# A Study of Cake Filtration Parameters Using the Constant Rate Process

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{10}and 7.4 × 10

^{11}m/kg for calcium carbonate and talc, respectively. These results do not agree with what would be predicted from an analytical equation for permeability, such as Kozeny-Carman. In addition, discontinuities were observed in all cases on the curves of filtrate volume with time for the initial stage of filtration. This behaviour is attributed to retarded packing compressibility (RPC) complicating the analysis of the filter medium resistance. RPC is an important component in determining the filter cake resistance and its functionality with cake forming pressure. It is found that there are additional effects that enhance the resistance to permeation in different cake materials, which is not recognised in the standard analytical approaches. These complexities can be related to shape, polydispersity, or agglomeration within the material sample and not to the experimental equipment or procedure. Furthermore, a complete and straightforward methodology is presented in this work for investigating the significance, or otherwise, of medium resistance on the later stages of the filtration.

## 1. Introduction

_{m}). The reliability of this data is greater than what is normally obtained from the alternative CPF, where a negative intercept on the classical t/V plot used for data analysis is often found [5,6].

_{av}) and the average concentration (C

_{av}) respectively with pressure, are given in Equations (1) and (2)

_{c}is the pressure drop across the cake (Pa), α

_{0}is the specific cake resistance (m/kg) for zero applied pressure, while C

_{0}is the bed concentration (m

^{3}/m

^{3}) for the same conditions. The power law exponents n and m are other constants for the material, called its compressibility coefficients (relating to the cake resistance and concentration, respectively). Values for α

_{av}can be determined experimentally by performing a series of filtrations at different pressures, allowing the resistance exponent n to be evaluated in a relative straight forward manner. While the same is possible for the cake concentration, to find the exponent m, it is more difficult to determine accurately experimentally. However, it can be normally assessed to the required degree, simply from measuring the change in cake height.

^{2}), C is the solid concentration by volume (m

^{3}/m

^{3}) and ${\rho}_{s}$ is the density of solids (kg/m

^{3}).

_{v}is the specific surface area per unit volume of the particles, $6/{x}_{sv}$ (1/m) and K is the Kozeny coefficient, where a value of 5 is used as is conventionally assumed in text book for packed beds [28,29]. The permeability may be converted to specific resistance using Equation (3), but attempts to predict specific resistance from particle size analysis data usually fail by at least one order of magnitude [5] with the resistance being greater than that predicted. The choice of any particular diameter for characterisation of an irregular particle depends on the intended application in most cases [30,31]. When the particle is non-spherical, the determination of a single particle size is not a simple task. Either particle orientation or direction affects the measurement of common statistical diameters [32]. Rhodes [33] reported a number of different geometrical sizes as well as different dynamic ones together with the statistical diameters. In this work, the Sauter mean diameter (${x}_{Sv}=6/{S}_{v}$) being the diameter of a sphere with the same average specific surface area per unit volume was used. This diameter is applicable for systems where the surface area is the determining factor and particle samples have relatively narrow size distributions.

## 2. Materials and Methods

#### 2.1. Materials

_{2}) layer sandwiched between two sheets of silica (SiO

_{2}). These layers of talc are approximately 19 Å thick and they are held together by weak Van der Waal’s forces. The edges of the talc are predominantly hydrophilic hydroxide groups, whereas the faces have hydrophobic groups. Hence, the measured Zeta potential is the net charge over the differing surfaces and the mineral can form structures commonly known as a ‘house of cards’ that are similar to clays.

_{10}, D

_{50}and D

_{90}of particle diameters over time for talc and calcium carbonate based on volume distribution. Three different shear rates induced by increasing the pump speed (P.Sp.) were used during the LD analysis. The samples were subjected to various shear rates and sonication energy, in order to monitor the formation of aggregates. Ideally, in situ cluster analysis is required during hindered sedimentation. However, the relatively high concentration of dispersed phase and the consequent light scattering problems made this difficult; hence alternatively, clustering at low concentrations was tested. The results presented in Figure 2 demonstrates no evidence of significant aggregation or clustering on talc or calcium carbonate under the prevailing conditions. It also demonstrates that both materials have very similar median particle size and distributions as determined by the usually reported values of D

_{10}, D

_{50}, and D

_{90}, based on a volume (or mass) particle size distribution.

_{0}, C

_{0}, n, and m from Table 2 in Equations (2) and (3), respectively.

#### 2.2. Methodology

_{m}) and the filtrate rate (dV/dt) are obtained from the intercept and differential of the polynomial fit respectively. Additionally, the collected filtrate versus total pressure is plotted (e.g., Figure 4) where again, a second order polynomial is fitted to smooth the data (Equation (7)) and to enable simple calculation of the differential, allowing $a=\frac{\mu c\alpha}{{A}^{2}}$ and $b=\frac{\mu {R}_{m}}{A}$ constants to be determined, which ultimately enable the membrane and cake resistances to be measured.

^{3}), ${V}_{0}$ is the initial filtrate volume (m

^{3}), $\Delta {P}_{T}$ is the total measured pressure (Pa) and $\Delta {P}_{m}$ is the pressure over membrane (Pa). This data-smoothing procedure was adopted to remove random fluctuations in measurements, and it was found necessary to apply different polynomial equations to different regions as discussed in more detail within the results section.

## 3. Results and Discussion

_{m}). Firstly, it was assumed that R

_{m}is equal to zero (R

_{m}= 0 at time 0 s). Secondly, the value of R

_{m}was determined by applying Darcy’s law to the first data point measured, neglecting any resistance due to the filter cake (R

_{m}= N

_{1}) at the first measurement of filtrate. For example, for conditions of 0.21 v/v and a 5 rpm pump speed (representing data shown in Figure 10) N

_{1}is 1.07 × 10

^{10}m

^{−1}for talc and 6.8 × 10

^{8}m

^{−1}for calcium carbonate. The third approach is to consider the last time before the first ‘plateau region’ occurs and again determine R

_{m}by assuming that all the pressure drop is over the filter medium by applying Darcy’s law (R

_{m}= N

_{2}at the first point of the filtrate plateau/stop). In the case of the talc filtration at 0.01 v/v solids (Figure 6a), R

_{m}= N

_{2}= 1.4 × 10

^{11}m

^{−1}, which occurs after 60 s into the filtration. While, in the case of the calcium carbonate filtration at 0.09 v/v solids (Figure 6c) R

_{m}= N

_{2}= 1.3 × 10

^{9}m

^{−1}would be at 10 s into the filtration. These provided three values for R

_{m}that could be used in the later analysis for pressure forming the filter cake. The true value of R

_{m}would, of course, likely be somewhere between these extreme values.

_{m}, hence three different values of cake forming pressure, to determine three different values of cake resistance for all the filtrations.

_{m}. There is some small variation in the talc resistance with pressure curve at lower pressure, within the region of cake formation (Figure 10a), but it is reasonably consistent after a pressure of 80 mbar has been reached. Thus, for the purpose of determining the constitutive relation between cake resistance and cake forming pressure (Equation (2)) the data illustrated in Figure 10: (a) for talc, from 80 mbar onwards can be used and (b) for calcium carbonate all pressure values can be used.

^{−14}and 3.6 × 10

^{−14}m

^{2}at 5 rpm pump rates) are actually similar to calcium carbonate (e.g., 1.64 × 10

^{−14}and 5.42 × 10

^{−14}m

^{2}at 5 pm). Hence, average specific cake resistance values (using Equation (3)) for talc are much greater, due the lower bed concentrations, giving an overall average for all conditions of 5.9 × 10

^{10}and 7.4 × 10

^{11}m/kg for calcium carbonate and talc, respectively. Even allowing for a low value of sphericity for talc (0.22) compared to calcium carbonate (0.81) [39] would not explain the (at least) an order of magnitude higher specific resistance displayed by the talc cakes compared to the calcium carbonate. Therefore, there is clearly additional effects that are enhancing the resistance to permeation in the talc cakes compared to the calcium carbonate ones.

## 4. Conclusions

_{m}with the initial layers of filter cake. It is then possible to use the determined values of R

_{m}for later analysis of cake specific resistance with pressure. Measurements of the specific cake resistance and overall cake permeability were compared to estimates from the Kozeny-Carmen analytical model (using the Sauter mean diameters). It was found that the K-C model considerably underestimated the resistance to filtration in all cases, but most significantly for the talc systems. It was assumed this was likely to due to the ‘enhanced’ resistance arising from particle shape effects, the presence of the finer particles within the distribution, or agglomeration.

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Particle diameter variation (D

_{10}, D

_{50}and D

_{90}) over time for three different shear conditions (P.Sp.) 1, 3 and 6 rpm, during analysis by laser diffraction using Horiba LA-920 for (

**a**) talc and (

**b**) calcium carbonate, based on volume distribution.

**Figure 3.**The Flow chart of calculation steps for the constant rate filtration (CRF) technique to determine specific resistance.

^{(1)}Parameters: a, b are fit constants, t is the process time, V is the filtrate volume, ∆P

_{m}is the membrane and ∆P

_{c}is the pressure over the cake. Also used in analysis, µ is the liquid viscosity, A the cake surface area, R

_{m}the membrane resistance, R

_{c}the total cake resistance and α

_{av}the average specific cake resistance. Initial data points (e.g., stage 1 in Figure 5) which are used to calculate R

_{m}, should not be included in the overall data trendline, except in the case of an incompressible material.

^{(2)}A series runs at different pressure are required in order to calculate C

_{0}and m.

**Figure 4.**Measured filtration pressure during filtration, at three starting concentrations: 0.01, 0.03, and 0.05 v/v of talc with 30 rpm pump speed.

**Figure 5.**The two stages (1 and 2, indicated in the figures by the two dashed vertical red lines) during talc filtration, (

**a**) 0.01 v/v and (

**b**) 0.05 v/v feed suspension concentration and calcium carbonate filtration, (

**c**) 0.09 v/v and (

**d**) 0.21 v/v feed suspension concentration at 5 rpm pump speed.

**Figure 6.**The initial stages of filtration test for talc, (

**a**) 0.01 and (

**b**) 0.05 (v/v), and for calcium carbonate, (

**c**) 0.09 and (

**d**) 0.21 (v/v), at 5 rpm pump speed.

**Figure 7.**Investigation on the variation of pump speed and rotation time during the filtration of talc at initial concentration of 0.05 v/v, using a pump speed of 5 rpm.

**Figure 8.**Filtration flow rate with time and pressure for talc using 5 rpm pump speed, (

**a**) 0.01 v/v and (

**b**) 0.05 v/v (2nd polynomial equation, Equation (6) ( ̶ ) and central difference (○) Equation (8)).

**Figure 9.**Filtration flow rate with time and pressure for calcium carbonate using 5 rpm pump speed, (

**a**) 0.09 v/v and (

**b**) 0.21 v/v (2nd polynomial equation, Equation (6) ( ̶ ) and central difference (○), Equation (8)).

**Figure 10.**Comparison between calculated specific cake resistances using R

_{m}= 0 (○), R

_{m}= N

_{1}(●) and R

_{m}= N

_{2}(∆), (

**a**) for talc at 0.05 v/v and (

**b**) for calcium carbonate at 0.21 v/v, both using a pump speed of 5 rpm.

Talc | Calcium Carbonate | |
---|---|---|

Natural pH | 9.4 | 8.0 |

Zeta potential (mV) | −16.0 | +7.5 |

IEP point | pH = 2.0 | pH = 10.3 |

S.M.D ^{a} (µm) | 6.0 | 5.0 |

Density ^{b} (g.cm^{−3}) | 2.978 | 2.790 |

^{a}(S.M.D) Sauter mean diameters were measured using a Horiba LA-920-wet Laser diffraction.

^{b}A Multivolume Pycnometer was used to determine materials densities.

Talc | Calcium Carbonate | |
---|---|---|

α_{0} (m kg^{−1}) | 1.1 × 10^{10} | 6.6 × 10^{9} |

n | 0.47 | 0.19 |

C_{0} (v/v) | 0.17 | 0.25 |

m | 0.21 | 0.08 |

**Table 3.**Test results for the different initial solid concentration suspensions and various pump suction of both solids: (a) talc and (b) calcium carbonate.

(a) | P.Sp. 5 rpm | P.Sp 10 rpm | P.Sp. 30 rpm | |||||||||

Ci ^{(1)} (v/v) | C.H (mm) | C.C (v/v) | RSD ^{2} ±(%) | α × 10^{−10} (m/kg) | C.H (mm) | C.C (v/v) | RSD ^{2} ±(%) | α × 10^{−10} (m/kg) | C.H (mm) | C.C (v/v) | RSD ^{2} ±(%) | α × 10^{−10} (m/kg) |

0.01 | 5.57 | 0.191 | 12.98 | 10.49 | 5.46 | 0.194 | 13.09 | 12.25 | 4.74 | 0.224 | 11.07 | 16.98 |

0.02 | 11.72 | 0.181 | 6.99 | 8.66 | 10.62 | 0.199 | 4.21 | 9.01 | 8.53 | 0.249 | 6.46 | 11.83 |

0.03 | 17.38 | 0.183 | 7.38 | 7.23 | 14.67 | 0.217 | 8.07 | 6.23 | 12.13 | 0.263 | 7.81 | 8.95 |

0.04 | 20.27 | 0.209 | 3.86 | 5.59 | 19.43 | 0.219 | 6.47 | 5.47 | 16.12 | 0.263 | 5.32 | 7.39 |

0.05 | 26.58 | 0.199 | 4.03 | 4.65 | 24.69 | 0.215 | 3.89 | 4.50 | 20.09 | 0.264 | 6.41 | 6.38 |

(b) | P.Sp. 5 rpm | P.Sp. 10 rpm | P.Sp. 30 rpm | |||||||||

Ci ^{(1)} (v/v) | C.H (mm) | C.C (v/v) | RSD ^{2} ±(%) | α × 10^{−10} (m/kg) | C.H (mm) | C.C (v/v) | RSD ^{2} ±(%) | α × 10^{−10} (m/kg) | C.H (mm) | C.C (v/v) | RSD ^{2} ±(%) | α × 10^{−10} (m/kg) |

0.09 | 20.12 | 0.475 | 2.45 | 4.77 | 19.31 | 0.494 | 2.32 | 5.41 | 18.28 | 0.522 | 2.23 | 4.43 |

0.11 | 24.36 | 0.479 | 1.69 | 1.49 | 21.21 | 0.55 | 1.58 | 9.11 | 19.85 | 0.588 | 1.38 | 3.25 |

0.13 | 31.35 | 0.440 | 2.64 | 2.63 | 27.76 | 0.497 | 2.53 | 1.52 | 26.61 | 0.518 | 2.24 | 5.51 |

0.15 | 36.38 | 0.438 | 1.21 | 2.23 | 32.33 | 0.492 | 1.15 | 4.00 | 30.34 | 0.525 | 1.02 | 4.72 |

0.17 | 44.22 | 0.408 | 1.42 | 6.46 | 35.48 | 0.508 | 1.37 | 4.16 | 34.72 | 0.52 | 1.10 | 5.35 |

0.19 | 45.02 | 0.448 | 0.71 | 1.66 | 39.30 | 0.513 | 0.67 | 3.44 | 37.86 | 0.533 | 0.58 | 5.94 |

0.21 | 51.81 | 0.430 | 1.20 | 1.59 | 44.68 | 0.499 | 1.13 | 3.85 | 42.09 | 0.528 | 0.98 | 4.87 |

^{1}(Ci) Initial concentration, (P.Sp.) Pump speed, (C.H) Cake height, (C.C) Final cake concentration and (α) Cake resistance from filtration equation, Equation (1).

^{2}Relative standard deviation (RSD) based on cake height measurement variability from five points for each test, leading to uncertainty value for cake concentration.

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**MDPI and ACS Style**

Mahdi, F.M.; Hunter, T.N.; Holdich, R.G.
A Study of Cake Filtration Parameters Using the Constant Rate Process. *Processes* **2019**, *7*, 746.
https://doi.org/10.3390/pr7100746

**AMA Style**

Mahdi FM, Hunter TN, Holdich RG.
A Study of Cake Filtration Parameters Using the Constant Rate Process. *Processes*. 2019; 7(10):746.
https://doi.org/10.3390/pr7100746

**Chicago/Turabian Style**

Mahdi, Faiz M., Timothy N. Hunter, and Richard G. Holdich.
2019. "A Study of Cake Filtration Parameters Using the Constant Rate Process" *Processes* 7, no. 10: 746.
https://doi.org/10.3390/pr7100746