# On the Origin of the Magnetic Concentration Gradient Force and Its Interaction Mechanisms with Mass Transfer in Paramagnetic Electrolytes

## Abstract

**:**

## 1. Introduction

^{1/2}Sc

^{1/3}, where K describes the flow configuration [5,9].

## 2. Materials and Methods: The Concept of Minimum Reluctance Principle

#### 2.1. Magnetic Field Theory Applied to a Simplified Magnetic Circuit

_{r}, and magnetic susceptibility (χ) are much greater than 1, depending on the magnetization. Paramagnetic materials are “weak magnetic materials”, with μ

_{r}$\approx $ 1 and 0 < χ < 1. Examples are electrolytes such as iron(III) chloride, nickel(II) chloride, and cobalt(II) sulfate. Diamagnetic materials are nonmagnetic materials with μ

_{r}= 1 and −1 < χ < 0. Examples are air, water, and electrolytes such as zinc chloride and potassium chloride [22,41].

_{r}, as shown in Figure 1; the rotor with magnetic pole pieces at its ends can rotate around its axis directed perpendicular to the paper plane between the pole pieces of the stator. The rotor is the moving part of the magnetic circuit.

_{s}and l

_{r}, respectively. The length of each airgap between the stator and rotor is denoted as l

_{g}. The cross-sectional area throughout the magnetic circuit is denoted as A. Suppose a current, I, is flowing through the magnetizing coil with N turns and no fringning.

_{r}= 1 + χ

_{mol}, where χ

_{mol}is the magnetic molar susceptibility for the material.

_{r}= 1 for air, and l

_{stat}and l

_{rot}are the mean lengths of the stator and rotor, respectively. In this case, the air gaps have their minimum length, denoted l

_{g}min, the cross-section area for the airgaps and rotor is denoted A

_{min}, and the reluctance is denoted $\mathcal{R}$

_{min}.

_{g max}. The area of the rotor side facing the flux is denoted A

_{max}. The effective cross-section area for the airgaps and rotor is denoted A

_{eff}. The width of the rotor is denoted l

_{w}. Suppose μ

_{r}is the same for case A and case B. The reluctance denoted $\mathcal{R}$

_{max}, is given by:

_{eff}= ½ (A

_{min}+ A

_{max}).

_{g min}and l

_{g max}with minimum values at θ = 0, π, 2π …, and maximum values at θ = π/2, 3π/2, 5π/2… Corresponding variations occur with the reluctance of the magnetic circuit. An illustrative example is given in the following.

_{stat}= 0.26 m, l

_{rot}= 0.04 m, l

_{w}= 0.02 m, l

_{g min}= 0.001 m, l

_{g max}= 0.01 m,

_{min}= 0.0004 m

^{2}, A

_{max}= 0.0008 m

^{2}, μ

_{r}= 2000, and μ

_{0}= 4π·10

^{−7}H/m.

_{eff}= ½(A

_{min}+ A

_{max}) = 0.0006 m

^{2}.

_{min}, and when the magnetic axis of the stator is perpendicular to that of the rotor (θ = π/2), $\mathcal{R}$

_{max}.

^{6}H

^{−1}

^{6}H

^{−1}

_{r}= 2000) is much higher than for air (μ

_{r}= 1).

_{min}. When the magnetic axis of the stator is perpendicular to that of the rotor (θ = π/2, 3π/2, 5π/2…), $\mathcal{R}$(θ) = $\mathcal{R}$

_{max}. Using values for $\mathcal{R}$

_{min}and $\mathcal{R}$

_{max}from the example above, and a small value of θ, for example θ = 12°, the reluctance is calculated by Equation (7):

^{6}≈ 9·10

^{6}H

^{−1}

#### 2.2. The Minimum Reluctance Principle

- The attraction mechanism, where a ferromagnetic, or paramagnetic, moving part is attracted into the magnetic field;
- The hindrance mechanism, where the moving part is hindered from leaving the magnetic field.

## 3. Results and Discussion

#### 3.1. Diffusion of a Paramagnetic Electrolyte Through a Thin Membrane

^{−6}with average velocity of the liquid solution 10

^{−6}m/s, pore diameter 0.22 × 10

^{−6}m, and kinematic viscosity of the liquid 10

^{−6}m

^{2}/s [39]. The concentration gradient is assumed to be limited to the membrane. The magnetic susceptibility of the electrolyte is proportional to its concentration; therefore, the magnetic susceptibility is higher in chamber 1 than in chamber 2. The gradient of the magnetic susceptibility, and consequently, the magnetic concentration gradient force, is assumed to be limited to the membrane as well. The reluctance of the electrolyte is inverse proportional to its magnetic susceptibility (Equation (4)); therefore, the reluctance is lower in chamber 1 than in chamber 2, and shows a gradient in the membrane as well, as shown in Figure 3C.

_{a}) is just the magnetic concentration gradient force, which has the same positive x-direction as in Figure 3, and is given by [39]:

_{h}) has the opposite direction from the positive x-direction in Figure 3, and is given by [39]:

_{a}and F

_{h}, are diminished. At equilibrium, there is no concentration gradient, i.e., no reluctance gradient, in the membrane, and consequently, no magnetic concentration gradient force. Therefore, the magnetic concentration gradient force is elusive, which is in accordance with the obtained experimental results [19,31].

#### 3.2. Deposition of Paramagnetic Ions on an Electrode Surface

^{2+}respective Cu

^{2+}, is flowing into the diffusion layer under influence of the magnetic field, it is attracted by the magnetic concentration gradient force which tends to fill the diffusion layer with the paramagnetic electrolyte in order to decrease the reluctance. The attraction force (F

_{a}), which is based on the minimum reluctance principle, has the same direction as the positive x-direction and is given in Equation (10) and shown in Figure 4.

_{h}), which is based on the minimum reluctance principle, has the opposite direction as the positive x-direction and is given in Equation (11) and shown in Figure 4.

_{4}solutions in magnetic fields. A homogeneous magnetic field, up to 13 T, was introduced perpendicular to the plane electrode surface. Two types of cell geometry were used: a flat embedded electrode and a wall electrode which was expected to influence the convection close to the surface. The principle setup is shown in Figure 5.

#### 3.3. Mass Transfer of Paramagnetic Ions from an Electrode Surface

_{3}CN solution of nitrobenzene (NB). NB was reduced by the electrode into the red colored paramagnetic ${\mathrm{NB}}^{-}.$ Cyclic voltammetry was used. A homogeneous magnetic field, 3.3 T, was introduced perpendicular to the end of the horizontal orientated cylinder, as shown in Figure 6.

^{+}at the electrode [23].

^{−}ions tends to flow out of the diffusion layer by free convection under influence of the magnetic field, it is hindered by the magnetic concentration gradient force in order to hinder an increase in the reluctance in the diffusion layer. The hindrance force (F

_{h}) which is based on the minimum reluctance principle, has the opposite direction as the positive x-direction, and is given in Equation (11) and shown in Figure 7. According to Ragsdale and White [51], the magnetic concentration gradient force and free convection are of the same order of magnitude.

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A | Area (m^{2}) |

B | Magnetic flux density (T) |

c | Electrolyte concentration (mol/m^{3}) |

F | Force (N) |

I | Electric current (A) |

K | Constant (dimensionless) |

l | Length (m) |

N | Number of turns (dimensionless) |

r | Position vector (m) |

$\mathcal{R}$ | Reluctance (H^{−1}) |

Re | Reynolds number (dimensionless) |

Sc | Schmidt number (dimensionless) |

Sh | Sherwood number (dimensionless) |

T | Torque (Nm) |

x | Position (m) |

θ | Angle (°) |

μ | Magnetic permeability (H/m) |

Φ | Magnetic flux (Wb) |

χ | Magnetic susceptibility (dimensionless) |

0 | Reference value |

1 | Chamber 1 |

2 | Chamber 2 |

a | Attraction |

eff | Effective |

g | Gap |

h | Hindrance |

max | Maximum |

min | Minimum |

mol | Molar |

r | Relative |

rot | Rotor |

stat | Stator |

total | Total |

w | Width |

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**Figure 1.**A simplified magnetic circuit consisting of a stator and rotor. The rotor’s axis is directed perpendicular to the paper plane. (

**A**) The magnetic axis of the stator and that of the rotor is aligned. (

**B**) The magnetic axis of the rotor is perpendicular to the magnetic axis of the stator. (

**C**). There is an angle θ between the magnetic axis of the stator and the rotor [42].

**Figure 2.**A segment of the magnetic circuit shown in Figure 1, consisting of the rotor and a small part of the stator. (

**A**) The rotor and stator are aligned. (

**B**) The rotor is turned to an angle θ and then released. (

**C**) The rotor is forced to an angle − θ. The blue lines represent the magnetic flux lines. Positive rotation direction is clockwise.

**Figure 3.**(

**A**) The diffusion process of an unstirred paramagnetic electrolyte from chamber 1 to chamber 2 through a thin, inert membrane under influence of a homogeneous static magnetic field, $\overrightarrow{B}$. (

**B**) Illustration of the concentration profile from chamber 1 to chamber 2 through the membrane. (

**C**) Illustration of the reluctance profile from chamber 1 to chamber 2 through the membrane [39].

**Figure 4.**Illustration of concentration profile (black) [49,50] and reluctance profile (red) for deposition of the paramagnetic metal ions on an electrode surface. F

_{a}and F

_{h}(green) represents the magnetic concentration gradient force. The magnetic field, $\overrightarrow{B}$, was applied perpendicular to the electrode surface. Positive x-direction is indicated.

**Figure 6.**Photos of the horizontal orientated cylinder electrode in the nitrobenzene (NB) solution with (

**B**) and without (

**A**) exposure of the magnetic field. The direction of the magnetic field is indicated [31].

**Figure 7.**Illustration of concentration profile (black) and reluctance profile (red) in the diffusion layer. F

_{h}(green) represents the magnetic concentration gradient force. The magnetic field, $\overrightarrow{B}$, was applied perpendicular to the electrode surface. Positive x-direction is indicated.

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**MDPI and ACS Style**

Waskaas, M.
On the Origin of the Magnetic Concentration Gradient Force and Its Interaction Mechanisms with Mass Transfer in Paramagnetic Electrolytes. *Fluids* **2021**, *6*, 114.
https://doi.org/10.3390/fluids6030114

**AMA Style**

Waskaas M.
On the Origin of the Magnetic Concentration Gradient Force and Its Interaction Mechanisms with Mass Transfer in Paramagnetic Electrolytes. *Fluids*. 2021; 6(3):114.
https://doi.org/10.3390/fluids6030114

**Chicago/Turabian Style**

Waskaas, Magne.
2021. "On the Origin of the Magnetic Concentration Gradient Force and Its Interaction Mechanisms with Mass Transfer in Paramagnetic Electrolytes" *Fluids* 6, no. 3: 114.
https://doi.org/10.3390/fluids6030114