A rigorous cake filtration theory

The currently accepted concepts of the parabolic filtration correlation stipulate that the filtration rate decreases inversely as the flow resistance which is proportional to the extent of the filter cake; that non-parabolic behavior is characteristic of only compressible cakes which exhibit an average resistance which is not constant but depends on slurry concentration, filtrate rate and applied pressure; that the septum resistance is separate from, but additive to, the cake resistance and is obtained from the intercept of the V_F^? vs V_F line; that the average porosity of compressible cakes decreases during a filtration which causes the exit flow rate to exceed the entrance flow rate; that internal variations can only be obtained via compression-permeability simulations; and that the local resistance decreases monotonically from the septum to the cake surface.A filtration theory based on the rigorous, multiphase equations of change finds, contrary to the concepts listed above, that the least permeable part of the cake at the cake-septum interface (K₀) controls the filtrate rate and a decreasing pressure gradient across this minimum K₀ causes the filtrate rate to decrease; that deviations from parabolic behavior are independent of cake compressibility (incompressible cakes are rare, if they exist at all) but can be expected when ?_α*, s, -ΔP_c, K₀ and (dP*/dξ)|_ξ=0 are not constant; that the septum resistance is intrinsically included in K₀ which appears in the slope and not the intercept of V_F^?1 vs V_F; that the average porosity is constant and the exit flow rate exceeds the entrance flow rate due to a difference between the surface porosity and the average porosity; that internal variations can be obtained directly from filtration data alone and do not require a compression-permeability simulation; and that the local filtration resistance can exhibit a minimum as a result of using a non-linear transformation from cake length to cake mass.     

Cake properties of nanocolloid evaluated by variable pressure filtration associated with reduction in cake surface area

A potential method has been developed for evaluating simultaneously both the average specific resistance and average porosity of the filter cake formed in unstirred dead‐end ultrafiltration of nanocolloids such as bovine serum albumin solution and silica sol. The method consists of variable pressure filtration followed by constant pressure filtration. The relation between the average specific cake resistance and the pressure drop across the cake was determined from the evolution of the filtration rate with time in the course of the variable pressure filtration period, based on the compressible cake filtration model. The average porosity was evaluated from the significant flux decline caused by a sudden reduction in the cake surface area in the middle of the constant pressure filtration period. The pressure dependences of both the average specific cake resistance and average cake porosity were obtained from only two runs which differed from each other in the pressure profiles. © 2014 American Institute of Chemical Engineers AIChE J, 60: 3869–3877, 2014     

Nanocolloid cake properties determined from step‐up pressure filtration with single‐stage reduction in filtration area

A sophisticated method was developed for evaluating simultaneously and accurately both the average specific resistance and average porosity of the filter cake formed in unstirred dead‐end ultrafiltration of nanocolloids such as protein solution and nanosilica sol. In the method, a step‐up pressure filtration test was conducted by using a filter with a single‐stage reduction in the effective filtration area. The influence of the pressure drop across the cake on not only the average specific cake resistance but also on the average cake porosity of highly compressible filter cake was evaluated using only flux decline data in one dead‐end filtration test, taking advantage of the decrease in the cake thickness caused by the pressure increase. As a result, the cake properties were easily determined for a variety of nanocolloids. Constant pressure dead‐end ultrafiltration data obtained under various pressures and concentrations were well evaluated based on the method proposed. © 2015 American Institute of Chemical Engineers AIChE J, 61: 4426–4436, 2015     

纤维素助滤剂的预敷过滤性能

对高黏度物料加入纤维素助滤剂的预敷过滤性能进行了研究,说明不同的预敷条件对预敷层的过滤性能会有很大影响。通过改变预敷压力和浓度,对纤维素滤饼层的过滤比阻、可压缩性系数和孔隙率进行了研究。实验表明,纤维素预敷层的比阻随压力增加而增加,随浓度增加而减小;孔隙率随压力增加而减小,随浓度增加而增加;可压缩性系数随浓度增加而增加。纤维素助滤剂为中等可压缩性物料,孔隙率大。研究结果可为高黏度物料用纤维素预敷过滤的工程应用提供参考。     

Filtration Behavior of Slurries with Varying Compressibilities

Abstract

A novel filtration apparatus allows simultaneous measurements of filtrate volume, hydraulic pressure and cake thickness using slurry volumes on the order of 100 cm³ Differences in interparticle interactions were studied by varying the barium chloride concentration of 0.38-μm polystyrene latex and filtering at pressures between 2 and 100 psi. Cakes formed from these slurries are highly compressible for concentrations between 0.01M and 0.10M, moderately compressible for the 0.005M concentration, and incompressible for the 0.001M concentration. Plots of filtrate volume versus cake thickness were linear for the incompressible cakes, whereas the compressible cakes showed significant deviations, which were pressure dependent. The pressure distribution for the incompressible cake was found to be essentially linear as predicted from the resistance plots assuming constant α and ε. For the highly compressible cakes, most of the pressure drop appears to occur near the cake/medium Interface with only small changes occurring at the top of thp cakp.     

Effect of Cake Compression on the Performance of Centrifugal Dewatering

The effect of cake compression on the centrifugal dewatering is investigated under various rotating speeds and different kinds of cakes. A numerical method is proposed to estimate the capillary pressure and cake permeability under various cake saturations. The most dewatering in this study are operated at the funicular state of cake. The cake permeability decreases, while the capillary pressure increases rapidly, with the decrease of cake saturation during a centrifugal dewatering. The cake saturation can also be simulated once the relations among capillary pressure, cake permeability, and cake saturation are known. The deviations between simulated saturations and experimental data for compressible talc cakes are less than 2%. The centrifugal dewatering of compressible cake includes two mechanisms, the water squeeze due to cake compression and the centrifugal drainage due to pressure difference. The role of cake compression is dominant at the initial 500 s of the dewatering period, and these two factors are almost equal in magnitude as well as in importance when the cake approaches its equilibrium saturation. The porosity and compressibility of cake play the major roles in determining the equilibrium cake saturation. The equilibrium saturation of the compressible talc cake with a lower porosity is much higher than that of incompressible Al₂O₃ cake under a fixed rotating speed. An increase in rotating speed (centrifugal effect) results in a lower equilibrium cake saturation for either compressible or incompressible cakes; however, the increase in rotating speed is more efficient for an incompressible cake in the centrifugal dewatering.     

Effect of Cake Compression on the Performance of Centrifugal Dewatering

The effect of cake compression on the centrifugal dewatering is investigated under various rotating speeds and different kinds of cakes. A numerical method is proposed to estimate the capillary pressure and cake permeability under various cake saturations. The most dewatering in this study are operated at the funicular state of cake. The cake permeability decreases, while the capillary pressure increases rapidly, with the decrease of cake saturation during a centrifugal dewatering. The cake saturation can also be simulated once the relations among capillary pressure, cake permeability, and cake saturation are known. The deviations between simulated saturations and experimental data for compressible talc cakes are less than 2%. The centrifugal dewatering of compressible cake includes two mechanisms, the water squeeze due to cake compression and the centrifugal drainage due to pressure difference. The role of cake compression is dominant at the initial 500 s of the dewatering period, and these two factors are almost equal in magnitude as well as in importance when the cake approaches its equilibrium saturation. The porosity and compressibility of cake play the major roles in determining the equilibrium cake saturation. The equilibrium saturation of the compressible talc cake with a lower porosity is much higher than that of incompressible Al₂O₃ cake under a fixed rotating speed. An increase in rotating speed (centrifugal effect) results in a lower equilibrium cake saturation for either compressible or incompressible cakes; however, the increase in rotating speed is more efficient for an incompressible cake in the centrifugal dewatering.     

Theory of Filtration of Ceramics: I, Slip Casting

The theory of colloidal filtration in slip casting of ceramics as developed by Aksay and Schilling for incompressible cakes has been extended to compressible materials. The rate of cake deposition in the mold depends upon the capillary pressure and permeability of the mold. Assuming that the capillary pressure is inversely proportional to an average diameter typifying the mold and that the permeability is directly proportional to the square of the diameter, an optimum diameter exists for production of maximum pressure drop across the cake and maximum rate of deposition.     

Filtration of non-Newtonian fluids

Equations of motion characterizing the flow of incompressible, time-independent non-Newtonian fluids, exhibiting the anomalous surface effect, through compressible porous media are developed. Approximations are made to arrive at workable filtration equations for slurries of non-Newtonian (power-law) fluids. The constant-pressure and constant-rate filtration relationships developed are verified experimentally using slurries of calcium carbonate in water and dilute CMC solutions. The anomalous surface effect is found to exist in the filtration of the non-Newtonian fluids. The specific cake resistance in the case of the non-Newtonian sludge and the ratio of the effective slip velocity to the pore velocity are found to be functions of both the CMC concentration and the pressure drop across the filter bed.     

Filtration in coal liquefaction. Influence of filtration conditions in non-hydrogenated systems

A series of experiments has been carried out to study the effects of filtration conditions upon the rate of filtration of non-hydrogenated coal digests. The results show the dependence of cake resistivity on both the filtration temperature and pressure. Filter cakes were found to be compressible, resulting in smaller increases in rate with increasing pressure than with incompressible cakes. The filtration temperature determines the packing of residual solids in the cake which in turn affects the cake resistivity. An empirical relation has been derived between filtration temperature and resistivity. With increasing temperature there is an increase in filtration rate due to the reduced viscosity, but a reduction owing to a higher packing density of solids in the filter cake.     

Porosity distribution in highly compressible cake: Experimental and theoretical verification of the dense skin

The rate of filtration and the water content of cake are influenced by the existence of a dense skin in a highly compressible cake. The phenomenon of the dense skin has been rarely studied, and its existence has not been verified experimentally. In this study, the porosity variation in a very compressible cake is measured by using a new experimental apparatus, and with this the existence of dense skin has been established experimentally. ‘Unified theory on solid-liquid separation’, a recently developed theory, is utilized for calculating the porosity variation in a very compressible cake.     

The Effect of Particle Sedimentation on Gravity Filtration

ABSTRACT

Simulation of cake formation of mono-sized and dual-sized particles under gravitational sedimentation and filtration is presented. The dynamic analysis proposed by Lu and Hwang in 1993 is applied to examine the local cake properties formed under a falling head by considering the hindered settling effect of particles in the slurry and the variation of the pressure drop across the filter septum. Results of this study show that, at a given position in a cake, the solid compressive pressure reaches a maximum value and then decreases for a gravity filtration due to the decrease in the driving head. A cake constructed with dual-sized particles has a more compact structure than does one with mono-sized particles, and larger particles will form looser packing than will smaller ones for mono-sized particles. A dual-dispersed suspension with a lower fraction of large particles will result in the lowest cake porosity and the highest specific filtration resistance of cake. Comparison of the porosity distribution in filter cake formed by means of gravity filtration and constant head filtration shows that the porosity near the filter septum of gravity filtration has a convex behavior while that of constant head filtration has a tendency toward concavity. This discrepancy is mainly due to the change in the driving head during the filtration process. Both theoretical and experimental results show that the uniformity of particle size distributions in the filter cake will be much better when the relative settling velocity between large and fine particles is reduced.     

The Permeability Distribution (PD) Method for Filter Media Characterization

Cake filtration is frequently used for the removal of particulate solids from fluids in industrial processes. The build up of a filter cake is usually accompanied by a decrease in overall permeability of the filter leading to an increased pressure drop over the filter medium. For an incompressible filter cake that builds up on a homogeneous filter cloth (surface filtration mode), a linear pressure drop profile is expected over time. However, occasionally experiments show curved pressure drop profiles. Whereas pressure drop profiles with increasing slope are generally ascribed to cake compression and/or depth filtration, pressure drop profiles with decreasing slopes are only ascribed to inhomogeneities in the filter. Such inhomogeneities can arise due to filter cake patches and/or an inhomogeneous filter cloth itself. In this work a method is proposed that transforms the pressure drop profile of a filter into a permeability distribution (PD) of the filter medium, thus accounting for possible inhomogeneities of the medium. The determination of the PD is looked at as an inverse problem of an integral transformation. The method is applied to experimental filter pressure drop data of laboratory scale jet-pulse cleaned bag filter plants. It is found that even clean filter media can exhibit a significant permeability profile.     

The Role of Polysaccharide on the Filtration of Microbial Cells

The role of polysaccharide on the “dead-end” microfiltration of microbial cells is studied. Yeast and blue dextran are used as typical samples of microbial cells and polysaccharide, respectively. The filter cake becomes more compressible in the early periods of filtration and exhibits much higher filtration resistance when dextran molecules co-exist in the cake structure. A cake compression mechanism is proposed to explain the pressure effects on the cake structure and filtration resistance. For the two-component cakes, the dextran molecules deform easily even under a pressure as low as 30 kPa, while significant yeast deformation is observed when pressure is higher than 100 kPa. It is attributed to the depletion of most solid compressive pressures by deformed dextran molecules. The cake porosity data indicate that the cake compressibility is higher under low filtration pressure, and blue dextran plays a significant role on the cake structure and occupies a considerable volume in the cake. A resistance model is also derived for understanding the relationship between the average specific cake filtration resistance and filtration pressure. The cake filtration resistance is determined by the effective volume fraction of each component in cake and nearly the same as that of blue dextran under low pressure.     

The effect of melt compressibility on a high-speed wire-coating process

The effects of melt compressibility on a wire-coating process have been investigated, assuming that the compressible behavior of a polymeric melt obeys the Spencer-Gilmore equation of state. The compressible model is distinctly different from the incompressible model in two ways: (1) it has substantially lower pressure build-up within the die, and (2) the location of the maximum velocity is closer to the traveling wire position. As a result, the velocity profile within the die may change from a parabolic shape to a shape somewhat similar to that observed in a drag-flow case; and the shear stress generated by the fluid on the moving wire is no longer constant. Calculations indicate that the effect of melt compressibility during wire coming may not be neglected if the wire speed Ls greater than 50 cm/s (100 ft/min). In addition, the relationships between processing parameters and product coating thickness for both compressible and incompressible fluids are quite different.     

The role of P<Subscript>i</Subscript>, P<Subscript>o</Subscript>, and P<Subscript>f</Subscript> in constitutive equations and new boundary conditions in cake filtration

The constitutive equations proposed by Tiller and Shirato were analyzed and a new constitutive equation originating from the sediment thickness was proposed. A new boundary condition of the filter cake based on the solid compressive pressure of the first solid layer,p _f , was also proposed. Accurate average specific cake resistances at various pressures and the thickness of cake were calculated with the new constitutive equation and boundary conditions. The influence ofp _f on the cake thickness and average porosity was studied theoretically. Using three constitutive equations, it was proved that the compressibility n obtained from filtration results instead of CPC (compression-permeability cell) of very compressible cake could not have an exact value.     

Analysis of Filtration Characteristics for Compressible Polycrystalline Particles by Partial Least Squares Regression

Crystal size and morphology have been varied by changing the initial supersaturation ratio and the temperature in reactive crystallization experiments. The influence of the chord length distribution, average cake porosity, and filtration pressure difference on the average cake resistance of polycrystalline particles of an industrially produced aromatic amine has been investigated by means of partial least squares (PLS) regression and sensitivity analysis. Analysis of the results has disclosed that wider chord length distributions as well as lower values of the measured average porosity lead to higher values for the average cake resistance. However, PLS regression and sensitivity analysis have identified the applied pressure difference itself as the most significant parameter influencing the magnitude of the cake resistance. This unexpected behavior is accounted for by compression of the filter cake occurring predominantly in small layers above the filter cloth characteristic for highly compressible cakes.     

Theory of Filtration of Ceramics: II, Slip Casting on Radial Surfaces

The theory of filtration of slip casting of incompressible beds of ceramic materials in planar molds is extended to deposition on internal and external cylindrical surfaces. Formulas are developed for (a) calculating the variation of the liquid pressure in both cake and mold as a function of the radius and (b) determining the time to produce a given thickness of a consolidated body. Assuming that the mold permeability K _m and capillary suction P _cap are related by an inverse parabolic relation, K _m P _cap ²= J , there is a mold with a specific permeability that produces a maximum rate of deposition from the slip. The theory presented in this paper strictly applies only to incompressible cakes and should be used cautiously with moderately compactible beds. For highly flocculated slurries yielding highly compressible bodies, the equations would not be expected to yield reliable results. Radial geometry leads to mathematical complexities not encountered in analyses of planar molds.     

Evaluation of compression-permeability characteristics of microbial cake based on microfiltration data

A method was developed for evaluating the compression-permeability characteristics of microbial cake based on the microfiltration data on microbial suspension. The parameters in newly proposed equations, which were applicable to extremely-high compressible cake, were determined to fit the data of the average specific resistance and average void ratio of the filter cake as functions of the filtration pressure. The calculations based on proposed equations were in good agreement with the compression-permeability cell data such as the local specific resistance and local porosity of the cake represented as functions of the local compressive pressure in all microorganisms used in this study.