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Influence of moisture content on reflectance of granular
materials. Part II: optical measurements and modelling
Hélène Garay, Amandine Monnard, Dominique Lafon-Pham
To cite this version:
Hélène Garay, Amandine Monnard, Dominique Lafon-Pham. Influence of moisture content on
re-flectance of granular materials. Part II: optical measurements and modelling. Granular Matter,
Springer Verlag, 2016, 18 (3), �10.1007/s10035-016-0649-6�. �hal-02078364�
I n fl u e n c e of m oist u re c o nt e nt o n re fl e ct a n c e of g r a n ul a r m at e ri als.
P a rt II: o pti c al m e as u re m e nts a n d m o d elli n g
H el e n e G a r a y 1 · A m a n di n e M o n n a r d 1 · D o mi ni q u e L af o n- P h a m 1
A bst r a ct I n s o m e fi el ds, it’s i m p ort a nt t o c o ntr ol t h e m ois-t ur e c o nois-t e nois-t of gr a n ul ar m aois-t eri als wiois-t h o uois-t a n y i n v asi v e m et h o d. M oist ur e c o nt e nt is s us p e ct e d t o pl a y a r ol e i n t h e o c c urr e n c e of s o m e u n w a nt e d p h e n o m e n a li k e r e m o bili z a-ti o n of p ara-ti cl es o n r o c k w alls i n c a v es f or e x a m pl e. T h e h y gr ot h er m al c o n diti o ns ar e i m p ort a nt as t h e y c a n m o dif y t h e m oist ur e c o nt e nt of t h e gr a n ul ar m at eri als o n t h e w all l e a di n g t o a h u mi di fi c ati o n or at t h e c o ntr ar y t o a dr yi n g of t h e p o w d er. I n t his st u d y, o pti c al m e as ur e m e nts ar e us e d t o c h ar a ct eri z e a w et p o w d er d uri n g its dr yi n g pr o c ess. Diff er-e nt st er-e ps of t his d y n a mi c dr yi n g pr o c er-ess ar er-e m o d er-ell er-e d a n d t h a n ks t o a n a d a pt ati o n of t h e e xisti n g M el a m e d’s m o d el of s c att eri n g of li g ht b y a p arti c ul at e m e di u m, si m ul ati o n r es ults ar e c o m p ar e d wit h i nstr u m e nt al m e as ur es. T h e m o d elli n g of s u c h a s yst e m all o ws t h e esti m ati o n of m oist ur e c o nt e nt of a gr a n ul ar m at eri al a n d bri n gs i nf or m ati o n a b o ut t h e l o c ali z a-ti o n of w at er i n t h e p arti c ul at e m e di u m.
Ke y w o r ds M oist ur e c o nt e nt · Gr a n ul ar m at eri al · O pti c al m e as ur e m e nts · M el a m e d’s m o d el · Dr yi n g pr o c ess
I nt r o d u cti o n
C ol o ur is a c o m pl e x pr o p ert y of m at eri als t h at c a n i n s o m e c as es b e c o nsi d er e d as a m ar ker of a n ot h er pr o p ert y m or e dif fi c ult t o m e as ur e: e x a m pl es c a n b e f o u n d i n t h e t e xtil e i n d ustr y [1 ,2 ] or f o o d i n d ustr y [3 ,4 ]. I n t his st u d y, pr es e nt e d i n t w o p arts ( p art I a n d p art II, h er e), t h e ai m is t o s h o w h o w o pti c al m e as ur e m e nts c a n h el p t o esti m at e t h e m oist ur e
B
H el e n e G ar a yh el e n e. g ar a y @ mi n es- al es.fr
1 C 2 M A – E c ol e d es mi n es d’ Al ès, 6 a ve n u e d e Cl a vi èr es, 3 0 3 1 9 Al es C e d e x, Fr a n c e
c o nt e nt of gr a n ul ar m at eri als. T h e o pti c al m e as ur e m e nts pr o-p os e d h a ve t h e a d va nt a g e of b ei n g n o n-i n vasi ve, a n ess e nti al pr o p ert y i n s o m e fi el ds s u c h as h erit a g e c o ns er vati o n.
I n t h e c as e of a t hi c k l a y er of a gr a n ul ar m at eri al, c ol o ur d e p e n ds o n p ar a m et ers s u c h as t h e n at ur e of t h e p arti cl es, t h eir si z e a n d s h a p e, a n d t h eir p a c ki n g. T h e pr es e n c e of wat er is als o a p ar a m et er t h at will c h a n g e t h e vis u al as p e ct as it c a n b e c o nsi d er e d as a n ot h er c o nstit u e nt of t h e s a m pl e.
I n t his sit u ati o n, t h e m ai n p h e n o m e n o n is t h e s c att eri n g of li g ht b y a c o n c e ntr at e d m e di u m i. e. m ulti pl e s c att eri n g. S o m o d els f o c us e d o n t h e s c att eri n g of li g ht b y a si n gl e p arti cl e s u c h as Mi e’s t h e or y c a n n ot b e us e d i n t his st u d y. A n ot h er t y p e of m o d el d es cri b e d i n t h e lit er at ur e c o nsi d ers t h e s a m-pl e as s u c c essi ve i n fi nit esi m al l a y ers of h o m o g e n o us m at eri al a n d d eri ves s o m e e q u ati o ns fr o m t h e r a di ati ve tr a nsf er e q u a-ti o n [5 ]: t h e N- fl u x m o d els [6 ] ar e t h e m ost pr e cis e of t h es e m o d els b ut t h e c al c ul ati o ns ar e fasti di o us. At t h e o p p osit e e xtr e m e, t h e t w o- fl u x m o d el, k n o w n as t h e K u b el k a- M u n k m o d el [ 7 ,8 ] is t h e e asi est of t h es e m o d els t o us e b ut t h e p ar a-m et ers of t his a-m o d el ar e t h e s c att eri n g ( S) a n d a bs or pti o n ( K) c o ef fi ci e nts of t h e gl o b al s yst e m. M or e r e c e ntl y a m o d el f or a m o n ol a y er of m o n o- dis p ers e d p arti cl es o n a s u bstr at e h as b e e n pr o p os e d [ 9 ] a n d t h e n e xt e n d e d f or a m o n ol a y er of p ar-ti cl es wit h a p arar-ti cl e si z e distri b uar-ti o n [1 0 ]. D.J. D a h m et al. c o nsi d er a s a m pl e t hr o u g h a r e pr es e nt ati ve l a y er. Wit h s o m e c o n diti o ns t h e y s h o w e d t h at it e xists a r el ati o n b et w e e n t h e a bs or pti o n a n d r e fl e cti o n of t h e w h ol e s a m pl e a n d t h e v ol-u m e fr a cti o n a n d t h e fr a cti o n of cr oss s e cti o n r es p e cti vel y of t h e r e pr es e nt ati ve l a y er. M el a m e d’s m o d el [1 1 ,1 2 ] is a n ot h er t y p e of s c att eri n g m o d el, as M el a m e d c al c ul at es t h e r e fl e ct e d, r efr a ct e d, a bs or b e d a n d tr a ns mitt e d p arts f or si n gl e p arti cl e a n d t h e n d e d u c es t h e b e h a vi o ur of t h e gl o b al s yst e m ass u m-i n g t h at t h e s p h erm-i c al p artm-i cl es ar e p a c ke d m-i n a h e x a g o n al f or m. G ar a y et al. [1 3 ] s h o w e d t h at t his m o d el h as s o m e a bilit y t o b e a d a pt e d f or a n ot h er s h a p e of p arti cl es: t h e y c
al-c ul at e d t h e eff e al-ct of t a ki n g elli ps oi ds of diff er e nt as p e al-ct r ati o a n d s h o w e d t h at t h e r es ult is cl os es t o t h e r es ult o bt ai n e d wit h s p h er es. T h at’s a n ot h er r e as o n t o c h o os e t his m o d el f or t his st u d y. T h e p art I of t h e st u d y s h o ws t h e e x p eri m e nts a n d t h e o pti c al m e as ur e m e nts m a d e t o tr a c k t h e dr yi n g pr o c ess of a mi n er al gr a n ul ar s a m pl e. T h e ai m is t o tr y t o tr a nsf er t his m et h o d t o t h e pr o bl e m of t h e c o ntr ol o ver ti m e of t h e pr e-hist ori c dr a wi n gs, pr o bl e m f or w hi c h t h e e x c h a n g e of wat er b et w e e n a m bi e nt air a n d p ai nts is cr u ci al. T h e st u d y pr es e nt e d i n t his p a p er is p art II of t h e gl o b al st u d y, f o c usi n g o n t h e m o d-elli n g of t h e dr yi n g pr o c ess. O n e st e p i n t his pr o c ess n e e ds p arti c ul ar d e vel o p m e nt: t h e wat er s urr o u n di n g t h e p arti cl es ( n ot e d p arti cl e @ wat er, as i n t h e d o m ai n of or g a ni c c h e mistr y w h e n a c or e p arti cl e is s urr o u n d e d wit h a c o ati n g). T his sit u-ati o n is m o d ell e d c o nsi d eri n g t h e i d e al c as e of a c or e p arti cl e s urr o u n d e d b y a s h ell of wat er of c o nst a nt t hi c k n ess.
1 P res e nt ati o n of M el a m e d’s m o d el
M el a m e d’s m o d el is a t h e or eti c al m o d el t h at d es cri b es t h e s c att eri n g of li g ht b y a p arti c ul at e m e di u m i n air. T h e p arti-cl es ar e m u c h l ar g er t h a n t h e i n ci d e nt li g ht wa vel e n gt h, ar e s p h eri c al, a n d ar e arr a n g e d i n a h e x a g o n al c o m p a ct m a n n er [1 1 ]. T h e r estri cti o n o n t h e si z e of p arti cl es is d u e t o t h e us e of t h e g e o m etri c al o pti cs r ul es.
M el a m e d e x pr ess es t h e r e fl e ct a n c e R as a f u n cti o n of n a n d k, r es p e cti vel y t h e r e al a n d i m a gi n ar y p art of t h e c o m-pl e x r efr a cti ve i n d e x, d, t h e di a m et er of t h e p arti cl es, λ t h e wa vel e n gt h of t h e i n ci d e nt li g ht, a n d a p ar a m et er x u, s p e ci fi c
t o t his m o d el, w hi c h t a kes i nt o a c c o u nt t h e arr a n g e m e nt of t h e p arti cl es ( E q. 1 ). T his e x pr essi o n i n t er ms of t h e p h ysi c al p ar a m et ers of t h e p o w d er is t h e m ai n i nt er est of t his m o d el as it b e c o m es p ossi bl e t o us e t h e m o d el t o pr e di ct r e fl e ct a n c e w h e n c ert ai n c o n diti o ns ar e c h a n g e d. R = f(n, k, d , λ, xu) ( 1) T h e f or m alis m of t h e m o d el is r e pr es e nt e d i n Fi g. 1 . F or gr e at er cl arit y, t h e t o p l a y er is s e p ar at e d fr o m t h e r est of t h e s a m pl e.
M el a m e d c o nsi d ers u nit ar y r a di ati o n i n ci d e nt o n t h e m e di u m. Aft er m ulti pl e i nt er-r e fl e cti o ns, a p art of t his i ni-ti al r a di ani-ti o n e m er g es u p war d. T h e ai m of t h e m o d el is t o q u a ntif y t his p art. T h e f u n cti o n f of E q. 1 c a n n ot b e writt e n s o si m pl y. I n fa ct t h e vari a bl es of t his e q u ati o n ar e us e d t o c al c ul at e ot h er c o ef fi ci e nts, s p e ci fi c t o t h e m o d el. S o m e of t h es e c o ef fi ci e nts ar e c al c ul at e d f or o n e p arti cl e:
• m e a n d m iar e t h e c o ef fi ci e nts of r e fl e cti o n, r es p e cti vel y
e xt er n al a n d i nt er n al, w hi c h d es cri b e t h e wa y t h e i n ci d e nt r a y i nt er a cts wit h t h e s urfa c e. T h e y d e p e n d o n b ot h t h e p arti cl e a n d m e di u m r efr a cti ve i n d e x.
• M is t h e c o ef fi ci e nt c orr es p o n di n g t o t h e m e a n att e n u a-ti o n of a r a y cr ossi n g t h e p ara-ti cl e o n c e (t h e att e n u aa-ti o n
Fi g. 1 F or m alis m of M el a m e d’s m o d el: f or gr e at er cl arit y, t h e u p p er l a y er of p arti cl es is s e p ar at e d fr o m t h e b ul k of t h e p arti c ul at e m e di u m
will d e p e n d o n t h e l e n gt h of t h e p at h i n t h e p arti cl e, a n d t h er ef or e o n t h e a n gl e of t h e i niti al r a y c o m p ar e d wit h t h e n or m al t o t h e p arti cl e s urfa c e). T h e c o ef fi ci e nt M d e p e n ds o n k, t h e a bs or pti o n c o ef fi ci e nt a n d o n d, t h e di a m et er of t h e p arti cl e.
• T r e pr es e nts t h e p art of t h e i nt er n al r a di ati o n t h at will b e “tr a ns mitt e d ” b y t h e p arti cl e aft er m ulti pl e i nt er-r e fl e cti o ns ( h eer-r e t h e t eer-r m “ter-r a ns missi o n ” t a kes i nt o a c c o u nt all t h e r a ys t h at l e a ve t h e p arti cl e w h at e ver t h eir dir e cti o n). It d e p e n ds o n mia n d M.
A fift h p ar a m et er c all e d x c orr es p o n ds t o t h e pr o b a bilit y t h at a r a y e m er g es a b o ve t h e s a m pl e. It d e p e n ds o n xu, k, d a n d T. It is t his p ar a m et er t h at e n a bl es t h e arr a n g e m e nt of t h e p arti cl es t o b e t a ke n i nt o a c c o u nt. T h e r e fl e ct a n c e R, a c c or di n g t o M el a m e d, c a n b e writt e n as: R = 1 + m e( A + B )+ A C − (1 + me( A + B )+ A C ) 2− 4 (m e+ C ) ( A + B ) 2 (m e+ C ) ( 2) A, B a n d C ar e a u xili ar y t er ms f or c al c ul ati o n. T h e y c a n b e e x pr ess e d as: A = 2 x me B = x (1 − 2 x me) T C = (1 − x ) (1 − m e) T ( 3) T h e r e fl e ct a n c e c al c ul at e d b y t h e m o d el t a kes i nt o a c c o u nt all t h e r a ys r es ulti n g fr o m t h e i nt er a cti o n of t h e i n ci d e nt r a di-ati o n wit h t h e m e di u m m a d e u p of p arti cl es a n d air. T his i nt er a cti o n c o nsists i n a s p e c ul ar p art ( n ot e d 2 x me o n Fi g. 1 )
a n d i n t h e r es ults of w h at h a p p e ns i n t h e b ul k of t h e s a m pl e (s c att eri n g a n d a bs or pti o n).
T his m e a ns, i n or d er t o c o m p ar e r es ults of m o d el c al c ul a-ti o ns wit h r e al m e as ur es, t h at t h e t ot al si g n al r e fl e ct e d b y t h e s a m pl e h as t o b e c oll e ct e d.
B ut m e as uri n g t h e r e fl e ct a n c e of p o w d ers is n ot as si m pl e as f or s oli d pl a n e m at eri als. T h e m e as ur e will d e p e n d o n t h e st at e of t h e s a m pl e s urfa c e, w hi c h c o nsists of t h e u p p er l a y er of p arti cl es. I n t h e c as e of p o w d ers, it is h ar d t o us e a n i nt e-gr ati n g s p h er e as t h e p o w d ers c a n p oll ut e t h e i nsi d e c o ati n g of t h e s p h er e. S o, o n e p ossi bilit y is t o us e a s p e ctr o g o ni o p h o-t o m eo-t er o-t o c h ar a co-t eri z e o-t h e a n g ul ar diso-tri b uo-ti o n of o-t h e si g n al, a n d t h e n, b y s p ati al i nt e gr ati o n of t h e si g n al, t o c al c ul at e t h e t ot al e missi o n. T h at e x p eri m e nt al w or k is d et ail e d i n p art I of t his st u d y.
T h e ai m of t his p art of t h e st u d y is t o d et er mi n e h o w M el a m e d’s m o d el c o ul d h el p t o d es cri b e t h e st at e of a p o w-d er w-d uri n g t h e w h ol e w-dr yi n g pr o c ess, at l e ast q u alit ati vel y, i. e. c a n t his m o d el e x pl ai n t h e vari ati o n m e as ur e d b y t h e s p e ctr o-g o ni o p h ot o m et er t h a n ks t o its p ar a m et ers ? T h e p h e n o m e n o n of dr yi n g is d y n a mi c as t h e wat er c o nt e nt vari es d uri n g t h e pr o c ess: wat er e va p or at es fr o m t h e s urfa c e a n d li q ui d wat er diff us es fr o m t h e b ul k of t h e s a m pl e t o t h e t o p of t h e s a m pl e, w hi c h is l ess s at ur at e d d u e t o e va p or ati o n.
T h e p ositi o n of wat er i n a p arti c ul at e m e di u m will d e p e n d o n t h e p h ysi c- c h e mi c al pr o p erti es of t h e p arti cl es, t h e wat er c o nt e nt, a n d t h e p ositi o n i n t h e s a m pl e ( n e ar t h e s urfa c e or i n t h e b ul k).
2 M o d elli n g of t h e d r yi n g p r o c ess
T h e e x p eri m e nt al p art is d et ail e d i n p art I of t his st u d y, t o w hi c h t h e i nt er est e d r e a d er is i n vit e d t o r ef er. T h e s a m pl e
st u di e d is a p ast e c o m p os e d of a y ell o w o c hr e a n d wat er all o w e d t o dr y u n d er a m bi e nt c o n diti o ns.
T h e dr yi n g pr o c ess of t his s a m pl e is d y n a mi c b ut s o m e st e ps c a n b e i d e nti fi e d. T h e y ar e s c h e m ati c all y r e pr es e nt e d i n Fi g. 2 :
• At t h e b e gi n ni n g of t h e o bs er vati o n, t h e p arti cl es ar e c o v-er e d b y a fil m of wat v-er.
• T h e wat er pr o gr essi vel y e va p or at es at t h e s urfa c e a n d t h e fil m is alr e a d y l ess s m o ot h.
• Air a p p e ars i n t h e b ul k of t h e s a m pl e, firstl y j ust u n d er t h e p arti cl es/ air i nt erfa c e, a n d t h e n, d u e t o diff usi o n of wat er fr o m t h e m ost c o n c e ntr at e d m e di u m t o war ds t h e l ess c o n c e ntr at e d m e di u m, i n t h e b ul k.
• “ Dr y ” s a m pl e: t h e s a m pl e is c o nsi d er e d as dr y w h e n its w ei g ht is c o nst a nt u n d er a m bi e nt c o n diti o ns.
St a g e 1 c a n b e c o nsi d er e d as “ p arti cl es i n wat er ” ( as i n st a g e 2) b ut wit h a fil m of wat er a b o ve. S o t h e pr es e n c e of t his fil m c a n b e m o d ell e d as a n o n-s c att eri n g, n o n- a bs or bi n g a n d s m o ot h m e di u m. T his c o at will h a ve a s p e c ul ar b e h a vi o ur.
St a g e 3 is t h e m ost c o m pli c at e d t o m o d el. It’s dif fi c ult t o esti m at e t h e t hi c k n ess of t h e c o at of wat er ar o u n d t h e p arti cl es: it’s pr o b a bl y n ot u nif or m all ar o u n d t h e p arti cl es, d u e t o t h e e xist e n c e of c a pill ar y b o u n ds. H o w e ver, as a first a p pr o xi m ati o n w e c a n c o nsi d er t his c o at u nif or m a n d t hi c k e n o u g h t o a p pl y M el a m e d’s m o d el.
T h e a d a pt ati o n of M el a m e d’s m o d el f or t his st e p is d et ail e d i n t h e f oll o wi n g s e cti o n.
3 M o d elli n g of t h e p a rti cl e @ w at e r st e p
L o o ki n g at t h e vari o us p ar a m et ers of t h e m o d el, t h e m ost a p pr o pri at e p ar a m et er f or m o di fi c ati o n is T, c orr es p o n di n g
Fi g. 2 S c h e m ati c r e pr es e nt ati o n of f o ur st a g es i n t h e dr yi n g pr o c ess: a Pr es e n c e of a fil m of wat er a b o ve t h e s urfa c e; b t h e fil m b e gi ns t o dr y a n d is n ot s m o ot h a n y m or e; c air a p p e ars i n t h e b ul k of t h e s a m pl e a n d d t h e s a m pl e is dr y. T h e cl assi c al M el a m e d’s m o d el c orr es p o n ds t o St a g e 4 ( Fi g. 2 ). St a g e 2 is r at h er si m pl e t o m o d el wit h M el a m e d’s m o d el: o nl y t h e p arts of r e fl e cti o n a n d r efr a cti o n at t h e i nt erfa c e p arti cl e/ air will b e m o di fi e d a n d t his
m o di fi c ati o n c a n b e esti m at e d b y c al c ul ati n g me a n d mi, r es p e cti vel y e xt er n al a n d i nt er n al r e fl e cti o ns c o ef fi ci e nts, wit h r el ati ve r efr a cti ve i n d e x np arti cl e/ nm e di u m
t o t h e p art of r a di ati o n i nsi d e a p arti cl e t h at will g o o ut of t h at p arti cl e aft er m ulti pl e r e fl e cti o ns i nsi d e it. I n t h e c as e of t h e p arti cl e @ wat er s yst e m, T will b e r e c al c ul at e d usi n g t h e h y p ot h es es t h at wat er d o es n ot a bs or b a n d t h e c o ati n g is t hi c k e n o u g h t o us e t h e l a ws of g e o m etri c al o pti cs. T his is a str o n g h y p ot h esis b e c a us e w e k n o w t h at, d uri n g t h e dr yi n g pr o c ess, t h e t hi c k n ess of t his c o at ar o u n d t h e p arti cl es will n ot b e u nif or m d u e t o t h e e xist e n c e of c a pill ar y b o u n ds. M or e o ver, c o nsi d eri n g t h at wat er d o es n’t a bs or b visi bl e li g ht, will n ot l et t a ke i nt o a c c o u nt t h e vari ati o n of t hi c k n ess of t his c o at as a p ar a m et er. Wit h t his t y p e of m o d elli n g, it will b e i m p ossi bl e t o f oll o w all t h e dr yi n g pr o c ess.
To c al c ul at e T, t h e c o ef fi ci e nt M is n e e d e d: i n t h e c al c ul a-ti o n, t h e c o ef fi ci e nt M is c al c ul at e d f or o n e p ara-ti cl e as if n o wat er is us e d.
All t h e r a ys t h at e m er g e o ut of t h e p arti cl e @ wat er s ys-t e m h a ve ys-t o b e ys-t a ke n i nys-t o a c c o u nys-t ( Fi g. 3 ). I n ys-t his s ysys-t e m, c o nsi d eri n g g e o m etri c al o pti cs, e a c h r a y w hi c h m e ets a n i nt erfa c e is p artl y r e fl e ct e d a c c or di n g t o Fr es n el’s l a w a n d p artl y r efr a ct e d. All t h e e m er gi n g r a ys c o m e fr o m t h e m ulti-pl e r e fl e cti o ns i nsi d e t h e wat er c o ati n g. T w o c o ef fi ci e nts mi,
m e a n i nt er n al r e fl e cti o n c o ef fi ci e nts, ar e d e fi n e d: firstl y m i 1at
t h e p arti cl e/ wat er i nt erfa c e a n d s e c o n dl y, mi 2, at t h e wat er/ air
i nt erfa c e. C o nsi d eri n g me 1, t h e m e a n e xt er n al r e fl e cti o n c o
ef-fi ci e nt, t h e c o ntri b uti o n of e a c h r a y c a n t h e n b e c al c ul at e d. L et us c o nsi d er a n i n ci d e nt r a y ( n ot e d “ a0” i n Fi g. 3 ). a1
is r e fl e ct e d at t h e s urfa c e i n t h e wat er c o at. a0b1 is r efr a ct e d
i nt o t h e p arti cl e. T h e str at e g y is t o f o c us o n t h e r a ys c o m-i n g dm-ir e ctl y fr o m a0 a n d c al c ul at e t h eir c o ntri b uti o ns aft er
s u c c essi ve cr ossi n gs of t h e p arti cl e. T h e c o ntri b uti o ns of t h e ot h er r a ys c o mi n g fr o m i nt er a cti o ns of a0 wit h t h e wat er c o at
will b e d e d u c e d fr o m t h e c al c ul ati o n of t h e c o ntri b uti o n of a0b1.
T h e i nt er a cti o ns c a n b e writt e n as: a1 = m e 1a0 a3 = m ∗i 2a1 = m ∗i 2m ∗e 1a0 a2 = a1 − a3 = m e 1(1 − m i 2) a0 a5 = (m e 1)2 m i 2(1 − m i 2) a0 a8 = (m e 1)3 (m i 2)2 (1 − m i 2) a0 a n d s o o n ( 4) T h e s u m of t h e r a ys e m er gi n g fr o m t h e i nt er a cti o n of a1
wit h t h e p arti cl e @ wat er s yst e m is
a2 + a5 + a8+· · · It c a n b e e x pr ess e d as a s u m of t h e t er ms of a g e o m etri c al pr o gr essi o n i n m∗ e 1m i 2. Wit h t h e n ot ati o ns of Fi g. 3 , t his s u m c a n b e writt e n as a3 i+ 2 = a1 1 − m1 − m i2 e 1m i2 ( 5)
t h e w at er c o at, its “ pri m ar y ” c o ntri b uti o n (i. e. wit h o ut a n y f urt h er cr ossi n g of t h e p arti cl e) c a n b e e x pr ess e d as
∞ j= 0
x3 j+ 2 = x1 ∗ 1 − m1 − m i2
e 1m i2 ( 6)
F or a1, E q. 5 r e pr es e nts t h e dir e ct c o ntri b uti o n f or T. B ut
t h e r a ys a0b1, a3b1, a6b1, a9b1, … w hi c h ar e r efr a ct e d i nt o t h e
p arti cl e a n d cr oss it, h a ve t o b e c o nsi d er e d. T h e i nt e nsiti es of t h es e r a ys c a n b e writt e n as: a0b1 = a0 (1 − m e 1) a3b1 = a0 (1 − m e 1) m e 1m i 2 a6b1 = a0 (1 − m e 1) m 2e 1m 2i 2 a9b1 = a0 (1 − m e 1) m 3e 1m 3i 2 ( 7)
All t h es e r a ys will h a ve a si mil ar p at h i n t h e s yst e m s o, f or e a c h r a y, t h e c o ntri b uti o n is a u ni q u e f u n cti o n of a0b1, a3b1,
a6b1, a9b1, r es p e cti vel y.
We c a n n ot e t h at, w h at e ver n > 0 , a3 nb1 = a∗3 (n − 1 )m e 1m i 2.
S o if f( a0b1) is t h e c o ntri b uti o n of a0b1, t h e t ot al c o
ntri-b uti o n of a0b1, a3b1, a6b1, a9b1 c a n b e writt e n as t h e s u m of
t his g e o m etri c al pr o gr essi o n i n me 1m i 2i. e.
T a0b1, a3b1, a6b1, a9b1 ,... = f (a0b1)∗ (1/ (m e 1m i 2)) ( 8)
T h e c o ntri b uti o n of a0b1 i n t h e s a m e wa y c a n b e di vi d e d
i n t w o p arts:
• firstl y t h e dir e ct c o ntri b uti o n of a0b2c2, a0b2c5, a0b2c8, ….
A c c or di n g t o E q. 6 , t his c o ntri b uti o n c a n b e writt e n as
a0b2c3 j+ 2 = a0b2c11 − m1 − m i2
e 1m i2 ( 9)
• A n d s e c o n dl y t h e c o ntri b uti o n of a 0b3, a0b3c3, a0b3c6,
a0b3c9, r efr a ct e d i n t h e p arti cl e c a n b e writt e n, as i n E q. 8 :
T (a0b3, a0b3c3, a0b3c6, a0b3c9 ,...)
= f(a0b3)∗(1 /( m e 1m i 2)) ( 1 0)
T his c al c ul ati o n, f o c usi n g o n a0b5, a n d t h e n o n a0b7 a n d
… a 0b2 n + 1 c a n b e c o nti n u e d.
S o, f or t h e c al c ul ati o n of t h e c o ef fi ci e nt T, all t h e c o ntri-b uti o ns h a ve t o ntri-b e t a ke n i nt o a c c o u nt, T = a11 − m1 − m i2 e 1m i2 + 1 1 − m e 1m i2 f (a0b1) ( 1 1) T h e s a m e c al c ul ati o n c a n b e m a d e f or e a c h r a y si mil ar t o a1 ( a0b2c1 or a0b4c1, et c., o n Fi g. 3 ). S o, w h at e ver t h e val u e of t h e r a y ( n ot e d x1) c o mi n g fr o m t h e p arti cl e a n d cr ossi n g
a a a a a a a a a0 a1 a0b3 a3b1 a0b3c3 a6b1 a9b1 a0b2c1 a0b2c3 a0b2c2 a0b2c4 a0b2c7 a0b2c8 a0b2c5 a0b2c6 a0b3c6 a0b3c9 a4 a2 a5 a3 a6 a8 a7 a9 a0b1 a0b2 a0b4 a0b4c1 a0b4c5 a0b4c4 a0b4c3 a0b4c2 a0b4c6 a0b5c3 a0b5c6 a0b5 a0b6 a0b6c1 …..
Fi g. 3 Pat h of a n i niti al r a y a0i nsi d e t h e p arti cl e @ wat er s yst e m
T = a11 − m1 − m i2 e 1m i2 + 1 1 − m e 1m i2 a0b2c11 − m1 − m i2 e 1m i2 + 1 1 − m e 1m i2 f (a0b3) ( 1 2) C o nti n ui n g t h e c al c ul ati o n of (a0b5): T = a11 − m1 − m i2 e 1m i2 + 1 1 − me 1m i2 a0b2c1 1 − m i2 1 − m e 1m i2 + 1 − m1 e 1m i2 a0b4c1 1 − m i2 1 − m e 1mi2+ 1 1 − m e 1m i2 f (a0b5) ( 1 3)
C o nti n ui n g t his c al c ul ati o n,
T (n ) = a11 − m1 − mi2 e 1mi2 + 1 − m i2 (1 − m e 1m i2)2a0b2c1 + 1 − m i2 (1 − m e 1m i2)3a0b4c1 + 1 − mi2 (1 − m e 1m i2)4a0b6c1 + · · · + 1 − mi2 (1 − m e 1m i2)n + 1a0b2 nc1 + f (a0b2 n + 1) 1 (1 − m e 1m i2)n + 1 ( 1 4) li mn → ∞ T (n ) = m e 1a01 − m1 − m i2 e 1m i2 + M (1 − m e 1) (1 − m i1) (1 − m i2) a0 n m n − 1 i1 M n (1 − m e 1m i2)n + 1 ( 1 5) as li mn → ∞ f (a0b2 n + 1)(1 − m e 11mi2)n + 1 = 0 b e c a us e 0 < m e 1m i2 < 1 a n d f (a0b2 n + 1) is a fi nit e n u m b er.
4 C al c ul ati o n a n d c o m p a ris o n wit h m e as u res
T h e m o d elli n g m et h o d d es cri b e d i n t his st u d y h as b e e n us e d t o c al c ul at e r e fl e ct a n c e of a s a m pl e w h os e c h ar a ct eristi cs ar e:
M e di a n p arti cl e si z e: d = 5 µ m. R efr a cti ve i n d e x: n = 2 .3 [ 1 4 ]
T h es e c h ar a ct eristi cs ar e t h os e of t h e o c hr e s a m pl e us e d i n t h e e x p eri m e nts d es cri b e d i n Part I of t his st u d y. I n t his s e cti o n, t h e m e as ur e m e nts a n al ys e d i n Part I ar e c o m p ar e d wit h t h e m o d elli n g r es ults.
A wa vel e n gt h is s el e ct e d: 6 2 0 n m. As it c orr es p o n ds t o a z o n e of t h e s p e ctr u m wit h hi g h r e fl e ct a n c e f or t h e s a m-pl e, t h e si g n al/ n ois e r ati o is m a xi mi z e d. T h e esti m ati o n of k, t h e a bs or pti o n c o ef fi ci e nt, is dif fi c ult, es p e ci all y f or gr a n ul ar m at eri als. We k n o w t h at r e fl e ct a n c e is a m o n ot o n e d e cr e as-i n g f u n ctas-i o n of t h e a bs or ptas-i o n c o ef fi cas-i e nt k. S o as-it’s p ossas-i bl e t o s ol ve n u m eri c all y — b y di c h ot o m y —t h e M el a m e d’s e q u a-ti o n i n a r e vers e f or m (i. e. k i n f u n ca-ti o n of R, n, d, x u a n d λ) . I n t h at c as e, R is t h e r e fl e ct a n c e val u e m e as ur e d f or t h e s a m pl e i n t h e dr y st at e [1 5 ]. T h e val u e of k o bt ai n e d t h a n ks t o t his c al c ul ati o n is t h e n us e d t o m o d el t h e ot h er st e ps i n t h e dr yi n g pr o c ess. T his is p ossi bl e b e c a us e t h e m o d elli n g of t h e ot h er st e ps of t h e dr yi n g pr o c ess is m a d e wit h t h e s a m e p arti cl es wit h, eit h er wat er, or a mi xt ur e of wat er a n d air i n t h e i nt er- p arti c ul ar s p a c e s o w e c a n c o nsi d er t h e s a m e val u e of k f or all t h e p arti cl es.
At 6 2 0 n m, t h e m e as ur e d val u e f or t h e r e fl e ct a n c e of t h e dr y s a m pl e is: R = 0 .5 3
S ol vi n g n u m eri c all y E q. 2 i n a r e vers e f or m gi ves k = 6 .5 1 .1 0− 4
T h e n k n o wi n g k, n, d, xu, a n d λ , t h e r e fl e ct a n c e, a c c or di n g
t o E q. 2 wit h s o m e a d a pt ati o ns d es cri b e d i n t h e S e cts. 2 a n d 3 of t his st u d y, is c al c ul at e d. T h e r es ults of t h es e c al c ul ati o ns ar e pr es e nt e d i n Ta bl e 1 .
T h e val u e of t h e r e fl e ct a n c e at 6 2 0 n m d uri n g t h e dr yi n g pr o c ess is e xtr a ct e d fr o m t h e m e as ur e m e nt d at a s et a n d pr e-s e nt e d i n Fi g. 4 , w h er e t h e val u ee-s fr o m Ta bl e 1 ar e a d d e d o n t h e c ur ve. It’s gi ve n i n t er ms of m oist ur e c o nt e nt e x pr ess e d i n dr y b asis ( w d. b.): w d .b .= 1 0 0m w m− m d w d W h er e m w is t h e w ei g ht of t h e h u mi d p o w d er a n d md is t h e w ei g ht of t h e dr y p o w d er. Ta bl e 1 C al c ul ati o n of r e fl e ct a n c e Dr yi n g pr o c ess st e ps C al c ul at e d R Parti cl es i n wat er wit h a fil m of wat er u psi d e 0. 6 7 9 Parti cl es i n wat er wit h o ut t h e fil m u psi d e 0. 6 1 3 Parti cl es @ wat er s yst e m 0. 3 5 3
0 % 2 0 % 4 0 % 6 0 % 8 0 % 1 0 0 % 1 2 0 % 1 4 0 % 0 % 2 0 % 4 0 % 6 0 % 8 0 % 1 0 0 % 1 2 0 % Re fl ec te d i nt en sit y ( %) M oi st ur e c o nt e nt ( % d. b.) I R efl 6 2 0 n m I R efl m el a m e d Fi g. 4 M e as ur e d r e fl e ct a n c e at 6 2 0 n m al o n g t h e dr yi n g pr o c ess (c ur v e ) a n d c al c ul at e d val u es f or s o m e p arti c ul ar st e ps (s q u ares). d. b. m e a ns “ dr y b asis ” as e x pl ai n e d b ef or e
T h e c al c ul at e d val u es ar e c o h er e nt wit h t h e m e as ur e-m e nts: t h er e is a e-m o e-m e nt i n t h e dr yi n g pr o c ess w hi c h c orr es p o n ds t o t h e c al c ul at e d r e fl e ct a n c e val u es a n d t his m o m e nt is i n a c c or d a n c e wit h o ur d es cri pti o n of t h e s yst e m m a d e t o d e fi n e t h e m o d elli n g h y p ot h esis.
T h e dr yi n g pr o c ess is d y n a mi c a n d o ur m o d el f o c us es o n s p e ci fi c all y i d e nti fi e d sit u ati o ns. N e vert h el ess, t his m o d el gi ves s o m e i nf or m ati o n a b o ut t h e wat er c o nt e nt of a gr a n ul ar s a m pl e wit h o ut a n y i n vasi ve n or d estr u cti ve e x p eri m e nt.
I n p arti c ul ar, t h e mi ni m u m of t h e c ur ve ( at 3 0 % of m ois-t ur e d. b.) c a n b e m o d ell e d, i. e. ois-t h e c al c ul aois-t e d val u es f oll o w ois-t h e tr e n d of t h e m e as ur e d c ur ve e ve n if w e k n o w t h at o ur s yst e m is pr o b a bl y n e ver t h e s a m e as t h e o n e w e m o d el. T his st u d y s h o ws t h e fl e xi bilit y of t h e M el a m e d’s m o d el w hi c h, t h a n ks t o its p hil os o p h y b as e d o n c al c ul ati o ns m a de c o n c er ni n g t h e p arti cl es, c a n b e a d a pt e d f or m or e c o m pli c at e d s yst e ms t h a n i d e al s p h er es s u c h as t h e p arti cl e @ wat er s yst e m c o nsi d er e d i n t his p art of t h e st u d y.
C o n cl usi o n
T h e st u d y pr es e nt e d i n t his p a p er s h o ws t h at M el a m e d’s m o d el f or t h e s c att eri n g of li g ht b y a p arti c ul at e m e di u m pr es e nts a r e al fl e xi bilit y as it gi ves t h e p ossi bilit y t o m o d-if y s o m e p ar a m et ers of t h e m o d el t o e n a bl e us t o t a ke i nt o a c c o u nt t h e wat er c o nt e nt of a gr a n ul ar m at eri al.
Wor ki n g o n vari o us c o ef fi ci e nts of t h e m o d el, it c a n c al c u-l at e t h e r e fl e ct a n c e of p arti cu-l es i n ot h er m e di a s u c h as wat er,
a n d c al c ul at e t h e r e fl e ct a n c e of p arti cl es s urr o u n d e d b y a c o ati n g of wat er i n air.
T h e dr yi n g pr o c ess of s u c h a gr a n ul ar s yst e m is d y n a mi c a n d t his m o d elli n g wit h M el a m e d’s m o d el f o c us es o n s o m e w ell- d e fi n e d sit u ati o ns. T h e m o d elli n g pr o c ess f oll o ws t h e vari ati o ns i n r e fl e ct a n c e wit h t h e wat er c o nt e nt i n t h e s a m-pl e. I n p arti c ul ar, it c a n i d e ntif y t h e mi ni m u m p oi nt of t h e c ur ve of r e fl e ct a n c e at 6 2 0 n m vers us m oist ur e c o nt e nt. T his r es ult is of i nt er est f or e x a m pl e t o gi ve i nf or m ati o n a b o ut c o m pl e x m e di a s u c h as p o w d ers a p pli e d t o a r o c k y s urfa c e. H y gr ot h er m al c o n diti o ns c a n m o dif y t h e wat er c o nt e nt of t h e p o w d er a n d t h e r es ults of t his st u d y c a n h el p t o h a ve a b ett er k n o wl e d g e of t h e wat er c o nt e nt a n d its l o c ali z a-ti o n i n s u c h a s a m pl e. T his b ett er k n o wl e d g e will b e h el pf ul f or st u d yi n g s u c h s yst e ms i n l o c ati o ns w hi c h ar e dif fi c ult t o a c c ess, m e a ni n g t h at o nl y p ort a bl e i nstr u m e nts c a n b e us e d.
R ef e re n c es
1. C a o, J., W u, X.: Parti al c orr el ati o n a n al ysis of c ol or r e fl e ct a n c e of d y n a mi c m oist ur e tr a nsf er of fa bri c a n d st ati c p h ysi c al i n di c at ors of fa bri c. A p pl. M e c h. M at er. 1 7 0 – 1 7 3 , 2 9 8 7 – 2 9 9 0 ( 2 0 1 2) 2. D alt o n, P. M., N o b bs, J. H., R e n n ell, R. W.: T h e i n fl u e n c e of m
ois-t ur e c o nois-t e nois-t o n ois-t h e c ol o ur a p p e ar a n c e of d y e d ois-t e xois-til e m aois-t eri als. I. D y ei n g m et h o ds a n d r e fl e ct a n c e m e as ur e m e nts o n w o ol. J. S o c. D y ers C ol o ur. 1 1 ( 9), 2 8 5 – 2 8 7 ( 1 9 9 5)
3. Fi gi el, A.: Dr yi n g ki n eti cs a n d q u alit y of b e etr o ots d e h y dr at e d b y c o m bi n ati o n of c o n ve cti ve a n d va c u u m- mi cr o wa ve m et h o ds. J. F o o d E n g. 9 8 ( 4), 4 6 1 – 4 7 0 ( 2 0 1 0)
4. R o m a n o, G., Ar g yr o p o ul os, D., N a gl e, M., K h a n, M. T., M üll er, J.: C o m bi n ati o n of di git al i m a g es a n d l as er li g ht t o pr e di ct m oist ur e c o nt e nt a n d c ol or of b ell p e p p er si m ult a n e o usl y d uri n g dr yi n g. J. F o o d E n g. 1 0 9 ( 3), 4 3 8 – 4 4 8 ( 2 0 1 2)
5. C h a n dr as e k h ar, S.: R a di ati ve Tr a nsf er. D o ver, N e w- y or k ( 1 9 6 0) 6. M a h e u, B., L et o ul o u z a n, J. N., G o u es b et, G.: F o ur- fl u x m o d els t o
s ol ve t h e s c att eri n g tr a nsf er e q u ati o n i n t er ms of L or e n z – Mi e p ar a-m et ers. A p pl. O pt. 2 3 ( 1 9), 3 3 5 3 ( 1 9 8 4)
7. K u b el k a, P.: N e w c o ntri b uti o ns t o t h e o pti cs of i nt e ns el y li g ht-s c att eri n g m at eri alht-s, p art I. J. O pt. S o c. A m. 3 8 , 4 4 8 – 4 5 7 ( 1 9 4 8) 8. K u b el k a, P., M u n k, F.: Ei n B eitr a g z ur O pti k d er Far b a nstri c h e. Z.
Te c h. P h ys. 1 2 , 5 9 3 – 6 0 1 ( 1 9 3 1)
9. G ar cí a- Val e n z u el a, A., G uti érr e z- R e y es, E., B arr er a, R. G.: M ulti pl e-s c att eri n g m o d el f or t h e c o h er e nt r e fl e cti o n a n d tr a ns-missi o n of li g ht fr o m a dis or d er e d m o n ol a y er of p arti cl es. J. O pt. S o c. A m. A O pt. I m a g e S ci. Vis. 2 9 ( 6), 1 1 6 1 – 1 1 7 9 ( 2 0 1 2) 1 0. V á z q u e z- Estr a d a, O., G ar cí a- Val e n z u el a, A.: O pti c al r e fl e cti vit y
of a dis or d er e d m o n ol a y er of hi g hl y s c att eri n g p arti cl es: c o h er e nt s c att eri n g m o d el vers us e x p eri m e nt. J. O pt. S o c. A m. A O pt. I m a g e S ci. Vis. 3 1 ( 4), 7 4 5 – 7 5 4 ( 2 0 1 4)
1 1. M el a m e d, N.: O pti c al pr o p erti es of p o w d ers. Part I. O pti c al a bs or p-ti o n c o ef fi ci e nts a n d t h e a bs ol ut e val u e of t h e diff us e r e fl e ct a n c e. J. A p pl. P h ys. 3 4 ( 3), 5 6 0 ( 1 9 6 3)
1 2. M a n d elis, A., B or o u m a n d, F., va n d e n B er g h, H.: Q u a ntit ati ve dif-f us e r e fl e ct a n c e a n d tr a ns mitt a n c e s p e ctr os c o p y odif-f l o os el y p a c ke d p o w d ers. S p e ctr o c hi m. A ct a A M ol. S p e ctr os c. 4 7 ( 7), 9 4 3 – 9 7 1 ( 1 9 9 1)
1 3. G ar a y, H., Et err a d ossi, O., B e n h ass ai n e, A.: S h o ul d M el a m e d’s s p h eri c al m o d el of si z e- c ol o ur d e p e n d e n c e i n p o w d ers b e a d a pt e d t o n o n-s p h eri c p arti cl es ? P o w d er Te c h n ol. 1 5 6 ( 1), 8 – 1 8 ( 2 0 0 5) 1 4. htt p:// w w w. n at ur al pi g m e nts. c o m/ y ell o w- o c h er-j cl. ht ml . A c c ess
e d 1 2 N o v 2 0 1 5
1 5. G ar a y, H., Et err a d ossi, O., B e n h ass ai n e, A.: Pr e di cti n g c h a n g es i n t h e c ol or of p o w d ers: d o es M el a m e d’s m o d el fit t o r e al i n d ustri al p o w d ers ? C ol or R es. A p pl. 2 9 ( 6), 4 1 3 – 4 1 9 ( 2 0 0 4)
