1 st Meetng of EcosimPro Users, UNED, Madrid, 3-4 May 2001 C7 MODELLING AND SIMULATION OF DISTRIBUTED PROCESSES: DIFFUSERS IN THE SUGAR INDUSTRY

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1 C7 MODELLING AND SIMULATION OF DISTRIBUTED PROCESSES: DIFFUSERS IN THE SUGAR INDUSTRY Merino Gómez, Alejanro Centro e Tenología Azuarera. Universia e Vallaoli C/ Real e Burgos. Eifiio Alfonso VIII. Planta baja s/n Vallaoli Spain Teléfono: FAX: alejanro@ta.uva.es Abstrat This paper esribes how a iffuser use in the sugar inustry has been moelle an simulate with EosimPro. Beause of its physial an operating harateristis, the mathematial moel of this equipment an be esribe by partial erivatives whih requires the isretization of the system. This work shows in a pratial way how these types of problems an be resolve using EosimPro. Key wors: Simulation, EosimPro, istribute systems, mass an energy transfer. 1 INTRODUCTION The objetive of this work is to show how to moel a system whose mathematial moel of behaviour is represente by ifferential equations in partial erivatives. First of all, the physial system is esribe in orer to offer a better unerstaning of the proess. Later, an explanation will be given of the methoology followe to transform the EDP moel into an ODE moel. This will be applie to a partiular ase: a iffuser in the sugar inustry. The use of simulation for these types of systems will be justifie an the mass an energy balanes neee to moel the system will be establishe. Lastly, the results obtaine through the simulation with EosimPro will be presente. 2 DESCRIPTION OF THE PROCESS Diffusion is one of the first stages in the beet sugar proution proess. The objet of this stage is to extrat the maximum amount of sugar from the beet after it has been washe an ut into thin strips, alle ossettes. A ounterurrent flow of hot water extrats the sugar from the ossettes. It is essential that this stage be well esigne beause the maximum amount of sugar must be extrate using as little water as possible. This is why a metho of ounterurrent extration is use, in whih water with less sugar onentration is put in ontat with ossettes with less sugar onentration so that the onentration graient, whih fores the mass transfer, is maximise at all times. These operations are arrie out in equipment known as iffusers. Although the proess is governe by the same physial priniples, there are ifferent types of iffusers with ifferent operating harateristis. In this ase we are going to use the RT iffuser whih is wiely use in Spain. Figure 1. Elements omprising an RT 2 iffuser [1] 1 an 2 helioial plates whih form two separate hannels for the juie; 3 transversal plate; 4 hutes for the ossettes; 5 transverse sreens. 7-1

2 This type of iffuser is omprise of some large rotating rums whih are separate into ells by a propeller attahe to the inner surfae. As the rum with the propeller rotates, the juie that is left on the bottom is transporte from the top of the iffuser to the bottom. In this way it is the ell that moves, although it woul be more pratial to onsier that eah ell is loate at eah rotation of the propeller. If the srew has 30 threas then the rum will have to rotate 30 times for the juie to travel from one sie of the iffuser to the other, an it is onsiere that the iffuser has 30 ells. There are sreens fixe to the yliner an, as the rum rotates, they move the ossettes along until they rop into the next ell. In this way the juie an the ossettes move in opposite iretions. These iffusers are very big (up to 45 m long an 7 m in iameter) an very expensive, whih means that the majority of fatories therefore have only one. They are also inispensable in the sugar proution proess whih makes empirial work with these types of systems very iffiult. Consequently, the use of simulation for these kins of systems is espeially attrative. 3 DISCRETISATION OF A DISTRIBUTED PARAMETER SYSTEM Global parameter systems are those in whih the properties an the state variables of the system an be onsiere to be uniform throughout the system. In reality, all the systems are istribute beause there are always spatial variations in the properties or variables. However, these variations are often small an an be ignore an we an therefore onsier the moel as a global parameter moel. On the other han, there is a series of systems in whih the spatial istribution of the properties requires the use of a istribute parameter moel. They are usually systems in whih the length-iameter ratio is very high an therefore the epenene of the variables on the length is very important. This is the ase of a rotating iffuser in a sugar fatory. The mathematial equations that moel the proesses represente by a istribute parameter moel are as follows: T (v z T) + = Transporte + Generaión (1) t z The problem is that EosimPro oes not support ifferential equations in partial erivatives. We therefore transform the istribute parameter moel by means of isretisation with respet to one of the variables an obtain an equation as follows: Tn (Tn 1 Tn ) = vz + Transporte + z Generaión (2) This is equivalent to a series of global moels onnete together [2]. The onnetion of numerous global parameter moels is mae in suh a way that the output from one element oinies with the input of the next element with a etermine elay. Figure 2 We therefore ivie the iffuser into elements. Determining the number of elements is a matter of having to ompromise between the preision of the alulation (the larger the number of elements, the loser the solution to that of istribute parameters) an the simulation time. The number of elements was etermine so that the final solution woul not vary signifiantly when another element was put in or taken out. The use of this tehnique turns out to be almost the same as the use of finite ifferenes in the solution of the moel. 4 DESCRIPTION OF THE MODEL The iffuser is a iffiult element to moel beause the phenomena that take plae insie a iffuser are of a very omplex nature. There are simultaneous mehanisms whih transfer mass by iffusion owing to the onentration graient, reverse osmosis phenomena an physial rushing phenomena. In aition, the mass an energy transfers are relate beause the iffusion onstant is affete by the temperature an the temperature is affete by the spee of iffusion. With respet to mass transfer, it was eie to opt for utmost simplifiation an to assume that mass transfer existe ue only to the ifferene in the onentration of the juie an the onentration of the solution. Apart from that explaine above, it is assume that the iffuser is ivie into 20 equal ells in whih the transfer of sugar, of non-sugars an of water takes plae. It is also assume that: 7-2

3 The ells are onnete in suh a way that the output from one ell oinies exatly with the input to the next ell, in aorane with the iretion of flow in eah phase. The total mass transfer from the iffuser is governe only by the expressions iniate hereinafter. Eah ell ontains a perfet mix an, therefore, the value of the variables at the outlet will be equal to the value of the variables insie eah ell. There is a ounterurrent flow between juie an ossettes throughout the whole iffuser. No onsieration is given to axial ispersion. Eah step is elaye with respet to the previous step following a elay of the first orer. The flow moel therefore omprises 20 tanks that ontain a perfet mix an are arrange in series, between eah of whih there is a elay of the first orer. The ossettes suffer an enthalpy variation whih is ue, on the one han, to the ifferene in temperature (if it exists) between the ossette an the juie an, on the other han, to the ossette/juie mass transfer. The number of ifferential elements that exist has been parametrise by means of the net onstant so that elements an be ae or remove at will. The first an last elements are onnete to the outsie by means of ports so that the input onitions are impose by the environment an the output onitions are impose by iffuser operation. 4.1 MASS BALANCES Mass transfer between the ossettes an the juie takes plae in eah of the iffuser ells. To moel the mass transfer, we will use the riving fore linearity hypothesis. Water transfer is brought about by replaing the sugar whih is remove uring the first stages of the proess. The water variation insie a ell an therefore be alulate with the following equation: W agt = k W (3) ag azt The transfer of sugar an non-sugars is alulate in aorane with the esign equation: W = k (C C ) (4) t The mass transfer onstants are the result of the prout of a mass transfer onstant times an exhange area times the ensity. The esign of this iffuser also takes into aount a series of fators whih affets the transfer of sugar an non-sugars. The following are the fators taken into onsieration: - Temperature - ph - State of the ossettes - Size of the ossettes - Permeability of the ossettes - Effet of the raft The following is a shemati of the iffuser: entraa e osetas Difusor2 salia e jugo Figure 3 salia e osetas agua fria agua e prensas To moel the effets of these fators, it has been onsiere that they have a ifferent effet on the mass transfer oeffiient. A value has therefore been efine for the mass transfer onstant multiplie by a series of fators that affets the iffusion. These fators will have a maximum value of 1 when their value is optimum an will erease when the effet of the fator is suh that mass transfer is impee. As an example, we are going to see how the effet of the temperature woul be introue into the moel. A simple esription of how it operates woul be that the extration water enters on the right-han sie of the iffuser an flows to the left, extrating the surose an other unwante substanes as it flows through the ossettes whih are moving from left to right. The equations for eah of the elements are given below: Effet of the temperature The effet of the temperature is lear the higher the temperature, the better the iffusion. In aition, a temperature greater than 50ºC is require to ahieve enaturalisation, an high temperatures also provie protetion against bateria. The upper temperature limit is 75 C. At temperatures any higher than this, we run the risk of beet egraation. 7-3

4 The following graph shows the values of the fator at ifferent temperatures: Fa to r 1,2 1 0,8 0,6 0,4 0,2 0 Influene of Temperature Temperature Figure 4 The introution of these influene funtions into EosimPro is easy. A series of funtions is reate, into whih we introue known experimental ata on the effets of ifferent fators on the extration an use the interpolation funtion linearinterp1d. The following is an example of one of the influene funtions introue into the moel FUNCTION REAL Inf_temp (IN REAL T) DECLS REAL Itemp CONST TABLE_1D inf_temp= {{0, 30, 50, 60,\ 63, 67, 70, 71, 72,73,74, 75, 76, 77, 78,\ 85,90,100}, {0.,0.05,0.2,0.7,0.85,0.95,0.99,1.,1.,\ 1, 0.99,0.8,0.7,0.6,0.5,0.,0.,0.}} BODY Itemp=linearInterp1D(inf_temp,T) RETURN Itemp END FUNCTION SUGAR BALANCE In the ossettes m W m az aztransf az = W C W C W (5) e az aze s ( C C ) az az azs aztransf = k (6) = m C (7) az In the solution m m az az = W C W C + W (8) e az aze s azs aztransf = m C (9) NON-SUGARS BALANCE In the ossettes m W m naz naztransf naz = W C W C W (10) e naz naze s nazs ( C C ) naz naz naztransf = k (11) = m C (12) In the solution m m naz naz e naz = W C W C + W (13) naz naze s nazs = m C (14) WATER BALANCE In the ossettes m W m ag agtransf ag e age s ags agtransf naztransf = W C W C W (15) = k W (16) ag ag aztransf = m C (17) In the solution maz = We Cage Ws Cags + Wagtransf (18) m = m C (19) az az 4.2 FLOWS The mass flows of ossettes an solution proue at eah stage are evaluate assuming that they are proportional to the masses that exist in eah element, multiplie by a onstant an by the spee of the rotor in rpm. In the variables that flow through the system we have introue a elay between stages so that the elay in transport that takes plae insie the iffuser is taken into aount an the hanges in an element take a ertain amount of time to affet the subsequent stages. This elay has been moelle with a funtion of the first orer, as follows: X je τ = X ( j 1) s X je (20) Where X je is the value of the variable X at the inlet of stage j, X (j-1)s is the value of the same variable at the outlet of the previous stage an τ is the time onstant, it is a measurement of elay. 7-4

5 4.3 ENERGY BALANCES In the same way as esribe in the previous setion, we alulate the energy balanes for eah ifferential element of the iffuser. As explaine before, the enthalpy variation whih the ossette suffers is ue, on the one han, to the ifferene in temperature (if it exists) between the ossette an the juie an, on the other han, to the ossette/juie mass transfer. We therefore have a simultaneous phenomenon of energy an mass transfer. The energy balane equations are those shown below: Cossette ( m H ) Juie ( m H ) = W e = W e H e H W e s W H s s H H s tm + H + H tm t H (21) t (22) The following are the equations to alulate the flow of heat ue to mass transfer: H tm = (m + m + m ) H (23) azt agt mart That is, the prout of the mass whih is being transferre, an the enthalpy of the ossette at its temperature. By using this expression, any other type of solution enthalpy is isregare. The energy transferre by the ifferene in temperatures is alulate by means of the following expression: H t = k (T T ) (24) Where k is a onstant whih enompasses the overall ossette-solution heat transmission oeffiient an the area of exhange. 5 MODEL DEVELOPMENT IN ECOSIMPRO All these equations have been inorporate into EosimPro, along with other neessary relate equations. Programming these types of systems is easy beause the statements EXPAND or EXPAND_BLOCK allow the insertion of numerous equations base on a parameter whih is moifie. This has two avantages: it ereases the amount of oe an we an parameterise the size of the set of equations whih is reate. The latter is espeially useful in this ase beause isretisation an be expane or ontrate with the use of a single parameter. The following is an example of the use in the iffuser moel of EXPAND_BLOCK EXPAND_BLOCK(j IN 1,net) mazt[j] mazt[j] Caz[j] = zona(maz[j],m[j],0.) Caz[j] = zona(maz[j],m[j],0.) mazt[j] = max((k1_real[j] * (Caz[j] - Caz[j])),0) maz[j]' = We[j] * Caze[j] - Ws[j] * Cazs[j] - --"Balane al azúar in the ossettes" maz[j]' = We[j] * Caze[j] - Ws[j] * Cazs[j] + --"Balane al azúar in the isoluión" tauw * Cazs[j]' = Caz[j] - Cazs[j] tauw * Cazs[j]' = Caz[j] - Cazs[j] END EXPAND_BLOCK The iffuser ommuniates with the exterior by means of five ports IN solio (Mix=ossette) in_os OUT solio (Mix=ossette) out_os IN f_juie (Mix=juie) in_agua_prens IN liquio (Mix=H2O) in_agua_aport OUT f_juie (Mix=juie) out_is IN analog_signal u_rot OUT analog_signal v_ There are two IN ports for the solution whih will extrat the surose, one OUT port for the surose-rih juie, one OUT port for the exhauste pulp, an one analogue IN port an OUT port whih are onnete to a ontroller that regulates the iffuser rotation spee. 6 SIMULATION OF THE DIFFUSER In the following paragraphs we will see how the iffuser respons to ifferent isturbanes. A number of stages equal to 20 has been use in the simulation. In the first plae, the parameters of the moel have been ajuste so that the responses are realisti. The proess is then simulate an, from the points of view of mass flows an onentrations, an analysis is mae of the results obtaine by means of simulation as ompare with the real ata that we have. The following are the values of the extration juie obtaine at the outlet of the iffuser: 7-5

6 Substane Outlet Juie % weight simulation % weight theoretial Sugar Non-sugar solubles Total ry substane Water ontent Outlet mass Table 1 While the values for the exhauste pulp that is isharge from the iffuser are: Substane Outlet Juie % weight simulation % weight theoretial Sugar Non-sugar solubles Frame Total ry substane Water ontent Outlet mass Table 2 To see how the sugar onentration evolves, the following graph shows the sugar onentrations in the juie an in the ossette along the length of the iffuser: 0,18 0,16 0,14 0,12 0,1 0,08 0,06 0,04 0, Figure 5 Sugar onentration in the juie in perentage weight vs. isretise length of iffuser. Sugar onentration in the ossette in perentage weight vs. isretise length of iffuser. Figure 5 shows a typial profile in the ase of ounterurrent extration, where the onentration ifferene is kept more or less onstant throughout the omplete extration proess, ereasing slightly as the ossette loses its sugar ontent. The following esribes how the sugar onentration in ifferent parts of the iffuser are affete by a variation in the flow of refresh water that is being ae. The water flow varies from kg/s up to kg/s. The graph shows isretisation points 1, 5, 10, 15 an 20. By efault in the experiments, the x axis represents the time in h Figure 6 Caz[1] Caz[5] Caz[10] Caz[15] Caz[20] It an be seen that as soon as there is any isturbane, the sugar onentration at the first point ereases. The hange at the remaining points takes plae with a ertain elay beause they are further away from the point at whih the isturbane takes plae. It an also be seen how the flow moel behaves if we a the erivative of the onentration with respet to time Figure 7 Cazs[1]' Cazs[5]' Cazs[10]' Cazs[15]' Cazs[19]' 7-6

7 It an be seen that the further the peaks are from the isturbane, the lower an more symmetrial they beome. This orrespons with the results obtaine if we onnet n tanks in series. Let us see what happens if, ue to heater failure, the ossettes enter at 15 C instea of 69 C. Figure 8 shows how the temperature of the juie ereases along the length of the iffuser as it makes ontat with the ol ossettes. We an arry out a three-imensional analysis of what happens. Figure 10 shows the variation in the sugar onentration with time an with istane as the quantity of water ae to the iffuser is inrease. Salto en la antia e agua a la entraa. Dese 71.3 hasta 100 kg/s 80 T[20] 70 T[15] 60 T[10] 50 T[5] T[1] Figure 10 Figure 8 It an be seen how the sugar onentration in the ossette inreases as the temperature goes own, espeially uring the first stages where the temperature is at its lowest. The sharp peaks show where there has been a suen hange in the temperature. In reality, however, these peaks woul not our in this fashion an the response woul be smoother. 7 PORTS AND PHYSICAL PROPERTIES To evelop this omponent we have reate a series of auxiliary libraries for physial properties an for ports. As an example of ports, the following is the juie port whih was reate to onnet the juie inlet an outlet of the iffuser with other omponents PORT f_juie (SET_OF(Chemial)Mix) Caz[1] Caz[5] Caz[10] Caz[15] Caz[20] SUM REAL W "Flujo másio (Kg/s)" SUM IN REAL Wi[Mix] "Flujo másio e aa omponente (Kg/s)" EQUAL OUT REAL C[Mix] "Conentraiones (%1 en peso)" EQUAL REAL P "Presión (bar)" Figure 9 REAL T EQUAL OUT REAL H SUM IN REAL f_energ REAL Pol REAL Pureza REAL Brix peso" REAL Rho REAL F "Temperatura (ºC)" "Entalpía espeífia KJ/Kg" "Flujo e entalpía (KJ/s)" "Polaria en %1 en peso" "Pureza en %1 en peso" Graos Brix en %100 en "Densia (Kg/m^3)" EQUAL OUT REAL ph 7-7

8 CONTINUOUS 1 = SUM (j IN Mix; C[j]) EXPAND(j IN Mix EXCEPT setofelem(mix,1)) Wi[j] = C[j] * W W=SUM(j IN Mix;Wi[j]) Pureza = zona(pol,max((brix/100),0.01),0) Pol = C[azuar] Brix = 100 * (C[azuar]+C[no_azuar]) H = H_juie(T,Brix,Pureza) f_energ = W * H Rho = Den_juie(T,Brix,Pureza) W=F*Rho END PORT The following physial properties were inlue: - Juie (onutivity, ensity, visosity, steam pressure, enthalpy) - Cossettes (enthalpy) - Water (visosity, ensity, enthalpy) 8 PROBLEM EXTENSION The iffuser is the entral part of the iffusion setion. Although this setion has not been esribe in this paper, it has been moelle in its entirety. The following figure shows an image of the iffusion setion obtaine through the graphi onnetion of ifferent omponents reate with EosimPro. Graphi onnetion was mae using the SmartSketh program whih allows ifferent omponents reate with EosimPro to be quikly onnete. It also generates the simulation oe of the omplete assembly. Figure 11 shows the result of the onnetion of all the omponents. Figure 11 Nomenlature C C ag C age C ags C az C aze C azs C naz C naze C nazs C C ag C age C ags C az C aze C azs C naz C nazs onentration in the ossette expresse as a ratio per water onentration in the ossettes insie the ifferential element, expresse as a ratio per water onentration in the ossettes at the inlet expresse as a ratio per water onentration in the ossettes at the outlet expresse as a ratio per sugar onentration in the ossettes insie the ifferential element expresse as a ratio per sugar onentration in the ossettes at the inlet of the element expresse as a ratio per sugar onentration in the ossettes at the outlet of the element expresse as a ratio per onentration of non-sugars in the ossettes insie the ifferential element expresse as a ratio per onentration of non-sugars in the ossettes at the inlet expresse as a ratio per unit weight. onentration of non-sugars in the ossettes at the outlet expresse as a ratio per unit weight. onentration in the solution expresse as a ratio per water onentration in the solution insie the ifferential element expresse as a ratio per water onentration in the solution at the inlet of the element expresse as a ratio per water onentration in the solution at the outlet of the element expresse as a ratio per water onentration in the solution insie the ifferential element expresse as a ratio per sugar onentration in the solution at the inlet of the element expresse as a ratio per sugar onentration in the solution at the outlet expresse as a ratio per onentration of non-sugars in the solution insie the ifferential element expresse as a ratio per onentration of non-sugars in the solution at the inlet of the element expresse as a ratio per 7-8

9 C azs H H e H s H j H je H js H t H tm onentration of non-sugars in the solution at the outlet of the element expresse as a ratio per speifi enthalpy of the ossette insie eah ell, in kj/kg. speifi enthalpy of the ossette at the inlet of eah ell, in kj/kg. speifi enthalpy of the ossette at the outlet of eah ell, in kj/kg. speifi enthalpy of the juie insie eah stage, in kj/kg. speifi enthalpy of the juie at the inlet of eah stage, in kj/kg. speifi enthalpy of the juie at the outlet of eah stage, in kj/kg. enthalpy transferre from the juie to the ossette ue to the effet of the temperature ifferene, in kj/kg. enthalpy transferre from the ossette to the juie ue to the effet of mass transfer, in kj/kg. k ag proportionality onstant, whih is aimensional. k az k naz m agt m azt m m ag m az m naz m m ag m az m naz m t W agtransf W aztransf mass transfer oeffiient for the sugar, in kg/s. mass transfer oeffiient for non-sugars, in kg/s. mass flow of water whih is transferre from the ossettes to the juie, in kg/s. mass flow of sugar mass whih is transferre from the ossettes to the juie, in kg/s. ossette mass insie the ifferential element, kg. water mass in the ossettes insie the ifferential element, kg. sugar mass in the ossettes insie the ifferential element, kg. non-sugar mass in the ossettes insie the ifferential element, kg. solution mass insie the ifferential element, kg. water mass in the solution insie the ifferential element, kg. sugar mass in the solution insie the ifferential element, kg. non-sugar mass in the solution insie the ifferential element, kg. mass of sugar or non-sugars whih is transferre, in kg. mass flow of water whih is transferre insie the element, kg/s. mass flow of sugar whih is transferre insie the element, kg/s. W e W s W e W s mass flow of ossettes whih enters the element, kg/s. mass flow of ossettes whih is isharge from the element, kg/s. mass flow of solution whih enters the element, kg/s. mass flow of solution whih is isharge from the element, kg/s. W naztransf mass flow of non-sugars whih is transferre insie the element, kg/s. Aknowlegements This work has been arrie out within the framework of the projet Simulation of sugar fatories to optimise the proess an personnel training (FEDER TAP 1FD ) arrie out at the Centro e Tenología Azuarera (Sugar Tehnology Centre) of the University of Vallaoli. The author thanks all the persons involve in the projet. Referenes [1] Sugar Tehnology. Beet an Cane Sugar Manufature P.W. van er Poel, H. Shiwek, T. Shwartz Bartens 1998 [2] Analysis an Simulation of Proesses Davi M. Himmelblau, Kenneth B. Bishoff Reverté S.A [3] Heat an Mass Transfer Ekert, E. R. G. y Dake, R.M. MGraw Hill, 2 n eition 1959 [3] Manual e energía para fábrias e azúar Tone Baloh, Enrique Wittwer E Bartens, 2ª Eiión [4] Beet-Sugar Tehnology R. A. M Ginnis Beet Sugar Development Founation 3 r eition 7-9

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