Inverse Model to Determine the Optimal Number of Drops of RDC Column Using Fuzzy Approach

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1 Inverse Moel to Determine the Optimal Number of Drops of RDC Column Using Fuzzy Approach 1 HAFEZ IBRAHIM, 2 JAMALLUDIN TALIB, 3 NORMAH MAAN Department of Mathematics Universiti Teknologi Malaysia UTM Johor Bahru, Johor, Malaysia MALAYSIA 1 hz_4@hotmail.com, 2 jtalib@utm.my, 3 normahmaan@utm.my Abstract: - Inverse moeling is natural in many real worl application incluing inustrial chemical engineering problems. This paper escribes the process of etermines optimal input an output of number of rops in various stage of rotating isc contactor column using fuzzy moel. An algorithm of the fuzzy moel is evelope to simulate the above process. Key-Wors: - Liqui-Liqui Extraction, RDC Column, Drop Distribution, Inverse Moel, Fuzzy Environment, Fuzzy Algorithm. 1 Introuction Liqui-Liqui Extraction is a process of separating components of a liqui using other liqui. The process of liqui-liqui extraction is commonly use in chemical, biochemical, biotechnology an Inustry. Many types of evices have been use over the years in solvent extraction processes, but the Rotating Disk Contactor (RDC) is the most wiely aopte since it was evelope by the Royal Dutch/Shell Group in the mile of the last century. The geometrical properties of RDC column consists of a series of stages separate by equally space horizontal stators [1, 2] (see Fig.1 [9]).The RDC column has several avantages such as simplicity in construction, high throughput an low power consumption [2]. The mathematical moeling of RDC column is usually concern with the rops istribution an rops iffusion. Both forwar moel an inverse moel are use to preict the istribution of rops, we can escribe the forwar moel in general as a process in obtaining values of outputs of rops istribution by using input of average rop iameter. The output obtaine in the forwar moel is use in the inverse moel to etermine the value of the input parameters for a esire value of the output parameters [3]. Fig.1 Rotating Disc Contactor Column. ISSN: Issue 4, Volume 10, April 2011

2 The basic principles of fuzzy moelling were lai by Zaeh [4] in There are three principles in eveloping a fuzzy moel, the use of linguistic variables in place of or in aition to numerical variables, the characterization of simple relation between variables by conitional fuzzy statements an the characterization of complex relation by fuzzy algorithms. These principles from the basic of two methos use for fuzzy moelling, namely the irect approach an system ientification. In the first metho, the system in first escribe linguistically using term from natural language an then translates into the formal structure of a fuzzy system. The secon metho is evelope from structure ientification to the parameters ientification. 2 Drop size istribution This section iscusses the proceure of the Expecte Value Metho (EVM) to etermine rop size istribution [5]. The rop size istribution is later use as an input value to our fuzzy moel. The Expecte value metho epens on four steps which are the bulk breaking process, the etermination of number of rops broken, number of aughters prouce from broken rops an number of aughter in each class. The etaile proceure is given below 2.1. General We efine the stage of RDC column as collection of classes between stators an rotators. Shown in Fig.2 Fig.2 Stages an Classes Tables 1 an 2 show the geometrical properties of RDC column an physical properties of system use. Table 1: The Geometrical Properties of RDC Column Geometrical properties of RDC Number of stages 23 Height of compartment(m) Diameter of rotor isk (m) Diameter of column (m) Diameter of stator ring(m) Rotor spee (rev/s) 4.2 Table 2: Physical Properties of the System Use. Physical properties of the system (cumene/isobutyric aci /water) Continuous Phase: isobutyric aci in water Disperse Phase: isobutyric aci in cumene Viscosity of continuous phase( kg / ms ) Viscosity of isperse phase( kg / ms ) Density of continuous phase ( 3 / ) Density of isperse phase ( 3 / ) 0.100E E-3 kg m 0.100E+4 kg m 0.862E+3 Interfacial tension (mn/m) 31.7 The initial rop size will be use the basis for the calculation of the classes size [5]. If the initial rop iameters is 0, an the require number of classes is N cl, then the size of the class j, c j which will hol rops of class j with iameters between an is E, j E, j+ 1 j E, j+ 1 E, j 1 c Where j 1,..., Ncl an 0 ( j ) j N ( E, j 1, 1,..., cl 2) N cl In many mass transfer an flow processes it is ISSN: Issue 4, Volume 10, April 2011

3 esirable to work only with average iameters instea of the complete rop size istribution (Mugele an Evans [1951]). Since the volume of a rop is important in the calculation of the mass transfer performance, the average iameter is obtaine by averaging the volume of the rops, rather than taking the arithmetic average iameter. Assuming that the rops are ranomly prouce an the iameter of the rops are uniformly istribute in each class, the volume average iameter av, j of rops in class j, whose iameter ranges from E, j to E, j + 1, j 1,..., Ncl can be written as 1 av, j E, j +, 1 E j 4 Then the critical rop size is obtaine by ( E j + )(, j + E, + 1) ( 3) cr WeDw, ReDw, 4 D r Where We Re Dw, Dw, ρω c D γ ρω c D μ 2 3 r c 2 r cr 1 ( 5) ( 6) Equations (5) an (6) are isc angular Weber number an a Reynols number respectively. An foun correlation for critical rop size an rotor spee [6], suitable for their wie ranges of experimental ata in ifferent size column (152,300,600 mm): Equivalent to: 0.7 γ ω cr ρc μc D r Where ω 2π N the angular rotor spee an ω enote the critical angular rotor spee cr appropriate to rop size. To obtain the probability of breakage for classes with average rop size larger than the critical rop size. Cauwenberg [7] use a moifie angular Weber number in equations (8) an (9) 5 for laminar ( ReD, ω 10 ) 5 ( Re D, ω 10 ) respectively : 0.258We P We 1.16 D, ω, m 1.16 D, ω, m We P We Where 1.01 D, ω, m 1.01 D, ω, m < an turbulent ( ) D ( 8) ( 9) c c cr r D,, m 10 ρ μ ω ω We ω γ for (laminar) ( ) c c cr r D,, m 11 ρ μ ω ω We ω γ for (turbulent) 2.2. Simulation result D In this section we can escribe Expecte Value Metho process, as two parts the first for first stage is calle Bulk Breaking Process see Fig.3 an the secon part for all stage also using BBP for all classes of previous stage to get new stage. In the simulation of rops break-up using this metho, the bulk breaking process will be applie to the swarm of rops from the moment the swarm of rops enters the column. As the swarm of N rops hits the first rotor isc, since the iameters of all rops entering from the istributer are larger than the critical rop size, they will unergo all the steps of the bulk cr breaking process, which prouces number of aughter rops in the classes of the first stage. After obtaining the rops in each classes of the first stage, the bulk breaking process is use again on the swarm of rops in each class. Assuming that the average iameter of rops in some of the classes of the first stage is greater than the critical rop size cr, then as the swarm of aughter rops in each class of the first stage hit the secon rotor isc, only those rops in a class with average iameter greater than the ISSN: Issue 4, Volume 10, April 2011

4 critical rop size will unergo the bulk breaking process, which prouces is turn number of aughter rops in the classes of the secon stage. The rops in the class with average iameter less than critical rops size will just move to next stage without breaking. In general, the number of aughter rops in the classes of the ith stage can be obtaine by applying the bulk breaking process, class by class to the swarm of rops in the (i-1)th stage. Table 4: Drop Distribution (Stage5) Class No. of rops Fuzzy Flow Chart This flow chart introuce by Ahme [8] to escribe input an output process using fuzzy approach. The three steps processes are Fuzzification, Fuzzy environment an Defuzzification. Fuzzification Input Fuzzy Environment Output Fig.3 Bulk Breaking Process After use all equations an using Expecte value Metho we get the simulation of istribution of Number of rops for all stages, we can show in Table 3 the number of rop istribution in each class of stage 1 an also in table 4 shows the number of rops of each class of stage 5. Table 3: Drop Distribution (Stage1) Class No. of rops Fuzzification Defuzzification Fig.4 Fuzzy Flow Chart Variables involve in an engineering esign are usually referre to as parameters which are classifie as input, output an performance parameters. In our problem the input parameters are the geometrical configuration an physical properties, an the input number of rops. The geometrical configurations are the iameters of rotor isc, the column, the rotor spee etc. Meanwhile the physical properties are the viscosity an the ensity of the continuous an isperse phase etc. The output parameters are the number of rops. ISSN: Issue 4, Volume 10, April 2011

5 In this moel, we assume that the input parameters of the geometrical configurations an physical properties are fixe for certain values. These values are taken from experimental ata (see Tables 1 an 2).The performance parameter is also assume to be fixe. The actual input parameters of the moel are the input number of rops. On the other han the output parameters are the output number of rops Fuzzy Environment The fuzzifie input parameters from fuzzification phase are then use to etermine the inuce output parameters. This process can be one by assuming that all the fuzzy sets (taken from the previous phase), FPi,express preferences of all input parameters pi Pi with Pi R + to be etermine, normalize an convex, P is a close interval positive real number. In this phase, the input, output an performance parameter must be etermine. We shoul also be able to specify the moel or function (Expecte Value Metho) use which map the input parameters to the output parameters Defuzzification In this phase the optimal combination of the input parameters will be etermine. First we must etermine the α cut for F Pi of the f corresponing. After this value has been etermine, all 4 combinations of the enpoints of the intervals representing α f cut must be generate. These four combinations of inputs values are actually the possible solutions of the problem. In the fuzzy flow chart, the process of input an output base on algorithm one by Maan(2005), the number of rops in stage 5 (classes 5 an 6) is obtaine using input value of number of rops in stage 1 (classes 5 an 6). The optimal number of rops an errors are also shown. 4 Fuzzy Algorithm Step 1: input parameters with omain an output parameters with omain. Step 2: select appropriate value of such that α, α,..., α 0,1]. 1 2 P i k ( α cut Step 3: for each (input parameters) etermine the en points of all α cut, F i 1, 2. k Step 4: for each Q i (output parameters) etermine the en points of all αk cut, for preferre output parameters F i 1, 2. QP Pi Step 5: generate all 2 combinations of all enpoints of intervals representing αk cut. Each combination is an m-tuple (in this problem m2). r j Step 6: etermine (using EVM ) for all combination from step 5, j 1,2,...,2 m. Step 7: for each α cut, etermine the inuce output parameters, F in by taking the min value an max value of each element of i, let Fin min rj max r j for all j 1, 2 Step 8: set F QP F in an fin the fuzzy number of f sup( FQP Fin ) m. Step 9: fin the α cut of FPi corresponing value of f. for α an Step 10: repeat steps 5 an 6 for enote the corresponing output parameters as r for each j 1, 2,...,2 m. j f Step 11: etermine the optimal combination of input parameters an stop. ISSN: Issue 4, Volume 10, April 2011

6 Now applie fuzzy algorithm on stage 1 (classes 5 an 6) an stage 5 (classes 5 an 6) to show the effect. Table 5: Preferre Input Value Input parameters Domain Suggeste value Class 5 [45 75] 60 Class 6 [62 106] 84 Table 6: Preferre Output Value output parameters Domain Suggeste value Class 5 [ ] 247 Class 6 [ ] 299 These values in tables 5 an 6 are now reay to be fuzzifie by the triangular membership function. Figures 4 an 5 show the triangular fuzzy number of the preference input an output parameters respectively. The two limits of omain will have fuzzy value of zeroes whereas the suggeste value will be assigne a fuzzy value equal to one. Fig.6 Triangular Fuzzy Number of Output Parameters. Now using suitable α cut as 0, 0.2, 0.4, 0.6, 0.8 an 1. all the input an output parameters obtaine from Figures 4 an 5, for example take α 0.2, then the α 0.2 cut for preferre input is { i i μa( i) } Aα 0.2 p P p 0.2 If i1, an α 0.2 cut is { μa 0.2} ( Aα 0.2 p1 P1 p , ) [48,72] The same proceure is use to calculate the α cut for each values are the liste in tables 7 an 8. The next step is to generate all the possible combinations of the enpoints of the interval representing each α cut for the input parameters. We have 4 combination of 2-tuple for each α cut. Shown in table 9. Fig.5 Triangular Fuzzy Number of Input Parameters. ISSN: Issue 4, Volume 10, April 2011

7 Table 7: α cut Values for Input Parameters Input parameters α cut values Class 4 [ ] [ ] [ ] [ ] [ ] Class 5 [ ] [ ] [ ] [ ] [ ] Table 8: α cut Values for Output Parameters Output parameters α cut values Class 4 [ ] [ ] [ ] [ ] [ ] Class 5 [ ] [ ] [ ] [ ] [ ] Table 9: the Combination for Each α cuts Values Parameter Combination of input (class 4 & class5) α cut values st combination ( ) ( ) ( ) ( ) ( ) 2 n combination ( ) ( ) ( ) ( ) ( ) 3 r combination ( ) ( ) ( ) ( ) ( ) 4 th combination ( ) ( ) ( ) ( ) ( ) Now using EVM to get the output parameters, show in Table 10 Table 10: the Output of Each Combination of Each α cuts Combination of input (class 4 & class5) α cut values Class4 class5 Class4 class5 Class4 class5 Class4 class5 Class4 class5 1 st combination n combination r combination th combination ISSN: Issue 4, Volume 10, April 2011

8 After that fin the inuce output parameters. The inuce output parameters can be F in obtaine by taking the minimum an maximum value (enpoints of interval) for each element of output parameters. As an example, in Table 10, the output of number of rops is equal to 68.4 an 74.4(class 4) for each input combination of α 0.2 in this proceure take the minimum value is 68.4 an maximum value of This inicates that α 0.2 -cut for the first inuce output parameter is [68.4, 74.4].The same process is repeate for ifferent values of alpha to obtain the corresponingα cut. Now we can show the intersection between inuce an preferre output in Figures 6 an 7 best possible combination of the input parameters in orer to prouce the output parameters which are close to the esire output value. Each of the four combinations of the enpoints of the interval is etermine an these values are then use to calculate the output parameters. All the ata are given in Tables 11 an 12. Table 11: Input Combination with Fuzzy Value z Combination of input Input value 1 st combination ( ) 2 n combination ( ) 3 r combination ( ) 4 th combination ( ) Table 12: Input Combination with Fuzzy Value z Combination of input Input value 1 st combination ( ) 2 n combination ( ) 3 r combination ( ) Fig.7 Intersection between Inuce an Preferre Output for Class 5. 4 th combination ( ) For the four combinations of the input parameters given in Tables 11 an 12 we have to choose only one combination which can provie the optimal solution. Also to output Table 13: Output Combination with Fuzzy Value z Combination of output output value Fig.8 Intersection between Inuce an Preferre Output for Class 6. The intersection point z an z997 respectively. These f values are then processe in the efuzzification phase. In this phase, efuzzification is carrie out to get the 1 st combination ( ) 2 n combination ( ) 3 r combination ( ) 4 th combination ( ) ISSN: Issue 4, Volume 10, April 2011

9 Table 14: Output Combination with Fuzzy Value z fuzzy algorithm gives error between inuce value an preferre value less than 1%. Combination of output output value Nomenclature: 1 st combination ( ) 2 n combination ( ) 3 r combination ( ) 4 th combination ( ) For the four combinations of the output parameters given in Tables 13 an 14 we have to choose only one combination which can provie the optimal solution. Table 15: Optimize Input Parameters (Stage 1) Input parameters Calculate input value Preferre value Error(%) Class Class Table 16: Optimize Output Parameters (Stage 5) D r 0 E,j c,r av,j W e D,m W e D,ω We D,ω,m Re γ μ c ρ c ω ω cr D,w Rotor iam eter initialrop iameters D rop size L im it for each rop size class. C ritical rop size average iam eter M oifie isc W eber number isc angular W eber num ber MoifieiscangularWebernumber Reynols number Interfacial tension Continuous phase viscosity Continuous phase ensity A n gu lar velocity Critical angular rotor velocity Output parameters Calculate output value Preferre value Error(%) References: Class Class The values of the input parameters for esire values of the output parameters are successfully etermine. The values are shown in Tables 15 an 16. The input values iffer from the preferre values with an error of 0.025% an 0.026% respectively. The percentage error for each of the output parameters of the system are 0.025% an 0.026% respectively. 5 Conclusions This work escribes the inverse moel ientify the number of rops of each class. The optimal number of rops epens on the result obtaine by the Expecte Value Metho. An effective [1] Hoa Molavi, Sima Hoseinpoor, Hossein Bahmanyar, H. An Investigation On The Effect of Continuous Phase Height On The First an Secon Critical Rotor Spees in a Rotary Disc Contactor, Worl Acaemy of Science, Engineering an Technology 61,(2010) [2] Maria A Moris, Fernano V Diez, Jose Coco. Hyroynamics of Rotating Disc Conactor, Separation an Purification Technology 11 ( 1997) pp: [3] Maan, N. Mathematical Moelling of Mass Transfer in Multi-Stage Rotating Disc Contactor Column. Ph.D. Thesis, Universiti Teknologi Malaysia, Malaysia; [4] Zaeh, L.A. Outline of A New Approach to The Analysis of Complex Systems an Decision Processes. IEEE Trans. On Systems, Man an Cybern (1): ISSN: Issue 4, Volume 10, April 2011

10 [5] Talib, J. Mathematical moelling of A Rotating Disc Contactor Column, Ph.D. Thesis. Brafor University, Brafor, U.K; [6] Java Saien Ghalehchian. Preiction of the Hyroynamic of Rotating Disc Contactors Base on a New Mote-Carlo Simulation Metho for Drop Breakage, journal of Chemical Engineering of Japan, Vol 35, No.7,2002, pp [7] Cauwenberg, V., J.Degreve an M.J.Slater; The Interaction of Solute Transfer Contaminants an Drop Break-up in Rotating Disc Contactor: part 1. Correlation of Drop Breakage Probabilities an Part 2. The Coupling of Mass Transfer an Breakage Process via Interfacial Tension, Ca.j.Ch.Eng., Vol 75,2002, pp [8] Ahme, T. Mathematical an Fuzzy Moelling of Interconnection in Integrate Circuit. Ph.D. Thesis. Sheffiel Hallam University, Sheffiel, U.K.; [9]Extraction Technology Group, liqui-liqui Extraction Column Design. Retrieve at URL ISSN: Issue 4, Volume 10, April 2011