Mathematical modeling of gelatine production processes

Transcription

1 Matheatial odeling of gelatine prodution proesses KAREL KOLOMAZNIK, DAGMAR JANACOA, JAROSLA SOLC and LADIMIR ASEK Departent of Autoati and Control Engineering Toas Bata Universit in Zlín, Fault of Applied Inforatis ná. T. G. Masarka 5555, 76 Zlín CZECH REPUBLIC Abstrat: - Reentl, gelatine or gelatinous protein has been a ver valuable raw aterial for pharaeutial, food and osetial industries. It is ade fro b-produts leather proessing industr b liing. This proess usuall takes fro 4 to 6 weeks aording the abient teperature and the ontent of aidi nitrogen, Following deliing operation, onsists in neutralization of aliu hdroxide b aids or suitable aid salts. Ke-Words: - Deliing, atheatial odeling, optiization, diffusion odel, ost funtion Introdution Delied hide splits, alost free of aliu salts, are read for extration b water at inreased teperature, at whih adequate water solutions of gelatinous gel are obtained. Although onsiderable researh has been devoted to the searh for ore environentall friendl deliing agents, rather less attention has been paid to the onstrution of atheatial odel whih an be used for the optiization of the proess. The ai of the present paper is to develop and verif a atheatial odel whih desribes the reoving of lie fro hide splits. Theor In order to siulate the proess and enable its optiization we introdue a deliing effiien () defined as a ratio between reoved weight of aliu hdroxide and its initial weight. Regarding to a ultiple step operation we relate the effiien to an initial ontent ( ) or to an ever initial ontent ( ) before a partiular step. Lied hide splits enter the deliing tehnolog with following paraeters a i,s - weight fration of aliu hdroxide in dr atter a s,k - weight fration of dr atter in hide splits a i, - axiu allowable weight fration of aliu hdroxide after finishing of a deliing proess olue onentration ( i ) is given: i = ρ ai, s as, k () where ρ is densit of input hide splits and it is further assued that the densit does not hange in a whole oplex of the de-liing tehnolog. Overall required effiien of the deliing proess s is given b su of partiular steps. ip ik s = = + + () ip where ip, ik are volue onentration of aliu hdroxide on the beginning ( ip ) and the end ( ik ) of deliing operation,,, are effiienies in partiular steps related to the initial ontent (volue onentration) of aliu hdroxide. Surfae aliu hdroxide is reoved during the first washing step. The effiien of its reoval ) an be alulated fro Eq. (). ( SUR SOL = () SUR Where SUR is volue onentration of surfae aliu hdroxide and SOL is its solubilit at given teperature. We suppose here, that after washing [4] of surfae aliu hdroxide is stopped, dissolved aliu hdroxide is not reoved. For the effiien related to initial ontent ( ) equation (4) is valid. ISBN:

2 SUR = (4) ip Where ( ) is effiien deterined aording to eonoial aspet. During the seond step unbound and partiall also bound aliu hdroxide is reoved b ultiple step bath washing. There are different riteria for optial effiien and in our ase we hose eonoial aspet, when operation osts of washing b pure water are the sae like those of washing b water solution of a deliing heial. Further ontinuation of washing b pure water is not effetive beause operation osts are inreasing extreel. Critial eonoi effiien, when washing b pure water is stopped and is followed b the heial deliing operation, depends on phsial paraeters (absorption paraeters of Languir isother (A, B), porosit (), and effetive diffusion oeffiient of aliu hdroxide in the inner volue of hide splits (D), tehnologial paraeters (initial onentration of aliu hdroxide, the thikness of hide splits, load of hide splits in a washing ahine, a ratio between volue of hide splits and water soaking nuber (Na) and input power of the eletri engine of blender) and eventuall eonoial paraeters (unit ost of washing water, water solution of deliing heial and ost of eletri energ). Effiien of washing ( ) related to initial weight of aliu hdroxide is then given: = ( ip SUR ) ip (5) Where ( ) is effiien deterined aording to eonoial aspet. During the following operation aidifing an equivalent aount of a deliing agent is added to water solution hanging aliu ollagenate to ollagen and releasing a neutral aliu salt, whih is reoved b washing out b pure water in a following step. The deliing agent is usuall dosed at 5-% surplus. The neutral aliu salt is reoved b washing out b pure water in the last step. Required effiien ( ) is given b a differene between total effiien and su of effiienies in the first and seond step. ( ) = + S (6) Effiien related to initial weight before last step ( ) is alulated fro aterials balane of oplete deliing proess. ip = = + ( ) ip ip ip (7) A quantitative relation between the equilibriu onentration of bound aliu hdroxide in hide splits ( A ) and unbound in surrounding liquid ( ) an be desribed b a following equation (Languir isother) A A = (8) B Deterination of effetive diffusion oeffiient Theoretial value of the diffusion oeffiient of the aliu salt an be alulated fro the Nernst equation, whih is valid for eletroltes at infinite dilute. Where: D = 8.9. o + o o l+. l z+ z T. o Λ z+. z (9) D - Theoretial value of the diffusion oeffiient, [ s - ] T - therodnaial teperature of eletrolte, [ s - ] l - Equivalent ondutivit of anions at + infinite dilute, [K] - Equivalent ondutivit of ations at l - Λ infinite dilute, [S/ekv.] - Equivalent ondutivit of ations at infinite dilute, [S/ekv.] z + - alene of ation z - - alene of anion In ase of sall onentrations and ver sall diffusion oeffiients we an assue, that onentration profile of laer where heial deliing was done is linear (quasistead ase). The situation is shown on the following Fig.. ISBN:

3 where Na = (4a) τ Na = Na + & (4b) Na is also diensionless onsuption of washing water. The effiien of surfae aliu hdroxide reoving is then given: Fig. Quasistead odel of aidifiation ( i onentration of hdrohlori aid, j onentration of aliu salt, x thikness of the laer where heial deliing took plae) τ ( τ ) τ + ( τ) & C d = p B integration Eq. (5) we reeive: (5) Diffusion of hdrohlori aid (i) for great Na is desribed b Eq. () S pj dx d SD pi τ = () x B integration we reeive: x D τ pi = () pj We an then approxiatel evaluate the diffusion oeffiient of aliu salt fro equation (), whih is valid for the deliing b hdrohlori aid. We an reeive the value of the effetive diffusion oeffiient fro the slope of the straight line resulting fro plotting x. Theoretial diffusion oeffiient of aliu hdroxide at infinite dilute alulated fro equation () is equal s -. Do =.8.. s 9 Flow-through washing, when aliu hdroxide is reoved, an be desribed b a following differential equation. dc dc + + C& () dτ dτ C Na Na = () + Na Na + psur ( τ ) exp Na Na = Na exp (6) + Na + Diensionless onsuption of washing water Na with using Eq. (6) is equal: + Na Na = Na ( Na+ ) ln ( ) (7) We alulate b appliation equation Eq. (). =.85 The effiien () related to a total initial aliu hdroxide ontent is given b Eq. (4). =.7 We deterine diensionless onsuption of washing water Na fro Eq. (7). We hoose Na equal Na =.88 As being said, optial value of effiien, when unbound and partiall bound aliu hdroxide is reoved b n-step bath washing, an be alulated fro eonoial and ass balane for i- step of bath washing as for washing b pure water, as for washing b a solution of a deliing heial,5. ISBN:

4 For deterination of the effiien of bath washing we used following entr data: Strength of bond A: 66 [] Absorption onstant B: 57 [ /kg] Initial onentration s :.84 [kg/ ] Half thikness b:. [] olue of hide splits : [ ] Porosit :.5 [] Soaking nuber Na: [] Diffusion oeffiient D:. - [ /s] Cost of water K v : [ / ] Cost of energ K e :. [ /kwh] Cost of heial K : [ / ] Input power of washing ahine blender P: 5 [kw] Effiien related to initial ontent of aliu hdroxide is equal aording Eq. (5): =.4 Operating tie of one bath washing an be estiated b using of Fourier riterion (Fo). The equilibriu is approxiatel reahed when D τ Fo = = τ = b b D B entering we reeive: τ =.7hours Real value is usuall -5% higher. Required final effiien is given b Eq. () =.999 s The ontent of aliu hdroxide after three washing les b pure water ( ) is equal ( ) = pi (8) B entering we reeive = 9.4 kg. Fig. Relation between ain operation osts and deliing effiien for washing b pure water and washing b a solution of a deliing heial,4. Fig. shows relation between ain operation osts and deliing effiien for washing b pure water and washing b a solution of a deliing heial. A ritial point (intersetion of both urves) represents a point, when it is useful fro eonoial point of view to replae siple washing b pure water for washing b a solution of heial. Eah point on urves represents one bath washing step. Conretel, in our ase after three bath washing b water, when effiien =.58 is reeived, it is neessar to ontinue in washing b a solution of a deliing heial. The load of of the washing ahine is tons of hide splits, whih represents volue approxiatel and beause the soaking nuber Na=, the aount of liquid is the the sae like the aount of hide splits. It is neessar to neutralize 9.4 kg of aliu hdroxide. In ase of using 5% hdrohlori aid it is neessar to use kg HCl at 5% surplus, whih represents.868 kols % HCl. ( ) + = + Ca OH HCl CaCl H O Tie of aidifing an be estiated fro Eq. () pik τ = D po x, (9) where D is effetive diffusion oeffiient in hide splits, ik is volue olar onentration of aliu hdroxide in hide splits, ISBN:

5 p is initial onentration of HCl in aidifing solution in kol/. B entering known values to (9) we reeive τ =.76hours A real value is usuall higher in pratie, beuase onentration of hdrohlori ais is dereasing as a result of ontinuing neutralization and so the driving fore of diffusion is dereasing too. After operation aidifing the neutral aliu salt is washed b bath washing so that rest value of the aliu salt expressed like a aliu hdroxide was.% in the dr atter of hide splits. Corresponding total effiien related to initial ontent of aliu hdroxide ( ) is given b Eq. (6). =.7 and effiien related to initial ontent of lie (Ca(OH) ) is given b Eq. (7) =.997 Aknowledgeent: This work was supported b the Ministr of Eduation, Youth and Sports of the Czeh Republi under the Researh Plan No. MSM 7885 and b the European Regional Developent Fund under the projet CEBIA - Teh No. CZ..5/../.89. Referenes: [] J.,M.,Constantin, Jala, 76, 98, p. 4. [] W. Nernst, Zur kinetik der losung befindlihen Körper. Z. Phsik Che.,, p. 6-6, 888. [] K. Koloazník, et al., Matheatial siulation for optiization of tanning proesses; deliing of white hide. In Proeedings of the 8th WSEAS International Conferene on Applied Inforatis and Couniations, Rhodes, Greee, 8, p. 74. [4] D. Janáčová et al., Matheatial Model of Raw Hides Proper Soaking, Tehniká univerzita v Košiiah, 8 th International Carpathian Control Conferene, Košie 8, 5-8, ISBN [5] D. Janáčová, K. Koloaznik et al, Matheati Modeling of Biolog Changes in the Cured Hides Soaking,, WSEAS Transations on Sstes, Australia, 7, ISSN Nuber of bath washing steps (n) at Na i =. (inreased b volue of added HCl) in one bath washing le is given b following equation ln n = ln ( ) ( + ) Na i.4 Tie of one deantation steps an be estiated fro unit value of Fourier riterion τ =.95hours alue of effetive diffusion oeffiient of CaCl was estiated fro Nernst equation (). 4 Conlusion Results fro this stud showed that onsuption of water and energ during deliing proess an be signifiantl redued. As a result operation osts and ipats on environent are iniized. Model alulation based on real data was presented in order to show how onstruted atheatial odel an be used for the optiization of deliing proess. ISBN: