Magnetically Assisted Filtration for Solid Waste Separation and

Ind. Eng. Chem. Res. 2005, 44, 9199-9207

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SEPARATIONS Magnetically Assisted Filtration for Solid Waste Separation and Concentration in Microgravity and Hypogravity Thana Sornchamni,‡ James E. Atwater,† James R. Akse,† Richard R. Wheeler, Jr.,† and Goran N. Jovanovic*,‡ Department of Chemical Engineering, Oregon State University, Corvallis, Oregon 97331, and UMPQUA Research Company, P.O. Box 609, Myrtle Creek, Oregon 97457

Development of efficient means for the recovery of valuable resources from solid waste materials is a critical requirement for future advanced life-support systems that will be needed to support long-duration manned missions in space. Of particular importance are technologies that may be used in hypogravity and microgravity environments. Gradient magnetically assisted fluidizedbed (G-MAFB) technology, which uses magnetic forces to compensate for the absence of gravity, is under development to serve as an operating platform for fluidized-bed and related operations in the space environment. In this study, the G-MAFB is used as a renewable filter in which granular ferromagnetic filtration media are magnetically consolidated into a packed bed. The filtration media attains a substantially different structure from that of an ordinary packed bed, when consolidated under the influence of the magnetic field gradient. This directly influences filtration rates. The fully loaded bed is then regenerated by magnetically controlled fluidization, thereby releasing a concentrated slug of solids, which may be further processed by a variety of treatment schemes. Filtration experiments have shown that G-MAFB-based methods can successfully separate suspended inedible plant biomass waste particles from a recirculating liquid stream. A mathematical model that describes the filtration process in the G-MAFB is presented. Correlations for estimation of the accumulation and detachment coefficients, based on hydrodynamic conditions and geometry of the filtration system, have also been developed. Model predictions are compared with experimental results obtained in the G-MAFB while operated under Earth’s gravity conditions (1 g). The experimental data are in good agreement with the theoretical predictions. Introduction The development of systems capable of operating effectively in the absence of gravity (microgravity) and under reduced gravity conditions (hypogravity) is an essential requirement for future long-duration manned space missions such as a lunar outpost, Mars transit, or a Mars base. Included among future environmental control and life support requirements are the development and validation of solid waste handling methods and hardware for the management of a variety of materials, including inedible plant biomass, paper, plastics, food waste, feces, etc. In all cases, solid waste management scenarios must protect the crew from potential harm by containment and stabilization, with respect to the growth of potential pathogens. Additional mission specific requirements may include volume reduction, the recovery of potentially valuable resources (i.e., CO2, H2O, CH4, and H2), and protection of the pristine planetary or lunar environment. Our research has focused on the use of magnetic methods for the † UMPQUA Research Company. * To whom correspondence should be addressed. Fax: (541) 737-4600. E-mail: [email protected]. ‡ Oregon State University.

separation, concentration, and gasification of comminuted solid wastes.1-6 In the absence of gravity, magnetic forces may be used to provide a direct body force, which, in properly designed systems, can compensate for the absence of gravitational acceleration. Similarly, these magnetic forces can also be used to augment gravity under low-gravity conditions such as those that prevail on the surfaces of the moon (0.165 g) and Mars (0.376 g), where g is the gravitational acceleration constant. Here, we report the development of magnetic filtration methods for the separation and concentration of inedible plant biomass particles, which are suspended in a recirculating aqueous stream. After the waste materials have been sufficiently concentrated, the magnetically consolidated filter bed is fluidized, thereby releasing a concentrated slug of solids in a manner which is compatible with further processing using methods such as supercritical water oxidation,7 fluidized-bed incineration,8 or gradient magnetically assisted fluidized bed (G-MAFB) gasification.3,6 The latter method, which is currently under investigation by our research team, provides the highest degree of microgravity and hypogravity performance capability. Magnetic forces may be used to control the degree of fluidization (or conversely, consolidation) of magnetically susceptible particles subjected to either gas or

10.1021/ie0501374 CCC: $30.25 © 2005 American Chemical Society Published on Web 10/22/2005

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liquid flow. The application of an external magnetic field in fluidized-bed operations using ferromagnetic particles is well-known in conventional two-phase gas-solid or liquid-solid fluidization.9-15 Application of a uniform external magnetic field significantly changes the fluidized-bed fluid dynamic behavior, and, if creatively used, may enhance bed performance. Magnetically stabilized fluidized beds have been used in a variety of applications, including biotechnology,16-20 chromatography,21-25 environmental remediation,26 and chemical synthesis.27,28 High-temperature applications are limited by the Curie temperature of ferromagnetic catalysts and fluidization media; however, this problem has been at least partially alleviated by the preparation of novel cobalt-based materials.4,5 Similar techniques in which bed expansion and consolidation are controlled by magnetic forces have also been applied to achieve filtration of solids from flowing gas and liquid streams.29-36 In the absence of gravity, normal fluidization is not possible, because there is no countervaling force with which to balance drag. Recently, in microgravity flight experiments that were conducted onboard NASA’s KC135 aircraft, we have demonstrated stable gradient magnetically assisted fluidization in the weightless environment.2 In these experiments, the direct magnetic body forces imposed upon granular ferromagnetic media by a magnetic field and field gradient were used as a substitute for gravity. With increasing magnetic force, for a given flow rate, granular ferromagnetic media undergo a decreasing degree of fluidization until fully consolidated packed bed behavior is evident.37,38 The geometry of the resulting packed bed varies noticeably from that of a typical packed bed, because of the formation of chainlike structures39 which originate from interparticle magnetic forces.40 In the current study, we use magnetic force to consolidate otherwise unconfined ferromagnetic spheres into a tightly packed filter bed. This forms the basis for a regenerable filter, in which solids are discharged in high concentration, via reduction in a magnetic field that is sufficient to achieve stable fluidization. These solids may then pass to a secondary treatment process. In the current study, we demonstrate the utility of magnetic control methods to achieve filtration of suspended wheat straw particles from a recirculating aqueous stream and provide a useful model for use as a design tool. Magnetically Assisted Fluidization and Consolidation in Microgravity. In conventional fluidized beds, fluidization conditions result from the interaction of three primary forces: the gravitational force (Fg), the buoyancy force (Fb), and the drag force (Fd). Two types of magnetic forces can be generated when a fluidized bed that contains ferromagnetic particles is placed in a magnetic field: (i) interparticle magnetic force and (ii) external (field-to-particle) direct magnetic body force. In a uniform magnetic field, only magnetic interparticle forces can be generated, whereas in a nonuniform (i.e., gradient, magnetic field) both interparticle and direct magnetic body forces are generated. In a uniform magnetic field, such as that typically used in magnetically stabilized fluidized beds, ferromagnetic fluidization media are magnetized, resulting in interparticle attractive and repulsive forces (Fi) that promote the formation of particle chains and clusters, which behave as more-massive particles.39 Detailed derivation of the expression describing attractive and repulsive interparticle forces is given by Pinto-Es-

Figure 1. Balance of forces acting on a fluidized particle containing ferromagnetic material in (a) a fluidized bed in microgravity in the absence of a magnetic field and (b) a gradient magnetically assisted fluidized bed in microgravity. V and U represent velocities of particles and fluid, respectively.

pinoza.47,48 The most significant effect of the magnetic interparticle forces is the change of the bed structure, which is due to the formation of multiparticle chainlike clusters (see Figure 2, presented later in this work). An additional direct body magnetic force (Fm) may be imposed on granular ferromagnetic media by a magnetic field gradient,

Fm ) µ0M∇H

(1)

where µ0 is the permeability of free space, M the magnetization, and H the magnetic field strength. The magnitude and orientation of this force is dependent primarily on the direction, strength, and gradient of the magnetic field, and on the magnetic susceptibility of the fluidization particles. Simple quantitative analysis of these forces suggests some obvious consequences. For example, to sustain the same quality of fluidization due to an additional magnetic force in the same direction as the gravitational force, one must increase fluid velocity, to create sufficient drag force to balance the magnetic force. This, in turn, will increase the relative velocity of particles and fluid, which readily increases mass transfer.41 The additional magnetic body force is essential to the operation of a fluidized bed in microgravity, and it is also beneficial in other gravitational environments. Also, by increasing the magnitude of the magnetic force beyond that required to balance drag, the ferromagnetic media may be consolidated into a packed bed. Under microgravity conditions, the gravitational force is no longer significant; however, the drag force still acts on the fluidized particle, as shown in Figure 1a. Under these conditions, the balance of forces no longer exists, and the particles will be immediately swept away in the direction of the fluid flow, unless otherwise confined. In microgravity, stable fluidization conditions can be restored by introducing an additional force, such as a magnetic force Fm, to oppose the drag force and thereby reinstate the balance of forces on the fluidized particle. Such a magnetic force, acting on the ferromagnetic particles, can be created simply by placing magnetically susceptible particles into a nonuniform magnetic field, as shown in Figure 1b. In previous work, we have shown that, given a suitable variable field profile, the resulting magnetic field gradient can create sufficient magnetic force, acting on the ferromagnetic particles to replace or supplement the gravitational force.1,2 Therefore, the application of suitably designed magnetic field gradients makes feasible a fluidization operation in the absence of gravity, or a creative enhancement of fluidized-bed performance in normal or variable gravity. We call this

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Figure 2. Chain formation in a magnetically assisted fluidized bed. In the absence of the field (left), all particles are “free” and randomly distributed. When the field is applied (right), the magnetically susceptible particles form chains. White, nonmagnetic particles are added to enhance contrast.

variation of the magnetically stabilized fluidized bed “the gradient magnetically assisted fluidized bed” (GMAFB). These methods can be used to control the degree of fluidization or, conversely, the degree of consolidation of granular ferromagnetic media in the absence of gravity. The arrangement of ferromagnetic particles in fluidized beds subjected to a magnetic field differs from that in a conventional fluidized bed. When ferromagnetic particles are magnetized and fluidized in the presence of a homogeneous or gradient magnetic field, these particles have a tendency to form chains oriented along the magnetic field lines.39 These chains are produced by attractive interparticle forces (Fi). In this study, the additional direct body force (Fm) is produced by a magnetic field gradient. The magnetic field is applied by a series of electromagnetic coils that is designed to have a field intensity that is the strongest at the bottom of the bed and decreases gradually toward the top of the bed. This configuration orients the magnetic body force in the direction opposite to the fluid flow and, therefore, opposes the drag force (Fd). By varying the intensities of the magnetic field and field gradient, and by adjusting the fluid flow rate, it is possible to achieve any desirable degree of fluidization or bed compaction. In our application, we are concerned with the filtration of suspended particulate matter from a recirculating aqueous stream containing inedible plant biomass. Solids are trapped within the magnetically consolidated depth filter. The ferromagnetic particles have a tendency to align themselves along magnetic field lines and to form particle chains. The structure of the packed bed that arises from particle chains resembles a more highly ordered particle architecture, as opposed to the mostly random distribution of particles in ordinary packed beds. A collinearly oriented chain structure presents less opportunity for the filtration of suspended particles; thus, we expect that the filtration rates observed in magnetically consolidated packed beds will differ considerably from ordinary packed-bed filtration. Beds in which interception and inertia are the predominant filtration mechanisms are particularly susceptible. Figure 2 illustrates the formation of chains in an incipiently fluidized bed that consists of white (for contrast) nonferromagnetic and black ferromagnetic particles. After the filter is sufficiently loaded with solids, the filter can be regenerated by reducing the magnetic field and field gradient intensities to evoke magnetically assisted fluidization, thereby releasing a concentrated slug of solids suitable for decomposition by a variety of

Figure 3. Schematic illustration of the gradient magnetically assisted filtration bed (G-MAFB) system, with respect to filtration model development.

methods. When the G-MAFB system is free from nonferromagnetic solids, the next filtration cycle can be performed. In this way, by carefully controlling the degree of bed expansion using magnetic forces, the G-MAFB system serves as a renewable filter for the separation and concentration of solid waste materials. Filtration Mechanisms. The primary mechanisms by which a collection of waste particles (filtration) may occur include interception, inertial impaction, Brownian diffusion, and electrostatic attraction.42,43 For the case in which all the forces acting on a particle (relevant to filtration) in a fluid stream are negligible, whenever the streamline along which the particle approaches a filter element passes within a distance of one-half the particle diameter from the element, interception of the particle by the filter element will occur. The relative importance of interception as a mechanism for particle deposition is determined by the ratio of diameters of particle and filtration media:

NR )

Dp dp

(2)

where Dp and dp are diameters of solid waste and filtration media, respectively. The relative importance of inertial impaction, as a mechanism for particle deposition in the filter, is described by the Stokes number (Nst):

Nst )

FstrUDp2 18µfdp

(3)

where Fstr, µf, and U symbolize density of the biomass, the fluid viscosity, and the nominal fluid velocity, respectively. The Peclet number (NPe) is used to characterize the importance of Brownian motion in a filtration process:

NPe )

Udp Dp

(4)

If a particle and the filtration medium carry electrostatic charges, the electrostatic force between the particles may also influence the filtration process. In this study, we determined that electrostatic charges were insignificant.44 Filtration Process. A schematic representation of the G-MAFB filtration process is shown in Figure 3. System boundary I represents the region of the experimental apparatus, including holding tank, pump, and flow meter, where only the solid waste particles are

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System boundary I: FC*(L,t) - FC*(0,t) ) Vtank

∂C*(0,t) (10) ∂t

C*(0,0) ) C0

(11)

where Vtank is the volume of the holding tank and F is the volumetric flow rate of the liquid.

System boundary II: ∂C*(x,t) ∂σ ∂(C*(x,t)) - U0 - a′ ) ∂x ∂t ∂t

Figure 4. Solid waste accumulated in the G-MAFB consolidated filter bed.

present. It is assumed that, in this volume, the fluid is very well mixed and, hence, the suspended solids concentration is uniform. System boundary II encompasses the G-MAFB filter bed, where filtration occurs. During the filtration process, the ferromagnetic media are tightly consolidated via the applied magnetic body force Fm. The biomass waste particles (comminuted wheat straw) in the recirculating water stream are captured within the intergranular spaces between particles, forming pockets of captured solids. Figure 4 shows an accumulation of solids as seen through the wall of the column in one of the experiments performed in this study (electromagnets are removed). We represent the kinetics of attachment as a first-order rate process with the characteristic filtration rate constant k1,

ratt ) -

dC*(x,t) ) - k1C*(x,t) dt

(5)

and with the initial conditions

C*(x,0) ) 0; t ) 0, 0 < x e L

(6)

C*(0,0) ) C0; t ) 0, x ) 0

(7)

where ratt, C*, t, and L respectively represent rate of attachment, biomass concentration in the filter bed, time, and length. Concurrently, some portion of the previously captured solids may detach and re-enter the recirculating fluid stream. We represent the rate of detachment (rdet) as a first-order process with respect to the concentration of captured solids on the filter media surface, σ(x,t) with the characteristic detachment rate constant k2,

rdet ) -

dσ(x,t) ) k2σ(x,t) dt

(8)

and the initial conditions

σ(x,0) ) 0; t ) 0, 0 e x e L

C*(x,0) ) 0; t ) 0, 0 < x e L

(13)

C*(0,0) ) C0; t ) 0, x ) 0

(14)

C*(0,t) ) C(t); t > 0, x ) 0

(15)

where a′ represents particle surface area per unit filtration bed volume, U0 the superficial fluid velocity, and a local bed voidage. Note that is a function of position and time, i.e., (x,t). Equation 16 may then describe the rate of filtration,

∂σ k1C*(x,t) ) - k2σ ∂t a′

(16)

with the corresponding initial conditions

σ(x,0) ) 0; t ) 0, 0 e x e L

(17)

C*(x,0) ) 0; t ) 0, 0 < x e L

(18)

As a first approximation, we assume that, after the solids are deposited on the surface of the particle filtering media, they form a thin layer around the surface of the filtration media; hence, the effective filter particle diameter increases as deposited solids accumulate. This variation of media diameter can be expressed as

dp(x,t) ) dp(x,0) +

σ(x,t) Fstr

(19)

Thus, the voidage of the filter bed at any given time can be represented as

dp3(x,t) ) 1 - (1 - 0) 3 dp (x,0)

(20)

where 0 is initial voidage distribution in the bed and Fstr is the density of the accumulated solid deposition. As we have shown in previous work,2 is also related to the magnetic field strength by the relation

3 - φ1(1 - ) - φ2 - φ3Hz3 ∂ ) ∂z/L φ Y3

(9)

The material balances for both system boundaries can be written in the form of partial-differential equations in the axially symmetric filter bed, as

(12)

4

where

(21)


150µfU0

φ1 ) φ2 ) φ3 )

(22a)

dp2(Fp - Ff)g 1.75FfU02

(22b)

dp(Fp - Ff)g µ0χ

∂Hz (Fp - Ff)g ∂z

(22c)

1 (Fp - Ff)gL

(22d)

φ4 )

with the boundary conditions )1 at z/L ) 1. U0 represents the superficial fluid velocity, and Fp, and Ff represent the densities of the filtration media (ferromagnetic particles) and fluid, respectively. In the absence of gravity (microgravity, ∼0 g), one can obtain

∂ ) ∂z

- η1(1 - ) - η2 - η3H3 Y3

∂H ∂z

(23)

where

η1 ) η2 )

150µfU0 dp2

1.75FfU02 dp

η3 ) µ0χ

(24a)

(24b) (24c)

with the boundary conditions )1 at z/L ) 1. Y represents the apparent elastic modulus of the ferromagnetic filtration media, and χ is the magnetic susceptibility of the filtration media. Details related to the evaluation of the elastic modulus of the filtration media in G-MAFB have been presented by Sornchamni.44 Equations 10-20 constitute a self-consistent model of the filtration process and are solved numerically using an Integrated Development Environment (IDE) application tool (Integrated Development Environment, Absoft), and IMSL Fortran Numerical Library v5.0 (Visual Numerics). Equations 21 and 22 are then used to evaluate necessary magnetic field strength and magnetic field gradient that are required to create voidage distribution for given operating conditions and material properties. The accumulation and detachment coefficients (k1 and k2, respectively) are evaluated from experimental data using a multiparameter optimization algorithm from the IMSL Fortran Numerical Library. Experimental Section The filtration experiments were conducted in a closed recirculating G-MAFB system with a constant magnetic field gradient. A schematic illustration of the experimental apparatus is shown in Figure 5. Solid waste particles are suspended in a well-mixed holding tank. An initially particle-laden aqueous stream recirculates from the holding tank to the G-MAFB column and back to the holding tank. The optical density of suspended particles in the liquid, as a function of time, is monitored using a laser-photodiode detector in which the output voltage is inversely proportional to the concentration of

Figure 5. Gradient-magnetically assisted fluidization/filtration bed (G-MAFB) apparatus used during filtration experiments.

solid waste particles traversed by the laser beam. The G-MAFB system is surrounded by a series of electromagnetic coils and contains spherical ferromagnetic filtration media consisting of ferrite-impregnated calcium alginate beads.2 The bed may be fluidized or compacted, depending on the applied magnetic field and field gradient intensities. The G-MAFB column is composed of Plexiglas, which allows for direct visual observation through the wall. The column has an inside diameter of 5.04 cm and an outside diameter of 5.80 cm. The distributor plate, which consists of a plastic mesh with 2-mm-square openings, is located at the bottom of the bed and can be easily removed or repositioned to any location along the column. The distributor plate material and the hole size were carefully chosen to prevent undesired particle deposition within the distributor. The magnetic field generator is composed of three direct-current (DC) power supplies connected to six parallel sets of electromagnet coils. These electromagnets can be positioned at any location along the GMAFB column. At any given location within the bed, the overall magnetic field intensity is the superposition of the magnetic field intensities produced by each individual coil. The magnetic field intensity is strongest at the bottom of the bed and decreases gradually and linearly toward the top of the bed. This produces a magnetic body force on the ferromagnetic media oriented downward toward the distributor plate. The degree of consolidation or fluidization of the contained media is controlled by the strength of the applied field and field gradient, which is easily varied by adjusting the current to each set of coils. In this study, a constant axial magnetic field gradient was produced by adjusting the currents and the spacings between the six electromagnets. The magnetic field profile along the vertical axis used for filtration with the G-MAFB is shown in Figure 6. Ferromagnetic fluidization/filtration media were produced by the dropwise extrusion of a mixture of zinc-manganese ferrite and sodium borosilicate glass hollow microspheres suspended in sodium alginate solution, into a calcium chloride cross-linking solution.2 Two types of ferromagnetic particles were used in the filtration experiments and their properties are summarized in Table 1. During the filtration process, the bed is kept in a tightly consolidated (packed bed) condition using magnetic forces directed toward the G-MAFB distributor

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Ind. Eng. Chem. Res., Vol. 44, No. 24, 2005 Table 2. Comminuted Wheat Straw Particle Size Distributions (Ferret Method45) Batch No. 1

Batch No. 2

particle cumulative cumulative diameter weight weight weight weight range (µm) percent (%) percent (%) percent (%) percent (%)

Figure 6. Axial (vertical) magnetic field profile used within the G-MAFB system during filtration experiments. Table 1. Summary of Ferromagnetic Particle Properties Value property

particle A

particle B

diameter density ferrite loading glass microsphere content gellan gum content 1.75% alginate solution content minimum fluidization velocity, Umf magnetic susceptibility, χ

2.5 mm 1351 kg/m3 35 wt % 3.5 wt % 0.3 wt % 61.2 wt % 0.0123 m/s 5.80

3.5 mm 1351 kg/m3 35 wt % 3.5 wt % 0.3 wt % 61.2 wt % 0.0174 m/s 5.80

plate. During filtration, biomass (wheat straw) waste particles are deposited in the void spaces among the ferromagnetic spheres. Experiments consisted of tracking the particle density within the holding tank as a function of filtration time. Two different size ranges of inedible plant biomass waste particles (wheat straw) were used. The smaller biomass particles (batch 1) included those particles that passed through a 180-µm standard sieve but were retained on a 149-µm standard sieve. The larger biomass particles (batch 2) passed through and were retained by 295-µm and 180-µm standard sieves, respectively. These lignocellulosic particles exhibit a long thin fibrous shape; therefore, the mesh fractions indicated above do not reflect particle diameters. In previous work, three different size measurement methods were evaluated for comminuted wheat straw.44 It was determined by Hinds45 that the characteristic length defined by the Ferret’s diameter provides the most useful description of particle size for this highly irregular material. The Ferret’s diameter is defined as the distance between the extreme left and right tangents that are perpendicular to the reference line. Table 2 summarizes the particle size distributions for each batch, based on this criterion, as determined by microscopic examination. The average diameters of the solids particles in batches 1 and 2 are 430 and 609 µm, respectively.

1000-1050 950-1000 900-950 850-900 800-850 750-800 700-750 650-700 600-650 550-600 500-550 450-500 400-450 350-400 300-350 250-300 200-250 151-200

0.00 0.00 0.00 0.00 1.76 0.80 1.52 1.29 4.98 6.83 15.69 14.43 19.41 16.25 12.06 3.58 1.40 0.00

100.00 100.00 100.00 100.00 100.00 98.24 97.44 95.92 94.63 89.65 82.82 67.13 52.70 33.29 17.04 4.98 1.40 0.00

1.16 0.88 3.68 3.87 6.40 11.85 8.70 6.90 15.56 13.96 7.19 11.97 4.72 1.44 1.18 0.54 0.00 0.00

100.00 98.84 97.96 94.28 90.41 84.01 72.16 63.46 56.56 41.00 27.04 19.85 7.88 3.16 1.72 0.54 0.00 0.00

given magnetic field intensity and gradient, the filtration rate increases as the fluid superficial velocity increases. In each of these experiments, the concentration of the solid waste particles in the holding tank decreased substantially during the filtration process. However, the concentration of solid waste particles remains constant after equilibrium between the attachment and detachment rates in the bed is attained. At this point, the rate of attachment of particulate matter

Figure 7. Time-varying suspended wheat straw biomass concentration in the holding tank (with conditions of dp ) 2.5 mm, Dp ) 430.4 µm, dHz/dz ) - 38 817 A/m2), under packed-bed conditions over a range of superficial fluid velocities (U0).

Results and Discussion A series of filtration experiments was conducted using a fixed magnetic field gradient and nominal (superficial) flow velocities varying over a range of 5.41-13.36 mm/ s. Comminuted wheat straw particles suspended in an aqueous stream were recirculated between the wellmixed holding tank and the magnetically consolidated filter bed. Loading of the filter was indirectly monitored by the reduction in particulate concentration within the holding tank over time, as determined by changes in optical density of the suspension. Results for these filtration experiments are presented in Figures 7-9. It is evident from these data that, at a




Figure 10. Effects of pH on time varying suspended wheat straw biomass concentration in the holding tank (with conditions of dp ) 3.5 mm, Dp ) 430.4 µm, dHz/dz ) - 38 817 A/m2), under packedbed conditions, with U0 ) 8.25 mm/s.

within the filter bed equals the rate of detachment. According to our model, the kinetics of the magnetically assisted filtration process can be characterized by the rate of accumulation of solids within the bed and the rate of detachment of previously filtered particles. The results of all filtration experiments were fitted to the filtration model to determine numerical values for the accumulation and the detachment coefficients. These two parameters are related to several mechanisms involved in the deposition of straw particles onto the surface of the magnetic beads, including diffusion deposition, direction interception, and inertial impaction. Both coefficients are expected to vary as a function of hydrodynamic and geometric factors that characterize the filtration system, such as fluid velocity and the size distributions for both entrapped solids and filtration media. Based on experimental results, we conclude that the accumulation coefficient is related to two mechanisms that are involved in the deposition of solids within the filter: direct interception and inertial impaction. The effect of the electrostatic attraction mechanism was evaluated by adjusting the pH of the recirculating deionized water stream (pH ) 3.6, 7.0, and 9.8) within the filtration system. These changes in pH varied the effective surface electrical charges of the particles. The experimental results shown in Figure 10 indicate that there is no difference in the filtration rate over this pH range. Hence, there is no evidence of an electrostatic mechanism for this filtration process. With respect to a

Figure 11. Correlation of accumulation coefficient (k1) with superficial fluid velocity. Table 3. Values and Statistics for Dimensionless Constants Used in the Correlation Equations for the Accumulation and Detachment Coefficients parameter

estimated value

standard error

T statistic

P value

ln a b c ln e f g

-1.27 2.77 -0.41 -10.35 0.65 -0.25

0.34 0.25 0.09 2.77 0.15 0.048

-3.71 11.29 -4.35 -5.67 1.29 -2.23

0.0008 0.0001 0.0001 0.0001 0.02 0.02

possible Brownian diffusion mechanism, Black42 concluded that, if the value of the Peclet number is >100, the filtration of solids by this mechanism can be ignored. Based on this criterion, the effects of a Brownian diffusion deposition mechanism are negligible in our experiments, since the corresponding Peclet numbers are ∼106 in magnitude. A correlation was derived by applying Buckingham’s π theorem46 and dimensional analysis, which relates the accumulation coefficient (k1) to hydrodynamic and geometric parameters of the filtration system:

( )[

k 1 Dp Dp )a Uint dp

b

]

Uint(Fstr - Ff)Dp2 µfdp

c

(25)

where Uint is the local/interstitial fluid velocity, which is defined as

Uint )

U0

(26)

The values of the dimensionless constants a, b, and c are dependent on characteristic length scales for the suspended solids and the filtration media (Dp and dp, respectively). (All values are given in the international system of units.) The values of a, b, and c, within the range of our experiments, are presented in Table 3. The correlation for values of k1, as a function of the nominal (superficial) fluid velocity, is shown in Figure 11. The analysis leading to the derivation of the correlation expression for the detachment coefficient (k2) stems from the experimental observation that the deposited waste biomass occurs in the form of particle clumps and/ or films, which surround the filtration media. Consequently, we assume that the nominal diameter of waste particles (Dp) can be neglected in the correlation for the detachment mechanism. Local fluid flow conditions, which are characterized by the local velocity Uint and

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Figure 12. Correlation of detachment coefficient (k2) with superficial fluid velocity.

the size of the filtration media (dp), are much more likely to have a significant role in the detachment mechanism. Again, the application of Buckingham’s π theorem and dimensional analysis led to the correlation encasing the interdependence among the coefficient k2, Uint, and dp:

( )(

k2dp dp )e Uint Dcol

f

)

dpUintFf µf

g

(27)

This correlation provides reasonable agreement with the experimental data, as illustrated in Figure 12. The nominal diameter of the solid waste particles is not included in this correlation. The standard errors for coefficients e, f, and g (Table 3) are larger than the corresponding errors obtained in the correlation for the accumulation coefficient k1. This is because the accuracy of the correlation for the detachment coefficient k2 is predominantly dependent on the accuracy of data at the end of the filtration process. Because the concentration of suspended solids in the holding tank at the end of the filtration process was typically very low, the relative experimental error for these measurements was relatively high. Detailed derivations leading to the correlations presented in eqs 25 and 27 are presented elsewhere.44 To test the predictive power of the filtration process model and the previous mentioned correlations, a test was performed in pilot-scale equipment (Dcol ) 7.0 cm, V ) 8000 cm3, dHz/dz ) -23 280 A/m2), which was much larger than the equipment used in previously described experiments. Figure 13 shows the experimental results obtained in this test and the prediction of the filtration process model. The coefficients of attachment and detachment were calculated from correlations in eqs 25 and 27. Excellent agreement between experimental data and model predictions was achieved. Conclusion The feasibility of solid waste filtration using a magnetically consolidated depth filter has been successfully demonstrated. Methods based on gradient magnetically assisted fluidized bed (G-MAFB) technology were used to separate inedible plant biomass waste particles from a recirculating liquid stream. Within the range of fluid velocities investigated in the filtration experiments, we observed that the rate of filtration increased with fluid velocity. A mathematical model was developed that described the filtration process and correlation equations were derived for the estimation of attachment and

Figure 13. Predicted changes of solid waste mass in the G-MAFB system with time (with the following conditions: dp ) 2.7 mm, Dp ) 430.4 µm, C0 ) 0.782 mg/cm3, Dcol ) 7.0 cm, Vtank ) 8000 cm3, U0 ) 0.01957 m/s, L ) 0.35 m, and dHz/dz ) -23 280 A/m2).

detachment coefficients, based on hydrodynamic and geometrical factors. Predictions of the model and correlation equations provided good agreement with the results of filtration experiments. This work forms the basis for the development of microgravity- and hypogravity-compatible solid waste separation and concentration methods in support of future long-duration manned spaceflight and extraterrestrial planetary habitation. Acknowledgment This work was sponsored by the U.S. National Aeronautics and Space Administration (under Grant Nos. NAG9-1181 and NAG9-1472). The authors thank Mr. John W. Fisher (NASA-Ames Research Center) for his helpful advice and support. Literature Cited (1) Sornchamni, T.; Jovanovic, G. N.; Reed, B. P.; Atwater, J. E.; Akse, J. R.; Wheeler, R. R., Jr. Operation of Magnetically Assisted Fluidized Beds in Microgravity and Variable Gravity: Experiment and Theory. Adv. Space Res. 2004, 34, 1494. (2) Jovanovic, G. N.; Sornchamni, T.; Atwater, J. E.; Akse, J. R.; Wheeler, R. R., Jr. Magnetically Assisted Liquid-Solid Fluidization in Normal and Microgravity Conditions: Experiment and Theory. Powder Technol. 2004, 148, 80. (3) Jovanovic, G. N.; Pinto-Espinoza, J.; Sornchamni, T.; Reed, B. P.; Wheeler, R. R., Jr.; Atwater, J. E.; Akse, J. R. Development of Enabling Technologies for Magnetically Assisted Gasification of Solid Wastes; SAE Technical Paper Series No. 2003-01-2374; Society of Automotive Engineers (SAE): Warrendale, PA, 2003. (4) Atwater, J. E.; Akse, J. R.; Jovanovic, G. N.; Wheeler R. R., Jr.; Sornchamni, T. Porous Cobalt Spheres for High Temperature Gradient Magnetically Assisted Fluidized Beds, Mater. Res. Bull. 2003, 28, 395. (5) Atwater, J. E.; Akse, J. R.; Jovanovic, G. N.; Sornchamni, T. Preparation of Metallic Cobalt and Cobalt-Barium Titanate Spheres as High Temperature Media for Magnetically Stabilized Fluidized Bed Reactors. J. Mater. Sci. Lett. 2001, 20, 487. (6) Jovanovic, G. N.; Sornchamni, T.; Yoo, S.; Atwater, J. E.; Akse, J. R.; Dehart, J. L.; Fisher, J. W. Magnetically Assisted Gasification of Solid Waste; SAE Technical Paper Series No. 199901-2183; Society of Automotive Engineers (SAE): Warrendale, PA, 1999. (7) Fisher, J. W.; Pisharody, S. A.; Abraham, M. A. Particle Size Effect on Supercritical Water OxidationsWheat Straw Particles; SAE Technical Paper Series No. 951739; Society of Automotive Engineers (SAE): Warrendale, PA, 1995. (8) Fisher, J. W.; Pisharody, S.; Wignarajah, K.; Lighty, J. S.; Burton, B.; Edeen, M.; Davis, K. A. Waste Incineration for Resource Recovery in a Bioregenerative Life Support System; SAE Technical

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Received for review February 4, 2005 Revised manuscript received September 12, 2005 Accepted September 15, 2005 IE0501374

Magnetically Assisted Filtration for Solid Waste Separation and

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