This article was downloaded by: [Fondren Library, Rice University ] On: 08 May 2013, At: 10:12 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Drying Technology: An International Journal Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/ldrt20 Physical Interpretation of Solids Drying: An Overview on Mathematical Modeling Research Wei Wang a , Guohua Chen a & Arun S. Mujumdar b a Department of Chemical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China b Department of Mechanical Engineering, National University of Singapore, Kent Ridge Crescent, Singapore Published online: 23 Apr 2007. To cite this article: Wei Wang , Guohua Chen & Arun S. Mujumdar (2007): Physical Interpretation of Solids Drying: An Overview on Mathematical Modeling Research, Drying Technology: An International Journal, 25:4, 659-668 To link to this article: http://dx.doi.org/10.1080/07373930701285936 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.
Transcript
This article was downloaded by: [Fondren Library, Rice University ]On: 08 May 2013, At: 10:12Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK
Drying Technology: An International JournalPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/ldrt20
Physical Interpretation of Solids Drying: An Overviewon Mathematical Modeling ResearchWei Wang a , Guohua Chen a & Arun S. Mujumdar ba Department of Chemical Engineering, Hong Kong University of Science and Technology,Clear Water Bay, Kowloon, Hong Kong, Chinab Department of Mechanical Engineering, National University of Singapore, Kent RidgeCrescent, SingaporePublished online: 23 Apr 2007.
To cite this article: Wei Wang , Guohua Chen & Arun S. Mujumdar (2007): Physical Interpretation of Solids Drying: AnOverview on Mathematical Modeling Research, Drying Technology: An International Journal, 25:4, 659-668
To link to this article: http://dx.doi.org/10.1080/07373930701285936
PLEASE SCROLL DOWN FOR ARTICLE
Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions
This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form toanyone is expressly forbidden.
The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses shouldbe independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims,proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly inconnection with or arising out of the use of this material.
Physical Interpretation of Solids Drying: An Overview onMathematical Modeling Research
Wei Wang,1 Guohua Chen,1 and Arun S. Mujumdar2
1Department of Chemical Engineering, Hong Kong University of Science and Technology,Clear Water Bay, Kowloon, Hong Kong, China2Department of Mechanical Engineering, National University of Singapore,Kent Ridge Crescent, Singapore
There are successive interests in developing various mathematicmodels for the solids drying process. Such efforts have been madefor nearly three decades since Luikov’s system and Whitaker’stheory. This article gives some physical interpretations of solid dry-ing by reviewing more than 70 published papers in the past decadesbased on studies done by researchers. Classical transport theory inporous media is sufficient for dealing with the solids drying processat a fundamental level. Current numerical technique and computertechnology are able to solve problems for any material geometry.These advancements seem to go faster than one’s understanding ofprocesses and materials. Almost all available models for solids dry-ing can only be used for a specific material system. Generality of amodel still remains a major problem due to the lack of fundamentaldata. Measurements of unknown properties and experimentalvalidation of models need to be improved. The design, operation,and optimization of industrial dryers call for a comprehensivemathematic model that gives attention to accuracy, generality,and complexity/simplicity.
Mathematical modeling is the use of mathematicallanguage to describe the behaviors of a process or a system.In addition to providing predictions and redescriptions ofthe process of interest, the cognitive modeling allows oneto extract more information to gain new insights into theprocess. For example, the modeling of a freeze-dryingprocess can help not only predict appropriate operatingconditions but also analyze the transport mechanisms, elim-inating guesswork that would normally be used to establishequipment geometry and process conditions.[1] The modelcan also serve such an important purpose in advancing
one’s understanding, revealing some new phenomena in aprocess and pushing the field forward. A mathematicalmodel on a solids drying process can be used to determineprofiles of various variables that are quite difficult to bemeasured by direct or indirect experimental techniques.For example, temperature distribution measurement islimited by the number of thermocouples that are insertedin the multidimensional space occupied by the materialbeing dried, while moisture distribution measurement isalmost impossible now.
Development of a mathematical model depends on howmuch one understands the process of interest. For a solidsdrying model, anyone who is familiar with transport the-ories can easily write the heat and mass as well as momen-tum transfer equations. The key point in developing such amodel is the identification of various transport mechanismsbased on some well-developed theories. Another importantpart of the modeling is the evaluation of an acquiredmodel. For example, how does one know if a mathematicalmodel works well? Researchers have proposed a number ofqualitative and quantitative criteria for model evalua-tions.[2] These can be summarized as accuracy, generality,and complexity=simplicity if focusing on the quantitativeissue only. It is clear that a model must match the practicalobservation as closely as possible no matter whether it istheoretical or empirical. In other words, the model shouldadequately show the relations between measured variablesof a process or a system. Generality refers to the ability of amodel to be used not only in an individual process but alsoin a family of similar processes. Usually, a model based onthe substantial theoretical foundation has a good gener-ality, although a system error may exist, corresponding tothe random error from experiments, due to the limitationof the understanding of a process. For an equation of state(EOS) in thermodynamics, for instance, it is preferable touse properties of pure materials to predict those of a mix-ture, which relies on experimental measurement as less as
Correspondence: Guohua Chen, Department of ChemicalEngineering, Hong Kong University of Science and Technology,Clear Water Bay, Kowloon, Hong Kong, China; E-mail: [email protected]
Drying Technology, 25: 659–668, 2007
Copyright # 2007 Taylor & Francis Group, LLC
ISSN: 0737-3937 print/1532-2300 online
DOI: 10.1080/07373930701285936
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possible. On the premise of accuracy and generality, amodel should be less complex. Therefore, some assump-tions have to be made to simplify the process. However,a model must possess certain complexity to capture theregularity of the process. Intuitively, the accuracy of amodel can be improved by increasing model complexity,whereas additional complexity will reduce the generality.Both high accuracy and wide generality would ensure anappropriate degree of complexity. It is not easy to balancethis intricate relationship among accuracy, generality, andcomplexity=simplicity.
In many branches of engineering—for example, chemicalengineering, food engineering, material science, as well asbiological engineering—different reactions of materialsystems undergoing external and=or internal loading haveto be studied and described precisely in order to be ableto predict the responses of these systems. A solids dryingmodel is actually a coupled heat and mass transfer math-ematical model governing the independent intensive vari-ables such as temperature and pressure, consequentlyresulting in the extensive variables and other dependentintensive variables. In the past few decades, hundreds ofsuch models were developed in an attempt to describe thebehaviors of solid drying processes. Most of them are basedon the Luikov’s system[3] and Whitaker’s theory,[4] regard-less of whether the moisture is solid or liquid. Bothresearchers have provided a conceptual starting point forall subsequent modeling studies for the solids dryingprocess. However, due to the limitation of cognition towardthe process, some parameters in a solids drying model stillneed to be determined by experiments without firm theor-etical foundation. It seems that the establishment of somefundamental theories drops behind the development ofmathematical models. For example, the adsorption-desorp-tion of moisture in a porous medium is far from complete.Therefore, the curve-fitting expression sometimes is neces-sary when the theoretical development is insufficient.
The purpose of the present article is to give some physi-cal interpretations of solids drying through reviewingmathematical models published in the past decades. Thearticle covers a variety of aspects, including solid materialas a porous medium, contributions of Luikov and Whi-taker, transport mechanisms, equilibrium relationship,volumetric heat source, drying induced deformation, basicassumptions, and current state. Heat and mass transferprocesses in solids drying are interpreted and areas forimprovement are identified. The authors sincerely hopethat this article will be a valuable addition to processunderstanding and model development for future modelingwork of the solid drying process.
SOLID MATERIAL AS A POROUS MEDIUM
Once dealing with a porous medium, the most importantpoint of investigation is to first determine the composition
of the material (or body), because one must knowphysically and chemically different materials that consti-tute the system under consideration. Most solids materialssubjected to a drying process can be treated as hygroscopicporous media with the multiphase transport of heat andmass. The material system in the field of engineering canbe composed in various ways. For a moist porous material,the pore space is filled with moisture (liquid water or ice) ifthe material is completely saturated or with moisture andair if it is partially saturated. Because the exact descriptionof the location of the pore (empty or filled with moistures)and solid material is nearly impossible, the heterogeneouscomposition can be investigated by using volume fractionconcept.[5]
The volume fraction concept assumes that the poroussolid always models a control space and that only the moist-ure (ice, liquid, vapor, and=or gas) contained in pores canleave the control space. Furthermore, it assumes that poresare statistically distributed and that an arbitrary volumeelement in the reference and its actual placement are com-posed of the volume elements of the real media.[5] The latterstatement means that neither a geometrical interpretation ofthe pore structure nor the exact location of the individualcomponents of the material is considered. The volume frac-tion concept results in the effect of the smeared substitutecontinuum, which can then be treated by the mixturetheory. The combination of the mixture theory with the vol-ume fraction concept involves microscale analysis. Thequestion arises as to what scale the mechanical or ther-modynamic investigations should be performed on, themacroscale or microscale? In principle, both strategies arepossible. The micromechanical approach, with all its aver-age processes, can reveal some important mechanical rela-tions. However, this approach is quite complicated. Sincethe macroscopic governing equations volume-averagedfrom the microscopic equations are sufficient to deal withsolids drying processes, only the macromechanicalapproach will be discussed in the following sections, as inmost of current modeling studies.
CONTRIBUTIONS OF LUIKOV AND WHITAKER
There are two milestone works, Luikov’s system andWhitaker’s theory, related to the modeling of a solid dryingprocess since the acceptance of solid materials as porousmedia. Luikov[3] derived macroscopic heat and mass trans-fer governing equations through volumetric summation oftransport equations of each species in each phase on thebasis of flux expressions of phenomenological relation-ships. By choosing independent variables such as tempera-ture, moisture content, or gas pressure, the generalizedgoverning equations are
q0
@ui
@t¼ �r � ji þ _IIi ð1Þ
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cq0
@T
@t¼ r � qþ
Xi
hi_IIi þ
Xi
jicirT ð2Þ
_IIi ¼ riq0
@ui
@tð3Þ
where i denotes mass capacities of vapor, liquid, inert gas,and solid and r is the phase conversion factor. Such atreatment makes the process very clear.
For the simple case without total pressure gradientinside the porous body, Luikov suggested that both con-centration gradient and temperature gradient contributeto the mass transfer flux as
ji ¼ �airui þ aidirT ð4Þwhere d is the thermal gradient coefficient. In the case ofintense moisture evaporation inside capillary-porous body,there is an increase in total pressure that results in a fil-tration pattern of vapor movement. The mass flux is onlycontributed by the pressure gradient; i.e., Darcy’s law.
However, the model has several drawbacks. Some coef-ficients like r and d are not the physical properties but pro-cess-related variables that depend on phenomenologicalrelationships. These coefficients are difficult to obtain inthe governing equations. For the latter case, the contribu-tions of the gas-phase diffusion and liquid bulk flow arenot included in the mass flux. Although a phase conversionfactor provides the solution with some simplicity, itsassumed value makes the solution semi-empirical.
Another well-known theory for the porous media trans-port is the work by Whitaker.[4] He started from heat andmass conservation equations of each phase (solid, liquid,and vapor plus inert gas) at the microscopic level (REV,representative element volume), and then volume-averagedthem to develop a mechanistic model at the macroscopiclevel as displayed in Fig. 1. Through some simplificationand rearrangement toward the volume-averaged energyand continuity equations, the forms of the governing equa-tions for heat and mass transfer were given by followingequations:
ðqcpÞeff
@T
@tþr �
Xi
ðjihiÞ ¼ r � ðkeffrTÞ � DH _II þ _qq ð5Þ
@ðeiqiÞ@t
þr � ji ¼ _IIi ð6Þ
If the heat conduction is only concerned in the porousmedium to achieve a further simplification, the energyequation can be expressed in Eq. (7) by dropping the con-vective term on the left-hand side of Eq. (5) as:
ðqcpÞeff
@T
@t¼ r � ðkeffrTÞ � DH _II þ _qq ð7Þ
The major assumptions in his derivations were the localthermal equilibrium, valid Darcy’s law, Fickian diffusion,
and filtrational flow in gas transport; capillary flow inliquid transport (assuming negligible effect of gas pressuregradient on liquid movement); rigid structure; and theabsence of bound water. Advantages of this mechanisticmodel are that the assumptions are very clear, the physicsof the model are better understood, and the parametersare well defined. However, nearly all models developedhereafter based on Whitaker’s theory were not derivedthrough the factual volumetric averaging scheme but thevolumetric summation on the macroscopic scale, or whatis called the explicit—or direct—use of Whitaker’s theoryas dipicted in Eqs. (5), (6), and (7).
Both Luikov’s and Whitaker’s works lay the foundationfor all subsequent process modeling studies of the porousmedium drying. However, neither Luikov’s system norWhitaker’s theory takes the presence of bound water intoaccount.
TRANSPORT MECHANISMS
Transport in a porous medium can follow variousmechanisms, and the most dominant ones are Fick-formdiffusion and bulk convection. Fick-form diffusion com-prises of molecular diffusion, Knudsen diffusion, and=orcapillary diffusion. Bulk convection refers to filtration flowor Darcy flow.
Water vapor in a porous media can move by molecularor Fickian diffusion if pores are sufficiently large (Knudsennumber, Kn<<1). Fick’s law is described as:
j ¼ �D@C
@xð8Þ
In fact, Fick’s law was obtained by an analogy of Fourier’slaw for heat propagation, rather than by the establishedmechanic principle. When the pore size is smaller thanthe molecular mean free path, Knudsen diffusion
FIG. 1. Different phases in porous media.
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dominates the process. It is commonly encountered in thefreeze-drying process. Knudsen diffusivity is[6]
D ¼ 97:0�rr
s
ffiffiffiffiffiffiffiffiT
Mw
rð9Þ
Transition-type diffusion occurs where both molecular-molecular and molecular-wall collision are involved, parti-cularly when the molecular mean free path and porediameter are comparable. The diffusivity expression isgiven by a harmonic mean of molecular and Knudsendiffusivities as:[7]
D ¼ 1
1=DM þ 1=DKð10Þ
Capillary flow is due to the difference of relative attrac-tions between liquid–liquid molecules and those betweenliquid–solid molecules. In a porous solid, the liquid isattracted or held more tightly when there is less liquid.Conversely, it is held less tightly when there is more liquid.Due to the differences in capillary attraction, liquid flowcan occur from locations in the solid having more waterto locations having less water; i.e., from higher concen-tration to lower concentration of water. This is referredto as unsaturated flow and is extremely important in dryingof a material with liquid moisture. Capillary diffusionexpression is the same as that of Fickian diffusion. How-ever, it is sometimes difficult to distinguish the contributionof a particular individual diffusion. In practice, the diffu-sivity generally includes effects of all possible mechanismsof moisture transport in both liquid and vapor form.
Like Fourier’s and Fick’s laws, Darcy’s law is anempirical relationship. It describes movement of fluid in aporous material. Darcy’s law written in terms of pressuregradient is
j ¼ �qK
l@P
@xð11Þ
Although this mathematical relationship was originallydiscovered to govern the flow of groundwater throughgranular media, or the flow of other fluids through per-meable material, it is widely used for the convection termin mass flux to quantify the bulk velocity. The permeability,K is the most important physical property of a porousmedium in much the same way as the porosity is the mostimportant geometrical property. It measures quantitativelythe ability of a porous medium to conduct fluid flow andis related to the connectivity of void spaces and to the grainsize of solids. At present, permeability is mainly obtainedthrough experimental measurements, except that thesolid matrix in a porous medium is composed of rigidspheres.[8]
In summary, the porous solid drying process can begoverned by Fourier’s law, Fick-form law, and Darcy’slaw. However, most of the solid materials encountered in
drying processes are extensively inhomogeneous and aniso-tropic materials. Inhomogeneity implies that materialproperties are closely related to the spatial position, whichhas been widely recognized in the solids drying process.Anisotropy suggests that the properties depend on thedirection. That means that the diffusivity, D, and per-meability, K, as well as thermal conductivity, k, are thesecond-order symmetric tensors in the three-dimensionalspace as below:
C ¼C11 C12 C13
C21 C22 C23
C31 C32 C33
24
35 ð12Þ
where C refers to any of three properties mentioned above.In the principle axis coordinate system, i.e., one of threecommonly used orthogonal coordinate systems, the tensorcan be expressed in a diagonal tensor with the diagonal ele-ments being non-zero while others zero through the coordi-nates transformation, which is called the orthogonallyanisotropic material.[9]
C ¼C11 0 00 C22 00 0 C33
24
35 ð13Þ
It seems plausible that most anisotropic materials insolid drying are at least treated as transversely isotropicmaterials.[4] Currently, nearly all solid materials under-going the drying process are treated as isotropic porousmaterials.
EQUILIBRIUM RELATIONSHIP
In nonhygroscopic porous media, the amount of physi-cally bound water is negligible. The internal surface vaporpressure for such a material equals that of the pure compo-nent and is only a function of temperature in a thermo-dynamic sense. It can be adequately represented by theClapeyron equation. In hygroscopic porous media, there isa certain amount of physically bound water. Transportphenomena in such a material would cause some additionalcomplications. There is most likely a level of moisture con-tent above which the material behaves nonhygroscopically.Below this level, the internal vapor pressure is a function ofboth temperature and moisture content in the form, whichis called the equilibrium moisture isotherm or the adsorptionequilibrium relationship. Consequently, the simple Clapeyronequation is not valid for hygroscopic porous media. Hence,the simple thermodynamic equilibrium relationship shouldbe replaced by an adsorption equilibrium relationship.Unfortunately, such a relationship is neither developed theo-retically nor experimentally due to complexities of porousmedia, which becomes a barrier to modeling investigations.
There are only a few papers addressing the issue ofbound water removal. In drying of materials with liquidmoisture, both models by Stanish et al.[10] and Ni et al.[11]
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used the curve-fitting relationships from dehydrationexperiments of wood and potato, respectively. Wang andChen’s model[12–14] and Chen et al.’s model[15] usedKelvin’s equation, which is an equilibrium relationshipfor capillary flow. In terms of freeze drying, Wang andShi’s model[16,17] as well as Wang and Chen’s model[18]
are those for unbound water removal with a required zeromoisture content in the dried region. In Millman et al.’smodel,[19] an adsorption rate equation of the first orderwas applied to describe the adsorption–desorption effect.Liapis and Bruttini’s model[1] used Langmuir’s equationwith the Langmuir constant determined by experiments.Compared to some semi-empirical relationships, a curve-fitting expression may have more accuracy in describingthe equilibrium relationship.
Under the drying circumstances where the ambientpressure is at atmospheric value or below, the water vaporand inert gas behave as perfect gases. The perfect gas law isvalid throughout the overall drying process. When theliquid moisture contains some solutes, such as salts, sugars,proteins, etc., as encountered in drying of foods, it cannotbe regarded as an ideal solution. Knowledge of thermody-namics involved in such a system is therefore very impor-tant to relate the concentration of water to its partialpressure.[20] Concentration of water must be replaced bythe activity of water. For homogeneous mixtures ofnon-polarity or weak polarity, NRTL (non-Random two-liquid) and UNIQUAC (universal quasi-chemical) equa-tions are sufficiently applied.
VOLUMETRIC HEAT SOURCE
Volumetric heat is generated by dielectric heating, e.g.,microwave or RF (radio frequency) heating, during a dry-ing process. Characteristics of microwave heating due tothe volumetric dissipation of energy can be beneficial,depending on the application. In drying of materials withliquid moisture, the selective heating of microwaves isextremely useful. Because the loss factor of water is muchgreater than that of most solids, regions of higher moisturecontent within the material will absorb more microwaveenergy. In microwave-freeze drying, ice crystals hardlyabsorb microwave energy since it has a very small lossfactor. Solids in the frozen material therefore act as aninternal heat source, supplying energy needed for drying.
The volumetric heating rate or the microwave powerdissipation is given as:[21]
_qq ¼ 2pf e0er00E2
rms ð14ÞFor pure material at a constant frequency, the permittivityis only a function of temperature as:
erðTÞ ¼ er0ðTÞ � jer
00ðTÞ ð15ÞHowever, this type of data does not exist for most solids
in an appropriate range of temperature except for liquid
water. In general, the loss factor of a porous material inEq. (15) can be written in the form of volume averagingthat includes all the phases
½er00ðS;TÞ�a ¼
Xi
biðSÞ½eri00ðTÞ�a ð16Þ
The exponent a is an empirical constant called the modeldegree, ranging from 0 to 1.[22]
When the material size is much smaller than the pen-etration depth of the microwaves, the electric field strengthcan be considered uniformly distributed within the particle.If a more precise simulation is expected, distribution of theelectric field strength in a porous medium should be knownthrough solving the wave equation within materials. Thewave equation is derived from Maxwell’s equations thatdescribe the behavior of both the electric and magneticfields. To obtain the wave equation, it is important tounderstand the nature of the medium in the drying process.A material undergoing drying is dielectric, nonmagnetic,linear, nondispersive, inhomogeneous, and isotropic. Forsuch a material without free charges and currents, the gen-eralized wave equation for electric field strength in the par-tial differential form is given as:
l0
@2ðeEÞ@t2
¼ r2E þr 1
eE � re
� �ð17Þ
In the electromagnetic field, the rate of change of E ismuch higher than e because of high frequency. Further-more, the gradient of e is quite small compared to E, orthe medium can be treated to be locally homogeneous,which means that e varies slowly in the space domain.Equation (17) can be simplified to the following form aftersome mathematical manipulation:
l0e@2E
@t2¼ r2E ð18Þ
Turner and his coworkers,[22–26] as well as Ratanadechoet al.,[27] devoted their efforts solving Eq. (18) to determinethe distribution of electric field strength in a material dur-ing drying. However, only the time-harmonic plane wave(for example, TEM wave) is applicable, and the mediumgeometry is limited to be cubic. More importantly, bound-ary conditions were so excessively simplified that someboundary effects were not taken into account.
In microwave-freeze drying, when the solid material hasa low loss factor, and absorbs a little microwave energy, themicrowave heating effect can be considered insignificant.Recent studies[18,28,29] have shown that the enhancementof microwave-freeze drying is quite possible with the aidof dielectric material. The key point of the dielectricmaterial assisted by microwave-freeze drying is that adielectric sphere or bar is first frozen with the solution asshown in Fig. 2; then the frozen material is freeze-driedwith microwave heating.
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DRYING-INDUCED DEFORMATION
In addition to heat and mass transfer, solid materials,especially plant foods, which are composed of cellulartissues, subjected to drying may more or less suffer a defor-mation due to shrinkage or swelling. To avoid the compu-tational complexities of a model, deformation is generallyignored except in studies where the focus is paid to thestrain–stress development and crack of dried products.[30]
When the physical structure of the solid material substan-tially changes during drying and has a strong effect onthe internal transport and product quality, drying-induceddeformation or strain–stress of materials has to be takeninto account.
The expansion or contraction of a material subjected todrying mainly results in two types of internal strain–stresses:the thermo-strain due to the temperature gradient and thehydro-strain due to the moisture gradient. There are alsocreep strain, mechanosorption strain, and stress strain.For highly deformable materials, the deformation doesnot fulfill the geometrical compatibility, and a strain tensorrelated to stresses would be generated. The strain tensor isdeduced from the displacement, and the stress tensor mustsatisfy the local mechanical equilibrium and boundary con-ditions. In the generalized analysis of the drying-inducedstrain–stress, the strain tensor should include all possiblestrain components.
The deformation of a solid material during drying can bereversible or irreversible. Knowledge of material mechanicsis necessary to determine if the material is elastic, viscoelas-tic, or elastoplastic. Then the constitutive equation, whichrepresents the mechanical behavior of a material relatedto the strain tensor and the stress tensor, must be formu-lated in order to determine the displacement of each pointwithin the solid matrix. For a linear elastic and isotropicporous material saturated with water, general equationsfor the displacement and the fluid pressure are[31]
Gr2U þ G
1� 2nre�rpw ¼ 0 ð19Þ
@e
@t¼ K
lr2pw ð20Þ
Considering a porous material as linear elastic is so farthe simplest and relatively well-developed approach. Other
treatments are comparatively complex and still under devel-opment because different materials have different mechan-ical responses to the drying-induced deformation.
In the last decade, researchers paid much attention todrying induced strain–stress. Many papers were publishedwith different assumptions of material deformation beha-viors.[32–37] Kowalski and his coworker presented a ther-momechanical approach to shrinking and crackingphenomena in drying based on their previous works.[38–42]
BASIC ASSUMPTIONS
Assumptions are made solely to reduce the computa-tional difficulties and to make the problem treatable. Animportant assumption made in a majority of multiphaseporous media models is that the solid (including ice),liquid, and gas (vapor and air) at any location are inthermal equilibrium with each other, so that there is onlyone temperature at any given location. By assuming thelocal thermal equilibrium, the isotherm relating vaporpressure as a function of moisture and temperature canbe used. Fourier’s law, Darcy’s law, and Fick’s law (ormass diffusion written in the form of Fick’s law) aresupposed to be valid throughout the space domain. Thegaseous phase is a perfect gas, as discussed previously inthe section on equilibrium relationships. The solid matrixis usually homogeneous, regardless of the existence ofdeformation of material, and it is rigid if ignoring the dry-ing-induced strain–stress. Meanwhile, the solid matrix isgenerally considered isotropic due to the lack of detailedinformation about most materials. This assumption isnecessary for most analytical and numerical solutionsexcept for some naturally formed materials such as woods.Transport mechanisms such as the Soret effect (mass fluxdue to temperature gradient) and the Dufour effect (heatflux due to concentration gradient) are generally consideredsmall;[30] hence, these are not taken into account. Propertydata involving Soret or Dufour effects are generally unavail-able anyway. Due to the internal evaporation (sublimation),convection heat transfer within a material is quite smallcompared to heat conduction. The gravity factor is ignoredin most studies. Perhaps this can be justified by the fact thatthe solid material is generally unsaturated where capillaryforce is much stronger than that of gravity.
CURRENT STATE
Since the 1970s, the modeling and simulation of solidsdrying has been the subject of interest for researchers. Obvi-ously, the field is truly interdisciplinary, with nearlyhundreds of papers published in top engineering, agricul-ture and forestry science, and mathematical journals. Thesepublications cover a wide range of topics, including thederivation of the physical and mechanical formulation,the development of analytical and numerical solutions, thephysical and mechanical characterization of media, and
FIG. 2. Schematic diagram of dielectric material–assisted microwave
freeze drying.
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the process experiments carried out on both laboratory andindustrial scales. Nowadays, as a result of ever-increasingcomputational power, numerical simulation has rapidlybecome a very powerful tool to study the drying processat a fundamental level. The results so obtained can thenlead many new and innovative drying operations into theindustrial sector.
Turner and Perre[42] have made a clear categorization ofavailable mathematical models for the solids drying pro-cess. The first fundamental difference between modelsdepends on the number of independent variables used todescribe the process. Besides the spatial and time variables,there are generally four variables in total governing a dry-ing process; i.e., moisture content, temperature, gaseousconcentration (density), and gaseous pressure. Threeoptions are possible, which can be known as one-variable,two-variable, or three-variable models. The one-variablemodel typically uses moisture content as the primary vari-able or an equivalent variable such as the saturation orpotential. The two-variable model uses moisture contentand temperature, T, or an equivalent variable such asenthalpy in the formulation. The most sophisticatedthree-variable model uses moisture content, temperature,and gaseous pressure, Pg, or an equivalent variable suchas gaseous density.
The one-variable model is the simplest model and is usedin drying at relatively low temperature or ignoring theeffect of temperature. This kind of model should beavoided because it does not account for the very importantcoupling effect between mass and heat transfer that existsduring drying. A two-variable model is appropriate formost drying conditions encountered in industry when theinternal pressure is not the primary concern. Both moisturecontent and temperature are chosen to correspond to themass and heat transfer equations. Such a model is suitablefor vacuum drying or vacuum-freeze drying since the vaporcan be regarded as the only component in gaseous phase.The partial differential equation of gaseous pressure, Pg,is included in the macroscopic equation set of modelbecause, from the author’s point of view, two more vari-ables, the gaseous density and partial pressure of the inertgas, would appear when inert gas exists. Among the sixvariables (T, S, Pv, Pin, qv, and qin), only three of themare independent, because there are three restrictionequations; i.e., ideal gas laws for both water vapor andinert gas and an equilibrium relationship for vapor. Theequation of gaseous pressure is necessary in the dryingprocess at atmospheric level.
The second fundamental difference between thereported models lies in the number of spatial dimensionused to describe the process.[43] A one-dimensional (1-D)model uses the thickness of a large plate sample or theradius of a sample with a cylindrical or spherical geometryas the only spatial dimension. To understand a process, it is
often more useful to develop one-dimensional models, tobe followed by multidimensional models that are closerto real situations. A 1-D model is relatively easy to solve.The important transport phenomena can be capturedbecause variation trends of process parameters are inde-pendent of the dimension number and coordinate system.Actually, a 1-D spherical model can deal with the three-dimensional (3-D) problem due to symmetry; a 2-D cylin-drical model can deal with the 3-D problem for the samereason. The multidimensional model is needed for situa-tions where the sample has an irregular geometry or ageometry that is unacceptable for the degradation of thenumber of dimension. More importantly, when the porousmedia composed of sample is anisotropic, the multidimen-sional model has to be taken into account.
Publications over the past two decades have reporteddrying of various materials, including fruits such as apple,vegetables such as carrot, wood, etc. Some researchers havedeveloped heat and mass transfer equations for porousmedia starting from conservation equations and mechan-istic flux models.[10,11,44–49] Perre, Turner, and their cowor-kers have contributed their efforts to modeling of wooddrying.[50–60] They performed mathematical modeling,experimental verification, and optimization. Wang andChen[12–15,61] presented their mathematical model ofcoupled heat and mass transfer in drying of porous media.Apple was chosen as the representative solid material infixed-bed drying and fluidized-bed drying with and withoutmicrowave heating.
Freeze drying is a dehydration technique that removesmoisture by sublimation. It plays an irreplaceable role indrying of heat-sensitive materials such as foods, pharma-ceuticals, and biological materials because of no air andlow processing temperature. Besides fruits and vegetables,which have a naturally formed porous structure, materialsfrozen from solutions can be porous. In recent decades, oneof the earlier modeling works on freeze drying with micro-wave heating was reported by Ma and Peltre.[62] Theirsimulations found that drying rates are essentially a func-tion of the microwave power input. Low operating vacuumchamber pressure is necessary to ensure that the tempera-ture of the frozen core is as low as possible. They foundthe optimum pressure to be about 6.67 Pa. Ang et al.[63]
presented a two-dimensional model considering anisotropyof frozen material, which was very similar to that of Maand Peltre.[62] Some researchers devoted their efforts tothe exact solutions for the sublimation problem. Basedon a series of strict assumptions, Lin[64–66] obtained theexact solutions of temperature and moisture distributionsas well as the position of the moving sublimation front.Similarly, Fey and Boles[67–71] performed analytical studiesby solving a set of formulas of heat and mass transfer forvacuum sublimation in an initially partially filled frozenporous medium. Liapis and his coworkers[1,19,72–75] made
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many efforts in modeling vial and tray freeze drying ofaqueous solutions with radiation heating over the material.A mathematical model, called the sorption-sublimationmodel, was constituted considering removal of free waterand bound water during the freeze-drying process. Wangand Chen[18,28,29,76] developed a model for dielectricmaterial–assisted microwave-freeze drying and gave adetailed discussion of transport mechanisms throughsolving their model.
SUMMARY
Modeling work of the solids drying process has beenunderway for many decades. Various mathematical modelshave been developed or are developing toward new andinnovative drying operations in the industrial sector. Now-adays, generalization of the formulation versus sophisti-cation of the process is the characteristics of the existingmodels that must be properly balanced in order to beeventually useful in practice. An effective combination ofclassical transport theory with porous media theory is suf-ficient for dealing with the drying process at a fundamentallevel. As a result of significant advancements in numericaltechniques together with the ever-increasing power of com-puters, numerical simulations have become a strong tool,able to deal with a comprehensive formulation with anymaterial geometry.
It should be pointed out, however, that currentmathematical models could only be used to address specificproblems. Some fundamental data are still lacking due toeither lack of understanding of the process or limitationof measurements. Structure properties of the solid matrix,phase equilibrium relationship, transport mechanisms inporous media, as well as the exact interaction between lossymaterial and electric field are still undetermined. Under-standings of the material deformation caused by drying-induced strain–stress are not in good agreement, and driedproduct properties are also unavailable. Such knowledge isextremely important for constructing a precise mathemat-ical model of solids drying and eliminating the guessworkthat has been done in the past. In particular, it is the opi-nion of the authors that the generality of a model shouldbe emphasized, which has the same importance as accu-racy. Finally, any mathematical model must be validatedon both laboratory and industrial scales. This requires aclose cooperation between drying theoreticians and practi-tioners. Modeling of the solids drying process still remainslargely an art.
NOMENCLATURE
a Diffusivity in Luikov’s system (m2=s)C Mass concentration (kg=m3)c Heat capacity of wet material (J=[kg��C])cp Constant pressure heat capacity (J=[kg��C])D Diffusivity (m2=s)
DK Knudsen diffusivity (m2=s)DM Molecular diffusivity (m2=s)E Electric field strength (V=m)Erms Root mean square electric field strength (V=m)e Element of strain tensorf Frequency of electric field (Hz)G Shear modulus (Pa)h Enthalpy (J=kg)DH Latent heat (J=kg)_II Internal evaporation intensity (kg=[m3�s])j Mass flux (kg=[m2�s])K Permeability (m2)k Thermal conductivity (J=[m�s�K])Mw Molecular weight of water (kg=kg-mol)P Pressure (Pa)pw Water pressure (Pa)q Heat flux by conduction (J=[m2�s])_qq Internal heat generation intensity (W=m3)�rr Average pore radius (m)S Moisture saturationT Temperature (K)t Time (s)U Displacement (m)u Moisture content in Luikov’s systemx Spatial axis (m)
Greek Letters
a Model degreeb Volume fractiond Thermogradient coefficient in Luikov’s systeme Permittivity (F=m)e0 Permittivity (F=m)e0 Dielectric constante00 Dielectric loss factorC Extensive diffusivity (may be D, K, or k)l Dynamic viscosity (kg=[m�s])l0 Magnetic permeability in free space (H=m)n Poisson’s ratioq Density (kg=m3)q0 Density of a perfectly dried material (kg=m3)r Phase conversion factor from liquid to vapors Tortuosity
Subscripts
eff Effectivei Solid (ice), liquid, vapor, or inert gas in porous
mediumin Inert gasg Gasr Relative
ACKNOWLEDGEMENT
Financial support by RGC grant of HKUST6038=00Pand RGC600704 is gratefully acknowledged.
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