Computer simulation and verification of deliming process mathematical model

Transcription

1 Computer simulation and verifiation of deliming proess mathematial model Hana Charvátová, Dagmar Janáčová, ladimír ašek, Pavel Mokrejš, Karel Kolomazník Abstrat The paper deals with hemial deliming proess that is one of operations of natural hides treatment. Nowadays, this proess is haraterized by great onsumption of water, hemial agents and eletrial energy needed for suffiient derease of lime in proessed hides. Therefore we searh optimal tehnologial proedure that will be advantageous from both eonomi and eologial aspets. We desribe method that we used to determination of fixing power whih is lime bound to ollagen fibers of pelt. This data enabled us to verify mathematial model that we formulated for desription of removing of lime from pelt during deliming in the bath system. Keywords Mathematial model, deliming proess, pelt, sorption oeffiient determination, helatometri titration. D I. INTRODUCTION eliming belongs to tannery operations of natural hides treatment. Deliming prepares pelts (hides prepared for deliming) for following tannery operations. The purposes of deliming an be sumarized as []: removing of lime (alium hydroxide) from pelts, lowering the ph in the pelt suitable for biohemial reation of protease (bating), redution of swelling of pelt. Lime is in the pelt partly bound to ollagen fibers. Therefore deliming is realized in two steps. First the unbound lime is mehanially removed by pure water. In the seond step the bound lime is removed by use of hemial agents (aids and aid salts). In pratie, these operations are haraterized by great onsumption of water, hemial agents and eletrial energy This work was supported by the Ministry of Eduation, Youth and Sports of the Czeh Republi under the Researh Plan No. MSM and by the European Regional Development Fund under the projet CEBIA-Teh No. CZ..05/..00/ H. Charvátová, Tomas Bata University in Zlín, Faulty of Applied Informatis, Department of Automation and Control Engineering, nám. T. G. Masaryka 5555, Zlín, Czeh Republi ( harvatova@fai.utb.z) D. Janáčová, Tomas Bata University in Zlín, Faulty of Applied Informatis, Department of Automation and Control Engineering, nám. T. G. Masaryka 5555, Zlín, Czeh Republi; phone: ; fax: ; janaova@fai.utb.z. ašek, Tomas Bata University in Zlín, Faulty of Applied Informatis, Department of Automation and Control Engineering, nám. T. G. Masaryka 5555, Zlín, Czeh Republi ( vasek@fai.utb.z) P. Mokrejš, Tomas Bata University in Zlín, Faulty of Tehnology, Department of Polymer Engineering, nám. T. G. Masaryka 5555, Zlín, Czeh Republi ( mokrejs@ft.utb.z) K. Kolomazník, Tomas Bata University in Zlín, Faulty of Applied Informatis, Department of Automation and Control Engineering, nám. T. G. Masaryka 5555, Zlín, Czeh Republi ( kolomaznik@fai.utb.z). needed for suffiient derease of lime in proessed hides. Therefore we searh optimal tehnologial proedure that will be advantageous from both eonomi and eologial aspets []. II. MATHEMATICAL MODEL OF DELIMING PROCESS By reason of deliming liquid onsumption lowering, deliming proess is mostly realized by washing of pelts in the bath without inflow and outflow of deliming liquid. By formulation of mathematial model of deliming we assumed that all lime will solute in deliming liquid and that pure water will be used as deliming liquid. This proess an be desribed by one-dimensional seond Fik s law [], [3], [4] D ( x, τ ) ( x, τ ) = A + x τ, τ > 0, 0 x b () Initial uniform distribution of lime in the pelt is given by equation () ( ) x,0 = p Equation (3) gives an initial onentration of alium ions in deliming liquid 0 ( 0) = 0 (3) Diffusion of alium ions from pelt into the deliming liquid as symmetri problem an be desribed by ondition (4) x ( τ ) 0, = 0 Equation (5) assumes a perfetly mixing of liquid phase (, ) = ε ( τ ) b t (5) 0 Boundary balane ondition (6) denotes the equality of the diffusion flux at the boundary between the solid and the liquid phases with the speed of aumulation of the diffusing element in the surrounding. b x D S d dτ 0 0 (, τ ) = ( τ ) () (4) (6) Issue 7, olume 6, 0 845

2 Fixing power of alium ions in the pelt represented by sorption oeffiient A in diffusion equation () an be determined from Langmuir sorption isotherm (7) A A = B + (7) For very small onentrations of alium ions an be written A = K (8) where K A. For simplifiation of mathematial solving of abovementioned model we introdued dimensionless parameters. Parameter C is dimensionless onentration of alium ions in the pelt (9) C = (9) p C 0 is dimensionless onentration of alium ions in the deliming liquid (0) C = 0 0 (0) p Symbol X is a dimensionless spae oordinate () x X = () b F o is dimensionless time of proess (Fourier riterion) () F o D t = b + ( A) Symbol Na is dimensionless volume of deliming liquid (3) () Na q = tan( q) ε + ( A) III. EXPERIMENTAL DETERMINATION OF DELIMING PROCESS COURSE (5) As we mentioned in previous setion, the fixing power of alium ions on ollagen fibers of pelt belongs to main parameters whih influene deliming proess ourse. Total quantity of alium ions in deliming system an be omputed as a sum of quantity of the alium ions both free and fixed in the pelt, and also quantity of alium ions, whih moved into the bath [] { S = total weight of alium ions in the system { A + { weight of alium weight of unbound weight of alium ions bound to alium ions ions in the deliming the pelt in the pelt liquid By substitution of the equation (8) into the balane equation (6) we obtained equation suitable for estimation of the oeffiient K or A: S 0 = ε ( K + ) + Na (7) Laboratory testing of deliming proess ourse onsisted in determination of bound omponent onentration A on unbound omponent onentration in the pelt. We used linear equation (7) for determination of sorption oeffiient A. Tested attle pelts were supply by Tarex Ltd, Otrokovie, Czeh Republi. For determination of dependene / 0 on Na, we put utup pelt samples of various weight into the wash bottles, whih had idential volumes of deliming liquid, so that soak numbers were between and 0 (Fig., Fig. ). Then we kept them in the nitrogen atmosphere till equilibrium state ahievement. (6) Na = 0 (3) By means of Laplae transformation we obtained analyti solution desribing dimensionless onentration field C ( X, F o ) in the pelt [5]: (, ) C X F Na o n= ε ( + A) = ε ( + A) + Na os( qx n )exp( qn Fo ) ( A) ( A) os q ε + ε + sin q Na q sin q q ( ) ( ) ( ) n n n n n where q n are omputed aording to equation (5) (4) Fig.. Pelt samples prepared for laboratory testing Issue 7, olume 6, 0 846

3 Fig.. Laboratory apparatus for determination of deliming proess ourse We determined Conentration of alium ions in the deliming liquid ( 0 ) by helatometri titration. The method priniple is based on addition of Complexone III of defined onentration into deliming liquid samples. Murexide was used as an indiator. Chemial reation of total alium ions with Complexone III aused olor hange of tested samples (Fig. 3) [6]. Fig.3. Colour hange of tested samples by helatometri titration sample before titration (on the left) and sample after titration (on the right) The Fig. 4 depits determined sorption isotherm of alium oxide by washing omputed for tested samples of pelt. alue of the sorption oeffiient alulated aording equation () was K = Supposed porosity of the pelt sample was ε = 0.5 [], [5]. Fig. 4 Sorption isotherm of alium oxide by washing before deliming (washing time: 48 hours) I. SIMULATION OF DELIMING PROCESS For omputer simulation of deliming proess ourse we used speial software appliation programmed in Matlab user environment (Fig. 5) [3] Issue 7, olume 6, 0 847

4 Fig. 5 User interfae of software appliation for modeling of deliming proess We used the experimentally determined value of fixing power for omputing of onentration field of the washed lime in the pelt (0). The other needed parameters as are diffusion oeffiient and porosity we took over from previous testing [], [5] of deliming proess ourse: olume of deliming liquid : m 3 olume of pelt in bath 0 : 3 m 3 Thikness of pelt b: 3.5 mm Effetive diffusion oeffiient D: m s Porosity of pelt material ε: 0.5 Fixing power A: 3.56 Fig. 6 and Fig 7 show omputed temperature fields in tested pelt. The results proved that pure water as deliming liquid has washed only about 30 % of the total mass of lime from tested samples. It is lear, that remaining lime have to be removed by some hemial deliming agent. Issue 7, olume 6, 0 848

5 Fig. 6 Computed dimensionless onentration field in the pelt Fig. 7 Computed onentration fields in the pelt in speifi dimensionless times of deliming. DELIMING PROCESS OPTIMIZATION Aording to solution of mathemati model of deliming washing proess (4), it is possible to find the optimum of deliming liquid of proess to be suessful ourse of the proess respetively, and that all from the orresponding the ost funtion [8], [0], []. To determination of the ost funtion for the material bath washing we assumed that we are able to eliminate omponent from the material by the deliming liquid and that the main proesses osts N C of onsidered proess are given by the sum of the onsumed eletri energy to the drive of mahinery osts N E and the onsumed deliming liquid osts N Issue 7, olume 6, 0 849

6 NC = N + N (8) E whereas the onsumed eletri energy osts are given by the produt of the eletri power unit prie K E, the time t and the eletromotor input P to the drive of mahinery N E = K P t (9) E The osts of the deliming liquid requirements N are given by the produt of unit prie of deliming liquid K and the deliming liquid volume 0 N = K (0) 0 We supposed as well that the inreasing deliming liquid requirements ause the dereasing of deliming liquid pollution during the washing whereby the effetiveness of washing proess inreases. Thereby the time interval, neessary to the drive of mahinery is shorter, hene the eletri energy osts are dereasing beause these are linearly inreasing with dependene on time. This implies that the sum of the deliming liquid requirements osts and the eletri energy in dependene on the deliming liquid requirements keeps a minimum. If we want to determine the total osts in dependene on the total dimensionless deliming liquid requirements then first it is neessary to determine the dependene of the washing degree y, whih determines the effiieny of the washing proess in dependene on the dimensionless time F o and that for the orresponding soak number Na. Dependene of the washing degree y, on the dimensionless time F o is given by equation () [5] CNa 0 y = = + A Na Na exp( Fo qn ) = ( + ) + ( + ) ε A Na ε A n= qn Na ε ( + A) + + Na ε ( + A) () In the Fig. 4 is depited dependene of the washing degree on the time. The time is omputed from Fourier number aording to equation () Determination of time to reahing demanded washing degree of given soak number enable to ompute ost funtion aording to equation () N = K Na + C KE P F b + A o ( ) () D Minimum of the funtion (3) determines optimal soaking number for the proess, whih is ratio of volume of deliming liquid to volume of pelt (3). I. DETERMINATION OF DELIMING PROCESS COST FUNCTION For determination of optimal soaking number, we omputed first dependene of the washing degree y, on the dimensionless time F o (Fig. 7). Conditions of the proess : olume of deliming liquid : m 3 olume of pelt in bath 0 : 3 m 3 Thikness of pelt b: 3.5 mm Effetive diffusion oeffiient D: m s Porosity of pelt material ε: 0.5 Fixing power A: 3.56 Input of eletromotor to the drive of mahinery P: kwh Eletri power unit prie K E : kw - h Unit prie of washing liquid K :.5 m -3 Required washing degree y 0 : 0.35 The optimal soaking numbers we omputed for deliming degrees 0.,3, 0.4, 0.5, 0.6. The omputed data we show in Fig. 9 Fig. 3. As you an see, the optimal soaking number inrease with inreasing required washing degree. By washing degree 0.3 the optimal soaking number is.9. By washing degree 0.4 the optimal soaking number is about.85.. By washing degree 0.5 the optimal soaking number is about 4. and by washing degree 0.6 the optimal soaking number is about 5.9. Issue 7, olume 6, 0 850

7 Fig. 8 Dependene of the washing degree y on the dimensionless time F o Fig. 9 Computed ost funtion for washing degree 0.3 Fig. 0 Computed ost funtion for washing degree 0.4 Fig. Computed ost funtion for washing degree 0.5 Fig. Computed ost funtion for washing degree 0.6 Issue 7, olume 6, 0 85

8 II. CONCLUSION In the paper we presented results whih we obtained by verifiation of mathematial model of deliming proess by use of pure water as deliming liquid. By experimental testing of deliming proess ourse of attle pelts we have determined fixing power of lime in the pelt about By mathematial modeling of the proess ourse for this value of fixing power, we have found out that pure water as deliming liquid has washed only about 30 % of total mass of lime from the tested samples. Computed ost funtion enables to find the optimum of deliming liquid. The obtained results along with other testing of deliming proess ourse will lead to finding an optimal tehnologial proedure that will be advantageous from both eonomi and eologial aspets. is given by equation. REFERENCES [] H. Charvátová, Modeling of pelt hemial deliming, Dissertation work. Tomas Bata University in Zlin, Zlin, 007. (in Czeh) [] D. Janáčová, Modeling of Extration Proesses, Habilitation work, TBU Zlin, 003. (in Czeh) [3] J. Šebestík, Mathematial modeling of bound omponent washing in the pelt, Diploma work. Tomas Bata University in Zlin, Zlin, 007. (in Czeh) [4] K. Kolomazník, D. Janáčová, Z. Prokopová, Modeling of Raw Hide Soaking, in WSEAS Transations on Information Siene and Appliations, Helleni Naval Aademy, Ostrava Poruba, 005. [5] K. Kolomazník, et al., Modeling of dynamial systems, Brno: UT Brno, 988. (in Czeh) [6] Chelatometriké (komplexometriké) titrae. Available from: >. [ ]. [7] J. Crank, The Mathematis of Diffusion, nd Ed. Clarendon Press, Oxford 977. [8] D. Janáčová, et al., Washing Proesses Optimization, in International Union of Leather Tehnologists and Chemists Soieties, London, 997. [9] K. Kolomazník, T. Fürst, D. Janáčová, M. Uhlířová,. ašek, Three Dimensional Transport Model Using in Soaking Proess, in WSEAS Transations on Computer Researh, WSEAS World Siene and Engineering Aademy and Siene, Queensland, 007. [0] K. Kolomazník, D. Janáčová,. ašek, M. Uhlířová, Control algorithms in a minimum of main proessing osts for prodution amaranth hydrolyzates, in WSEAS Transations on Information Siene and Appliations, WSEAS World Siene and Engineering Aademy and Siene, Athens, 006. [] J. Dolinay, et al., New Embedded Control System for Enzymati Hydrolysis, in Proeedings of the 8th WSEAS International Conferene on Applied Informatis and Communiations, Rhodes, Greee, 008, p. 74. III. LIST OF SYMBOLS Symbol Meaning Unit volume of pelt m 3 0 volume of washing liquid m 3 t time s volume onentration of alium ions in pelt kg m -3 o volume onentration of alium ions in bath kg m -3 p initial onentration of alium ions in pelt kg m -3 A volume onentration of alium ions bound into pelt kg m -3 D effetive diffusion oeffiient m s - Symbol Meaning Unit x position oordinate m b half thikness of pelt m ε porosity of pelt Na soaking number q n n -th root of a ertain transendent equation A sorption oeffiient (fixing power) B sorption oeffiient m 3 kg - S area of pelt m F o Fourier number C dimensionless volume onentration of alium ions in pelt C 0 dimensionless volume onentration o falium ions in bath X dimensionless spae oordinate y washing degree y 0 required washing degree P input of eletromotor to the drive of mahinery kwh K E eletri power unit prie - kw - h K unit prie of washing liquid m -3 N E eletri energy to the drive of mahinery osts N deliming liquid osts N C the main proesses osts Hana Charvátová is a researh worker at the Department of Automation and Control Engineering, Faulty of Applied Informatis, of Tomas Bata University in Zlín. Her researh ativities inlude reyling tehnology and modeling of natural and syntheti polymers treatment. Dagmar Janáčová is an Assoiate Professor in the Department of Automation and Control Engineering, Faulty of Applied Informatis, of Tomas Bata University in Zlín. Her researh ativities inlude: modeling of treatment proesses of natural polymers, transport proesses, reyling of tannery wastes, and optimization and eologial approah of tannery proesses. She has reeived the following honors: Diploma of England, XXIII IULTCS Congress, London, 4 September, 997; Gold Medal - EUREKA EU Brussels 997; Speial Prize, Ministry of Agriulture, Belgium, 997. ladimír ašek is a Professor in the Department of Automation and Control Engineering,Faulty of Applied Informatis, of Tomas Bata University in Zlín. His researh ativities inlude: miroomputer appliations in tehnology proesses, omputer monitoring and ontrol systems, and disrete deterministi ontrollers approah of tannery proesses. He has reeived the following honors: Diploma of England, XXIII IULTCS Congress, London, 4 September, 997; Gold Medal - EUREKA EU Brussels 997; Speial Prize, Ministry of Agriulture, Belgium, 997. Pavel Mokrejš is an Assoiate Professor in the Department of Polymer Engineering, Faulty of Tehnology of Tomas Bata University in Zlín. His researh ativities inlude: treatment of liquid and solid waste form tannery industry, treatment of leather waste from shoe industry, utilization protein hydrolysates, biodegradable asings and films, and isolation of proteins from untraditional soures. Karel Kolomazník is a Professor in the Department of Automation and Control Engineering, Faulty of Applied Informatis, of Tomas Bata University in Zlín. His researh ativities inlude: modeling of biopolymers treatment, hemial engineering transport proesses, reyling of proteins, optimization and eologization of tanning proesses, and turning of vegetable and animal fats into biodiesel. He has reeived the following honors: Germany, Amerian, England Leather Assoiations, XXIII IULTCS Congress, Friedrihshafen, May 5 0, 995; England, XXIII IULTCS Congress, London, 4 September, 997; USA, ALCA 997, Annual Meeting, Regent Resort, NJ; Gold Medal - EUREKA EU Brussels 997. Issue 7, olume 6, 0 85