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Drying of granular medium by hot air and microwaves. Modeling and prediction of internal gas pressure and binder distribution
Powder Technology 286 (2015) 636–644
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Powder Technology journal homepage: www.elsevier.com/locate/powtec
Drying of granular medium by hot air and microwaves. Modeling and prediction of internal gas pressure and binder distribution Lamine Hassini a, Roman Peczalski b,⁎, Jean-Louis Gelet c a b c
University of Tunis El Manar, Faculté des Sciences de Tunis, Laboratoire d'Energétique et des Transferts Thermique et Massique (LETTM), Tunisia University of Lyon, Université Claude Bernard Lyon 1, Laboratoire d'Automatique et Génie des Procédés (LAGEP) UMR CNRS 5007, La Doua Campus, CPE Building, Villeurbanne, 69622, France MERSEN Company, Saint Bonnet de Mûre, France
a r t i c l e
i n f o
Article history: Received 8 March 2013 Received in revised form 31 August 2015 Accepted 5 September 2015 Available online 7 September 2015 Keywords: Drying Micro-waves Heat and mass transfer Porous granular medium Internal pressure Mineral binder
a b s t r a c t With the aim of enhancing the industrial drying (hot air and microwaves) of inserts made of agglomerated sand and to assess product quality, a comprehensive internal heat and mass transfer model has been proposed. In this model, liquid and vapor transfer by filtration, liquid expulsion at the surface and binder transfer by advection were directly accounted for. This model was validated on the basis of the experimental average water and binder content and core temperature curves for drying trials at different operating conditions. Then, it was used for comparing the drying time, the internal pressure and the binder content calculated for several test processes. It was demonstrated that the simultaneous application of hot air and micro-waves provided a real possibility to dry faster and to maintain quite the same internal overpressure and overall binder content as for the reference (damage free) process. The heating power should be increased gradually during the process. The decrease of the drying time was around 30% with regard to the reference mild process. © 2015 Elsevier B.V. All rights reserved.
1. Introduction 1.1. Industrial and scientific context The context of this study is the drying of wet agglomerated sand used as the filling material for high current industrial electrical fuses. The wet agglomerate is composed of calibrated sand and of the aqueous solution of a mineral binder (from the silicate family). This material is compacted and dried in a nearly parallelepipedic ceramic container (shell) which is roughly 10 cm high, 5 cm wide and 1 cm thick (see Fig. 1). During drying the container is opened to ambient air only by two small orifices on its top part, each of few millimeters diameter. For patent registration reasons, the exact composition and geometry of the product will not be revealed. The drying process is aimed at removing water almost entirely while preserving some significant amount of the binder. To accelerate the drying of the fuses' interiors, it is intended to substitute the standard hot air process by a new process with combined micro-waves and hot air heating. The great advantage of micro-waves heating as regards the convective or contact one is that the heat penetrates instantaneously and deeply into the product and that heat is ⁎ Corresponding author. E-mail address: [email protected] (R. Peczalski).
http://dx.doi.org/10.1016/j.powtec.2015.09.009 0032-5910/© 2015 Elsevier B.V. All rights reserved.
dissipated mainly in the wet part of the product [1]. In this way, the core temperature of a product dried by micro-waves is higher than its surface temperature and can quickly rise over the water boiling point [2]. The inversion of the temperature gradient enables the vapor thermo-diffusion to be directed from the core to the surface alike the ordinary diffusion [3]. The intense water vaporization in the core of the product induces a strong raise of the internal gas phase (vapor and air) pressure, which pressure literally pushes the water within and out of the product. The water transfer by filtration dominates then the transfer by diffusion and that increases very significantly the drying rate. At high micro-waves power, the internal pressure may be high enough to expulse water directly in the liquid phase at the product surface (‘water pumping’) providing a kind of mechanical dewatering with non energy consumption for evaporation. All of these phenomena lead to substantially higher water fluxes from the core to the product surface and thus to higher drying rates than in a purely convective process for the same average product temperature. Barba et al. [4] have experimentally studied the microwaves assisted drying of cellulose derivative (HPMC) powder. Using a customized microwaves power protocol where the power was decreased as the product water content decreased, they obtained drying kinetics as fast as for high temperature convective process, while keeping internal product temperature at a lower safe level which is of high importance for thermo-sensitive pharmaceutical products.
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Fig. 1. Layout of the product within the container (axial section).
As concerns process modeling, the classical ‘water diffusion’ approach would be acceptable only in the case of soft drying conditions. Malafronte et al. [5] proposed a model for the combined convectivemicrowave assisted drying process of food (potatoes). The model accounted for the dependence of dielectric properties on product's temperature and water content and coupled the heat and mass transport equations with the Maxwell's electromagnetic field equation in transient regime. As concerns water transfer, only liquid and vapor diffusion were considered. No water expulsion at the sample surface was reported. This can be explained by the fact that for the drying conditions considered the core sample temperature did not exceed 80 °C (so was below the boiling point of water) and the gas pressure build-up within the sample and the consecutive internal liquid water flow were insignificant. For the above mentioned effects may be accounted of, substantial modifications of a classical diffusive kind drying model are required. First, the water transfer due to gas pressure gradient must be included in the model. Perré and Moyne [6] and independently Turner and Joly [7] were the first to propose this. Second, the boundary conditions must be adapted in a way enabling the liquid water expulsion. This adaptation is generally omitted or not explicated in the literature and at our best knowledge only Constant et al. [8] considered it and gave some precisions about its implementation. The adjunction of a micro-waves source in the dryer can reduce the processing times of clay [3] or wood [9] up to 50%. However, the important internal overpressure which exerts a stress on the material can be a cause of cracking and a compromise must be sought between the drying rate and the pressure raise while defining the operating conditions. In the case of clay drying, Kowalski and Mielniczuk [10] had suggested that the micro-waves should be applied only in the first stage of drying when the product is nearly saturated and that hot air should be applied in the second period where the product is already partially dry. They calculated that for this combined process the stress level was much lower than for a purely micro-waves process with a similar processing time. This observation was the start point of our first study [11] which concerned only heat and water internal transport modeling and was focused on possible product damage due to the gas phase overpressure.
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However, in the case of mineral agglomerate considered here, there is an additional issue concerning the binder's distribution. The mineral binder secures the mechanical strength of the final product. Its content must not be reduced below a critical value needed for a particle to stick each to other after drying. The problem is that the binder is dissolved in water and, as water is removed during drying, some part of the binder is removed either. This is particularly true for micro-wave assisted or very hot air drying when the water is literally expulsed in the liquid phase during the first period of the process. This leads to very high water removal rates but also binder depletion rates. The proper rate at which the drying will be driven by means of supplied heating power must then be dictated by the binder content evolution within the product. This is the main subject of the present study and to our best knowledge there are no scientific publications on this problem. As concerns more generally the issue of thermal extraction of liquid from granular media, several studies were devoted to the removal of a thermoplastic organic binder from an injection molded ceramic compact prior to sintering [12,13]. During the thermal processing, the organic binder melts and the volatile components evaporate thus enabling internal transfer in the liquid and gaseous phase. In a study concerning a composite made of alumina particles and paraffin wax (pentacosane) submitted to high temperature processing (over 150 °C), Stangle and Aksay [12] had developed a comprehensive internal transfer model very similar to the one used in this study. However due to the low vapor tension of the binder, the evaporative and transfer phenomena driven by the gaseous phase pressure were much less important than in the case of the removal of an aqueous solution considered in this study and the results were not really comparable. 1.2. Aim of this study The goal was to predict the product state evolution for different combined micro-waves and hot air drying schedules and to find out the best operating conditions ensuring a fast water removal with non material cracking. The product damage was supposed to be entirely due to the gas phase (vapor and air) overpressure and to the mineral binder depletion. The risk associated with binder depletion was evaluated by considering the final average binder content and by considering the binder content heterogeneity i.e. the difference between the bottom and surface content in the sample. The work was carried out in three steps. First, an internal heat and mass transfer model was developed. In this model, the internal gas phase pressure effect was made perfectly explicit, especially the phenomena of liquid and vapor transfer by filtration and of liquid expulsion at the surface. A special equation was introduced describing the advective binder transfer within the product. A particular attention was also paid to the boundary conditions (see next section). The boundary conditions accounting for water expulsion were already introduced in [11] but the binder transfer description is an original contribution of this study. Second, the model was validated by comparing experimental and simulated average global water and binder content and core temperature curves. The drying trials were realized on a specifically designed dual heating mode (micro-waves and hot air) pilot dryer. The simulations were carried out using COMSOL ‘Multiphysics’ software. Finally, the model was applied for testing different drying schedules. The simulations were realized for two simple single heating mode processes and two combined dual heating mode processes. It will be shown that a simultaneous application of micro-waves and hot air allows for process speeding up without increasing the internal pressure or decreasing the binder content as compared to a safe reference process. As compared to the exploratory simulations presented in [11], in this paper more complex schedules with lower micro-waves power input were investigated which are closer to those recommended finally for manufacturing practice. These simulation results are original contributions of this study.
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(5) The heat is transferred by conduction and advection associated to the liquid phase flow (the heat advected with water vapor is neglected). (6) The vapor moves by filtration due to the gas phase pressure gradient, by thermo-diffusion due to temperature gradient and by ordinary diffusion due to the water content gradient. (7) If the surface is saturated, the liquid phase (water and binder) leaves the product directly, without water evaporation, being expulsed by the gas phase overpressure. (8) The capillary liquid water flow evaporates at the surface and leaves the product by convection. The water vapor leaves the product directly by convection.
2. Methods 2.1. Dual heating mode pilot dryer For the purpose of this study, a drying oven was specifically designed and realized. This oven consisted of: a parallelepipedic chamber with stainless steel walls (730 mm wide, 610 mm high and 610 mm deep); a micro-waves generator of 2450 Mhz frequency and with a power tuneable up to 1.2 kW; a hot air generator of 5 kW maximal power, with a temperature adjustable up to 200 °C and a velocity up to 3 m/s (in the center of the chamber); a circular plate (sample holder) made of mecanite (500 mm wide) rotated by an electric motor and suspended by a wire to an electronic balance (range of 15 kg, precision of 1 g) fixed on the top of the chamber; three optical temperature probes, two inserted in the samples and one for air temperature control; and a hygrometer. During periodic weighing, rotation of the plate and air blowing were stopped. The setting of operating conditions within the chamber and the acquisition of the mass and the temperature of the product were operated by an industrial programmable controller. The concept outline of the dual mode dryer is drawn on Fig. 2. This dryer was used for experimental trials needed for model validation. 2.2. Internal transfer model The internal heat and mass transfer model was based on the equations of Constant et al. [8] and Perré and Turner [9]. For the purpose of this study, the mineral binder transfer equation was added and the water transfer equations and corresponding boundary conditions were modified. The final general form of the equations' system is given in Section 2.2.2. A separate Section (2.2.3) is devoted to the boundary conditions which deserve special attention because of the ‘water pumping’ phenomenon. The main model assumptions are given below: (1) The material is multiphase and heterogeneous at the microscopic scale but is represented by an equivalent homogeneous medium with continuously varying state variables and macroscopically defined properties. (2) The different phases are in local thermodynamic equilibrium as concerns the temperature and vapor pressure. (3) The liquid phase (solution of binder in water) flows by filtration due to the gas phase pressure gradient and by capillarity due to the liquid water content gradient (the influence of the temperature gradient on the capillary flow is neglected). (4) The liquid water and binder specific fluxes are purely advective ones associated to the liquid phase flow at the pro rata of their respective fractions (the binder ordinary diffusion in water is neglected).
2.2.1. Binder and water transfer In the present case the mineral binder is dissolved in liquid water and the resulting solution is the liquid phase of the granular medium. This liquid phase flows by filtration due to the gas phase pressure (Pg) gradient and by capillarity due mainly to the liquid water content (X) gradient (the influence of the temperature gradient on the capillary flow is neglected). The liquid flux is thus given by: nl ¼ −
Keq Keq Pc l gradX gradPg þ l νl νl X
ð1Þ
where Kl is the product liquid permeability, νl is the liquid viscosity, Pc is the capillary pressure (a function of the water content X). If the binder ordinary diffusion in water is neglected, the liquid water and binder specific fluxes are purely advective ones associated to the liquid phase flow at the pro rata of their respective fractions. These fluxes are thus respectively given by: nw ¼
X n XþY l
ð2Þ
nb ¼
Y n XþY l
ð3Þ
The transfer equations are obtained by inserting the above fluxes' expressions in the local mass balance equations and are written as follows: ρ0
∂X ¼ −divðnw þ nv Þ ∂t
ð4Þ
ρ0
∂Y ¼ −divðnb Þ ∂t
ð5Þ
In Eq. (4) nv is the water vapor flux. The vapor is supposed to move by filtration due to the gas phase pressure gradient, by thermo-diffusion due to temperature gradient and by ordinary diffusion due to the water content gradient. This flux is hence given by: nv ¼ k41 gradT þ k42 gradX þ k43 gradPg
ð6Þ
In order to reduce the length of this equation, the numerous physical parameters representing the medium properties were grouped into global coefficients (kmn) that are given in Appendix A.
Fig. 2. Layout of the dual mode micro-waves and hot air pilot drying oven.
2.2.2. Full equations' system The model consists finally of five PDEs: the heat transfer equation determining the temperature (T), the liquid water transfer equation determining the liquid water content (X), the dry air transfer equation determining the gas pressure (Pg), the water vapor transfer equation determining the mass of liquid water vaporized per unit volume (mvap) and the mineral binder transfer equation determining the binder
L. Hassini et al. / Powder Technology 286 (2015) 636–644
content (Y). The full set of equations describing the heat and water transfer within the product are written as: Heat transfer m11
∂T ∂mvap ¼ divðk11 gradTÞ−nl Cpl gradT þ φdiss þ m14 ∂t ∂t
ð7Þ
Liquid water transfer m22
∂X ¼ div k21 gradT þ k22 gradX þ k23 gradPg ∂t
ð8Þ
Dry air transfer ∂Pg ∂T ∂X þ m32 þ m33 ¼ div k31 gradT þ k32 gradX þ k33 gradPg m31 ∂t ∂t ∂t ð9Þ
Water vapor transfer m44
∂mvap ¼ div k41 gradT þ k42 gradX þ k43 gradPg ∂t
ð10Þ
Binder transfer m55
∂Y ¼ div k51 gradX þ k52 gradPg ∂t
ð11Þ
As stipulated in the introduction section, the gas pressure gradient appears clearly as one of the transfer driving forces. The influence of the binder amount on the volume of the void space available for gas transfer (effective porosity) and thus on the gas concentration within the product (Eq. (9)) was neglected. The term ∂mvap/∂t in Eqs. (7) and (10) represents the liquid to vapor phase change rate term. This term is not the accumulation term for vapor mass. The accumulation of vapor mass in a control volume was neglected. The exact expressions of the m (‘capacities’) and k (‘conductances’) coefficients pertaining to the above five transfer equations are given in Appendix A. The volumic thermal power generated within the product by dissipation of micro-waves energy was simply calculated by a semiempirical formula: φdiss ¼ αsat
Φmw X Φmw þ αdry V Xsat V
ð12Þ
where Φmw is the power of the micro-waves generator, V is the sample volume, αsat is the absorption coefficient measured for the water saturated product, Xsat is the saturated product water content and αdry is the absorption coefficient measured for the dry product (the same value is applied for the ceramic shell). Eq. (12) is based on the assumption that the electric field is uniform within the product and that the wet product has a dissipation factor which is directly proportional to the water content and that the dry product has a constant dissipation factor. A similar approach was used by Constant et al. [8] and Kowalski et al. [3]. Eqs. ((7) to (11)) are not only coupled by the gradients of the state variables (X, T and Pg) but also by the m and k coefficients. These coefficients are in fact combinations of product properties (material densities, thermal conductivities, fluid permeabilities, species diffusivities) which depend on the state variables. The semiempirical expressions describing these properties have to be known with some precision and be consistent with each other in order to obtain realistic results without numerical problems. This concerns particularly the expressions of gas and liquid relative permeabilities [2], of the capillary suction and of the sorption isotherm. For the purpose
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of this study, the hydraulic product properties (capillary suction, liquid permeability) were experimentally determined and fitted by an empirical model [14]. The gas phase transport properties (gas permeability, vapor pressure and diffusivity) were adapted from the literature for similar materials [2,8]. The values or expressions for all relevant material properties used in the simulations are given in Appendix B. The equations of our model were solved by means of the ‘COMSOL Multiphysics’ software (v. 3.5) but using the open ‘PDE coefficient form' mode and not the predefined heat and mass transfer modules. The calculation domain included the product and its shell (see next section for further details). 2.2.3. Boundary conditions The product (agglomerated sand) is confined in a ceramic shell (see Fig. 1). The shell is air tight excepted two circular openings at the top. A perfect thermal contact was assumed on all product faces in contact with the shell which is directly exposed to the hot air flow. The heat transfer within the shell was described by a classical heat diffusion equation with convective boundary conditions and with a volumic heat source due to dissipation of micro-waves energy. In this section, a focus on the mass and heat transfer at the product surface in direct contact with the air (in the holes) will be done as it is one of the original elements of our model. In order to account for the important phenomenon of liquid water expulsion (also called ‘water pumping’ in the literature), the water balance equation at the opened product surface was written in the following way:
ðnw þ nv Þjsurf ¼ hm
Keq Patm Mv Patm −Pva Xsurf l ∂Pg ln − RTa Patm −Pvsurf Xsurf þ Ysurf νl ∂n surf ð13Þ
where n is the water flux (w for liquid, v for vapor) arriving at the surface (see Eqs. (2) and (3)), hm is the convective mass transfer coefficient, Kl is the product liquid permeability, νl is the liquid viscosity, Pv is the vapor pressure (a for air, surf for surface), and Ta is the air temperature. The novelty of this boundary equation lies in the second term on the right side which represents the liquid water flux arriving at the surface by filtration due to gas pressure gradient. Including this term means that liquid water arriving at the surface by filtration leaves the product directly as liquid. Consequently, the liquid water arriving by capillary flow in fact evaporates at the surface and leaves the product by convection (diffusion in the boundary layer) as vapor and evidently the water vapor arriving at the surface leaves the product by convection in the same form. The additional boundary liquid filtration (water expulsion) term is effective if the gas pressure in the product sub surface layer is higher than the atmospheric pressure and if its water content is close to saturation. Constant et al. [8] have suggested a similar approach formulating a criterion for water expulsion by means of the liquid pressure (Pl = Pg−Pc) and not the gas pressure. However, if the water content approaches the saturation limit, the capillary pressure tends to zero and the two formulations are equivalent. Besides, in reference [8], the mathematical expression of the flux of expulsed liquid water is not explicitly given. An additional boundary condition is needed for the binder transfer equation. It is written as:
nb jsurf ¼ −
Keq Ysurf l ∂Pg Xsurf þ Ysurf νl ∂n surf
ð14Þ
This formula stands that the only way for the binder to leave the product is to be washed out by liquid water, the latter being expulsed by the gas phase overpressure.
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According to the above water transfer boundary condition (Eq. (13)), the heat balance equation at the product free boundary was written as: ∂T þ Cpl ðTsurf −Tre f Þnl surf ∂n surf Keq Xsurf l ∂Pc ∂X ¼ Cpl ðTsurf −Tre f Þ þ Δhvap Xsurf þ Ysurf νl ∂X ∂nsurf Keq ∂Pg þ Cpl ðTsurf −Tre f Þ l − hðTa −Tsurf Þ νl ∂n surf
−λeq
ð15Þ
where λ is the product thermal conductivity, nl is the total (capillary flow and filtration) liquid phase (water and binder) flux arriving at the surface (the heat advected with water vapor was neglected), Cpl is the specific heat of the liquid phase, and h is the convective heat transfer coefficient at the product surface. This equation separates clearly the heat flux due to evaporation of capillary water (first term on the left side) and the heat flux due to liquid water expulsion (second term on the left side). For the sake of completeness, the boundary condition for the gas transfer has to be mentioned. This condition stands simply that the gas pressure at the product surface equals the atmospheric pressure. 3. Results and discussion 3.1. Experimental curves and model validation In order to check the validity of our model, the experimental and simulated drying curves were compared for the three following cases: 1) a high air temperature process labeled HAH; 2) a medium microwaves power process labeled MWM, and 3) a high micro-wave power process labeled MWH. The values of the operating parameters pertaining to each process are given in Table 1. The average water (and binder) content of the product versus time curves are plotted on Fig. 3 and the core temperature ones on Fig. 4. The experimental data were processed by a moving average filter what explains the smoothness of the curves. In general, the simulated curves corresponded well to the experimental ones and the model could be considered valid. Some minor but noticeable differences should be attributed to the low accuracy of the semi-empirical expression representing the micro-waves energy dissipated within the product. It must be emphasized that with a classical formulation of boundary conditions (without the liquid filtration flux) for water transfer, it was not possible to obtain an agreement between experience and simulation for high temperature or high microwaves power even by artificially far increasing the superficial mass transfer coefficient (hm). In order to illustrate this point, the liquid content curves calculated with classical boundary conditions (Eqs. (13) and (15) without the second term on the right hand side) and a usual value (given in Appendix B) of the hm coefficient were also presented on Fig. 3 (see the dotted lines). As can be observed, the simulated drying times were roughly multiplied by 4 in the case of strong operating conditions (MWH and HAH) where the water filtration and expulsion in liquid phase were actually a major contribution to overall water elimination.
Fig. 3. Average water and binder content of the product versus time for different drying conditions. The gray curves correspond to experimental data, the black continuous curves correspond to simulations realized considering water expulsion at the product opened boundary, and the dotted black curves correspond to simulations without water expulsion.
One can observe that for significant water removal acceleration a high temperature or a high micro-waves power have to be applied. As shown on Fig. 4, these intensive drying conditions correspond to internal temperature raise up to and beyond the boiling temperature (100 °C) and consecutive internal pressure build-up and water transfer by filtration accompanied by liquid expulsion at the surface. The high power micro-waves process allowed for faster drying than the high temperature one, this lasts being the present industrial practice. But the micro-waves power must be moderated to avoid too much overpressure within the sample or too much binder expulsion (see next section). One can also observe on Fig. 3 that the drying rate is truly low for the medium micro-waves power process. The surprisingly slow drying for moderate operating conditions is due to the particular geometry of the electrical fuse considered here. In fact, for technological reasons, the agglomerated sand is enclosed in a tight container which is opened only by two small orifices on its top part. The surface area for mass exchange with ambient air is thus extremely small and it is not possible to dry this product in reasonable time without using very intensive conditions like micro-waves or high temperature convection. 3.2. Instant view of internal heat and mass transfer In order to highlight the differences between simple hot air drying (process HAM, see Table 2) and simple micro-waves drying (process MWM, see Table 2), the spatial profiles of the state variables (X, Pg, T,
Table 1 Operating conditions for model validation. Ta, °C
RH, %
Φmw, W
Label
2
0
HAH
19
400
MWM
18
800
MWH
160 50 50
Fig. 4. Core temperature of the product versus time, measured (gray curves) and simulated (black curves) for different drying conditions.
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Table 2 Operating conditions for exploratory simulations.
1 2 3 1 2 3 4
Δt, min
Ta, °C
Φmw, W
Label
1000 1000 300 300 400 240 180 180 400
120 50 50 120 160 80 80 120 160
0 400 400 0 0 200 400 400 0
HAM MWM MWM-300/HAM-300/HAH-400
MWL-HAL-240/MWM-HAL180/MWM-HAM-180/HAH-400
Y) for these two processes are compared on Figs. 5 and 6. The processing time is 5000 s which corresponds to the early stage of drying when the product is still quite wet and water is moving essentially in the liquid state. The profiles are plotted within the product's sample limits along a vertical axis z passing through the holes in the shell, so that the maximal coordinate z corresponds to the product surface in contact with ambient air (free boundary). The temperature profile governs mainly the gas phase pressure profile (by the contribution of vapor saturation pressure) and temperature maximums coincide with the pressure ones. The gas phase pressure maximum is situated at the bottom of the sample for process MWM while it is situated near the top of the sample for process HAM (see Fig. 5). The liquid water is pushed away from this pressure maximum so that in this early stage of drying the water content is higher at the top than at the bottom. Consequentially the top layer is enriched in binder, especially for process HAM, while the core is depleted. The bottom temperature is higher for the MWM process than for the HAM one due to the uniformity of dielectric heating (see Fig. 6). The important temperature drop at the top boundary of the sample (free boundary) is due to intense evaporation of capillary liquid water. 3.3. Exploratory simulations of combined processes It is reminded that in our approach the product quality was assessed by means of the following parameters: the internal gas pressure, the binder content and the binder distribution. Excessive pressure rise or binder depletion may cause the fracture of the product. Some requirements for a process that guarantees an acceptable final binder content can be formulated on a theoretical basis. According to our model, the binder flow within the product is strictly linked to the liquid phase diffusion (capillary flow) and filtration within the product and the binder flow out of the product is exclusively linked to the liquid phase filtration. In order to slow down the binder removal, the transfer in the gas phase should be promoted rather than in the liquid phase and
Fig. 5. Simulated distributions of water content and gas phase pressure along the product vertical axis at the beginning of the process for different drying modes.
Fig. 6. Simulated distributions of binder content and temperature along the product vertical axis at the beginning of the process for different drying modes.
as concerns specifically the liquid transfer, the capillary flow should be promoted rather than filtration. Practically it means that: (1) the heating should start at low power to avoid the liquid water expulsion (pumping) and should be slowly increased afterward to limit gas pressure rise, (2) the material should not be initially saturated and have small pores to increase capillary pressure and to decrease the product permeability. These recommendations are evidently opposite to the ones aimed at a very fast drying rate and a compromise solution must be sought for to assure both the product mechanical cohesion and short drying cycles. Some preliminary simulations have clearly shown that high levels of temperature or micro-waves power led systematically to prohibited levels of internal gas pressure or binder loss. For that reason, only medium or low operating conditions were considered for the final analysis presented below. In order to find out a safe but as fast as possible drying process, combinations of hot air and micro-waves heating steps were applied. In this section, the simulation results for four drying schedules will be presented: (1) a simple one stage medium temperature hot air process (labeled HAM) which has been tried on a pilot dryer and leads sometimes to cracks, (2) a simple one step medium power micro-waves process (labeled MWM, already used for validation) which has proved to be quite cracks free but is very long,
Fig. 7. Simulated average water and binder content of the product versus time for different drying conditions.
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Fig. 8. Simulated internal gas phase pressure versus time for different drying conditions.
Fig. 10. Simulated binder content at the bottom (b) and at the surface (s) of the sample versus time for different drying conditions.
(3) a combined three stage process where micro-waves and hot air are applied in an alternate way starting with the micro-waves and finishing with hot air (labeled MWM-300/HAM-300/HAH400), (4) a combined four stage process where micro-waves and hot air are applied simultaneously from the beginning and with gradually increasing intensity, excepted the last stage where only hot air is applied (labeled MWL-HAL-240/MWM-HAL-180/MWMHAM-180/HAH-400).
but to speed-up the process on the other hand. As shown on Figs. 7 and 8, both the three step and four step combined processes allowed for reducing the drying time by 30% with an internal pressure practically unchanged (with regard to the reference process MWM). But the four step process led to a higher final binder content and lower binder content gradient as compared to the three step one and can be thus considered as safer. Besides, the curves on Figs. 7, 9 and 10 illustrate very well the specific features of binder transfer. As the binder is displaced with liquid water only, the liquid water content and binder content evolutions are closely linked during the period when the transfer in the liquid phase is prevalent, but they become independent when the transfer in the vapor phase will take over. In this case, at the end of the drying process, the binder remains trapped in the porous structure of the granular media and its content gets a nearly constant value.
The values of the operating parameters pertaining to each process are given in Table 2. The corresponding water (and binder) loss versus time curves are plotted on Fig. 7, the internal gas pressure curves are plotted on Fig. 8, the average binder content evolutions are plotted on Fig. 9 and the local (bottom and surface) binder content curves are plotted on Fig. 10. One can easily observe that the process HAM is the fastest but also has the highest internal pressure rise and the deepest binder depletion. The cracking risk associated to such a pressure and binder levels is too high as empirically demonstrated by tests on the pilot dryer (sample damage). The pressure and binder content levels corresponding to the cracking free process (MWM) were then considered as reference safe values and the operating conditions of the new combined processes were adjusted in a way to match these levels on one hand
4. Conclusions In order to simulate the combined hot air and micro-waves drying of agglomerated sand, a comprehensive internal heat and mass transfer model has been proposed. In this model, the internal gas phase pressure effect was made perfectly explicit, especially the phenomena of liquid and vapor transfer by filtration and of liquid expulsion at the surface. The internal transfer of the mineral binder by advection with the liquid phase was incorporated. This model was validated on the basis of the experimental mean water and binder content and core temperature curves for drying trials at different operating conditions. Then, it was used for comparing the drying time, the internal pressure and the binder content calculated for four selected processes: a strong process with hot air applied all over the time, a mild process with micro-waves applied all over (the reference one for product quality) and two processes which combined the two heating modes. It was demonstrated that the simultaneous application of hot air and micro-waves provided a real possibility to dry faster and to maintain quite the same internal overpressure and overall binder content as for the reference process. The heating power should be increased gradually during the process. The decrease of the drying time was around 30% with regard to the reference mild process.
Acknowledgments
Fig. 9. Simulated average binder content of the product versus time for different drying conditions.
This study was a part of the project ‘ACOIFF’ aimed at innovation in electrical fuses manufacturing and supported financially by the French Ministry of Industry (FUI grant) (082906467).
L. Hassini et al. / Powder Technology 286 (2015) 636–644 Pg
Appendix A The coefficients of the model presented in this paper are given below: m11 ¼ ρ0 Ceq p ; m14 ¼ Δhvap þ Δhdes ; m22 ¼ ρ0
m31
2 3 ∂Pv T −P þ P g v 7 ρ Ma 6 6 ∂T 7 ¼ − ϕ0 −X 0 5 ρl R 4 T2
m32 ¼ −
m33
Pv R t T X Y
643
Total pressure of the gas phase (air and vapor) within the product Water vapor pressure within the product Ideal gas constant Time Product temperature Product water content dry basis Product binder content dry basis
Greek symbols Δhdes Specific differential enthalpy of water desorption Δhvap Specific differential enthalpy of water vaporization ϕ0 Dry product total porosity φdiss Volumic thermal power dissipated by micro-waves within the product eq λ Equivalent thermal conductivity of the product νg Kinematic viscosity of the gas phase νl Kinematic viscosity of the liquid phase ρ0 Dry product apparent density ρl Liquid phase density
Ma ∂Pv ρ ρ ϕ0 −X 0 þ 0 Pg −Pv RT ∂X ρl ρl
ρ Ma ¼ ϕ0 −X 0 ρl RT
Pa J.mol−1.K−1 s °C, K kg.kg−1 kg.kg−1
J.kg−1 J.kg−1 W.m−3 W.m−1.°C−1 m−2.s−1 m−2.s−1 kg.m−3 kg.m−3
m44 ¼ 1; m55 ¼ ρ0 ; k11 ¼ λeq k21 ¼
Mv Pv RT
1−
k22 ¼
Mv Pv RT
1−
! Pv ðMv −Ma Þ ∂Pv ∂T Mv Pv þ Ma Pg −Pv
Appendix B The values and expressions of material properties used in the simulations are given in the Table below.
! eq Pv ðMv −Ma Þ ∂Pv X Kl ∂Pc − ∂X X þ Y νl ∂X Mv Pv þ Ma Pg −Pv eq
k23 ¼
k31 ¼ −
k32
Mv Deq v RT
Mv Deq v ¼− RT
1−
! Pv ðMv −Ma Þ ∂Pv ∂T Mv Pv þ Ma Pg −Pv
! Pv ðMv −Ma Þ ∂Pv 1− ∂X Mv Pv þ Ma Pg −Pv
k33 ¼
eq
K Mv Pv M Deq Pv Ma X Kl g − v v þ X þ Y νl RT Mv Pv þ Ma Pg −Pv Mv Pv þ Ma Pg −Pv νg
Keq g
Ma Pg −Pv þ Mv Pv þ Ma Pg −Pv νg
k41 ¼ −
k42 ¼ −
Mv Deq v RT Mv Deq v RT
Mv Deq v RT
Pv ðMv −Ma Þ ∂Pv ∂T Mv Pv þ Ma Pg −Pv Pv ðMv −Ma Þ ∂Pv ∂X Mv Pv þ Ma Pg −Pv eq
k43 ¼ −
K Mv Pv M Deq Pv Ma g þ v v RT Mv Pv þ Ma Pg −Pv Mv Pv þ Ma Pg −Pv νg
k51 ¼ −
Y Kl ∂Pc Y Kl ; k52 ¼ X þ Y νl ∂X X þ Y νl
eq
eq
The parameters Pc, Pv, Dv, Kl, Kg, λ, and ν are product specific functions of state variables X and T. Their expressions are given in Appendix B. The symbols used in the equations are given below. Ceq p Cpl Deq v h hm Keq g Keq l Ma Mv n Patm Pc
ϕ0 = 0.35 τ0 = 2 K0 = 1.10−3 m2 ρ0 = 1757 (kg/m3) ρl = 988 (kg/m3) Cpl=4185 (J/kg/K) Ceq p = 900 + 4185 X (J/kg/K)
Pv Ma Mv Pv þ Ma Pg −Pv
!
1−
Value or correlation
Total porosity of the dry product Tortuosity of the dry product Specific permeability of the product Apparent density of the dry product Density of the liquid phase Specific heat capacity of the liquid phase Equivalent specific heat capacity of the product Relative permeability of the liquid phase Relative permeability of the gas phase Capillary pressure in the product
!
1−
Material property
Equivalent specific heat capacity of the product Specific heat capacity of the liquid phase Equivalent water vapor diffusivity in the product Heat transfer coefficient at the product surface Mass transfer coefficient at the product surface Equivalent gas phase permeability of the product Equivalent liquid phase permeability of the product Air molar mass Water vapor molar mass Mass flux (w—liquid water, v—water vapor, b—binder) Atmospheric pressure Capillary pressure within the product
J.kg−1.K−1 J.kg−1.K−1 m2.s−1 W.m−2.K−1 m.s−1 m2 m2 kg.mol−1 Pa kg.s−1.m−2 Pa Pa
Equivalent conductivity of the product Equivalent water vapor diffusivity of the product Diffusivity of water vapor in air Specific differential enthalpy of water vaporization Specific differential enthalpy of water desorption Partial pressure of water vapor in the product Activity of water in the product Saturation pressure of water vapor Hygroscopic moisture content Saturation moisture content Kinematic viscosity of the liquid phase Kinematic viscosity of the gas phase Overall heat transfer coefficient at the shell surface Heat transfer coefficient at the product air opened surface Mass transfer coefficient at the product air opened surface Absorption coefficient for the water saturated product Absorption coefficient for the dry product (and for the shell) Initial water content of the product Initial binder content of the product Thermal diffusivity of the shell
Krl = Sγ[1 − (1 − S1/m)m]2m Krg = (1 − S)γ [1 − (1 − S1/m)m]2m Pc = (ρlg/α) (S−1/m − 1)1/n (Pa)
n = 5.2 m = 1−1/n g = 9.81 m.s−2 α = 0.029 γ = 0.25 S = X/Xsat
λeq = 0.9 (W/m/K) 2 Deq v = (ϕ0/τ0)KrlDva (m /s)
Dva = 4.343 ⋅ 10−11T2.334 (m2/s) Δhvap = 4187(597.3 − 0.592(T − 273) (J/kg) Δhdes = 30107 exp(−482.1859X) (J/kg) Pv = awPvsat (Pa) aw = (X/Xhyg)(2 − X/Xhyg) if (X ≤ Xhyg) else aw = 1 Pvsat = exp(25.270 − 5123/T) (Pa) Xhyg = 0.05 (kg/kg) Xsat = ϕ0(ρl/ρ0) (kg/kg) vl = 2.4 ⋅ 10−510(247.8/(T − 140))/ρl (m2/s) vg = 4.184 ⋅ 10−4T2.5/Pg(100 + T) (m2/s) h = 15 (W/m2/K) h = 8 (W/m2/K) h = 0.01 (m/s) αsat = 0.35 αsat = 0.15 Xi = 0.1 (kg/kg) Yi = 0.02 (kg/kg) 2.05 ⋅ 10−6 (m2/s)
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References [1] I.W. Turner, J.R. Puiggali, W. Jomaa, A numerical investigation of combined microwave and convective drying of a hygroscopic porous material: a study based on pine wood, Chem. Eng. Res. Des. 76 (2) (1998) 193–209. [2] P. Salagnac, P. Glouannec, D. Lecharpentier, Numerical modeling of heat and mass transfer in porous medium during combined hot air, infrared and microwaves drying, Int. J. Heat Mass Transf. 47 (2004) 4479–4489. [3] S.J. Kowalski, K. Rajewska, A. Rybicki, Stresses generated during convective and microwave drying, Dry. Technol. 23 (2005) 1875–1893. [4] A.A. Barba, A. Dalmoro, M. d'Amore, Microwave assisted drying of cellulose derivative (HPMC) granular solids, Powder Technol. 237 (2013) 581–585. [5] L. Malafronte, G. Lamberti, A.A. Barba, B. Raaholt, E. Holtz, L. Ahrné, Combined convective and microwave assisted drying: experiments and modeling, J. Food Eng. 112 (4) (2012) 304–312. [6] P. Perré, C. Moyne, Processes related to drying: part II: use of the same model to solve transfers in saturated and unsaturated porous media, Dry. Technol. 9 (5) (1991) 1153–1179.
[7] I.W. Turner, C. Jolly, Combined microwave and convective drying of a porous material, Dry. Technol. 9 (5) (1991) 1209–1269. [8] T. Constant, C. Moyne, P. Perré, Drying with internal heat generation: theoretical aspects and application to microwave heating, AICHE J. 42 (2) (1996) 359–368. [9] P. Perré, W. Turner, Microwave drying of softwood in an oversized waveguide: theory and experiment, AICHE J. 43 (10) (1997) 2579–2586. [10] S.J. Kowalski, B. Mielniczuk, Analysis of effectiveness and stress development during convective and microwave drying, Dry. Technol. 26 (1) (2007) 64–77. [11] L. Hassini, R. Peczalski, J.-L. Gelet, Combined convective and microwave drying of agglomerated sand. Internal transfer modeling with the gas pressure effect, Dry. Technol. 31 (8) (2013) 898–904. [12] G.C. Stangle, I.A. Aksay, Simultaneous momentum, heat and mass transfer with chemical reaction in a disordered porous medium: application to binder removal from a ceramic green body, Chem. Eng. Sci. 45 (7) (1990) 1719–1731. [13] M.R. Barone, J.C. Ulicny, Liquid-phase transport during removal of organic binders in injection-molded ceramics, J. Am. Ceram. Soc. 73 (11) (1990) 3323–3333. [14] D. Wildenschild, K.H. Jensen, Laboratory investigations of effective flow behaviour in unsaturated heterogeneous sands, Water Resour. Res. 35 (1) (1999) 17–27.
Contents lists available at ScienceDirect
Powder Technology journal homepage: www.elsevier.com/locate/powtec
Drying of granular medium by hot air and microwaves. Modeling and prediction of internal gas pressure and binder distribution Lamine Hassini a, Roman Peczalski b,⁎, Jean-Louis Gelet c a b c
University of Tunis El Manar, Faculté des Sciences de Tunis, Laboratoire d'Energétique et des Transferts Thermique et Massique (LETTM), Tunisia University of Lyon, Université Claude Bernard Lyon 1, Laboratoire d'Automatique et Génie des Procédés (LAGEP) UMR CNRS 5007, La Doua Campus, CPE Building, Villeurbanne, 69622, France MERSEN Company, Saint Bonnet de Mûre, France
a r t i c l e
i n f o
Article history: Received 8 March 2013 Received in revised form 31 August 2015 Accepted 5 September 2015 Available online 7 September 2015 Keywords: Drying Micro-waves Heat and mass transfer Porous granular medium Internal pressure Mineral binder
a b s t r a c t With the aim of enhancing the industrial drying (hot air and microwaves) of inserts made of agglomerated sand and to assess product quality, a comprehensive internal heat and mass transfer model has been proposed. In this model, liquid and vapor transfer by filtration, liquid expulsion at the surface and binder transfer by advection were directly accounted for. This model was validated on the basis of the experimental average water and binder content and core temperature curves for drying trials at different operating conditions. Then, it was used for comparing the drying time, the internal pressure and the binder content calculated for several test processes. It was demonstrated that the simultaneous application of hot air and micro-waves provided a real possibility to dry faster and to maintain quite the same internal overpressure and overall binder content as for the reference (damage free) process. The heating power should be increased gradually during the process. The decrease of the drying time was around 30% with regard to the reference mild process. © 2015 Elsevier B.V. All rights reserved.
1. Introduction 1.1. Industrial and scientific context The context of this study is the drying of wet agglomerated sand used as the filling material for high current industrial electrical fuses. The wet agglomerate is composed of calibrated sand and of the aqueous solution of a mineral binder (from the silicate family). This material is compacted and dried in a nearly parallelepipedic ceramic container (shell) which is roughly 10 cm high, 5 cm wide and 1 cm thick (see Fig. 1). During drying the container is opened to ambient air only by two small orifices on its top part, each of few millimeters diameter. For patent registration reasons, the exact composition and geometry of the product will not be revealed. The drying process is aimed at removing water almost entirely while preserving some significant amount of the binder. To accelerate the drying of the fuses' interiors, it is intended to substitute the standard hot air process by a new process with combined micro-waves and hot air heating. The great advantage of micro-waves heating as regards the convective or contact one is that the heat penetrates instantaneously and deeply into the product and that heat is ⁎ Corresponding author. E-mail address: [email protected] (R. Peczalski).
http://dx.doi.org/10.1016/j.powtec.2015.09.009 0032-5910/© 2015 Elsevier B.V. All rights reserved.
dissipated mainly in the wet part of the product [1]. In this way, the core temperature of a product dried by micro-waves is higher than its surface temperature and can quickly rise over the water boiling point [2]. The inversion of the temperature gradient enables the vapor thermo-diffusion to be directed from the core to the surface alike the ordinary diffusion [3]. The intense water vaporization in the core of the product induces a strong raise of the internal gas phase (vapor and air) pressure, which pressure literally pushes the water within and out of the product. The water transfer by filtration dominates then the transfer by diffusion and that increases very significantly the drying rate. At high micro-waves power, the internal pressure may be high enough to expulse water directly in the liquid phase at the product surface (‘water pumping’) providing a kind of mechanical dewatering with non energy consumption for evaporation. All of these phenomena lead to substantially higher water fluxes from the core to the product surface and thus to higher drying rates than in a purely convective process for the same average product temperature. Barba et al. [4] have experimentally studied the microwaves assisted drying of cellulose derivative (HPMC) powder. Using a customized microwaves power protocol where the power was decreased as the product water content decreased, they obtained drying kinetics as fast as for high temperature convective process, while keeping internal product temperature at a lower safe level which is of high importance for thermo-sensitive pharmaceutical products.
L. Hassini et al. / Powder Technology 286 (2015) 636–644
Fig. 1. Layout of the product within the container (axial section).
As concerns process modeling, the classical ‘water diffusion’ approach would be acceptable only in the case of soft drying conditions. Malafronte et al. [5] proposed a model for the combined convectivemicrowave assisted drying process of food (potatoes). The model accounted for the dependence of dielectric properties on product's temperature and water content and coupled the heat and mass transport equations with the Maxwell's electromagnetic field equation in transient regime. As concerns water transfer, only liquid and vapor diffusion were considered. No water expulsion at the sample surface was reported. This can be explained by the fact that for the drying conditions considered the core sample temperature did not exceed 80 °C (so was below the boiling point of water) and the gas pressure build-up within the sample and the consecutive internal liquid water flow were insignificant. For the above mentioned effects may be accounted of, substantial modifications of a classical diffusive kind drying model are required. First, the water transfer due to gas pressure gradient must be included in the model. Perré and Moyne [6] and independently Turner and Joly [7] were the first to propose this. Second, the boundary conditions must be adapted in a way enabling the liquid water expulsion. This adaptation is generally omitted or not explicated in the literature and at our best knowledge only Constant et al. [8] considered it and gave some precisions about its implementation. The adjunction of a micro-waves source in the dryer can reduce the processing times of clay [3] or wood [9] up to 50%. However, the important internal overpressure which exerts a stress on the material can be a cause of cracking and a compromise must be sought between the drying rate and the pressure raise while defining the operating conditions. In the case of clay drying, Kowalski and Mielniczuk [10] had suggested that the micro-waves should be applied only in the first stage of drying when the product is nearly saturated and that hot air should be applied in the second period where the product is already partially dry. They calculated that for this combined process the stress level was much lower than for a purely micro-waves process with a similar processing time. This observation was the start point of our first study [11] which concerned only heat and water internal transport modeling and was focused on possible product damage due to the gas phase overpressure.
637
However, in the case of mineral agglomerate considered here, there is an additional issue concerning the binder's distribution. The mineral binder secures the mechanical strength of the final product. Its content must not be reduced below a critical value needed for a particle to stick each to other after drying. The problem is that the binder is dissolved in water and, as water is removed during drying, some part of the binder is removed either. This is particularly true for micro-wave assisted or very hot air drying when the water is literally expulsed in the liquid phase during the first period of the process. This leads to very high water removal rates but also binder depletion rates. The proper rate at which the drying will be driven by means of supplied heating power must then be dictated by the binder content evolution within the product. This is the main subject of the present study and to our best knowledge there are no scientific publications on this problem. As concerns more generally the issue of thermal extraction of liquid from granular media, several studies were devoted to the removal of a thermoplastic organic binder from an injection molded ceramic compact prior to sintering [12,13]. During the thermal processing, the organic binder melts and the volatile components evaporate thus enabling internal transfer in the liquid and gaseous phase. In a study concerning a composite made of alumina particles and paraffin wax (pentacosane) submitted to high temperature processing (over 150 °C), Stangle and Aksay [12] had developed a comprehensive internal transfer model very similar to the one used in this study. However due to the low vapor tension of the binder, the evaporative and transfer phenomena driven by the gaseous phase pressure were much less important than in the case of the removal of an aqueous solution considered in this study and the results were not really comparable. 1.2. Aim of this study The goal was to predict the product state evolution for different combined micro-waves and hot air drying schedules and to find out the best operating conditions ensuring a fast water removal with non material cracking. The product damage was supposed to be entirely due to the gas phase (vapor and air) overpressure and to the mineral binder depletion. The risk associated with binder depletion was evaluated by considering the final average binder content and by considering the binder content heterogeneity i.e. the difference between the bottom and surface content in the sample. The work was carried out in three steps. First, an internal heat and mass transfer model was developed. In this model, the internal gas phase pressure effect was made perfectly explicit, especially the phenomena of liquid and vapor transfer by filtration and of liquid expulsion at the surface. A special equation was introduced describing the advective binder transfer within the product. A particular attention was also paid to the boundary conditions (see next section). The boundary conditions accounting for water expulsion were already introduced in [11] but the binder transfer description is an original contribution of this study. Second, the model was validated by comparing experimental and simulated average global water and binder content and core temperature curves. The drying trials were realized on a specifically designed dual heating mode (micro-waves and hot air) pilot dryer. The simulations were carried out using COMSOL ‘Multiphysics’ software. Finally, the model was applied for testing different drying schedules. The simulations were realized for two simple single heating mode processes and two combined dual heating mode processes. It will be shown that a simultaneous application of micro-waves and hot air allows for process speeding up without increasing the internal pressure or decreasing the binder content as compared to a safe reference process. As compared to the exploratory simulations presented in [11], in this paper more complex schedules with lower micro-waves power input were investigated which are closer to those recommended finally for manufacturing practice. These simulation results are original contributions of this study.
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(5) The heat is transferred by conduction and advection associated to the liquid phase flow (the heat advected with water vapor is neglected). (6) The vapor moves by filtration due to the gas phase pressure gradient, by thermo-diffusion due to temperature gradient and by ordinary diffusion due to the water content gradient. (7) If the surface is saturated, the liquid phase (water and binder) leaves the product directly, without water evaporation, being expulsed by the gas phase overpressure. (8) The capillary liquid water flow evaporates at the surface and leaves the product by convection. The water vapor leaves the product directly by convection.
2. Methods 2.1. Dual heating mode pilot dryer For the purpose of this study, a drying oven was specifically designed and realized. This oven consisted of: a parallelepipedic chamber with stainless steel walls (730 mm wide, 610 mm high and 610 mm deep); a micro-waves generator of 2450 Mhz frequency and with a power tuneable up to 1.2 kW; a hot air generator of 5 kW maximal power, with a temperature adjustable up to 200 °C and a velocity up to 3 m/s (in the center of the chamber); a circular plate (sample holder) made of mecanite (500 mm wide) rotated by an electric motor and suspended by a wire to an electronic balance (range of 15 kg, precision of 1 g) fixed on the top of the chamber; three optical temperature probes, two inserted in the samples and one for air temperature control; and a hygrometer. During periodic weighing, rotation of the plate and air blowing were stopped. The setting of operating conditions within the chamber and the acquisition of the mass and the temperature of the product were operated by an industrial programmable controller. The concept outline of the dual mode dryer is drawn on Fig. 2. This dryer was used for experimental trials needed for model validation. 2.2. Internal transfer model The internal heat and mass transfer model was based on the equations of Constant et al. [8] and Perré and Turner [9]. For the purpose of this study, the mineral binder transfer equation was added and the water transfer equations and corresponding boundary conditions were modified. The final general form of the equations' system is given in Section 2.2.2. A separate Section (2.2.3) is devoted to the boundary conditions which deserve special attention because of the ‘water pumping’ phenomenon. The main model assumptions are given below: (1) The material is multiphase and heterogeneous at the microscopic scale but is represented by an equivalent homogeneous medium with continuously varying state variables and macroscopically defined properties. (2) The different phases are in local thermodynamic equilibrium as concerns the temperature and vapor pressure. (3) The liquid phase (solution of binder in water) flows by filtration due to the gas phase pressure gradient and by capillarity due to the liquid water content gradient (the influence of the temperature gradient on the capillary flow is neglected). (4) The liquid water and binder specific fluxes are purely advective ones associated to the liquid phase flow at the pro rata of their respective fractions (the binder ordinary diffusion in water is neglected).
2.2.1. Binder and water transfer In the present case the mineral binder is dissolved in liquid water and the resulting solution is the liquid phase of the granular medium. This liquid phase flows by filtration due to the gas phase pressure (Pg) gradient and by capillarity due mainly to the liquid water content (X) gradient (the influence of the temperature gradient on the capillary flow is neglected). The liquid flux is thus given by: nl ¼ −
Keq Keq Pc l gradX gradPg þ l νl νl X
ð1Þ
where Kl is the product liquid permeability, νl is the liquid viscosity, Pc is the capillary pressure (a function of the water content X). If the binder ordinary diffusion in water is neglected, the liquid water and binder specific fluxes are purely advective ones associated to the liquid phase flow at the pro rata of their respective fractions. These fluxes are thus respectively given by: nw ¼
X n XþY l
ð2Þ
nb ¼
Y n XþY l
ð3Þ
The transfer equations are obtained by inserting the above fluxes' expressions in the local mass balance equations and are written as follows: ρ0
∂X ¼ −divðnw þ nv Þ ∂t
ð4Þ
ρ0
∂Y ¼ −divðnb Þ ∂t
ð5Þ
In Eq. (4) nv is the water vapor flux. The vapor is supposed to move by filtration due to the gas phase pressure gradient, by thermo-diffusion due to temperature gradient and by ordinary diffusion due to the water content gradient. This flux is hence given by: nv ¼ k41 gradT þ k42 gradX þ k43 gradPg
ð6Þ
In order to reduce the length of this equation, the numerous physical parameters representing the medium properties were grouped into global coefficients (kmn) that are given in Appendix A.
Fig. 2. Layout of the dual mode micro-waves and hot air pilot drying oven.
2.2.2. Full equations' system The model consists finally of five PDEs: the heat transfer equation determining the temperature (T), the liquid water transfer equation determining the liquid water content (X), the dry air transfer equation determining the gas pressure (Pg), the water vapor transfer equation determining the mass of liquid water vaporized per unit volume (mvap) and the mineral binder transfer equation determining the binder
L. Hassini et al. / Powder Technology 286 (2015) 636–644
content (Y). The full set of equations describing the heat and water transfer within the product are written as: Heat transfer m11
∂T ∂mvap ¼ divðk11 gradTÞ−nl Cpl gradT þ φdiss þ m14 ∂t ∂t
ð7Þ
Liquid water transfer m22
∂X ¼ div k21 gradT þ k22 gradX þ k23 gradPg ∂t
ð8Þ
Dry air transfer ∂Pg ∂T ∂X þ m32 þ m33 ¼ div k31 gradT þ k32 gradX þ k33 gradPg m31 ∂t ∂t ∂t ð9Þ
Water vapor transfer m44
∂mvap ¼ div k41 gradT þ k42 gradX þ k43 gradPg ∂t
ð10Þ
Binder transfer m55
∂Y ¼ div k51 gradX þ k52 gradPg ∂t
ð11Þ
As stipulated in the introduction section, the gas pressure gradient appears clearly as one of the transfer driving forces. The influence of the binder amount on the volume of the void space available for gas transfer (effective porosity) and thus on the gas concentration within the product (Eq. (9)) was neglected. The term ∂mvap/∂t in Eqs. (7) and (10) represents the liquid to vapor phase change rate term. This term is not the accumulation term for vapor mass. The accumulation of vapor mass in a control volume was neglected. The exact expressions of the m (‘capacities’) and k (‘conductances’) coefficients pertaining to the above five transfer equations are given in Appendix A. The volumic thermal power generated within the product by dissipation of micro-waves energy was simply calculated by a semiempirical formula: φdiss ¼ αsat
Φmw X Φmw þ αdry V Xsat V
ð12Þ
where Φmw is the power of the micro-waves generator, V is the sample volume, αsat is the absorption coefficient measured for the water saturated product, Xsat is the saturated product water content and αdry is the absorption coefficient measured for the dry product (the same value is applied for the ceramic shell). Eq. (12) is based on the assumption that the electric field is uniform within the product and that the wet product has a dissipation factor which is directly proportional to the water content and that the dry product has a constant dissipation factor. A similar approach was used by Constant et al. [8] and Kowalski et al. [3]. Eqs. ((7) to (11)) are not only coupled by the gradients of the state variables (X, T and Pg) but also by the m and k coefficients. These coefficients are in fact combinations of product properties (material densities, thermal conductivities, fluid permeabilities, species diffusivities) which depend on the state variables. The semiempirical expressions describing these properties have to be known with some precision and be consistent with each other in order to obtain realistic results without numerical problems. This concerns particularly the expressions of gas and liquid relative permeabilities [2], of the capillary suction and of the sorption isotherm. For the purpose
639
of this study, the hydraulic product properties (capillary suction, liquid permeability) were experimentally determined and fitted by an empirical model [14]. The gas phase transport properties (gas permeability, vapor pressure and diffusivity) were adapted from the literature for similar materials [2,8]. The values or expressions for all relevant material properties used in the simulations are given in Appendix B. The equations of our model were solved by means of the ‘COMSOL Multiphysics’ software (v. 3.5) but using the open ‘PDE coefficient form' mode and not the predefined heat and mass transfer modules. The calculation domain included the product and its shell (see next section for further details). 2.2.3. Boundary conditions The product (agglomerated sand) is confined in a ceramic shell (see Fig. 1). The shell is air tight excepted two circular openings at the top. A perfect thermal contact was assumed on all product faces in contact with the shell which is directly exposed to the hot air flow. The heat transfer within the shell was described by a classical heat diffusion equation with convective boundary conditions and with a volumic heat source due to dissipation of micro-waves energy. In this section, a focus on the mass and heat transfer at the product surface in direct contact with the air (in the holes) will be done as it is one of the original elements of our model. In order to account for the important phenomenon of liquid water expulsion (also called ‘water pumping’ in the literature), the water balance equation at the opened product surface was written in the following way:
ðnw þ nv Þjsurf ¼ hm
Keq Patm Mv Patm −Pva Xsurf l ∂Pg ln − RTa Patm −Pvsurf Xsurf þ Ysurf νl ∂n surf ð13Þ
where n is the water flux (w for liquid, v for vapor) arriving at the surface (see Eqs. (2) and (3)), hm is the convective mass transfer coefficient, Kl is the product liquid permeability, νl is the liquid viscosity, Pv is the vapor pressure (a for air, surf for surface), and Ta is the air temperature. The novelty of this boundary equation lies in the second term on the right side which represents the liquid water flux arriving at the surface by filtration due to gas pressure gradient. Including this term means that liquid water arriving at the surface by filtration leaves the product directly as liquid. Consequently, the liquid water arriving by capillary flow in fact evaporates at the surface and leaves the product by convection (diffusion in the boundary layer) as vapor and evidently the water vapor arriving at the surface leaves the product by convection in the same form. The additional boundary liquid filtration (water expulsion) term is effective if the gas pressure in the product sub surface layer is higher than the atmospheric pressure and if its water content is close to saturation. Constant et al. [8] have suggested a similar approach formulating a criterion for water expulsion by means of the liquid pressure (Pl = Pg−Pc) and not the gas pressure. However, if the water content approaches the saturation limit, the capillary pressure tends to zero and the two formulations are equivalent. Besides, in reference [8], the mathematical expression of the flux of expulsed liquid water is not explicitly given. An additional boundary condition is needed for the binder transfer equation. It is written as:
nb jsurf ¼ −
Keq Ysurf l ∂Pg Xsurf þ Ysurf νl ∂n surf
ð14Þ
This formula stands that the only way for the binder to leave the product is to be washed out by liquid water, the latter being expulsed by the gas phase overpressure.
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According to the above water transfer boundary condition (Eq. (13)), the heat balance equation at the product free boundary was written as: ∂T þ Cpl ðTsurf −Tre f Þnl surf ∂n surf Keq Xsurf l ∂Pc ∂X ¼ Cpl ðTsurf −Tre f Þ þ Δhvap Xsurf þ Ysurf νl ∂X ∂nsurf Keq ∂Pg þ Cpl ðTsurf −Tre f Þ l − hðTa −Tsurf Þ νl ∂n surf
−λeq
ð15Þ
where λ is the product thermal conductivity, nl is the total (capillary flow and filtration) liquid phase (water and binder) flux arriving at the surface (the heat advected with water vapor was neglected), Cpl is the specific heat of the liquid phase, and h is the convective heat transfer coefficient at the product surface. This equation separates clearly the heat flux due to evaporation of capillary water (first term on the left side) and the heat flux due to liquid water expulsion (second term on the left side). For the sake of completeness, the boundary condition for the gas transfer has to be mentioned. This condition stands simply that the gas pressure at the product surface equals the atmospheric pressure. 3. Results and discussion 3.1. Experimental curves and model validation In order to check the validity of our model, the experimental and simulated drying curves were compared for the three following cases: 1) a high air temperature process labeled HAH; 2) a medium microwaves power process labeled MWM, and 3) a high micro-wave power process labeled MWH. The values of the operating parameters pertaining to each process are given in Table 1. The average water (and binder) content of the product versus time curves are plotted on Fig. 3 and the core temperature ones on Fig. 4. The experimental data were processed by a moving average filter what explains the smoothness of the curves. In general, the simulated curves corresponded well to the experimental ones and the model could be considered valid. Some minor but noticeable differences should be attributed to the low accuracy of the semi-empirical expression representing the micro-waves energy dissipated within the product. It must be emphasized that with a classical formulation of boundary conditions (without the liquid filtration flux) for water transfer, it was not possible to obtain an agreement between experience and simulation for high temperature or high microwaves power even by artificially far increasing the superficial mass transfer coefficient (hm). In order to illustrate this point, the liquid content curves calculated with classical boundary conditions (Eqs. (13) and (15) without the second term on the right hand side) and a usual value (given in Appendix B) of the hm coefficient were also presented on Fig. 3 (see the dotted lines). As can be observed, the simulated drying times were roughly multiplied by 4 in the case of strong operating conditions (MWH and HAH) where the water filtration and expulsion in liquid phase were actually a major contribution to overall water elimination.
Fig. 3. Average water and binder content of the product versus time for different drying conditions. The gray curves correspond to experimental data, the black continuous curves correspond to simulations realized considering water expulsion at the product opened boundary, and the dotted black curves correspond to simulations without water expulsion.
One can observe that for significant water removal acceleration a high temperature or a high micro-waves power have to be applied. As shown on Fig. 4, these intensive drying conditions correspond to internal temperature raise up to and beyond the boiling temperature (100 °C) and consecutive internal pressure build-up and water transfer by filtration accompanied by liquid expulsion at the surface. The high power micro-waves process allowed for faster drying than the high temperature one, this lasts being the present industrial practice. But the micro-waves power must be moderated to avoid too much overpressure within the sample or too much binder expulsion (see next section). One can also observe on Fig. 3 that the drying rate is truly low for the medium micro-waves power process. The surprisingly slow drying for moderate operating conditions is due to the particular geometry of the electrical fuse considered here. In fact, for technological reasons, the agglomerated sand is enclosed in a tight container which is opened only by two small orifices on its top part. The surface area for mass exchange with ambient air is thus extremely small and it is not possible to dry this product in reasonable time without using very intensive conditions like micro-waves or high temperature convection. 3.2. Instant view of internal heat and mass transfer In order to highlight the differences between simple hot air drying (process HAM, see Table 2) and simple micro-waves drying (process MWM, see Table 2), the spatial profiles of the state variables (X, Pg, T,
Table 1 Operating conditions for model validation. Ta, °C
RH, %
Φmw, W
Label
2
0
HAH
19
400
MWM
18
800
MWH
160 50 50
Fig. 4. Core temperature of the product versus time, measured (gray curves) and simulated (black curves) for different drying conditions.
L. Hassini et al. / Powder Technology 286 (2015) 636–644
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Table 2 Operating conditions for exploratory simulations.
1 2 3 1 2 3 4
Δt, min
Ta, °C
Φmw, W
Label
1000 1000 300 300 400 240 180 180 400
120 50 50 120 160 80 80 120 160
0 400 400 0 0 200 400 400 0
HAM MWM MWM-300/HAM-300/HAH-400
MWL-HAL-240/MWM-HAL180/MWM-HAM-180/HAH-400
Y) for these two processes are compared on Figs. 5 and 6. The processing time is 5000 s which corresponds to the early stage of drying when the product is still quite wet and water is moving essentially in the liquid state. The profiles are plotted within the product's sample limits along a vertical axis z passing through the holes in the shell, so that the maximal coordinate z corresponds to the product surface in contact with ambient air (free boundary). The temperature profile governs mainly the gas phase pressure profile (by the contribution of vapor saturation pressure) and temperature maximums coincide with the pressure ones. The gas phase pressure maximum is situated at the bottom of the sample for process MWM while it is situated near the top of the sample for process HAM (see Fig. 5). The liquid water is pushed away from this pressure maximum so that in this early stage of drying the water content is higher at the top than at the bottom. Consequentially the top layer is enriched in binder, especially for process HAM, while the core is depleted. The bottom temperature is higher for the MWM process than for the HAM one due to the uniformity of dielectric heating (see Fig. 6). The important temperature drop at the top boundary of the sample (free boundary) is due to intense evaporation of capillary liquid water. 3.3. Exploratory simulations of combined processes It is reminded that in our approach the product quality was assessed by means of the following parameters: the internal gas pressure, the binder content and the binder distribution. Excessive pressure rise or binder depletion may cause the fracture of the product. Some requirements for a process that guarantees an acceptable final binder content can be formulated on a theoretical basis. According to our model, the binder flow within the product is strictly linked to the liquid phase diffusion (capillary flow) and filtration within the product and the binder flow out of the product is exclusively linked to the liquid phase filtration. In order to slow down the binder removal, the transfer in the gas phase should be promoted rather than in the liquid phase and
Fig. 5. Simulated distributions of water content and gas phase pressure along the product vertical axis at the beginning of the process for different drying modes.
Fig. 6. Simulated distributions of binder content and temperature along the product vertical axis at the beginning of the process for different drying modes.
as concerns specifically the liquid transfer, the capillary flow should be promoted rather than filtration. Practically it means that: (1) the heating should start at low power to avoid the liquid water expulsion (pumping) and should be slowly increased afterward to limit gas pressure rise, (2) the material should not be initially saturated and have small pores to increase capillary pressure and to decrease the product permeability. These recommendations are evidently opposite to the ones aimed at a very fast drying rate and a compromise solution must be sought for to assure both the product mechanical cohesion and short drying cycles. Some preliminary simulations have clearly shown that high levels of temperature or micro-waves power led systematically to prohibited levels of internal gas pressure or binder loss. For that reason, only medium or low operating conditions were considered for the final analysis presented below. In order to find out a safe but as fast as possible drying process, combinations of hot air and micro-waves heating steps were applied. In this section, the simulation results for four drying schedules will be presented: (1) a simple one stage medium temperature hot air process (labeled HAM) which has been tried on a pilot dryer and leads sometimes to cracks, (2) a simple one step medium power micro-waves process (labeled MWM, already used for validation) which has proved to be quite cracks free but is very long,
Fig. 7. Simulated average water and binder content of the product versus time for different drying conditions.
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Fig. 8. Simulated internal gas phase pressure versus time for different drying conditions.
Fig. 10. Simulated binder content at the bottom (b) and at the surface (s) of the sample versus time for different drying conditions.
(3) a combined three stage process where micro-waves and hot air are applied in an alternate way starting with the micro-waves and finishing with hot air (labeled MWM-300/HAM-300/HAH400), (4) a combined four stage process where micro-waves and hot air are applied simultaneously from the beginning and with gradually increasing intensity, excepted the last stage where only hot air is applied (labeled MWL-HAL-240/MWM-HAL-180/MWMHAM-180/HAH-400).
but to speed-up the process on the other hand. As shown on Figs. 7 and 8, both the three step and four step combined processes allowed for reducing the drying time by 30% with an internal pressure practically unchanged (with regard to the reference process MWM). But the four step process led to a higher final binder content and lower binder content gradient as compared to the three step one and can be thus considered as safer. Besides, the curves on Figs. 7, 9 and 10 illustrate very well the specific features of binder transfer. As the binder is displaced with liquid water only, the liquid water content and binder content evolutions are closely linked during the period when the transfer in the liquid phase is prevalent, but they become independent when the transfer in the vapor phase will take over. In this case, at the end of the drying process, the binder remains trapped in the porous structure of the granular media and its content gets a nearly constant value.
The values of the operating parameters pertaining to each process are given in Table 2. The corresponding water (and binder) loss versus time curves are plotted on Fig. 7, the internal gas pressure curves are plotted on Fig. 8, the average binder content evolutions are plotted on Fig. 9 and the local (bottom and surface) binder content curves are plotted on Fig. 10. One can easily observe that the process HAM is the fastest but also has the highest internal pressure rise and the deepest binder depletion. The cracking risk associated to such a pressure and binder levels is too high as empirically demonstrated by tests on the pilot dryer (sample damage). The pressure and binder content levels corresponding to the cracking free process (MWM) were then considered as reference safe values and the operating conditions of the new combined processes were adjusted in a way to match these levels on one hand
4. Conclusions In order to simulate the combined hot air and micro-waves drying of agglomerated sand, a comprehensive internal heat and mass transfer model has been proposed. In this model, the internal gas phase pressure effect was made perfectly explicit, especially the phenomena of liquid and vapor transfer by filtration and of liquid expulsion at the surface. The internal transfer of the mineral binder by advection with the liquid phase was incorporated. This model was validated on the basis of the experimental mean water and binder content and core temperature curves for drying trials at different operating conditions. Then, it was used for comparing the drying time, the internal pressure and the binder content calculated for four selected processes: a strong process with hot air applied all over the time, a mild process with micro-waves applied all over (the reference one for product quality) and two processes which combined the two heating modes. It was demonstrated that the simultaneous application of hot air and micro-waves provided a real possibility to dry faster and to maintain quite the same internal overpressure and overall binder content as for the reference process. The heating power should be increased gradually during the process. The decrease of the drying time was around 30% with regard to the reference mild process.
Acknowledgments
Fig. 9. Simulated average binder content of the product versus time for different drying conditions.
This study was a part of the project ‘ACOIFF’ aimed at innovation in electrical fuses manufacturing and supported financially by the French Ministry of Industry (FUI grant) (082906467).
L. Hassini et al. / Powder Technology 286 (2015) 636–644 Pg
Appendix A The coefficients of the model presented in this paper are given below: m11 ¼ ρ0 Ceq p ; m14 ¼ Δhvap þ Δhdes ; m22 ¼ ρ0
m31
2 3 ∂Pv T −P þ P g v 7 ρ Ma 6 6 ∂T 7 ¼ − ϕ0 −X 0 5 ρl R 4 T2
m32 ¼ −
m33
Pv R t T X Y
643
Total pressure of the gas phase (air and vapor) within the product Water vapor pressure within the product Ideal gas constant Time Product temperature Product water content dry basis Product binder content dry basis
Greek symbols Δhdes Specific differential enthalpy of water desorption Δhvap Specific differential enthalpy of water vaporization ϕ0 Dry product total porosity φdiss Volumic thermal power dissipated by micro-waves within the product eq λ Equivalent thermal conductivity of the product νg Kinematic viscosity of the gas phase νl Kinematic viscosity of the liquid phase ρ0 Dry product apparent density ρl Liquid phase density
Ma ∂Pv ρ ρ ϕ0 −X 0 þ 0 Pg −Pv RT ∂X ρl ρl
ρ Ma ¼ ϕ0 −X 0 ρl RT
Pa J.mol−1.K−1 s °C, K kg.kg−1 kg.kg−1
J.kg−1 J.kg−1 W.m−3 W.m−1.°C−1 m−2.s−1 m−2.s−1 kg.m−3 kg.m−3
m44 ¼ 1; m55 ¼ ρ0 ; k11 ¼ λeq k21 ¼
Mv Pv RT
1−
k22 ¼
Mv Pv RT
1−
! Pv ðMv −Ma Þ ∂Pv ∂T Mv Pv þ Ma Pg −Pv
Appendix B The values and expressions of material properties used in the simulations are given in the Table below.
! eq Pv ðMv −Ma Þ ∂Pv X Kl ∂Pc − ∂X X þ Y νl ∂X Mv Pv þ Ma Pg −Pv eq
k23 ¼
k31 ¼ −
k32
Mv Deq v RT
Mv Deq v ¼− RT
1−
! Pv ðMv −Ma Þ ∂Pv ∂T Mv Pv þ Ma Pg −Pv
! Pv ðMv −Ma Þ ∂Pv 1− ∂X Mv Pv þ Ma Pg −Pv
k33 ¼
eq
K Mv Pv M Deq Pv Ma X Kl g − v v þ X þ Y νl RT Mv Pv þ Ma Pg −Pv Mv Pv þ Ma Pg −Pv νg
Keq g
Ma Pg −Pv þ Mv Pv þ Ma Pg −Pv νg
k41 ¼ −
k42 ¼ −
Mv Deq v RT Mv Deq v RT
Mv Deq v RT
Pv ðMv −Ma Þ ∂Pv ∂T Mv Pv þ Ma Pg −Pv Pv ðMv −Ma Þ ∂Pv ∂X Mv Pv þ Ma Pg −Pv eq
k43 ¼ −
K Mv Pv M Deq Pv Ma g þ v v RT Mv Pv þ Ma Pg −Pv Mv Pv þ Ma Pg −Pv νg
k51 ¼ −
Y Kl ∂Pc Y Kl ; k52 ¼ X þ Y νl ∂X X þ Y νl
eq
eq
The parameters Pc, Pv, Dv, Kl, Kg, λ, and ν are product specific functions of state variables X and T. Their expressions are given in Appendix B. The symbols used in the equations are given below. Ceq p Cpl Deq v h hm Keq g Keq l Ma Mv n Patm Pc
ϕ0 = 0.35 τ0 = 2 K0 = 1.10−3 m2 ρ0 = 1757 (kg/m3) ρl = 988 (kg/m3) Cpl=4185 (J/kg/K) Ceq p = 900 + 4185 X (J/kg/K)
Pv Ma Mv Pv þ Ma Pg −Pv
!
1−
Value or correlation
Total porosity of the dry product Tortuosity of the dry product Specific permeability of the product Apparent density of the dry product Density of the liquid phase Specific heat capacity of the liquid phase Equivalent specific heat capacity of the product Relative permeability of the liquid phase Relative permeability of the gas phase Capillary pressure in the product
!
1−
Material property
Equivalent specific heat capacity of the product Specific heat capacity of the liquid phase Equivalent water vapor diffusivity in the product Heat transfer coefficient at the product surface Mass transfer coefficient at the product surface Equivalent gas phase permeability of the product Equivalent liquid phase permeability of the product Air molar mass Water vapor molar mass Mass flux (w—liquid water, v—water vapor, b—binder) Atmospheric pressure Capillary pressure within the product
J.kg−1.K−1 J.kg−1.K−1 m2.s−1 W.m−2.K−1 m.s−1 m2 m2 kg.mol−1 Pa kg.s−1.m−2 Pa Pa
Equivalent conductivity of the product Equivalent water vapor diffusivity of the product Diffusivity of water vapor in air Specific differential enthalpy of water vaporization Specific differential enthalpy of water desorption Partial pressure of water vapor in the product Activity of water in the product Saturation pressure of water vapor Hygroscopic moisture content Saturation moisture content Kinematic viscosity of the liquid phase Kinematic viscosity of the gas phase Overall heat transfer coefficient at the shell surface Heat transfer coefficient at the product air opened surface Mass transfer coefficient at the product air opened surface Absorption coefficient for the water saturated product Absorption coefficient for the dry product (and for the shell) Initial water content of the product Initial binder content of the product Thermal diffusivity of the shell
Krl = Sγ[1 − (1 − S1/m)m]2m Krg = (1 − S)γ [1 − (1 − S1/m)m]2m Pc = (ρlg/α) (S−1/m − 1)1/n (Pa)
n = 5.2 m = 1−1/n g = 9.81 m.s−2 α = 0.029 γ = 0.25 S = X/Xsat
λeq = 0.9 (W/m/K) 2 Deq v = (ϕ0/τ0)KrlDva (m /s)
Dva = 4.343 ⋅ 10−11T2.334 (m2/s) Δhvap = 4187(597.3 − 0.592(T − 273) (J/kg) Δhdes = 30107 exp(−482.1859X) (J/kg) Pv = awPvsat (Pa) aw = (X/Xhyg)(2 − X/Xhyg) if (X ≤ Xhyg) else aw = 1 Pvsat = exp(25.270 − 5123/T) (Pa) Xhyg = 0.05 (kg/kg) Xsat = ϕ0(ρl/ρ0) (kg/kg) vl = 2.4 ⋅ 10−510(247.8/(T − 140))/ρl (m2/s) vg = 4.184 ⋅ 10−4T2.5/Pg(100 + T) (m2/s) h = 15 (W/m2/K) h = 8 (W/m2/K) h = 0.01 (m/s) αsat = 0.35 αsat = 0.15 Xi = 0.1 (kg/kg) Yi = 0.02 (kg/kg) 2.05 ⋅ 10−6 (m2/s)
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