel’s permeability (m2 ) capillary length = T·L (m) compressibility coefficient pressure drop (Pa) total pressure drop (Pa) cake pressure drop (Pa) pressure due to liquid phase (Pa) pressure due to solid phase (Pa) pressure due to walls (Pa) filter resistance time (s) tortuosity (>1) flow rate of associated liquid (m3 /s) flow rate of associated liquid without solid (m3 /s) filtrate volume mass of dry cake per unit area of filter (kg/m2 ) specific cake resistance (m/kg) reference specific cake resistance (m/kg) porosity solidosity sphericity factor solid volume fraction fluid viscosity (Pa s) density of solid (kg/m3 ) density of cake (kg/m3 ) filter area (m2 )
U0 = −
˛ = 200
(2)
(1 − ε) • 1 ε3 ϕ2 · De2 · s
(3)
This relation, in a constant-pressure filtration process, leads to: w Rs ˛ t · V+ · = V 2PT ˝2 PT ˝
(4)
Eq. (4) is the basis of the entire filterability test. All the parameter values can be determined experimentally, and ˛ can then be extracted. The pressure effect on filter cake properties is given by many empirical equations describing the variation of ˛, such as the power law (5) (Tarleton and Wakeman, 2007a,b): ˛ = ˛0 · P n
(5)
Relation (5) can be linearized by a logarithmic approach. The compressibility coefficient, n, can be deduced and gives decisive information on the cake. The test presented in this study is based on the Kozeny–Carman model (Carman, 1937) since it is the simplest and its application is consistent with our assumptions. The Kozeny–Carman model requires a porosity lower than 0.7. For higher porosity, several models have been developed:
2. Theoretical approach 2.1. Standard filterability laws Considering the cake as a solid block (Wakeman and Vince, 1984), crossed by circular capillaries of the same diameter, d, the average velocity of a fluid moving in laminar flow in this medium is given by Poiseuille’s equation: ·U·l d2
1 dp · ˛ dw
known as the modified Darcy’s Law. Combining these equations, and using the hypothesis of the Kozeny–Carman model, the specific cake resistance can be given by:
enhance the dry solid content of the cake. Decision support in this area is guided in particular by measuring two key parameters; flow resistance and compressibility (Tarleton and Wakeman, 2007a,b). For technological choice this information is supplemented by the need for a scrubbing step. The temperature sensitivity of the product must also be taken into account among other aspects. The new filterability–compressibility test system presented here can measure both of these parameter values. The analytical method is based on a simplified adaptation of the AFNOR standard (AFNOR, 2006a,b) to nuclear requirements but can be adapted for a wide range of suspensions. The proposed simplification is to limit the size of the device and use an automatic system of straightforward design. Particular attention was paid to reducing the quantity of nuclear material necessary for the test.
P = 32
289
(1)
This law comes from the Navier–Stokes equation, with various assumptions: laminar and stationary flow, incompressible and Newtonian fluid.
• Brinkman’s model (Brinkman, 1947) considers the flow around spheres, and gives an empirical relation of permeability for high porosities:
KB = KDL
3 1+ 4
1−
8 −3
where KDL =
2d2/p 9
(6)
• Happel and Kubawara’s model (Benmachou, 2005) describes the porous medium as cells (Happel, 1959) with a free surface. It requires porosity higher than 0.6 (Kuwabara, 1959). KH = KDL
6 − 91/2 + 95/2 − 62 6 + 45/2
(7)
• Others models (Couturier, 2002), such as those proposed by Neale and Nader or by Howells and Hinch, exist for very high porosities above 0.9. However, in all these models, particles are considered as spherical. Models by Davies or Jackson and James describe the effect on permeability if the particle is considered as anisotropic (Ragueh, 2011). 2.2. Principles of cake formation, simulation possibilities 2.2.1. Cake growth and pressure During the cake growth, local properties fluctuate. The cake porosity, saturation, and particle size distribution are not homogeneous (Abboud, 1996), but the total pressure in the cake remains constant (Tien et al., 2001). The total pressure is equal to the sum of the liquid and solid pressure. This results in the following expression: dpl + dps = 0
(8)
290
J.P. Féraud et al. / Nuclear Engineering and Design 265 (2013) 288–293
Two variations of this equation have been proposed (Bai and Tien, 2005): dpl =y dps
(9)
where y stands for: y = −1 (type 1)
(10)
y=
−1 (type 2) (1 − εs )
(11)
y=
−εs (type 3) (1 − εs )
(12)
Each suspension better corresponds to one of the three equations; using the correct one will be helpful to better describe cake buildup and cake properties. 2.3. Cake collapse If the cake is compressible, rearrangements may occur if pressure is applied (Antelmi et al., 2001; Cabane et al., 2002). This phenomenon, called cake collapse, has various consequences including increased resistance and decreased porosity and thickness. The mechanism by which cakes collapse involves very small motions of particles in contact with one another. In some cases these motions leave the small-scale coordination unchanged, and in other cases the organization of particle–particle contact within the aggregates in the cake can be slightly modified. 2.4. Simulation The cake buildup may be simulated in two ways. Burger et al. (2001) has proposed an iterative approach in which the last layer formed is considered as the filtering medium for the next layer. Ni et al. (2006) describe a physical approach. A force balance is calculated for the particle, with short-range interactions such as the Van der Walls interaction, or electronic interactions. This approach is limited by the number of particles taken into consideration. All simulations are done on a theoretical suspension, and the models were not applied to an industrial facility (Tien, 2006). This is the real limitation of simulation, but it gives an approach to reality and to the difficulty of the filtration step. 2.4.1. Empirical relations for determination of cake resistance and variations Empirical relations between fluid velocity and cake resistance, and between fluid velocity and porosity, have been established (Ergun, 1952). These relations follow an a·Ub law, where the values of a and b fluctuate with the cake formation step. Other studies (Saleem and Krammer, 2007; Saleem et al., 2012) have shown that the cake resistance increases with velocity and pressure, and that the cake resistance decreases as the cake ages.
capillaries. These simplifications are associated with the Bernoulli law of conservation of energy of a liquid. Performing this test on a suspension, however, implies continual changes in the thickness L of the porous medium since the cake is formed as the test progresses. Two approaches can be adopted in this case. The test can be performed at a constant flow rate (Iritani et al., 2007) or at constant pressure. Working at constant pressure is particularly interesting here because it allows us to control the permeability of the porous medium, and thus investigate the compressibility of the cake versus the pressure under satisfactory conditions (Sorensen et al., 1996). The flow resistance ˛ (or Rg ) of the porous medium is a mean resistance value and cannot be directly compared with the resistance of the filter medium Rs . The two parameters do not have the same dimensions. In practice, the experimental curve of t/V versus V is a straight line whose slope can be used to determine ˛ and whose y-intercept can be used to calculate the resistance of the filter medium Rs . More information is given by Lee and Wang (2000) in a brief review of theories in the literature concerning transport processes in filter cakes during filtration and consolidation. 2.6. Wall effects Ergun and Orning (Ergun, 1952) developed a general equation relating pressure drop to fluid flow for fixed beds consisting of spheres: Pg 1 (1 − ε)2 (1 − ε) g · U 2 = 150 · U 2 · + 1.75 · L dp ε3 ε3 dp
(13)
In order to predict wall effects, Mehta and Hawley (1969) decided to modify the Ergun equation. It has been shown that if the column-to-particle diameter ratio is less than 50:1, wall effects have to be considered. As the present test lies outside this range of diameters, wall effects may be ignored. Another study (Tiller, Lu et al., 1972) demonstrated that compressible beds of small particles are strongly affected by the walls when the tests are carried out in compression-permeability cells. Wall effects are introduced as a new force in the pressure equation (9), modified as follows: ps + pl + p(wall) = papp
(14)
The effect of different materials of construction has also been studied. The PTFE-coated cylinder generates less friction than a stainless steel cylinder (Tiller et al., 1972). It also gives empirical relations for porosity and specific cake resistance as function of pressure, taking wall effects into consideration. Numerical simulations have been done to better understand the consolidation of the cake. They show that sidewall friction creates a region of lower compressive stresses and pore pressure, with a higher void ratio (Zhao et al., 2003). 3. Materials, experimental setup and operating conditions 3.1. Model substance
2.5. Filterability test The filterability test characterizes the flow resistance of a liquid in a porous medium (cake) in formation. The test is performed at four pressure levels to determine the compressibility or deformability of the solid. The description of this flow by the laws of physics is based on three initial assumptions: (i) the porous medium, or cake, is the result of a pileup of beads; (ii) the porous medium is nondeformable with pressure (implying constant permeability); (iii) the filtrate flow in the cake is laminar. In this case, the cake is associated with a pileup of N beads and/or an arrangement of n
The product investigated here is a mixed compound of U/Ce oxalate with a bulk relative density close to 1.3 and a particle size distribution centered on 20 m. This material is used to simulate reprocessing of the next generation of spent nuclear fuel. The solid is dispersed in mother liquors at a concentration of 5 g L−1 . The chemical composition is representative of industrial solutions. Mother liquors are aqueous nitric acid (<2 M), in these chemical conditions, viscosity is assumed to be the same as water, 10−3 Pa s. This product is obtained by precipitation route, at room temperature.
J.P. Féraud et al. / Nuclear Engineering and Design 265 (2013) 288–293
291
Fig. 1. Experimental setup.
3.2. Experimental setup The filterability test is based on experiments described by Roorda and van der Graaf (2005) and Couturier et al. (2007). The experimental setup designed by CEA is shown in Fig. 1. The technical design ensures reproducible flow resistance measurements. (i) The slurry tank guarantees the homogeneity of the suspension by preventing settling or ranking of the particles before the filterability test. It is equipped with a helical impeller mixer to limit shearing of the precipitate. (ii) The peristaltic pump is an Ismatec ISM 897® or Heidolph 5206® , depending on the pressure range, with a Eurotherm® 4 − 20 mA controller. This option avoids excessive flow pulsation and significantly limits shearing of the precipitate. (iii) The pressure sensor is a Wika S11® flushmounted PTFE diaphragm to prevent solid buildup on the sensor. The 4–20 mA output is connected to one of the analog inputs of the Eurotherm® controller. (iv) The controller is a Eurotherm 2408® with two analog inputs to recover 4–20 mA signals from the pressure switch and the balance, an analog output for PID control of the peristaltic pump, and an RS232 serial port to recover the environmental variables from a PC running Itools® software. (v) Valve V1 is a 3-way interconnected valve allowing fast reconfiguration from recirculation to filtration/measurement flow. Valve V2 is used to obtain the desired pressure. Finally the filter holder is a Gantois REPS® mesh with a 14 m cutoff threshold. The optimization of this component is detailed in Fig. 1.
Fig. 2. Filterability at 600 mbar.
measurement on the last 60 cm3 of filtrate. In these conditions, enough data are provided to accurately measure the t/V line slope and calculate the filterability. The cake obtained has a mean height of 1 cm. Figs. 2 and 3 show that the cake is high enough to allow cake specific resistance measurements. The average dryness is about 0.5.
4. Experimental results 4.1. Specific flow resistance and compressibility method Before investigating the effect of the filter holder diameter, the proper behavior of the pressure regulation loop at different values must be verified. Two experimental results at different pressures are shown in Figs. 2 and 3. The circled lines indicate the valve V1 opening and the PID control perturbation around the pressure value. The double arrows show the start of the filterability test. The black line is the filterability curve. Pressure oscillations always appear below 20 cm3 of filtrate volume. After that, a straight line pressure allowed a filterability
Fig. 3. Filterability at 1000 mbar.
292
J.P. Féraud et al. / Nuclear Engineering and Design 265 (2013) 288–293
Fig. 5. Cake compressibility of U/Ce oxalate between 300 and 1300 mbar. Fig. 4. Filterability at 980 mbar: example of cake fissuring.
When cake fissuring occurred during the measurement, the results remain reliable. Fig. 4 shows an example of cake fissuring. After 60 cm3 , fissuring appeared and a pressure drop was noted. This bed crack was monitored and was immediately compensated by the pressure regulation. The data remained consistent since the filterability was easily obtained by considering the first part of the t/V line. The compressibility of nuclear material is obtained easily with filterability values measured at four different pressures in the range of 300 to 1300 mbar. The curve is fitted with a logarithmic law from Eq. (5) (Figs. 5 and 6). The U/Ce oxalate compressibility factor, given by the slope value, is 0.39. This corresponds to a product with moderate compressibility (Leclerc, 1997). This compressibility behavior shows the energy loss due to porosity decreasing with pressure. 4.2. Effect of filter holder diameter The final item to optimize in the technological development stage concerns the smallest volume of suspension allowing a consistent filterability/compressibility test result. Limiting the reagent volume makes it easier to test expensive or hazardous compounds. The filter holder, which is used to measure the height of the cake, must be several centimeters high to facilitate the measurement. The only feature that can be reduced is the diameter of the filter
holder. A reduction in scale is possible to the extent that wall/cake interactions remain negligible. The study described here consisted in measuring the flow resistance of a nuclear material suspension at 600 mbar with filter holders 3 cm and 1.8 cm in diameter. The smaller diameter would allow testing with only one-third of the reagent necessary for the larger one. Four measurements were performed with each filter holder (see Fig. 2), The operating procedure takes into account the possible evolution of the slurry. The filtration tests were carried alternately on the two filter holders. Chronological distribution may highlight two different means among diameters tested. In fact, further statistical test (see Appendix A) shows that the measurements do not reveal any trend or significant effect of the filter holder diameter on the ˛ measurement. Downsizing the filter holder did not affect the consistency of the determination of the specific resistance ˛. Small variations in cake specific resistance measurement may have multiple explanations: little bubbles in pipes, pressure fluctuations, solid concentration fluctuations, bed formation and reorganization under pressure. The flow resistance may also be measured with a compressibility/permeability cell equipped with a piston which squeezes the cake at constant pressure. It is commonly accepted that this design requires a holder diameter of at least 5 cm in order to homogeneously feed the liquor onto the top of the cake during the filtration test. The necessary amount of slurry is about ten times greater than with the smaller filter holder developed in this study. This is why
Fig. 6. Chronological distribution of ␣ measurements according to the diameter d of the filter holder.
J.P. Féraud et al. / Nuclear Engineering and Design 265 (2013) 288–293
this new development is a real advantage when it is necessary to manipulate toxic or hazardous materials. 5. Conclusion
Appendix A. Statistical test A.1. Fisher test The measured standard deviations s1 (d = 3 cm) and s2 (d = 1.8 cm) are estimators of the same standard deviation. Can they be merged? H0 . Yes = Merge standard deviations s* to refine the statistics to = 7 degrees of freedom No = Calculate sd according to the Aspin-Welch equation:
Fobs =
s 2 1
s2
= 0.7 < Fcrit (9.3))
(15)
where Fcrit is the value obtained in the Fisher table with 3 degrees of freedom for a 95% confidence level. For the results shown here, hypothesis H0 is verified and the two standard deviations can therefore be merged according to Eq. (16):
s∗ =
v1 s12 + v2 s22 v1 + v2
(16)
where s* is the merged standard deviation (m kg−1 ), the degree of freedom, 1designates the large filter holder, and 2 the small filter holder. In this case: s1 = 3.16 × 109 m kg−1 and s2 = 3.77 × 109 m kg−1 → s* = 2.47 × 109 m kg−1 A.2. Student test Compare tobs with tcrit to evaluate the following assumptions. H0 .
Rg1 and Rg2 estimate the same mean.
H1 . Rg1 and Rg2 cannot be considered to estimate the same mean. Test:
Rg1 − Rg2 ∗ tobs = < tcrit
s∗
1 n1
Result: n1 = n2 = 4 → tobs = 2.36 < tcrit = 2.45 (Student’s tdistribution for a 95% bilateral confidence level and six degrees of freedom). References
A filterability test system was quickly developed. It allows accurate pressure monitoring and measurement of cake resistance and compressibility for a specific product. This operating mode allows accurate monitoring of both the pressure and t/V versus V. A particular effort was made to miniaturize the filter holder in order to limit the quantity of suspension necessary for the test. Fisher’s test validated the use of a filter holder only 1.8 cm in diameter, reducing the amount of suspension necessary by a factor of 3. The filter holder is the result of a specific development using transparent material to allow a direct indication of the cake height on the integrated graduations, making it unnecessary to dismantle the unit to obtain this value. Further improvement will concern a high-pressure design (7b) with a better glove box design to study the filterability of U/Pu oxalate coprecipitate. Other filter holder diameters would be tested while doing these improvements, in order to investigate wall effect at wider scale (1–5 cm).
H1 .
293
+
1 n2
(17)
Abboud, N.M., 1996. A theoretical study of pore pressure and porosity variations along the mud cake thickness. Comput. Meth. Water. Resour. III, 323–336. AFNOR NF EN 14701-2 (2006a). Characterization of sludges–Filtration properties–Part 2: Determination of the specific resistance to filtration: 14. AFNOR NF EN 14701-3 (2006b). Characterization of sludges–Filtration properties–Part 3: Determination of compressibility: 9. Antelmi, D., Cabane, B., et al., 2001. Cake collapse in pressure filtration. Langmuir 1722, 7137–7144. Arab-Chapelet, B., Grandjean, S., et al., 2007. Synthesis of new mixed actinides oxalates as precursors of actinides oxide solid solutions. J. Alloys Compd. 444, 387–390. Arab-Chapelet, B., Grandjean, S., et al., 2008. Synthesis and characterization of mixed An(IV)An(III) oxalates (An(IV) = Th, NP, U or Pu and An(III) = Pu or Am). J. Nucl. Mater. 373 (1–3), 259–268. Bai, R.B., Tien, C., 2005. Further work on cake filtration analysis. Chem. Eng. Sci. 60 (2), 301–313. Benmachou, K., 2005. Étude et modélisation du colmatage d’un filtre plissé. Borda, G., Brackx, E., et al., 2011. Use of a pulsed column contactor as a continuous oxalate precipitation reactor. Nucl. Eng. Des. 241 (3), 809–814. Brinkman, H.C., 1947. A Calculation of the Viscous Force Exerted by a Flowing Fluid on a Dense Swarm of Particles. Appl. Sci. Res. Sect. A: Mech. Heat Chem. Eng. Math. Methods 1 (1), 27–34. Burger, R., Concha, F., Karlsen, K.H., 2001. Phenomenological model of filtration processes: 1. Cake formation and expression. Chem. Eng. Sci., 4537–4553. Cabane, B., Meireles, M., et al., 2002. Cake collapse in frontal filtration of colloidal aggregates: mechanisms and consequences? Desalination 146 (1-3), 155–161. Carman, P.C., 1937. Fluid flow through granular beds. Chem. Eng. Res. Design 75, S32–S48. Charton, S., Duhamet, J., et al., 2012. Axial dispersion in pulsed disk and doughnut columns: a unified law. Chem. Eng. Sci. 75, 468–477. Couturier, S., 2002. Étude de la déshydratation mécanique assistée thermiquement. Couturier, S., Valat, M., et al., 2007. Enhanced expression of filter cakes using a local thermal supply. Sep. Purif. Technol. 57 (2), 321–328. Ergun, S., 1952. Fluid flow through packed columns. Chem. Eng. Prog. 48 (2), 89–94. Happel, J., 1959. Viscous flow relative to arrays of cylinders. AIChE J. 5, 174–177. Iritani, E., Katagiri, N., et al., 2007. Flux decline behaviors in dead-end microfiltration of activated sludge and its supernatant? J. Membr. Sci. 300 (1-2), 36–44. Kuwabara, S., 1959. The forces experienced by randomly distributed parallel circular cylinders or spheres in a viscous flow at small Reynolds number. J. Phys. Soc. Jpn. 14, 527–532. Lee, D.J., Wang, C.H., 2000. Theories of cake filtration and consolidation and implications to sludge dewatering. Water Res. 34 (1), 1–20. Léger, B. (2008). Étude de l’essorage en vue d’assister thermiquement le procédé, Université Bordeaux 1. Leclerc, D. (1997) Filtration sur support: aspects théoriques Techniques de l’ingénieur: J3501. Mehta, D., Hawley, M.C., 1969. Wall effect in packed columns. Ind. Eng. Chem. Process Design Dev. 8 (2), 280-282. Ni, L.A., Yu, A.B., Lu, G.Q., Howes, T., 2006. Simulation of the cake formation and growth in cake filtration. Minerals Eng. 19, 1084–1097. Ragueh, D.M., 2011. Filtration de silices précipitées: mise en évidence des relations entre propriétés macroscopiques et échelles locales caractéristiques des dépôts. Toulouse 3 Paul Sabatier. Roorda, J.H., van der Graaf, J.H.M., 2005. SUR test used for optimisation of membrane filtration plants treating wastewater effluents? Desalination 179 (1–3), 131–150. Saleem, M., Krammer, G., 2007. Effect of filtration velocity and dust concentration on cake formation and filter operation in a pilot scale jet pulsed bag filter. J. Hazard. Mater. 144 (3), 677–681. Saleem, M., Krammer, G., et al., 2012. Influence of operating parameters on cake formation in pilot scale pulse-jet bag filter. Powder Technol. 224:, 28–35. Sorensen, P.B., Agerbaek, M.L., et al., 1996. Predicting cake filtration using specific filtration flow rate. Water Environ. Res. 68 (7), 1151–1155. Tarleton, S., Wakeman, R. (Eds.), 2007a. Equipment selection and process design. Solid/liquid separation. Elsevier, Kidlington, Oxford, UK. Tarleton, S., Wakeman, R. (Eds.), 2007b. Principles of Industrial Filtration. Elsevier, Kidlington, Oxford, UK. Tien, C., 2006. Introduction to Cake Filtration. Tien, C., Teoh, S.K., et al., 2001. Cake filtration analysis: the effect of the relationship between the pore liquid pressure and the cake compressive stress. Chem. Eng. Sci. 56 (18), 5361–5369. Tiller, F.M., Lu, W.M., et al., 1972. Role of porosity in filtration.7 effect of side-wall friction in compression-permeability cells. AIChE J. 18 (1), 13-&. Wakeman, R. J., Vince A., 1984. Solid–Liquid Separation. London. Zhao, J., Wang, C.H., et al., 2003. Cake consolidation in a compression-permeability cell: effect of side-wall friction. J. Colloid Interface Sci. 262 (1), 60–72.
Contents lists available at ScienceDirect
Nuclear Engineering and Design journal homepage: www.elsevier.com/locate/nucengdes
New filterability and compressibility test cell design for nuclear products J.P. Féraud a , D. Bourcier a,∗ , D. Ode a , F. Puel b,c a b c
CEA Marcoule, DTEC/SGCS/LGCI, BP 17171, 30207 Bagnols-sur-Cèze, France Université Lyon 1, Villeurbanne, France CNRS, UMR5007, Laboratoire d‘Automatique et de Génie des Procédés (LAGEP), CPE-Lyon, 43 bd du 11 Novembre 1918, 69100 Villeurbanne, France
h i g h l i g h t s • • • • •
Test easily usable without tools in a glove box. The test minimizes the slurry volume necessary for this type of study. The test characterizes the flow resistance in a porous medium in formation. The test is performed at four pressure levels to determine the compressibility. The technical design ensures reproducible flow resistance measurements.
a r t i c l e
i n f o
Article history: Received 4 April 2013 Received in revised form 10 June 2013 Accepted 10 June 2013
a b s t r a c t Filterability and compressibility tests are often carried out at laboratory scale to obtain data required to scale up solid/liquid separation processes. Current technologies, applied with a constant pressure drop, enable specific resistance and cake formation rate measurement in accordance with a modified Darcy’s law. The new test cell design described in this paper is easily usable without tools in a glove box and minimizes the slurry volume necessary for this type of study. This is an advantage for investigating toxic and hazardous products such as radioactive materials. Uranium oxalate precipitate slurries were used to test and validate this new cell. In order to reduce the test cell volume, a statistical approach was applied on 8 results obtained with cylindrical test cells of 1.8 cm and 3 cm in diameter. Wall effects can therefore be ignored despite the small filtration cell diameter, allowing tests to be performed with only about one-tenth of the slurry volume of a standard commercial cell. The significant reduction in the size of this experimental device does not alter the consistency of filtration data which may be used in the design of industrial equipment. © 2013 Elsevier B.V. All rights reserved.
1. Introduction Nuclear fuel reprocessing with actinide oxalates requires a filtration step between precipitation and calcining. In this case, liquid/solid separation should be satisfactory with pure uranium or plutonium oxalates. A new uranium/plutonium coprecipitate now being investigated is composed of needle-shaped aggregates (ArabChapelet et al., 2007; Arab-Chapelet et al., 2008). This material exhibits poor filterability because of the particle shape and agglomerate structure. After coprecipitation (Charton et al., 2012) (Borda et al., 2011), the filtration step leaves a porous cake.
∗ Corresponding author at: CEA Marcoule, 30207 Bagnols-sur-Cèze, France. Tel.: +33 466 791 496. E-mail address: [email protected] (D. Bourcier). 0029-5493/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.nucengdes.2013.06.036
Evaluating the mechanical behavior of the cake during solid/liquid separation is compulsory for orienting technological options toward devices suitable for the solid product (Tarleton and Wakeman, 2007a,b). The evaluation consists in measuring the cake flow resistance and compressibility at small scale. The flow resistance is one of the main important parameters in solid/liquid separation. It defines the difficulty of the operation, and can be measured using the filtration device presented here. Porosity and compressibility result in a friable cake, and the residual liquid captured in it cannot be easily eliminated by draining because of the creation of preferential pathways. To improve the dryness of the final product, studies have been done on depth filtration theories and on cake filtration, in order to better understand and master the filtration operation. In this study, the valuable product is the solid phase of the suspension. The technological selection criterion should ultimately
J.P. Féraud et al. / Nuclear Engineering and Design 265 (2013) 288–293
The Kozeny–Carman model can determine the value of this resistance based on experimental data (Léger, 2008). It considers the cake as incompressible, consisting of spherical particles. The value of the resistance comes from an identification of Darcy’s law in filtration:
Nomenclature d dp De kB kDL kH l n dp, P PT PG pl ps pwall Rs t T U U0 V w ˛ ˛0 ε εs ϕ
ф s g ˝
capillary diameter (m) particle diameter (m) equivalent sphere diameter (m) brinkmann’s permeability (m2 ) Happel’s corrective parameter (m2 ) Happel’s permeability (m2 ) capillary length = T·L (m) compressibility coefficient pressure drop (Pa) total pressure drop (Pa) cake pressure drop (Pa) pressure due to liquid phase (Pa) pressure due to solid phase (Pa) pressure due to walls (Pa) filter resistance time (s) tortuosity (>1) flow rate of associated liquid (m3 /s) flow rate of associated liquid without solid (m3 /s) filtrate volume mass of dry cake per unit area of filter (kg/m2 ) specific cake resistance (m/kg) reference specific cake resistance (m/kg) porosity solidosity sphericity factor solid volume fraction fluid viscosity (Pa s) density of solid (kg/m3 ) density of cake (kg/m3 ) filter area (m2 )
U0 = −
˛ = 200
(2)
(1 − ε) • 1 ε3 ϕ2 · De2 · s
(3)
This relation, in a constant-pressure filtration process, leads to: w Rs ˛ t · V+ · = V 2PT ˝2 PT ˝
(4)
Eq. (4) is the basis of the entire filterability test. All the parameter values can be determined experimentally, and ˛ can then be extracted. The pressure effect on filter cake properties is given by many empirical equations describing the variation of ˛, such as the power law (5) (Tarleton and Wakeman, 2007a,b): ˛ = ˛0 · P n
(5)
Relation (5) can be linearized by a logarithmic approach. The compressibility coefficient, n, can be deduced and gives decisive information on the cake. The test presented in this study is based on the Kozeny–Carman model (Carman, 1937) since it is the simplest and its application is consistent with our assumptions. The Kozeny–Carman model requires a porosity lower than 0.7. For higher porosity, several models have been developed:
2. Theoretical approach 2.1. Standard filterability laws Considering the cake as a solid block (Wakeman and Vince, 1984), crossed by circular capillaries of the same diameter, d, the average velocity of a fluid moving in laminar flow in this medium is given by Poiseuille’s equation: ·U·l d2
1 dp · ˛ dw
known as the modified Darcy’s Law. Combining these equations, and using the hypothesis of the Kozeny–Carman model, the specific cake resistance can be given by:
enhance the dry solid content of the cake. Decision support in this area is guided in particular by measuring two key parameters; flow resistance and compressibility (Tarleton and Wakeman, 2007a,b). For technological choice this information is supplemented by the need for a scrubbing step. The temperature sensitivity of the product must also be taken into account among other aspects. The new filterability–compressibility test system presented here can measure both of these parameter values. The analytical method is based on a simplified adaptation of the AFNOR standard (AFNOR, 2006a,b) to nuclear requirements but can be adapted for a wide range of suspensions. The proposed simplification is to limit the size of the device and use an automatic system of straightforward design. Particular attention was paid to reducing the quantity of nuclear material necessary for the test.
P = 32
289
(1)
This law comes from the Navier–Stokes equation, with various assumptions: laminar and stationary flow, incompressible and Newtonian fluid.
• Brinkman’s model (Brinkman, 1947) considers the flow around spheres, and gives an empirical relation of permeability for high porosities:
KB = KDL
3 1+ 4
1−
8 −3
where KDL =
2d2/p 9
(6)
• Happel and Kubawara’s model (Benmachou, 2005) describes the porous medium as cells (Happel, 1959) with a free surface. It requires porosity higher than 0.6 (Kuwabara, 1959). KH = KDL
6 − 91/2 + 95/2 − 62 6 + 45/2
(7)
• Others models (Couturier, 2002), such as those proposed by Neale and Nader or by Howells and Hinch, exist for very high porosities above 0.9. However, in all these models, particles are considered as spherical. Models by Davies or Jackson and James describe the effect on permeability if the particle is considered as anisotropic (Ragueh, 2011). 2.2. Principles of cake formation, simulation possibilities 2.2.1. Cake growth and pressure During the cake growth, local properties fluctuate. The cake porosity, saturation, and particle size distribution are not homogeneous (Abboud, 1996), but the total pressure in the cake remains constant (Tien et al., 2001). The total pressure is equal to the sum of the liquid and solid pressure. This results in the following expression: dpl + dps = 0
(8)
290
J.P. Féraud et al. / Nuclear Engineering and Design 265 (2013) 288–293
Two variations of this equation have been proposed (Bai and Tien, 2005): dpl =y dps
(9)
where y stands for: y = −1 (type 1)
(10)
y=
−1 (type 2) (1 − εs )
(11)
y=
−εs (type 3) (1 − εs )
(12)
Each suspension better corresponds to one of the three equations; using the correct one will be helpful to better describe cake buildup and cake properties. 2.3. Cake collapse If the cake is compressible, rearrangements may occur if pressure is applied (Antelmi et al., 2001; Cabane et al., 2002). This phenomenon, called cake collapse, has various consequences including increased resistance and decreased porosity and thickness. The mechanism by which cakes collapse involves very small motions of particles in contact with one another. In some cases these motions leave the small-scale coordination unchanged, and in other cases the organization of particle–particle contact within the aggregates in the cake can be slightly modified. 2.4. Simulation The cake buildup may be simulated in two ways. Burger et al. (2001) has proposed an iterative approach in which the last layer formed is considered as the filtering medium for the next layer. Ni et al. (2006) describe a physical approach. A force balance is calculated for the particle, with short-range interactions such as the Van der Walls interaction, or electronic interactions. This approach is limited by the number of particles taken into consideration. All simulations are done on a theoretical suspension, and the models were not applied to an industrial facility (Tien, 2006). This is the real limitation of simulation, but it gives an approach to reality and to the difficulty of the filtration step. 2.4.1. Empirical relations for determination of cake resistance and variations Empirical relations between fluid velocity and cake resistance, and between fluid velocity and porosity, have been established (Ergun, 1952). These relations follow an a·Ub law, where the values of a and b fluctuate with the cake formation step. Other studies (Saleem and Krammer, 2007; Saleem et al., 2012) have shown that the cake resistance increases with velocity and pressure, and that the cake resistance decreases as the cake ages.
capillaries. These simplifications are associated with the Bernoulli law of conservation of energy of a liquid. Performing this test on a suspension, however, implies continual changes in the thickness L of the porous medium since the cake is formed as the test progresses. Two approaches can be adopted in this case. The test can be performed at a constant flow rate (Iritani et al., 2007) or at constant pressure. Working at constant pressure is particularly interesting here because it allows us to control the permeability of the porous medium, and thus investigate the compressibility of the cake versus the pressure under satisfactory conditions (Sorensen et al., 1996). The flow resistance ˛ (or Rg ) of the porous medium is a mean resistance value and cannot be directly compared with the resistance of the filter medium Rs . The two parameters do not have the same dimensions. In practice, the experimental curve of t/V versus V is a straight line whose slope can be used to determine ˛ and whose y-intercept can be used to calculate the resistance of the filter medium Rs . More information is given by Lee and Wang (2000) in a brief review of theories in the literature concerning transport processes in filter cakes during filtration and consolidation. 2.6. Wall effects Ergun and Orning (Ergun, 1952) developed a general equation relating pressure drop to fluid flow for fixed beds consisting of spheres: Pg 1 (1 − ε)2 (1 − ε) g · U 2 = 150 · U 2 · + 1.75 · L dp ε3 ε3 dp
(13)
In order to predict wall effects, Mehta and Hawley (1969) decided to modify the Ergun equation. It has been shown that if the column-to-particle diameter ratio is less than 50:1, wall effects have to be considered. As the present test lies outside this range of diameters, wall effects may be ignored. Another study (Tiller, Lu et al., 1972) demonstrated that compressible beds of small particles are strongly affected by the walls when the tests are carried out in compression-permeability cells. Wall effects are introduced as a new force in the pressure equation (9), modified as follows: ps + pl + p(wall) = papp
(14)
The effect of different materials of construction has also been studied. The PTFE-coated cylinder generates less friction than a stainless steel cylinder (Tiller et al., 1972). It also gives empirical relations for porosity and specific cake resistance as function of pressure, taking wall effects into consideration. Numerical simulations have been done to better understand the consolidation of the cake. They show that sidewall friction creates a region of lower compressive stresses and pore pressure, with a higher void ratio (Zhao et al., 2003). 3. Materials, experimental setup and operating conditions 3.1. Model substance
2.5. Filterability test The filterability test characterizes the flow resistance of a liquid in a porous medium (cake) in formation. The test is performed at four pressure levels to determine the compressibility or deformability of the solid. The description of this flow by the laws of physics is based on three initial assumptions: (i) the porous medium, or cake, is the result of a pileup of beads; (ii) the porous medium is nondeformable with pressure (implying constant permeability); (iii) the filtrate flow in the cake is laminar. In this case, the cake is associated with a pileup of N beads and/or an arrangement of n
The product investigated here is a mixed compound of U/Ce oxalate with a bulk relative density close to 1.3 and a particle size distribution centered on 20 m. This material is used to simulate reprocessing of the next generation of spent nuclear fuel. The solid is dispersed in mother liquors at a concentration of 5 g L−1 . The chemical composition is representative of industrial solutions. Mother liquors are aqueous nitric acid (<2 M), in these chemical conditions, viscosity is assumed to be the same as water, 10−3 Pa s. This product is obtained by precipitation route, at room temperature.
J.P. Féraud et al. / Nuclear Engineering and Design 265 (2013) 288–293
291
Fig. 1. Experimental setup.
3.2. Experimental setup The filterability test is based on experiments described by Roorda and van der Graaf (2005) and Couturier et al. (2007). The experimental setup designed by CEA is shown in Fig. 1. The technical design ensures reproducible flow resistance measurements. (i) The slurry tank guarantees the homogeneity of the suspension by preventing settling or ranking of the particles before the filterability test. It is equipped with a helical impeller mixer to limit shearing of the precipitate. (ii) The peristaltic pump is an Ismatec ISM 897® or Heidolph 5206® , depending on the pressure range, with a Eurotherm® 4 − 20 mA controller. This option avoids excessive flow pulsation and significantly limits shearing of the precipitate. (iii) The pressure sensor is a Wika S11® flushmounted PTFE diaphragm to prevent solid buildup on the sensor. The 4–20 mA output is connected to one of the analog inputs of the Eurotherm® controller. (iv) The controller is a Eurotherm 2408® with two analog inputs to recover 4–20 mA signals from the pressure switch and the balance, an analog output for PID control of the peristaltic pump, and an RS232 serial port to recover the environmental variables from a PC running Itools® software. (v) Valve V1 is a 3-way interconnected valve allowing fast reconfiguration from recirculation to filtration/measurement flow. Valve V2 is used to obtain the desired pressure. Finally the filter holder is a Gantois REPS® mesh with a 14 m cutoff threshold. The optimization of this component is detailed in Fig. 1.
Fig. 2. Filterability at 600 mbar.
measurement on the last 60 cm3 of filtrate. In these conditions, enough data are provided to accurately measure the t/V line slope and calculate the filterability. The cake obtained has a mean height of 1 cm. Figs. 2 and 3 show that the cake is high enough to allow cake specific resistance measurements. The average dryness is about 0.5.
4. Experimental results 4.1. Specific flow resistance and compressibility method Before investigating the effect of the filter holder diameter, the proper behavior of the pressure regulation loop at different values must be verified. Two experimental results at different pressures are shown in Figs. 2 and 3. The circled lines indicate the valve V1 opening and the PID control perturbation around the pressure value. The double arrows show the start of the filterability test. The black line is the filterability curve. Pressure oscillations always appear below 20 cm3 of filtrate volume. After that, a straight line pressure allowed a filterability
Fig. 3. Filterability at 1000 mbar.
292
J.P. Féraud et al. / Nuclear Engineering and Design 265 (2013) 288–293
Fig. 5. Cake compressibility of U/Ce oxalate between 300 and 1300 mbar. Fig. 4. Filterability at 980 mbar: example of cake fissuring.
When cake fissuring occurred during the measurement, the results remain reliable. Fig. 4 shows an example of cake fissuring. After 60 cm3 , fissuring appeared and a pressure drop was noted. This bed crack was monitored and was immediately compensated by the pressure regulation. The data remained consistent since the filterability was easily obtained by considering the first part of the t/V line. The compressibility of nuclear material is obtained easily with filterability values measured at four different pressures in the range of 300 to 1300 mbar. The curve is fitted with a logarithmic law from Eq. (5) (Figs. 5 and 6). The U/Ce oxalate compressibility factor, given by the slope value, is 0.39. This corresponds to a product with moderate compressibility (Leclerc, 1997). This compressibility behavior shows the energy loss due to porosity decreasing with pressure. 4.2. Effect of filter holder diameter The final item to optimize in the technological development stage concerns the smallest volume of suspension allowing a consistent filterability/compressibility test result. Limiting the reagent volume makes it easier to test expensive or hazardous compounds. The filter holder, which is used to measure the height of the cake, must be several centimeters high to facilitate the measurement. The only feature that can be reduced is the diameter of the filter
holder. A reduction in scale is possible to the extent that wall/cake interactions remain negligible. The study described here consisted in measuring the flow resistance of a nuclear material suspension at 600 mbar with filter holders 3 cm and 1.8 cm in diameter. The smaller diameter would allow testing with only one-third of the reagent necessary for the larger one. Four measurements were performed with each filter holder (see Fig. 2), The operating procedure takes into account the possible evolution of the slurry. The filtration tests were carried alternately on the two filter holders. Chronological distribution may highlight two different means among diameters tested. In fact, further statistical test (see Appendix A) shows that the measurements do not reveal any trend or significant effect of the filter holder diameter on the ˛ measurement. Downsizing the filter holder did not affect the consistency of the determination of the specific resistance ˛. Small variations in cake specific resistance measurement may have multiple explanations: little bubbles in pipes, pressure fluctuations, solid concentration fluctuations, bed formation and reorganization under pressure. The flow resistance may also be measured with a compressibility/permeability cell equipped with a piston which squeezes the cake at constant pressure. It is commonly accepted that this design requires a holder diameter of at least 5 cm in order to homogeneously feed the liquor onto the top of the cake during the filtration test. The necessary amount of slurry is about ten times greater than with the smaller filter holder developed in this study. This is why
Fig. 6. Chronological distribution of ␣ measurements according to the diameter d of the filter holder.
J.P. Féraud et al. / Nuclear Engineering and Design 265 (2013) 288–293
this new development is a real advantage when it is necessary to manipulate toxic or hazardous materials. 5. Conclusion
Appendix A. Statistical test A.1. Fisher test The measured standard deviations s1 (d = 3 cm) and s2 (d = 1.8 cm) are estimators of the same standard deviation. Can they be merged? H0 . Yes = Merge standard deviations s* to refine the statistics to = 7 degrees of freedom No = Calculate sd according to the Aspin-Welch equation:
Fobs =
s 2 1
s2
= 0.7 < Fcrit (9.3))
(15)
where Fcrit is the value obtained in the Fisher table with 3 degrees of freedom for a 95% confidence level. For the results shown here, hypothesis H0 is verified and the two standard deviations can therefore be merged according to Eq. (16):
s∗ =
v1 s12 + v2 s22 v1 + v2
(16)
where s* is the merged standard deviation (m kg−1 ), the degree of freedom, 1designates the large filter holder, and 2 the small filter holder. In this case: s1 = 3.16 × 109 m kg−1 and s2 = 3.77 × 109 m kg−1 → s* = 2.47 × 109 m kg−1 A.2. Student test Compare tobs with tcrit to evaluate the following assumptions. H0 .
Rg1 and Rg2 estimate the same mean.
H1 . Rg1 and Rg2 cannot be considered to estimate the same mean. Test:
Rg1 − Rg2 ∗ tobs = < tcrit
s∗
1 n1
Result: n1 = n2 = 4 → tobs = 2.36 < tcrit = 2.45 (Student’s tdistribution for a 95% bilateral confidence level and six degrees of freedom). References
A filterability test system was quickly developed. It allows accurate pressure monitoring and measurement of cake resistance and compressibility for a specific product. This operating mode allows accurate monitoring of both the pressure and t/V versus V. A particular effort was made to miniaturize the filter holder in order to limit the quantity of suspension necessary for the test. Fisher’s test validated the use of a filter holder only 1.8 cm in diameter, reducing the amount of suspension necessary by a factor of 3. The filter holder is the result of a specific development using transparent material to allow a direct indication of the cake height on the integrated graduations, making it unnecessary to dismantle the unit to obtain this value. Further improvement will concern a high-pressure design (7b) with a better glove box design to study the filterability of U/Pu oxalate coprecipitate. Other filter holder diameters would be tested while doing these improvements, in order to investigate wall effect at wider scale (1–5 cm).
H1 .
293
+
1 n2
(17)
Abboud, N.M., 1996. A theoretical study of pore pressure and porosity variations along the mud cake thickness. Comput. Meth. Water. Resour. III, 323–336. AFNOR NF EN 14701-2 (2006a). Characterization of sludges–Filtration properties–Part 2: Determination of the specific resistance to filtration: 14. AFNOR NF EN 14701-3 (2006b). Characterization of sludges–Filtration properties–Part 3: Determination of compressibility: 9. Antelmi, D., Cabane, B., et al., 2001. Cake collapse in pressure filtration. Langmuir 1722, 7137–7144. Arab-Chapelet, B., Grandjean, S., et al., 2007. Synthesis of new mixed actinides oxalates as precursors of actinides oxide solid solutions. J. Alloys Compd. 444, 387–390. Arab-Chapelet, B., Grandjean, S., et al., 2008. Synthesis and characterization of mixed An(IV)An(III) oxalates (An(IV) = Th, NP, U or Pu and An(III) = Pu or Am). J. Nucl. Mater. 373 (1–3), 259–268. Bai, R.B., Tien, C., 2005. Further work on cake filtration analysis. Chem. Eng. Sci. 60 (2), 301–313. Benmachou, K., 2005. Étude et modélisation du colmatage d’un filtre plissé. Borda, G., Brackx, E., et al., 2011. Use of a pulsed column contactor as a continuous oxalate precipitation reactor. Nucl. Eng. Des. 241 (3), 809–814. Brinkman, H.C., 1947. A Calculation of the Viscous Force Exerted by a Flowing Fluid on a Dense Swarm of Particles. Appl. Sci. Res. Sect. A: Mech. Heat Chem. Eng. Math. Methods 1 (1), 27–34. Burger, R., Concha, F., Karlsen, K.H., 2001. Phenomenological model of filtration processes: 1. Cake formation and expression. Chem. Eng. Sci., 4537–4553. Cabane, B., Meireles, M., et al., 2002. Cake collapse in frontal filtration of colloidal aggregates: mechanisms and consequences? Desalination 146 (1-3), 155–161. Carman, P.C., 1937. Fluid flow through granular beds. Chem. Eng. Res. Design 75, S32–S48. Charton, S., Duhamet, J., et al., 2012. Axial dispersion in pulsed disk and doughnut columns: a unified law. Chem. Eng. Sci. 75, 468–477. Couturier, S., 2002. Étude de la déshydratation mécanique assistée thermiquement. Couturier, S., Valat, M., et al., 2007. Enhanced expression of filter cakes using a local thermal supply. Sep. Purif. Technol. 57 (2), 321–328. Ergun, S., 1952. Fluid flow through packed columns. Chem. Eng. Prog. 48 (2), 89–94. Happel, J., 1959. Viscous flow relative to arrays of cylinders. AIChE J. 5, 174–177. Iritani, E., Katagiri, N., et al., 2007. Flux decline behaviors in dead-end microfiltration of activated sludge and its supernatant? J. Membr. Sci. 300 (1-2), 36–44. Kuwabara, S., 1959. The forces experienced by randomly distributed parallel circular cylinders or spheres in a viscous flow at small Reynolds number. J. Phys. Soc. Jpn. 14, 527–532. Lee, D.J., Wang, C.H., 2000. Theories of cake filtration and consolidation and implications to sludge dewatering. Water Res. 34 (1), 1–20. Léger, B. (2008). Étude de l’essorage en vue d’assister thermiquement le procédé, Université Bordeaux 1. Leclerc, D. (1997) Filtration sur support: aspects théoriques Techniques de l’ingénieur: J3501. Mehta, D., Hawley, M.C., 1969. Wall effect in packed columns. Ind. Eng. Chem. Process Design Dev. 8 (2), 280-282. Ni, L.A., Yu, A.B., Lu, G.Q., Howes, T., 2006. Simulation of the cake formation and growth in cake filtration. Minerals Eng. 19, 1084–1097. Ragueh, D.M., 2011. Filtration de silices précipitées: mise en évidence des relations entre propriétés macroscopiques et échelles locales caractéristiques des dépôts. Toulouse 3 Paul Sabatier. Roorda, J.H., van der Graaf, J.H.M., 2005. SUR test used for optimisation of membrane filtration plants treating wastewater effluents? Desalination 179 (1–3), 131–150. Saleem, M., Krammer, G., 2007. Effect of filtration velocity and dust concentration on cake formation and filter operation in a pilot scale jet pulsed bag filter. J. Hazard. Mater. 144 (3), 677–681. Saleem, M., Krammer, G., et al., 2012. Influence of operating parameters on cake formation in pilot scale pulse-jet bag filter. Powder Technol. 224:, 28–35. Sorensen, P.B., Agerbaek, M.L., et al., 1996. Predicting cake filtration using specific filtration flow rate. Water Environ. Res. 68 (7), 1151–1155. Tarleton, S., Wakeman, R. (Eds.), 2007a. Equipment selection and process design. Solid/liquid separation. Elsevier, Kidlington, Oxford, UK. Tarleton, S., Wakeman, R. (Eds.), 2007b. Principles of Industrial Filtration. Elsevier, Kidlington, Oxford, UK. Tien, C., 2006. Introduction to Cake Filtration. Tien, C., Teoh, S.K., et al., 2001. Cake filtration analysis: the effect of the relationship between the pore liquid pressure and the cake compressive stress. Chem. Eng. Sci. 56 (18), 5361–5369. Tiller, F.M., Lu, W.M., et al., 1972. Role of porosity in filtration.7 effect of side-wall friction in compression-permeability cells. AIChE J. 18 (1), 13-&. Wakeman, R. J., Vince A., 1984. Solid–Liquid Separation. London. Zhao, J., Wang, C.H., et al., 2003. Cake consolidation in a compression-permeability cell: effect of side-wall friction. J. Colloid Interface Sci. 262 (1), 60–72.