Separation of Heterogeneous Solid Fluid Mixtures IX. Filtration theory with new flowing models through packed beds

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1 Searation of Heterogeneous Solid Fluid Mixtures IX. Filtration theory with new flowing models through acked beds TIBERIU DINU DANCIU* EMIL DANCIU University Politehnica of Bucharest Chemical and Biochemical Engineering Deartment 1-7 Polizu Str Bucharest Romania Starting from geometrical model based on articles of ideal shae deosed as incomressible reciitate and on extreme degrees of comaction for sherical layers (hexagonal and cubic lattice resectively) new relations for filtration secific resistances calculus were roosed. The weight between the two extreme layouts exressed as X C gives for the modified traditional model very closed values to the geometrical model. For the engineering aroach a model exclusively based on void fraction ε is more convenient so corrections were suggested. Presented formulas are limited to the constant ressure filtration with incomressible reciitate consisting from articles of irregular size but with a calculable shae factor φ. Keywords: filtration theory filtration modeling real geometry of solid articles and reciitate ores secific resistances In a revious aer [1] two new overall models for laminar flow through layers formed by identical sherical articles were roosed in two extreme oosite settlings comact and loose resectively. The first model is based on the accurate ores geometry (the flowing cross section and tortuosity) and is further referred as the geometrical model while the second one is based on the ambiguous concet of void fraction (orosity ε) well-known from literature [1 5] but modified to fit the results of both our first model and exerimental data [6] it will be further referred as the modified traditional model. Recently a comarison between results obtained by conventional filtration theory through reciitate and filter material (suort) with other aroaches of exerimental data [9 14] was done by Chi Tien and Renbi Bai [15]. In this aer the extension of these two models for granular beds consisting of non-sherical articles of irregular size but having volumes (V) and surfaces (A) which could be calculated using geometrical aroximations and a shericity factor (ψ) is resented. In this aroach the degree of comaction for random settling could be estimated either using the mean diameter of articles or from exerimental data taking into account the balance between the two extreme tyes of layers order defined for sherical articles. Based on the adjusted flowing equations the secific resistances of both reciitate and filter material will be comuted to be substituted into the classic form of filtration rate exression. Flowing models through acked beds comosed of nonsherical articles In a solid liquid susension solid articles are nonsherical even when they arise from rocesses forming microcrystals with defined geometrical shaes which yields to incomressible reciitates. According to the revious aer [1] the sherical articles could be imagined sliding from an initial loose layout to a more comact one by occuying available gas in the neighborhood; when this comact layout is reached no further comressing is ossible. If the article geometry could be estimated using a unique characteristic dimension (i.e. the edge length of a known shae crystal say ) or directly seen (on a microscoe) various equivalent diameters and further the shericity and the orosity of the bed with random settling could be found (obviously using more exerimental data). Equivalent diameters and shericity for non-sherical articles For different geometrical shaes could be estimated: - the equivalent diameter of a article d ech by matching either the volume to a fictive shere (V = V S ) or the external surface resectively; it yields: 3 6 V V 3 V = dv = (1a) 6 π 2 A A =A da = (1b) π - the shae factor (shericity) exressed in two ways [3.186 and ]: 2 ψ= A or φ= d 6 V A This becomes by substituting relations (1): ( ) 2 6 V V V V 3 V 2 3 A π A A 6 V A dv A π π ψ= = = π = = =φ (3) - the absolute deviation from the sherical form of the article: 6 V φ dv = =ψ dv A (4) This quantity will be used further assuming the calculus of V and A from characteristic dimensions (eventually reduced to only one ). In table 1 results of calculus for d V ψ and φ d V / in some usual cases (reresentative geometrical shaes for some crystallograhic classes) are given. It could be concluded that articles having shaes far beyond from sherical have these values far below unity (ositions 4 7) while the quasi-sherical articles have these values close to the unity (but still below ositions (2) * tiberiu_danciu@yahoo.com 182 htt:// REV.CHIM.(Bucharest) 67 No

2 Table 1 EQUIVALENT DIAMETERS (d V ) AND SHAPE FACTORS (ψ φ) FOR PARTICLES WITH DEFINITE GEOMETRY 8 11). The articles having the characteristic dimension a far below the diameter of the circumscribed shere have the quantity φ d V > (ositions 12 14). It should be emhasized that φ d V will substitute the diameter d S in all relations reviously established for sherical layers. Average diameters for irregular size articles The solid hase of a susension is generally olydisersed so a samle of the dried and sieved reciitate is needed to be divided (screened) into N g granulometric grous each one being quasi-monodisersed. Many exressions for the average diameter d has been roosed [2-4 6]. For consistency the average diameter with resect to the article volume will be calculated d V with relation (1a). Starting with a samle having the mass m screened into N g fractions by sieving the j fraction having the mass m j and the diameter d j (the fraction being considered monodisersed) and the mass fraction w j = m j / m. The mass of a single article from the j fraction is denoted with m j and the mass of the average article from the samle with m. The number of articles from j fraction will be n j resectively for the entire samle N g N = nj It yields: 1 Ng Ng Ng N m g j w j w j m = m N = m nj = m = m m m = j= 1 j= 1mj j= 1mj j= 1mj j= 1 where: 1 3 N g V 6 w j m =ρ V =ρ = 6 3 j= 1 π ρ d j and using (1a) it follows: d V 13 N g w j 6 V = = 3 3 j= 1 d π j Finally after d V calculus with (5) using N g diameters d j several well-defined articles (as shae) from the fraction (5) having the closest diameter to d V and it yields with V and A quantities d V and φ the roduct φ d V being used further to characterize the behaviour of all articles from the granular bed corresonding to the diameter d S in case of a granular bed having the same size and the same sherical shae (i.e. with shericity 1). Void fraction ε degree of comaction X C (for the reciitate layer) tortuosity factor k tt For sherical articles of the same size and a uniform settling in the granular bed ε is not a function of the REV.CHIM.(Bucharest) 67 No htt:// 183

3 diameter d S. The degree of comaction defined as a ratio and denoted with X C lies between the two extreme disositions (limits) [1]: comact (ε = ε C = 25.95%) corresonding to X C = 1 and loose resectively (ε = ε a = 47.64%) corresonding to X C = 0: X C εa ε 6 ( π+ 6 ε) = = = ε ; <ε< ε ε π ( 2 1) a C ( ) (6) Using the ratio X C could be defined also a tortuosity factor k tt [1] bounded by 1 and k ttc resectively with exression: ( ) ktt = ktt C XC + ktt a (1 XC ) = π XC + 1 = XC + 1 (7) This corresonds to: so the weighted ore length in a layer with a given thickness δ: (8) It follows: ε=εc lor = l C = δ and ε=ε l = l =δ a or a For any intermediate value ε C <ε < ε a which corresonds to a degree of comaction X C it yields (in case of sherical articles layer) δ < or < C. For the deendence ε(d V ) exerimental data available in literature are so oor that no confident conclusions could be derived. Only a vague assertion that for a given geometry increasing the articles dimensions yields to a decrease of the secific surface σ and also to an increase value for ε was done [3]. If the first art is obvious the second one is justified only by intuition. From data resented in tables 2 and 3 it follows that even for sherical granular beds quasi-monodisersed there is no certain rediction for evolution of deendencies ε(d S ) or X C (d S ). From table 1 for layers containing sheres with diameters below 1 mm ε(d S ) is monotone decreasing and also σ C (d S ) resulting from secific surface exression: This assertion disagrees with table 2; according to data above the degree of comaction X C increase slowly with d S. Although non-exected for non-conductive little articles this tendency could be exlained when they are free oured in air by electrical reulsive charging. This follow to loose granular beds while the settling with 12 contact oints to other articles (comact) is switched at least artially to only 6 contact oints (i.e. by changing the hexagonal lattice to a cubic one). It could be stated that with decreasing of the diameter d S this tendency is much ronounced. From table 3 given data (with unsecified source) for sherical articles with d S > 1 mm indicate a slow and non-uniform increasing of the deendence ε(d S ) but (9) Table 2 LAYER OF BULK SPHERES * ε AFTER GELDART [7] Table 3 LAYER OF BULK CERAMIC SPHERES [3] 184 htt:// REV.CHIM.(Bucharest) 67 No

4 ε calc (d S ) as a result of secific surface data (σ) show a decreasing with come-backs so a ertinent conclusion is imossible (for ε > ε a X C < 0). Only extreme diameters resent some confidence; erhas beds vibrating eventually sub-merged in liquid and using frequencies according to articles size could yield to much accurate and reroducible data and easier to interret. Extension of above results to filtration (retention of solid articles from susension) is difficult because of their variable size and shae. In this case the hydrodynamic method for estimating the quantity ε is recommended [5]. One could anticiate that for articles of different size the smallest deose by filtration inside the gas left by the larger ones in the reciitate so a decreasing of ε(and an increasing of X C ) is exected during the time since the thickness of the layer (δ) is growing together with caillary length ( ) and their tortuosity (k tt ). Adjusted relations for non-sherical articles Inside the two roosed models for the laminar flow through granular beds containing only identical sherical articles [1] some adjustments are required for layers containing non-sherical articles with different sizes which yield to a sort of disorder inside the bed; reviously given relations should be corrected starting from known values for d i and w i continuing with average values d V from (5) then ε (exerimental) and further shericity φ and finally the roduct φ d V given by (4). The geometrical model [1] will be modified for a certain degree of comaction X C (ε) from (6) and a tortuosity factor k tt from (7) calculated with ε(fig. 1) as follows: - the caillary average diameter d or is obtained by weighting with X C the two extreme diameters d C and d a corresonding to comact and loose settling resectively: - the volumetric flow G V on the entire filter area A f : (14) The so-called modified traditional model (using the void fraction ε) [1] will be altered as shown: - the void fraction (weighted): - the modified equivalent diameter for the gas: (13) (15) (16) which could be assimilated with d g = X C. d gc + (1-X C ). dga where d gc and d ga refer to the sherical article s layer (φ. d V = d S ) this assimilation remains for next quantities; - the average flow rate inside the gas: ((17) - the Reynolds number and the laminar condition: (10) Obviously ε from X C should satisfy the condition ε C < ε < ε a. For small and quasi-sherical articles (d V < 1 mm) this condition is always satisfied. The diameter d S from all flowing relations should be relaced with the roduct φ d V given by (4) where V and A are averages for the size and shae of articles found by exeriment (eventually microscoic); - the average flow rate through ores having the length = k tt δ and the diameter d or yields from Hagen Poiseuille relation: [ (1 ) ] 2 tot or 2 tot C C C a µ ktt δ µ δ XC ktt C + XC ktt a d X d + X d v = = (1 ) (11) with tot = 0 l ± h ρ g (the h sign is given by gravity); δ=δ (t) is the time-variable (t) thickness of the reciitate;. - the ore number N or ; if they are aroximated to cylinders having all the same diameter d or and being regular distributed on the filtration area A f which is erendicular on the flowing direction: - the Reynolds number and the laminar condition: (12) - the volumetric flow on the entire filter area A f : where: REV.CHIM.(Bucharest) 67 No htt:// 185 (18) (19a) is the suerficial (fictive) velocity related to the area A f. It was shown that both methods give more or less equivalent results at least for identical sherical small articles (d S < 1 mm) with a deviation of maximum 7.5% [1] the first one being more rational (stated as a reference) while the last one is easier to aly. Filtration theory through incomressible reciitate and filter medium Filtration as a searation rocess of heterogeneous fluid solid mixtures has two major asects (unit oerations): de-dusting when the fluid is a gas and the roerly filtration when the fluid is a liquid resectively. The following assertions are available only for the last case. The main variables in filtration rocesses are the velocity (v) or the filtrate volumetric flow (G V ) and the ressure dro ( tot ) which is resonsible for the fluid flowing through ores of already settled reciitate and of the filter medium at the end. While the thickness of the reciitate (and consequently the ore length) increases during the

5 time of the rocess t filtration could occur at constant ressure dro tot = constant (usually) resectively at v (G V ) = constant when the ressure dro should grow with time so there is a deendence tot (t). Although filtration is a discontinuous rocess in deth as an unit oeration it could be realized discontinuous (gravitational) seudocontinuous (with constant feed of susension in ressfilters) or even quasi-continuous (having several successive stages like filtration washing squeezing discharging reconditioning) in cellular drum filters or band filters. Using the constant ressure dro technique the filtration seed could be increased via: - increasing the articles diameter (d ) using flocculants and/or coagulant additives; - increasing the filtrate fluidity i.e. decreasing the dynamic viscosity µ by susension heating; - increasing the ressure difference tot for incomressible reciitates by increasing the liquid column for gravitational filtration by increasing the ressure in the susension room and/or by decreasing the ressure in the filtrate room (this rocedure being limited by the vaor ressure of the liquid) by increasing the diameter and/or the rotative seed of the centrifuge etc. Filtration at tot = constant with deosition of incomressible reciitate using the geometrical model Filtration at a constant temerature of a susension comosed of a Newtonian fluid (µ=constant ρ= constant) is considered. It roduce an incomressible reciitate having solid articles with definite forms nondeformable characterized by ρ S f d V ; they are deosed on a suort (canvas orous late metallic rofiles) with d 0 the ore diameter and δ 0 the ore length equal with the thickness of the filter layer both constant during the rocess. Because in most cases 1 µm < d V < 1 mm and consequently the reciitate s ores diameter 0.1 µm < d < 0.1 mm it follows that flowing regime within caillary is laminar. The filter layer with δ 0 = constant and with an almost constant ε 0 (in first few moments a artial clogging occurs because of the little articles; during this eriod the filtrate is turbid and need to be recycled) yields to a constant resistance R f0. In the same time the reciitate thickness is variable δ(t) having also a variable reartition of ressure dro tot (fig. 1). The overall difference tot is maintained constant but the intermediate value i decreases fast in the beginning of filtration from = 0 to a value which remains almost constant after a eriod since the filter resistance doesn t changes further so o = i - l tends fast to a constant value. Another necessary quantity for the calculus is the orosity ε which could be assumed constant if the filter is continuously feed with susension having an uniform distribution of the solid (by mixing in rotative filters by uming in ress filters etc.). The void fraction ε could be found only by exeriment (taking an assay from the susension filtering in almost same conditions and drying the reciitate): after washing the reciitate and eventually relacing the retained liquid with a more volatile one (e.g. after washing with water relacing could be done with methanol) by weighing the wet reciitate (still having a very thin but visible liquid film) and afterwards the dried one the difference could yield (knowing the density of vaorized liquid) to the void volume and further to void fraction ε by division. Taking into account the trend of the solid articles to settle against the ores oening an advanced comaction could be exected and conditions ε C < ε < ε a (or 0 < X C < 1) are obeyed. According to figure1 and considering that = tot - o = o - i has only a small variation (i.e. i varies during a short eriod t) the ressure on the whole layer (x) is variable because of δ(t) during the entire filtration time. The ressure gradient on x direction against the flowing sense of the filtrate is ositive and could be estimated from Hagen Poiseuille equation written for a current value of t: (20) where the velocity v being indeendent on t could be found from (11) and should be used in (14) to calculate the momentary volumetric flow G v = d v / d t related to entire filter surface A f dv being the infinitesimal variation of filtrate volume from t till t + dt (corresonding with growing of reciitate thickness δ from x to x + dx). The filtrate volume V obtained in time t with resect to the filter surface A f assing through the entire reciitate thickness δ is the same for all elementary layers including the dx one located at the reciitate surface in contact with susension. To calculate the solid quantity at time t in the reciitate m solid dx should be relaced in relation (20) knowing the elementary volume dv r = A f. (1 - ε). dx while the elementary mass is dm solid = ρ solid. (1-ε). A f. dx with δ olid being the density of the deosit. By integration one could get: (21) Because the entire solid hase quantity m solid remains in the reciitate m r from mass balance it yields: (22) where w sus is the solid mass fraction in susension (measurable). The overall susension mass m sus could be obtained by adding the quantity assing into filtrate V ρ with the one remaining in the wet reciitate V r ε ρ: From relations (22) and (23) it yields: (23) Fig.1. Variation of (tx) in suort (0) and reciitate ( r ) (24) 186 htt:// REV.CHIM.(Bucharest) 67 No

6 and further: (25) One could assume that above ratio denoted with χ is invariant with duration so is the same at the end if the susension is well mixed (concentration w sus is constant in time and uniform in sace). From (25): (26) Filtration equation for incomressible reciitates derives from relation (14) for the volumetric flow G V. A suerficial velocity v 0 related to the entire filter surface A f is defined; for the variable thickness δ: secific resistance ρ el ). It follows that filtration flow is somehow analogous with the amerage in a circuit fed by a consumable battery (having a growing electrical resistance). From equation (29) one could find that secific filtration resistances deend on ores diameter (d 2 ). By rearranging and integration between obvious limits (for t from 0 till t f and for V between 0 and V f ) it yields: (32) which is the Guth relation for filtration with incomressible reciitate denoting with It follows: (27) resectively. Both constants are usually found from exerimental data linear fitting with relation (32) recisely quantities r and r 0 then using relations (28) and (29) where only size and shae of articles are imlied (φ d V ) and layers characteristics (ε and ε 0 ). All these quantities are involved in calculus of filtration constants a and b on industrial scale. (28a) The same momentary flow asses through the suort (the filter made from natural or synthetic canvas metallic net sintering glass or ceramic orous lates); by analogy: (29) (29) Relacing tot = + o and assuming the flow G V given by relation (14) asses through both resistances of the variable reciitate δ from equation (26) and of the constant suort: (30) In relations (28 30) were denoted with r and r 0 resectively secific resistances of the reciitate and suort (the last one includes the constants thickness δ 0 ). Switching to mass flow G m = G V. ρ which divided by A f reresents the mass flux: where is the internal resistance against the (31) filtrate flowing through the reciitate with variable δ while is the external resistance of the suort. Both have velocity dimensions (m/s) are given by the viscosity µ of the filtrate and they are against the filtrate flow. Relation (31) is very much alike the generalized Ohm law a roerty flux (here the mass flux q m similar to the current density q el = I / A in A/m 2 ) is exressed like a ratio between a otential gradient which roduce the transort (here of ressure analog with the voltage U) and a resistance to the transort (here R f / analog with electrical Filtration equation using the modified traditional model By far using the overall quantities like ε and σ defining the granular bed is more attractive but these dimensions could not describe the real flow through. It was shown [1] that not the whole section of ores articiates to the filtrate flow (at least for layers comosed of identical sherical articles) so a realistic geometrical model should take into account stagnant zones (i.e. the void section ε A f is not entirely available for flowing). This assumtion should modify the equivalent diameter d ech with relation (16) for the gas diameter d g which alters significant all flowing relations (17 19) so much closer results to the geometrical model could be obtained without any other corrections to fit exerimental data. In usual methods for resistances r and r 0 calculus various numerical (and yet emirical) coefficients for ressure dros in granular beds at a given velocity are resent. For examle in Kozeny Carman [4] relation for a sherical layer: (33) the coefficient 180 (i.e. 72 5/2) is nothing else but the theoretical coefficient of 72 from the Poiseuille relation adjusted with the factor of 5/2. In the best known relation for laminar flow through granular beds Ergun [8] suggest the correction factor of 25/12 which lead to a coefficient of 150. The definition of voids equivalent diameter for the granular bed (if only sherical articles are resent) after Kozeny is: (34) Since d ech is resent in the denominator of (33) as a square (relacing d S ) it yields the value of 72 for the factor (32 9/4). It follows that exerimental data claim a correction of this coefficient at least with a factor included between the two values (2.5 resectively 2.083). In other sources [4] even the value 2.78 could be reached. It could be concluded that definition (34) doesn t reresent the real flowing section through granular beds voids. An alternative definition for the diameter was introduced with relation (15) which drives the correction coefficient to 2.25 an REV.CHIM.(Bucharest) 67 No htt:// 187

7 average between traditional models. Furthermore the geometric model imoses for the effective free section of flowing the exression 2/3 ε A f (19) instead of ε A f so it s obvious that the current manner of estimating the flowing section gives always bigger values (because of ellicle stagnant zones). From all above assertions one could conclude that secific resistances are much closer to reality if they are calculated with relations involving ε and σ S = 6 / d V : in which secific surface σ for non-sherical layers (eventually non-uniform also) could be found from relation (9) written accordingly for them. In these exressions the numerical factor of 5 comes from 180 / 6 2 (with 6 from definition of δ S ). The modified model allows the definition of both secific resistances relacing directly the effective diameter of gas d g from (16) in exressions (29) for G V and (19) for G Vg. Proosed models coming from substitution of caillary diameters d (and d 0 ) for the geometrical model and d g (d g0 ) for the modified model could be written: (28b) resectively: (19b) It follows the calculus definition for both secific resistances r and r 0 for reciitate and suort each one of them in two ways: - for the reciitate layer using relations (28b) and (19b): (35) (36) - for the suort giving identical exressions but with the subscrit (0) characteristic for this layer (r 0 r g0 ε 0 k tt0 X C0 φ 0 d V0 ). To comare the two tyes of modeling is enough to calculate the deendence r g /r = f(ε) from: (37) Some significant values for these ratios were calculated all over the domain of ε C < ε < ε a. A grahical reresentation of G Vg /G V (X C ) and r g /r(x C ) is shown in figure 2. A fourth degree olynomial in X C could be obtained by regression: the correlation coefficient being unitary. (38) 188 htt:// REV.CHIM.(Bucharest) 67 No

8 Conclusions The reference relations are given by the geometrical model based on the ideal shae of deosed articles to form an incomressible reciitate and moreover on the degree of comaction for the layer. By weighting the two extreme degrees of comaction via X C the modified traditional model gives almost the same values like the geometrical model. Because a model based exclusively on the void fraction ε is more convenient (and nevertheless widely used already) the results obtained in this manner should be adjusted with the diagram from figure 2 or with relation (38). No other filtration techniques were analyzed (with variable ressure) because for incomressible reciitates the secific resistance r will be similar calculated. On the other hand the roblem of comressible reciitates remains oen since the usual relation r = r 1 ( ) s is urely emirical and resents a dimension deficit. An algorithm based on some relation of comaction growing during the filtration X C = f(t) is needed either with = constant or with G V = constant. The increasing degree of comaction could be exlained by articles shae changing (they become flat) and also by changing of the characteristic dimension (the average diameter decreases). To develo a theory exerimental data are necessary concerning the variation of void fraction ε with V (and t consequently) at different values of tot in order to found a non-linear deendence X C (ε). Nomenclature A surface (m 2 ); a b constants in filtration equation (s / m 6 s / m 3 ); d diameter (m); g gravity (9.81 m / s 2 ); G m mass flow (kg / s); G V volumetric flow (m 3 / s); h height (m); I amerage (A); k tt tortuosity factor; L side (characteristic) length (m); m mass (kg); n N number; (difference of) ressure (N / m 2 ); q el current density (A / m 2 ); q m mass flux (kg / m 2 s); r filtration secific resistance (m -2 ); Re Reynolds number; R f filtration resistance (m / s); S generating line (m); t time duration (s); U voltage (V); v flow rate (m / s); V volume (m 3 ); w mass fraction (kg / kg); x length (m); X C degree of comaction; Greeks: δ layer thickness (m); Fig. 2. Deendencies G Vg /G V (X C ) and r g / r(x C ) - - from equation ε void fraction (m 3 / m 3 ); µ dynamic viscosity (kg / m s); π invariant of circle; ρ density (kg / m 3 ); ρ el secific resistance (Ù m); σ secific surface (m 2 / m 3 ); φ shae factor; χ volumetric ratio of reciitate; ψ shericity. Subscrits: a loose; A area; C comact; calc calculated; cc circle; cil cylinder; con cone; dr rectangle; ech equivalent; f filter filtration; g voids gas; hex equiangular hexagon; i intermediate; j monodisersed fraction; article; at square; or ore; r reciitate; tg equiangular entagon; rb rhomb; sc circular sector; solid solid; strat layer; sus susension; S shere; t time; tech equilateral triangle; tot overall; V volume; 0 initial fictive filter material (suort) References 1. DANCIU T.D. DANCIU E. Rev. Chim.(Bucharest) 66 no BRATU Em.A. Oeratii unitare în ingineria chimica vol. I-II Ed. Tehnicã Bucuresti TUDOSE R.Z. Ingineria roceselor fizice din industria chimica vol. 1 Ed. Academiei Române Bucuresati JINESCU G. Procese hidrodinamice si utilaje secifice în industria chimicã Ed. Didactica si Pedagogica Bucuresti BIRD R.B. STEWART W.E. LIGHTFOOT E.N. Transort Phenomena (revised 2 nd ed.) John Wiley & Sons New York McCABE W.L. SMITH J.C. HARRIOTT P. Unit Oerations of Chemical Engineering 7 th ed. McGraw-Hill Education New York JINESCU G. VASILESCU P. JINESCU C. Dinamica fluidelor reale în instalatiile de roces Ed. SemnE Bucuresti ERGUN S. Chem. Eng. Progress 48 no TOSUN I. Chem. Eng. Sci. 41 no STAMATAKIS K. TIEN C. Chem. Eng. Sci. 46 no TIEN C. BAI R. RAMARUO B.M. A.I.Ch.E.J. 43 no TILLER F.M. LU R. KWON J.H. LEE D.J. Water Res. 33 no LEE D.J. JU S.P. KWON J.H. TILLER F.M. A.I.Ch.E.J. 46 no BURGER R. CONCHA F. KARLSSEN K.H. Chem. Eng. Sci. 56 no TIEN C. BAI R. Chem. Eng. Sci. 58 no RUTH B.F. Ind. Eng. Chem. 27 no Manuscrit received: REV.CHIM.(Bucharest) 67 No htt:// 189