T» a, ,<> v r» -o . . • Jy \s *< /. . • A <*. .V** 1 ** ** ^ % o V • AT ^ l v . • • • - <> '• • » • j. & ^ *'T77« A ;. ^. < ♦ A.* "^ * %^ -'^"- X/ :»fc: %,^ / aVaV ** * * • *<5^TV^*_ O v«o^ • • .■ ^ «b • Information Circular 8873 Computer Simulation Applied to the Separation of Porous Leach Residue Solids From Liquor by Horizontal Belt Filtration By Daniel T. Rogers and Roy T. Sorensen, Jr. UNITED STATES DEPARTMENT OF THE INTERIOR James G. Watt, Secretary BUREAU OF MINES Robert C. Horton, Director IC 8873 Bureau of Mines Information Circular/1982 Computer Simulation Applied to the Separation of Porous Leach Residue Solids From Liquor by Horizontal Belt Filtration By Daniel T. Rogers and Roy T. Sorensen, Jr. UNITED STATES DEPARTMENT OF THE INTERIOR ^ ;w ^f\0 i*^ As the Nation's principal conservation agency, the Department of the Interior has responsibility for most of our nationally owned public lands and natural resources. This includes fostering the wisest use of our land and water resources, protecting our fish and wildlife, preserving the environmental and cultural values of our national parks and historical places, and providing for the enjoyment of life through outdoor recreation. The Department assesses our energy and mineral resources and works to assure that their development is in the best interests of all our people. The Department also has a major responsibility for American Indian reserva- tion communities and for people who live in island territories under U.S. administration. This publication has been cataloged as follows Rogers, Daniel T Computer simulation applied to the separation of porous leach residue solids from liquor by horizontal belt filtration. (Information circular / Bureau of Mines ; 3873) Supt. of Docs, no.: I 28.27:8873- 1. Aluminum— Metallurgy— Mathematical models. 2. Aluminum- Metallurgy— Data processing. 3. JFilters and filtration— Mathematical models. 4. Filters and filtration— Data processing. 5. Aluminum oxide. 6. Hydrochloric acid. I. Sorensen, Roy T. II. Title. HI. Series: Information circular (United States. Bureau of Mines) ; 8873. TN295.U4 [TN775] 622s [669'.722] 81-607166 AACR2 For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C, 20402 Ill CONTENTS V) Page Abstract 1 Introduction 1 Belt filtration modeling 2 Background 2 Filtration and washing 2 Process description 2 Definition of washing efficiency 2 Prior material balance calculations 2 Prior horizontal belt filter modeling 3 Selection of a model 3 Purpose of model development 3 Ground rules for model development 3 The existing perfect-mixing-cells-in-series (PMCS) model 3 Description 3 Discussion 3 The diffusion model 3 Description 3 Discussion 3 The shrinking voids model 3 Description 3 Discussion 3 Derivation of equations for a single wash- filtration stage 4 Fundamental PMCS equations 4 Development of diffusion and shrinking voids model equations 4 Balance around the wash step 4 Diffusion model: migration of AI2O3 from cake voids 5 Page Shrinking voids model: hypothetical voids shrinkage 6 Completion of countercurrent calculations . . 7 Equations for liquor weight and volume balance 7 Summary of material balance procedure 7 Sample calculations 9 First material balance trial 9 Procedure following first balance trial 9 Evaluation by least squares error 9 Goodness of fit 9 Least squares fit 10 Model evaluation using miniplant data 10 Application of the shrinking voids model 12 Development of material balances from plant data 12 Prediction of hypothetical plant material balances 12 Optimization of horizontal belt filtration operations 12 Summary and conclusions 13 Appendix A.— Material balance by Fortran computer 14 Appendix B.— Material balance using a programmable calculator 21 Appendix C— Measurement of leach residue porosities and densities 25 Appendix D.— Cake liquor density and volume determinations 26 Appendix E.— Nomenclature 27 ILLUSTRATIONS 1. Schematic of continuous horizontal belt filtration circuit >. 2 2: Schematic for two mathematical molds (diffusion and shrinking voids) of the displace washing of the continuous belt filtration 4 3. Summary of equations used in calculating a material balance around a single wash step 5 4. Cake liquor solute concentration, Cf, as a function of diffusion time 6 5. Sample filtration circuit material balance form 8 B-1. Schematic of TI-59 program for material balance using the shrinking voids model balance 24 TABLES 1. Miniplant belt filtration data, test 1-3 7 2. Summary of a material balance calculation method, test 1-3 10 3. Summary of sums of squares of errors (SSE's) for test 1-3 material balance 10 4. Material balances for seven miniplant tests 11 5. Comparison of best-fit material balance with shrinking voids model material balance 11 6. Average filtration parameters for miniplant leach residues containing 123 lb of dry solids 12 7. Predicted AI2O3 filtration losses under various operating configurations 12 8. Predicted form filtrate concentrations under various operating configurations 13 C-1. Summary of 1-day-6ld leach residue densities and porosities 25 COMPUTERSIMULATIONAPPLIEDTOTHESEPARATIONOFPOROUS LEACH RESIDUE SOLIDS FROM LIQUOR BY HORIZONTAL BELT FILTRATION By Daniel T. Rogers 1 and Roy T. Sorenson, Jr. ABSTRACT The Bureau of Mines, in its alumina miniplant project to investigate alumina recovery from domestic, nonbauxitic ores, has conducted research on the use of a hydrochloric acid leaching, gas sparging crystallization technology. An important element of this research is the efficient separation of undissolved, siliceous residue from AICI 3 -bearing leach liquors by continuous, horizontal, countercurrent vacuum belt filtration. In an effort to calculate material balances quickly and to predict material balances based on different belt filtration configurations, the perfect-mixing-cells-in-series model (PMCS) for calculating material balances around belt filters was used. Because of the porous nature of the solids, the model produced erroneous material balances. Therefore a reliable model, the shrinking voids model, was developed postulating the presence of an unwashable voids liquor volume that decreases with decreasing liquor AICI 3 concen- tration. This volume decrease postulation is equivalent to assuming that dilute liquors flow more freely, causing more voids liquor volume to become washable. Least-squares based computer programs are provided, which are useful not only in producing material balances from plant data but also in predicting balances for untested configurations using the same feed materials. INTRODUCTION As the United States is dependent on foreign sources of bauxite for the alumina re- quired for aluminum production, domestic kaolinitic clay, hereafter referred to as clay, is being explored as an alternate source of aluminum. The Bureau of Mines is presently testing technology in which aqueous hydrochloric acid (HCI) is used to leach aluminum from calcined clay. This leaching produces an aqueous aluminum chloride (AICI 3 ) solu- tion that must be separated from the siliceous solids not dissolved by HCI. Two possi- ble methods for achieving this separation, vacuum filtration and classifier-thickener systems, were tested. Because of the time required to calculate material balances from the filtration data obtained in the alumina miniplant operation, and the need to predict material balances for different possible belt filtration configurations, an existing computer model, the perfect-mixing-cells-in-series model (PMCS), was used for these calculations. As this model did not result in accurate calculations because of the porous nature of the solids, a new model, the shrinking voids model, was developed to achieve the following objectives: 1. Calculate material balances rapidly from alumina miniplant data. 2. Predict material balances for belt filtration configurations different from those tested. 3. Predict filtration balances for slurries containing porous solids, when the existing model is not expected to be usable. 'Chemist. 'Metallurgist. Both authors are with the Boulder City Engineering Laboratory, Bureau of Mines, Boulder City, Nev. BELT FILTRATION MODELING BACKGROUND During investigations in the Bureau's alumina miniplant project for extracting alumina from clay by a hydrochloric acid leaching technology, separation of the undissolved, siliceous residue from the AICl3-bearing solution was found to be a major problem. The filtration was slow and the washing efficiencies were poor. Two systems for solids-liquid separation, a clast-ifier-thickener system and a belt filtration system, were investigated. The belt filtra- tion system has proven to be a satisfactory approach to the solids-liquid separation problem. In an effort to calculate accurate material balances quickly around the belt filtration system and to predict results from various belt filtration configurations, it was necessary to devise a new modeling technique. Filtration and Washing Process Description The method of solids-liquid separation covered in this report is horizontal vacuum belt filtration. The alumina miniplant filter consists of a moving belt onto which slurry (with or without the addition of flocculant solution) is deposited under the impetus of a vacuum. This deposition process separates the alumina-bearing liquor from the cake that the belt carries through one or more sprays (washes) for recovery of additional alumina values. The filtrate at each stage is then transferred countercur- rent to the direction of cake movement for use as a wash spray (see fig. 1). Countercurrent filtration and washing is generally considered the most efficient method for max- imum product recovery for a given amount of wash water added in the final cake wash. Definition of Washing Efficiency Ideally, after the cake is washed in the filtration pro- cess, the wash liquor will have displaced all of the more concentrated solute in the liquor with wash liquor to give perfect washing. However, since perfect washing is sel- dom achieved, a measure of washing efficiency is needed to evaluate the process. The R value (equal to 100 minus the efficiency) measures the percent of solute remaining in the cake after washing, and subtracts any solute added in the wash. 3 R = c 2- C v Ci - C u (100), (1) where C2 = solute concentration (pounds per gallon) in washed cake liquor, C1 = solute concentration (pounds per gallon) in feed cake liquor, and C w = solute concentration (pounds per gallon) in wash liquor. This R value will, since it is actually a function of the wash ratio, N, of wash liquor volume to cake liquor volume, change as N changes. This report will, however, refer to the residuals, R, only in passing because the ultimate goal is to determine solute losses through the final wash cake. Prior Material Balance Calculations Heretofore material balances for horizontal belt filtra- tion in the clay-HCI miniplant were made by laborious 3 Dahlstrom, D. A., and Silverblatt, C. E. Continuous Vacuum and Pressure Filtration. Ch. in Solid/Liquid Separation Equipment Scaleup, ed. D. B. Purchas. Upland Press Ltd., Croydon, England, 1977, p. 477. 2nd wash cake Farm filtrate 2nd wash filtrate FIGURE 1.— Schematic of continuous horizontal belt filtration circuit. calculations. A solids-liquid material balance was made from plant input flow rates and cake moisture contents. An alumina balance of the liquor stream was then made by best-fitting or equal-weighting the AI2O3 analysis of the liquor in the filtration test samples. Using these methods, material balances were made for seven horizontal belt filtration tests, and the balances were later used for evaluating the models developed in this report. Prior Horizontal Belt Filter Modeling Previously, in order to predict material balances for cir- cuits employing various wash liquor rates or number of washes, the following two assumptions were required: 1 . R values for additional stages were equal to the average R value in the first two stages. 2 . Cake moisture decreased in a straight line with the number of wash stages. SELECTION OF A MODEL Purpose of Model Development Miniplant filtration data are most easily applied when summarized in a mathematical model. Such models must, of course, predict stream compositions as close as possi- ble to those observed. The models should also predict reliable material balances for continuous filtration system configurations different from those tested in the miniplant. Such models facilitate selection of the most cost-effective filter system configuration. For example, both alumina losses through the final wash cake and the degree of product liquor dilution by wash water for any number of wash stages or any volume of wash water addi- tion are predicted rather than measured. Ground Rules for Model Development The models in this study are all guided by theoreti- cally logical constraints on the relationships among filtra- tion flow streams. These constraints are applied in such a manner that the final material balance stream composi- tions differ minimally from measured compositions— us- ing a least squares error procedure. The following logical constraints common to all these models are referred to as ground rules: 1 . No solid or liquid losses occur. 2 . The filtrates contain no solids. 3 . Liquor volume input equals liquor volume output for each washing-filtration stage. 4 . The ratio of solute in the aqueous phase to that in the solid phase remains constant (presumably no solute exists in the solid phase for the study material). 5 . Total cake liquor volume is constant from one washing stage to the next. The Existing Perfect-Mixing-Cells-in-Series (PMCS) Model Description The PMCS model 4 uses the previously mentioned ground rules. It assumes that cake washing is described 'Torniak, A. Predict Performance of Belt-Filter Washing. Chem. Eng. v. 86, No. 9, Apr. 23, 1979, pp. 143-146. Tomiak, A. Theoretical Recoveries in Filter Cake Reslurrying and Washing. AlChE J., v. 19, No. 1, January 1973, pp. 76-84. by the complete equilibration of cake liquor with wash li- quor as the later moves through each of a number of perfect-mixing cells in the cake. The number of these cells is an intensive property, j, which does not change with cake size. Discussion This model specifies that j must be an integer greater than zero. Testing of this model with alumina miniplant data reveals that unless j values are extrapolated to some value less than 1, the model fails completely. It fails because the leached solids are very porous, indicating an imperfect rather than a perfect mixing (PMCS model) con- dition. The Diffusion Model Description Because the PMCS model inadequately describes filtra- tion of the alumina miniplant solids, it was decided to develop a model that recognized the presence of a large voids volume inside the leached residue particles, which has been determined to be 54.6 pet (see appendix C). The model assumes the cake liquor to be composed of the following two separate fractions: Internal— liquor entrained in the pores of the solids. External— liquor outside the pores and between the solid particles. During a wash, the external fraction is washed from the cake solids according to the PMCS model (with j cells) and the internal fraction is untouched by the wash liquor. After excess liquor is filtered from the cake, the fractions dif- fuse into one another for a specified time (see fig. 2). Discussion The diffusion model is a considerable improvement over the PMCS model. However, it predicts that alumina resid- uals, R, are constant from one wash stage to the next. In the miniplant, however, R values actually decrease signif- icantly with decreasing liquor concentrations. Attempts to eliminate this discrepancy lead to the development of a better model. The Shrinking Voids Model Description This model assumes that washing efficiency changes as cake liquor concentration changes. Presumably, liquor viscosity decreases (fluidity increases) as the liquor becomes more dilute, which causes the solid particle voids liquors to be more readily washable (displaceable). This increased washing efficiency is expressed as a de- crease in the volume of nondisplaceable voids as the liquor concentration decreases (see fig. 2). Discussion This model will be shown to be the best of the three treated in this report. However, it has not been verified us- ing any other configuration than that in the miniplant two- stage washing and filtering system. It might be objected that a more realistic model would incorporate AICI 3 diffusion during, as well as after, washing. It is, however, felt that diffusion during washing is small, which makes the simplicity of the shrinking voids model very attractive. Wash liquor (I) New feed cake (Shrinking voids model) (4) Unwoshobie liquor in voids Washable liquor film Wash filtrate (2) New feed cake [Diffusion model) (4) FIGURE 2.— Schematic for two mathematical models (diffusion and shrinking voids) of the displace washing of acid- leached calcined kaolin residues during continuous belt filtration. DERIVATION OF EQUATIONS FOR A SINGLE WASH-FILTRATION STAGE Fundamental PMCS Equations The basic PMCS (with j perfect-mixing cells) equation describing the material balance around a single wash step is * = (N^-f^) /(N-f), (2) Vi \ V 2 M ) where N = V-|/V , the wash ratio, f = fraction of salt removed during a wash with salt-free wash liquor, e-jN j-1 gN)k and A n , V n = alumina weights and liquor volumes for the liquor stream, n, in figure 3 (n = o for feed cake, n = 1 for wash liquor, n = 2 for wash filtrate, n = 3 for initial wash cake, and n = 4 for equilibrated wash cake). Subscripts e and i attached to cake liquor volumes and alumina weights refer to external and internal fractions respectively. Development of Diffusion and Shrinking Voids Model Equations Balance Around the Wash Step ' Both the diffusion and shrinking voids models begin with the application of the PMCS equation 2. However, for this application, cake liquor will consist only of the displaceable external fraction. Therefore, the cake liquor alumina weight (Aq = A 0I + A^) and liquor volume (V = Voi + v pe) are temporarily considered equal to Aoe and V oi respectively. Thus, equation 2 becomes ^1 = ^.f^Wf). (3) This means the wash ratio is now N = V^Noe- Also, since cake volumes do not (ground rule 5) change during washing, Vi equals V2. Therefore, the wash liquor alumina weight can be determined from A1 = , V n A 2 V 1 — I • — i 1 Voe V! Aoe /(N-f), (A 2 - f AoeWN-f), _Vl Voe N(A 2 - f Aoe)/(N-f). (4) Feed C3ke,next step (4) Wash liquor (I) . _( A o, + A 3e) , A 4'>oi + V oe) V4> A 4e rA oi +A 3e~ A 4i ,, -v _ K ( A oe" A 3e) v 4i " v oi K (V +v , ( v oi +v oe) V 4e =V j +v oe -v 4j At=j£r o -XL C JZ "O o > (Shrinking voids model) Feed cake(O) < ' Washed cake(3) A j = specified Aoe= specified V j = specified V oe = specified f Was hing \ d V A-, = A "3i oi A 3e ; V^l -A 2 V 3l = V o, v 3e = V oe »■ y filtr jtion / >x Diffusion on i Feed cake, next step (4) < ' -tc , -tc A 4i =A ,e +V 0l (l-e A 4e =A oi +A 3e- A 4i v 4, =v o, v 4e ^v oe /A ,4A 3e \ Woi +V e / Wash filtrate (2) Ag r estimated V 2 ~ specified (Diffusion model) FIGURE 3.— Equations used in calculating a material balance around a single wash step. Now the internal liquor parameters, Aqj and V j, tem- porarily set to zero, are again permitted nonzero values. Then the external cake alumina weight follows from material balance A 3e = A i + Aoe + A-| - A 2 - A 3 j, and since A 3i = A oi , gives A 3e = A^ + At - A 2 - Diffusion Model: Migration of Al 2 3 From Cake Voids At this point, the external cake liquor will be highly defi- cient in Al 2 3 . It is, therefore, proposed that a specified amount of diffusion will occur to change this situation. The occurrence of this process can be expressed by a first order decay curve illustrated in figure 4. The equation is CfC eq = (Cj - C eq ) e tc , (5) where tc, a positive time constant, is actually the product of a diffusion time interval, t, and a diffusion rate constant, c (which must be positive). Cj, Cf, and C eq are the initial, final, and equilibrium alumina concentrations in the liquor. The equilibrium concentration is, of course, defined by alumina weight A^ + A^ C = eq liquor volume (6) where the subscript 3 refers to the initial product cake liquor. Then for the subscript 4, which refers to the final product cake liquor, the relationships Cj = A 3 j/V 3j , Cf = A4j/V 4 j, and C eq of equation 6 can be substituted into equa- tion 5 so that A 4i may be determined. This gives A4I A 3i + A^ V 4i V 3i + Vae A 3 j A 3i + A;je V 3j V 3i + Vae e tc .(7) Now, since cake volume does not change, the relation- ships V 4i = V 3i = VojandV^ = V^ = V^ must hold. Also, Aqj = A 3 j must hold, since no alumina has been removed from the particle interior during washing and before diffu- sion occurs. Thus, on rearrangement of equation 7, the final internal cake liquor alumina weight must be A 4i = A oi e-tc + V ,(1 - e-tcj Aoi + A 3e V oi + V oe (8) This diffusion model is not the best one for giving good predictions. The diffusion model does not, for example, predict the washing efficiency increases (that is, R decreases) observed in the minipiant on passing from one wash stage to a succeeding one. Such increases are con- trary to what is typically reported in the literature, where washing efficiency decreases are considered common. 5 'Dahlstrom, D. A. Predicting Performance of Continuous Filters. Chem. Eng. Prog., v. 74, No. 4, April 1978, pp. 69-74. < cr LlI o o o Cf 'eq Initial concentration TIME (t) FIGURE 4.— Cake liquor solute concentration, C f , as a function of diffusion time. Furthermore, the use of equilibrium equations is rendered inconsequential on discovering that application of the model to miniplant data always requires that the positive time constant approach infinity. Shrinking Voids Model: Hypothetical Voids Shrinkage This model introduces the concept of an unwashable voids fraction that systematically changes (decreases) as the cake passes from one wash to the next. The model thus proposes a voids volume decrease that is directly pro- portional to the AI2O3 concentration decrease, AC (concen- tration decreases being a crude measure of viscosity decreases). Specifically, the new voids volume, V 4i , is ex- pressed in terms of the old voids volume, V oi , and a voids shrinkage constant, k V 4i =V oi -kAC. (9) Using the notation of figure 3, AC can be expressed as AA \ ( A oe + Aoi) - (A3e + A 3i) AC = V oi + v c (10) Since A oi = A 3i , this gives Aoe - A 3e AC = V i + V oe /Aoe - A 3e giving V 4i = V oi -k - — — \ v oi " v oe (11) (12) Since alumina concentration is uniform (at equilibrium) throughout the cake, the new voids alumina weight must be the voids volume fraction, V 4i /(V oi + Voe), of tne tota ' cake alumina, A 3 j + A3 e , A 4 i = V 4 i (A 3 i + A^) Voi + Vo« (13) The quantities V^ and A^ are then determined from material and volume balance as in figure 3. Completion of Countercurrent Calculations The essential relationships, summarized in figure 3, are applied to countercurrent washing beginning with a form cake of known composition, an estimated filtrate alumina weight, A 2 , and a filtrate volume equal to the wash water volume used in the system (refer to fig. 1). These calcula- tions are repeated for each successive wash and filtration stage. Then, at this point, it is generally found that the alumina weight, A f , in the final wash liquor does not equal the wash water weight originally specified (Af is usually zero). Therefore, the first filtrate alumina concentration, A 2 , must be reestimated and the whole countercurrent system balance calculation done again. After this second set of calculations is complete, the correct A 2 value may be closely estimated by linear interpolation A 2 = A 2 „ (A f " • A f ) (A 2 " -A 2 Q (A f " - Af) (14) where the singly primed constants refer to wash water and first wash filtrate aluminas from the first material balance trial and the doubly primed numbers to those from the second trial. This interpolation always gives the correct A 2 value for the diffusion model and generally comes close for the shrinking voids model. In any case, the TI-59 6 calculator and Fortran programs used in doing these calculations do not assume that the equation yields exact answers, but rather use it to produce rapid convergence to the correct A 2 value. The foregoing calculations have presumed that the parameters, Vj, j, and tc (diffusion model) or V| and k (shrinking voids model) are known. If these are not known, the values must be varied in the search for another material balance that will cause the balance values to more closely approximate the correct values (measured during a filtration run). The closeness of fit will be opti- mized using a least squares method to be described at the end of the "Sample Calculations" section. Equations for Liquor Weight and Volume Balance Once the optimal set of parameters is established, liquor volumes and weight balances can be made. For- tunately, a simple method exists for converting alumina weight and liquor volume to liquor weight. For pure aqueous AICI 3 liquor, density is reliably calculated from lb \ / A lb Al 2 3 \ p — = 8.34 + 2.10 gal / \V gal liquor/ (15) With impurities present, the constant 2.10 may be in- creased. With density known, the liquor weight then must be (W lb) = (V gal liquor) L — V gal = 8.34 (V gal liquor + 2.10 (A lb Al 2 3 ). (16) SUMMARY OF MATERIAL BALANCE PROCEDURE To obtain the optimum material balance for the two- wash belt filter shown in figure 1, it is necessary to measure the concentrations and volumes of specified filtration streams. This information is presented in table 1 for miniplant test 1-3. The code 1 in the table specifies that the first wash filtrate is used to dilute the form filtration feed slurry. A code would indicate filtrate was not re- cycled. For the 123 pounds of dry leach solid passing through the filter each hour, the cake liquor volume can be analytically determined using the methods of appendix D. The parameters V, and k are merely trial values to use in the first set of shrinking voids model calculations. TABLE 1.— Miniplant belt filtration data, test 1-3 Number of washes AUOa, weight-percent: 1st wash filtrate Form cake liquor 2d wash filtrate 1st wash cake liquor Wash water 2d wash cake liquor Code Reactor discharge: Al 2^3 pounds Liquor 1 gallons Wash water do Cake liquor, Vj do Internal cake liquor, Vj do square gallons per pound 2.78 8.31 1.29 6.84 4.68 1 80.75 74.40 28.52 12.76 eg- e 5 •Reference to specific equipment does not imply endorsement by the Bureau of Mines. Voids shrinkage constant, k e Estimated. 'Includes 2 gal of flocculant water. The balance procedure assumes conservation of vol- ume as well as material. Therefore, all stream volumes (fig. 5) are readily specified from the values in table 1, from the assumption that all wash filtrate volumes are the same as those for wash water, and from the assumption that all cake volumes, V t = V| + V e , are the same. The calculation begins with the determination of the form slurry alumina weight: Aform slurry = A reaC f 0r discharge + (Aist wash filtrate) (Code). The alumina weight for the two form filtration liquor fractions is proportional to volume, Vf orm fjitrate "form filtrate = tj x Af rm slurry anc ' v form slurry Aform cake = Af orm slurry — Af orm filtrate- Form cake alumina fractions (internal and external) are then Aj form cake — rr Af orm ca |< e and v t Ae form cake = Af orm ca |< e — Aj f orm C ake- (1) Reactor dischorge (5)First woeh filtrate 7) Second wash filtrate (9) Wash water A = As As As W = W: W = Ws V: v= V= Vs ^^^^^ Optional ^\. ■ 2) Flocculent (3) Form slurry \j A= A = W= W = /ist A ( 2nd "\ I wash ) \wQ9k J V = V = \ Form ^* _^. — — filtr. ^^ ^ — — • ' \^^ = VOLCL IS UNREASONABLE, CAUSING CALCULATION PROBLEMS 2300 IF( VI .GE . VOLCL) VI= .99* VOLCL 2310 AWTC5)=8 2320 AWT5C=9 2330 WWC=ANWT C IW-1 ) +1 2340 AWT5E=AWT5C+1 2350 WWE=WWC+1 2360 LIM=0 2370 C-##### DOES 1 MATERIAL BALANCE FOR ASSUMED 1ST FILTRATE COMPOSITION 2380 799 LIM=LIM+1 2390 VIC=VI 2400 AWTC3)=AWTC1)+AWT(2)+AWTC5)*C0DE 2410 AWT(6)=VOLLIQC6)*AWTC3)/VOLLIQC3) 2420 AIWTC6)=VI*AWTC6)/VOLLIQC6) 2430 AEWTC6)=AWT(6)-AIWTC6) 2440 DO 900 K=7,IW,2 2450 WASHRA=VOLWW/(VOLCL-VIC) 2460 F=1-EXP(-WASHRA) 2470 AWTCK)=WASHRA*(AWTCK-2)-F*AEWTCK-l) )/CWASHRA-F) 2480 TEMP=AEWTCK-l)+AWTCK)-AWT(K-2) 2490 VIC=VIC-EKK* CAEWT(K-l)-TEMP)/VOLLIQCK-l) 2500 C-##### VIC >= VOLCL IS UNREASONABLE, CAUSING CALCULATION PROBLEMS 2510 IF( VIC .GE. VOLCL) GOTO 999 2520 AIWT(K+l)=VIC*(AIWTCK-l)+TEMP)/VOLLIQCK-l) 17 2530 AEWT(K44)=AIWT(K-1)+TEMP-AIWT(K+1) 2540 AWT(K+1)=AIWT(K+1)+AEWT(K+1) 2550 900 CONTINUE 2560 AWT5E=AWT5C 2570 AWT5C=AWT(5) 2580 WWE=WWC 2590 WWC=AWT(IW-1) 2600 ERR= ABS ( ANWT ( IW-1 ) -AWT ( IW-1 ) ) 2610 IF( ERR. LT. 0.0005) GOTO 950 2620 IF(LIM.GT.20) GOTO 999 2630 C-##### WASH WATER AL203 DIFFERS FROM TRUE AL203 BY MORE THAN 0.0005 2640 C-##### LBS. SO, CHOOSE NEW 1ST FILTRATE ESTIMATE 2650 AWT ( 5 ) =AWT5C-( WWC-ANWT ( IW-1 ) ) * ( AWT5C-AWT5E ) / ( WWC-WWE ) 2660 GOTO 799 2670 950 AWT(4)=AWT(3)-AWT(6) 2680 IT=IT+1 2690 IF(MODE.EQ.l) GOTO 963 2700 C-##### CALCULATES SUMS OF SQUARES OF ERRORS ON PCT. BASIS 2710 SSE=0 2720 NUM=0 2730 DO 960 11=5, IW 2740 IF(ANWT(II).EQ.0.0)GOTO 960 2750 SSE=SSE+((ANWT(II)-AWT(II))/ANWT(II))**2 2760 NUM=NUM+1 2770 960 CONTINUE 2780 ASE=SSE/NUM 2790 IF(ASE.GT.BASE) GOTO 962 2800 C-##### A LOWER SSE HAS BEEN FOUND 2810 BASE=ASE 2820 BVI=VI 2830 BKK=EKK 2840 962 IF(MODE.NE.l) GOTO 999 2850 C-##### PRINT THE FINAL RESULTS 2860 963 WRITE(6,964) 2870 964 FORMAT(/15X,26HTHE FINAL MATERIAL BALANCE) 2880 WRITE(6,965) VI,EKK 2890 965 FORMAT(/27H PARTICLE INTERNAL VOLUME »,F8.3, 2900 - 30H GALLONS, SHRINKAGE CONSTANT =,F8.3) 2910 WRITE(6,970) 2920 970 FORMAT (45H NO. LBS. AL203 LBS. LIQUOR GAL. LIQUOR) 2930 DO 980 M=1,IW 2940 C-=— == CALCULATES WEIGHT OF AQUEOUS ALCL3 SOLUTIONS 2950 WTLIQ(M)=8.34*VOLLIQ(M)+2.10*AWT(M) 2960 WRITE(6,985)M,AWT(M) ,WTLIQ(M) ,VOLLIQ(M) 2970 980 CONTINUE 2980 985 FORMAT(I4,F12.3,2F12.2) 2990 WRITE(6,990)ASE,SSE 3000 990 FORMAT(20H AVG. SQ. OF ERROR = ,F9.6,7H SSE =,F9.5) 3010 WRITE(6,995) IT 3020 995 FORMAT (19H NO. OF BALANCES = ,14) 3030 999 EKK=EKK+DKK 3040 1000 CONTINUE 3050 VI=VI+DVI 18 3060 1100 CONTINUE 3070 IF(MODE.EQ.l) GOTO 2000 # 3080 Q-mmn CHOOSES area OF VI - CONST matrix for next search 3090 DVI=DVI/2 3100 VI=BVI-(L-l)*DVI/2 3110 DKK=DKK/2 3120 EK=BKK-(L-l)*DKK/2 3130 IF(DVI.GT.O.Ol.OR.DKK.GT.O.l) GOTO 80 3140 C-##### BEST-FIT HAS BEEN POUND, PREPARE TO PRINT IT 3150 MODE=l 3160 L=l 3170 VI=BVI 3180 EK=BKK 3190 GOTO 80 3195 C-##### PREDICTOR MODE INITIALIZATION 3200 1200 READ(5,1700) VI, EK 3210 1700 FORMAT(2F8.3) 3220 WRITE(6,1800) VI, EK 3230 1800 FORMAT(23H INTERNAL CAKE VOLUME =,F8.3,17H GALLONS, CONST.=,F8.3) 3240 L=l 3250 JW=1 3260 AWT(IW-1)=0 3270 GOTO 20 3280 2000 STOP 3290 END Run in Material Balance Mode (Test 1-3) LIST BOLDAT #FILE (BCME)BOLDAT ON MRC 100 2 10.483 200 2.78 300 8.31 400 1.29 500 6.84 600 700 4.68 800 72.4 2. # RUN BELFIL ^RUNNING 0588 28.52 12.76 1. MODE 2 WASHES, STREAM STREAM STREAM STREAM STREAM 5 6 7 8 9 STREAM 10 REACTOR DISCH.AL203 CONC.= 2.780 PCT. AL203 8.310 PCT. AL203 1.290 PCT. AL203 6.840 PCT. AL203 0.000 PCT. AL203 4.680 PCT. AL203 10.483 VOLUMES IN GALLONS REACTOR DISCHARGE 72.40 FLOCCULANT 2.00 WASH WATER 28.52 CAKE LIQUOR CODE 12.76 1.00 19 The Final Material Balance PARTICLE INTERNAL VOLUME = 9.141 GALLONS, SHRINKAGE CONSTANT = 7.065 NO. LBS. AL203 LBS. LIQUOR GAL. LIQUOR 1 80.747 773.39 72.40 2 0.000 16.68 2.00 3 86.595 1040.20 102.92 4 75.859 911.24 90.16 5 5.848 250.14 28.52 6 10.736 128.96 12.76 7 3.212 244.60 28.52 8 8.100 123.43 12.76 9 0.000 237.86 28.52 10 4.888 116.68 12.76 AVG. SQ. OF ERROR = 0.009120 SSE = 0.04560 NO. OF BALANCES = 289 #ET=43.0 PT=1.5 10=0.3 Run in Predictor Mode (Four Washes) LIST BOLDAT #FILE (BCME) BOLDAT ON MRC 100 1 4 10.254 200 9.9 8.8 300 71.52 5. 20. 13.27 1. # RUN BELFIL ^RUNNING 0679 MODE 1 4 WASHES, REACTOR DISCH.AL203 CONC.= 10.254 INTERNAL CAKE VOLUME = 9.900 GALLONS, CONST .= 8.800 VOLUMES IN GALLONS REACTOR DISCHARGE FLCCCULANT WASH WATER CAKE LIQUOR CODE 71.52 5.00 20.00 13.27 1.00 20 The Final Material Balance PARTICLE INTERNAL VOLUME = 9.900 GALLONS, SHRINKAGE CONSTANT = 8.800 NO. LBS. AL203 LBS. LIQUOR GAL. LIQUOR 1 77.619 759.48 71.52 2 0.000 41.70 5.00 3 87.115 987.92 96.52 4 75.138 852.09 83.25 5 9.496 186.74 20.00 6 11.977 135.82 13.27 7 7.768 183.11 20.00 8 10.249 132.19 13.27 9 5.563 178.48 20.00 10 8.044 127.56 13.27 11 2.904 172.90 20.00 12 5.385 121.98 13.27 13 -0.000 166.80 20.00 14 2.481 115.88 13.27 VG. SQ. OF ERROR = 0.000000 SSE = 0.00000 0. OF BALANCES = 1 #ET=45.9 PT=0.5 IO=0.4 21 APPENDIX B.— MATERIAL BALANCE USING A PROGRAMMABLE CALCULATOR The balance form in figure 5 is used for this procedure. 1. Input composition is specified (A, W, V) for a. Heat exchanger (reactor) discharge b. Flocculant c. Wash water 2. All wash filtrate volumes are set equal to the wash water volume. 3. An average cake liquor volume is determined from the average value obtained in the miniplant run. See appendix C for methods used to determine this value. Set all cake volumes equal to this value V t . 4. Make estimates (guesses) for values of a. Internal cake liquor volume (Vj = to V t ) b. Voids shrinkage constant (k = any finite value) 5. Make up the following data table specifying the values and order of data entry into the TI-59 program- mable calculator: a. Number of washes (1 to 12) n b. First wash filtrate, analytical wt-pct Al 2 3 a-i c. First feed cake (= form cake), analytical wt-pct Al 2 3 a 2 d. First wash liquor, analytical Wt-pct AI2O3 83 e. Succeeding feed cakes (if applicable) f. Succeeding wash liquors (if applicable) g. Final feed cake, analytical wt-pct Al 2 3 a 4 h. Final wash liquor (that is, wash water), analytical wt-pct AI2O3 a 5 i. Final product cake, analytical wt-pct AI2O3 a6 j. Circuit code (0 = first wash filtrate not recycled, 1 = recycled) c k. Reactor discharge, pounds of Al 2 3 .... A 2 I. Reactor discharge, gallons of liquor .... V 2 m. Wash water, gallons of liquor V f n. Total cake liquor, average gallons of liquor V t o. Internal cake liquor, gallons (estimated) Vj p. Voids shrinkage constant, square gallons per pound k 6. The TI-59 program summarized at the end of this ap- pendix is run as follows: a. Push RST button b. Push R/S, enter first data value from step 5 (data table). Repeat until entire table has been entered. c. After a few minutes, the analytical weight balances for streams of steps 5b through 5i will be printed out in order. The final SSE value represents the sum of the percent least squares errors for the previous analytical balance values. 7. To find a least squares SSE, different values of Vj and k must be tried. To change these values, enter them in order after the last SSE is read and another material balance will be printed out. Continue to do this until SSE reaches a minimum value. To save printer paper, keep printer off until it has been determined that a minimum SSE has been reached. Then turn the printer back on and enter the appropriate Vj and k values to obtain the optimal Al 2 3 balance values. At this time, one may want to predict a balance for a filtration system having different values for different reactor discharge alumina (memory 31), reactor li- quor volumes (memory 30), wash water volumes (memory 29), cake liquor volumes (memory 28), wash water alumina weights (memory 2), and numbers of wash stages (memory 42). If so, enter these new values into the corresponding memories. If the first wash filtrates in memories 34 and 36 are equal, change the values of one of them. Then enter the number of washes into the register, push A, push A', and enter values for steps 5o and 5p into calculator and wait for the calculator to stop. Then push RCL 40 to obtain the final cake alumina losses. Figure B-1 provides a schematic of the program for material balance using the shrinking voids model. Program Memory Bank 1 A ca k e analysis (final) 2 A wash ii quor analysis (wash water) 3 A cake analysis (next to last) 4 A wasn |jq U or analysis (next to last) 5-24 Alternate A ca ke analysis ar| d A wasn |jq UOr analysis until first stage analysis is reached 25 Not assigned 26 Af j rS ( W ash liquor guess 27 Vjnternal cake liquor guess 28 Vjotal cake liquor 29 V wa sh water 30 V rea ctor discharge + flocculant 31 A reac tor discharge + flocculant 32 Circuit code 33 A r ', wash water (earlier calculation) 34 A2', first wash filtrate (earlier calculation) 35 Af ", wash water (current calculation) 36 A 2 ", first wash filtrate (current calculation) 37 A t otai cake liquor (previous wash) 38 A ex t e rnal feed cake 39 Asternal feed cake 40 A t otal cake liquor (present wash) 41 k, voids shrinkage constant 42 Number of washes 43 N, wash ratio 44 f, pure water wash recovery 45 Vj n t e rnal cake liquor 46 I(error)2 47 Current wash calculation number 48 Memory where current A stored 49 Location of highest numbered A 22 Belt-Filter Material Balance Modeling TI-59 Program 000 22 INV 050 72 ST* 100 36 36 150 65 X 200 43 RCL 001 58 FIX 051 00 00 101 42 STO 151 43 RCL 201 44 44 002 91 R/S 052 69 OP 102 26 26 152 32 32 202 95 = 003 11 A 053 30 30 103 25 CLR 153 95 = 203 42 STO 004 22 INV 054 71 SBR 104 42 STO 154 42 STO 204 35 35 005 86 STF 055 45 Y* 105 47 47 155 40 40 205 43 RCL 006 01 01 056 43 RCL 106 43 RCL 156 65 X 206 42 42 007 42 STO 057 28 28 107 45 45 157 43 RCL 207 32 X:T 008 00 00 058 95 = 108 42 STO 158 27 27 208 43 RCL 009 42 STO 059 72 ST* 109 27 27 159 55 4 209 47 47 010 36 36 060 00 00 110 94 +/- 160 43 RCL 210 67 EQ Oil 85 + 061 97 DSZ 111 85 + 161 28 28 211 13 C 012 01 1 062 00 00 112 43 RCL 162 95 = 212 87 IFF 013 95 = 063 61 GTO 113 28 28 163 42 STO 213 00 00 014 42 STO 064 76 LBL 114 95 = 164 39 39 214 38 SIN 015 34 34 065 16 A» 115 35 1/X 165 94 +/- 215 43 RCL 016 76 LBL 066 91 R/S 116 65 X 166 85 + 216 35 35 017 15 E 067 42 STO 117 43 RCL 167 43 RCL 217 19 D' 018 91 R/S 068 45 45 118 29 29 168 40 40 218 76 LBL 019 72 ST* 069 04 4 119 95 = 169 95 = 219 38 SIN 020 00 00 070 02 2 120 42 STO 170 42 STO 220 43 RCL 021 97 DSZ 071 02 2 121 43 43 171 38 38 221 35 35 022 00 00 072 04 4 122 94 +/- 172 43 RCL 222 48 EXC 023 15 E 073 69 OP 123 22 INV 173 40 40 223 26 26 024 91 R/S 074 04 04 124 23 LNX 174 87 IFF 224 94 +/- 025 42 STO 075 43 RCL 125 94 +/- 175 00 00 225 85 + 026 32 32 076 45 45 126 85 + 176 14 D 226 43 RCL 027 91 R/S 077 69 OP 127 01 1 177 19 D' 227 26 26 028 42 STO 078 06 06 128 95 = 178 76 LBL 228 85 + 029 31 31 079 02 2 129 42 STO 179 14 D 229 43 RCL 030 91 R/S 080 06 6 130 44 44 180 01 1 230 38 38 031 42 STO 081 69 OP 131 43 RCL 181 44 SUM 231 85 + 032 30 30 082 04 04 132 31 31 182 47 47 232 43 RCL 033 91 R/S 083 91 R/S 133 85 + 183 43 RCL 233 39 39 034 42 STO 084 42 STO 134 43 RCL 184 26 26 234 95 = 035 29 29 085 41 41 135 26 26 185 75 — 235 48 EXC 036 91 R/S 086 69 OP 136 65 X 186 43 RCL 236 40 40 037 42 STO 087 06 06 137 43 RCL 187 44 44 237 42 STO 038 28 28 088 86 STF 138 32 32 188 65 X 238 37 37 039 43 RCL 089 00 00 139 95 = 189 43 RCL 239 75 - 040 49 49 090 43 RCL 140 65 X 190 38 38 240 43 RCL 041 42 STO 091 49 49 141 43 RCL 191 95 = 241 40 40 042 00 00 092 42 STO 142 28 28 192 65 X 242 95 = 043 76 LBL 093 48 48 143 55 4 193 43 RCL 243 65 x 044 61 GTO 094 25 CLR 144 53 ( 194 43 43 244 43 RCL 045 71 SBR 095 42 STO 145 43 RCL 195 55 T 245 41 41 046 45 YX 096 46 46 146 30 30 196 53 ( 246 55 « 047 43 RCL 097 76 LBL 147 85 + 197 43 RCL 247 43 RCL 048 29 29 098 12 B 148 43 RCL 198 43 43 248 28 28 049 95 = 099 43 RCL 149 29 29 199 75 — 249 95 = 23 250 22 INV 304 67 BQ 358 48 EXC 412 46 46 466 65 x 251 44 SUM 305 14 D 359 36 36 413 01 1 467 02 2 252 27 27 306 03 3 360 48 EXC 414 22 INV 468 85 + 253 43 RCL 307 06 6 361 35 35 415 44 SUM 469 02 2 254 29 29 308 03 3 362 48 EXC 416 48 48 470 95 = 255 55 • 309 06 6 363 33 33 417 76 LBL 471 42 STO 256 53 ( 310 01 1 364 48 EXC 418 52 EE 472 49 49 257 43 RCL 311 07 7 365 35 35 419 92 RTN 473 92 RTN 258 28 28 312 69 OP 366 43 RCL 420 76 LBL 259 75 — 313 04 04 367 36 36 421 17 B' 260 43 RCL 314 43 RCL 368 75 - 422 43 RCL 261 27 27 315 46 46 369 53 ( 423 35 35 262 95 = 316 69 OP 370 43 RCL 424 99 PRT 263 42 STO 317 06 06 371 35 35 425 01 1 264 43 43 318 61 GTO 372 75 — 426 22 INV 265 94 +/- 319 16 A' 373 43 RCL 427 44 SUM 266 22 INV 320 76 LBL 374 02 02 428 48 48 267 23 LNX 321 13 C 375 54 ) 429 61 GTO 268 94 +/- 322 22 INV 376 55 4 430 38 SIN 269 85 + 323 87 IFF 377 53 ( 431 76 LBL 270 01 1 324 00 00 378 43 RCL 432 45 Y x 271 95 = 325 17 B' 379 35 35 433 01 1 272 42 STO 326 58 FIX 380 75 — 434 85 + 273 44 44 327 03 03 381 43 RCL 435 93 • 274 43 RCL 328 43 RCL 382 33 33 436 00 275 40 40 329 02 02 383 54 ) 437 02 2 276 65 X 330 52 EE 384 65 X 438 00 277 43 RCL 331 22 INV 385 53 ( 439 07 7 278 27 27 332 52 EE 386 43 RCL 440 09 9 279 55 4 333 32 X:T 387 36 36 441 65 X 280 43 RCL 334 43 RCL 388 75 — 442 73 RC* 281 28 28 335 35 35 389 43 RCL 443 00 00 282 95 = 336 52 EE 390 34 34 444 45 yx 283 42 STO 337 22 INV 391 95 = 445 01 1 284 39 39 338 52 EE 392 42 STO 446 93 • 285 94 +/- 339 22 INV 393 36 36 447 01 1 286 85 + 340 58 FIX 394 61 GTO 448 95 = 287 43 RCL 341 22 INV 395 12 B 449 65 X 288 40 40 342 67 EQ 396 76 LBL 450 73 RC* 289 95 = 343 47 CMS 397 19 D' 451 00 00 290 42 STO 344 22 INV 398 87 IFF 452 65 X 291 38 38 345 86 STF 399 01 01 453 93 • 292 87 IFF 346 00 00 400 52 EE 454 00 293 00 00 347 43 RCL 401 99 PRT 455 08 8 294 14 D 348 36 36 402 75 — 456 03 3 295 43 RCL 349 19 D' 403 73 RC* 457 04 4 296 40 40 350 61 GTO 404 48 48 458 65 X 297 19 D* 351 12 B 405 95 = 459 92 RTN 298 43 RCL 352 76 LBL 406 55 • 460 76 LBL 299 42 42 353 47 CMS 407 73 RC* 461 11 A 300 32 X:T 354 48 EXC 408 48 48 462 86 STF 301 43 RCL 355 36 36 409 95 = 463 01 01 302 47 47 356 48 EXC 410 33 X2 464 42 STO 303 22 INV 357 34 34 411 44 SUM 465 42 42 24 StepeO-15 Reads No. of IncHollres mottriol balance mode X LBLE Step* 16-23 Reads Al203conc. T Steps 24-42 Reods Circuit code Input slurry Al2 O3 wt. Input slurry volume Wosri woter volume Cake total liquid volume _£ LBL GTO Steps 43-63 Coverts AlgQ, Cone to wt. LBL A Steps 64-96 Reads Coke internal liq vol Voids shrink const. K I LBL B Steps 97-176 Calculates f ,n,ond form cake AI2O3 Step 177 Print s Form cake AI2O3 J i 1 L LBL D Steps 178-211 Calculates Wash hq Al 2 3 Steps 212-214 | , ' Checks balance flog Steps 215-217 Prints Wash liq Al 2 3 LBL sin Stepe 218-294 Calculates f,n, and cake liq AI2O3 Steps 295-305 Prints Washed cake Al 2 3 Steps 306-319 Prints SSE 1 LBL C Steps 320-325 Checks balance flog Steps 326-343 Checks for system material balance LBL D' Steps 396-400 Subroutine Checks predictor flag 31 | 1 1 Steps 401-416 Prints : Al 2 3 wt 1 a Sums (error)2 ' LBL EE Steps 417-419 Return LBL B ' Steps 420-430 Prints : Wash water Al 2 3 LBL Y » Steps 431-459 Subroutine Converts AI2 O3 Cone, to wt. LBL A Steps 460-473 Subroutine Initializes -No. of washes -Predictor mode balance flog set Steps 344-351 Sets balance flag LBL CMS Steps 352-395 Choose new 1st wash filtrate AI2O3 FIGURE B-1.— Schwmtfc of TI-59 program for material balance using the shrinking voids model. 25 APPENDIX C.-MEASUREMENT OF LEACH RESIDUE POROSITIES AND DENSITIES The porosity of calcined kaolin clay that has been leached in a stoichiometrically 5-pct-excess aqueous 26 wt-pct HCI was determined as follows: A sample of the leached solids (still saturated in the leach liquor) was first dried with a cloth dish towel to remove external moisture. Then a weighed amount, W s , of these solids was dropped into a tared 25-ml pycnometer which was then filled to the mark with leach liquor of known density, p|j qu0 r- Then using the weight, Wg, of leach liquor added, the volume, V s , of the original towel-dried solids was found to be V s = 25 Pliquor milliliters. (C-1) The solids were then thoroughly washed with distilled water and dried in an oven at 125° C to obtain a dry solids weight, W ds . From this weight, the weight, Wj, and volume V|, of internal liquor (inside the particle pores) were found to be Wj = W s - Wds grams and Wj V, = milliliters. (C-2) (C-3) Pliquor This information is then used to find the percent poros- ity, P, by volume P = 100 (C-4) V, Table C-1 summarizes these porosities along with the wet particle densities W s p s = — grams per milliliter, V s (C-5) and an absolute solids density W ds Pds = — grams per milliliter. (C-6) V ds TABLE C-1.— Summary of 1 -day-old leach residue densities and porosities (20° C) Wet particle density' g/cm 3 Dry particle absolute density, g/cm 3 Dry particle porosity, volume percent Feed size, mesh: Minus 10 plus 14 Minus 14 plus 20 Minus 20 plus 28 1.687 1.619 1.728 2.212 2.146 2.181 56.24 53.40 54.14 Average 1.68 2.18 54.6 Vliquor = 1-2776. 26 APPENDIX D.— CAKE LIQUOR DENSITY AND VOLUME DETERMINATIONS The original wet cake is weighed, W ws , and washed with a weight of wash water, W ww . The repulp liquor obtained has a density, p R , and weight-percent Al 2 3 , P R , as does the original liquor in the cake, density, p^, and weight- percent, P . The dry washed cake has a weight, W ds . The relationship between density and weight-percent Al 2 3 is p = 1 + 0.02079 (P ) 11 , p R = 1 + 0.02079 (Pr)1-1. The weight of the original liquor is w L = w ds + w ws . The weight of alumina in the original liquor is Pr W A = — (W L + W^). 100 The percent alumina in the original liquor is W A P = 100 (D-1) (D-2) (D-3) (D-4) (D-5) Wi To obtain the original liquor density, P in equation D-5 is substituted into equation D-1 p = 1 + 0.02079 100 W A V-1 W (D-6) Substitution of the W L and W A values of equations D-3 and D-4 gives / /W ws - W ds + W WW \\1-1 P = 1 + 0.02079 P R ] . (D-7) \ \ W ws - W ds // Solving equation D-2 for P R gives 0.02079 (D-8) Finally, substitution of this value into equation D-7 gives /pr - A 1/1:1 p = 1 + 0.02079 \0.02079j f W ws - W ds + W v 1.1 - w ds w ws - w ds + WwwV' 1 (D-9) W ws - W ds With this density, the liquor volume per weight of cake solids must then be V, = (W ws - W ds )/ Po . (D-10) 27 APPENDIX E.-NOMENCLATURE A = Pounds Al 2 3 in liquor stream (usually subscripted). A f = Pounds Al 2 3 in wash water stream (value usually zero). C eq = Equilibrium AI2O3 concentration in liquor, pounds per gallon. Cf = Final AI2O3 concentration in liquor, pounds per gallon. Cj = Initial AI2O3 concentration in liquor, pounds per gallon. C w = Solute concentration in wash liquor, pounds per gallon. C-| = Solute concentration in feed cake liquor, pounds per gallon. C 2 = Solute concentration in washed cake liquor, pounds per gallon. e = Natural logarithm base, 2.71828. . . f = Fraction of cake AI2O3 removed during wash with salt-free wash liquor. j = Number of perfect mixing cells. k = Voids shrinkage constant, square gallons per pound. N = The wash ratio, Vi/V . R = Residual, a theoretical measure of solute remaining in cake after washing. tc = Time constant in diffusion model. V = Gallons of liquor (usually subscripted). W = Pounds of liquor in stream. Greek letters A = Change in the quantity that follows, p = Liquor density, pounds per gallon. Subscripts used with A and V e = External liquor fraction of cake particle, i = Internal liquor fraction of cake particle, t = Total liquor fraction of cake particle. = Feed cake. 1 = Wash liquor. 2 = Wash filtrate. 3 = Washed cake. 4 = New feed cake. •U.S. GOVERNMENT PRINTING OFFICE l 1982 0-367-'l 1 t8/626 wmtssaaasssssm^mm mmxssmv"' "-'•- -*sssg*&msB£ ■ H 17^ 82 O > ■ l > ■" . ■ » ■ v ^„ : **"** :wm?; &^ '.®b§t.' A * ** wW- ^ ^ '.WW/ ** ** °.v/mw ^>* '* .^L'. % ^ , %Zt«*P* ^ *•» 'WTO >r. * o ***«■ v^^'y ^-^v v^/ v^v v^^/ %' '<»^ , 4l^ * 3> ♦ ' • 4? V « » • • -V /.-i-^i-\ c°*.^ait*«- /.vi^.-% * '&k*°-> A /.--^. V 5^a ^o< o V sP^ V^\/ V^%°° V^*>' °v^V V : -'\^' *** O * ^ v W ^ i&' ^ „ >S' ~^ -ww ** *^ -ask* «** ^ -ys^> ** ^ • * J? > V ^6* ^^ ^^ * .Nflfetf. V ^ .V >°> J* *bV' **o« :flr^': "ov^ :^lla-. *^o^ :jfl^*: *b^ :^^»'-. 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