**3.3.2 Numerical approach**

218 Acoustic Waves – From Microdevices to Helioseismology

ratio between the ends. The analysis is based on an extension of Ensminger's (Ensminger,

l l <sup>2</sup> <sup>1</sup> l

x 0

Ensminger studied the propagation of a wave in extensional mode in a cone with no loss of which the lateral dimensions were short in comparison with the length. In the case of a

<sup>1</sup> <sup>0</sup>

+ +=

v is the velocity of the particles, ω is the pulsation and c is the velocity of the longitudinal

On the basis of the dimensions given in figure 3, the solution to this differential equation leads to an approximate velocity amplification ratio between the two extremities (Nassar,

ω

*v*

10 mm

32 mm

∂ +∂ (2)

2 2 2 2 1

( ) *v v*

1997); |v(0) /v(L)| = 1/ 0.46 = 2.16 for a resonance frequency: f = 60 KHz.

V

*x x xxc* ∂ ∂

l*2* 

l

*1*  Fig. 3. Basic shape. Triangular sensor of thickness e = 1 mm, L = 32 mm, 1 = 2 mm and

S e x = ⋅ ( ) Then S e x x /x =⋅ + 11 1 ( ) (1)

*e* 

1960) theory.

According to figure 2, the x section is written:

x1 L

triangular shape, this equation takes the following form:

wave in the material making up the vibratory element.

Fig. 2. Basic analytical shape

Where:

<sup>2</sup> = 16 mm

While an analytical study can only take into consideration one particular mode of vibration of the triangular part of the sensor, a numerical study based on the finite elements method can determine all the vibrating modes of these parts as well as those of the realised sensor.

For a real structure; whole sensor included a binding rod and a triangular truncated part (Figure 4), the displacement differential equations were solved with a continuous regime, taking into account the boundary conditions at the surfaces. The materials were defined by Young's modulus E, Poisson's coefficient ν and density ρ. The results presented below were applied without loss and they were compared to the characteristics of the longitudinal mode determined by the analytical calculations. This comparison was also made for the triangular sensor which was studied as a whole.

For our study, the ANSYS analysis software was used. The sensors used were made essentially of piezoelectric material. A source of excitation was engraved in the general structure of the vibrating element (triangle part), providing mechanical continuity without any break (Figure 4a). This type of engraving was considered as it has been demonstrated (Nassar, 1997) that for the same longitudinal mode, the amplification ratio at the ends is **71** times bigger when there is one engraved source providing mechanical continuity with the vibrating element so that one source can impact the structure by gluing (Figure 3b).

Fig. 4. From left to right: (a) engraved source; (b) embedded source; (c) elongation mode

Table 1 show the resonance frequencies and the vibration velocity transformation ratio (|v(0) /v(L)|) and table 2 present the difference rate of this ratio according to the position of the excitation source




Table 2. Impact of the nature and the location of the source

Low Frequency Acoustic Devices for Viscoelastic Complex Media Characterization 221

The time of flight dt(ns) of the signal, measured in distilled temperature-controlled water at 30±0.1°C (reference medium) remains stable. The precision obtained was 1ns over a global reply time of 10µs, given a relative precision of 10-4 per measurement at the zero-crossing

The reliability of a measurement system resides in its reproducibility and its faculty to follow all the stages of the gelation process. In the standard conditions using 12 grams of skimmed milk powder dissolved in 100 ml of distilled water and in accordance with the literature (Noël & al., 1989), the sol-gel transition clotted between 15 and 16 min. At a regulated ambient temperature at 30 °C similar to the one of the medium under test, as shown in figure 6, the time of flight of the signal decreased indicating an increase in the mechanical resistance of the product. This variation had not reached a plateau value,

> 0 20 40 60 80 100 120 Time (min)

Fig. 6. Typical curves of the gelation process of two media prepared in the same conditions at a reaction temperature of 30.1 °C observed using the ultrasonic technique. The difference between the two curves provides a quantitative estimation of the global dispersion of the

Figure 6 also provides a qualitative estimation of the reproducibility of the ultrasonic measurements. The curves show the progress of the action of the rennet in two media prepared in the same conditions. The maximum dispersion of the measurements was 5 ns

ultrasonic measurement due to the electronics and sample preparation

due to the electronic parts and the milk reconstitution process.

Estimation of clotting time

Dispersion of the measurements

**3.4.2 Ultrasonic monitoring of gelation: measurement of the variation in the time-of-**

**flight of the wave** 

0

5

10

15

20

Variation of time of flight dt (ns)

25

30

35

40

point

**3.4.2.1 Measurement stability** 

**3.4.2.2 Measurement reliability in gelation process** 

indicating that the medium was still changing.

The results show a good correlation between the frequencies determined by the calculations and those determined numerically or using impedance measurements.

A significant increase in the amplitude of vibration was observed resulting from the design of the electrode on an active element.

### **3.4 Application for monitoring changes in state 3.4.1 Pointed sensors for sol-gel transition**

The milk gelation can be considered as an aggregation of different sized molecules (Walstra& Vliet, 1986; Fox, 1989; Dalgleish, 1993). This model was explored for several reasons: the available knowledge, the experimental conditions that are known and relatively easy to conduct, the complex medium with the physical properties of liquid and gel states in close contact.

As the reaction progresses, the average mass of each aggregate increases and the number of molecules in the medium tested decreases. The aggregation process results in a giant macromolecule defining the gel.

This process was examined using two identical ultrasonic sensors near-field coupled through the medium to be characterized. The working frequency was 60 kHz.

Figure 5 presents a schematic diagram of the measuring device. The emitter is driven by a sharp electrical pulse lasting 15 µs. These conditions provide a longitudinal vibration mode at the end of the sensor which behaves like a point source. This phenomenon generates a divergent ultrasonic wave in the medium, one part of which was measured using a receiver located at a constant distance from the transmitter by the first the zero-crossing of the wave.

The propagation of the wave in the medium is more or less a compressional wave, as suggested by the time of flight corresponding to a velocity of 1600 m/s in reconstituted milk samples at 25 °C.

Fig. 5. Diagram of the measuring cell. Tus is the temperature of the product at "sensor level", Tcw is the temperature of the container walls and Ta is the ambient temperature

### **3.4.2 Ultrasonic monitoring of gelation: measurement of the variation in the time-offlight of the wave**

### **3.4.2.1 Measurement stability**

220 Acoustic Waves – From Microdevices to Helioseismology

The results show a good correlation between the frequencies determined by the calculations

A significant increase in the amplitude of vibration was observed resulting from the design

The milk gelation can be considered as an aggregation of different sized molecules (Walstra& Vliet, 1986; Fox, 1989; Dalgleish, 1993). This model was explored for several reasons: the available knowledge, the experimental conditions that are known and relatively easy to conduct, the complex medium with the physical properties of liquid and gel states in

As the reaction progresses, the average mass of each aggregate increases and the number of molecules in the medium tested decreases. The aggregation process results in a giant

This process was examined using two identical ultrasonic sensors near-field coupled

Figure 5 presents a schematic diagram of the measuring device. The emitter is driven by a sharp electrical pulse lasting 15 µs. These conditions provide a longitudinal vibration mode at the end of the sensor which behaves like a point source. This phenomenon generates a divergent ultrasonic wave in the medium, one part of which was measured using a receiver located at a constant distance from the transmitter by the first the zero-crossing of the wave. The propagation of the wave in the medium is more or less a compressional wave, as suggested by the time of flight corresponding to a velocity of 1600 m/s in reconstituted milk

through the medium to be characterized. The working frequency was 60 kHz.

*Receiver*

Fig. 5. Diagram of the measuring cell. Tus is the temperature of the product at "sensor level", Tcw is the temperature of the container walls and Ta is the ambient temperature

*Water flux*

*Electric signal*

*Calculator*

*Electrical and mechanical protection*

*Sensor*

and those determined numerically or using impedance measurements.

of the electrode on an active element.

macromolecule defining the gel.

*Transmiter*

*Ta*

close contact.

samples at 25 °C.

*Sharp electrical pulse*

**3.4 Application for monitoring changes in state 3.4.1 Pointed sensors for sol-gel transition** 

The time of flight dt(ns) of the signal, measured in distilled temperature-controlled water at 30±0.1°C (reference medium) remains stable. The precision obtained was 1ns over a global reply time of 10µs, given a relative precision of 10-4 per measurement at the zero-crossing point

### **3.4.2.2 Measurement reliability in gelation process**

The reliability of a measurement system resides in its reproducibility and its faculty to follow all the stages of the gelation process. In the standard conditions using 12 grams of skimmed milk powder dissolved in 100 ml of distilled water and in accordance with the literature (Noël & al., 1989), the sol-gel transition clotted between 15 and 16 min. At a regulated ambient temperature at 30 °C similar to the one of the medium under test, as shown in figure 6, the time of flight of the signal decreased indicating an increase in the mechanical resistance of the product. This variation had not reached a plateau value, indicating that the medium was still changing.

Fig. 6. Typical curves of the gelation process of two media prepared in the same conditions at a reaction temperature of 30.1 °C observed using the ultrasonic technique. The difference between the two curves provides a quantitative estimation of the global dispersion of the ultrasonic measurement due to the electronics and sample preparation

Figure 6 also provides a qualitative estimation of the reproducibility of the ultrasonic measurements. The curves show the progress of the action of the rennet in two media prepared in the same conditions. The maximum dispersion of the measurements was 5 ns due to the electronic parts and the milk reconstitution process.

Low Frequency Acoustic Devices for Viscoelastic Complex Media Characterization 223

8b). When p increases, bigger and bigger masses are formed (Figure 8c). For a certain critical value of p, pc, a giant chain appears (continuous connectivity of the space from one side to the other A ↔ B ; Figure 8d) defining the gel point. Above the threshold pc, the medium in the gel phase has the macroscopic behavior of a viscoelastic solid. For p = 1 all the units

Initial product p << pc

Additive

Rennet product (chymosin)

**(a) (b)**

<sup>p</sup> → pc

**(c) (d)**

<sup>p</sup> →<sup>1</sup>

**(e)** 

Fig. 8. Schematic representation of the sol-gel transition. a) Initial phase, b) Suspension of molecules of finite sizes, c) Agglomeration and formation of macromolecules of large masses, d) Critical connection phase: p = pc, e) Network continuity connection to give a

As the properties of a gelling medium are proportional to the reaction progress, it is possible to represent the behavior of the viscoelasticity in terms of the connectivity rate p according

• For p < pc, the system is a liquid whose viscosity increases as the gel point approaches.

In order to relate the theoretical aspect to the experimental results, the following curve (Figure 9) has been divided into five different stages. The phenomenon describes the gelation process when the ambient temperature was different from the product

According to figure 9, "stage (a)" could be interpreted as a proteolytic phase characterized by the appearance of two polypeptides resulting from the effect of the rennet product. This stage is followed by the formation of aggregates of finite size (Figure 8b). This molecular reorganization might be related to the change of slope of the curve (stage b) reflecting a decrease in the time of flight, which means an increase in the velocity in the sol medium.

• For p > pc, the medium becomes a solid gel whose elasticity increases with p.

temperature. It was undetectable when these temperatures were the same.

**A**

**B**

p = pc A↔B

p = 1

belong to the giant mass (Figure 8e).

single giant macromolecule, the gel

to the following cases (Figure 8):

• For p = pc, elastic behavior appears.

p < pc

p > pc

### **3.4.2.3 Evolution of the molecular network**

Figure 7 presents the variations in the time of flight resulting from the variation of the ultrasonic wave velocity during the milk gelation process at different ambient temperatures.

Fig. 7. Ultrasonic sensor responses during milk gelation at 30.1 °C, but at different ambient temperatures (from 17 °C to 30 °C)

The curve observed at an ambient temperature of 30 °C, similar to the temperature of the milk, was similar to those obtained by measurements using other physical methods (McMahon & Brown, 1984). When the ambient temperature was significantly different from the product temperature, ultrasonic curves showed a specific pattern.

According to Dalgleish (Dalgleish, 1982), gelation process was defined like a transition phenomenon in which a soluble suspension made of macromolecules (liquid phase) becomes insoluble when a giant mass forms. We can assume that basic macromolecules are synthesized by linking monomers (building units) via covalent bonds. This chemical reaction is due to the presence of functional groups of the monomers that are able to form chemical bonds with other functional groups of the monomers (Mercier & Marechal, 1993). The network formation occurs if the functionality of the units is greater than two (Mercier & Marechal, 1993). As the reaction progresses, the conversion status of the system is characterized by the connectivity rate p (Flory, 1953; Stockmayer, 1943). Below the gelation threshold, the viscosity of the medium increases as the connectivity rate p approaches the critical advancement rate pc. This phenomenon is known as a critical connectivity transition. Above the threshold, the medium ceases to flow and the gel phase develops some elasticity. This phenomenon introduces structural changes in the physical properties and more particularly in the mechanical behavior of the medium, thus resulting in the transition from a liquid state to a viscoelastic solid state.

To illustrate this process schematically, let us consider an initial solution containing units that can link together. At the beginning, the medium behaves like a viscous solution in a sol phase due to the presence of a single type of finite-size masses: in this case, p is low (Figure

Figure 7 presents the variations in the time of flight resulting from the variation of the ultrasonic wave velocity during the milk gelation process at different ambient temperatures.

> 0 20 40 60 80 100 120 140 Time (min)

Fig. 7. Ultrasonic sensor responses during milk gelation at 30.1 °C, but at different ambient

The curve observed at an ambient temperature of 30 °C, similar to the temperature of the milk, was similar to those obtained by measurements using other physical methods (McMahon & Brown, 1984). When the ambient temperature was significantly different from

According to Dalgleish (Dalgleish, 1982), gelation process was defined like a transition phenomenon in which a soluble suspension made of macromolecules (liquid phase) becomes insoluble when a giant mass forms. We can assume that basic macromolecules are synthesized by linking monomers (building units) via covalent bonds. This chemical reaction is due to the presence of functional groups of the monomers that are able to form chemical bonds with other functional groups of the monomers (Mercier & Marechal, 1993). The network formation occurs if the functionality of the units is greater than two (Mercier & Marechal, 1993). As the reaction progresses, the conversion status of the system is characterized by the connectivity rate p (Flory, 1953; Stockmayer, 1943). Below the gelation threshold, the viscosity of the medium increases as the connectivity rate p approaches the critical advancement rate pc. This phenomenon is known as a critical connectivity transition. Above the threshold, the medium ceases to flow and the gel phase develops some elasticity. This phenomenon introduces structural changes in the physical properties and more particularly in the mechanical behavior of the medium, thus resulting in the transition from

To illustrate this process schematically, let us consider an initial solution containing units that can link together. At the beginning, the medium behaves like a viscous solution in a sol phase due to the presence of a single type of finite-size masses: in this case, p is low (Figure

the product temperature, ultrasonic curves showed a specific pattern.

**Ta = 20 °C** 

**Ta = 23 °C** 

**Ta = 30 °C** 

**Ta = 17 °C** 

**3.4.2.3 Evolution of the molecular network** 

0

temperatures (from 17 °C to 30 °C)

a liquid state to a viscoelastic solid state.

10

20

30

Variation of time of flight dt (ns)

40

50

60

8b). When p increases, bigger and bigger masses are formed (Figure 8c). For a certain critical value of p, pc, a giant chain appears (continuous connectivity of the space from one side to the other A ↔ B ; Figure 8d) defining the gel point. Above the threshold pc, the medium in the gel phase has the macroscopic behavior of a viscoelastic solid. For p = 1 all the units belong to the giant mass (Figure 8e).

Fig. 8. Schematic representation of the sol-gel transition. a) Initial phase, b) Suspension of molecules of finite sizes, c) Agglomeration and formation of macromolecules of large masses, d) Critical connection phase: p = pc, e) Network continuity connection to give a single giant macromolecule, the gel

As the properties of a gelling medium are proportional to the reaction progress, it is possible to represent the behavior of the viscoelasticity in terms of the connectivity rate p according to the following cases (Figure 8):


In order to relate the theoretical aspect to the experimental results, the following curve (Figure 9) has been divided into five different stages. The phenomenon describes the gelation process when the ambient temperature was different from the product temperature. It was undetectable when these temperatures were the same.

According to figure 9, "stage (a)" could be interpreted as a proteolytic phase characterized by the appearance of two polypeptides resulting from the effect of the rennet product. This stage is followed by the formation of aggregates of finite size (Figure 8b). This molecular reorganization might be related to the change of slope of the curve (stage b) reflecting a decrease in the time of flight, which means an increase in the velocity in the sol medium.

Low Frequency Acoustic Devices for Viscoelastic Complex Media Characterization 225

of the medium could dominate the remaining part of the reaction, due to the weak thermal variation (≈ 2 °C) resulting from the difference in temperature between the medium and the

During "stage (e), the gel strengthened. The gel was stronger when the ambient temperature was very close to the temperature medium (Figure 7). This phenomenon can be shown experimentally by an increase in Δdt(ns) resulting from a decrease in time of flight, whereas theoretically it was explained by the evolution of the connectivity rate p towards 1 following the establishment of continuous connections of finite size masses on the giant molecule, the

**3.4.3 Monitoring the formation dynamics of the cohesion forces in a fractionated** 

A key step often met in agro-industry processes is the formation of the matrix of the final product. In cheese-making, this phase involves the cohesion of the elements making up the medium. Generally, it is the conversion of the matter from a heterogeneous state (made up of overlapping grains) to a homogenous state. In this particular case, the cohesion of the curd grains, essential step in the process, varies according to the process conditions as well as the enzymatic and bacterial activities in the medium. It is thus necessary to take these into

During draining, very different physical states are involved in the conversion of the curd grains from a heterogeneous medium to a more homogenous medium. It is therefore difficult to describe the interaction between the ultrasonic wave and the curd grains during draining using just one physical model. So, for a better evaluation of the different phases in

1. From moulding and for a very short period of time the grains are touching and cohesive links begin to form between the contact surfaces, thus forming a skeleton containing

connected porosities through which the whey continues to drain (Figure 10a). 2. When the whey evacuation channels become blocked (Figure 10b), this phase is described by the multilayer model by Brekovskikh (Brekovskikh, 1980). It is equivalent to a material made up of layers of grains and whey of which the thickness is equivalent to the size of the grains as well as whey evacuation channels. The layers of whey become thinner and thinner until they disappear (Figure 10c), producing a homogenous medium (final phase). This approximation is valid insofar as the main signal beam is confined to a narrow area of the medium, which is the case in our measurements.

Indeed, the zone of interest is comparable to the size of the grains (Figure 11). The evolution of the medium throughout the entire draining phase in the mould was described using these two models: the outflow of the whey and the cohesion of the grains.

 *a b c*

**medium: measurement of the variation in the wave amplitude** 

environment (ambient temperature).

account in the description of the cohesion.

Fig. 10. Evolution of the medium over time

the processing of the medium:

**Analysis of the medium during the cohesion process** 

gel.

Fig. 9. The specific ultrasonic response during milk gelation when the ambient temperature was different from the temperature of the milk

The temperature inside the medium remained constant during this stage and resulted from the propagation of heat by pure and free convection. The formation of more or less voluminous masses in the medium (stage (c)) induced the transition from a viscous state to a viscoelastic state, slowing down the free convection. This led to a slight temperature decrease in the medium at the level of the sensors (TUS in Fig. 4) to reach a new equilibrium where heat was mainly transmitted through the container walls (regulated temperature: Tcw°C), by conduction. The changes in the medium during this stage could be interpreted in the following manner:


These two phenomena make "stage (c)" a competition between :


The connectivity rate p, reached a critical value pc in "stage (d)", at the maximum of the curve when the existence can be assumed of a giant macromolecular chain linking the two extreme sides of the considered space (Figure 8d). During this stage, the mechanical aspect

Ultrasonic measurement Thermal measurement

0 10 20 30 40 50 60 70

Time (min)

Fig. 9. The specific ultrasonic response during milk gelation when the ambient temperature

The temperature inside the medium remained constant during this stage and resulted from the propagation of heat by pure and free convection. The formation of more or less voluminous masses in the medium (stage (c)) induced the transition from a viscous state to a viscoelastic state, slowing down the free convection. This led to a slight temperature decrease in the medium at the level of the sensors (TUS in Fig. 4) to reach a new equilibrium where heat was mainly transmitted through the container walls (regulated temperature: Tcw°C), by conduction. The changes in the medium during this stage could be interpreted in

1. Due to a thermal conduction phenomenon, a slight temperature gradient appears in the medium, between the container walls (Tcw = 30.1°C) and the center of the vat, at the location of the measuring point; TUS°C (following on the ambient temperature). The

2. The time of flight decreased as the reaction progressed. This decrease can be attributed to the development of an elastic modulus resulting from the formation of macromolecules, changing the medium from a viscous liquid state to a viscoelastic solid state. The phenomenon was expressed physically by the evolution of the connectivity

2. A decrease in the time of flight resulting from the appearance of an elastic component

The connectivity rate p, reached a critical value pc in "stage (d)", at the maximum of the curve when the existence can be assumed of a giant macromolecular chain linking the two extreme sides of the considered space (Figure 8d). During this stage, the mechanical aspect

temperature decrease induced an increase of the time of flight.

1. An increase of the time of flight resulting from a decrease in temperature.

These two phenomena make "stage (c)" a competition between :

0

the following manner:

was different from the temperature of the milk

Stage (e) Stage (a) Stage (b) Stage (c) Stage (d)

rate p towards its critical value pc .

in the changing medium.

5

10

15

Variation of time of flight dt(ns)

20

25

30

28.2 28.4 28.6 28.8 29 29.2 29.4 29.6 29.8 30 30.2

Medium temperature variation (°C) at the

sensors level

of the medium could dominate the remaining part of the reaction, due to the weak thermal variation (≈ 2 °C) resulting from the difference in temperature between the medium and the environment (ambient temperature).

During "stage (e), the gel strengthened. The gel was stronger when the ambient temperature was very close to the temperature medium (Figure 7). This phenomenon can be shown experimentally by an increase in Δdt(ns) resulting from a decrease in time of flight, whereas theoretically it was explained by the evolution of the connectivity rate p towards 1 following the establishment of continuous connections of finite size masses on the giant molecule, the gel.

### **3.4.3 Monitoring the formation dynamics of the cohesion forces in a fractionated medium: measurement of the variation in the wave amplitude**

A key step often met in agro-industry processes is the formation of the matrix of the final product. In cheese-making, this phase involves the cohesion of the elements making up the medium. Generally, it is the conversion of the matter from a heterogeneous state (made up of overlapping grains) to a homogenous state. In this particular case, the cohesion of the curd grains, essential step in the process, varies according to the process conditions as well as the enzymatic and bacterial activities in the medium. It is thus necessary to take these into account in the description of the cohesion.

### **Analysis of the medium during the cohesion process**

During draining, very different physical states are involved in the conversion of the curd grains from a heterogeneous medium to a more homogenous medium. It is therefore difficult to describe the interaction between the ultrasonic wave and the curd grains during draining using just one physical model. So, for a better evaluation of the different phases in the processing of the medium:


The evolution of the medium throughout the entire draining phase in the mould was described using these two models: the outflow of the whey and the cohesion of the grains.

Fig. 10. Evolution of the medium over time

Low Frequency Acoustic Devices for Viscoelastic Complex Media Characterization 227

The characterisation of media using ultrasounds is often limited by the heterogeneous nature of the matrix which can, in the case of cosmetic, pharmaceutical and agro-food products, be viscoelastic and heterogeneous (foam or emulsion for example). Wave attenuation in such media is mainly due to viscous absorption and scattering from heterogeneities. The higher the frequency the greater the attenuation, hence the necessity to

The search for a compromise between the analysis frequency and the volume of the medium to be characterised led us to propose specific sensor geometries associated with specific

In order to manage this constraint, a very low frequency acoustic technique was adapted so as to communicate sufficient energy to a particularly absorbent sample. This was achieved by mechanical excitation caused by a shock. An electrical image of this excitation is obtained

The sensor proposed is illustrated in figure 15. In the shape of a thin disc, its structure is

A structure like this offers the advantage of being able to work at resonances lower than those of the piezoelectric disc and thus several resonance modes can be used of which the main ones are flexion and radial modes. This type of sensor also offers a large area of contact with the medium studied, which, in the case of soft and aerated materials, can be

The resonance modes of a circular structure, notably those of a disc or a thin ring, have been studied for many years by several authors (Aggarwal, 1952a, 1952b; Moseley, 1960; Vogel & Skinner, 1965; Leissa, 1969; Blevins, 1979; Irie & al., 1984; Lee & Singh, 1994) . The main modes of resonance of a disc are radial modes in the disc plane and flexion modes outside the disc plane (Tables 3, 4 & 5). The tables show a good correlation between the theoretical,

analyse the media using low frequencies in order to characterise the evolving matter.

using a second identical sensor used as the synchronisation reference (Figure 14).

**3.5 Case of highly absorbent matter** 

excitation conditions.

Fig. 14. Schematic diagram

advantageous.

**3.5.1 Principle of the sensor proposed** 

numerical and experimental analyses.

made up of a ring with an embedded piezoelectric disc.

Fig. 11. Estimation of the propagation zone of the main signal beam

Heated to a constant temperature of 35 °C, the experimental mould was instrumented with a transmitter and two ultrasonic receivers spread out so as to integrate the signals transmitted through two paths presenting enough interfaces between the grains. This thus reduced measurement dispersion linked to the random number of interfaces (Figure 12).

Fig. 12. On the left: The experimental measuring device with a horizontal cross-section of the mould at sensor level and on the right: omni-directional, optimised sensor in elongation mode at 246 kHz

Figure 13 shows that the amplitude of the ultrasonic signal is a parameter that is sensitive to variations in the properties of the medium under investigation.

Fig. 13. Typical curve reflecting the cohesion phenomenon as seen by the variation in the ultrasonic amplitude
