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Magnetic separation
Magnetic separation
Magnetic separation
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Magnetic separation is the process of separating components of mixtures by using a magnet to attract magnetic substances.[1] The process that is used for magnetic separation separates non-magnetic substances from those which are magnetic. This technique is useful for the select few minerals which are ferromagnetic (iron-, nickel-, and cobalt-containing minerals) and paramagnetic.[2] Most metals, including gold, silver and aluminum, are nonmagnetic.

A large diversity of mechanical means are used to separate magnetic materials.[2] During magnetic separation, magnets are situated inside two separator drums which bear liquids. Due to the magnets, magnetic particles are being drifted by the movement of the drums. This can create a magnetic concentrate (e.g. an ore concentrate).[2]

History

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Michael Faraday discovered that when a substance is put in a magnetic environment, the intensity of the environment is modified by it.[3] With this information, he discovered that different materials can be separated with their magnetic properties. The table below shows the common ferromagnetic and paramagnetic minerals as well as the field intensity that is required in order to separate those minerals.[3]

Common Ferromagnetic and Paramagnetic Minerals
Mineral Formula Field Strength (kG)
Ferromagnetic Magnetite 1
Pyrrhotite 0.5 - 4
Paramagnetic Ilmenite 8 - 16
Siderite 9 - 18
Chromite 10 - 16
Hematite 12 - 18
Wolframite 12 - 18
Tourmaline 16 - 20

In the 1860s, magnetic separation started to become commercialized. It was used to separate iron from brass.[3] After the 1880s, ferromagnetic materials started to be magnetically separated. Among others, Thomas Edison tried to commercialize the magnetic enrichment of poor iron ores but failed. In the 1900s, high intensity magnetic separation was inaugurated which allowed the separation of pragmatic materials.[3]

After the Second World War, systems that were the most common were electromagnets. The technique was used in scrap yards. Magnetic separation was developed again in the late 1970s with new technologies being inaugurated.[2] The new forms of magnetic separation included magnetic pulleys, overhead magnets and magnetic drums.

In mines where wolframite was mixed with cassiterite, such as South Crofty and East Pool mine in Cornwall or with bismuth such as at the Shepherd and Murphy mine in Moina, Tasmania, magnetic separation is used to separate the ores. At these mines, a device called a Wetherill's Magnetic Separator (invented by John Price Wetherill, 1844–1906)[4] was used. In this machine, the raw ore, after calcination was fed onto a conveyor belt which passed underneath two pairs of electromagnets under which further belts ran at right angles to the feed belt. The first pair of balls was weakly magnetized and served to draw off any iron ore present. The second pair were strongly magnetized and attracted the wolframite, which is weakly magnetic.[4] These machines were capable of treating 10 tons of ore a day.

Common applications

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Magnetic separation can also be used in electromagnetic cranes that separate magnetic material from scraps and unwanted substances.[1] This explains its use for shipment equipments and waste management. Unwanted metals can be removed from goods with this technique. It keeps all materials pure.[1] Recycling centres use magnetic separation often to separate components from recycling, isolate metals, and purify ores.[1] Overhead magnets, magnetic pulleys, and the magnetic drums were the methods used in the recycling industry.[1]

Magnetic separation is also useful in mining iron as it is attracted to a magnet.[3]

Another application, not widely known but very important, is to use magnets in process industries to remove metal contaminants from product streams.[1] This takes a lot of importance in food or pharmaceutical industries.

Magnetic separation is also used in situations where pollution needs to be controlled, in chemical processing, as well as during the benefaction of nonferrous low-grade ores.[1]

Magnetic separation is also used in the following industries: dairy, grain and milling, plastics, food, chemical, oils, textile, and more. N52 magnets are used in magnetic separation for food processing, recycling, and manufacturing.[5] They improve food safety, enhance recycling quality, and protect equipment in manufacturing, ensuring efficiency and high standards across these industries.[6][7]

Magnetic cell separation

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Magnetic cell separation is on the rise. It is currently being used in clinical therapies, more specifically in cancers and hereditary diseases researches.[8] Magnetic cell separation took a turn when, Zborowski, an Immunomagnetic Cell Separation (IMCS) pioneer, analyzed commercial magnetic cell separation. Zborowski uncovered crucial revelations that were then used, and are still used today, in the human understanding of cell biology.[8] Today, the manufacture of therapeutic products concerning cancers and genetic diseases, are being innovated due to these discoveries.[8]

In microbiology

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DNA purification using a GE MagRack 6 and magnetic beads with a coating that attaches to the substance of interest. The beads are visible accumulated on the top left of the solution surface.

Magnetic separation techniques are also used in microbiology. In this case, binding molecules and antibodies are used in order to isolate specific viable organisms, nucleic acids, or antigens.[9] This technology helps isolating bacterial species to identify and give diagnostics of genes targeting certain organisms.[9] When magnetic separation techniques are combined with PCR (polymerase chain reaction), the results increase in sensitivity and specificity.[9]

Low-field magnetic separation

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Low-field magnetic separation is often in environmental contexts such as water purification and the separation of complex mixtures.[10] Low magnetic field gradients are field gradients that are smaller than one hundred tesla per meter.[10] Monodisperse magnetite () and nanocrystals () are used for this technique.[10]

Magnetic filters are fitted on the boiler's pipework to collect magnetite from the circulating water before it has a chance to build up and lower the efficiency of the heating system. The water circulating around the heating system picks up bits of sludge (or magnetite) which can build up. The magnetic filter attracts all these bits of debris with a strong magnet as the water flows around it, preventing a build-up of sludge in the pipework or in the boiler.[11]

Weak magnetic separation

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Weak magnetic separation is used to create cleaner iron-rich products that can be reused.[12] These products have low levels of impurities and a high iron load. This technique is used as a recycling technology.[12] It is coupled with steelmaking slag fines as well as a selection of particle size screening.[12]

Magnetic Separation Force Calculations

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It can be shown[13] that magnetic force per unit volume on a permeable particle with relative permeability mu sub (pr) is proportional to the spatial gradient of the square of the magnetic flux density. The formula can be used in magnetic finite element analysis software to compute force densities on a wide variety of practical examples, obtaining results agreeing with Oberteuffer's paper.

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Magnetic separation

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from Grokipedia
Magnetic separation is a physical separation technique that utilizes differences in the magnetic properties of materials to isolate magnetically susceptible particles—such as ferromagnetic or paramagnetic substances—from non-magnetic ones in a , typically by applying an external non-uniform . This method, which originated in the mid-19th century for concentrating low-grade iron ores, has since developed into a versatile process applicable across various industries due to advancements in generation and particle handling. The core principle of magnetic separation relies on magnetophoresis, the motion of magnetic particles under the influence of a magnetic force that is proportional to the particle's , the strength of the (H), and its gradient (dH/dx), as described by the equation F = μ₀ χ V H · ∇H, where χ is susceptibility, V is volume, and μ₀ is permeability of free space. Materials respond differently: ferromagnetic particles (e.g., ) are strongly attracted and saturate at low fields, paramagnetic ones (e.g., ) require higher fields for separation, while diamagnetic materials (e.g., ) are weakly repelled. Competing forces like gravity, inertia, and fluid drag must be overcome, often achieved through optimized field gradients in wet or dry processes. Magnetic separators are categorized by intensity and configuration, including low-intensity or belt types (typically 500–1,500 gauss) for strongly magnetic minerals and high-intensity induced-roll or wet high-intensity magnetic separators (WHIMS, up to 20,000 gauss) for weakly magnetic ores, with capacities ranging from 1–150 tons per hour. Key applications include mineral beneficiation to upgrade ores like , , and by removing impurities; metal to recover and non-ferrous fractions from waste; environmental uses such as heavy metal ion removal from ; and bioprocessing for purifying proteins, cells, and biomolecules in diagnostics and food production, achieving high yields.

Fundamentals

Definition and Basic Principles

Magnetic separation is a physical process that exploits the differences in of materials to separate ferromagnetic, paramagnetic, or diamagnetic components from a using an external . This technique relies on the attraction or repulsion of particles based on their inherent response to , allowing selective isolation without chemical alteration. At its core, the method draws from fundamental , where materials exhibit behaviors such as strong attraction in ferromagnetics like iron, weak attraction in paramagnetics like aluminum oxide, or repulsion in diamagnetics like , providing the basis for differential separation. The basic operational mechanism involves applying an external to induce in susceptible particles within the , leading to their migration toward regions of higher field intensity while non-magnetic or oppositely responsive particles remain unaffected or move differently. Separation occurs through this differential motion, often in a flowing medium or across a matrix, where the magnetic force competes with gravitational, hydrodynamic, and inertial forces to direct particles along distinct paths. Key influencing factors include the magnetic field strength (denoted as B), which determines the overall intensity of ; the field (grad B), essential for generating the that drives particle movement; and intrinsic particle properties such as , which quantifies the material's responsiveness, along with , which affects the balance of forces and separation efficiency. A simple conceptual setup illustrates these principles through devices like drum separators, where a rotating embedded with permanent or electromagnets passes a mixture over its surface, attracting magnetic particles to adhere while non-magnetic ones continue onward. Similarly, belt configurations employ a moving conveyor with overhead magnets to lift and collect magnetic components, demonstrating how field application and geometry enable practical separation in bulk processes. These basic configurations highlight the reliance on controlled field exposure to achieve clean partitioning based on magnetic behavior.

Magnetic Properties of Materials

Materials are classified into three primary categories based on their magnetic response to an external : ferromagnetic, paramagnetic, and diamagnetic. Ferromagnetic materials, such as iron and , exhibit strong attraction to magnetic fields due to the alignment of atomic magnetic moments, resulting in a positive and high (χ > 0). Paramagnetic materials, like , show weak attraction with a small positive susceptibility (χ ≈ 10^{-3} to 10^{-5}), where individual atomic moments align temporarily with the field but randomize without it. Diamagnetic materials, including , are weakly repelled by magnetic fields, displaying a negative susceptibility (χ < 0, typically on the order of -10^{-5} to -10^{-6}). χ quantifies this response as the ratio of induced magnetization M to the applied magnetic field H (χ = M/H), serving as a key parameter in determining a material's separability in magnetic fields. In ferromagnetic materials, the magnetic behavior is characterized by hysteresis and remanence, which arise from the irreversible alignment of magnetic domains. Hysteresis refers to the lag in magnetization behind changes in the applied field, forming a loop in the B-H curve that indicates energy loss during magnetization cycles. Remanence is the residual magnetization (B_r) that persists after the external field is removed, due to the stable domain structure, making these materials useful for permanent magnets but requiring demagnetization in separation processes. For example, magnetite (Fe₃O₄) has a high susceptibility of approximately 3–6 in SI units and significant remanence, enhancing its separability in low-field applications. In contrast, ilmenite (FeTiO₃), with χ ≈ 3 × 10^{-3} SI, behaves paramagnetically with negligible hysteresis. Superparamagnetism emerges in ferromagnetic nanoparticles (typically <20 nm), where thermal energy overcomes the energy barrier for domain reversal, leading to rapid magnetization without hysteresis or remanence at room temperature. This property is advantageous in biomedical magnetic separation, as superparamagnetic iron oxide nanoparticles (SPIONs) can be easily magnetized and demagnetized, preventing aggregation and enabling targeted applications like cell sorting. Factors influencing separability include particle size, shape, and concentration in mixtures. Smaller particles experience weaker magnetic forces relative to Brownian motion, reducing separation efficiency, while elongated shapes can alter dipole interactions; higher concentrations may lead to agglomeration, complicating induced dipole formation in paramagnetic materials versus the permanent dipoles in ferromagnets. For instance, in mineral mixtures, optimal separability occurs for particles around 100-200 μm, balancing force and flow dynamics. Hematite (Fe₂O₃), with χ ≈ 10^{-3} SI, exemplifies weak induced dipoles that require higher fields for effective separation from diamagnetic quartz (χ ≈ -10^{-5} SI).

Historical Development

Early Innovations

The foundational observations for magnetic separation emerged in the 19th century through studies of how materials interact with magnetic fields. Michael Faraday's experiments in 1845 demonstrated that all substances modify the intensity of a magnetic environment, introducing the concepts of diamagnetism and paramagnetism to differentiate materials based on their magnetic responses. This work provided the theoretical basis for exploiting magnetic characteristics to separate substances, though practical applications followed later. Commercialization began in the 1860s with the first successful use of magnetic separators to isolate iron from brass filings and turnings, marking an initial industrial application beyond laboratory settings. By the 1880s, the technique expanded to ferromagnetic ores in mining, driven by inventors like , who developed an electromagnetic ore separator in 1880 to concentrate iron from crushed rock and tailings. Edison's device used a hopper to feed material over a powerful electromagnet, diverting magnetic particles into one receptacle while non-magnetic ones fell into another, achieving early ore enrichment. Early equipment in mining included simple electromagnetic separators, such as the Swedish Wenstrom drum-type machine introduced by 1887, which rotated a magnetized drum to attract iron ore particles. That same year, Edison conducted ore concentration experiments demonstrating the feasibility of electromagnetic separation for low-grade deposits, building on prior patents like his 1880 U.S. Patent No. 228,329 for a magnetic ore-separator. Additional 1880s patents, including Edison's 1882 U.S. Patent No. 263,131, advanced wet separators that processed slurried ores, allowing magnetic attraction in liquid media to improve handling of fine particles. These innovations faced significant limitations, primarily due to the low magnetic field strengths of early electromagnets, which restricted separation to strongly ferromagnetic materials and required slow feed rates to avoid gangue contamination. Primitive setups, like Z. Palmer's 1854 alternating-polarity horsehoe magnet on a rotating log at Palmer Hill, highlighted labor-intensive operations and inefficiency for large-scale mining. Despite these challenges, the 1792 British patent by William Fullarton for magnetically attracting iron ore laid the groundwork, predating commercial efforts but underscoring the long-standing interest in the method.

20th Century and Modern Advances

During the 1940s, electromagnetic separation techniques were pivotal in the Manhattan Project, where they were employed for uranium isotope enrichment. This method, based on deflecting charged uranium ions in a magnetic field to separate the lighter U-235 from U-238, was proposed by Ernest Lawrence in 1941 and scaled up at the Y-12 Plant in Oak Ridge, Tennessee, with operations beginning in late 1943. The process, though energy-intensive and inefficient—yielding only about 1/5,825 parts as enriched product—demonstrated the potential of magnetic fields for precise particle separation on an industrial scale, influencing subsequent advancements in isotope and mineral processing. In the 1950s, high-intensity magnetic separators emerged to handle weakly magnetic ores, marking a significant evolution from earlier low-intensity methods. These devices, developed amid the growth of the taconite iron ore industry, enabled the recovery of fine, paramagnetic particles that were previously uneconomical to process, such as those in low-grade iron formations. By the late 1960s, innovations like the Jones wet high-intensity magnetic separator further refined this capability, allowing continuous operation for weakly magnetic minerals in mining applications. Post-1980s developments were driven by the introduction of rare-earth permanent magnets, particularly neodymium-based ones, which provided stronger, more stable magnetic fields without the need for electricity. Invented in 1984, these magnets revolutionized separator design by enabling compact, high-intensity units that outperformed traditional electromagnets, facilitating broader use in mineral processing and recycling. Concurrently, superconducting magnets advanced high-gradient magnetic separation in the 1990s, with liquid-helium-free systems expanding applications for ultra-fine particle capture, such as in kaolin purification and environmental remediation. These innovations shifted processes from batch to continuous modes, improving throughput and reducing energy costs. From 2000 to 2025, microfluidic magnetic separation gained prominence post-2010, integrating microscale channels with magnetic fields for precise handling of cells and particles in biomedical contexts. Nanomaterial integration, particularly superparamagnetic iron oxide nanoparticles, enhanced precision in biomedical separations, as highlighted in recent reviews emphasizing their role in targeted drug delivery and cell sorting without aggregation issues. In mining, AI-optimized field designs emerged in 2023–2025 patents and developments, using machine learning to predict and adjust magnetic gradients for better fine-particle recovery and operational efficiency. Overall, these advances have boosted separation efficiency for submicron particles by up to 90% in select systems and promoted continuous processing, minimizing waste and scaling industrial applications.

Types of Magnetic Separation

Low-Intensity Magnetic Separation

Low-intensity magnetic separation (LIMS) employs low-gradient magnetic fields typically ranging from 0.03 to 0.3 T to recover ferromagnetic materials, such as magnetite, from non-magnetic gangue in bulk processing applications. These fields are generated using permanent magnets, like strontium ferrite, or simple electromagnets, which create a uniform magnetic zone without requiring high gradients. Common equipment includes wet and dry drum separators, as well as cross-belt variants, designed for modular setups that handle particle sizes from fine slurries (<6 mm) to coarser dry feeds (up to 8 mm). In the process, a feed of ore slurry (for wet separation) or dry material is directed toward the magnetic drum or belt, where ferromagnetic particles are attracted and adhere to the surface due to the applied field. The drum rotates to carry these particles past the feed zone and out of the magnetic influence, allowing them to be discharged into a concentrate launder, while non-magnetic material follows the slurry flow or falls away. For dry operations, such as in cross-belt separators, the feed passes over a moving belt above stationary magnets, with magnetics captured and removed by a scraper or belt deflection outside the field. Wet variants, like concurrent or counter-current drums, balance magnetic forces against hydraulic drag to achieve separation, often processing at capacities exceeding 150 tons per hour per meter of drum width. LIMS offers high throughput and excellent selectivity for strongly magnetic particles, making it ideal for initial concentration stages in mineral processing, particularly for coarse feeds greater than 50 μm. Its simple design ensures ease of operation and low maintenance, with minimal energy consumption compared to higher-intensity methods. However, it is ineffective for paramagnetic or weakly magnetic materials, which require stronger fields, and wet processes can incur higher operational costs due to water usage and environmental considerations. A prominent example is its application in taconite processing on the Mesabi Range, where wet LIMS recovers magnetite from low-grade iron ore to produce pellet feed, often achieving concentrates with over 65% iron content in pilot and commercial plants. Equipment variants, such as cross-belt separators, are used for dry removal of tramp iron from ore feeds, enhancing downstream efficiency in bulk handling.

High-Intensity and High-Gradient Magnetic Separation

High-intensity magnetic separation employs magnetic fields typically ranging from 0.5 to 2 T, generated by rare-earth permanent magnets or electromagnets, to separate weakly magnetic materials that are unresponsive to lower field strengths. Dry variants, such as induced roll and rare-earth roll separators using permanent magnets, extend this to processing of paramagnetic minerals like ilmenite or chromite from gangue. High-gradient magnetic separation (HGMS), a specialized subset, enhances separation by incorporating a magnetizable matrix—such as steel wool, ferromagnetic wires, or spheres—within the magnetic field to create localized field gradients up to 10^4 T/m. These gradients amplify the magnetic force on particles, enabling the capture of paramagnetic species with low magnetic susceptibility, separating them from diamagnetic or non-magnetic ones. In HGMS processes, a slurry or fluid containing particles flows through a column or chamber packed with the matrix, where the applied uniform field magnetizes the matrix elements, inducing high gradients that attract and retain magnetic particles on their surfaces. Non-magnetic particles continue with the flow, achieving separation based on differential magnetic forces overcoming hydrodynamic drag. Captured particles are periodically released by demagnetizing the system or flushing with a reverse flow, allowing collection of concentrates. Variants include superconducting HGMS systems, which achieve fields exceeding 5 T using cryogenically cooled magnets, suitable for ultra-fine particle processing. This technique excels in recovering paramagnetic particles smaller than 10 μm, such as fine iron oxides or rare-earth minerals, where conventional methods fail due to insufficient force. However, it demands higher energy consumption for field generation and faces challenges like matrix clogging from particle aggregation, necessitating careful control of flow rates and slurry properties. A prominent example is the Jones wet HGMS, which uses a grooved matrix in a rotating carousel for continuous mineral purification, recovering over 90% of fine hematite in industrial settings. HGMS evolved significantly post-1970s, building on early matrix concepts from the 1950s to address fine iron recovery in taconite ores and environmental applications, with key advancements in superconducting integration by the 1980s.

Applications

Industrial and Environmental Uses

Magnetic separation plays a pivotal role in mining operations, particularly for ore beneficiation where it efficiently separates iron minerals like and from gangue materials. In iron ore processing, low-intensity magnetic separators are commonly employed to concentrate ferromagnetic particles, enabling the production of high-grade iron concentrates essential for steelmaking. This method accounts for a substantial share of global iron ore beneficiation, with applications in major producing regions such as those in the Asia Pacific, including and , which together with others account for over 70% of worldwide iron ore output as of 2024. As of 2025, innovations in magnetic separation are projected to boost mineral extraction efficiency by up to 30%, with emerging applications in recycling ash and slag residues for multi-element recovery. Additionally, magnetic separation is utilized in coal desulfurization to remove pyrite, a paramagnetic sulfur-bearing mineral, thereby reducing sulfur emissions during combustion and improving coal quality for cleaner energy production. In recycling, magnetic separation facilitates the recovery of valuable metals from waste streams, distinguishing ferrous metals like steel and iron from non-metallic debris in processes such as shredding and sorting. For non-ferrous metals, eddy current separators induce magnetic fields to repel materials like aluminum and copper, enhancing purity in recycled outputs. Since the early 2000s, these techniques have been increasingly applied to electronic waste (e-waste) processing, where high-intensity separators recover metals from circuit boards and components, supporting sustainable resource recovery amid rising e-waste volumes. Environmentally, magnetic separation aids in pollution control by treating water contaminated with heavy metals; high-gradient magnetic separation (HGMS) effectively captures arsenic and other ions using magnetized adsorbents, achieving removal efficiencies suitable for geothermal and industrial wastewater remediation. In soil remediation, the technique targets magnetic contaminants such as heavy metal-laden particles, allowing their extraction from polluted sites through wet high-intensity magnetic separation, which recovers metals like cadmium, copper, lead, and zinc while minimizing soil disturbance. Case studies highlight the efficiency of magnetic separation in industrial settings, such as magnetite processing where optimized wet drum separators achieve recovery rates exceeding 85%, with one reported instance yielding 86.46% magnetite recovery at 63.31% iron grade from low-grade ores. Economically, these methods offer lower operating costs compared to chemical beneficiation alternatives, as they reduce reagent usage and energy demands while enabling high-throughput processing without generating secondary waste.

Biomedical and Microbiological Applications

Magnetic cell separation techniques, particularly immunomagnetic methods, utilize antibody-coated superparamagnetic beads to target specific cell surface markers, enabling precise isolation from complex biological samples. Developed in the 1990s by , magnetic-activated cell sorting (MACS) employs high-gradient magnetic fields and nanosized particles (typically 50 nm) for efficient separation, serving as a cost-effective alternative to fluorescence-activated cell sorting. This approach supports positive selection, where labeled target cells such as CD34+ stem cells are retained for transplantation, and negative selection, which depletes unwanted cells like leukocytes to enrich rare populations including circulating tumor cells (CTCs) in cancer diagnostics, achieving purities exceeding 90% in systems like CellSearch®. These techniques preserve cell viability (>95%) due to the gentle magnetic forces and minimal perturbation from nanoscale beads. In microbiological applications, magnetic separation facilitates the rapid isolation of from food and environmental samples using functionalized with specific antibodies. For instance, immunomagnetic particles targeting can capture over 90% of at concentrations from 10² to 10⁴ CFU/mL in spiked or milk, enabling downstream detection while maintaining bacterial viability for viability assays. This method enhances sensitivity in diagnostics by concentrating low-abundance microbes, reducing matrix interference in complex samples like or products. Recent advances have integrated magnetic separation with microfluidic platforms for high-throughput biomedical processing since 2015, allowing on-chip MACS with reduced sample volumes and automated sorting of heterogeneous cell populations. These devices combine magnetic fields with for precise, label-free or affinity-based separation, achieving high accuracy in applications like rare CTC enrichment. In , superparamagnetic nanoparticles (SPIONs) conjugated with therapeutics have entered clinical trials from 2020 to 2025, such as NanoTherm® (NCT06271421, recruiting as of 2025) for recurrent , where magnetic guidance enables localized and drug release, improving survival rates to 23.2 months compared to 14.6 months with standard therapy. Key advantages of these biomedical applications include their non-destructive nature, which supports downstream functional analyses, and emerging label-free options like diamagnetophoresis using paramagnetic salts to exploit intrinsic cellular , avoiding labeling altogether. For example, MACS-based enrichment of + T-cells from peripheral blood mononuclear cells achieves 82% purity and 85% recovery with over 95% viability, facilitating research in monitoring by isolating subsets like +CD39+ regulatory T-cells for immune suppression studies.

Theory and Calculations

Magnetic Forces and Separation Efficiency

The magnetic force acting on a susceptible particle in an inhomogeneous arises from the interaction of the induced moment with the . For a linearly magnetizable particle of volume VV and χ\chi, the induced dipole moment is m=χVμ0B\mathbf{m} = \frac{\chi V}{\mu_0} \mathbf{B}, where B\mathbf{B} is the vector and μ0\mu_0 is the permeability of free space. The force on a is given by Fm=(m⋅∇)B\mathbf{F}_m = (\mathbf{m} \cdot \nabla) \mathbf{B}, which substitutes to yield the core equation Fm=χVμ0(B⋅∇)B\mathbf{F}_m = \frac{\chi V}{\mu_0} (\mathbf{B} \cdot \nabla) \mathbf{B}. This force directs paramagnetic particles (χ>0\chi > 0) toward regions of increasing field strength and diamagnetic particles (χ<0\chi < 0) toward decreasing strength. In practical separation processes, the magnetic force competes with hydrodynamic drag and gravitational sedimentation, determining particle deflection and capture. The drag force on a spherical particle at low Reynolds number follows Stokes' law: Fd=3πηdv\mathbf{F}_d = 3\pi \eta d \mathbf{v}, where η\eta is the fluid viscosity, dd is the particle diameter, and v\mathbf{v} is the relative velocity between the particle and fluid. The gravitational force (including buoyancy if in a fluid) is Fg=πd36(ρp−ρf)g\mathbf{F}_g = \frac{\pi d^3}{6} (\rho_p - \rho_f) \mathbf{g}, where ρp\rho_p and ρf\rho_f are the particle and fluid densities, respectively, and g\mathbf{g} is the acceleration due to gravity. Particle motion is governed by Newton's second law, mdvdt=Fm−Fd−Fg−Fbm \frac{d\mathbf{v}}{dt} = \mathbf{F}_m - \mathbf{F}_d - \mathbf{F}_g - \mathbf{F}_b, where mm is the particle mass and Fb\mathbf{F}_b accounts for other body forces if present; numerical integration or approximations predict trajectories, with capture occurring if the particle contacts the collection surface within the residence time. Separation efficiency is evaluated using performance metrics that quantify material recovery and purity. The recovery rate is defined as the ratio of the mass of target magnetic material collected in the concentrate to the total mass of magnetic material in the feed: recovery = mseparatedmtotal magnetic\frac{m_\text{separated}}{m_\text{total magnetic}}. The grade represents the fraction of magnetic material in the concentrate: grade = mmagnetic in concentratemtotal in concentrate\frac{m_\text{magnetic in concentrate}}{m_\text{total in concentrate}}. These depend on operational factors such as flow rate, which controls residence time and thus deflection distance, and matrix saturation in high-gradient setups, where accumulated particles reduce the effective field gradient and capture capacity. Optimal conditions balance high recovery (often >90% for amenable ores) with acceptable grade (>50% magnetic content).90012-0) The cutoff diameter dcd_c marks the smallest particle size separable under given conditions, derived by balancing forces to ensure sufficient deflection for capture. For drag-dominated regimes (common in wet separations of fine particles), equate the magnetic force to drag at the terminal migration velocity vmv_m, assuming a one-dimensional gradient ∇B=dBdz\nabla B = \frac{dB}{dz} and neglecting gravity: χVB∇Bμ0=3πηdvm\frac{\chi V B \nabla B}{\mu_0} = 3\pi \eta d v_m. With V=πd36V = \frac{\pi d^3}{6}, this simplifies to vm=χd2B∇B18μ0ηv_m = \frac{\chi d^2 B \nabla B}{18 \mu_0 \eta}. Capture requires vm≥kuv_m \geq k u, where uu is the superficial flow velocity and kk (typically 0.1–1) is a geometric factor relating deflection distance to channel dimensions and residence time t=L/ut = L/u ( LL is separator length). Thus, dc=18μ0η(ku)χB∇Bd_c = \sqrt{\frac{18 \mu_0 \eta (k u)}{\chi B \nabla B}}
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