Application of High-Gradient Magnetic Separation for the Recovery of Super-Paramagnetic Polymer Adsorbent Used in Adsorption and Desorption 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 (C_LF,i) and the fluid velocity (v₀) or volumetric flow rate (Q_LF) on the performance of the recovery of MPA via HGMS were assessed. The results indicated that the separation efficiency (η or P₀), breakthrough time (t_B) and exhaustion time (t_E) of HGMS reduce as C_LF,i, as well as v₀, increases. Further, the filter saturated capture capacity (σ_S) of HGMS also decreases with increasing v₀. The effect of v₀ on t_B proportional to 1/v₀² is more significant than that on σ_S 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 (F_m = μ₀ V_p M_p‧▽H). The other is the viscous drag force (viscous resistance force) (F_d), which can be obtained by the Stokes equation (F_d = 3πμ_f d_pv_p/f). In these notations, μ₀ is the magnetic permeability of the vacuum (= 4π × 10⁻⁷ H m⁻¹); V_p (= (1/6) π d_p³) is the volume of a magnetic particle; d_p is the particle diameter; M_p (= (χ_p − χ_f) H) is the magnetization of particles; χ_p and χ_f are, respectively, the magnetic susceptibilities of the particle and fluid; H is the magnetic field strength; μ_f is the fluid viscosity; v_p/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 d_p, the stronger the F_m, or the lower the μ_f, the weaker the F_d, 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 (C_LF,i) of 0.94–4.14 g L⁻¹ and volumetric flow rates (Q_LF) 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 (C_LF,e) were measured. The separation efficiency of MPA (η or P₀), breakthrough time (or effective separation time) (t_B), exhaustion time (or saturation time) (t_E) and filter saturated capture capacity (σ_S) for HGMS were examined and elucidated for different C_LF,i and Q_LF. 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 V_c = 37.78 cm³, height of filter chamber L = 7.85 cm, cell cross-section area A_c = 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 m_wm 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 ρ_wm 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 C_LF,i and volumetric flow rates Q_LF. For the influences of C_LF,i, the HGMS experiments were conducted at C_LF,i of 4.14, 3.21, 2.06, 1.51 and 0.94 g L⁻¹ with Q_LF of 0.833 L min⁻¹. As for the effects of Q_LF, the cases with a Q_LF 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 C_LF,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 C_LF,i and outflow concentration C_LF,e of the magnetic particles in the HGM separator can be expressed as the following equation:

P = C_LF,e/C_LF,i = exp [−fFR_caL/(πa)]

(1)

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; R_ca (= r_c/a) is the dimensionless normalized particle capture radius (r_c). R_ca can be expressed as:

R_ca = (1/2)(v_m/v₀)(a/a_B)²

(2)

where

(a_B/a)⁴ = (v_m/A₁)t + 1

(3)

R_ca = (1/2)(v_m/v₀)/[(v_m/A₁)t + 1]^1/2

(4)

A₁ = [β ρ_p a/(4 C_LF,i)]

(5)

In the above equations, β is the particle aggregation factor with a value of 0.1–0.18; ρ_p is the density of the magnetic particles (kg m⁻³); v_m is the magnetic velocity (a function of magnetic field strength (H₀) and magnetization of wire of matrix (M_wm)) (m s⁻¹); v₀ is the flow velocity (m s⁻¹); a_B 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.

$$(A_c dx {\epsilon }_F)\frac{\partial C}{\partial t}=A_c v_0 C(t,x) - A_c v_0 C(t + dt,x + dx) - \frac{\partial \sigma }{\partial t} A_{c}dx$$

(6)

This gives

$${\epsilon }_F\frac{\partial C}{\partial t}= -v_0\frac{\partial C}{\partial x}-\frac{\partial \sigma }{\partial t}$$

(7)

or

$${\epsilon }_F\frac{\partial C}{\partial t}+\frac{\partial \sigma }{\partial t} + v_0\frac{\partial C}{\partial x}=0$$

(8)

Assuming ε_F$\frac{\partial C}{\partial t}$ is negligible compared to the other terms, i.e., ε_F$\frac{\partial C}{\partial t}$ ~ 0, in the dx interval and defining τ = t − (ε_F x/v₀), Equation (8) can be simplified as Equation (9).

$$\frac{\partial \sigma }{\partial \tau } + v_0\frac{\partial C}{\partial x}=0$$

(9)

The time variation of σ increases with v₀, C and the term σ_S − σ, as described below.

$$\frac{\partial \sigma }{\partial \tau }= \lambda v_{0}C$$

(10)

λ = λ₀ [1 − (σ/σ_S)]

(11)

It can be calculated from Equations (9)–(11) to obtain the solution of the outlet concentration C_LF,e as:

C_LF,e/C_LF,I = [exp(C_LF,i v₀λ₀τ/σ_S)]/([exp(C_LF,i v₀λ₀τ/σ_S)] + [exp(λ₀L)] − 1)

(12)

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; σ_S is the saturated σ (kg m⁻³); λ is the characteristic length (m⁻¹); λ₀ is the characteristic length at τ = 0; ε_F is the void fraction of the magnetic matrix; v₀ is the flow velocity (m s⁻¹); A_c 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); C_LF,i and C_LF,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 (R_ca) and the wire radius of magnetic matrix (a). It can be represented by Equation (11) with Equation (13) as below.

λ₀ = (2/π) (1 − ε_F) (R_ca/a)

(13)

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 (t_η) at a set separation efficiency η based on Equations (1)–(5) can be obtained as follows.

t_η = A₁(A₂)²  L² (v_m/v₀²)/ln(η²)

(14)

For a given t_η, v₀ must satisfy:

v₀ ≦ (A₁)^{1/2 A}₂ L (v_m/t_η)^1/2/[ln(η²)]^1/2

(15)

where

A₂ = fF/(2πa)

(16)

v_m = 2μ₀ (χ_f − χ_p) M_wm H₀ r_p²/(9μ_f a)

(17)

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; M_wm is the magnetization of the wire of the matrix (emu g⁻¹ or emu cm⁻³); r_p is the radius of the paramagnetic particle (μm or nm); μ_f is the viscosity of the fluid (lb_m 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); χ_f is the magnetic susceptibility of the fluid (emu cm⁻³ Oe⁻¹); χ_p 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 (= C_LF,e/C_LF,i) with an accumulated volume of liquid filtrated (V_LF, expressed as cell volumes (V_c)) at different inlet MPA concentrations C_LF,i (4.14, 3.21, 2.06, 1.51 and 0.94 g L⁻¹), with a volumetric flow rate Q_LF = 0.833 L min⁻¹. A higher C_LF,i shows an easier saturation with a smaller V_LF. In the case with the highest C_LF,i of 4.14 g L⁻¹, the separation cell reaches P = 95% (denoted as P95), with the smallest V_LF of about 70.6 V_c. On the other hand, the lowest C_LF,i of 0.94 g L⁻¹ gives the largest V_LF of about 301.5 V_c at P95. For cases with C_LF,i of 1.51, 2.06 and 3.21 g L⁻¹, the corresponding V_LF at P95 is 166.7, 131.5 and 90.8 V_c, respectively.

From Equations (2)–(5) of the Watson and Gerber theory, it can be seen that the capture radius R_ca is related to the magnetic velocity v_m (thus, the magnetic field strength H₀ and the magnetization of the wire of the matrix M_wm, as indicated in Equation (17)), flow velocity v₀, and saturation magnetization M_Sp and concentration C_LF,i of magnetic particles. When the magnetic field strength and the volumetric flow rate are constant, the actual wire radius a_B of the magnetic media increases, while the capture radius R_ca decreases as C_LF,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 V_LF).

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

Figure 4 presents the variations of penetration P with V_LF at various volumetric flow rates of the liquid Q_LF (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; = Q_LF/A_c, A_c = 4.8125 cm²), with C_LF,i = 2.06 g L⁻¹. It shows that HGMS is more quickly saturated if the Q_LF is higher. For v₀ = 1.731 m min⁻¹, the V_LF at P95 is smallest at 131.5 V_c. As for v₀ = 0.866 of m min⁻¹, the V_LF at P95 is largest at 191.2 V_c. At v₀ = 1.132 of m min⁻¹, the V_LF at P95 is 157.3 V_c. The decrease of V_LF at P95 with increasing Q_LF is due to the cause that the capture radiuses R_ca of the magnetic matrix become shorter as the Q_LF 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 t_B (taken at P10 (C_LF,e/C_LF,i = 10%)) and the reciprocal of the square of the fluid velocity (1/v₀²). The results indicated that the effective separation time, i.e., t_B, 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 t_B.

Figure 6 presents the variation of the saturation capture capacity (σ_S) as computed at P95 with 1/v₀. The σ_S is proportional to 1/v₀. It is noted that the capture capacity (σ) at C_LF,e with V_LF is computed as follows.

$$\begin{gathered} \sigma =[C_{LF,i} V_{LF} V_c - {\displaystyle {\int }_{\hspace{0.33em}0}^{\hspace{0.33em}{V}_{LF}\hspace{0.33em}{V}_c}{C}_{LF,e}\hspace{0.33em}}d(V_{LF}\hspace{0.33em}{V}_c)]/{\mathrm{V}}_{\mathrm{c}} \\ =[C_{LF,i} V_{LF}- {\displaystyle {\int }_{\hspace{0.33em}0}^{\hspace{0.33em}{V}_{LF}\hspace{0.33em}}{C}_{LF,e}\hspace{0.33em}}d(V_{LF})] \end{gathered}$$

(18)

In order to achieve a higher σ_S, a lower v₀ is preferred. Comparing Figure 5 and Figure 6 further indicates that the effects of v₀ on the t_B at P10 and t_E at P95 proportional to 1/v₀² are more pronounced than those on σ_S 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 σ_S, v₀ and L inserted into Equation (12) at the MPA concentrations C_LF,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. Parameters for prediction of the operation of high-gradient magnetic separator (HGF-10 1).

Initial Concentration	Flow Velocity	Exhaustion Time at P95	Breakthrough Time at P10	Characteristic Length	Determination 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
4.14	1.731 or 0.0289	3.3 or 198	0.36 or 21.6	36.9	0.992
3.2	1.731 or 0.0289	4.4 or 264	0.53 or 31.8	37.71	0.985
2.06	1.731 or 0.0289	6.4 or 384	0.8 or 48	38.11	0.994
1.51	1.731 or 0.0289	9.3 or 558	1.7 or 102	43.65	0.986
0.94	1.731 or 0.0289	12.8 or 768	3 or 180	48.43	0.988
At various v0 with CLF,i = 2.06 g L−1
2.06	1.731 or 0.0289	6.4 or 384	0.8 or 48	38.11	0.994
2.06	1.132 or 0.0189	9.8 or 588	2.31 or 138.6	53.48	0.995
2.06	0.866 or 0.0144	14.8 or 888	4.75 or 285	63.72	0.990

R²: Determination coefficient, =$1-\left(\frac{\left[\sum {\left(y_e-y_c\right)}^2\right]}{[\sum {\left(y_e-y_m\right)}^2]}\right)$ , where y_e and y_c, and y_m are the experimental and predicted results and the average of experimental values of C_LF,e/C_LF,i, respectively. C_LF,e = outlet concentrations of MPA (M-PVAC-IDA). v₀ = 1.731, 1.132, 0.866 m min⁻¹ corresponding to Q_LF = 0.833, 0.545, 0.417 L min⁻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 C_LF,e/C_LF,i = 10% represents the t_B of the separation chamber of the magnetic matrix, then the values of t_B 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 Q_LF 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 C_LF,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 C_LF,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, t_B is one of the important parameters of the magnetic separation operation. The HGMS with a short t_B 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 t_B 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, t_B 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 R_ca with increasing MPA concentration or flow velocity was deduced. This was confirmed, as the t_B 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 R_ca 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 R_ca, 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 t_B, too. Thus, the flow velocity setting is important for high-gradient magnetic separation. Suppose that t_ON 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 t_ON, yielding

1 − η = 1 − P₀ = P = C_LF,e/C_LF,i ≥ [exp(C_LF,i v₀λ₀τ/σ_s)]/([exp(C_LF,i v₀λ₀τ/σ_s)] + [exp(λ₀τ)] − 1)

(19)

At the conditions of the given C_LF,i = 2.06 kg m⁻³, η (or P₀) = 0.9 and t_ON = 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 f_c (Equation (21)), yielding Equation (22), supposing r_h = 0.005 m, ρ_p = 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/f_c) g ((ρ_p − ρ_w)/ρ_w) d_p]^0.5

(20)

f_c = 64/Re = 64μ/(ρ_w v₀ 4r_h)

(21)

Equation (22) can be obtained, as Equation (20) is taken into the relationship of the friction coefficient of Equation (21).

v₀ ≥ [(γ r_h g (ρ_p − ρ_w)/(2ρ_w)) d_p]^0.5

(22)

In the above equations, r_h is the hydraulic radius, f_c is the friction coefficient, γ is the constant (0.03–0.06), ρ_p is the density of the magnetic particles, ρ_w is the density of the water, d_p 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 (C_LF,i), the higher concentration of M-PVAC-IDA leads to the shorter breakthrough time (t_B). The higher volumetric flow rate (Q_LF) also results in the shorter t_B.

2.

The saturation capture capacity (σ_S) is inversely proportional to the flow velocity (v₀); the breakthrough time is inversely proportional to the square of the flow velocity.

3.

If C_LF,i = 2.06 kg m⁻³, the separation efficiency η = 0.9 and the operation time t_ON = 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.