Atena Dehghani Kiadehi

Mikel Leturia
email: mikel.leturia@utc.fr

Franco Otaola

Aissa Ould-Dris

Khashayar Saleh

A novel technique for Residence Time Distribution (RTD) measurements in solids unit operations

Keywords: Residence Time Distribution, Bubbling Fluidized Bed, Phosphorescent Pigment, Optical Method

This work presents a novel technique with fast response for Residence Time Distribution (RTD) measurements in gas-solid unit operations (e.g., fluidized bed reactors). This technique is based on an optical method which eliminates the requirement of knowing the velocity and concentration profiles at the exit section of the system.

Experiments were carried out with SiC particles and a phosphorescent pigment used as a tracer. A concentration measurement system was developed to measure the tracer concentration in SiC/pigment mixtures. The corresponding pigment concentrations were evaluated at the bottom of this system using a photomultiplier. The pigment concentration was derived from the integral of the signal intensity received by the photomultiplier. Then, a calibration curve was established which provided the empirical relationship between the integral and pigment concentration. In order to validate this RTD measurement technique, a series of experiments was performed in a bubbling fluidized bed and the effect of the bed height was studied. It was shown that the experimental RTD curves were in good agreement with the theoretical RTD of bubbling fluidized beds. This solids RTD measurement technique can be used to provide a better understanding of the hydrodynamics of complex solids unit operations.

Introduction

In the study and design of solids unit operations, characterization of the overall flow behavior and mixing by means of solid Residence Time Distribution (RTD) are major issues. The solids flow patterns in these units are very complex. The solids in a continuous operation leave the system after a period of time approximating its mean residence time. Consequently, in solids unit operations there can be a broad range of residence times, which can be a serious concern if, as in many commercial applications, good control of solids mixing and uniform treatment of solids is desired [START_REF] Harris | A novel method for measuring the residence time distribution in short time scale particulate systems[END_REF].

The RTD does not carry complete information about the flow and structure of a particular unit operation. Nevertheless, it is possible to affirm that the RTD is a characteristic of the overall flow behavior and mixing which occur in the solids unit operations and are vital for solids unit operation design and scale-up, plant operation and optimization [START_REF] Harris | Particle residence time distributions in circulating fluidised beds[END_REF].

Fluidized bed reactors are present in many applications and gas-solid processes. These reactors are widely spread in numerous industries such as biomass gasification, reforming and chemical looping combustion [START_REF] Guio-Perez | Effect of Ring-Type Internals on Solids Distribution in a Dual Circulating Fluidized Bed System -Cold Flow Model Study[END_REF][START_REF] Peltola | One-dimensional modelling of chemical looping combustion in dual fluidized bed reactor system[END_REF][START_REF] Wei | Hydrogen production by supercritical water gasification of biomass: Particle and residence time distribution in fluidized bed reactor[END_REF]. The process efficiency in such processes depends on the contact efficiency and contact time between gas and solid particles. Therefore, knowledge of solid mixing and RTD is essential in order to describe, design and model fluidized bed reactors.

The residence time distribution function, E(t), and the cumulative distribution function, F(t) for an instantaneous tracer injection or Dirac pulse input are defined as [START_REF] Fogler | Elements of Chemical Reaction Engineering[END_REF]:

𝐸(𝑡) = 𝐶(𝑡) ∫ 𝐶(𝑡)𝑑𝑡 ∞ 0
Eq. ( 1)

𝐹(𝑡) = ∫ 𝐸(𝑡)𝑑𝑡 𝑡 0 Eq. ( 2
)
where C(t) is the tracer concentration at time t.

The mean residence time (tm), which corresponds to the average of the residence times, is given by the first moment of the distribution E(t):

𝑡 𝑚 = ∫ 𝑡 𝐸(𝑡) 𝑑𝑡 ∞ 0
Eq. ( 3)

The essential requirement of fluidized bed reactors is to have a uniform contact between gas and solid phases. It is commonly accepted that it behaves as a continuously stirred tank reactor (CSTR) for solid phase. For good mixing, the RTD of the solid phase should be similar to that of an ideal CSTR.

The residence time distribution function of an ideal CSTR (continuous stirred tank reactor) is defined as [START_REF] Fogler | Elements of Chemical Reaction Engineering[END_REF]:

𝐸(𝑡) = 1 𝜏 • 𝑒 -𝑡 𝜏 ⁄
Eq. ( 4)

where, τ is the space time (s) obtained by dividing solid volume in the reactor (V) by the solids volumetric flow rate entering the reactor (Q). In practice, it is calculated as the mass of solids in the fluidized bed (Ms) divided by the solids mass flow rate (𝑚̇):

𝜏 = 𝑉 𝑄 = 𝑀 𝑠 𝑚̇ Eq. (5)
Several tracer methods have been used for solids RTD measurement. These methods are summarized in Table 1. Generally, the difficulties encountered when solid RTD measurements are required are caused by the size and density distributions of particles. On the other hand, the solids concentration is not uniform in the outlet section and the gas and solid velocities may be different.

From the review of the reported experimental methods, several research groups [START_REF] Harris | A novel method for measuring the residence time distribution in short time scale particulate systems[END_REF][START_REF] Harris | Particle residence time distributions in circulating fluidised beds[END_REF][START_REF] Du | Lateral solids mixing behavior of different particles in a riser with FCC particles as fluidized material[END_REF][START_REF] Harris | The influence of the riser exit on the particle residence time distribution in a circulating fluidised bed riser[END_REF][START_REF] Huang | Solids mixing in a down-flow circulating fluidized bed of 0.418-m in diameter[END_REF][START_REF] Wei | Dispersion of lateral and axial solids in a cocurrent[END_REF][28][START_REF] Chen | On-line Measurement of Residence Time Distribution in a Twin-screw Extruder[END_REF] preferred to employ phosphorescent tracer particles, thanks to its simplicity and high accuracy. In these works, the properties of phosphorescent tracer particles were similar to the average properties of the fluidized bed particles.

Wei et al [START_REF] Wei | Dispersion of lateral and axial solids in a cocurrent[END_REF] made RTD measurements in a co-current down-flow circulating fluidized bed using phosphorescent tracer particles. Tracer mixtures were prepared with alumina, containing a small amount of fine phosphorescent particles as the tracer. Chen et al. [START_REF] Chen | On-line Measurement of Residence Time Distribution in a Twin-screw Extruder[END_REF] also chose a method of light transmission using fluorescence to study solids RTD in extruders. A He-Ne laser was used as a light source. From the experimental results, they concluded that the light transmission method gave good RTD curves with an uncertainty of 6% on the average residence time.

The optical method has been perfected by Harris et al. [START_REF] Harris | A novel method for measuring the residence time distribution in short time scale particulate systems[END_REF][START_REF] Harris | Particle residence time distributions in circulating fluidised beds[END_REF][START_REF] Harris | The influence of the riser exit on the particle residence time distribution in a circulating fluidised bed riser[END_REF]. In these phosphorescent tracer experiments, phosphorescent particles were used as both bulk and tracer particles. In fact, just a small amount of phosphorescent particles were illuminated and activated as a tracer. The detection was then carried out without disturbance of the hydrodynamic behavior by means of a light detector.

In these works, the phosphorescent tracer particle consumption was quite high due to the large amount of the fluidized particles (bulk particles). In order to avoid this problem, a pneumatic injection of phosphor tracer technique (PIPTT) was proposed by Du et al. [START_REF] Du | Lateral solids mixing behavior of different particles in a riser with FCC particles as fluidized material[END_REF]. A similar technique was undertaken by Huang et al. [START_REF] Huang | Solids mixing in a down-flow circulating fluidized bed of 0.418-m in diameter[END_REF]. An amount of phosphorescent tracer particles with properties similar to the bulk particles were injected into the fluidized bed, so wasteful amounts of phosphorescent particles were reduced significantly.

Since the intensity of the light emitted by the activated tracer decays with time, there is an upper residence time limit for phosphorescent tracer techniques (for example 3min for the tracer particle used by Harris et al. [START_REF] Harris | A novel method for measuring the residence time distribution in short time scale particulate systems[END_REF]). Therefore, the solids RTD measurements using this technique have been extensively studied in short solids residence time units (e.g., riser and downer of fluidized beds).

In summary, developing an accurate technique able to detect very low tracer concentrations is a nontrivial matter but is required to accurately characterize the solids RTD in unit operations with long residence times. Indeed, by detecting very low concentrations, the tracer consumption would be reduced, and the hydrodynamic pattern of the unit operation would be modified as little as possible.

It must be noted that such a technique also eliminates the requirement of knowing the velocity and concentration profiles at the exit section of the system and is therefore more accurate than conventional methods.

The main purpose of this study was to develop a technique able to accurately detect very low tracer concentrations, allowing experimental RTD studies with extended residence times to be carried out without disturbing the hydrodynamics of the studied system. This technique was applied to a bubbling fluidized bed reactor and the results were validated by comparison with a theoretical model.

Materials and Methods

Materials

Experiments were carried out with SiC (Silicon carbide) used as bulk particles and a phosphorescent pigment (Lumilux® Green SN-F50 WS) used as a tracer. The tracer particles were chosen to have fluidization properties similar to the SiC particles. The particle size distributions and densities were studied using a laser granulometer Mastersizer 2000 (Malvern Instruments) and a helium pycnometer Accupyc 1330 (Micrometrics), respectively. The properties of the SiC and pigment particles are summarized in Table 2. The median diameter D50 is defined as the diameter where half of the population lies below this value. Similarly, 90 percent of the distribution lies below the D90, and 10 percent of the population lies below the D10.

This pigment absorbs photons in the UV range and emits photons in the visible range. The peak of the excitation spectrum is in the UV zone (360-380nm) of the spectrum, whereas the emission peak is in the visible light (520nm) portion of the spectrum, having a yellowish-green light (Figure 1). There is usually an overlap between the higher wavelength end of the excitation spectrum and the lower wavelength end of the emission spectrum. When electrons go from the excited state to the ground state, there is a loss of vibrational energy. As a result, the emission spectrum has an increased wavelength when compared to the excitation spectrum. This phenomenon is known as Stokes law or

Stokes shift [START_REF] Yang | The origin of the Stokes shift: The line shapes of quantum well exciton absorption and photoluminescence spectra[END_REF]. The greater the Stokes shift, the easier it is to separate the excitation light from the emission light. Since the excitation light must not disturb the measurement sensor (photomultiplier, or PMT), the overlap between the excitation and emission wavelength is generally eliminated by an appropriate selection of optical filters.

In the present work, in order to achieve maximum intensity, the phosphorescent pigment particles was excited by the wavelength corresponding to the peak of the excitation curve, and the emission detection was selected at the peak wavelength of the emission curve. The selection of excitation and emission wavelengths was controlled by a UV lamp with a narrow wavelength range and an optical filter in the detection zone to eliminate the UV light that came from the lamp.

Experimental setup 2.2.1. Concentration measurement system

A "concentration measurement system" was developed to measure the tracer concentration in SiC/tracer mixtures. This system was composed of a vertical tube (PVC, ID=38mm, h=300mm) with two transparent sections for activation of the tracer particles and detection of the emission light. The lower section of the tube was equipped with a storage bin to collect the particles. This system was used under ambient temperature and in a darkroom. A schematic diagram of the setup is presented in Figure 2, where the feeding system, activation zone, vertical tube and detection zone are identified.

The activation zone:

In the activation zone, the sample goes through a funnel with an exit orifice of 1.5mm, using a vibrating motor attached to the funnel to ensure a steady and low particle mass flow rate (≈ 6 g/min).

A 25W UV-LED spot light source with an LED controller from Hamamatsu L11921-415, is directed right at the exit of the funnel. The emission spectrum of the UV light (365 nm +/-5nm) fits around the maximum of the excitation spectrum of the tracer. It is very important that the light source chosen for the pigment excitation has sufficient power especially for very low pigment concentrations. A ventilator was used under the UV-LED light in order to prevent overheating of the lamp during the experiment. The interior portion of the tube in the activation zone was covered by a reflective aluminum film in order to reflect the UV light toward the pigment particles and increase the activation efficiency of the system. As a consequence, all the particles had a proper excitation after leaving the activation zone.

The detection zone:

The detection zone is composed of four elements, a photomultiplier, a power supply system, an acquisition system and a UV filter. The photomultiplier (or PMT), Lynx Silicon Photomultiplier Module LynX-A-33-W50-T1-X, utilizes a 3×3mm 2 active area Silicon Photomultiplier (SiPM). The choice of this type of PMT is justified by its compatibility with the selected pigment because of its precision in the wavelength range from 350-950nm, and peak of detection at 510nm. The PMT was installed at the tube outlet at a distance of about 50mm from the tube wall. This distance is close enough for detection of the pigment intensities and at the same time far enough from the wall such that it is not overwhelmed by the particles closest to the wall, therefore detecting all particles in the flow. The detection efficiency according to the wavelength, the emission spectrum of the UV light and the absorption and emission wavelength of Lumilux® Green SN-F50 WS are presented in Figure 1.

The UV filter, HOYA UV(0) L39, was placed between the PMT and the tube wall to reduce the signal of the UV LED detected by the PMT. This UV filter only eliminates the wavelength of the UV light so it increases the sensitivity of the detection system and allows low tracer concentrations to be measured, without being masked by the UV LED [START_REF] Harris | A novel method for measuring the residence time distribution in short time scale particulate systems[END_REF].

Calibration curve

The calibration of the "concentration measurement system" was carried out in order to provide the empirical relationship between the integral of the detected signal and the pigment concentration in the SiC/pigment mixture. To establish the calibration curve, SiC/pigment mixtures with different compositions (ranging from 0.25wt.% to 6.25wt.%) were prepared. The upper limit of the sample concentration was set to 6.25wt.% since it saturates the PMT maximum sensitivity. As a low consumption of tracer is desired, the measurements were carried out with the highest PMT sensitivity and very low pigment concentrations. For the calibration, 20 g of the different samples (SiC/Pigment mixtures) with known compositions were injected into the concentration measurement system, through the funnel vibrating at a low mass flow rate (≈6 g/min). This low mass flow rate ensured that there was no particle superposition in front of the photomultiplier in the detection zone. After each measurement, the equipment was cleaned for 10 seconds with a vacuum cleaner.

It must be highlighted that this calibration curve focuses mainly on very low pigment concentrations, according to the objectives of this study: particles RTD measurement in solids unit operations with long residence time with minimum tracer consumption (minimal disturbance of the hydrodynamics of the studied system). Indeed, at higher pigment concentrations, the PMT sensor would reach saturation (maximum value). The PMT sensor could still be used for higher pigment concentrations but its sensitivity (which is adjustable) should be reduced to avoid saturation and a new calibration curve should be established under these conditions. However, for RTD measurements, low pigment concentrations are required in order to minimize the pigment consumption and to limit any disturbance of the hydrodynamics.

The measurements were repeated 5 times for each tracer concentration in order to estimate the repeatability and to ensure that there was no drifting of the concentration measurement system over time. The integral of the emitted light from each sample was related to the pigment concentration after eliminating the background noise from the raw signal (described in section 3.1)).

The operating parameters of the calibration procedure are reported in Table 3.

Methodology validation using RTD measurements in a bubbling fluidized bed

In order to validate the RTD measurement methodology, a bubbling fluidized bed was used to compare its experimental solids RTD with theoretical values. Figure 3 SiC particles were used as a bulk powder and phosphorescent tracer particles (Lumilux® Green SN-F50 WS) were used to determine the RTD of particles within the fluidized bed reactor. In order to measure the particles RTD in the fluidized bed, a given powder mass was initially introduced to the system such that the bed had a height to diameter ratio (H/D) between 1 and 2 (depending on the experiment), where H represents the height of the powder bed at fixed bed conditions (SiC and tracer particles) and D the diameter of the column. The tracer was premixed with the bulk powder (SiC) by fluidization during 5 minutes to ensure a homogenous mixture, with a total concentration of

6.25wt.%.

A bubbling fluidized bed was used in this study since it is widely accepted that this type of fluidized beds is well described by the ideal CSTR model (continuous stirred tank reactor). Therefore, the RTD of a SiC/pigment mixture inside the fluidized bed column is expected to be similar to a CSTR model. It should also be mentioned that no slugging of the fluidized bed was observed, even for high H/D values, so that the fluidized bed could be assimilated to a CSTR. The initial concentration was selected to be as low as possible in order to have the lowest tracer consumption without loss of detection sensitivity. It should be noted that the low tracer concentration was necessary because high concentrations would disturb the fluidization hydrodynamics and also, the tracer price could be an issue for the implementation of the technique in larger equipments.

In the present study, the pressure drop method was used to characterize the minimum fluidization velocity (Umf) of the SiC powder. Figure 4 shows the pressure drop diagram at increasing and decreasing gas superficial velocities. The minimum fluidization velocity (Umf) corresponds to the intersection between the upward sloping line (corresponding to a fixed bed) and the horizontal line (corresponding to a fluidized bed). When increasing the superficial velocity, it can be observed that the pressure rises slightly above the horizontal line corresponding to the fluidized state. This excess pressure is associated with the extra force required to overcome wall friction, interlocking of the particles or any "precompaction" of the powder (when filling the fluidization column). As a consequence, the minimum fluidization velocity (Umf) was determined based on the pressure drop diagram at decreasing gas superficial velocities. Based on the experimental results, the minimum fluidization velocity (Umf) of the SiC powder is approximately 6.3 mm/s (Figure 4).

Regarding the RTD characterization, three RTD curves were measured by varying the H/D ratio (1, 1.5 and 2, respectively) with the same fluidization velocity (𝑈/𝑈 𝑚𝑓 = 3). These measurements were then used for the validation of the RTD measurement methodology. It should be noted that the samples were collected at the exit of the fluidized bed and then, the concentration measurement system was used in order to measure the concentration of the collected samples. As explained previously, the concentration measurement system was calibrated at a low mass flow rate (6 g/min) in order to obtain a low solid fraction in front of the detector (photomultiplier). Consequently, the concentration of the samples collected at the exit of the fluidized bed was measured in the exact same conditions.

This procedure ensured that the same calibration curve could be used (because the solids fraction in front of the detector was the same for calibration and collected samples characterization).

Regarding the fluidization conditions (𝑈/𝑈 𝑚𝑓 = 3), a range of 2 < 𝑈/𝑈 𝑚𝑓 < 5 is typically used to obtain a smooth and regular bubbling fluidization behavior, with good mixing properties. Increasing the superficial gas velocity generally leads to better mixing properties, but also more elutriation (i.e., entrainment of the particles). From a practical point of view, increasing the superficial gas velocity generally leads to better gas-solid heat and mass transfer characteristics, which can be relevant in many processes. In the present paper, it will be shown that at 𝑈/𝑈 𝑚𝑓 = 3, a RTD very close to the theoretical RTD of a perfectly mixed reactor (ideal CSTR model) is already obtained. Consequently, increasing the superficial velocity above 𝑈/𝑈 𝑚𝑓 = 3 would lead to very similar results (i.e., similar RTD curves). For each measurement, a data post-treatment procedure consisting of two main steps was applied:

Results

Data post-treatment procedure

 First, a 1 st order Savitzky-Golay filter was applied to remove the noise from the measured signal. The smoothed signal was then used to calculate a baseline, from the instant when the PMT began detecting the emission of particles until no more emission was detected ;

 Secondly, the integral between the raw measured signal (without filtering) and the baseline was calculated. This integral is noted 𝐼 and will be used to characterize the tracer concentration of different SiC/pigment mixtures.

Calibration curve

For the calibration curve (Figure 6), the measurements were repeated 5 times for each concentration, intercalating between the measurements of the RTD. This was done to ensure that there was no drifting in the measurements and that the calibration curve was reliable over time. The error bars correspond to the 95% confidence intervals. A linear relation was obtained between the value of the integral of the PMT signal and the pigment concentration.

The linear fitting (passing through the origin) of the integral value with the tracer concentration in the mixture gives:

𝐼 = 38.62 × 𝐶 𝑃𝑖𝑔𝑚𝑒𝑛𝑡 Eq. ( 6
)
where 𝐼 is the integral of the detected signal and 𝐶 𝑃𝑖𝑔𝑚𝑒𝑛𝑡 is the pigment concentration (wt.%) in the SiC/pigment mixture (using a constant sample mass of 20 g).

It should be noted that the calculated value of the slope of the calibration curve (Eq. ( 6)) is valid under the conditions used in this work: minimum sensitivity of the PMT sensor, low and constant mass flow rate (same solids fraction in front of the detector), with a constant sample mass of 20 g (same signal length, as shown in Figure 5). Consequently, the concentrations of the samples collected at the exit of the fluidized bed (results presented in section 3.3) were measured in the exact same conditions to ensure that the same calibration curve could be used.

Generally speaking, the value of the slope could be affected by the mass flow rate, the sample mass and the sensitivity of the PMT sensor (which is adjustable). However, following the same procedure, a calibration curve corresponding to other conditions could easily be obtained.

RTD measurements

The present section describes and discusses the experimental results obtained from the fluidized bed RTD characterization. The RTD curves were determined experimentally by adding the tracer particles and measuring the variation in time of the pigment concentration at the exit. It should be recalled that the samples were collected at the output of the fluidized bed and then, the concentration measurement system was used in order to measure the concentration of the collected samples.

These experimental RTD results were then compared to the theoretical ideal CSTR model (Eq. ( 4))

which is generally accepted as a valid description of the particles RTD in bubbling fluidized beds.

First, the repeatability of the RTD measurement technique developed in this work was evaluated by repeating the same RTD experiment for three times, with identical conditions: 1422 g of SiC powder

(which corresponds to a height to diameter ratio H/D=1.5), introduction of 95 g of pigment, solid mass flow rate of 2.5 g/s. Figure 7 gives the F-curves (Eq. ( 2)) as a function of time for the three experiments. Vertical bars on the data points are the 95% confidence intervals for the cumulative distribution function (F-curves). These results show the good repeatability of the RTD characterization method proposed in this study.

Afterwards, the validity of this RTD measurement technique was evaluated by comparing the experimental results to the theoretical ideal CSTR model (Eq. ( 4)) for different H/D ratios of the fluidized bed. The operating conditions of these experiments are shown in Table 4. Figure 8 shows the experimental C-curves and F-curves as a function of time for the 3 values of the height to diameter ratio (H/D) of the fluidized bed. The theoretical curves corresponding to the ideal CSTR model (Eq. ( 4)) are also presented. The obtained data show a good agreement between the theoretical model and the experiments for all three H/D ratios. These results confirm that under the experimental conditions used in this study, the behavior of the experimental bubbling fluidized bed is very close to the theoretical ideal CSTR model (Eq. ( 4)), which is a commonly accepted result.

Moreover, Table 5 presents a comparison of the experimental and theoretical mean residence times (𝑡 𝑚 , Eq. ( 3)) and shows that the relative error is always lower than 6%. Consequently, these experiments confirm the validity of the RTD measurement technique developed in this work.

Finally, Table 6 reports the pigment mass balance, i.e., a comparison between the masses of introduced pigment and "measured pigment" (i.e., detected by the PMT), for the 3 values of the height to diameter ratio (H/D). The relative error is always close to 10% (maximum value = 12%) and it should be noted that the "measured pigment mass" is always lower than the introduced pigment mass. This could be explained by the elutriation (i.e., entrainment) of the pigment particles and also by their adhesion to the fluidized bed walls and concentration measurement system walls.

Consequently, the main source of this relative error corresponds to inherent characteristics of the process (elutriation, attrition, adhesion, etc.) rather than the precision of the concentration measurement system.

Overall, the results presented in this section confirm the validity, repeatability and reliability of the developed methodology, which can be used for the characterization of particles RTD in solids unit operations, especially with long residence times.

Conclusions

A novel concentration measurement technique was developed in this study for measuring the particles RTD in solids unit operations. The procedure which was developed is well suited to solids unit operations with long residence times. Also, the presented technique eliminates the requirement of knowing the velocity and concentration profiles at the exit section of the system and is therefore more accurate than conventional methods. A bubbling fluidized bed was studied and the experimental RTD results were then compared to the theoretical ideal CSTR model in order to validate this RTD measurement method.

The following conclusions can be drawn based on the results obtained in this study:

 The tracer concentration measurement system has the ability to detect very low pigment concentrations (even lower than 0.5wt.%). It is required to measure the RTD of solids unit operation with long residence times while maintaining a low consumption of tracer particles.

 The linear relationship between the integral of the PMT detected signal and pigment concentration was proven to be reliable.  The experimental values of the mean residence time were close to the theoretical mean residence time values, with small relative errors (less than 6%, as shown in Table 5).

Further investigation is required by integrating the tracer concentration measurement system at the exit of the fluidized bed reactor in order to perform continuous measurements. Also, the possibility of coating the bulk particles with a thin layer of phosphorescent material could be investigated, in order to avoid the difficulties of finding phosphorescent particles of the same physical properties as the bulk particles. Wavelength (nm) Integral (V.s)

Pigment concentration (wt.%)

  shows the schematic diagram of the experimental setup.This fluidized bed consists of six main parts: fluidization column, air distributor, air flow controller, solids feeder, acquisition system and exit of the fluidized bed. The fluidization column is a Pyrex cylinder of 100mm inner diameter and 480mm height. The solids feed control system (feeding system), which consists of a funnel with a vibrating motor and a soft tube with a valve, allows us to have a controlled and stable solids flow rate. A solids exit channel with a 4mm tube is located at the bottom of the fluidized bed at 30mm above the air distributor. The air distributor is made of a perforated plate (2mm* 176 holes, 7% open area). The fluidization gas (air at ambient temperature; 23±2 °C and the relative humidity of 7±1%) is supplied from the compressor. Before entering into the fluidized bed, the compressed air passes through a filter, pressure regulator and a calibrated mass flow controller.

Figure 5

 5 Figure 5 represents the measured raw signals of different SiC/pigment mixtures as a function of time.



  The experimental C-curves and F-curves of the fluidized bed reactor measured with the present RTD technique are in good agreement with the theoretical values (ideal CSTR model).

Figure 1 :

 1 Figure 1: Lumilux® Green SN-F50 WS, PMT, UV filter and UV LED efficiency spectra (according to the pigment, PMT, UV filter and UV-LED characteristics indicated in the corresponding datasheets).

Figure 2 :

 2 Figure 2: Schematic diagram of the concentration measurement system.

Figure 3 :

 3 Figure 3: Schematic diagram of the experimental fluidized bed.

Figure 4 :

 4 Figure 4: Pressure drop vs. superficial gas velocity (H/D=1.5; SiC mass=1422g).

Figure 5 :

 5 Figure 5: Measured raw signals for twenty grams of SiC/Pigment mixtures.

Figure 6 :

 6 Figure 6: Calibration curve (Tracer concentration (wt.%) vs Integral of the PMT signal).

Figure 7 :

 7 Figure 7: F(t) curves for identical conditions (H/D=1.5).

Figure 8 :

 8 Figure 8: C (t) and F (t) of the fluidized bed as a function of time, compared to theoretical results for three different H/D values: (a) H/D = 1, (b) H/D = 1.5 and (c) H/D = 2.

Table 1 :

 1 Summary of solids RTD methods.

	Method

Table 4 :

 4 Experimental conditions used for particle RTD measurements in the fluidized bed.

	Experiment Number	Height to diameter (H/D)	Solid mass flow rate (g/s)	SiC mass (g)	Initial pigment concentration (wt.%)
	1	1	2.5±0.2	948	6.25
	2	1.5	2.5±0.2	1422	6.25
	3	2	2.5±0.2	1896	6.25

Table 5 :

 5 Effect of the height to diameter ratio (H/D) on the mean residence time.

	Height to
	diameter (H/D)

Mean residence time (t m ) (min)

		Experimental	Theoretical	Relative error (%)
	1	5.1 ± 0.4	5.40	5.6
	1.5	10.1 ± 0.4	10.15	0.5
	2	12.4 ± 0.4	12.26	1.1

Table 6 :

 6 Pigment mass balance (comparison of the introduced and measured pigment masses).

	Height to		Pigment mass balance
	diameter (H/D)	Introduced	Measured	Relative error
		pigment (g)	pigment (g)	(%)
	1	63.3	56.5	10.7
	1.5	95.0	83.6	12.0
	2	126.4	112.1	11.3

Authors System

Details of the method Radioactive tracer Abellon et al. [START_REF] Abellon | A single radiotracer particle method for the determination of solids circulation rate in interconnected fluidized beds[END_REF] Four interconnected fluidized bed

Bed material: Glass beads Tracer: 24 Na and 192 Ir Ambler et al. [START_REF] Ambler | Residence time distribution of solids in a circulating fluidized bed: Experimental and modelling studies[END_REF] CFB Pulse tracer technique with Gallium-68 tracer Bhusarapu et al. [START_REF] Bhusarapu | Quantification of solids flow in a gas-solid riser: single radioactive particle tracking[END_REF] Riser of CFB Single radioactive particle ( 46 Sc) Lin et al. [START_REF] Lin | Hydrodynamics of a commercial scale CFB boiler-study with radioactive tracer particles[END_REF] CFB boiler 56 Mn as tracer Mahmoudi et al. [START_REF] Mahmoudi | The solids flow in the CFB-riser quantified by single radioactive particle tracking[END_REF] CFB riser Positron emission particle tracking using 18 F as tracer Pant et al. [START_REF] Pant | Residence time distribution study in a pilot-scale gas-solid fluidized bed reactor using radiotracer technique[END_REF] Pilot scale fluidized bed reactor

Gold-198 as tracer Chan et al. [START_REF] Chan | Particle velocities and their residence time distribution in the riser of a CFB[END_REF] Riser of fluidized bed Positron emission particle tracking using 18 F as tracer

Magnetic tracer

Avidan and Yerushalmi [START_REF] Avidan | Solids Mixing in an Expanded Top Fluid Bed[END_REF] Expanded top fluidized bed Ferromagnetic particle tracer Guío-Pérez et al.

[15], [START_REF] Guío-Pérez | Influence of ring-type internals on the solids residence time distribution in the fuel reactor of a dual circulating fluidized bed system for chemical looping combustion[END_REF] Dual circulating fluidized bed

Steel particles as tracer

Legile et al. [START_REF] Legile | Détermination du temps de séjour des particules dans un lit fluidisé gaz-solide par tracage magnétique[END_REF] Thin channel type gas-solid fluidized bed Ferromagnetic particle tracer

Chemical particle tracer

Andreux et al. [START_REF] Andreux | Hydrodynamic and solid residence time distribution in a circulating fluidized bed: Experimental and 3D computational study[END_REF] Riser of cold fluidized bed NaClcrystals injected in the connecting point between L-valve and riser Cui et al. [START_REF] Cui | Gas and solids mixing in a dynamically scaled fluid coker stripper[END_REF] Stripper of fluidized bed FCC particles were impregnated with a saturated solution of table salt and then dried to remove all moisture Rhodes et al. [START_REF] Rhodes | Effects of Operating Conditions on Longitudinal Solids Mixing in a Circulating Fluidized Bed Riser[END_REF] Riser of CFB NaClas tracer; Tracer sampling was performed in three locations in the riser Smolders and Baeyens [START_REF] Smolders | Overall solids movement and solids residence time distribution in a CFB-riser[END_REF] Riser of CFB NaCl tracer

Phosphor particle tracer

Wei and Du [START_REF] Du | Lateral solids mixing behavior of different particles in a riser with FCC particles as fluidized material[END_REF] Riser of CFB The tracer particles are made by coating of fin phosphor particles on the surface of different property particles with nitryl-varnish as the bond.

Harris et al. [1],

[2], [START_REF] Harris | Stochastic modelling of the particle residence time distribution in circulating uidised bed risers[END_REF]- [START_REF] Harris | The influence of the riser exit on the particle residence time distribution in a circulating fluidised bed riser[END_REF] Riser of CFB All particles have phosphorescent properties Huang et al. [START_REF] Huang | Solids mixing in a down-flow circulating fluidized bed of 0.418-m in diameter[END_REF] Down-flow CFB A pneumatic injection phosphor tracer technique (PIPTT) was used to determine the axial and lateral solids dispersion by measuring the solids RTD at same axial but different lateral positions. Wei et al. [START_REF] Wei | Dispersion of lateral and axial solids in a cocurrent[END_REF] CFB downer A unique phosphor tracer technique measures the solids tracer concentration with time at different radial and axial positions within the co-current down flow circulating fluidized bed.

Yan et al.[28]

Riser of CFB The solids mixing in a riser with a height of 10 m and 0.186 m inner diameter was investigated by using the pneumatic phosphor tracer technique. Chen et al. [START_REF] Chen | On-line Measurement of Residence Time Distribution in a Twin-screw Extruder[END_REF] Extruder A He-Ne laser as a light source with carbon black as the tracer

Operating parameters Values

Mass of the sample (g) 20

Pigment concentration (wt.%) 0 -6.25

Mass flow rate (g/min) 6±1

Tube inner diameter (mm) 38

Tube length (mm)


Seyyed Ehsan

Hashemi Petroodi

Amélie Beauville

Dit Eynaud

Nathalie Klement

Reza Tavakkoli-Moghaddam

Simulation-based optimization approach with scenario-based product sequence in a Reconfigurable Manufacturing System (RMS): A case study

Keywords: Simulation-Based Optimization, Reconfigurable Manufacturing System (RMS), Reconfigurability, Simulated Annealing (SA)

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Nowadays, rapid and global data transformation, open market and changeable demand compel manufacturing systems structure to change. According to the new technologies, new product variety and high demand fluctuation for different products, production systems should be more flexible. They must be able to react and response quickly in face of these changes. The traditional manufacturing systems were not able to handle these challenges. The Dedicated Manufacturing System (DMS) has a high throughput but very low flexibility to produce different product part families. The Flexible Manufacturing System (FMS) consists of full automated components with high flexibility to response a stable demand of products. The Reconfigurable Manufacturing System (RMS) can overcome the limitations of these systems. RMS is part of Industry 4.0, aiming to cover high flexibility of FMS and high throughput of DMS. It is also able to adjust rapidly in context of functionality and productivity by rearranging existing components. Because RMS is needed in the new generation of manufacturing systems, it is an important subject. The RMS is capable to cover a variable demand. Hence, this system is a dynamic system. According to the variable demand in real world systems, reconfigurability of machines and system structure is an effective point of RMS. Moreover, with respect to the extension of market competition and decreasing production costs, using RMS leads system to progress in this situation.

This study focuses on proposing a manufacturing system enabling assembly of two different types of products: diesel engines (abbreviated DVR in the following paragraphs) and gasoline engines (EB). A production planning problem of a RMS is investigated. To calculate total completion time of the system, four general scenarios for the product sequence are considered. Task assignment to the workstations and their sequence in each station are already known. Lower and upper bound for the number of machines in each station are defined.

LITERATURE REVIEW

According to the evolution of manufacturing systems, each of them have their own special advantages and disadvantages. In the dedicated manufacturing system (DMS), automation and structure are fixed. A possibility to produce in a high throughput is the main advantage of the DMS [START_REF] Koren | Design of reconfigurable manufacturing systems[END_REF]. Moreover, the fixed structure of DMS is unchangeable either to manufacture various products or to increase throughput of the system (Koren et al., 2017). The Flexible manufacturing system (FMS) had been introduced after DMS [START_REF] Katz | Design principles of reconfigurable machines[END_REF]. These systems can adapt to many manufacturing requirement easily and quickly [START_REF] Abele | Performance evaluation methods for mechanical interfaces in reconfigurable machine tools[END_REF]. FMS has been extended to make possibility structure to adjust and scale the capacity quickly within producing part families [START_REF] Elmaraghy | Reconfigurable process plans for responsive manufacturing systems[END_REF]. The RMS has been introduced by [START_REF] Koren | Reconfigurable manufacturing systems[END_REF]. The efficiency is high in the context of responsiveness to sudden changes of the market [START_REF] Battaïa | Decision support for design of reconfigurable rotary machining systems for family part production[END_REF]. The RMS is an adjustable system regarding capacity and functionality. In fact, the RMS covers advantages of both previous manufacturing systems [START_REF] Bi | Reconfigurable manufacturing systems: the state of the art[END_REF] and also overcome some disadvantages of the FMS such as high cost, obsolescence, unfavourable tools, and unreliability [START_REF] Mehrabi | Reconfigurable manufacturing systems: Key to future manufacturing[END_REF]. One of the main differences between FMS and RMS is the customized flexibility of RMS and the general flexibility of FMS. Customized flexibility means that the system can be changed whenever it is needed [START_REF] Wiendahl | Changeable manufacturing-classification, design and operation[END_REF]. A whole comparison of these three manufacturing systems have been proposed by [START_REF] Zhang | An analytical comparison on cost and performance among DMS, AMS, FMS and RMS[END_REF], [START_REF] Koren | Design of reconfigurable manufacturing systems[END_REF] and (Koren et al., 2017).

Researchers worked on process planning in RMS; [START_REF] Chaube | An adapted NSGA-2 algorithm based dynamic process plan generation for a reconfigurable manufacturing system[END_REF] proposed an NSGA-2 algorithm to solve an RMS process planning problem. Firstly, they assigned tasks to a set of reconfigurable machines, and then optimized the completion time and cost by scheduling these tasks. In this study, machines are not the same, and they have their own set of reconfigurations. In an other study, [START_REF] Prasoon | An algorithm portfolio approach to reconfigurable set-up planning[END_REF] worked on optimization of reconfigurable set-up plans in a dynamic production system by studying an algorithm portfolio approach.

Basically, simulation modelling is an efficient approach to handle the complex systems under uncertainty [START_REF] Borshchev | From system dynamics and discrete event to practical agent based modeling: reasons, techniques, tools[END_REF]. Simulation should be used as an approach to evaluate complex systems [START_REF] Juan | A review of simheuristics: Extending metaheuristics to deal with stochastic combinatorial optimization problems[END_REF]. Discrete Event Simulation is the procedure of modelling by considering different changes over time [START_REF] Chica | Why simheuristics? Benefits, limitations, and best practices when combining metaheuristics with simulation[END_REF]. On the other hand, optimization gives the possibility to find the best parameter combination in order to run the system efficiently. The right optimization method should be selected depending on the faced problem. Exact methods provide optimal solution for small size problems and metaheuristics can solve NP-hard problems (nondeterministic polynomial time problems), providing near optimal solution. For instance, [START_REF] Dou | Precedence graphoriented approach to optimise single-product flow-line configurations of reconfigurable manufacturing system[END_REF] proposed a genetic algorithm (GA) to find some of the best configurations among all the optimal configurations that had been obtained by some feasible generated sequences. A systematic approach to generate different feasible configurations for the single-product RMS was developed. Combination of simulation models and these optimization methods enhances the solution. Hybridization of simulation and metaheuristics provides interesting results because of its ability to provide high quality solutions for NP-hard real problems in reasonable calculation time [START_REF] Juan | A review of simheuristics: Extending metaheuristics to deal with stochastic combinatorial optimization problems[END_REF]. [START_REF] Gansterer | Simulationbased optimization methods for setting production planning parameters[END_REF] proposed a simulation-based optimization method to assess some parameters in production planning. [START_REF] Fu | Optimization for simulation: Theory vs. practice[END_REF] proposed a classification of hybridization approaches based on simulation and optimization, namely, simulation-based optimization and optimization-based simulation. Optimization and simulation phases connect with each other by different interfaces. These interfaces may be user-defined [START_REF] Dehghanimohammadabadi | Simheuristic of patient scheduling using a table-experiment approach: simio and matlab integration application[END_REF] [START_REF] Attar | A simulation-based optimization approach for free distributed repairable multi-state availability-redundancy allocation problems[END_REF] or general tools like Excel software [START_REF] Dehghanimohammadabadi | Simheuristic of patient scheduling using a table-experiment approach: simio and matlab integration application[END_REF]. Simul8 is a good software to implement a Discrete Event Simulation (DES) model (Carteni and De Luca, 2012). [START_REF] Imran | Cell formation in a cellular manufacturing system using simulation integrated hybrid genetic algorithm[END_REF] linked DES and different metaheuristics.

PROPOSED APPROACH

Problem Description

Among the different paradigms to build a reconfigurable production system, the integration of mobile robots on the production system has been selected in order to enable easy reconfiguration of the production line. This choice is made based on the assumption that there exists a safety system enabling the integration of a collaborative robot on a movable platform. The reconfiguration of the production system consists in the reallocation of the movable robots on workstations. This occurs when the system is subject to fluctuations of the economic context, leading to changes of the production demand in volume or product variety. This paper focuses on product variety.

In this study, we consider two objective functions aimed to minimize total completion time. The system contains m workstations with specific assigned tasks to produce p product types. The problem is about allocation of n identical mobile robots to the workstations. s = 1,2,…,m , r = 1,2,…,n and k = 1,2,…,p respectively are sets of stations, robots and product types. As parameters of the model, Pk and crs are respectively the price of the product k and cost of per percent utilization of robot r on station s. These parameters are fixed. On the other hand, two other parameters are obtained by simulation: the number of final products k (Nk) and the utilization percentage of assigned robots to station s (us). In the proposed model, two binary decision variables xrs and yrs are 1 if robot r is assigned to station s respectively for producing DVR and EB, and one continuous variable CT representing total completion time. Task sequence of each product and processing time are already known. Lower and upper bound are considered for the number of robots in the stations. To calculate completion time, different scenarios are defined for product arrival. The objective function maximizing profit is calculated by (1) and the objective function minimizing completion time is presented by (2).

𝑀𝑎𝑥 𝑧 1 = ∑ 𝑁 𝑘 𝑝 𝑘=1 𝑃 𝑘 -∑ 𝑢 𝑠 ∑ (𝑥 𝑟𝑠 + 𝑦 𝑟𝑠 ) 𝑐 𝑟𝑠 𝑛 𝑟=1 𝑚 𝑠=1
(1) 𝑀𝑖𝑛 𝑧 2 = 𝐶 𝑇

(2)

Methodology

To model the proposed problem, an efficient DES model was developed using Simul8 tool. The simulation model is running with respect to the demand of specific period, in our study during one week. Tasks sequence, tasks duration, product mix ratio, and generated robot allocation are input parameters of the simulation model. Utilization of robots (in percentage of the total simulated time) and the number of final products are output parameters of this model which are considered as input parameters of the optimization phase (Fig. 1). The optimization module generates a new resource allocation, which is used as input for the next simulation run, and so on.

Input parameters of the simulation model, as Excel file, are imported in Simul8. The simulation model is developed to provide insights into the workflow process, calculate the utilization percentage of resources, and obtain the number of final products. In the proposed SA, detailed in Fig. 3, a random generated solution is used as initial solution with an initial temperature, and neighbourhood solutions are generated by changing robot allocation on different stations according to a swap structure.

If the newly generated solution improves the objective function, this last one is saved. However, keeping a worse solution is authorized according to a predefined probability, which decreases at each iteration of the algorithm. The temperature of the SA is updated, which reduces the acceptation of a worse solution for the next step, and the cycle is repeated. After a predefined number of iterations, the algorithm stops.

SA is applied to solve the two objective functions, minimizing completion time and maximizing the profit value. [START_REF] Torabi | An interactive possibilistic programming approach for multiple objective supply chain master planning[END_REF] introduced the so-called TH method, based on a fuzzy approach, used to solve the bi-objective model.

APPLICATION TO A CASE STUDY

Case Study: an Automotive Assembly System

In this study, we consider the engine assembly system of an automotive company as a case study. The factory aims to improve its current system and make a reconfigurable manufacturing system. In this company, two types of engines, diesel and gasoline, are assembled. These engines are respectively called DVR and EB. Four assembly station are investigated in this study. The assigned tasks to workstations and their sequence is shown by Fig. 4. Reconfigurability is reached by having mobile collaborative robots on AGVs. These robots are the resources we target to assign to workstations, depending on the current product mix and product sequence. The considered problem is production planning and resource allocation problem for the reconfigurable system. Actually, we want to evaluate the effect of reconfigurability on the current system. Thereby, two models with two different objective functions are considered in optimization phase. The objective functions are maximizing the profit of company and minimizing the total completion time. The assumptions considered in this study are:

 Each robot should be assigned to only one workstation at the same time.  Conveyor starts moving when the last product is released in the stations.  The total completion time is calculated with respect to the four proposed scenarios and the number of assigned robots to the workstations.  There is an upper-bound for the number of assigned robots to the workstations. The maximum number of robots which can be assigned to each workstation is equal to the maximum number of parallel tasks in the workstation.  At maximum one robot can be assigned to workstation 10.  At maximum two robots can be assigned to workstation 11.  The maximum number of robots which can be used in workstations 12 and 13 are 5.

 At least one robot should be assigned to each workstation.  The maximum number of existing product in each workstation at each time is one product.  The assigned tasks to the workstations are already known.  DVRs and EBs have some common tasks and some specific tasks.  Tasks sequence for each type of product are already known. Different scenarios are considered for the product sequence in assembly line. For example, in a scenario DVRs are assembled and then EBs can be assembled (Fig. 5), and this scenario can be considered vice versa as second scenario (Fig. 6). In other scenarios, products might be assembled alternately, in which they can be started with a DVR (Fig. 7) or an EB (Fig. 8). In these scenarios the ratio of the number of DVR engines and EB engines can be different. In these figures, 1 robot in workstation 10, 2 robots in workstation 11, 5 robots in workstation 12 and 13 are assumed. Considering the relative ratio between the products is the main point in these scenarios. For example, the number of DVRs may be greater than EBs or vice versa. According to the number of assigned robots to the workstations, the defined scenarios and the processing time of tasks, total completion time of each station can be calculated. These values, presented in Table 1, are considered in 10 -2 min.

Results and Discussion

A SA algorithm is developed to solve separately each objective function. A feasible random initial solution is generated. After 2000 iterations of the SA algorithm, the best profit value is obtained. The solutions for robot allocation to the stations are shown for DVRs (Fig. 9) and EBs (Fig. 10). After 2000 iterations, the minimal completion time is 2269 min with a 50-50 ratio of the two types of products (Fig. 11).

Obtained solutions for robot allocation to the stations are shown for DVRs (Fig. 12) and EBs (Fig. 13). Scenario 2 is the best scenario for product sequence with respect to given number of products. To implement TH method for solving the bi-objective model, Z-positive ideal solution (Z PIS ) and Z-negative ideal solution (Z NIS ) for two objective functions are needed (Table 2). The best scenario of product sequence with respect to the demand variety can be selected. Regarding results depicted in Table 4, it is preferable to use scenario 1 (assembling all DVRs before EBs) when the EB ratio is higher, and use scenario 3 (assemble products alternately) when the DVR ratio is higher.

CONCLUSION AND FUTURE STUDY

In this paper, a general concept of hybridization approach based on simulation and optimization is implemented. It is developed to solve production planning and resource allocation problem in a reconfigurable manufacturing system (RMS). Solutions obtained in the optimization phase are then evaluated using the developed simulation model. In this approach, some parameters are obtained from simulation and imported into the optimization to give an efficient resource allocation structure. A metaheuristic (SA) has been applied on the problem to determine the best configuration to achieve a minimal completion time with maximization of the profit function. Consequently, it can be concluded that the proposed approach provides a good performance for design and production planning problems of RMS. Moreover, regarding defined scenarios for product sequence, the best scenario can be chosen with respect to the demand variety of product families.

As a future study, other optimization algorithms could be tested to solve this problem. Furthermore, the methodology could be applied to an extended use case with a higher number of workstations and resources, meaning a higher degree of complexity. The hybridization between simulation and optimization has also to be continued, as the communication between the different modules is not fully automatized yet.

Fig. 1 .

 1 Fig. 1. Schema of the proposed hybridization approach.A Simulated Annealing (SA) algorithm was developed in Python for the optimization phase, presented Fig.2.

Fig. 4 .

 4 Fig. 4. Precedence of the assigned tasks to the workstations.

Fig. 5 .

 5 Fig. 5. Assembly line for assembling all DVRs before EBs.

Fig. 6 .

 6 Fig. 6. Assembly line for assembling all EBs before DVRs.

Fig. 7 .

 7 Fig. 7. Assembly line for assembling products alternately, starting with DVR.

Fig. 8 .

 8 Fig. 8. Assembly line for assembling products alternately, starting with EB.

Fig. 9 .

 9 Fig. 9. Best robot allocation to the workstations maximizing profit value for assembling DVRs (X*).

Fig. 10 .

 10 Fig. 10. Best robot allocation to the workstations maximizing profit value for assembling EBs (Y*).

Fig. 11 .

 11 Fig. 11. Completion time values (10 -2 min).

Fig. 12 .

 12 Fig. 12. Best robot allocation to the workstations minimizing completion time for assembling DVRs (X*).

Fig. 13 .

 13 Fig. 13. Best robot allocation to the workstations minimizing completion time for assembling EBs (Y*).

Table 1 . Completion time of each station with respect to the number of assigned robots (Unit: 10 -2 min)

 1

	Nb of	WS 10	WS 11			WS 12
	robots	>=1	1 >=2 1	2	3	4 >=5
	DVR CT 42 56 31 85 51 34 34 17
	EB	CT 42	0	0	79 62 45 45 45
	Nb of				WS 13
	robots	1		2	3	4 >=5
	DVR CT 110 70 60 50 40
	EB	CT	0	0	0	0	0

Table 2 . ZPIS and ZNIS for minimizing completion time

 2

				Z PIS	Z NIS
	Second	model	(minimize	2269	3501
	completion time function) (min)

Table 3 . Values of the objective function with respect to different parameters of TH

 3

	𝝑	(𝝋 𝟏 , 𝝋 𝟐 )	𝝁 𝟏	𝝁 𝟐	𝒁 𝟐
					(min)
	0.2 (0.2, 0.8) 0.579 0.999	2269
	0.2 (0.5, 0.5) 0.610 0.999	2269
	0.2 (0.8, 0.2) 0.686 0.670	2668
	0.5 (0.2, 0.8) 0.642 0.999	2269
	0.5 (0.5, 0.5) 0.648 0.999	2269
	0.5 (0.8, 0.2) 0.650 0.670	2668
	0.8 (0.2, 0.8) 0.562 0.999	2269
	0.8 (0.5, 0.5) 0.668 0.999	2269
	0.8 (0.8, 0.2) 0.690 0.670	2668
	According to

Table 3 ,

 3 it can be concluded that if the importance of an objective function (𝝋 𝒊 ) increases, this objective function will be improved and vice versa. It also should be said that the satisfaction degree of each objective function will increase by changing 𝝑. Moreover, the Pareto frontier of the problem was provided.The objective function values for different product ratios are listed in Table4. These different values with respect to the different numbers of DVRs and EBs are shown Fig.14.

Table 4 . Completion time values for different number of products

 4

	Number	Number of	Completion time	Best
	of DVRs	EBs	value (min)	scenario
	0	5000	2251	1
	1000	4000	2301	1
	2000	3000	2351	1
	2349	2438	2269	2
	3000	2000	2061	3
	4000	1000	1031	3
	5000	0	182	3

Fig. 14. Completion time values for different number of products.

ACKNOWLEDGEMENTS

This work took place in the framework of the OpenLab 'Materials and Processing' combining ENSAM network, GeorgiaTech Lorraine, LIST Luxemburg and Groupe PSA company.