## Abstract

This work deals with the modeling of microwave heating of a water drop. A drop model is reduced to its electric dipoles, masses, and charges are constructed using the associating of COMSOL Multiphysics and Matlab software. The considered model proposes a microscopic point of view on microwave heating, which transforms electrical energy into heat.

### Keywords

- microwave heating
- water drop
- electric dipoles
- modeling

## 1. Introduction

Domestic microwave ovens first appeared in American homes at the end of World War II; the American company Raytheon, which manufactured the radars for the Liberty Ships, was redirecting its production towards civilian applications. This is the case in this chapter which deals with microwave heating. The present chapter is interested in a simple water drop reduced to its electric dipoles, masses, and charges. The drop contains 1536 of them in a 3.585 nm edge cube. These dipoles are in permanent interaction and move according to a brownian movement or similar. For these dipoles, receiving an electromagnetic wave means receiving an external force additional to those they receive from the other dipoles of the drop. Their movement generates friction and thus transforms part of the captured energy into heat. We must also keep in mind the adage: “a cold-cooked carrot is no longer raw”. Thus, heat produces irreversible changes. However, the cloud of dipoles similar to those of water is not a drop of water. It makes it easy to calculate its collective behavior in a microwave field, displacement, and speed of particles but does not put in thermal memory the sequence of events. Only a human program can transform the kinetic energy acquired by dipoles into thermal energy captured by water and invent a temperature scale that accumulates the sequence of events.

The expression “microwave heating” means that the material itself transforms into heat according to the equality * W = J.Q*, the electromagnetic energy it captures (1 calorie = 4.185 Joule, where

*is mechanical energy,*W

*is a universal constant, Q is a heat).*J

Internal movement of two electrical charges of the dipole also results in a current different from that of the free carriers in a metal or ions in the water itself, i. e., * J = σE* (where σ is electric conductivity and

*is an electric field) is displacement current,*E

during microwave heating, the thermal gradient is oriented from the inside to the outside; the material is not heated by its environment but heats it;

diversity of behavior of the volume is much greater than that of the surface. Moreover, the electrical and magnetic heterogeneity of the materials contributes to the diversity.

## 2. Electromagnetic waves in everyday life

### 2.1 Elementary experiments with induction stove and microwave oven

The active part of an induction hob is a spiral supplied with around 25 kHz and located just below the hob. The energy is not radiated, but it decreases exponentially as it moves away from the spiral. It generates a Foucault current in the surrounding conductors, which tends to move away from the inductor, i.e., from the plate.

To be convinced, one can place a sheet of household aluminum on the plate in operation. If the electrical force is greater than the weight of this aluminum sheet, the sheet will fly away and leaves the field. However, if the experimenter puts the finger on the sheet to hold it, he notices heat release. Heat generation is nature’s answer of last resort. It is interesting to tear the sheet and bring the two pieces together: an electric arc appears to ensure the current continuity. This arc sometimes denominates the extra-current of rupture.

To test a microwave oven, the experimenter can heat a soup plate slightly moistened and covered with a well-joined kitchen film. He will quickly see the film inflate like a balloon. Water evaporates and naturally finds with steam a mechanical application. Obviously, one should not use valuable plates or plates decorated with conductive paint.

The experimenter could dry a piece of bread put on the same plate but pre-cooled in a refrigerator and always covered with a kitchen film. He will extract drops of liquid water and then analyze them because they include a lot of salts and other components coming from the bread itself. It is indeed very useful to analyze the evaporated drop to highlight a possible peculiarity of microwave heating and try to find out if it is due to internal friction.

It is also interesting to test the widely marketed metal trays. If the tray is placed on the metal grid of the oven, an arc will appear with the vibrations during operation. The arc can also be maintained by directly capturing microwave energy. One must follow the experiment by standing a few meters from the oven because the door grid is not completely impermeable. However, it is made to see since there is a lamp in the oven!

It is instructive to experiment with metal trays filled with common foods to test the quality of heating. The considered domestic experiments also give an order of magnitude of the masses and volumes, power, and time encountered in microwave heating, for example, one liter of water, one microwave kW, and one minute of heating. These proportions are used in the simulation program presented below. Mass and volume are related to the number of dipoles chosen since 18 cm^{3} of water contains * N*, Avogadro’s number of molecules, and weighs 18 g. Time is measured by the period of the incident microwave field since period

*is the inverse of the chosen frequency.*T

### 2.2 Magnetron and power supply

The magnetron is an electromagnetic wave generator, very similar, in its operating principle, to a 50 Hz alternator but which emits at very high frequencies, for example, 2.45 GHz. The magnetron constitutes a cylindrical vacuum diode formed of an electron-emitting cathode and a stator receiving anode. A continuous magnetic field is applied in the axis to spin the electrons. The rotor appears because the anode generates an alternating circular electric field, sometimes in one direction, sometimes in the other to the group–ungroup the electrons in the beam and thus find the characteristics of a real alternator winding. The electrons finish their route and reach the anode by producing heat which must be eliminated by ventilation or circulation of fluid. The electron beam, in a vacuum, can rotate very quickly (much faster than a rotor) and thus generate a high-frequency current. The anode also has an antenna that collects and emits produced electromagnetic energy.

One can find magnetrons in industrial microwave heating and radars, on roads, and in kitchens. The kitchen magnetron has a special power supply: the first 50 Hz alternation (60 in some countries) charges a capacitor, and this charge will be added to the second. Therefore, the magnetron transmits half-time and even less if the user wishes to reduce the power of his oven. Thus, the heated food makes the average value and leaves a little of the thermal energy of the hot regions to diffuse towards the cold regions. In the elementary experiments proposed above, the observer should detect that the balloon above the plate inflates and deflates in time with the power supply.

## 3. Construction of the model and numerical results

The proposed model studies a water drop, of a cubic form for simplification, in an electromagnetic field. The water dipoles are replaced by point dipoles carrying the masses and electrical charges of hydrogen * H* and oxygen

*and mobile in the aqua (aqueous liquid), possessing all the other mechanical and thermal properties of water and the vacuum permittivity. Aqua and dipoles have the same density as water.*O

The model was constructed using the associating of COMSOL and Matlab software [1, 2]. The considered model is a cube with an edge length of * H*” and

*” (Figure 1) [3].*O

The following boundary conditions are chosen for modeling: for the faces, z = 0 and z = * aa* values of the potential

*= 0 and*V

*are applied, respectively, and for the other faces, the value*V = Vin

*, of each dipole is supposed to be fixed at the mass center of the tetrahedrons of meshing of the cube chosen so that the density of the dipoles is the same as that of water molecules in a liquid drop. The points*mc

*and*H

*linked to each*O

*are randomly oriented in the local spherical coordinates. The coordinates of all the*mc

*are grouped in the permanent matrix*mc

*(size 1536 × 3), generating two random matrices,*pmc

*and*pH

*of the same size. The formula below represents a part of the first stage of the modeling, i.e., random placement of the 1536 centers of gravity*pO,

*in a cube with edge*pmc

*= 3.585e–9 m:*aa

Taking into consideration length of the dipoles* , lm* = 1.755e-10 m, distances to the point

*(*H

*) and to the point*lH

*(*O

*) from the center of gravity respectively:*lO

the following code was written to determine random positions of oxygen * O* and hydrogen

*in the cube:*H

* for tt =* 1

:line.

* theta = pi*rand; phi =* 2

*pi*rand;

* pH*(

*1)*tt,

*(*= pmc

*1)*tt,

*(*+ lH*sin

*)*theta

*(**cos

*);*phi

* pH*(

*2)*tt,

*(*= pmc

*2)*tt,

*(*+ lH*sin

*)*theta

*(**sin

*);*phi

* pH*(

*3)*tt,

*(*= pmc

*3)*tt,

*(*+ lH*cos

*);*theta

* pO*(

*1)*tt,

*(*= pmc

*1)*tt,

*(*– lO*sin

*)*theta

*(**cos

*);*phi

* pO*(

*2)*tt,

*(*= pmc

*2)*tt,

*(*– lO*sin

*)*theta

*(**sin

*);*phi

* pO*(

*3)*tt,

*(*= pmc

*3)*tt,

*(*– lO*cos

*);*theta

Here, * theta* and

*are local spherical coordinates. The initial velocity of the atoms of hydrogen*phi

*and oxygen*H

*is chosen to be zero. These atoms carry mass and electric charges, interact with each other and are under the action of fields received from outside; their centers of gravity play no role.*O

For calculation, which presents the second stage of the modeling, COMSOL Multiphysics requires classifying the points * H* and

*according to a precise order: classification of the points according to their coordinates*O

*,*x

*and*y,

*. It is obviously necessary to put markers to reform the dipoles.*z

The actual calculation requires a certain number of loops “for,” which also serve as time markers. The main marker is the period * T* of the wave chosen for heating (note that

*, where*T = 1/f

*2.45 GHz is frequency). This period is divided into 60 equal parts. The COMSOL calculation is then carried out in each part using static analysis. This heating period is preceded and followed by a period of analysis of the initial situation and final situations.*f ≈

At the end of each part of the period, obtained results, fields, positions, and speeds replace the initial data, and certain values are stored for further analyses. This is the case with the test dipole numbered (named) “1234,” whose position and electric field values are written in an annexed memory. The third stage of modeling, as the main obtained results along with their interpretations, is presented below. Figure 2 gives values of static potential * V* between two opposite faces of the model cube. The intervals (0, 10) and (70, 80) on the horizontal axis make it possible to analyze the initial and final situations, i.e., the period just before and after applying the electric field. Figure 3 represents a variation of length (

*, m) of the dipole “1234” for two different potential amplitudes (which corresponded to the blue and the red line, respectively):*L

The variation in length around 1.755e–10 m shows that the water molecule undergoes compression and traction actions which are possible sources of internal friction and heat. This variation can also be a source of physicochemical modification analogous to the Kerr effect [4, 5].

Tribology shows that friction is a very general phenomenon that has useful results and others that are less. For example, prehistoric man domesticated fire and created a prototype of the violin, and conversely, the rolling of trucks and cars damages the treads of roads. So it is always prudent to look at the quality of the material heated in the microwave.

Figure 4 presents the electric field applied to dipole “1234” during its Brownian motion in the drop. In the Figure, the red line corresponds to the force received by the oxygen * O,* and the blue one corresponds to those received by the hydrogen

*of the considered dipole.*H

In Figure 5, the red line corresponds to the impressed heating field. Its coherence is transposed on the blue line presenting the * Ez* field submitted here on hydrogen (dipole “1234”). On the other hand, the two components in

*and*x

*(green line) remain erratic but do not heat up. In addition to heating, these fields could promote other phenomena such as hydrogen bonding. Accumulation of all dipoles’ kinetic energy of the considered water drop is shown in Figure 6.*y

The accumulation of kinetic energy denoting the existence of friction induces an accumulation of heat in the cube (the model of considered water drop) because it has no other possible use; the interval (70, 80) of the horizontal axis shows some descent due to the stopping of the heating.

From the macroscopic point of view, this property of heating can be treated as electronic conduction in metals by introducing the imaginary part

Figure 7 (** a**,

**, and**b

**) are produced by COMSOL at the times of heating (8 s, 20 s, and 25 s, respectively) during a period surrounded by two rest phases. These figures are 2D sections perpendicular to the**c

*-axis. The boundary conditions chosen for modeling are: for the face*x

*= 0 value of potential*z

*= 0 and for the face*Vin

*=*z

*=*aa – Vin

*(*sin

*); T is the period. For the faces,*t/T

*= 0 and*y

*=*y

*value of the applied potential is*aa

The color gives the distribution of electrical energy in the cube. Even in Figure (a), one can distinguish a difference in the distribution of the energy and, therefore, also of the potential.

The dots show the position of oxygen and hydrogen as a function of time. The poles are not always very close; on the sides, there is a string of very tight dipoles, but they are due to a function of Matlab, which is to return the dipoles which had come out of it into the cube.

## 4. Conclusion

The considered model of the water drop has proposed a microscopic point of view at the microwave heating being a response of the studying material and not a heat transfer. This fact has made it possible to understand the physical origin of the two permittivity components and envisage the production of new materials unimaginable with traditional heating means.

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