MATHEMATICAL MODELING OF A PLATE-AND-FRAME-PRESS









Proceedings
of the American Filtration and Separations Society National Technical Conference
April 21-24, 1996
Valley Forge, Pennsylvania




by






Scott A. Wells
Professor of Civil Engineering
Department of Civil Engineering
Portland State University
Portland, Oregon 97207-0751 USA
(503) 725-4276 FAX (503) 725-4298 scott@eas.pdx.edu









MATHEMATICAL MODELING OF A PLATE-AND-FRAME-PRESS

by

Scott A. Wells
Professor of Civil Engineering
Department of Civil Engineering
Portland State University
Portland, Oregon 97207-0751 USA
(503) 725-4276 FAX (503) 725-4298 scott@eas.pdx.edu






Abstract
Cake filtration in a plate and frame filter press was simulated mathematically. The process of operation of the plate-and-frame press was based on the following steps: (i) after emptying the cake from the prior filtration cycle, a suspension is pumped under pressure into a cell; (ii) during this period of filling the cell, some filtration occurs; (iii) after the filling period, cake filtration occurs. The model described in this study considered only filtration after the filling process. Governing equations for porosity were developed for the plate-and-frame press and were solved numerically using a two-dimensional spatial finite difference grid. Single-line relaxation with over-relaxation and alternating-direction-implicit numerical schemes were used to solve the governing equations. Determination of appropriate initial and boundary conditions were discussed based on characteristics of the plate-and-frame press.




INTRODUCTION



Plate-and-frame presses are used frequently in solid-liquid separation processes. After emptying the cake from a cell of a plate-and-frame press from a prior filtration cycle, a suspension is pumped under pressure into the empty cell. During this period, some filtration occurs. After filling the cell, filtration proceeds as the pump pressure increases. The model described in this study considered filtration occurring after the filling process. Figure 1 shows the layout of an individual plate-and-frame press cell used in the modeling study.

THEORETICAL BACKGROUND


Governing equations for cake filtration include solid and liquid continuity and the reduced forms of the solid and liquid momentum equations assuming that the inertial and gravity terms of the liquid and solid phase and the solid-solid shear stresses are negligible and that /x=0 (where x is the spatial coordinate into the plane of Figure 1):




1




2




3




4




5




where is the porosity [-], Vl is the liquid velocity [cm/s], Vs is the solid velocity [cm/s], ' is the effective stress [kPa], p is the porewater pressure [kPa], t is time [s], y and z are spatial coordinates [cm], p is the total applied pressure [kPa].

By taking the derivative of Equation 3 with respect to y and the derivative of Equation 4 with respect to x, adding the resulting equations and substituting Equation 1,
6




Using the definition of the constitutive property that and the definition of a partial differential such that
and , and assuming that /t >> (Vsy)/y and (Vsz)/z (Voroboyov, 1993), Equation 6 becomes




7



The boundary conditions for the domain shown in Figure 1 are








where i is the initial porosity of the suspension (constant over time if the feed solution is constant and no filtration occurs in the manifold to the individual filtration cells), o is the terminal porosity along the filter medium (a function of time because the applied pressure is changing as a function of time as the pressure output of the pump supplying the filter cells varies as a function of time), and initial is the initial porosity distribution in the filtration cell after the filling and initial filtration process.



MATHEMATICAL MODEL DESCRIPTION



The governing equation was solved by finite difference methods. The spatial domain was divided into equally spaced grid points in the y and z directions. A general implicit-explicit finite difference equation for Equation 7 would be






8



where
,

,

,



, and = 1 for fully implicit and = 0 for fully explicit.

Direct solutions of the non-linear, implicit equation are not used since they also require extensive computational time. Linearization of the term at the n+1 time step was accomplished by using a Taylor series expansion neglecting higher order terms such that





In this case, using an approximation for /t at the n time level by using a backward difference in time [such as, (n-n-1)/t] would eliminate the non-linearity, such that
. The governing equation was solved using an alternating-direction-implicit finite difference and a single-line over-relaxation technique. Details are shown in Wells (1994).

DETERMINATION OF THE TERMINAL POROSITY AS A FUNCTION OF TIME



The value of o, the boundary condition along the z=0 and z=H axes, is a function of time because of fluid pressure changes from the pump supplying the filter cells. As suggested by Voroboyov (1993) a typical characteristic curve for a pump is necessary as input to the model. The pump characteristic curve must be supplied in the following format:
9


where p is the porewater pressure supplied by the pump in kPa, V is the suspension velocity in cm/s, pa (kPa), pb (kPa/cm/s), and pc (kPa/cm2/s2) are empirical parameters.

The maximum applied pressure can be determined by taking the derivative of the above equation, dp/dV = 0, and solving for Vmax at pmax. Then
and pmax is determined by substituting this result in Equation 9.

In the model the velocity of the suspension is estimated based on the total filtrate production from all the filter cells. The suspension velocity is therefore the rate of filtrate production.





DETERMINATION OF THE LIQUID VELOCITY AND FILTRATE PRODUCTION



The technique to calculate these quantities was similar to those calculated by Wells (1991) where the momentum Equations 3 and 4 were inverted to solve for Vlz and Vly such as










These equations were put into finite difference form and solved from the porosity distribution. The total filtrate production, Qtotal in cm3, was calculated from





where W is the cell width in cm, N is the number of filter cells for the entire filter press, the "2" is to account for the filtrate production along the z=H boundary, and i is the number of model cells along the y-axis.




CONSTITUTIVE RELATIONSHIPS



The model used various functional forms of constitutive relationships. These relationships are between the cake porosity (or void ratio) and effective stress and between the cake porosity and permeability (or similarly between cake resistance and porewater pressure).
Relationships used by Wells (1991), such as




10




and
11




where ava [gm/cm/s2], avb [-], pka [cm2], and pkb [-] are empirical coefficients, were used in the model.

The terminal porosity o was determined by integrating the stress-strain relationship for the solid phase from Equation 10, such as









After integration and simplification, the porosity at any porewater pressure p is then 12




The terminal porosity is determined by setting p=0 kPa in the above equation.



POROSITY INITIAL CONDITION



The initial distribution should be based on a model of the dynamics of filling the filtration cell. Several initial porosity distributions were evaluated in Wells (1994). One of these initial pressure distributions used was a parabolic distribution in z from z=0 to z=H/2 (the centerline). This pressure distribution was of the form





where a [kPa], b [kPa/cm], and c [kPa/cm2] are empirical coefficients and z is the distance from z=0 in cm. Note that dp/dz = b + 2cz. To satisfy the boundary conditions that p=papplied (a fraction of the theoretical maximum pressure) at z=H/2 and p=0 at z=0, the coefficient a must be zero and (using H/2=h)





Because the filtrate production must always be non-zero and positive, the condition that dp/dz > 0 requires that b>0 and . The filtrate production at z=0 is then at the beginning of the model simulation
13




DETERMINATION OF MODEL PARAMETERS



Data required to run the model include the following: (1) relationship between permeability and porosity (such as parameters pka and pkb in Equation 11), (2) relationship between porewater pressure (or effective stress) and porosity (such as ava and avb as in Equation 10), (3) relationship between pump pressure and suspension velocity (pump characteristic curve, where pa, pb, and pc are curve parameters as in Equation 9), and (4) the pressure differential at the initiation of expression (pinitial, some fraction of p-total).

Once pinitial (at t=0) is known, the suspension velocity at t=0 can be calculated from inverting Equation 9, such that





After the suspension velocity at t=0 is known, then the parameter "b" in Equation 13 can be determined from





where k and e are evaluated at pinitial. Once the parameter b is known, then the initial parabolic distribution of porosity is known.

Then after each model time step, the filtrate velocity is computed and then the porosity at z=0 is determined based on the total new pressure differential.





MODEL RESULTS



Model results of the progression of filtration in a plate and frame press cell after 6 s and 20 s of filtration are are shown in Figures 2-5 for a kaolin suspension in water. Characteristics of this model run are shown in Table 1.




CONCLUSIONS



A numerical model of the dynamics of a plate-and-frame press was developed. Important research needs to be directed toward our improvement in understanding of slurry constitutive properties. This type of advance will lead to the use of complex mathematical models for evaluating and designing dewatering processes.




REFERENCES



Voroboyov, E. (1993) Personal communication, Sugar Research Institute, Kiev, Ukraine.
Wells, S. A. (1991) " Two-Dimensional, Steady-State, Modeling of Cake Filtration in a Laterally Unconfined Domain," Fluid/Particle Separation Journal, Vol. 4, No. 2, 107-116.
Wells, S. A. (1994) "Filter Press Dewatering: User's Manual for the Two-Dimensional Model," Technical Report, Department of Civil Engineering, Portland State University, Portland, Oregon.




Acknowledgments: Appreciation is expressed to Dr. Eugene Voroboyov for many helpful discussions at the Sugar Research Institute, Kiev, Ukraine while the author was a Fulbright Scholar at the Ukrainian State University of Food Technology. Those who contributed to the success of at the Ukrainian State University included Dr. Ivan Malejik, the chairman of the Food Processing Cathedra, Dr. Leonid Bobrivnik, Vice-Rector and Chief of Research, and Dr. Ivan Guhly, the Rector of the University.