ZLG-0.4型試驗(yàn)用凍干機(jī)的設(shè)計(jì)【含14張CAD圖紙】

資源目錄里展示的全都有，所見即所得。下載后全都有,請(qǐng)放心下載。原稿可自行編輯修改=【QQ：401339828 或11970985 有疑問可加】 Transp Porous Med (2007) 66:5976DOI 10.1007/s11242-006-9022-2ORIGINAL PAPERMultiscale aspects of heat and mass transfer duringdryingPatrick PerrReceived: 30 November 2005 / Accepted: 26 March 2006 /Published online: 30 August 2006 Springer Science+Business Media B.V. 2006AbstractThe macroscopic formulation of coupled heat and mass transfer has beenwidely used during the past two decades to model and simulate the drying of onesingle piece of product, including the case of internal vaporization. However, moreoftenthanexpected,themacroscopicapproachfailsandseveralscaleshavetobecon-sidered at the same time. This paper is devoted to multiscale approaches to transfer inporousmedia,withparticularattentiontodrying.Thechangeofscale,namelyhomog-enization, is presented first and used as a generic approach able to supply parametervalues to the macroscopic formulation. The need for a real multiscale approach isthen exemplified by some experimental observations. Such an approach is requiredas soon as thermodynamic equilibrium is not ensured at the microscopic scale. Astepwise presentation is proposed to formulate such situations.KeywordsChange of scale Computational model Drying Dual scale Homogenization Porous media Wood1 IntroductionThis paper focuses on multiscale modeling of coupled transfer in porous media. Now-adays, the comprehensive set of equations governing these phenomena at the macro-scopicleveliswellknownandhasbeenwidelyusedtosimulateseveralconfigurations,particularlythedryingprocess.However,thismacroscopicdescriptionhassomedraw-backs: it generates a dramatic demand in physical and mechanical characterizationandfailsinsome,notespeciallyunusualconfigurations.Thesedrawbacksareprobablythe main motivation for multiscale approaches. Different strategies, hence possibili-ties, can be applied. In the case of time scale separation, the coupling between scalesP. Perr (B)LERMAB (Integrated Wood Research Unit), UMR 1093 INRA/ENGREF/University H. Poincar Nancy I, ENGREF, 14, rue Girardet, 54 042 Nancy, Francee-mail: perrenancy-engref.inra.fr60Patrick Perris sequential: the multiscale approach reduces to a change of scale. When the timescales overlap, a concurrent coupling has to be treated: this is a real multiscale con-figuration, more demanding in computational resources and in applied mathematics.The following content is proposed in this paper:Sequential coupling: Techniques are available that allow macroscopic properties tobe computed using the properties and morphology of the so-called unit cell. Homoge-nization is a part of these techniques and can be applied successfully on actual porousmediasuchaswood,fibrousmaterials,solidfoams,etc.,providedtherealmorphologyis taken into account. Examples of mechanical properties of oak, including shrink-age, will be considered. Finally, it has to be noted that homogenization assumes thatboth scales are independent, which allows the solution to be computed only once andsubsequently used in the macroscopic set of equations.Concurrent coupling: The previous assumption often fails in real life situations.In such cases, the scale level cannot be considered as independent and a multiscaleapproach becomes necessary. Some formulations are presented here to explain howseveral scales can be considered simultaneously, from a simple coupling betweenmicroscopic phases to a comprehensive formulation in which the time evolutions ofthe macroscopic values and macroscopic gradients are considered over the Repre-sentative Elementary Volume. Such strategies are much more demanding in terms ofdevelopment and computational time. Some configurations have already been com-puted and are used here to picture the equations. However, the reader should beaware that this is a new and open field, especially in the domain of coupled heat andmass transfer, which is the subject of ongoing research work.In the following, the macroscopic scale always refers to the scale we are interestedin, whereas all smaller scales are referred to as microscopic scales. This indicationis therefore independent of the real size of these scales. For example, when predict-ing shrinkage of a wood tissue, the macroscale is the cellular arrangement (typicallysome hundreds of micrometers) and the microscales are the cell wall (typically somemicrometers) and the scale of the macromolecules (some tens of nanometers). At theopposite end, when dealing with a stack approach, the macroscale is the stack size(some meters) and the microscale is the board section (some centimeters).2 Macroscopic formulationSeveralsetsofmacroscopicequationsareproposedintheliteratureforthesimulationof the drying process. However, this part will just focus on the most comprehensiveset of equations used at the macroscopic level, which describes the system using threeindependent state variables. At present, researchers using a three-variable modelagree with the formulation to be used. The set of equations, as proposed below, orig-inates for the most part from Whitakers (1977) work with minor changes requiredto account for bound water diffusion and drying with internal overpressure (Perrand Degiovanni 1990). In particular, the reader must be aware that all variables areaveraged over the REV (Representative Elementary Volume) (Slattery 1967), hencethe expression “macroscopic”. This assumes the existence of such a representativevolume, large enough for the averaged quantities to be defined and small enough toavoid variations due to macroscopic gradients and non-equilibrium configurations atthe microscopic level.Multiscale aspects of heat and mass transfer during drying61Water conservation t?ww+ gv+ b?+ ?w vw+ v vg+ bvb?= ?gDeffv?.(1)Air Conservation t?ga?+ ?a vg?= ?gDeffa?.(2)Energy conservation t?wwhw+ g(vhv+ aha) + bhb+ ohs gPg?+ ?whwvw+ (vhv+ aha)vg+ hbbvb?= ?gDeff(hvv+ haa) + effT?+ ?,(3)where the gas and liquid phase velocities are given by the Generalised Darcy Law:v?= K?k?,?= P? ?g, where ? = w,g.(4)The quantities are known as the phase potentials and is the depth scalar. All othersymbols have their usual meaning.Boundary conditionsFor the external drying surfaces of the sample, the boundary conditions are assumedto be of the following formJw|x=0+ n = hmcMvln?1 xv1 xv|x=0?,Pg?x=0+= Patm,Je|x=0+ n = h(T|x=0 T),(5)whereJwandJerepresentthefluxesoftotalmoistureandtotalenthalpyatthebound-ary, respectively, x denotes the position from the boundary along the external unitnormal and xvthe molar fraction of vapor.In all these equations, subscript eff denotes the “effective” property that has to bedetermined experimentally or by using a predictive scaling approach (see the nextsection). The averaged value of variable , indicated by a bar, is defined as =1VREV?REV dV.(6)A more detailed description of these equations and related assumptions can be foundelsewhere(Perr1996,1999).Sincethisformulationtakescareoftheinternalpressurethrough the air balance (Eq. 2), the set of equations proved to be very powerful andable to deal with numerous configurations involving intense transfers: high-tempera-ture convective drying, vacuum drying, RF/vacuum drying, IR/vacuum drying, etc.For example, the simulation of convective drying at high temperature, with super-heated steam or moist air, can be predicted with good accuracy when drying lightconcrete (Perr et al. 1993). The most important mechanisms and trends are alsowell predicted in the case of wood, in spite of its strong anisotropy and its biological62Patrick Perr00.10.20.30.4Pressure05101520Width (cm)024Thickness505254565860Temperature05101520Width (cm)024Thickness051015Power05101520Width (cm)0001234Thickness0.51Moistur econtent05101520Width (cm)24ThicknessDrying time : 20 hoursFig. 1Example of drying simulation: comprehensive modeling of convective drying with Radio-Frequency heating (Perr and Bucki 2004)variability. Among the specific behaviors of wood, the internal gaseous pressure gen-erated by internal vaporization is able to drive moisture in the longitudinal direction.This effect is easily observed and proved experimentally (by the endpiece tempera-ture) and was simulated by using this set of equations in the drying model (Perr etal. 1993; Perr 1996).As another example of intricate physical mechanisms, Fig.1 depicts the vari-able fields (volumetric power, temperature, moisture content, and internal pressure)obtained after 20hours of convective drying of an oak section with radio-frequencyheating. In this case, the power field is computed by solving Maxwells equations.This field depends on the dielectric properties of wood, hence on the temperatureand moisture content fields. Similarly, the temperature and moisture content fieldsdepend on the power field. To solve this two-way coupling efficiently, requires a trickycomputational strategy.3 HomogenizationAlthough the set of macroscopic equations presented in the previous section is apowerful foundation for computational simulation of drying, it requires knowledgeof several physical parameters, most of them being a function of both temperatureand moisture content. Consequently, supplying the computer model with all physicalcharacterizations is a tedious task, which restrains the use of modeling. The first goalfor a multiscale approach is to use modeling to predict one part of the parametersMultiscale aspects of heat and mass transfer during drying63Fig. 2Principle of double co-ordinates system used in periodic homogenization (after Sanchez-Palencia 1980)required at the macroscale. Homogenization is one of the mathematical tools, whichallowsthemacroscopicpropertiestobepredictedfromthemicroscopicdescriptionofa heterogeneous medium (Sanchez-Palencia 1980; Suquet 1985; Hornung 1997). Theprinciple of the method will be explained hereafter, using a simple parabolic equationas reference problem:ut= (a u) + fin 0,T ?,u = 0on 0,T ?,u(0,x) = (x) L2(?),(7)where ? is the (bounded) domain of interest, a(x) the diffusion coefficient (order 2tensor), u(t, x) the variable field (i.e. temperature for thermal diffusion or moisturecontent for mass diffusion) f a source term and the initial field.Let us now consider a heterogeneous and periodic medium, which consists of ajuxtaposition of unit cells (Fig.2). A small parameter denotes the ratio between themacroscopic scale, denoted by vector x, and the microscopic scale, denoted by vectory. x is used to locate the point in the macroscopic domain ? (i.e. points M and M?inFig.2) whereas y is used to locate the point within the unit cell Y (i.e. points M and M?in Fig.2). With this new configuration, the reference problem becomes a multiscaleproblem:ut= ?a u?+ fin 0,T ?,u= 0on 0,T ?,u(0,x) = (x) L2(?),(8)where u(t,x) = u(t,x,x) = u(t,x,y) and a(x) = a?x,x?= a(x,y), a uniformly ellip-tic, bounded and Y-period set of functions in ?n(n =number of spatial dimensions).Although the homogenization procedure can be derived with the diffusion coefficient64Patrick Perrdependent on x, the unit cell is supposed to be dependent on y only in the followinga(x) = a(y).(9)The homogenization theory tells us thatu0u0weakly in H10(?),(10)where u0is the solution of the homogenized problemu0t= ?A0 u0?+ fin 0,T ?,u0= 0,on 0,T ?,u0(0,x) = (x) L2(?).(11)The macroscopic property, A0is given byA0ij=?aij(y)?+n?k=1?aik(y)jyk(y)?,Homogenizedcoefficient=Average of themicroscopic coefficient+Correctiveterm.(12)In Eq. 12, the functions jare solutions of the following problems, to be solved overthe unit cell Yn?i=1yi?n?k=1aik(y)jyk(y)?= n?i=1aij(y)yi,j = 1,.,n.(13)Equation 12 tells us that the macroscopic property consists of two contributions:the average of the microscopic properties, which accounts for the proportion andvalues of each phase in the unit cell;a corrective term, which accounts for the morphology of the constituents withinthe unit cell, thanks to the solution of the cell problems. This term might bevery important. For example, the macroscopic stiffness of the earlywood part ofsoftwoodisonly510%oftheaveragedvalueofthemicroscopicproperties,whichmeansthatthecorrectivetermremoved9095%ofthisaveragedvalue(Farruggia1998).Equations 12 and 13 can be derived either by the method of formal expansionor, more rigorously, by using modified test functions in the variational form of (8)(Sanchez-Hubert and Sanchez-Palencia 1992). To derive the limit problem in a for-mal way, the unknown function uis developed as the following expansionu(t,x,y) = u0(t,x) + u1(t,x,y) + 2u2(t,x,y) + (14)Due to the rapid variation of properties inside Y, two independent space derivativesexist(u) = x(u) +1y(u).(15)Applying this derivative rule to problem (8) leads to a formal expansion of the para-bolic equation in powers of . The term with 2tells us that u0does not depend on y.Multiscale aspects of heat and mass transfer during drying65Using this result, the term with 1givesy?a(y)yu1(t,x,y)?= y (a(y)xu0(t,x).(16)Equation 16 is linear, so u1can be expressed as a linear expression of the derivativesof u0with respect to xyu1(t,x,y) =n?j=1yj(y)u0(t,x)xj.(17)In Eq. 17, j(y) H1per(?) are Y-periodic functions, solutions of the following prob-lemsy?a(y)yj(t,x,y)?= y?a(y)ej?.(18)Note that Eq. 18 is just Eq. 13 written with the derivative notation defined in Eq. 15.ejis the unit vector of axis j.Finally, the term with 0readsu0t+ x?a(y)?xu0+ yu1?+ y?a(y)?xu1+ yu2?= f.(19)Averaging Eq. 19 over the unit cell Y allows the Y-periodic terms to vanishu0t+ x?(?a(y)? + ?b(y)?)xu0?= fwith bij(y) =n?k=1aik(y)j(y)yk.(20)Equation 20 is just the homogenized problem and the rule to obtain the homogenizedproperty A0, as already formulated in Eqs. 11 and 12, respectively.The detail of the formal expansion in powers of for the mechanical prob-lems encountered in drying can be found elsewhere: for elasticity (Sanchez-Palencia1980; Ln 1984; Sanchez-Hubert and Sanchez-Palencia 1992), for thermo-elasticity(LHostis 1996) and for shrinkage (Perr 2002; Perr and Badel 2003).The homogenization formulation results in classical PDE problems. Moreover,owing to the assumption that the microscopic and macroscopic scales are indepen-dent, together with the simple physical formulation used in this work (elastic constit-utive equation and shrinkage proportional to the change of moisture content), theproblems are steady-state, linear, and uncoupled.However, the computer model must be able to handle any geometry and deal withproperties that vary strongly in space. This is why the FE method is among the appro-priate numerical strategies. Finally, the mesh must represent the real morphology ofthe porous medium as closely as possible. The best strategy able to fulfil this require-ment consists in building the mesh directly from a microscopic image of the porousmedium. To address this demand, two numerical tools have been developed:(1)MeshPore: software developed to apply image-based meshing (Perr 2005);(2)MorphoPore: code based on the well-known FE strategy, specifically devotedto solving homogenization problems, namely to deal with all kinds of boundaryconditions encountered in solving the cell problems.66Patrick PerrThese products are written in Fortran 95, but they run on a PC as classical Win-dows applications, thanks to a graphical library used for pre- and post-processing(Winteracter 5.0).Figure 3 depicts one typical set of solutions, which allows the macroscopic proper-ties (stiffness and shrinkage) of one annual ring of oak to be calculated. In this figure,the solid lines represent the initial position of boundaries between different kindsof tissues (vessels, parenchyma cells, fiber, and ray cells), while the colored zonesrepresent the deformation of these zones as calculated for each elementary solution(an amplification factor is applied so that the deformation field can be observed eas-ily). These solutions emphasize the complexity of the pore structure of oak, and itsimplication on mechanical behavior. For example, the fiber zones are strong enoughto impose this shrinkage on the rest of the structure (problem w) and the ring porouszone is a weak part unable to transmit any radial forces (in this part, the vesselsenlargement is obvious, problem 11) and prone to shear strain (problem 12).w 11 2212radialtangentialxxxFig. 3Periodic displacement fields computed for the four problems to be solved over the representa-tive cell. (W) shrinkage problem ; (11) and (22) stiffness problems in radial and tangential direction.(12) corresponds to the shear problem. Solid lines represent the initial contours of tissues (Perr andBadel 2003). Color code, from dark to light : ray cells, fiber zones, parenchyma cells, and vesselsMultiscale aspects of heat and mass transfer during drying67This approach allowed us to quantify the effect of fiber proportion and fiber zoneshape on the macroscopic values or to predict the increase of rigidity and shrinkagecoefficients due to the increase of the annual ring width (Perr and Badel 2003).Keepinginmindthatthishomogenizationprocedurewasobtainedbyletting tendtowards zero, the time variable undoubtedly disappears within the unit cell Y. Con-sequently, the macroscopic property has to be calculated only once and subsequentlyused in the homogenized problem. This is a typical sequential coupling.Up to here, we have presented a set of macroscopic equations that allows numer-ous transfer and drying configurations to be computed and a mathematical method,homogenization, that allows macroscopic properties (the “effective” parameters ofthe macroscopic set) to be computed from the constitution of the material at themicroscopic level (concept of unit cell, or REV, Representative Elementary Volume).Although consistent and comprehensive, this approach is not relevant for certainconfigurations. The last part of this paper is therefore devoted to concurrent coupling.4 Dual-scale methodsFigure 4 depicts two examples for which the previous approach fails: soaking samplesof hardwood with water. In such a process, a typical dual scale mechanism occurs: thewaterflowsveryrapidlyinthosevesselsthatareopenandconnected.Inoak,thiseasytransfer happens in the vessels without (or with a low amount of) thyloses, whereasthe early part of each annual growth ring is very active in the case of beech (Fig.4).Then, moisture needs more time to invade the remaining part of the structure, byliquid transfer in low-permeable tissues or by bound water and water diffusion. Thephotographs in Fig.4 show that the local thermodynamic equilibrium cannot alwaysbe assumed. This occurs as soon as the macroscopic and the microscopic time scaleshavethesameordersofmagnitude.Asaresult,themicroscopicfieldnotonlydependson the macroscopic field, but also on the history of this macroscopic field. Such mediamanifest microscopic storage with memory effects. Due to this phenomenon, forexample, transient diffusion in a sheet of paper presents a non-Fickian behavior: thecharacteristic time does not increase as the thickness squared (Lescanne et al. 1992).Obviously, the previous formulations (macroscopic formulation and change of scale)must be discarded.Fig. 4Absence of local thermodynamic equilibrium when soaking wood samples, oak (left) andbeech (right). View of the outlet face (S. GhazilLERMAB)68Patrick PerrDifferent strategies have been imagined, from simple global formulations to morecomprehensive ones:(1)relatively simple global formulations are able to account for the microscopicdelay but have to be fed by experimental knowledge;(2)a mesoscopic model consists in dealing with the microscopic detail at the mac-roscopic level. This is just a continuous model for which the physical parameterschange rapidly in space. Such models are obviously able to catch all subtlemechanisms and interaction between scales, but are very demanding in terms ofcomputer resources. They must be limited to cases for which the ratio betweenthe microscale and the macroscale () remains close to unity;(3)finally, homogenization can be extended to these configurations without localequilibrium. To do this, the microscopic tran