MATHEMATICAL MODELS FOR THE PREDICTION OF THE ENCAPSULATION BEHAVIOR IN FOOD SYSTEMS

Transcription

1 tud ş Cercetăr Ştnţfce Chme ş Ingnere Chmcă, Botehnolog, Industre Almentară 2010, 11 (2), pp IN X ORIGINAL REEARCH PAPER MATHEMATICAL MODEL FOR THE PREDICTION OF THE ENCAPULATION BEHAVIOR IN FOOD YTEM IulanaVntlă* Dunarea de Jos Unversty Galaţ, Food cence and Engneerng Faculty, Domneasca 111, Galaţ, Romana *Correspondng author: Receved: February 10, 2010 Accepted: Aprl 8, 2010 Abstract: The smulaton of the encapsulaton behavor n the multphase complex system such food structure s based on the mathematcal models constructed n respect wth the Non-equlbrum thermodynamcs Theory, Flory-Huggns Free Volume Theory (FHFV) and Complex Dspersed ystems (CD). The present research paper presents the dfferental equatons descrbng the evoluton n tme of the multphase dvdng surfaces and the excess quanttes such as surface densty, surface momentum, surface energy and surface entropy assocated wth the dvdng surfaces. The new completed theory of bo-polymers phase transtons co-jonted from Interfacal Transport Phenomena (ITP), FHFV and CD wll be valdated wth the nverse analyss method. Keywords: encapsulaton, ITP, FHFV, CD, bo-polymers 265

2 VINTILĂ INTRODUCTION Encapsulaton s a process n whch thn flms, generally of polymerc materals, are appled to lttle sold partcles, lqud or gases droplets. Ths method s used to trap actve components and release them under controlled conons. everal materals have been encapsulated n the food ndustry, among others, amno acds, vtamns, mnerals, antoxdants, colorants, enzymes and sweeteners [1]. Encapsulaton s used to protect compounds from oxdaton, evaporaton and off-flavor. The encapsulated compounds can be presented ether wth sphercal geometry (snglepartcle structure) or wth rregular geometry (aggregated structure) [2]. The most used mechansms n encapsulaton are extruson, coacervaton and complexaton. Developng nanoencapsulaton of multple bologcally actve trace ons n a slca-based system for controlled-release s openng up applcaton possbltes for regeneratve bomedcne. The quanttatve macroscopc analyss of the complex food systems behavor n course of the encapsulaton process s descrbed n the scentfc lterature [1]. The mathematcal models for the dynamcs of the encapsulaton process respect the non-equlbrum thermodynamcs unversal theory, n respect of stress-deformaton behavor of the mult-phases nterfaces. Food systems wth complex nterfaces wll be characterzed wth nonlnear constructve equatons for stress-deformaton behavor dependency. The dynamc behavor of the complex encapsulaton system s descrbed based on the Interfacal Transport Phenomena [3]. Interfaces are modeled by the Gbbs law dvdng surface model and excess quanttes such as surface densty, surface momentum, surface energy and surface entropy are assocated wth the dvdng surface [4]. The Flory-Hggns Free Volume (FHFV) theory s developed by Vrentas [5] and can predct the behavor of the synthetc polymers system n soluton state. The valdaton of the new mathematcal model for descrpton of the encapsulaton dynamc process wll be valdated wth the nverse analyss method [6]. REULT AND DICUION The dfferental equatons descrbng the evoluton n tme of the multphase system nterface and the excess quanttes assocated are as followngs: (1) Jump Mass Balance Equaton: dsρ + ρ dv v + ρ( v v ) ξ = 0 (1) (2) Jump peces Mass Balance: dω ρ + dv j + jξ + ρ( ω ω )( v v ) ξ = 0 (2) (3) Jump Momentum Balance: N dv ρ dvt ρ τ + ρ( ω ω )( v v ) ξ Tξ = 0 (3) = t. Cerc. t. CICBIA (2)

3 MATHEMATICAL MODEL FOR THE PREDICTION OF THE ENCAPULATION BEHAVIOR IN FOOD YTEM (4) Jump Energy Balance ρ ρ d E (5) Jump Entropy Balance d ρ N = T : v + j τ dvq + ρ Q = 1 (4) 1 ( E E ) + v v ( v v ) ξ ( v v ) Tξ + qξ 2 φ + dv Q ρ T T where: v - the surface velocty; ρ - the phases densty; ξ - the surface normal vector; ω - the mass fracton of speces; j - the mass flux vector of speces; T τ + ρ 1 ( )( v v ) ξ + T φξ 0 - the surface stress tensor and T stress tensor of phases; - the force per unt mass actng on the nterface speces; E - the surface energy per unt mass; q - the surface energy flux vector; Q - the rate of radant energy transfer per unt mass; φ T - the surface thermal energy flux vector; - the surface entropy per unt mass; - the nterface temperature. The boldface bracket represents the transfer of mass, momentum, energy and entropy from the bulk phase to the nterface (6). (1) (1) (2) (2 ψξ = ψ ξ + ψ ξ (6) [ ] ) (1) (2) where ψ and ψ are the values of the ψ descrbed as the unt mass, momentum, energy and entropy of the adjonng bulk phases (1) and (2) n the ITP. The deformaton behavor n the encapsulaton process of the W/W emulsons depends on 6 factors: (1) mass transfer coeffcent across the nterfaces ; (2) nterfaces surface tenson (Helfrch expanson); (3) nterfaces bendng rges; (4) nterfacal permeablty; (5) nterfaces dlatatonal coeffcent; (6) surface rheologcal parameters. The rheologcal behavor of an emulson nterface stablzed wth low molecular peptdes s descrbed by the Boussnesq law [2]. The algnments of the partcles at the nterfaces affect the nterfacal rheologcal parameters. The ncreasng of the deformaton rate decreases the shear vscosty and surface dlatatonal vscosty. (5) t. Cerc. t. CICBIA (2) 267

4 The surface chemcal potental energy F (7): VINTILĂ μ of speces depend on total surface Helmholtz free F μ =, j ρ F T,ρ = F ( ρ,, ρ )d j 1 L The Flory-Hggns Free Volume (FHFV) theory s developed by Vrentas [5] and can predct the behavor of the synthetc polymers system n soluton state. Based on the FHFV theory expressed n equaton (8), we can desgn the phase dagram n the transton state of all types of bo-polymers as a dependency of water actvty vs. temperature. μ 1 w = 2 ln( 1 φ) + 1 φ + λφ F( φ) RT N + (8) where: μ w - s the actvty of water absorbed n the bo-polymer structure; T - s the absolute temperature; φ - s the volume fracton of dssolved bo-polymer/fracton of bo-polymer; F (φ) - s the free volume term; N - s the rato of the molar volume of solvate bo-polymer and water. In adon, the state dagram can be completed wth the Complex Dspersed ystem data n order to nvestgate the dfferent transton states of the bo-polymers n the encapsulaton process. We propose also the evaluaton of the encapsulaton performance ndex (9) wth the followng expresson: F W Encapsulat on Performance Index (%) = C V 100 (9) C V where: C - the concentraton of compound from the ntal nanopartcle extract; W - the theoretcal amount of compound added; V - the volume of compound from the nanopartcle extract; - the concentraton of compound n the un-encapsulated porton. F We propose the valdaton of the new complex mathematcal model for the descrbng of the encapsulaton dynamcs to be realzed wth the nverse analyss method (Fgure 1). N (7) 268 t. Cerc. t. CICBIA (2)

5 MATHEMATICAL MODEL FOR THE PREDICTION OF THE ENCAPULATION BEHAVIOR IN FOOD YTEM Expermental Data Computaton Model Operatng Varable s Measurements F exp Optmzaton ntalzatons Parameters Fle ntalzaton s Incremental Computaton Computed Data Fle F c Gauss Newton Mn Φ ( P, F c, F exp ) P P+ Δ P NO tagnaton YE End Fgure 1. The prncple of nverse analyses method [6] CONCLUION The encapsulaton phases transtons can be descrbed wth the thermodynamcs equatons and the chemcal potental of the coexstng phases. The complete model of the encapsulaton dynamcs can also predct the phase transtons wth the constructon of a state dagram. The state dagram allows the selectng of the optmum processng system n order to create a certan food structure. REFERENCE 1. hahda, F., Han, X: Encapsulaton of food ngredents, Crtcal Revews n Food cence and Nutrton, 1993, 33, ; 2. Pedrolas, I.R.: Almentos mcroencapsulados. Partculardades de procesos para la mcroencapsulacon de almentos para larvas de especes acuıcolas, n: VI Internatonal mposum of Aquatc Nutrton. (Eors: Cruz uarez, L.E., Rcque-Mare, D., Tapa alazar, M., Gaxola- Cortes, M.G., mes N.), Mexco, 2002, ; 3. ags, L.M.C.: Dynamcs of controlled release systems based on water-n-water emulsons: A general theory, J. Controlled Release, 2008, 131 (1), 5-13; t. Cerc. t. CICBIA (2) 269

6 VINTILĂ 4. lattery,.c., ags, L.M.C.: Interfacal Transport Phenomena, prnger, New York, 2007; 5. Vrentas, J.., Vrentas C.M.: Evaluaton of the sorpton equaton for polymer-solvent system, Journal of Appled Polymer ystem,1994, 51 (10), ; 6. Gavrus, A: Identfcaton des paramètres rhéologques par analyse nverse. In: Ph.D. Thess, Pars, t. Cerc. t. CICBIA (2)