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Article

Application of High-Gradient Magnetic Separation for the Recovery of Super-Paramagnetic Polymer Adsorbent Used in Adsorption and Desorption Processes

1
Graduate Institute of Environmental Engineering, National Taiwan University, Taipei 106, Taiwan
2
Department of Occupational Safety and Health, China Medical University, Taichung 404, Taiwan
3
Department of Environmental Science and Engineering, Tung-Hai University, Taichung 407, Taiwan
4
Department of Chemical Engineering and Biotechnology, National Taipei University of Technology, Taipei 106, Taiwan
5
Department of Environmental Engineering, National I-Lan University, Yi-Lan 260, Taiwan
6
Department of Bioengineering and Biosystems, Karlsruhe Institute of Technology, 76344 Eggenstein-Leopoldshafen, Germany
*
Author to whom correspondence should be addressed.
Processes 2023, 11(3), 965; https://doi.org/10.3390/pr11030965
Submission received: 18 February 2023 / Revised: 17 March 2023 / Accepted: 19 March 2023 / Published: 21 March 2023
(This article belongs to the Special Issue Advanced Liquid Waste and Gas Waste Treatment Processes)

Abstract

:
This study examined the application of high-gradient magnetic separation (HGMS) for recycling of super-paramagnetic polymer adsorbent (MPA), namely, polyvinyl acetate-iminodiacetic acid. The HGMS can be incorporated with the adsorption and desorption processes (ADPs) with fresh or regenerated desorbed MPAs and exhausted adsorbed MPAs, respectively. This combines the permanent magnet’s advantage of low running costs with the easy operation using the solenoid to flush the filter in place. The effects of the inlet concentration of MPA in solution (CLF,i) and the fluid velocity (v0) or volumetric flow rate (QLF) on the performance of the recovery of MPA via HGMS were assessed. The results indicated that the separation efficiency (η or P0), breakthrough time (tB) and exhaustion time (tE) of HGMS reduce as CLF,i, as well as v0, increases. Further, the filter saturated capture capacity (σS) of HGMS also decreases with increasing v0. The effect of v0 on tB proportional to 1/v02 is more significant than that on σS proportional to 1/v0. A kinetic model of HGMS shows good agreements for the experimental and predicted breakthrough results, with determination coefficients of 0.985–0.995. The information obtained in this study is useful for the rational design and proper operation of a HGMS system for the recycling and reuse of MPA in ADPs.

1. Introduction

In the past, magnetic separation (MS) was used for magnetic mineral separation in the metallurgical industry [1], the de-ironization of kaolin in the paper industry [2] and pyrite removal in the coal mining industry [3]. Recently, due to the use of magnetic particles as a carrier with the affinity ligand on the surface, MS technology has been widely applied in biochemistry [4], biomedical engineering [5,6,7] and environmental engineering, including protein purification [8], separation of lactoferrin [9], cell separation [10], immunoassay [11], enzyme immobilization [4], affinity separation [12,13,14,15,16] and waste water treatment [17,18,19,20,21,22,23,24]. MS uses a magnetic driving force to separate magnetic and non-magnetic materials. In order to increase its efficiency, high-gradient magnetic separation (HGMS) technology has been rapidly developed [8,9]. By filling magnetic matrices into the MS chamber, it can generate a very high magnetic field strength (or magnetic field intensity, H) and gradient with a strong capture ability for magnetic particles.
The motion of magnetic particles in a fluid can generally be controlled by two forces. One is the magnetic force (Fm = μ0 Vp Mp‧▽H). The other is the viscous drag force (viscous resistance force) (Fd), which can be obtained by the Stokes equation (Fd = 3πμf dpvp/f). In these notations, μ0 is the magnetic permeability of the vacuum (= 4π × 10−7 H m−1); Vp (= (1/6) π dp3) is the volume of a magnetic particle; dp is the particle diameter; Mp (= (χpχf) H) is the magnetization of particles; χp and χf are, respectively, the magnetic susceptibilities of the particle and fluid; H is the magnetic field strength; μf is the fluid viscosity; vp/f is the particle velocity relative to the fluid; ▽H is the magnetic field strength gradient at the location of particle. It can be seen from the above equations that the larger the dp, the stronger the Fm, or the lower the μf, the weaker the Fd, resulting in the easier separation of magnetic particles from the fluid. The increase of the H and gradient of H can promote effective separation. The commonly used MS devices can be classified into four types, namely, intermittent [25,26], mobile [10,27,28], high gradient [29,30,31,32] and fluidized bed [33,34].
Previous works studied the syntheses of the magnetic polymer adsorbent (MPA) of polyvinyl acetate-iminodiacetic acid (noted as M-PVAC-IDA) and the chemical modification to enhance the affinity of the adsorbent to adsorb copper (II) from an aqueous solution [17,18]. Meanwhile, the adsorption isotherms were set up. In this study, the HGMS was applied for the recycling and reuse of MPA during the adsorption and desorption processes (ADPs). The use of NdFeB (neodymium-iron-boron) magnets in the yoke allows high magnetic inductions, leading to the efficient and fast separation of magnetic adsorbent particles used in ADPs. It may offer the high recycling efficiency of MPA with low operating costs. Another advantage of HGMS is that the non-magnetic impurities can be excluded from MPA during the recovery of MPA. The novel design using a rotary permanent magnet leads to an “on-off” characteristic of the magnetic field in the separation zone. The HGMS of M-PVAC-IDA (denoted as MPA hereafter) was conducted with various inlet concentrations of MPA (CLF,i) of 0.94–4.14 g L−1 and volumetric flow rates (QLF) of 0.417–0.833 L min−1 (or flow velocities v0 of 0.866–1.731 m min−1). The corresponding outlet concentrations of MPA (CLF,e) were measured. The separation efficiency of MPA (η or P0), breakthrough time (or effective separation time) (tB), exhaustion time (or saturation time) (tE) and filter saturated capture capacity (σS) for HGMS were examined and elucidated for different CLF,i and QLF. In addition, a kinetic model was applied to describe the breakthrough behaviors of the effluent of HGMS and compared with the experimental results.

2. Materials and Methods

2.1. Materials

The MPA (i.e., M-PVAC-IDA) used was synthesized via suspension polymerization using super-paramagnetic Fe3O4 gel and specific chemicals. These included the following. Oleic acid, epichlorohydrin, divinylbenzene and vinyl acetate (VAC) were purchased from Aldrich (Sigma-Aldrich Inc., St. Louis, MO, USA). Iminodiacetic acid (IDA) was obtained from Sigma (Sigma-Aldrich Inc., St. Louis, MO, USA). The MPA is nearly non-porous, with an insignificant porosity of 0.003 and a specific area of the external surface of 12.9 m2 g−1. The density of the MPA is about 1.63 g cm−3. The size of the MPA particles made is about 500 nm to 2 μm and mostly about 1 μm in number. For the details of the synthesis and other properties of MPA, refer to the previous studies [17,18].

2.2. Device

A HGMS incorporated with adsorption and desorption operations is illustrated in Figure 1. A laboratory-type (Steinert HGF-10 1, Cologne, Germany), permanent, magnet-based HGM separator is used for this study. It is a new type of HGMS separator designed using switchable permanent magnets. The separator operates in a cyclic fashion, making it suitable for suspensions with low and moderate concentrations of magnetic particles. The magnetic flux density (B) is 0.3 Tesla. The pole gap is 25 mm, with a pole shoe area of 100 × 80 mm2. The total weight is 75 kg. The motor power is 0.12 kW.
A filter chamber with size 2.75 × 1.75 × 7.85 cm3 (volume of filter chamber or cell volume Vc = 37.78 cm3, height of filter chamber L = 7.85 cm, cell cross-section area Ac = 4.8125 cm2), containing magnetic matrices, is placed vertically between the poles of the magnet. The matrices are surrounded by a sheet metal housing forming chambers for flow distribution. The matrix filling factor F with all received matrices of mass mwm of 34.5466 g filled in the chamber is 0.1157. The supply and discharge connectors are placed at the ends of the flow distribution chambers. The density of the magnetic wire of the matrix ρwm is 7900 kg m−3.

2.3. Experimental Conditions

The MPA recovery performance was investigated by monitoring the breakthrough curve of MPA containing fluid. The MPA particles were kept in suspension by continuously mixing the fluid in the batch HGM separator. During operation, the suspension was pumped through the filter chamber until a certain pressure drop was reached or as MPA-containing fluid reached complete breakthrough. After the breakthrough of the MPA, the MPA particles captured by the filter were recovered by a short intensive rinsing in the counter flow direction, with the magnet system switched to “off” until the effluent was clean. Then, the filter chambers were taken out of the magnet system, dismantled and thoroughly cleaned. Finally, the cleaned filter chambers were re-installed, and the magnet was switched to “on” again to start a new filtration cycle.
About 20 g MPA were used to perform the experiments of MS under the following conditions, with various inlet particle concentrations CLF,i and volumetric flow rates QLF. For the influences of CLF,i, the HGMS experiments were conducted at CLF,i of 4.14, 3.21, 2.06, 1.51 and 0.94 g L−1 with QLF of 0.833 L min−1. As for the effects of QLF, the cases with a QLF of 0.833, 0.545 and 0.417 L min−1, respectively corresponding to the fluid velocities v0 of 1.731, 1.132 and 0.866 m min−1, were examined at CLF,i of 2.06 g L−1.

2.4. Analytical Methods

The properties of the MPA were measured by a Superconducting Quantum Interference Device (MPMS7, Quantum Design, Inc., San Diego, CA, USA). The concentration of MPA in liquid was monitored by UV/VIS spectrophotometer (Cintra-20, GBC Scientific Equipment Pty Ltd., Braeside VIC 3195, Australia) at a specific spectrum wave length of 245 nm for assaying the optical density (OD). Additionally, dry weight (DW) measurements were carried out from pooled samples collected during the throughput operation in order to check the relationship of OD/DW via a calibration curve.

2.5. Theoretical Background

2.5.1. Watson and Gerber Theory

There are two ways generally used to describe a HGMS. Firstly, a microscopic level focuses on the forces responsible for the attraction and capture of a single particle on the magnetized wire [35,36]. The other approach uses a macroscopic view, describing the whole filter based on the breakthrough behavior of the filter [37,38].
According to the theory of MS established by Watson [35] and Gerber and Lawson [36], the relationship between the inlet concentration CLF,i and outflow concentration CLF,e of the magnetic particles in the HGM separator can be expressed as the following equation:
P = CLF,e/CLF,i = exp [−fFRcaL/(πa)]
In Equation (1), P is the penetration of the magnetic particles in fluid; (1 − P = η) is the separation efficiency); f is the arrangement factor of magnetic filter media and 4/3 for random filters; F, a and L are the filling factor (or filling density), wire radius and height (or thickness) of magnetic media in the separation chamber, respectively; Rca (= rc/a) is the dimensionless normalized particle capture radius (rc). Rca can be expressed as:
Rca = (1/2)(vm/v0)(a/aB)2
where
(aB/a)4 = (vm/A1)t + 1
Rca = (1/2)(vm/v0)/[(vm/A1)t + 1]1/2
A1 = [β ρp a/(4 CLF,i)]
In the above equations, β is the particle aggregation factor with a value of 0.1–0.18; ρp is the density of the magnetic particles (kg m−3); vm is the magnetic velocity (a function of magnetic field strength (H0) and magnetization of wire of matrix (Mwm)) (m s−1); v0 is the flow velocity (m s−1); aB is the actual wire radius of the magnetic media with the buildup of particles (m); t is the filtration time (also called operation time or separation time) (s).

2.5.2. Mass Balance Approach for Kinetic Models of HGMS

This method is based on a mass balance for the particle suspension carried out over the whole filter volume. For deep-bed filtration, the mass balance of the particles in the MS chamber can be derived and represented with the following equations.
( A c   d x   ε F ) C t = A c   v 0   C ( t , x )     A c   v 0   C ( t   +   d t , x   +   d x )     σ t   A c d x
This gives
ε F C t =   v 0 C x σ t
or
ε F C t + σ t   +   v 0 C x = 0
Assuming εF C t is negligible compared to the other terms, i.e., εF C t ~ 0, in the dx interval and defining τ = t − (εF x/v0), Equation (8) can be simplified as Equation (9).
σ τ   +   v 0 C x = 0
The time variation of σ increases with v0, C and the term σS − σ, as described below.
σ τ =   λ   v 0 C
λ = λ0 [1 − (σ/σS)]
It can be calculated from Equations (9)–(11) to obtain the solution of the outlet concentration CLF,e as:
CLF,e/CLF,I = [exp(CLF,i v0λ0τ/σS)]/([exp(CLF,i v0λ0τ/σS)] + [exp(λ0L)] − 1)
In the above equations, σ is the capture capacity of a magnetic particle on the magnetic media in a magnetic separation cell at time τ; σS is the saturated σ (kg m−3); λ is the characteristic length (m−1); λ0 is the characteristic length at τ = 0; εF is the void fraction of the magnetic matrix; v0 is the flow velocity (m s−1); Ac is the section area of the separation chamber of the magnetic media (m2); x and L, respectively, stand for the position at a certain point in the x direction and the total length of the magnetic media separation chamber (m); C is the concentration at a certain point at the magnetic separation time t (mg L−1); CLF,i and CLF,e are the inflow and outflow concentrations, respectively. The detailed solution of Equation (12) is described in Appendix A.
If the matrix is considered as a single wire, the characteristic length (λ) is related to the capture radius (Rca) and the wire radius of magnetic matrix (a). It can be represented by Equation (11) with Equation (13) as below.
λ0 = (2/π) (1 − εF) (Rca/a)

2.5.3. Separation Time

Equations (1) and (12) indicate that the penetration of the magnetic particles in the fluid, P, increases with time. The separation time (tη) at a set separation efficiency η based on Equations (1)–(5) can be obtained as follows.
tη = A1(A2)2  L2 (vm/v02)/ln(η2)
For a given tη, v0 must satisfy:
v0 ≦ (A1)1/2 A2 L (vm/tη)1/2/[ln(η2)]1/2
where
A2 = fF/(2πa)
vm = 2μ0 (χfχp) Mwm H0 rp2/(9μf a)
In the above equations, H0 is the magnetic field strength, or the magnetic field intensity (Oe) applied in the x direction perpendicular to the filter; Mwm is the magnetization of the wire of the matrix (emu g−1 or emu cm−3); rp is the radius of the paramagnetic particle (μm or nm); μf is the viscosity of the fluid (lbm ft−1, or kg m−1s−1 or N s m−2); μ0 is the magnetic permeability of the vacuum (=4π × 10−7 H m−1); χf is the magnetic susceptibility of the fluid (emu cm−3 Oe−1); χp is the magnetic susceptibility of the particle (emu cm−3 Oe−1).

3. Results and Discussion

3.1. Hysteresis Curves of M-PVAC-IDA before and after ADPs

An atom of any substance has its magnetism; therefore, it can be said that all substances are the magnets, and the magnetic extent is dependent on the magnetic strength. Figure 2 shows the comparison of the hysteresis curves of different types of M-PVAC-IDA. The saturation magnetization of Cu(II)-adsorbed M-PVAC-IDA decreases only slightly compared to that of fresh M-PVAC-IDA. Thus, the effect of adsorbed copper on the magnetism of M-PVAC-IDA is insignificant.
However, long-time storage of M-PVAC-IDA resulted in the partial oxidation state of iron and, thus, a significant decline of its saturation magnetization. Figure 2 indicates that the saturation magnetization of M-PVAC-IDA after 2 years of storage time decreased nearly 50%. Therefore, ensuring the preservation of M-PVAC-IDA to maintain its saturation magnetization is an important work for the recycling of ferrite magnetic material or magnetite.

3.2. HGMS at Different MPA Concentrations

Figure 3 illustrates the variation of penetration P (= CLF,e/CLF,i) with an accumulated volume of liquid filtrated (VLF, expressed as cell volumes (Vc)) at different inlet MPA concentrations CLF,i (4.14, 3.21, 2.06, 1.51 and 0.94 g L−1), with a volumetric flow rate QLF = 0.833 L min−1. A higher CLF,i shows an easier saturation with a smaller VLF. In the case with the highest CLF,i of 4.14 g L−1, the separation cell reaches P = 95% (denoted as P95), with the smallest VLF of about 70.6 Vc. On the other hand, the lowest CLF,i of 0.94 g L−1 gives the largest VLF of about 301.5 Vc at P95. For cases with CLF,i of 1.51, 2.06 and 3.21 g L−1, the corresponding VLF at P95 is 166.7, 131.5 and 90.8 Vc, respectively.
From Equations (2)–(5) of the Watson and Gerber theory, it can be seen that the capture radius Rca is related to the magnetic velocity vm (thus, the magnetic field strength H0 and the magnetization of the wire of the matrix Mwm, as indicated in Equation (17)), flow velocity v0, and saturation magnetization MSp and concentration CLF,i of magnetic particles. When the magnetic field strength and the volumetric flow rate are constant, the actual wire radius aB of the magnetic media increases, while the capture radius Rca decreases as CLF,i increases. Thus, the saturation of the magnetic particles on the wire of the matrix with a higher concentration will be reached quicker (with a smaller VLF).

3.3. The Effect of Volumetric Flow Rate of Liquid on HGMS

Figure 4 presents the variations of penetration P with VLF at various volumetric flow rates of the liquid QLF (0.833, 0.545 and 0.417 L min−1) or the liquid flow velocity v0 (1.731, 1.132 and 0.866 m min−1; = QLF/Ac, Ac = 4.8125 cm2), with CLF,i = 2.06 g L−1. It shows that HGMS is more quickly saturated if the QLF is higher. For v0 = 1.731 m min−1, the VLF at P95 is smallest at 131.5 Vc. As for v0 = 0.866 of m min−1, the VLF at P95 is largest at 191.2 Vc. At v0 = 1.132 of m min−1, the VLF at P95 is 157.3 Vc. The decrease of VLF at P95 with increasing QLF is due to the cause that the capture radiuses Rca of the magnetic matrix become shorter as the QLF or v0 increases, as indicated in Equation (4), resulting in a lower saturation capture capacity and quicker breakthrough and exhaustion.
In addition, Figure 5 depicts the relationship between the breakthrough time tB (taken at P10 (CLF,e/CLF,i = 10%)) and the reciprocal of the square of the fluid velocity (1/v02). The results indicated that the effective separation time, i.e., tB, is proportional to 1/v02. This implies that the operation time is proportional to 1/v02 if the separation efficiency η (= 1 − P) is given. A higher v0 leads to a quicker breakthrough by a shorter tB.
Figure 6 presents the variation of the saturation capture capacity (σS) as computed at P95 with 1/v0. The σS is proportional to 1/v0. It is noted that the capture capacity (σ) at CLF,e with VLF is computed as follows.
σ = [ C L F , i   V L F   V c     0 V L F V c C L F , e d ( V L F V c ) ] / V c = [ C L F , i   V L F   0 V L F C L F , e d ( V L F ) ]
In order to achieve a higher σS, a lower v0 is preferred. Comparing Figure 5 and Figure 6 further indicates that the effects of v0 on the tB at P10 and tE at P95 proportional to 1/v02 are more pronounced than those on σS proportional to 1/v0. The results of Figure 5 and Figure 6 are consistent with those of Figure 4.

3.4. Modeling of the Kinetics HGMS

With σS, v0 and L inserted into Equation (12) at the MPA concentrations CLF,i of 0.94, 1.51, 2.06, 3.21 and 4.14 g L−1 and the fluid velocity of 0.866, 1.132 and 1.731 m min−1, the non-linear regressions were conducted. The simulated results are listed in Table 1 and plotted in Figure 3 and Figure 4. The results show good fits with the experimental data with R2 of 0.985–0.995.
Taking that the CLF,e/CLF,i = 10% represents the tB of the separation chamber of the magnetic matrix, then the values of tB are 3, 1.7, 0.8, 0.53 and 0.36 min at the respective concentrations of 0.94, 1.51, 2.06, 3.21 and 4.14 g L−1 with QLF at 0.833 L min−1, and are 4.75, 2.31 and 0.8 min at respective v0 of 0.866, 1.132 and 1.731 m min−1 with CLF,i at 2.06 g L−1. These values also exhibit the same tendency with the experimental data, indicating that the higher the MPA concentration, as well as the fluid velocity, the shorter the breakthrough time. From a comparison of the initial characteristic length λ0 for the cases with the different MPA concentrations studied, it indicates that the value of λ0 of the kinetic models equation decreases with increasing CLF,i. According to Equation (13), λ0 is related to the capture radius. Thus, it can be illustrated that the higher MPA concentration has the shorter capture radius due to that MPAs block some lines of the magnetic field, causing the shorter initial characteristic length λ0. For cases of the different flow velocities examined, it was observed that the value of λ0 decreases as v0 increases. It can be interpreted that the higher flow velocity has the shorter capture radius because of the larger inertial impact force, resulting in the shorter initial characteristic length.

3.5. Operation Practice

In this study, the feasibility of using HGMS as a recovery method was examined for the recycling and reuse of MPA in ADPs. In this system, the operation time of HGMS should be short, and the energy consuming should, therefore, be reduced. Thus, tB is one of the important parameters of the magnetic separation operation. The HGMS with a short tB indicates that the operation of the magnetic separation chamber should be better when periodic. In one cycle, the HGMS of the magnetic particles must be stopped before reaching tB to avoid a great loss of the magnetic particles in the effluent of the separation chamber of the HGMS. According to the results of previous sections, tB is not only affected by the particle concentration, but also influenced by the fluid velocity. In the Watson and Gerber theory, a decrease of the Rca with increasing MPA concentration or flow velocity was deduced. This was confirmed, as the tB of the HGMS of MPA was shortened when the MPA concentration, as well as the flow velocity, increased. It was explained by the cause that the Rca is influenced by drag force and magnetic force. By using the deep-bed filtration model, a good qualitative description of the results in this study was obtained. In this model, a decrease of the initial characteristic length λ0 with an increasing MPA concentration or flow velocity was observed. In a single wire, λ0 is directly proportional to Rca, based on Equation (13). Thus, it is also attributed to fluid drag force and magnetic force. It also describes the steepness of the particle breakthrough curve. Therefore, the lower λ0 means the HGMS has a lower recovery ability of MPA.
The flow velocity is also indeed an important factor in high-gradient magnetic separation. In order to increase the treatment volume and to shorten the treatment time in operation, what is usually done is increasing the flow velocity. However, if the flow velocity is too high, it will dramatically decrease the separation efficiency and greatly shorten the effective tB, too. Thus, the flow velocity setting is important for high-gradient magnetic separation. Suppose that tON is the necessary operation time interval for the separation of magnetic media to facilitate the switching of the magnetic field. Equation (19) can be deduced from Equation (12) under the given separation efficiency η (or P0) and operation time tON, yielding
1 − η = 1 − P0 = P = CLF,e/CLF,i ≥ [exp(CLF,i v0λ0τ/σs)]/([exp(CLF,i v0λ0τ/σs)] + [exp(λ0τ)] − 1)
At the conditions of the given CLF,i = 2.06 kg m−3, η (or P0) = 0.9 and tON = 120 s, then v0 can be calculated from the equation to be lower than 0.01 m s−1.
The lower the flow velocity, the higher the separation efficiency is. However, it has been thought that the flow velocity cannot decrease without any limitation. It must be higher than the speed of sweeping and scouring to avoid the deposit of the magnetic particles on the tube wall. As calculated from Equation (20), which, with the substitution of the friction coefficient fc (Equation (21)), yielding Equation (22), supposing rh = 0.005 m, ρp = 1.629 kg cm−3 in Equation (22), a flow velocity above 2.47 × 10−3 m s−1 is thought to be more suitable. This means that the flow velocity must overcome the friction shearing stress caused by the gravity of the magnetic particles to avoid the deposit of magnetic particles on the tube wall, which, however, increases the particle loss of M-PVAC-IDA.
v0 ≥ [(8γ/fc) g ((ρpρw)/ρw) dp]0.5
fc = 64/Re = 64μ/(ρw v0 4rh)
Equation (22) can be obtained, as Equation (20) is taken into the relationship of the friction coefficient of Equation (21).
v0 ≥ [(γ rh g (ρpρw)/(2ρw)) dp]0.5
In the above equations, rh is the hydraulic radius, fc is the friction coefficient, γ is the constant (0.03–0.06), ρp is the density of the magnetic particles, ρw is the density of the water, dp is the particle diameter of the magnetic particles, Re is the Reynolds number and g is the acceleration due to gravity.
Noting the low running cost and small space requirement of the new magnetic separation system, the broad applications involving HGMS will attract more attention in the future. The information obtained in this study is useful for the rational design and proper operation of a HGMS system for the recycling and reuse of MPA in ADPs.

4. Conclusions

Through the results and discussion elucidated above, we can draw the following essential conclusions.
1.
Due to fact that the capture radius of a HGM separator is inversely proportional to the inlet concentration of MPA (CLF,i), the higher concentration of M-PVAC-IDA leads to the shorter breakthrough time (tB). The higher volumetric flow rate (QLF) also results in the shorter tB.
2.
The saturation capture capacity (σS) is inversely proportional to the flow velocity (v0); the breakthrough time is inversely proportional to the square of the flow velocity.
3.
If CLF,i = 2.06 kg m−3, the separation efficiency η = 0.9 and the operation time tON = 120 s, then v0 can be calculated from the kinetic equation to be lower than 0.01 m s−1. The flow velocity exceeding 2.47 × 10−3 m s−1 is considered more proper in order to prevent the deposit of the magnetic particles on the tube wall.
4.
The experimental data show a good fit with the mass balance approach model equation with a R2 of 0.985–0.995.
5.
The value of the characteristic length (λ0) of the kinetic model equation decreases with an increasing flow velocity.

Author Contributions

Conceptualization, J.-Y.T. and C.-Y.C.; methodology, J.-Y.T. and C.-Y.C.; validation, C.-W.T., D.-R.J. and B.-L.L.; formal analysis, J.-Y.T. and C.-W.T.; verification, J.-Y.T. and C.-Y.C.; discussion, suggestions and resources, C.-Y.C., C.-C.C., C.-F.C., Y.-H.C., J.-L.S., M.-H.Y. and M.F.; writing—original draft preparation, J.-Y.T. and C.-Y.C.; writing—review and editing, M.-H.Y. and C.-Y.C.; supervision, C.-Y.C.; funding acquisition, M.-H.Y. and C.-Y.C. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the National Science and Technology Council of Taiwan (formerly National Science Council), NSC 97-2221-E-002-113-MY3 and NSTC 111-2622-E-039-001.

Data Availability Statement

Data sharing not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

Nomenclature

AcCross section area of filter or separation cell, cm2 or m2; = 4.8125 cm2 in this study
A1Defined in Equation (5), cm or m
A2Defined in Equation (16), cm−1 or m−1
aRadius of ferromagnetic wire of matrix, cm or m; placed axially along z axis
aBRadius of particle buildup profile (actual wire radius of magnetic media with build up of particles), m
BMagnetic flux density, or magnetic induction, G; B(G) = μ0H (Oe)
CConcentrations of MPA (i.e., M-PVAC-IDA) in solution, mg L−1 or g L−1
CLF,i, CLF,eInlet and outlet concentrations of MPA (i.e., M-PVAC-IDA) in solution, mg L−1 or g L−1
dxDifferential thickness of filter, m
dpDiameter of paramagnetic particle, μm or nm
FFilling factor (or filling density) of matrix, -; =Lwx πa2/(dx Ac) or Lwm πa2/(L Ac) (= Vwm/Vc = 4.37/37.78 = 0.1157 filled with all matrix of 34.5466 g received in this study)
FcLimiting F, -; if wires in filter are separated by a distance greater than 2Xc, than particles interact with only one wire at a time, i.e., wires act independently; ~3π/(4Xc2) [35]
FdViscous resistance force (viscous drag force), N
FmMagnetic force, N
Fm/vMagnetic force per unit volume, N m−3
fMatrix arrangement factor in Equation (1), -
fcFriction coefficient in Equation (21), -
g° Acceleration due to gravity
HMagnetic field strength, or magnetic field intensity, Oe; applied in x direction
H0H applied in x direction perpendicular to filter, Oe
▽HMagnetic field strength gradient at the location of particle; i.e., grad (H)
LHeight of separation chamber housing magnetic filter media, m or cm; =7.85 cm in this study
LATAdsorption treated liquid
LaNormalized L, -; =L/a
LRFFresh liquid for regeneration of aged magnetite
LRAAged regeneration liquid (liquid after regeneration of aged magnetite)
LWLWaste liquid
LwmLength of wire of matrix, m
LwxLwm in a thickness dx of filter, m; Lwx/Ac = Fdx/(πa2)
LwxeEffective Lwx, m; =(2/3)Lwx, assuming approximately 1/3 of Lwx is parallel to H0 and ineffective in filtering process [35]
MMagnetization, emu g−1 or emu cm−3
MAAtomic mass
MadMolecular weight of adsorbate
MMAAged magnetite
MMFFresh or freshed (regenerated) magnetite
MpMagnetization of particles, emu g−1 or emu cm−3 or A m2 cm−3; =(χpχf) H
MpSSaturation magnetization of magnetic particles, emu g−1
MrResidual magnetization, emu g−1
MSSaturation magnetization, emu g−1
MSpMS of particles, emu g−1
MwmMagnetization of wire of matrix, emu g−1 or emu cm−3
mwmMass of wire of matrix, g or kg; = 34.5466 g for all matrix as received in this study
NiNumber of particles per unit volume of incident on filter, m−3
NeNumber of particles per unit volume of outlet fluid from filter, m−3; =Ni exp(−4FRcaL/(3πa)) for cases (1) in streamline flow with a filling factor less than Fc, and (2) in the limit of extreme turbulent [35]
PPenetration of magnetic particles in fluid, -; =CLF,e/CLF,i = 1 − η
P0 or ηSeparation efficiency, -; =1 − P
P10 P = 10%
P95P = 95%
QLFVolumetric flow rate of liquid, L min−1
RcaDimensionless normalized particle capture radius, -; =rc/a; if position of particle with initial coordinate (y/a)i for large x/a satisfies Rca ≧ (y/a)i ≦ − Rca, then the particle will be captured; Rca ~(1/2)vm/v0 for small values of vm/v0 (say, ≦1), however Rca ~(1/4)vm/v0 for vm/v0 =10 [35]
ReReynolds number, -; =ρw v0 4rh/μw
R2Determination coefficient
rRadius of circular pipe, m
rcParticle capture radius, m
rhHydraulic radius, m; =ratio of cross-section area/wetted perimeter (=πr2/(2πr) = r/2 for circular pipe)
rpRadius of paramagnetic particle, μm or nm
tFiltration time (also called separation time or operation time), s
tBBreakthrough time (or effective separation time) for HGMS, s; estimated as at P = CLF,e/CLF,i = 10%
tEExhaustion time (or saturation time) for HGMS, s; estimated as at P = CLF,e/CLF,i = 95%
tONNecessary operation time interval for separation of magnetic media to facilitate the switching of magnetic field, s
tηSeparation time at a set separation efficiency η (=1 − P) for HGMS, s
Vc or CV Volume of filter chamber or cell volume, cm3 or m3; =37.78 cm3 in this study
VLFAccumulated volume of liquid filtrated expressed as CV
VpVolume of a magnetic particle (=(4/3) π rp3 for spherical particle), nm3
VwmVolume of wire of matrix, cm3 or m3; =mwmwm = 34.5466 g/7900 (kg/m3) = 0.00000437 m3 for all matrix as received in this study
vVelocity, m s−1
vmCharacteristic magnetic velocity, m s−1
vp/fVelocity of magnetic particle relative to fluid, m s−1
v0Fluid velocity (superficial velocity), m s−1; the magnetic wire is placed axially along z axis with the uniform magnetic field strength applied in x direction while fluid flowing past the wire in negative x direction.
Xx/a, -
XcDimensionless distance from wire at which particle has changed y/a coordinate by 5% from initial value of (y/a)i, - [35]
x, y, zx, y and z directions of rectangular coordinates
Yy/a, -
(y/a)i Initial coordinate of position of particle
βPacking factor of the buildup (i.e., particle aggregation factor), -; =0.1−0.18
γConstantin in Equation (20), -; =0.03−0.06
εFFilter void fraction, (-); =(VcVwm)/Vc = 1 − F (=0.8843 with all received matrix of 34.5466 g filled in the chamber)
η or P0 Separation efficiency of MP for HGMS, -; =1 − P
λCharacteristic length, m−1; =λ0 [1 − (σ/σS)]
λ0Characteristic length at τ = 0, m−1; =(2/π)(1 − εF)(Rca/a)
μfViscosity of fluid, kg m−1s−1 or N s m−2
μrMagnetic permeability of medium relative to that of vacuum, -, =μ’/μ0
μwViscosity of water, kg m−1s−1 or N s m−2
μ0Magnetic permeability of vacuum, H m−1, =4π × 10−7 H m−1
μ’Magnetic permeability, H m−1
ρpDensity of magnetic particles, kg m−3
ρwDensity of water, kg m−3
ρwmDensity of wire of matrix, g/cm3 or kg/m3; =7900 kg/m3 in this study
σCapture capacity of magnetic particle on the magnetic media in magnetic separation cell at time t, kg m−3
σSSaturated σ, kg m−3
σEiSurface charge density of layer with ions i
τTime defined by τ = t − (εF x/v0), s
χMagnetic susceptibility, emu cm−3 Oe−1 = Mp/H
χpMagnetic susceptibility of particle, emu cm−3 Oe−1
χfMagnetic susceptibility of fluid, emu cm−3 Oe−1
Subscripts
f Fluid
p Particle
Abbreviations
A-MPAsExhausted adsorbed MPAs
ADPsAdsorption and desorption processes
CVCell volume
D-MPAs Fresh or Regenerated desorbed MPAs
DWDry weight
HGMSHigh-gradient magnetic separation
IDAIminodiacetic acid
NdFeBNeodymium-iron-boron
M-PVA-IDAMPA of polyvinyl acetate-iminodiacetic acid
MPAMagnetic polymer adsorbent or super-paramagnetic polymer adsorbent
MSMagnetic separation
ODOptical density
PVAPolyvinyl acetate
VACVinyl acetate

Appendix A

The solution of the outlet concentration CLF,e (Equation (12)) was obtained as follows:
Figure A1. Sketch of mass balance approach for kinetic models of HGMS.
Figure A1. Sketch of mass balance approach for kinetic models of HGMS.
Processes 11 00965 g0a1
σ τ + V 0 C x = 0
σ τ = λ V 0 C
λ = λ 0 1 σ σ S
τ = t ε F x V 0
with the conditions:
x = 0, C = C0
τ = 0, σ = 0
Combining Equations (A1)–(A3) gives
σ τ = V 0 C x = λ 0 1 σ σ S V 0 C
d C C = λ 0 σ λ 0 σ S d x
d C C = σ λ 0 σ S λ 0 d x
Integration of Equation (A9) leads to
ln C = λ 0 σ s 0 x σ d x λ 0 x + k τ
Appling condition (x = 0, C = C0) gives ln C0 = k(τ). Then, Equation (A10) becomes
ln C C 0 = σ λ 0 x with σ = λ 0 σ S 0 x σ d x
Substituting condition τ = 0, σ = 0 gives
ln C C 0 = λ 0 x
Equation (A11) can be partially differentiated by τ. This leads to
1 C d C d τ = λ 0 σ S 0 x σ τ d x
Substitution of σ τ by V 0 C x from Equation (A1) results in
1 C d C d τ = λ 0 σ S 0 x V 0 C x d x = λ 0 V 0 σ S C 0 C d C = λ 0 V 0 σ S C C 0
Integrating Equation (A14) leads to
C C 0 = [ exp A ] U x ( [ exp A ] U x ) 1
where
A = λ 0 V 0 C 0 σ S τ
At τ = 0, Equation (A15) gives
C C 0 τ = 0 = U x U x 1
Comparison of Equations (A12) and (A17) leads to
C C 0 τ = 0 = exp λ 0 x = U x U x 1
Then
U x = exp λ 0 x exp λ x 1
Equation (A15) is divided by U(x) to give
C C 0 = exp A [ exp A ] U 1 x
where A is defined by Equation (A16) and
U 1 x = [ exp λ 0 x ] 1 exp λ 0 x = 1 exp λ 0 x
Thus
C C 0 = exp A exp A + exp λ 0 x 1 = exp σ S 1 λ 0 C 0 V 0 τ exp σ S 1 λ 0 C 0 V 0 τ + exp λ 0 Z 1
The outlet concentration C at x = L is then obtained as
C C 0 x = L = exp σ S 1 λ 0 C 0 V 0 τ exp σ S 1 λ 0 V 0 C 0 τ + exp λ 0 L 1

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Figure 1. Application of high-gradient magnetic separation (HGMS) incorporated with adsorption and desorption operations. (a) Adsorption of waste liquid (LWL) via fresh or freshed (regenerated) magnetite (MMF) and recovery of aged magnetite (MMA); LAT: adsorption treated liquid. (b) Desorption/regeneration of MMA via fresh regeneration liquid (LRF) and recovery of freshed (regenerated) magnetite MMF; LRA: aged regeneration liquid.
Figure 1. Application of high-gradient magnetic separation (HGMS) incorporated with adsorption and desorption operations. (a) Adsorption of waste liquid (LWL) via fresh or freshed (regenerated) magnetite (MMF) and recovery of aged magnetite (MMA); LAT: adsorption treated liquid. (b) Desorption/regeneration of MMA via fresh regeneration liquid (LRF) and recovery of freshed (regenerated) magnetite MMF; LRA: aged regeneration liquid.
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Figure 2. Magnetization curves (M vs. H) of M-PVAC-IDA particles. ○, □, △: M-PVAC-IDA, Cu(II)-adsorbed M-PVAC-IDA, oxidized M-PVAC-IDA.
Figure 2. Magnetization curves (M vs. H) of M-PVAC-IDA particles. ○, □, △: M-PVAC-IDA, Cu(II)-adsorbed M-PVAC-IDA, oxidized M-PVAC-IDA.
Processes 11 00965 g002
Figure 3. Breakthrough curves for the solution containing M-PVAC-IDA using HGM separator (HGF-10 1) at various CLF,i with QLF = 0.833 L min−1 (v0 = 1.731 m min−1). ◇, ○, □, △, *: CLF,i = 4.14, 3.21, 2.06, 1.51, 0.94 g L−1. CLF,e, CLF,i: Outlet and inlet concentrations of M-PVAC-IDA in solution. VLF: Accumulated volume of liquid filtrated expressed as cell volume. QLF: Volume flow rate. v0: Flow velocity. —: Prediction.
Figure 3. Breakthrough curves for the solution containing M-PVAC-IDA using HGM separator (HGF-10 1) at various CLF,i with QLF = 0.833 L min−1 (v0 = 1.731 m min−1). ◇, ○, □, △, *: CLF,i = 4.14, 3.21, 2.06, 1.51, 0.94 g L−1. CLF,e, CLF,i: Outlet and inlet concentrations of M-PVAC-IDA in solution. VLF: Accumulated volume of liquid filtrated expressed as cell volume. QLF: Volume flow rate. v0: Flow velocity. —: Prediction.
Processes 11 00965 g003
Figure 4. Breakthrough curves for the solution containing M-PVAC-IDA using HGM separator (HGF-10 1) at various QLF with CLF,i = 2.06 g L−1. ○, □, △: QLF = 0.833, 0.545, 0.417 L min−1 (v0 = 1.731, 1.132, 0.866 m min−1). —: Prediction. CLF,e, CLF,i, VLF, QLF: As specified in Figure 3.
Figure 4. Breakthrough curves for the solution containing M-PVAC-IDA using HGM separator (HGF-10 1) at various QLF with CLF,i = 2.06 g L−1. ○, □, △: QLF = 0.833, 0.545, 0.417 L min−1 (v0 = 1.731, 1.132, 0.866 m min−1). —: Prediction. CLF,e, CLF,i, VLF, QLF: As specified in Figure 3.
Processes 11 00965 g004
Figure 5. tB vs. 1/v02 for the solution containing M-PVAC-IDA using HGM separator (HGF-10 1). tB: Breakthrough time. v0: Fluid velocity.
Figure 5. tB vs. 1/v02 for the solution containing M-PVAC-IDA using HGM separator (HGF-10 1). tB: Breakthrough time. v0: Fluid velocity.
Processes 11 00965 g005
Figure 6. σs vs. 1/v0 for the solution containing M-PVAC-IDA using HGM separator (HGF-10 1). σs: Saturated capture capacity of HGM separator (HGF-10 1). v0: Fluid velocity.
Figure 6. σs vs. 1/v0 for the solution containing M-PVAC-IDA using HGM separator (HGF-10 1). σs: Saturated capture capacity of HGM separator (HGF-10 1). v0: Fluid velocity.
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Table 1. Parameters for prediction of the operation of high-gradient magnetic separator (HGF-10 1).
Table 1. Parameters for prediction of the operation of high-gradient magnetic separator (HGF-10 1).
Initial ConcentrationFlow
Velocity
Exhaustion Time at P95 Breakthrough Time at P10Characteristic LengthDetermination Coefficient
CLF,i
g L−1
v0
m min−1 or m s−1
tE
min or s
tB
min or s
λ0
m−1
R2
At various CLF,i with QLF of 0.833 L min−1
4.141.731 or 0.02893.3 or 1980.36 or 21.636.90.992
3.21.731 or 0.02894.4 or 2640.53 or 31.837.710.985
2.061.731 or 0.02896.4 or 3840.8 or 4838.110.994
1.511.731 or 0.02899.3 or 5581.7 or 10243.650.986
0.941.731 or 0.028912.8 or 7683 or 18048.430.988
At various v0 with CLF,i = 2.06 g L−1
2.061.731 or 0.02896.4 or 3840.8 or 4838.110.994
2.061.132 or 0.01899.8 or 5882.31 or 138.653.480.995
2.060.866 or 0.014414.8 or 8884.75 or 28563.720.990
R2: Determination coefficient, =   1 y e y c 2 [ y e y m 2 ] , where ye and yc, and ym are the experimental and predicted results and the average of experimental values of CLF,e/CLF,i, respectively. CLF,e = outlet concentrations of MPA (M-PVAC-IDA). v0 = 1.731, 1.132, 0.866 m min−1 corresponding to QLF = 0.833, 0.545, 0.417 L min−1.
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Tseng, J.-Y.; Chang, C.-C.; Tu, C.-W.; Yuan, M.-H.; Chang, C.-Y.; Chang, C.-F.; Chen, Y.-H.; Shie, J.-L.; Ji, D.-R.; Liu, B.-L.; et al. Application of High-Gradient Magnetic Separation for the Recovery of Super-Paramagnetic Polymer Adsorbent Used in Adsorption and Desorption Processes. Processes 2023, 11, 965. https://doi.org/10.3390/pr11030965

AMA Style

Tseng J-Y, Chang C-C, Tu C-W, Yuan M-H, Chang C-Y, Chang C-F, Chen Y-H, Shie J-L, Ji D-R, Liu B-L, et al. Application of High-Gradient Magnetic Separation for the Recovery of Super-Paramagnetic Polymer Adsorbent Used in Adsorption and Desorption Processes. Processes. 2023; 11(3):965. https://doi.org/10.3390/pr11030965

Chicago/Turabian Style

Tseng, Jyi-Yeong, Chia-Chi Chang, Cheng-Wen Tu, Min-Hao Yuan, Ching-Yuan Chang, Chiung-Fen Chang, Yi-Hung Chen, Je-Lueng Shie, Dar-Ren Ji, Bo-Liang Liu, and et al. 2023. "Application of High-Gradient Magnetic Separation for the Recovery of Super-Paramagnetic Polymer Adsorbent Used in Adsorption and Desorption Processes" Processes 11, no. 3: 965. https://doi.org/10.3390/pr11030965

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