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Simulation of the Magnetic Freezing Process Applied to Foods

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Magnetic field (MF) freezing of foods is an innovative process already developed at the industrial level. However, the real interaction among the variables taking place in the process is not yet given in the literature because the MF distribution inside the volume of the processing cell is still not well studied. This fact jeopardizes both drawing correct conclusions about the synergy produced by the programmed combination of physical agents and achieving an optimal design of the processing device. Three different representative magnetic freezing processes applied to foods are here dealt representing the modeling of the distribution of the MF around the foods. One of them is based on the application of a static MF produced from magnets in a laboratory freezer. Another one is promoted by the action of an oscillatory magnetic field (OMF) produced by an iron core inductor device disposed also in a laboratory freezer. The third one also is based on the action of an OMF produced by using air core inductor coils simulating the performance of a commercial food freezer. With the aim of understanding the interactions between foods and MF necessary to support beneficial actions of its presence during freezing, analytical, numerical modeling, and experimental procedures have been studied and are here reviewed. Taking into account the accuracy afforded by using modeling and considering also the difficulty of performing experimental determinations of MF inside the volume of the device during a real freezing process, the modeling procedure seems to be a very suitable and practical tool to know about the MF distribution in it. Due to the very different behaviors of the MF vectors that can be found depending on the design of the freezer and the location of the sample inside, it is recommended to perform a previous specific mathematical modeling for each planned MF freezing process applied to foods.

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This work was supported by the Spanish Ministry of Economy and Competitiveness (MINECO) through the project AGL2012-39756-C02-01.


A.C. Rodríguez acknowledges the predoctoral contract BES-2013-065942 from MINECO, jointly financed by the European Social Fund, in the framework of the National Program for the Promotion of Talent and its Employability (National Sub-Program for Doctors Training). J. Sánchez-Benítez acknowledges support from MINECO for the Ramon y Cajal (RyC-2010-06276) fellowship

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Correspondence to A. C. Rodríguez.



Analytical description of the stationary MF in the laboratory freezer

In order to assess the respective accuracies of the proposed FEM model and the experimental measurements, a comparison with the results provided by an analytical method is employed in this work. By analyzing the different analytical approaches exposed in the literature to calculate the MF generated by permanent magnets, they can be typically divided into two groups [59, 60]: one is the Coulombian model in which two fictitious charge densities (positive and negative), distributed on the north and south faces of the magnet, would be responsible for generating a magnetic scalar potential V m (measured in A), of which \( \overrightarrow{B} \) (T) could be extracted by applying Eqs. (1) and (4) [61, 62]; the other group is the Amperian model which makes use of fictitious currents generating a MF equivalent to that of the magnet. In this case, \( \overrightarrow{B} \) is calculated as the curl of the magnetic vector potential \( \overrightarrow{A} \) (T m) or based on Biot-Savart’s law [63]. Based on the later model, we have obtained an expression for the MF \( \overrightarrow{B} \) on the axis passing through the centers of both cylindrical magnets. Starting from a cylindrical magnet with radius R (m) and height L (m), as represented in Fig. 17, a current uniformly distributed throughout its lateral surface is supposed in such a way that the induced MF is equivalent to that of the magnet. This is accepted as a valid model for a cylindrical magnet, considering that its magnetization is sufficiently uniform [64]. In this model, the currents must flow in the pertinent direction so that the lines of induced \( \overrightarrow{B} \) follow a trajectory going from the north to the south poles of the magnet.

The total equivalent current I eq (A) would be given by

$$ {I}_{\mathrm{eq}}=\frac{p}{\pi {R}^2} $$

where p is the net magnetic moment of the magnet (A/m2). Taking a differential element of surface as a circular loop of height dz and radius R (both in m), the differential current, di, circulating through this loop would be

$$ di=\frac{I_{\mathrm{eq}} dz}{L}=\frac{P}{\pi {R}^2L} dz $$

Given a point P on the axis of the cylindrical magnet, this differential current would contribute to MF with a d\( \overrightarrow{B} \) only in the direction of that axis, since MF components parallel to the magnet basis are cancelled out by symmetry. By applying the Biot-Savart’s law, the expression for the differential MF d\( \overrightarrow{B} \) (T) in a material medium is as follows:

$$ d\overrightarrow{B}=\frac{\mu {R}^2 di}{2{\left(\sqrt{R^2+{z}_P^2}\right)}^3}\overrightarrow{k}=\frac{\mu p\ dz}{2\pi L{\left(\sqrt{R^2+{z}_P^2}\right)}^3}\overrightarrow{k} $$

in which z P is the distance (m) from the point P to the differential current loop, μ is the magnetic permeability of the medium (H/m), and \( \overrightarrow{k} \) is the unit vector in the direction of axis going out from the magnet north pole. Considering the angle θ formed by the relative position vector from the differential current loop to P and the magnet axis, the change of variable R = z P · tan(θ) allows working out the integral on the lateral surface to find the MF \( \overrightarrow{B} \):

$$ \overrightarrow{B}=-\frac{\mu p}{2\pi {LR}^2}{\int}_{\theta_1}^{\theta_2}\sin \left(\theta \right) d\theta \overrightarrow{k}=\frac{\mu p}{2\pi {LR}^2}\left(\cos \left({\theta}_2\right)-\cos \left({\theta}_1\right)\right)\overrightarrow{k} $$

Magnetization \( \overrightarrow{M} \) is defined in terms of the net magnetic moment of the magnet divided by its volume, and multiplied by μ 0, thus resulting in the magnetic remanence \( \overrightarrow{B_r} \) (T). Thus, the MF on the axis of a cylindrical magnet can be expressed as a function of \( \overrightarrow{B_r} \), the relative permeability μ r of the material medium (H/m), distance z to the magnet (m) and dimensions of the cylinder by applying the known Eq. [65] to a medium different from vacuum, according to the geometric variables shown in Fig. 17:

$$ \overrightarrow{B}={\mu}_r\frac{\overrightarrow{B_r}}{2}\left(\frac{z+L}{\sqrt{{\left(z+L\right)}^2+{R}^2}}-\frac{z}{\sqrt{z^2+{R}^2}}\right) $$
Fig. 17
figure 17

Derivation of MF \( \overrightarrow{B} \) on the axis of a cylindrical magnet by using Biot-Savart’s law

Nevertheless, looking for an expression of MF off the axis of the magnet, there is not such a simple solution to this problem requiring, in general terms, the use of elliptic integrals as shown in [64, 66,67,68]. In the present work, the MF out of the magnet axis has been determined by using the equations given in [69] for the case of a ring-shaped magnet with axial magnetic polarization, conveniently adapted to a material medium. In order to calculate the MF created by a solid cylinder, the inner radius of the ring needs to be zero. Those authors used a Coulombian approach to find the field, introducing fictitious positive and negative magnetic charge densities (+ σ* and − σ*, measured in T) in both superior and inferior bases, respectively. The total field at any point can be calculated as a sum of two terms depending on each charge distribution. The equations given in terms of the MF intensity \( \overrightarrow{H} \) (A/m) have been expressed here in terms of the MF \( \overrightarrow{B} \) (T), considering only the volume out of magnets, i.e., without the magnetization term in the constitutive equation, to simplify those formulas. So according to Babic and Akyel manuscript, in cylindrical coordinates (ρ, φ, z 0), taking the center of the south pole circular base as the origin of coordinates, there are two terms for the radial component:

$$ {B}_{\rho}^{+}\left(\rho, \varphi, {z}_0\right)={\mu}_r\frac{\sigma^{\ast }}{\pi }{\sum}_{n=1}^2{\left(-1\right)}^{n-1}\frac{1}{k_n^{+}}\sqrt{\frac{r_n}{\rho }}\left[E\left({k}_n^{+}\right)-\left(1-\frac{k_n^{+2}}{2}\right)K\left({k}_n^{+}\right)\right] $$
$$ {B}_{\rho}^{-}\left(\rho, \varphi, {z}_0\right)={\mu}_r\frac{-{\sigma}^{\ast }}{\pi }{\sum}_{n=1}^2{\left(-1\right)}^{n-1}\frac{1}{k_n^{-}}\sqrt{\frac{r_n}{\rho }}\left[E\left({k}_n^{-}\right)-\left(1-\frac{k_n^{-2}}{2}\right)K\left({k}_n^{-}\right)\right] $$

where the functions K and E are the complete elliptic integrals of the first and second kinds, respectively. And the two terms for the vertical component are as follows:

$$ {\displaystyle \begin{array}{c}\hfill {B}_{z_0}^{+}\left(\rho, \varphi, {z}_0\right)={\mu}_r\frac{\sigma^{\ast }}{4\pi \left({z}_0-L\right)}{\sum}_{n=1}^2{\left(-1\right)}^{n-1}\frac{k_n^{+}}{\sqrt{\rho {r}_n}}\left(\left[\sqrt{\rho^2+{\left({z}_0-L\right)}^2}-\rho \right]\left[\sqrt{\rho^2+{\left({z}_0-L\right)}^2}-\right.\right.\hfill \\ {}\hfill \left.{r}_n\right]\left.\varPi \left({h}_1^{+},{k}_n^{+}\right)+\left[\sqrt{\rho^2+{\left({z}_0-L\right)}^2}+\rho \right]\left[\sqrt{\rho^2+{\left({z}_0-L\right)}^2}+{r}_n\right]\varPi \left({h}_2^{+},{k}_n^{+}\right)\right)\hfill \end{array}} $$
$$ {\displaystyle \begin{array}{c}\hfill {B}_{z_0}^{-}\left(\rho, \varphi, {z}_0\right)={\mu}_r\frac{-{\sigma}^{\ast }}{4\pi {z}_0}{\sum}_{n=1}^2{\left(-1\right)}^{n-1}\frac{k_n^{-}}{\sqrt{\rho {r}_n}}\left(\left[\sqrt{\rho^2+{z}_0^2}-\rho \right]\left[\sqrt{\rho^2+{z}_0^2}-{r}_n\right]\Pi \left({h}_1^{-},{k}_n^{-}\right)+\right.\hfill \\ {}\hfill \left.\left[\sqrt{\rho^2+{z}_0^2}+\rho \right]\left[\sqrt{\rho^2+{z}_0^2}+{r}_n\right]\Pi \left({h}_2^{-},{k}_n^{-}\right)\right)\hfill \end{array}} $$

where Π is the complete elliptic integral of the third kind. There is neither azimuthal component of \( \overrightarrow{B} \) , in accordance with Eq. (6), nor dependence on this azimuthal coordinate because of axial symmetry of MF, and its cylindrical components at any point off the magnets result in the following:

$$ {B}_{\rho}\left(\rho, \varphi, {z}_0\right)={B}_{\rho}^{+}\left(\rho, \varphi, {z}_0\right)+{B}_{\rho}^{-}\left(\rho, \varphi, {z}_0\right) $$
$$ {B}_{\varphi}\left(\rho, \varphi, {z}_0\right)=0 $$
$$ {B}_{z_0}\left(\rho, \varphi, {z}_0\right)={B}_{z_0}^{+}\left(\rho, \varphi, {z}_0\right)+{B}_{z_0}^{-}\left(\rho, \varphi, {z}_0\right) $$

It can be easily proved [62] that the sum of the magnetic charge densities of both magnet poles has to be equal to the magnetization of the magnet. Then, the value of σ* which appears in Eqs. (17) to (20) is half that of the magnet remanence \( \overrightarrow{B_r} \). Some other parameters were introduced in those equations to shorten them, namely

$$ {\displaystyle \begin{array}{cc}\hfill {k}_n^{+}=\sqrt{\frac{4\rho {r}_n}{{\left(\rho +{r}_n\right)}^2+{\left({z}_0-L\right)}^2}}\hfill & \hfill {k}_n^{-}=\sqrt{\frac{4\rho {r}_n}{{\left(\rho +{r}_n\right)}^2+{z}_0^2}}\hfill \\ {}\hfill {h}_1^{+}=\frac{2\rho }{\rho +\sqrt{\rho^2+{\left({z}_0-L\right)}^2}}\hfill & \hfill {h}_2^{+}=\frac{2\rho }{\rho -\sqrt{\rho^2+{\left({z}_0-L\right)}^2}}\hfill \\ {}\hfill {h}_1^{-}=\frac{2\rho }{\rho +\sqrt{\rho^2+{z}_0^2}}\hfill & \hfill {h}_2^{-}=\frac{2\rho }{\rho -\sqrt{\rho^2+{z}_0^2}}\hfill \\ {}\hfill {r}_1={r}_{\mathrm{in}}=0\hfill & \hfill {r}_2={r}_{\mathrm{out}}=R\hfill \end{array}} $$

It is worth noting that on doing r 1 = 0, as this would be the diameter of the cylindrical hollow of ring-shaped magnets (not present in these solid cylindrical magnets), the first terms of the summations in Eqs. (17 to 20) are cancelled. These equations result in indeterminations for any point on the axis, that is when ρ = 0. In this case, Eq. (16) is adequate for the purpose of determining the MF.

The formulas presented in this analytical approach are referred to only one cylindrical magnet, so the superposition principle will be applied to obtain the solution for the case with two magnets, calculating the MF \( \overrightarrow{B} \) at any point as the vector addition of the fields created by both magnets. In this particular case, given the arrangement of the magnets with their axes coinciding, there will not be any azimuthal component of the MF either.

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Rodríguez, A.C., Sánchez-Benítez, J. & Sanz, P.D. Simulation of the Magnetic Freezing Process Applied to Foods. Food Eng Rev 9, 271–294 (2017).

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