Modelagem de um Secador Pneumático python scipy

Any correspondence concerning this service should be sent to the repository administrator staffoataoinptoulousefr Identification number DOI 101016jpowtec201308019 Official URL httpdxdoiorg101016jpowtec201308019 This is an authordeposited version published in httpoataounivtoulousefr Eprints ID 9723 To cite this version Aubin Antoine and Ansart Renaud and Hemati Mehrdji and Lasuye Thierry and Branly Marc Modeling and simulation of drying operations in PVC powder production line Experimental and theoretical study of drying kinetics on particle scale 2014 Powder Technology vol 255 pp 120133 ISSN 0032 5910 Open Archive Toulouse Archive Ouverte OATAO OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible Modeling and simulation of drying operations in PVC powder production line Experimental and theoretical study of drying kinetics on particle scale Antoine Aubin ab Renaud Ansart ab Mehrdji Hemati ab Thierry Lasuye c Marc Branly c a Université de Toulouse INPT UPS Laboratoire de Génie Chimique 4 Allée Emile Monso F31030 Toulouse France b CNRS Laboratoire de Génie Chimique F31030 Toulouse France c INEOS ChlorVinyls France Chemin des Soldats F62670 Mazingarbe France a b s t r a c t Keywords PVC Convective drying Fluidization Drying kinetic model Pneumatic dryer model An experimental method to determine drying kinetic at a particle scale and a pneumatic dryer model are presented The particle scale kinetics were obtained by immersion of a fixed mass of wet PVC particles cake in a batch dense fluidized bed containing inert hot particles glass bead It appears clearly that the PVC drying is controlled by a competition between internal and external transfers The drying kinetic was described by a shrinking core type model and integrated in a onedimensional steadystate model simulating a pneumatic dryer A twophase continuum model was used to describe the steadystate flow of a diluted dispersed phase wet PVC powder and a continuous phase humid air through dryer The model takes into account the convec tive heat mass and momentum transfers The numerical results are compared with industrial experimental data The results show that the inlet temperature is the most important parameter in the operation 1 Introduction This study has been realized in the Chemical Engineering Laboratory in Toulouse in partnership with INEOS ChlorVinyls In PVC powder pro duction line after the polymerization step a suspension composed of water and PVC particles is obtained Most of this suspension water is eliminated during a centrifugation step leading to a wet porous pow der called cake with a humidity between 02 and 035 kg of waterkg of dry PVC The cake drying occurs essentially in a pneumatic dryer coupled with a fluidized bed dryer with a role of eliminating the residual humidity of PVC less than 005 kg of waterkg of dry PVC This opera tion which consumes a lot of energy between 800 and 1500 kJkg dry PVC in accordance to the operating conditions and the PVC grade rep resents 30 of the production cost With the purpose of reducing drying cost it is important to understand the different physical phenomena involved mass transfer heat transfer hydrodynamic phenomena spe cific to the dryers This study focuses firstly on the acquisition of the kinetics data in a batch dense fluidized bed and secondly on the model ing of the dehydration of PVC particles in a pneumatic dryer The drying kinetic is obtained by immersing a small amount of PVC particles mean diameter 143 μm in a hot fluidized bed filled with glass beads mean diameter 324 μm These bigger particles role is to suppress the agglomeration phenomena between PVC particles and to realize the drying in isothermal conditions The experimental results show that the drying kinetic is strongly dependent on air temperature and humidity These experimental results were represented according to a kinetic model based on heat and mass exchanges between PVC and air external transfers and heat and mass diffusion inside the par ticle intern transfers That kinetic model is then included in a steadystate onedimensional pneumatic dryer model which simu lates the industrial dryer 2 Background Porous media drying is a complex problem and still not well understood despite of the numerous studies that can be found in the literature due to the coupled exchanges mass heat and momentum transfers between the gas and the particles 1 depending on solid par ticles morphological parameters and watersolid affinity As shown in Fig 1 during the drying of a porous particle two periods can be dis tinguished The first period consists in the elimination of free water located at its surface In this period the evaporation is controlled by external transfers which can be in accordance to the dryer technology eg conductive convective or radiative In our case pneumatic and flu idized bed dryer these transfers are convective and depend on the local relative velocity between air and particles called slip velocity These transfers depending on the drying technology have been widely studied and the literature provides a lot of correlations sometimes Corresponding author at Université de Toulouse INPT UPS Laboratoire de Génie Chimique 4 Allée Emile Monso F31030 Toulouse France Tel 33 5 34 32 36 93 Email address antoineaubinensiacetfr A Aubin httpdxdoiorg101016jpowtec201308019 contradictory for its estimation 24 In the second period the water located in the particles pores is eliminated the drying is con trolled by a combination between internal and external transfers The internal transfers depend on the solid properties and structure The literature shows that many elementary phenomena may occur during drying Such phenomena are complex and depend on the structure of the solid particle the watersolid liaison and the water composition These elementary mechanisms represented in Fig 2 can be 56 capillary forces depending on liquid wetting on the solid surface γLV cosθ temperature and particle pore size and distribution These last properties affect the saturation water vapor tension at the solidliquid interface and the internal liquid movement in the pores vapor diffusion through the pores under the influence of tempera ture and water vapor partial pressure solid diffusion corresponding to bound water migration on the solid surface This mechanism occurs in the case of a solid partially dissolved in water and with a strong liaison between water and solid chemical interaction hydrogen liaisons Thus the modeling of a wet particle drying by hot air has to consider water evaporation at the particle surface during constant drying rate period and the movement of an evaporation zone between the wet core and the dry crust of the particle This zone progresses from the par ticle surface to its core Several numerical simulations were realized on porous material 78 The thickness of that drying zone is due to the wetting properties of the liquid in the pores if the wetting is perfect the liquid can spread on the pores internal surface and affect the ther mal gradient 9 In the present case the wetting properties of PVC which depends on the superficial tension γLV of the aqueous solution introduced during the polymerization step and the affinity between the solid particles and this solution are represented by the contact angle θ The low values of liquid superficial tension and high values of the contact angle between 80 and 947 1011 lead to suppose that the drying zone could be modeled by a surface of discontinuity Concerning the capillarity effect the poresize distribution of a PVC particle presented in Fig 3 shows that the minimal pore size is about 30 nm The Kelvin law expresses water activity pwp as a function of pore radius and shows that for pore radius higher than 10 nm the water activity is equal to 1 Hence the capillary effect does not influence the liquidwater equilibrium in this study Concerning the solid and liquid diffusion previous analysis on INEOSs PVC shows that it is non soluble in water and that the PVC has a small affinity Fig 1 Different phases of the particle drying Fig 2 Internal transfer in a porous particle 23 Fig 3 Pore size distribution of a PVC particle determined by mercury porosimetry Micrometrics Autopore IV with water For example in an atmosphere with 75 of relative hu midity the equilibrium humidity of PVC is 24 g of waterkg of dry PVC at 50 C Hence the liquid and solid diffusion can also be neglected in this study and the only internal transfer remaining is the vapor diffu sion This mechanism is influenced by temperature and water vapor partial pressure gradients The SEM picture of a PVC particle Fig 4 shows a typical pellet grain structure 12 indeed each particle is a primary particle stack With such a structure two pore families can be distinguished Fig 3 the space between the primary particles pore size superior to 250 nm which is macroporosity the average size of these pores is 33356 nm These pores are dried in the pneumatic dryer 93 of pore volume the primary particles own porosity pore size inferior to 250 nm which is mesoporosity the average size of these pores is 816 nm These pores are dried in the fluidized bed dryer 7 of pore volume A shrinkingcore type model is used by numerous authors 1316 to represent the evaporation front evolution with particles having a pelletgrain structure In this kind of model the particle is divided into two zones a wet core which shrinks over time and a dry crust through which the evaporated water diffuses to the particle surface The elementary mechanisms involved here are conductive heat transfer from particle surface to its core and water vapor diffusion through the pores If the heat transfer leads the drying phenomena the problem is analogous to the classical Stefan problem for drop evaporation or freez ing 1718 Concerning the pneumatic dryer this technology is commonly used in order to eliminate free or barelybound water A pneumatic dryer is a continuous convective dryer with a diluted solid transport This technology presents numerous advantages 1920 such as the short resident time of particles allows the drying of thermosensible product in parallel flow high temperature can be applied at the inlet resulting in a high thermal efficiency the dryer can be used as a transport system From a hydrodynamic point of view pneumatic drying is analogous to pneumatic conveying Hence several pneumatic conveying models 2122 have inspired the pneumatic drying models presented in the literature These models 42325 consider the interactions mass heat and momentum transfer between a gas phase and a dispersed phase coupled with internal mechanisms 3 Material and methods The experiments were realized with a cake provided by INEOS Chlorvinyls France the properties are presented in Table 1 The particle size analysis realized on dry particles shows a Gaussiantype dis tribution with a mean particle diameter of 144 μm The characteristic sizes presented in Table 1 are determined by laser granulometry Malvern Mastersizer 2000Sirocco 2000 01 μm The particle density porosity tortuosity and poresize distribution Fig 3 were deter mined by mercury porosimeter on dry product Micrometrics Autopore IV 01 nm To determine the cake humidity different samples were placed in a ventilated oven at 50 C during 24 h the cake initial humidity varies between 0195 and 0205 kg of waterkg of dry PVC Preliminary experiments realized in a convective dryer have shown that the drying kinetic strongly depends on the external conditions sample weight and thickness air flowrate The study of the physical phenomena at a particle scale is then difficult to realize Hence just as several studies realized in our laboratory 2627 the drying kinetic is studied by immersing a small sample of wet PVC about 3 g in a hot fluidized bed containing 13 kg of glass beads with a mean particle diameter of 324 μm see properties in Table 1 The experimental pilot is represented in Fig 5 The dryer is com posed of a 10 cm diameter and 40 cm height glass column The presence of an extended section on the top prevents the fine parti cles to be elutriated The fluidization air is preheated in an electri cal heater and when it is necessary mixed up with water vapor produces by an evaporator as shown in Fig 4 A water flowrate is spread on an electrical heater The water is instantaneously evapo rated and carried into the warm air stream The fluidization flowrates velocities are chosen in agreement with the character istic velocities minimal fluidization and terminal settling velocity of PVC particles and glass beads These values determined at ambi ent temperatures are presented in Table 1 The air velocity range chosen is from 25 to 40 cms which corresponds to 3 and 45 times the minimal fluidization velocity of the glass beads The dryer is equipped with several temperatures 01 C and pressure 01 mbar sensors and two capacitive hygrometers temperature and humidity sensors As shown in Fig 4 a vacuum pump collects the wet air to measure its humidity The measure ment system whose role is to ensure that the humidity measurements are realized at constant conditions air flowrate and temperature is equipped with a filter to prevent particles to damage the sensor a thermostatic bath to maintain a constant temperature and prevent con densation phenomena the hygrometer and a rotameter to maintain a constant flowrate This hygrometer measures the air dew point with an accuracy of 01 C This hygrometer measurement is controlled according to a second hygrometer which measures the relative humid ity with an accuracy of 01 placed at the cyclone outlet To avoid Fig 4 SEM picture of a PVC particle Table 1 Powder PVC and glass bead physical properties experimental measurement 01 cms Properties PVC Glass beads d50 μm 144 324 d10 μm 104 233 d90 μm 199 450 d32 μm 139 314 p kgm3 1175 2500 τ 30115 χ 377 λp Wm K 017 Cp Jkg K 1670 Umf cms 12 88 Ut cms 653 2425 a General scheme b Evaporator c Humidity measurement system mw0 were determined These informations enable the calculation of the initial X0 and the instantaneous X solid humidity with m0 as the dry sample weight X ¼ mw0mw m0 and X0 ¼ mw0 m0 ð4Þ The calculated initial humidity corresponds to the initial humidity measured in the ventilated oven at 50 C during 24 h with an error inferior to 5 The temporal evolution of the normalized solid humidity XX0 represented in Fig 7 is used to determine characteristic times t75 t50 t25 and the drying time tdry correspond to the time necessary for the normalized solid humidity to reach respectively 075 050 025 and 005 5 Preliminary results The cake wet PVC is a very cohesive powder so the particles tend to agglomerate themselves This phenomenon reduces external and internal transfers Indeed the bigger the particles are the lower the con vective transfer is In the same way the internal transfer is reduced with particle agglomerates as the water vapor has a longer diffusion path So the evolution of outlet air humidity during batch drying is evaluated with cake samples of different masses The initial calculated humidity was compared to the real samples humidity This study showed that the optimal sample mass is between 2 and 4 g of cake Indeed with inferior mass the sensor precision is too low and with higher mass the sample is too sticky to disperse and the results are not reproducible To reduce this phenomenon the sample is mixed up with glass beads at ambient temperatures before being introduced in the fluidized bed This proceeding breaks the bounds between wet particles and so improves the sample dispersion in the bed Fig 8 shows that the glass bead mass ratio has an influence until 25 For higher values the drying time stays constant Hence the following experiments are realized with a glass bead mass ratio of 30 Fig 9a and b shows an example of the results obtained with the experimental conditions summarized in Table 2 Experiment 1 It can be observed that the drying time is about 30 s which is close to the response time of the hygrometers there is an initial 3 s lag due to the sample injection and the time needed for the wet air to reach the hygrometer the absolute air humidity and the drying flowrate have a similar evolution the bed temperature decrease during the experiments stays below 1 C which shows that the drying is effectuated in isothermal conditions In Fig 8 the evolutions of the normalized solid humidity versus time for different experiments realized in the same conditions Exps 1 2 3 are presented This figure shows that our experimental system is reproducible and the variance of t50 is less than 1 s 6 Experimental results Table 2 sums up the experimental data bed temperature inlet air absolute humidity and air velocity and characteristic times t25 t50 a Comparison between numerical and experimental results b Hygrometer transfert function Fig 6 Hygrometer response to a step signal Fig 7 Definition of the characteristic times Fig 8 Evolution of solid humidity for different fraction of glass beads Air velocity 323 cms Tbed 342 C cake weight 29 g a Absolute air humidity drying flowrate evolution Exp1 b Solid humidity and bed temperature evolution Exp 1 c Reproductibility results Evolution of normalised solid humidity Exp 1 2 3 Table 2 List of experiments response time These results also show that even in a strongly stirred dryer as the fluidized bed the external transfer resistance is not neglectable These resistances controlled by the hydrodynamic parameters of the system determine the energy and mass flow exchange between the dryer atmosphere and the PVC particles Indeed the experimental results in Fig 10 show that a rise of the air flowrate from 92 to 124 kgh decreases the drying time from 53 to 36 s The higher the air velocity is the higher the relative velocity between the air and the particles is Hence the mass and heat transfers are increased and the drying time is shorter Regarding the bed temperature effect the results presented in Fig 11 show that a rise from 35 to 55 C strongly decreases the drying time from 40 to 24 s Indeed a bed temperature increase will improve the heat transfer between the fluidized bed and the particles but also the mass transfer driving force Y Y Fig 12 shows the influence of air humidity on the drying kinetic and drying characteristic times and presents the evolution of the normalized solid humidity versus time These results show that a rise of the air absolute humidity from 16 to 25 g of waterkg of dry air increases the drying time from 55 to 84 s This can be explained by the effect of the drying driving force a rise of initial humidity results to an increase of Y and so there is a drying driving force reduction All these results show that the drying kinetic of PVC particles is directed by both external and internal humidity transfer 7 Kinetic model a Evolution of drying flowrate and normalised solid humidity versus time b Evolution of characteristic times versus air mass flowrate a Evolution of drying flowrate and normalised solid humidity versus time b Evolution of characteristic times versus air temperature 71 Hypothesis and equations Knowing the low affinity of water with the PVC surface the drying inside the particle can be represented by the moving of an evaporation front from the surface to the core of the particle Fig 13 The shrinking core model principle is based on the following hypotheses The particles are considered perfectly spherical and uniform The evaporation occurs on the evaporation front The fluidized bed is considered as a stirred tank The wet core humidity and the dry crust humidity are respectively considered equal to the particle initial humidity X0 and null The mass transfer is directed by the water vapor diffusion through the dry crust and the convection at the particle surface The heat is transferred from the air to the particle by convection and is diffused in the dry crust by conduction The sample weight is low enough to assure the bed isothermicity The particle temperature is considered uniform This hypothesis can be justified by the low value of the thermal Biot number In addition this hypothesis has been justified with an analog model calculating the temperature profile in the dry crust The instantaneous profile of the air humidity in the dry crust is obtained with the quasistationary state hypothesis The equations of the model solved with the variable steps RungeKutta method are presented below Water vapor flux density at particle surface with BiM the mass Biot number defined as Mass balance on the particle Instantaneous solid humidity Wet core radius evolution Heat balance on the particle a Evolution of the normalised solid humidity versus time b Evolution of characteristic times versus inlet air absolute humidity Fig 12 Evolution of the normalized solid humidity and characteristic times for different initial relative humidity Exps 9 to 14 with Cph the specific heat of the wet particle defined as Cph Cpp X Cpl 12 To solve these equations the apparent diffusion and the mass and heat transfer coefficients are calculated using the below correlations The apparent diffusion coefficient is calculated from the binary molecular diffusion coefficient Fig 13 Shrinking core model Dapp DairH2 O X τ 13 with χ and τ given in Table 1 The binary molecular diffusion coefficient is calculated from the Fuller Schettler and Giddings correlation 28 DairH2 O 11757 109 Tg175 14 The heat transfer coefficient is calculated from the Baeyens correlation 29 Nu 015 Re 15 The mass transfer coefficient is calculated from the Chilton and Colburn analogy which is classically used in drying h ky Cph Sc Pr 23 16 72 Model validation In Figs 14 and 15 typical numerical results are presented These results show that the real drying time is about 11 s which is four a Evolution of air absolute humidity and PVC humidity Fig 14 Modeling results inherent kinetic Operating parameters Y 0 kgkg T 416 C Fg 987 kgh X0 0167 kgkg T0 15 C m0 2487 g b Evolution of particle temperature a Evolution of air absolute humidity versus time b Evolution of PVC normalised humidity versus time Fig 15 Comparison between experimental and numerical results Drying of 29 g of wet PVC in fluidized bed at 416 C with an air velocity of 323 cms times faster than the time observed with the hygrometer and that the drying operation occurs in three phases The sample warmingup period from 0 to 3 s phase 1 The constant drying rate period from 3 to 105 s phase 2 The falling drying rate period from 105 to 13 s phase 3 In order to compare the numerical results with the experimental data the hygrometer transfer function is integrated in the model The theoretical results are compared with the experimental finding Exps 1 2 and 3 Fig 15 indicates a good concordance between the model predictions and the experimental data To have a better understanding of the relative influence of the internal and external resistances to mass transfer the sensitivity of the mass Biot number was studied Its expression is remembered below BiM ky RDapp ρg kyρgDappR 17 internal resistanceexternal resistance In the previous simulation conditions the Biot number estimated by the model is equal to 144 which means a competition between both transfer resistances Fig 16 shows the outlet air absolute humidity evolution versus time for different Biot number values from 01 to 10 times the Biot number calculated by the model These results show that the drying is longer and more difficult with the high Biot number So Fig 16 Air absolute humidity evolution versus time for different Biot number values in our case the external resistance is not negligible in front of internal resistance This phenomenon can be explained by the morphological parameters of PVC particles The low internal resistance can be attributed to the evaporation of water inside the macroporosity of small particles and to the low affinity with water The model considers a monodisperse granulometry while in reality the PSD shows a Gaussiantype distribution d10 d50 and d90 presented in Table 1 Fig 17 shows that in our case the particle diameter has a small influence on the characteristic times So the monodisperse hypothesis does not alter significantly the model results Fig 18 shows the comparison between the characteristic times given by the model and by the experimental data These results show that the model predicts correctly the effect of bed temperature Fig 17 inlet air humidity between 0 and 16 g of waterkg of dry air Fig 17 and fluidization air flowrate up to 11 kgs Fig 17 on the characteristic times Besides at lower air fluidization air flowrate and at higher inlet air humidity 16 g of waterkg of dry air the model underestimates the characteristic times especially t25 This phenomenon can be explained by different reasons The effect of inlet air humidity and fluidization air flowrate on the cake dispersion Indeed a rise of inlet humidity and a decrease of fluidization flowrate disfavor this phenomenon The effect of air humidity on the local equilibrium which is not taken into account And by considering the PVC particle as a homogeneous media ie a uniform pore size Indeed an increase of air humidity can lead to a slow down of the drying rate Moreover the presence of mesoporosity in the PVC particles can explain the gaps observed especially at the Fig 17 Influence of particle diameter on the characteristic times Fig 18 Comparison of experimental and numerical characteristic times a Influence of the bed temperature b Influence of the air inlet humidity c Influence of the dry air flowrate Fig 18 Comparison of experimental and numerical characteristic times end of the drying operations In that case transfers are controlled by a combination between molecular and Knudsen diffusion 8 Pneumatic dryer model The aim of this study is to model a pneumatic dryer macroscopic model using the particle scale kinetics previously elaborated As shown in Fig 1 the drying occurs in two periods the removal of surface water which is governed by a convective mechanism the removal of pore water which is driven by convective and diffusive mechanisms This period is modeled with the shrinking core model described in the previous parts 81 Hypothesis and equations In this model a twophase continuum model was used to describe the steadystate flow of a diluted dispersed phase wet powder and a continuous phase through a pneumatic dryer The model is based on the following assumptions Mass energy and momentum balances occur between the two phases The friction forces between the dispersed phase and the wall can be neglected The continuous phase composed of a mixture of water vapor and other gas is considered as an ideal gas The particles are spherical and composed of a homogeneous porous matrix The particle size distribution is considered as monodisperse Electrostatic forces and surface tension effects are neglected The wall heat loss is neglected The mass energy and momentum balances developed in onedimensional steadystate for the kphase and the transport equations for solid and gas humidity deducted from the mass balances are presented below In this work the kphase can be either the gas phase k g or the particle phase k p Mass balance ddz αk ρk Uk Γk 18 with Γk the volume rate of mass transfer of the kphase expressed from the water vapor flow density Nw Γp Γg αp sp Nw 19 where αp is the local volume fraction of solid and sp the specific surface area of a particle The water vapor flux density can be expressed as the product of a global masstransfer coefficient and the drying driving force Nw Ky Y Y 20 The global masstransfer coefficient is expressed as a function of the mass Biot number as seen in the previous part in Eq 6 043 Ky ky B iM X X0 131 1 1 X0Xc 21 Eq 21 is valuable for solid humidity less than or equal to the critical humidity Xc Xc ρwater X ρp 22 This case is commonly observed when the pneumatic dryer is fed by a cake obtained by an efficient centrifugation In opposite when the initial humidity is greater than Xc the evaporation takes place essentially on a continuous liquid film at the particle surface convective transfer The reduction of this film thickness during the evaporation decreases the drying surface In this case the global mass transfer coefficient can be estimated by Ky ky ρp ρwater X 1 X 23 for X Xc 23 Energy balance Fk0 dHk dz Qqk Qwk Ac Γk Hg w 24 with Qqk as the energy exchanged from the qphase to the kphase and Qwk is the heat loss of the kphase Qgp Qpg αp Ac sp h Tg Tp 25 where Ac is the pipe cross section h is the heattransfer coefficient and Tp and Tg are the solid and gas phase temperatures respectively As the solid flow is very diluted the heat loss of the solid phase is neglected for the gas phase it is expressed as Qwg hwall Twall Tg 26 Momentum balance d dz αk ρk Uk 2 αk dP dz αk ρk g Iqk Up Uk Γk Ff wk 27 The first two terms on the righthand side of this equation represent respectively the influence of pressure and gravity on the flows Iq k represents the interactions between the phases and Ffw k represents the friction forces between the wall and the kphase In this model the only interaction taken into account is the drag Igp Ipg 3 αp ρg Ur 2 Cd 4 dp 28 where Ur is the slip velocity Ur Ug Up The drag coefficient Cd is calculated by the Wen and Yu correlation 30 Cd 24 Re 1 015 Re 0687 αg 17 29 The friction force between the pipe and the continuous phase is estimated by Ff w g f ρg 2 Dpipe Ug 2 30 The friction factor f is calculated by the Blasius formula f 64 Re pipe if Re pipe 2100 31 f 00791 Re 14 pipe if Re pipe 10 000 32 As the solid flow is much diluted the particle friction forces are neglected Humidity transport equations dX dz Ac Γp F0p 33 dY dz Ac Γg F0g F0p F0g dX dz 34 with F0p and F0g as the dry flowrate of the solid and gas phases Transfer coefficients Some dimensionless empirical correlations can be used to calculate the heattransfer coefficient in a pneumatic dryer Baeyens correlation 29 see Eq 15 De Brandt correlation 29 Nu 016 Re1 3 Pr0 67 35 Bandrowski correlation 31 Nu 000114 αp 0 5984 Re08159 36 The Chilton and Colburn analogy see Eq 16 is used in order to calculate the mass transfer coefficient Baeybes and De Brandt correlations predict the transfer coefficient in a diluted phase In order to model the effects of collisions between particles on the transfer rate Bandrowski correlation 31 expressed the Nusselt number as a function of the solid volume fraction αp This relation indicates that in the inlet section of a pneumatic dryer the important value of αp can counterbalance the high values of slip velocity In other words it takes into account the effect of solid dispersion in the inlet zone 9 Results and discussion Firstly the model predictions are compared with the experimental measurements effectuated by Baeyens 29 on an industrial pneumatic dryer Secondly they are confronted with our preliminary industrial results obtained at INEOS ChlorVinyls company The simulation data PVC properties geometrical and operating parameters are resumed in Table 3 91 Comparison with literature findings In the case of the industrial experiments published by Baeyens 29 the inlet PVC humidity 026 is greater than the critical humidity 0134 so the drying rate is controlled by a two stage process The first stage corresponds to the surface water evaporation and the second to the pore water evaporation Fig 19 represents the theoretical results obtained using the different correlations previously presented and the experimental measurements effectuated by Baeyens 29 on an industrial process Fig 18 represents the evolution of solid humidity versus Table 3 Simulation data 29 Data Baeyens INEOS Particle properties Mean particles diameter μm 180 160 Solid density kgm3 1116 1016 Inlet PVC humidity kgkg 026 033 Porosity 015 0367 Tortuosity 80 32562 Dryer geometry Pipe diameter m 125 17 Pipe length m 25 22 Operating parameters Inlet air temperature C 127 180 Inlet air humidity kgkg 0003 0004 Dry air flowrate th 4648 45 Dry PVC flowrate th 6670 8545 Thermal loss kWm 25 25 pipe length while Fig 18 represents the evolution of air temperature versus pipe length The differences observed between experimental data and theo retical results can be explained by the worst dispersion of the wet co hesive PVC in the gas phase at the pipes inlet zone The theoretical results which fit the best the experimental data were obtained using the Bandrowski correlation This can be explained by the effect of the slip velocity on the Reynolds number and so on the heat and mass transfer coefficients As shown in Fig 20 this magnitude is re ally important at the inlet and progressively decreases along the dryer This leads to very important variations of the transfer coeffi cients see Fig 21 In the Bandrowski correlation this effect is atten uated by the solid volume fraction of the dispersed phase 92 Comparison with INEOS ChlorVinyls results To establish the humidity and temperature profiles along the pneumatic dryer INEOS industrial dryer is equipped with several hygrometers and thermocouples Fig 22 shows the comparison be tween industrial data and numerical results obtained using the Bandrowski correlation It can be noticed that the temperature pro file is correct while some differences appear on the drying flowrate evolution in the industrial dryer the humidity is barely constant in the first meters while in the model results humidity rises strongly This is due to the dispersion phenomenon not taken into account in the model which slows down the drying in the acceleration zone 93 Parameter study The results of the following simulations were obtained using the Bandrowski correlation and Baeyens parameters see Table 3 The effect of dry air mass flowrate was examined between 25 and 464 th As shown in Fig 23 this parameter has a small influence on the drying rate above 35 th The air velocity influences the drying rate only in the acceleration zone then the slip velocity is equal to the terminal settling velocity At the dryer outlet the air is far to be saturated so the decrease of the dry air flowrate does not affect the dry ing driving force significantly eg a decrease of dry air flowrate from 46 th to 35 th increases the outlet solid humidity from 0054 to 0067 kg waterkg dry PVC and a decrease of dry air flowrate from 35 th to 25 th increases the outlet solid humidity from 0067 to 0097 kg waterkg dry PVC Concerning the inlet air temperature Fig 24 shows a dominant effect eg an increase of inlet air temperature from 96 C to 156 C decreases the outlet solid humidity from 0097 to 0014 kg waterkg dry PVC Indeed increasing the temperature leads to a more important heattransfer driving force But as the equilibrium humidity depends on the temperature it will also increase the masstransfer driving force Fig 21 Evolution of global mass transfer coefficient versus solid humidity for different correlations Fig 22 Comparison between numerical results and INEOS industrial data Fig 20 Evolution of slip velocity versus pipe length a Evolution of solid humidity versus time for different correlations b Evolution of air temperature versus time for different correlations Fig 19 Comparison between experimental and numerical results for different correlations The inlet air humidity effects were investigated between 0 and 00105 kg waterkg dry air The last value corresponds to an atmo spheric air 15 C almost saturated 984 of relative humidity As shown in Fig 25 such an increase of inlet air humidity increases the outlet solid humidity from 0052 to 0054 kg waterkg dry PVC This increase shows that the inlet air humidity which is a sustained parameter slightly influences the drying driving force It can be noticed that the solid temperature is influenced by the dry air flowrate during the falling drying rate period only the air humidity during the constant drying rate period only and the air temperature influences it during both periods In fact the temperature stage value of the constant drying period is controlled by the drying driving forces As shown in Eqs 19 and 20 this force depends only on the air humidity Y and the equilibri um humidity Y which is directly related to the solid temperature In the falling drying rate period the solid temperature increases until the thermal equilibrium with the air this equilibrium is controlled by the dry air flowrate and temperature 10 Conclusion An experimental and numerical protocol was developed to study the drying kinetic of porous media at the particle scale The results show the significant influence of the operating parameters essentially air temperature and humidity It appears clearly that the PVC drying is controlled by a competition between internal and external transfers The use of a shrinkingcore type model to simulate the particlescale drying was confirmed by the experimental results PVC is essentially constituted of macropores dpore b 250 nm The volume fraction of mesopores 10 b dpore b 250 nm is inferior to 10 So we choose a shrinking core type model supposing that the par ticles can be represented like a spheric pseudohomogeneous sphere ie uniform porosity and pore size This model permits the drying kinetic to be simulated correctly The gaps observed at the end of the drying operation is probably related to the transfers in the mesoporosity primary particles A onedimensional steadystate model for pneumatic dryer was established This model is applied for the drying process of wet PVC powder The drying rate is controlled by the convective transfer in the first period and by the convective and diffusive transfers in the second The first period corresponds to the surface water drying besides the second corresponds to the evaporation of the water in the pores simu lated by a shrinking core model The model takes into account the con vective heat and mass and momentum transfers The parameter study shows that the inlet temperature is the most important parameter in the operation But this model does not take into account the wet powder dispersion which is certainly the limiting step of this process Nomenclature Roman symbols Ac Pipe crosssection m2 Cd Drag coefficient Cp Specific heat J kg1 K1 dp Particle diameter m d32 Sauter diameter m d50 Median diameter m Dapp Apparent diffusion coefficient m2 s1 DH2O Water vaporair binary diffusion coefficient m2 s1 Dpipe Pipe diameter m f Friction factor Ffw k Wall friction force on k phase kg m2 s2 g Acceleration of gravity m s2 h Convective heat transfer coefficient W m2 K1 Hk kphase mass enthalpy J kg1 Hg w Water vapor mass enthalpy J kg1 Iq k Interface forces from q to k phase kg m2 s2 ky Convective mass transfer coefficient kg m2 s1 Ky Global mass transfer coefficient kg m2 s1 m0 Dry sample mass kg mw Mass of evaporated water at t time kg mw0 Total mass of evaporated water kg Nw Mass transfer rate kg m2 s1 p Saturated water vapor pressure Pa pw Water vapor partial pressure Pa P Absolute pressure Pa Qq k Heat transfer from q to k phase W m1 Qw k kphase wall heat loss W m1 rc Wet core radius m Fig 24 Evolution of solid humidity and temperature versus pipe length for different inlet temperatures 126 96 and 156 C Fig 25 Evolution of solid humidity and temperature versus pipe length for different inlet air humidity Fig 23 Evolution of solid humidity and temperature versus pipe length for different dry air flowrates 464 35 and 25 th Roman symbols R Particle radius m RH Air relative humidity sp Particle Specific Surface m2 m 3 tdry Time when X reaches 005 X0 s t25 Time when X reaches 025 X0 s t50 Time when X reaches 050 X0 s t75 Time when X reaches 075 X0 s Tk kphase temperature C Uk kphase velocity m s1 Umf Minimal fluidization velocity m s1 Ur Relative velocity m s1 Ut Settling velocity m s1 Wdrying Drying flowrate kg s1 X 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