Numerical Study of Focusing Effects of Microwaves inside Wood Due to Timber Ring Structure

16 Pages·2017·4.69 MB·English

Article
Numerical Study of Focusing Effects of Microwaves
inside Wood Due to Timber Ring Structure
RocioSanchez-Montero* ID,Pablo-LuisLopez-Espi ID,JuanAntonioMartinez-Rojas,
JesusAlpuente-HermosillaandCristinaAlen-Cordero
DepartmentofSignalTheoryandCommunications,EscuelaPolitecnicaSuperior,UniversidaddeAlcala,
CampusUniversitario,Ctra.deMadridaBarcelonakm33.600,AlcaladeHenares28805,Spain;
[email protected](P.-L.L.-E.);[email protected](J.A.M.-R.);[email protected](J.A.-H.);
[email protected](C.A.-C.)
* Correspondence:[email protected];Tel.:[phone]
Received:15January2018;Accepted:26February2018;Published:28February2018
Abstract: The aim of this study is the detailed calculation of microwave propagation inside raw
timberincylindricalconfigurations. Twodifferentapproacheshavebeenused. Thefirstoneuses
anexactformulationandanalyticalapproximationsinordertoexploretheelectromagneticfield
distributioninsidedrywood.Theintroductionofconductivityintheexactmodelmakesitsocomplex
thattheequationsareunsuitableforanalyticalmanipulation. Inordertofurtherexploretheeffectof
moistureincylindricalwoodstructures,afullscalenumericalsimulationusingcommercialsoftware
hasbeenperformed. Theresultsshowthatformicrowavefrequenciesinthe3GHzrangeandfor
typicalwoodparameters,acylindricallogbehavesasakindofFresnellens. Thisworkhasimportant
applicationsinmicrowavetreatmentandsensingofwood.
Keywords: microwave propagation in wood; non-destructive sensing of wood; physical wood
treatments;analyticalanalysis;numericalanalysis
1. Introduction
Microwave treatment and sensing of wood is increasingly important. A review of different
microwave wood testing techniques is given in [1] and these include propagation modelling,
measurementtechniques,hardwareimplementation,anddeterminationofwoodproperties.According
to[2,3],microwaveheating,forexample,offersseveraladvantagesoverotherdryingmethodsand
chemical procedures because its technique is eco-friendlier and causes less damage in the timber
samplecomparedtoemployingtoxicchemicals. Besides,microwaveheatingmethodisfasterand
gentlerincomparisontolivesteam. Asacontactlessandnondestructivetestingtechnique,microwave
scanningisabletoestimateseveralimportantwoodparametersincludingdensity,moisturecontent,
andgrainangle[4–13]. FollowingBucur[14],othernon-destructivetechniquesuseX-rays,gamma
rays,thermalimaging,ultrasound,nuclearmagneticresonance,andneutronimaging. Microwave
based nondestructive techniques have been applied to a variety of applications, including in situ
microwavetomography,moisturecontentdetermination,anisotropymeasurement,onlineimaging,
internalinspectionoflogs,mechanicalgrading,anddefectdetection.Microwavesandultrasoundhave
similarresolution,butmicrowavesarenotlimitedbythecontactnatureofultrasoundtransducers.
Asfarasweknow,mostmodelsofmicrowavepropagationinsidewoodarebasedonrectangular
configurations,althoughsomestudiesinmoregeneralgeometriesarepublished[15,16]. In[17],the
effectofannualgrowthringsonnondestructivemeasurementsisexplicitlystudied. Thislimitsthe
applicationofmicrowavetechniquesinareassuchasmeasurementoflivingtreesinsitu.
Importanteffortshavebeenmadetodevelopgeneralizedmodelsforwatercontentpredictionin
woodusingitsdielectricproperties,butdetailedmodelsofelectromagneticpropagationinsidewood
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aremoremodest. Duetothecomplexity,mostmodelsdonotconsidertheeffectsofthegeometryor
thedielectricvariationandoscillationscausedbytheringstructureoftimber.
This work studies microwave propagation and electric field distribution at 3 GHz inside
cylindrical wood structures, following the procedures described in [18–20]. From our previous
work [21] with Bragg fibers, it is expected that important focusing effects of microwaves inside
woodduetoitsringstructuremaybeobserved. Thesefocusingphenomenacouldbeveryimportant
fortheinterpretationoftestingdataandforheatingapplications,wheresomezonescouldbeburned
whiletheotherswouldbealmostcold. Besides,thequickproductionofveryhotwatervaporand
volatilesinsmallareasoftheirradiatedwoodcouldproduceexplosions,deformations,andcracks.
Asmicrowaveheatingandtreatmentofwoodisaneconomicalalternativetoothermethods,itisvery
importanttoknowthedistributionofradiationinsidethepiecesandtheperturbationsintroducedby
thecomplexstructureofthisnaturalmaterial.
Two complementary approaches have been used. First, an exact model based on a complete
formulationoftheMaxwellequationsincylindricalcoordinateshasbeendeveloped. Thismodelis
onlyanalyticallytractableinthelimitofzeroconductivity. Thisimpliesthattheimaginarypartof
thewoodpermittivityisnegligible,whichisonlycorrectfordrywood. However,someinteresting
similaritiesaresharedbythedryandmoistwoodmodels. Thissemianalyticalmodelincludesthe
effectofgrainanglebymeansofacorrectionoftheeffectivepermittivityinsidewood,avoidingthe
fulluseofhelicoidalcoordinateswhichproduceequationstoocomplexforthiscase. Theresulting
equationsarethensolvedbymeansofanimplicitRunge–Kutta(RK),whichprovidestheunconditional
stabilityneededtocalculatethevaluesofthecorrectedpermittivity.
The second model uses a full scale numerical simulation of a cylindrical, ring shaped, wood
sample.Moreover,theanalysisoftheinfluenceofthenumberofringsandthegeometryoftherings,by
definingunequalringsgeometry,intheelectricfielddistributionhasbeendone. Thesimulationswere
performedbymeansofcommercialsoftware,CSTMicrowaveStudio®(2017version,Framingham,
MA, USA) [22], which uses finite-integral technique (FIT) algorithms for the computation of the
electromagneticfield. Othernumericalsimulationapproachestothestudyofmicrowavepropagation
inwoodcanbeseenin[23].
Bothmodelsagreeinthelimitofzeroconductivityandpredictimportanteffectsduetothering
structureofcylindricalwood. Theresultsshowanimportantfocusingoftheradialdistributionofthe
electricfieldwhichdependsonthenumberandthicknessofthegrowthrings. Also,stationarywave
patternsareobservedduetothefinitelengthofwoodlogs. Theimportantpointofthisworkisto
showtheeffectoftheconcentricringstructureofwoodonthemicrowavepropagationinsidetimber.
ThiseffectissimilartoaBraggfiber,producinginterestingfocusingphenomenaaswillbeshown.
The rest of the paper is organized as follows: mathematical formulation involving both
semianalyticalandnumericalmodelsisexplainedinSection2,thenresultsarepresentedanddiscussed
indetailinSection3. Finally,relevantconclusionsarepresentedinSection4.
2. MathematicalFormulation
2.1. SemianalyticalModel
Rawcylindricaltimbercanbeconsideredastackofconcentricallydielectriclayers,soamodel
witharbitraryperiodicvariationofdielectricpermittivityisthemostaccurate. Considerablesymbolic
manipulationoftheMaxwellequationsincylindricalcoordinatesisnecessaryforarbitrarypermittivity
functionswhichcandependonlyontheradiusofthecylinder. Theequationsfortheexactmodelare
derivedfromourpreviousworks[24,25]. Theyarereproducedhereforcompleteness.
Theequationswereobtainedasfollows. First,wewritethecompletesetofMaxwellequationsin
cylindricalcoordinates
1∂(rDr) + 1∂Dφ + ∂Dz = ρ (1)
r ∂r r ∂φ ∂z
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1∂(rBr) + 1∂Bφ + ∂Bz =0 (2)
r ∂r r ∂φ ∂z
1∂Ez − ∂Eφ + ∂Br =0 (3)
r ∂φ ∂φ ∂t
∂Er − ∂Ez + ∂Bφ =0 (4)
∂z ∂r ∂t
(cid:0) (cid:1)
1∂ rEφ − 1∂Er + ∂Bz =0 (5)
r ∂r r ∂φ ∂t
1∂Hz − ∂Hφ − ∂Dr −J =0 (6)
r
r ∂φ ∂z ∂t
∂Hr − ∂Hz − ∂Dφ −J =0 (7)
φ
∂z ∂r ∂t
(cid:0) (cid:1)
1∂ rHφ − 1∂Hr − ∂Dz −J =0 (8)
z
r ∂r r ∂φ ∂t
Performingsomealgebraicmanipulations,thefullsetofMaxwellequationscanbereducedtoa
pairofcoupledordinarydifferentialequations. Thefollowingansatzisinsertedintothecylindrical
MaxwellEquations(1)–(8),whereristheradius,zisthelengthofthewoodlog,andphiistheangle
measured from the radius. E is the electric field strength, H is the magnetic field, D is the electric
displacementfield,Bisthemagneticfluxdensity,omegaistheangularfrequency,tisthetime,andlis
theangularmomentnumber.
E = E e (r)e(ilφ)×e(iβz)×e−(iωt) (9)
r,φ,z 0[r,φ,z] r,φ,z
H = H h (r)e(ilφ)×e(iβz)×e−(iωt) (10)
r,φ,z 0[r,φ,z] r,φ,z
Thisproducesasetofseparableequations,assumingρ =0(chargedensity)and J =0(current
density),whichimpliesthatconductivityisnotconsidered. EquationswhosetermsareintheformβE
orβHareselected. TheseareEquations(1),(2),(4),(6)and(7). Now,theEquations(4),(6)and(7)can
bealgebraicallysolvedwithrespecttotheradialcomponentse (r),e (r),h (r),andh (r),
r φ r φ
(cid:18) e (r)(cid:19) (cid:16) (cid:17)−1
e (r) = − H lµ h (r)ω−iE βr z × ε ε (r)µ E rω2−E β2r (11)
r 0z 0 z 0z 0 r 0 0r 0r
r
(cid:18) h (r) (cid:19) (cid:16) (cid:17)−1
e (r) = − iH µ r z ω+E lβe (r) × ε ε (r)µ E rω2−E β2r (12)
φ 0z 0 0z z 0 r 0 0φ 0φ
r
(cid:18) h (r)(cid:19) (cid:16) (cid:17)−1
h (r) = − ε ε (r)E le (r)ω+iH βr z × ε ε (r)µ H rω2−H β2r (13)
r 0 r 0z z 0z 0 r 0 0r 0r
r
(cid:18) e (r) (cid:19) (cid:16) (cid:17)−1
h (r) = ε ε (r)E re (r) z ω−H lβh (r) × ε ε (r)µ H rω2−H β2r (14)
φ 0 r 0z z 0z z 0 r 0 0φ 0φ
r
Substituting this solution into Equations (1) and (2), another pair of ordinary differential
equations is obtained. In order to analyze these equations, it is convenient to decoupled them.
Usinge (r) = e (r)+ie (r);h (r) = h (r)+ih (r)andtakingbothrealandimaginarypartsinthe
z zR zI z zR zI
resultingequations,thefollowingexpressionsarefinallyobtained
e(cid:48)(cid:48) (r)+A e(cid:48) (r)+A e(cid:48) (r)+A h(cid:48) (r)+A h(cid:48) (r)+A e (r)+
zR 11 zR 12 zI 13 zR 14 zI 15 zR (15)
A e (r)+A h (r)+A h (r)+A =0
16 zI 17 zR 18 zI 19
e(cid:48)(cid:48) (r)+A e(cid:48) (r)+A e(cid:48) (r)+A h(cid:48) (r)+A h(cid:48) (r)+A e (r)+A e (r)+
zI 21 zR 22 zI 23 zR 24 zI 25 zR 26 zI (16)
A h (r)+A h (r)+A =0
27 zR 28 zI 29
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h(cid:48)(cid:48) (r)+A e(cid:48) (r)+A e(cid:48) (r)+A h(cid:48) (r)+A h(cid:48) (r)+A e (r)+A e (r)+
zR 31 zR 32 zI 33 zR 34 zI 35 zR 36 zI (17)
A h (r)+A h (r)+A =0
37 zR 38 zI 39
h(cid:48)(cid:48) (r)+A e(cid:48) (r)+A e(cid:48) (r)+A h(cid:48) (r)+A h(cid:48) (r)+A e (r)+A e (r)+
zI 41 zR 42 zI 43 zR 44 zI 45 zR 46 zI (18)
A h (r)+A h (r)+A =0
47 zR 48 zI 49
where A (i = 1, 2, 3 and j = 1, ..., 9) coefficients try to simplify the writing of the final equations,
ij
becausethecompleteformulaeareratherlong. Thesecoefficientsmultiplytheradialcomponents,
bothrealandimaginary,oftheaxialelectricandmagneticfieldinsidethecylindricalwoodsample.
Thevalueofcoefficients A definedintheEquation(15)aredetailedinthefollowingequations
ij
(fromEquations(19)–(22))
(cid:16) (cid:17) (cid:16) (cid:17)−1
A = ε ε2(r)µ ω2−β2rε(cid:48)(r)−β2ε (r) × ε ε2(r)µ rω2−β2rε2(r) (19)
11 0 r 0 r r 0 r 0 r
(cid:16) (cid:17)
A = ε ε (r)µ r2ω2−β2r2−l2 r−2 (20)
15 0 r 0
A = (cid:0)ε(cid:48)(r)µ H lβω(cid:1)(cid:16)ε ε2(r)µ E rω2−ε (r)rE β2(cid:17)−1 (21)
18 r 0 0z 0 r 0 0z r 0z
andwhere
A = A = A = A = A = A =0 (22)
12 13 14 16 17 19
A (i=2andj=1,...,9)coefficientsemployedintheEquation(9)aredefinedintheEquations
ij
(23)–(26)
(cid:16) (cid:17) (cid:16) (cid:17)−1
A = ε ε2(r)µ ω2−β2rε(cid:48)(r)−β2ε (r) × ε ε2(r)µ rω2−β2rε2(r) (23)
22 0 r 0 r r 0 r 0 r
(cid:16) (cid:17)
A = ε ε (r)µ r2ω2−β2r2−l2 r−2 (24)
26 0 r 0
A = −(cid:0)ε(cid:48)(r)µ H lβω(cid:1)(cid:16)ε ε2(r)µ E rω2−ε (r)rE β2(cid:17)−1 (25)
27 r 0 0z 0 r 0 0z r 0z
andwhere,
A = A = A = A = A = A =0 (26)
21 23 24 25 28 29
In the following equations from Equations (27)–(30) the values of the A (where i = 3 and
ij
j=1,...,9)usedintheEquation(17)aredetailed.
(cid:16) (cid:17)(cid:16) (cid:17)−1
A = − ε ε(cid:48)(r)µ rω2−ε ε (r)µ ω2+β2 ε ε (r)µ rω2−β2r (27)
33 0 r 0 0 r 0 0 r 0
A = −(cid:0)ε ε(cid:48)(r)E lω(cid:1)(cid:16)ε ε (r)µ H rω2−H β2r(cid:17)−1 (28)
36 0 r 0z 0 r 0 0z 0z
(cid:16) (cid:17)
A = ε ε (r)µ r2ω2−β2r2−l2 r−2 (29)
37 0 r 0
and,
A = A = A = A = A = A =0 (30)
31 32 34 35 38 39
IntheEquations(31)–(34)thevalueofthecoefficients A (wherei=4andj=1,...,9)whichhave
ij
beendefinedintheEquation(18),aredetailed.
(cid:16) (cid:17)(cid:16) (cid:17)−1
A = − ε ε(cid:48)(r)µ rω2−ε ε (r)µ ω2+β2 ε ε (r)µ rω2−β2r (31)
44 0 r 0 0 r 0 0 r 0
A = (cid:0)ε ε(cid:48)(r)E lβω(cid:1)(cid:16)ε ε (r)µ H rω2−H β2r(cid:17)−1 (32)
45 0 r 0z 0 r 0 0z 0z
(cid:16) (cid:17)
A = ε ε (r)µ r2ω2−β2r2−l2 r−2 (33)
48 0 r 0
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andwhere,
A = A = A = A = A = A =0 (34)
41 42 43 46 47 49
wherex’denotesdx/dr,β = 2πεreff1/2 andε istheeffectivepermittivityparameter.
λ reff
Wecanobservethatthissystemofequationsisnotcompletelycoupled. Equations(15)and(18)
formapairofmutuallycoupledequationsine (r)andh (r). ThesamecanbesaidforEquations(16)
zR zI
and(17). Eachpairofmutuallycoupledequationshaveasimilarmathematicalstructure. Then,it
ispossibletoreducetheproblemofmicrowavepropagationinsidecylindricalwoodsampleswith
arbitraryradialrelativepermittivitiestoapairofordinarydifferentialequations. Weshallchoose
Equations(15)and(18)forconvenience. Theresultswerecalculatedandsimplifiedbymeansofthe
Maximaprogram[26].
Inordertosolvethefinalequations,animplicithighorderRunge–Kuttaalgorithmwasused.
ImplicitRKalgorithmsaremorestablethantheirexplicitcounterparts[27]andthisfacteliminates
someproblemsrelatedwiththeperiodicnatureofthedielectricstructureofwoodandtheextreme
sensibility of the solutions to the value of the effective permittivity. Although the core equations
ofthisproblemarethesameastheequationsforaBraggfiberinourpreviouswork[24],thereare
importantconceptualandcomputationaldifferencesbetweenbothproblems. Firstofall,therelative
permittivitiesofadjacentlayersinthewoodmodelarelargerthaninaBraggfiberandconductivityis
notpresentinopticalfibers. Moreimportantevenistheconsiderationofthegrainangleinthewood
modelwhichintroducesnewcomplexitiesandnewpropagationmodes. Thisgrainanglewouldbe
similartoaBraggfiberwithtorsion,whichwasnotconsideredinourpreviouswork. Also,theratio
betweenthewavelengthandthedimensionsoftheringsisdifferent,producingsubtleeffectsthathave
forcedustodevelopanewnumericalapproachtosolvethefinalequations. Whileinourprevious
calculations, collocation methods were successfully used, in this case an implicit, unconditionally
stable,RKalgorithmhasbeendevelopedinordertoavoidlargenumericalinstabilitiesproducedby
themodesplittingintroducedbythegrainangledependenceontherelativepermittivity.
2.2. NumericalSimulation
Duetomathematicalconstraints,conductivitycannotbeeasilyintroducedintheexactmodel,
becausetheMaxwellequationscannotbedecoupled. Therefore,adetailednumericalsimulationofan
idealizedcylindricalsampleofwoodhasbeenperformedinordertostudytheeffectofmoistureon
microwavepropagation. ThecommercialsoftwareCSTMicrowaveStudio® [22],whichimplementsa
finite-integraltechnique(FIT)approach,wasused.
Thesamplewasformedbyconcentricallylayersoftwodifferentpermittivitiescorrespondingto
typicalepsilonvaluesofpoplar[27]usingthedielectricCartesiantensorfacilityofCST(Computer
Simulation Technology). The exact values are shown in Figure 1. Summer and spring rings are
independentlyconsidered. Experimentaldielectricandconductivitydataforbothdryandwetwood
wereobtainedfrom[28–31]. Thethicknessofgrowthringsinsidepoplarhasbeentakenfrompictures
ofseveralsamplesfromtheMadridareaandaroundedmeanvaluehasbeenusedforthesimulations.
Thus, thisstudymustbeconsideredasageneraldescriptionofmicrowavepropagationinsidean
actualringstructuredwoodbutnotadetailedanalysisofaparticularwoodsample.
Theselectedfrequencyforourstudyhasbeen3GHzfornumericalconvenienceinthecaseofthe
semianalyticalcalculations. Thepermittivityvaluescanbeconsideredalmostunalteredfromthemore
common2.45GHzaccordingto[27]. However,accordingtotheelectricfielddistributionshownin
Figures2and3fordryandwetpoplarsample,theeffectofthemicrowavefrequencyisnoticeable
inspiteoftheconstantvalueofthepermittivityatthosefrequencies. Animportantconsequenceof
theseresultsistherequirementofseveralfrequenciestoachieveamorehomogenouselectricfield
distributioninsidewood.
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Figure 1. Structural configuration of dielectric tensors and excitation used for CST (Computer
Figure 1. Structural configuration of dielectric tensors and excitation used for CST (Computer
Simulation Technology) analysis.
SimuFlaigtiuorne T1e. cShtnruoclotugrya)l acnonafliygsuisra.tion of dielectric tensors and excitation used for CST (Computer
Simulation Technology) analysis.

Figure 2. Electric field distribution inside dry poplar. (a) f = 2.45 GHz 50% permittivity variation ; (b)
f = 3 GHz 50% permittivity variation; (c) f = 2.45 GHz 100% permittivity variation; and (d) f = 3 GHz
Figure 2. Electric field distribution inside dry poplar. (a) f = 2.45 GHz 50% permittivity variation; (b)
Figur1e002%. pEelermctirtitcivfiiteyl vdardiaistitorinb. utioninsidedrypoplar. (a)f =2.45GHz50%permittivityvariation;
f = 3 GHz 50% permittivity variation; (c) f = 2.45 GHz 100% permittivity variation; and (d) f = 3 GHz

(b)f =1030%G Hpezrm50it%tiviptye vrmariiatttiivonit.y variation; (c) f = 2.45 GHz 100% permittivity variation; and
( d)f =3GHz100%permittivityvariation.
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Alimitationofourstudyisthatrealwoodhasdifferentgrowthringthicknesses,heterogeneous
A limitation of our study is that real wood has different growth ring thicknesses, heterogeneous
zoneswithdefects,moisturegradients,gaps,etc.,whichcouldbeintroducedinfuturestudies.
zones with defects, moisture gradients, gaps, etc., which could be introduced in future studies.

Figure 3. Electric field distribution inside wet poplar. (a) f = 2.45 GHz 50% permittivity variation;
Figure3. Electricfielddistributioninsidewetpoplar. (a)f =2.45GHz50%permittivityvariation;
(b) f = 3 GHz 50% permittivity variation; (c) f = 2.45 GHz 100% permittivity variation; and (d) f = 3
(b)f =3GHz50% permittivity variation; (c) f = 2.45 GHz 100% permittivity variation; and
GHz 100% permittivity variation.
(d)f =3GHz100%permittivityvariation.
3. Results and Discussion
3. ResultsandDiscussion
In both models, analytical and numerical, axial excitation and propagation of microwaves are
studied. The timber is excited by an incident plane wave along the z-axis, the electric field is
Inbothmodels,analyticalandnumerical,axialexcitationandpropagationofmicrowavesare
perpendicular to the cylinder axis, and so the magnetic field, as shown in Figure 1. The relative
studied. The timber is excited by an incident plane wave along the z-axis, the electric field is
permittivity function used for the semianalytical analysis is shown in Figure 4. The semianalytical
perpendicular to the cylinder axis, and so the magnetic field, as shown in Figure 1. The relative
model shows a slow modulation of the amplitude of the electric field due to the periodic nature of
permittivityfunctionusedforthesemianalyticalanalysisisshowninFigure4. Thesemianalytical
the dielectric layers of wood, as can be seen in Figures 5 and 6. The behavior of this permittivity
modelshowsaslowmodulationoftheamplitudeoftheelectricfieldduetotheperiodicnatureofthe
function is modelled on [28]. For simplicity and easier convergence of the RK algorithm, uniform
dielecgtrroicwltahy reirnsg otfhiwckonoedss, aasndca an lboweesre evnariinatFioing ubreetsw5eeann ddif6f.erTehnet wbeohodav dioierleocftrtihc ivsaplueersm iint ttihveit ryadfuianlc tion
ismoddierellcetdiono nw[a2s8 ]u.sFedo.r Tsihmisp ilsi ccilteyaralny da elaimsiietratcioonn voef rtgheen sceemoifanthaelyRticKala mlgoodrietlh amnd,u onniefo orfm thger omwatinh ring
thicknreeasssoanns dfora ulosiwnge ralvsaor aia ftuilol nnubmetewriecealn sdimifufelarteinont wtooolo. dThduies,l ethcetr ricesvualtlsu aerse innotth eexarcatdlyi atlhde israemctei oans was
used.thTeh oinseiss ccallecaurllaytead luimsinitga etixopneroimftehnetasl epmopilaanr adlayttai.c almodelandoneofthemainreasonsforusing
For the previous reasons, the semianalytical method is restricted to only 5 rings, while the full
alsoafullnumericalsimulationtool. Thus,theresultsarenotexactlythesameastheonescalculated
numerical simulations range from 5 growth rings to 20. The thickness of growth rings are taken form [27].
usingexperimentalpoplardata.
The semianalytical results can be seen as a validation technique for this study. The length of the
Forthepreviousreasons,thesemianalyticalmethodisrestrictedtoonly5rings,whilethefull
sample is 1 m in both methods. The semianalytical and numerical simulations have been done at
numerical simulations range from 5 growth rings to 20. The thickness of growth rings are taken
3 GHz because the wavelength at that frequency is comparable to the diameter of the wood sample.
form[27]. Thesemianalyticalresultscanbeseenasavalidationtechniqueforthisstudy. Thelengthof

thesampleis1minbothmethods. Thesemianalyticalandnumericalsimulationshavebeendoneat
3GHzbecausethewavelengthatthatfrequencyiscomparabletothediameterofthewoodsample.
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MoreoMvoerre,otvheart, fthreaqt ufreenqucyenicsys iism siimlairlatro ttoh tehev valaulueew whhiicchh iiss eemmpploloyyeded ini ncocmommemrceiarcl imalicmroiwcraovwe advryeindgr ying
Moreover, that frequency is similar to the value which is employed in commercial microwave drying
ovenso[v3e2n,s3 3[3]2.,33].
ovens [32,33].

Figure 4. Relative permittivity function used for the calculation of the electric field in the exact model.
FigurFei4g.uRree 4la. Rtievleatpiveer mpeirtmtivitittiyviftyu nfucnticotinonu suesdedf oforrt thhee ccaallccuullaattiioonn oof fthteh eeleecletrcitcr fiicelfide ilnd tihne tehxeacetx macotdmel.o del.

Figure 5. Particular solution with analytical analysis of the electric field inside a cylindrical wood
Figure 5. Particular solution with analytical analysis of the electric field inside a cylindrical wood
Figursea5m.pPlae rwtichuenla trhseo cluortiroenctiwveit fhacatnoar lfyotri ctahlea enffaelcytsiviseo rfeltahteiveel epcetrrmicifttieivldityin iss i0d.e70a acnydli nthder gicraalinw aonogdles iasm ple
sample when the corrective factor for the effective relative permittivity is 0.70 and the grain angle is
whenαth =e 4c5o°.r rectivefactorfortheeffectiverelativepermittivityis0.70andthegrainangleisα=45◦.
α = 45°.
The periodic structure produces the modification of the values of the relative permittivity,
ThepTehreio pdeirciosdtriuc csttururectpurroe dpurcoedsutchese mtheo dmifiocdaiftiicoantioonf tohfe tvhael uveasluoefs tohfe trheel arteivlaetipvee rpmeirtmtiivttiitvyi,twy, hich,
which, in turn, modifies the electromagnetic field modes inside the wood cylinder. In order to apply
in turwmnhi,circmohw,o idanvi tfieu eprsnro, tmphaeogdaeitfliieoecnst rtthooe mt healeeg cdnteretoatmiilceadgfi nesetlutdidc ymfi eoolfdd i nemtseoridnneassli idrniensigdt hest etrhuwec twouoroedo dofc c ywylloiinnodddee, rrh.. iIgnhI onerrd oferrerd qteour eanptocpielays p ply
microwave propagation to the detailed study of internal ring structure of wood, higher frequencies
microuwpa tvo etepnrso opf aGgHazti (omnictrootwhaevdese itna itlheed msmtu drayngoef)i anrtee nrnecaelsrsianryg. sHtrouwcetvuerre, aotf swucoho sdh,ohrti gwhaevrelfernegqtuhes,n cies
up to tens of GHz (microwaves in the mm range) are necessary. However, at such short wavelengths,
uptottheen esfofefcGt oHf zlim(mitiecdr oswkina vdeespitnh tahned minmcreraasnegde a)ttaerneunaeticoens ssahroyu.ldH boew setvuedri,edat csaurecfhulslhy.o Brteswidaevse, ltehne gths,
the effect of limited skin depth and increased attenuation should be studied carefully. Besides, the
theefcfoemctpouftalitmionitaeld desmkiannddse parteh faanr dhiginhcerre, asos etdhisa twteilnl ubae tcioonnssidheoruedld inb feustutured wieodrkcsa.r Ieff suulclyh. aBcecusirdaceys ,the
computational demands are far higher, so this will be considered in future works. If such accuracy
compcuotualtdio bnea lpdraecmticaanlldy saacrheiefvaerdh, iag hneorn,-sinovtahsiivsew dielltebremcinoantsioidn eoref dtreine faugteus rweowuoldrk bse. Ipfossusicbhlea, cfcour racy
could be practically achieved, a non-invasive determination of tree ages would be possible, for
example.
couldbepracticallyachieved,anon-invasivedeterminationoftreeageswouldbepossible,forexample.
example.
The solutions are valid only for certain values of the effective relative permittivity (permittivity
ThesTohlue tsioolnustioanres avrael ivdaloidn olynlfyo rfocre cretratianinv vaalulueess ooff tthhee eeffffeecctitvive ererlealtaivtiev peeprmeritmtivitittiyv (iptyer(mpietrtimviittyt ivity
eigenvalues) due to the existence of discrete propagation modes inside the cylindrical material. These
eigeneviagleunevsa)ludeus)e dutoe ttoh tehee exxiisstteennccee oof fdidscisrectree tperoppraogpataigona tmioondems iondsiedse tihnes icdyelintdhreiccayl mlinatderriiacla. lThmeasete rial.
eigenvalues are calculated by means of an iterative procedure which runs the RK algorithm searching
Theseeiegiegnevnavluaelus aersea craelccualalcteudl abtye dmebaynsm oef aann siteorfaatinvei tperroacteidvuerep wrohciecdh ururensw thhei cRhK raulgnosritthhemR seKaraclhgiongri thm
for the correct behavior of the solutions. In this case, the azimuthal number of the selected
searchfpoirnro gptahfgoea rtcitoohrner emccoto rdbreee chista lvb =ieo h1r. a Tovfhi oetrrheoef ofsrteoh,l euetvsieoornlyus .tm ioIonnd set.h iIsins c htchaarissaec,c taethsrieez ,eatdhz ibemya uazt ihdmaelfu intnhiutaeml vnbaeulrum eo bof ef trthhoeef estfhefleeeccsttieevldee cted
propagation mode is l = 1. Therefore, every mode is characterized by a definite value of the effective
propagationmodeisl=1. Therefore,everymodeischaracterizedbyadefinitevalueoftheeffective
relativ epermittivity, whichcanbeexpressedasacorrectionfactormultiplyingthemeaneffective

relativepermittivity.
Forests 2018, 9, x FOR PEER REVIEW 9 of 16
Forestsr2e0la18ti,v9e, 1p0e6rmittivity, which can be expressed as a correction factor multiplying the mean effective9 of16
relative permittivity.

Figure 6. A particular solution with analytical analysis of the electric field inside a cylindrical wood
Figure6.Aparticularsolutionwithanalyticalanalysisoftheelectricfieldinsideacylindricalwoodsample
sample when the corrective factor for the effective relative permittivity is 0.66 and the grain angle is
whenthecorrectivefactorfortheeffectiverelativepermittivityis0.66andthegrainangleisα=30◦.
α = 30°.
TheeTxhaec texmacotd meloadlello awllsowthse thvea vriaartiiaotinono foft htheeg grraaiinn aannggllee,, αα. .TThhe egrgariani annagnleg ilse diesfdineefidn aesd thaes atnhgelea ngle
betwebeentwtheeent itmheb teimrabxeris aaxinsd anthde thwe owoododlo lgosg.s.I tIti sis tthhee lloonnggiittuuddininaal lararrarnagnegmeemnet noft wofowodo toimdbtiemrsb oerr tshoe rthe
pattern resulting from this. From an analytical perspective, if the grain angle had been inserted
patternresultingfromthis. Fromananalyticalperspective,ifthegrainanglehadbeeninserteddirectly
directly in the cylindrical Maxwell’s equations, they would not be separable. In fact, helicoidal
inthecylindricalMaxwell’sequations,theywouldnotbeseparable. Infact,helicoidalcoordinates
coordinates should be used, but the model was too complex for the present study. Instead of a full
should be used, but the model was too complex for the present study. Instead of a full helicoidal
helicoidal model, we have made the hypothesis that the effect of the grain angle can be considered
model,wehavemadethehypothesisthattheeffectofthegrainanglecanbeconsideredusingonlyits
using only its variation inside the dielectric permittivity profile.
variationinsidethedielectricpermittivityprofile.
The numerical simulation of the grain angle would imply a helicoidally arrangement of different
Twhoeondu emlemereicnatsl, ssiom tuhalat tpioanraomfetthriec gfurnacintioannsg slheowulodu blde uimsepdl yinsaidhee ltihceo Cidaarltleysiaarnr daniegleecmtreicn tteonfsodri foffe rent
woodtheele CmSeTn stosf,tswoarthe.a Itnp tahrisa mregeatrridc, fthuen acntiaolnytsicsahl omuolddebl eisu mseodre ignesniderealt,h aeltChoaurgtehs icaonndduicetlievcittyri ceftfeecntsso rof
theCSarTe shoafrtwd atore .inItnrotdhuiscer,e gifa radd,dtihtieonaanl ahlyytpicoathlemseosd aerlei snmoto mreagdeen. eDruael, atolt hthoeu glihmcitoantidounsc toivf itbyotehf fects
arehaarpdptrooainchtreos,d wuec eh,aivfea odpdtietdio fnoar lah cyopmobtihneesde sstaurdeyn, cootmmpaadrien.gD thuee rteosuthltes floimr ait naetiuotnrasl ogfrabiont ahnagplep arnoda ches,
dry wood. Then, the effect of moisture is studied using numerical analysis and the grain angle
wehaveoptedforacombinedstudy,comparingtheresultsforaneutralgrainangleanddrywood.
variation uses the analytical model.
Then,theeffectofmoistureisstudiedusingnumericalanalysisandthegrainanglevariationusesthe
In Figures 5 and 6, we can compare the results for α = 45° and 30°, respectively. It must be noted
analyticalmodel.
that the correction factor for the effective relative permittivity changes with the value of the grain
InFigures5and6,wecancomparetheresultsforα=45◦ and30◦,respectively. Itmustbenoted
angle, so that the corrections are 0.70 for α = 45° and 0.66 α = 45°. The ability to calculate the grain
thattahnegcleo irsr eimctpioorntafnatc, tboercafuosret ghreaienf faencgtliev ies rreellaatteidv ewpitehr tmhei tmtievcihtyancichaal nstgreesngwthi tohf wthoeodv.a Tluhee eofffetchtieveg rain
anglev,asloueth oaft tthhee rceolartrievcet ipoenrsmaitrteiv0it.y7 0dfuoer tαo =the4 5g◦raainnd an0.g6l6e αco=rre4c5ti◦o.nT ihs ecaalbciulliatytedto bcya lmcuelaantse othf eang rain
angleiitseriamtivpeo rbtiasnect,tiboenc aalugsoerigthrmai nwahnicghle fiinsdres lathtee dcowrriethct tahseymmpectohtaicn ibceahlasvtiroern goft hthoef owsociolldat.inTgh efieelfdfesc tive
valueionfsitdhee wreolaotdiv aeftpeerr ma inttuimvibtyerd oufe rtiongths.e Tghraeisne acnaglcluelactoirornesc taioren tiismcea lccuonlastuemdibnyg maneda nnsuomfearniciatlelyra tive
bisectuionnstaablgleo, reivthenm uwsinhgic thhefi inmdpslitchite Rcuonrrgeec–tKaustytam aplgtoortiitchmbe fhoarv tihoer porfopthaegaotsiocnil lsatetipn. gDfiueel tdos tihniss,i donelyw ood
two decimal figures and five rings are considered in Figures 5 and 6, while the numerical model—
afteranumberofrings. Thesecalculationsaretimeconsumingandnumericallyunstable,evenusing
which discards the grain angle influence—can be calculated for a larger number of rings.
theimplicitRunge–Kuttaalgorithmforthepropagationstep. Duetothis,onlytwodecimalfigures
The CST simulations have been done without considering the grain angle. The numerical values
andfiveringsareconsideredinFigures5and6,whilethenumericalmodel—whichdiscardsthegrain
for the simulations were: length = 1 m, number of cylindrical layers = from 5 to 20, radius = from 35
angleinfluence—canbecalculatedforalargernumberofrings.
to 145 mm. Growth ring thickness are different depending on the season. One type has been set to 10
Tmhme CanSdT tshiem outhleart itoon 5s mhmav. Tehbee veanludeos noef twhei tdhioeluecttcroicn csoindsetarinnt goft heaechg rlaayinera hnagvlee b.eTehne tankuemn ferroimca [l2v7]a lues
forthaensdim thuelya tairoen sshwowerne i:nl eFnigguthre =1.1 m,numberofcylindricallayers=from5to20,radius=from35to
145mm. Growthringthicknessaredifferentdependingontheseason. Onetypehasbeensetto10mm
andth eotherto5mm. Thevaluesofthedielectricconstantofeachlayerhavebeentakenfrom[27]
andtheyareshowninFigure1.
ThenumericalFITsimulationallowsthestudyofalayerofgrowthrings. Ingeneral,theresults
inFigures7–12provethecriticaleffectofthenumberofringsontheelectricfielddistributioninside
ofpoplarwood. Besides,thiseffectisnottrivialaccordingtotheradialelectricfieldvariationseen
inFigures13–18. Theelectricfieldmaximavarieswiththenumberofthegrowthringsboth,inthe
Forests 2018, 9, x FOR PEER REVIEW 10 of 16
The numerical FIT simulation allows the study of a layer of growth rings. In general, the results
in Figures 7–12 prove the critical effect of the number of rings on the electric field distribution inside
Forests 2018,9,106 10of16
of poplar wood. Besides, this effect is not trivial according to the radial electric field variation seen in
Figures 13–18. The electric field maxima varies with the number of the growth rings both, in the radial
radialandlongitudinaldirections. Adirectconsequenceoftheseresultsforheatingapplicationsisthat
and longitudinal directions. A direct consequence of these results for heating applications is that the
thethermaldistributioninsidewoodwouldvarycriticallyfromonesampletoanotherwithdifferent
thermal distribution inside wood would vary critically from one sample to another with different
thicknesses. Forthecalculations,twoextremecasesofthepermittivityvariationbetweengrowthrings
thicknesses. For the calculations, two extreme cases of the permittivity variation between growth
(50%and100%),accordingtotheliterature[28],havebeenstudied. ComparingFigure7,Figure9,
rings (50% and 100%), according to the literature [28], have been studied. Comparing Figures 7, 9,
andFigure11(50%dielectricvariation)withFigures8,10and12(100%dielectricvariation)itcanbe
and 11 (50% dielectric variation) with Figures 8, 10 and 12 (100% dielectric variation) it can be
concludedthattheeffectofthisvariableisveryimportant. Forexample,inFigures7and8itcanbe
concluded that the effect of this variable is very important. For example, in Figures 7 and 8 it can be
oobbsseerrvveedd tthhaatt tthhee mmoosstt rreemmaarrkkaabbllee cchhaannggeess iinn tthhee eelleeccttrriicc fifieelldd ddiissttrriibbuuttiioonn,, aarree pprroodduucceedd bbeettwweeeenn
tthhee 55--rriinngg ssaammpplleess aanndd tthhee 1155--rriinngg ssaammpplleess,, iinn bbootthh,, tthhee rraaddiiaall aanndd tthhee aaxxiiaall ddiirreeccttiioonn.. TThhiiss iimmpplliieess
tthhaatt,, eevveenn iiff ttwwoop poopplalarrs asammpplelsesh havaeveth tehes asmameed idmimenesniosinosn,sth, tehireierl eecltercitcraicn adntdh etrhmeramlbael hbaevhiaovricoarn cbane
cboem copmletpeldetdedif fdeirfefnert.ent.
IInn FFiigguurreess 99––1122 tthhee iinnflfluueennccee ooff mmooiissttuurree oonn tthhee ddiissttrriibbuuttiioonn ooff tthhee eelleeccttrriicc fifieelldd iinnssiiddee ppooppllaarr
wwoooodd iiss eexxpplloorreedd.. AAccccoorrddiinngg ttoo tthheessee rreessuullttss,, tthheerree aarree ttwwoo mmaaiinn ccoonncclluussiioonnss ooff tthhiiss:: tthhee eelleeccttrriicc fifieelldd
ddiissttrriibbuuttiioonn iiss ccoommpplleetteellyy aalltteerreedd bbyy tthhee pprreesseennccee ooff mmooiissttuurree aanndd tthhee nnuummbbeerr ooff rriinnggss cchhaannggeess tthhiiss
bbeehhaavviioorr iinn aa nnoonn--pprreeddiiccttaabbllee wwaayy.. FFoorr eexxaammppllee,, iinn tthhee fifivvee rriinnggss ccaassee,, mmooiissttuurree pprroodduucceess aa nnoottiicceeaabbllee
cceennttrraall ffooccuussiinngg eeffffeecctt mmaakkiinngg tthhee wwoooodd ssaammppllee ssiimmiillaarr ttoo aa mmeettaalllliicc wwaavveegguuiiddee.. OOnn tthhee ccoonnttrraarryy,, tthhee
1100--rriinngg ccaassee sshhoowwss aa ccoommpplleetteellyy ooppppoossiittee bbeehhaavviioorr,, ffrroomm aa cceennttrraall ddiissttrriibbuuttiioonn iinn tthhee ddrryy ccaassee ttoo aa
mmuucchh aatttteennuuaatteedd rraaddiiaall ddiissttrriibbuuttiioonn.. TThhee aaxxiiaall ddiissttrriibbuuttiioonn iiss aallssoo nnoottiicceeaabbllyy aalltteerreedd.. TThhee iimmpplliiccaattiioonnss
ffoorr hheeaattiinngg aapppplliiccaattiioonnss aarree cclleeaarr..
AA ddeettaaiilleedd ssttuuddyy ooff tthhee rraaddiiaall eelleeccttrriicc fifieelldd ddiissttrriibbuuttiioonn iiss sshhoowwnn iinn FFiigguurreess 1133––1188.. IInn tthhiiss sseett ooff
fifigguurreess,, tthhee ccoommbbiinneedd aanndd ccoommpplleexx iinntteerraaccttiioonn aammoonngg tthhee aannaallyyzzeedd vvaarriiaabblleess aarree mmoorree cclleeaarrllyy sseeeenn..
TThhee mmaaxxiimmuummo offt htheee leelcetcrtircicfi feiledldis ids idspislpaclaecdedfr ofrmomth ethcee ncteenrtetor atod aif fdeirfefenrtepnot spitoiosnitidoenp denepdeinngdionngt ohne
nthuem nbuemrobferth oef gthroew grthowritnhg rsi.nAglss. oA,ltshoe, sthhae psehaopfeth oef ethleec terlieccfitreicld fideilsdt rdibisutrtiiobnutiisognr iesa gtlryeaatflfye catfefde.cted.

Figure 7. Electric field distribution inside dry poplar wood with different values of the number of
Figure7. Electricfielddistributioninsidedrypoplarwoodwithdifferentvaluesofthenumberof
growth rings with 50% permittivity variation at 3 GHz.
growthringswith50%permittivityvariationat3GHz.