**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 *O* 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.

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 *aa* ¼ 3*:*585*e* � 9 m containing 1536 electric dipoles. The electric charges of the dipoles are �2 � 1*:*606*e* � 19 C, and their masses are 2 � 0*:*16*e* � 26 kg for the point labeled "*H*" and 16 � 0*:*16*e* � 26 kg for the point labeled "*O*" (**Figure 1**) [3].

The following boundary conditions are chosen for modeling: for the faces, z = 0 and z = *aa* values of the potential *V* = 0 and *V = Vin* are applied, respectively, and for the other faces, the value *<sup>∂</sup>V=∂<sup>n</sup>* <sup>¼</sup> 0 is applied. The modeling contains three main stages: (1) random placement of the dipoles, (2) calculation of the electric fields by COMSOL Multiphysics and repositioning of the dipoles, and (3) exploitation of the obtained results. The mass center, *mc*, 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 *H* and *O*

**Figure 1.** *Model of a water drop reduced to its electric dipoles.*

linked to each *mc* are randomly oriented in the local spherical coordinates. The coordinates of all the *mc* are grouped in the permanent matrix *pmc* (size 1536 � 3), generating two random matrices, *pH* and *pO,* 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 *pmc* in a cube with edge *aa* = 3.585e–9 m:

$$pmc = aa^\*rand(\mathbf{1536,3)}.$$

Taking into consideration length of the dipoles*, lm* = 1.755e-10 m, distances to the point *H* (*lH*) and to the point *O* (*lO*) from the center of gravity respectively:

$$lH = lm^\*mH/(mH+mO),\\ lO = lm^\*mO/(mH+mO),$$

the following code was written to determine random positions of oxygen *O* and hydrogen *H* in the cube:

> *for tt =* 1*:line. theta = pi\*rand; phi =* 2*\*pi\*rand; pH*(*tt,*1) *= pmc*(*tt,*1) *+ lH\*sin*(*theta*)*\*cos*(*phi*); *pH*(*tt,*2) *= pmc*(*tt,*2) *+ lH\*sin*(*theta*)*\*sin*(*phi*); *pH*(*tt,*3) *= pmc*(*tt,*3) *+ lH\*cos*(*theta*); *pO*(*tt,*1) *= pmc*(*tt,*1) *– lO\*sin*(*theta*)*\*cos*(*phi*); *pO*(*tt,*2) *= pmc*(*tt,*2) *– lO\*sin*(*theta*)*\*sin*(*phi*); *pO*(*tt,*3) *= pmc*(*tt,*3) *– lO\*cos*(*theta*); *end*

Here, *theta* and *phi* are local spherical coordinates. The initial velocity of the atoms of hydrogen *H* and oxygen *O* 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.

For calculation, which presents the second stage of the modeling, COMSOL Multiphysics requires classifying the points *H* and *O* according to a precise order: classification of the points according to their coordinates *x*, *y,* and *z*. It is obviously necessary to put markers to reform the dipoles.

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 *T = 1/f*, where *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.

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

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].

**Figure 2.** *Values of static potential* V *between two opposite faces of the cube.*

**Figure 3.** *Variation of dipole length "1234" for two different potential amplitudes.*

**Figure 4.** *Electric field applied to dipole "1234" during its Brownian motion in the drop.*

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 *H* of the considered dipole.

**Figure 5.** *Heating field and thermal agitation.*

**Figure 6.** *Accumulation of the kinetic energy of all the dipoles.*

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 *x* and *y* (green line) remain erratic but do not heat up. In addition to heating, these fields could promote

**Figure 7.** *Distribution of electric potential obtained by modeling for different values of calculation time.*

*One Model of Microwave Heating of Water Drop DOI: http://dx.doi.org/10.5772/intechopen.104949*

other phenomena such as hydrogen bonding. Accumulation of all dipoles' kinetic energy of the considered water drop is shown in **Figure 6**.

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 *ε*<sup>00</sup> added to the real part *ε*<sup>0</sup> of the permittivity. This heat accumulation is the consequence of temperature rise that varies according to the different materials and is characterized by its specific heat. The direction of the current does not matter. **Figure 7(a)–(c)** show the distribution of the electric potential obtained by modeling for various chosen values of calculation time.

**Figure 7** (**a**, **b**, and **c**) 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 *x*-axis. The boundary conditions chosen for modeling are: for the face *z* = 0 value of potential *Vin* = 0 and for the face *z* = *aa – Vin* = *sin*(*t/T*); T is the period. For the faces, *y* = 0 and *y* = *aa* value of the applied potential is *<sup>∂</sup>V=∂<sup>n</sup>* <sup>¼</sup> 0.

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.
