(ELLE 21: smmw M i.§QLl"‘- EEG-U72 CHROMAT OGRAPHiC CQLUMNS “r ' g . ,. a..." .- 1595: .c». ‘3 351;, as». i: . j' “‘53“; {r‘ Ara: flfii“; 3‘, I, xfim -s .1 [mini-.iu‘wii‘il‘ § .53. . l “4%) ‘ $.11}: IE 6“ ’ «p 5- '3 Feéia:§§-."z k:- “3- 5 "“3“ fllJIHHIIHI‘llHJHllthllU*IIHWIHJNIWIIW “BEAM ‘- 293 10571 2248 Michigan§mw Umvcrszty This is to certifg that the thesis entitled SOLUTE DISPERSION IN LIQUID-SOLID CHROMATOGRAPHIC COLUMNS presented by Martin C. Hawley has been accepted towards fulfillment of the requirements for .Ph.D_._ degree in WMineering flfl/Z Major professor/ Date June 12, 1961+ 0-169 —-——_ -—‘II‘ Rum“ Wm? ABSTRACT SOLUTE DISPERSION IN LIQUID-SOLID CHROMATOGRAPHIC COLUMNS by Martin C. Hawley Improvement of the scale-up procedure for liquid-solid chromatographic columns was the overall objective of this research. It was assumed that an axial dispersed, plug flow model with adsorp- tion occurring would describe this unsteady state process. Theoretically, it was shown that the spread of a pulse of solute, as it passed between two measuring points, was the sum of the spreads of the adsorption and the dispersion process. It was thought that the unreliability of the scale-up procedure was due to the liquid velocity effects and not dependent upon the adsorption process. Therefore, an eXperimental investigation of the Operating variables in the absence of the adsorption was undertaken. The effects of the variables (column diameter and length, particle diameter, and pulse viscosity) were evaluated by their effect on the axial dispersion coefficient. The pulse input method was utilized for the calculation of axial dispersion coefficients. These coefficients were evaluated from concentration versus time data of a tracer at two positions in a bed of glass spheres. The solute was a dilute aqueous sodium chloride solution and the concentration-time data were obtained at two positions in the bed by Martin C. Hawley electrolytic conductance. The column diameter had no effect on the axial dispersion coefficient for interstitial velocities between 0.1 and 20 ft/hr and in this same velocity range. a high pulse viscosity did not affect the dispersion coefficient. It was also shown that the axial diapersion coefficient is proportional to the particle diameter, 0.00M? and 0.0071 inch diameter glass spheres. ud The average Peclet Number (Pe = -529 was 0.22 for modified Reynolds Numbers between 10'3 and 10'1. SOLUTE DISPERSION IN LIQUID-SOLID CHROMATOGRAPHIC COLUMNS By Martin C. Hawley A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemical Engineering 196% To Donna and Valerie ACKNOWLEDGMENT The author gratefully acknowledges the Upjohn Company and Consumers Power Company for financial support of this work. Appreciation is extended to Dr. R. A. Zeleny and Dr. D. K. Anderson for their guidance during the course of the project. Appreciation is also extended to Mr. W. B. Clippinger for his help in the construction of the apparatus. The understanding and patience of the author's wife, Donna, is sincerely appreciated. ii TABLE OF CONTENTS ABSTRACT ACHOJWMEN‘I‘ TABLE OF comma LIST OF FIGURES LIST OF TABLES MODUCTIOH STATEMENT (I? THE PROBLm Axial Dispersion With No Adsorption Adsorption in the Bed with No Axial Dispersion Infinitely Long Bed with Both Axial Dispersion and Adsorption AXIAL DISPERSION IN PACE BEDS Diffusion Model Solutions to the Diffusion Equation The Mixing Cell Model The Random-Walk Model Comparison of Models TEE mm PBCGRAM Viscous Fingering End Effects Objectives APPARATUS AID PROCEDURE The Flow System The Tracer iii Page 111 vi 15 l6 25 33 35 39 1&1 141 142 #2 1L2 iv Packing material Conductance Cells The MBasuring Circuit Packing Procedure Run Procedure Calculation of the Dispersion Coefficient from the Data RESULTS AND DISCUSSION Column Diameter Pulse Viscosity Particle Diameter Column Length Comparison'With Other Work CONCLUSIONS APPENDIX I - Reynold's Mathod of Treating Turbulence or Eddies APPENDIX II - Explanation of Pulse Skewneas APPENDIX III - The Solution to Equation (10h) NOMENCLATURE BIBLIOGRAPHY 1+5 1+5 1+9 51 51 53 56 57 57 61 61 66 67 69 72 82 8. 9. 10.» 12. 13. 1h. 15. 16. 17. 18. LIST OF FIGURES Diagram of the Adsorbing and Non-adsorbing Bed Increment of Bed Side-stepping Model of Dispersion Comparison of The Concentration Profiles of the Diffus ion and Random-Walk Models Schemtic of Flow Equipment Specific Conductance of Sodium Chloride at 25’ C. Specific Conductance of Sucrose with a Trace Amount of NaCl at. 25’ C. my... Conductivity son for One-half Inch Column Six-Inch Conductivity Cell Electrical Circuit for Conductance Measurements Schemtic of Rectifying Amplifying Circuit Conductance versus Time Plot of a Run D/u‘y versus luau/11v for Column Diameter Comparison D versus u for Pulse Viscosity Couparison D versus u for Particle Diameter Comparison n/n, versus hue/11v for Particle Diameter Comparison udP/D versus limp/p. for Comparison With Other Work Plot of (NM versus g 1; to Show Skewnees Page 37 1&3 1L6 1.7 AB 52 55 58 59 62 63 7o I. II. III- IV- VI- Data for beads; 3 Data for beads; 2 Data for heads; 2 Data for beads; 3 Data for beads; 2 Data for Trace of LIST OF TABLES l/2-inch column; .0071 inch glass cells equally spaced; 12 inches long l/2-inch column; .0071 inch glass cells 18 inches apart l/2-inch column; .ooh7 inch glass cells 18 inches apart 6-inch column; .0071 inch glass cells 6 and 12 inches apart 6-inch column; .0071 inch glass cells 12 inches apart 100 Centipoise Sugar Solution with a NaCl as Pulse vi Page 7h 75 76 77 78 79 INTRODUCTION At the present time the scaling up of liquid-solid chromato- graphic columns is unreliable. A chromatographic separation may be accomplished in a small laboratory column, but when the process is scaled to industrial size, it does not always work as predicted by the laboratory model. A liquid-solid chromatographic process is a separation technique which is effected in a packed bed. The separation is due to the difference in adsorptivities of the components being separated by the solid material. To visualize this unsteady state process, consider a bed packed with a solid which has an affinity for the components to be separated. For simplicity, the separation of two components will be considered. A solvent is continually flowing through the packed bed and at time zero the two components to be separated are introduced uniformly as a pulse across the top of the bed. As they move through the bed, these two components become separated from each other because they have different adsorptivities. The magnitude and the relative difference in the adsorptivities of the two components prescribes the length of bed which will effect the separation. Chromatography is used primarily as an analytic tool in the laboratory, but in the pharmaceutical industry it is sometimes used as an industrial separation technique. The process is a batch type Operation and normally low flow rates are used. It is not usually considered an economical means of separation, but the pharmaceutical industry, on the other hand, deals with valuable compounds, so when other more conventional separating techniques fail, sometimes chromato- graphy is used. unfortunately, the nature of the components common to the pharmaceutical industry (their adsorptivities, diffusivities, etc.) is usually unknown. Since the properties of the components are unknown, one has to try various combinations of solvents and packings before a suitable combination is found. After a combination of solvent and.packing material is found which will separate the components, the operating variables are analyzed in the laboratory. Therefore, the usual scale-up procedure is to first obtain the desired separation in a small laboratory sized column and then increase the size of the column according to the amount of material to be separated. In order to keep the time required for the separation short and the pressure drop low, it is usually desirable to have the column as short as possible. In the laboratory the length of column is optimized with respect to the pressure drop and the time required for the separation. Theoretically, it is Justifiable to increase the cross sectional area of the column in proportion to the amount of material to be separated with- out changing the length. In practice, this is the scale-up method along with increasing the length.a certain amount only for good measure. Since this method of scale-up does not always work, a fundamental approach to the scale-up prdblem will be given here, such that the scale- up procedure will be more reliable. Chromatography can be thought of as being composed of two separate processes superimposed on each other. They are: 1. Fluid flowing in a packed bed 2. .A sorption process. It is proposed here that the unreliability in the current scale-up procedure is due to the fluid flow part of the process. Therefore, there must be a factor or a combination of factors which change with the column diameter, such that the separation achieved in columns of different diameters is not the same. Theoretically, this investigation will show how the two processes, fluid flow and adsorption, can be separated. An experimental investi- gation of some factors which may change the flow characteristics with column diameter is also reported. STATEMENT OF THE PROBLEM Consider a packed bed with two concentration measuring devices, one at X0 the other at Xm (Figure l). - m ‘—- x'O Tm X p + (I) 1 flow adsorbing nonuadsorbing —-> section section b Db Figure 1. Diagram of the Adsorbing and Non-adsorbing Bed. A shot of tracer is injected upstream from X0 into the bed in which a solvent is continually flowing with volumetric flow rate W. The surface of the packing has an affinity for the solute, hence unsteady state adsorption takes place inside the bed. The bed is constructed with an adsorbing section followed by a non-adsorbing section. The two sections are further distinguished by different effective axial dispersion coefficients, D and Db respectively. Three assumptions made for this process are: l. The tracer is injected uniformly across the cross section of the bed. 2. The flow is assumed uniform over the cross section of the bed; as a result radial diffusion can be neglected. 3. The adsorption isotherm is assumed to be linear. This process may be characterized by two mass transport processes, convection and axial dispersion. The concentration CA is the average value for species A across a section of this column and the amount of transfer by dispersion is assumed to be proportional to the axial concentration gradient :gééi. This is analogue to Fick's law for molecular diffusion. It is shown in Appendix I that the axial dispersion coefficient D is the sum of the molecular diffusivity and an eddy diffusivity. A material balance of component A is made on an incremental length of packed bed as illustrated in Figure 2. X xelax liquid WCA -—————- _————-—‘WCA KXaA AX(CA - CA1) ‘ -DAEBC '__‘A ------'~ 4 —————v--DA an. 3X 5—- solid Figure 2. Increment of Bed. The terms which describe the input and the output of material for each component are similar to those of component A. This tacitly assumes there are no interactions (i.e., component B does not affect the isotherm of A and so on). The equations which describe the process of this system are the equations of conservation of component A (Liquid Phase) W BCA - Kxa. (C - C ) .—. .1. BCA (1) “'De'K ‘57 “De A A1 D “‘at for X < X1 w acA _ 1 acA W '5?" - — ? f°r X 2 X1 (2’ b b b b (Solid Phase) aCA’5 (CA- CA1) = ‘5?- for x _<_ x1 (3) A8 = o for x _>_ x1 (4) CA. a no“ (5) assume that where a cross-sectional area of tube :- solids transfer area per volume a concentration of species A in the liquid - concentration of species A at the solid-liquid interface a concentration of A on the solid . axial dispersion coefficient a void fraction of bed =- mss transfer coefficient :- equilibrium constant defined by equation (5) =time 2 ll volumetric flow rate >4 u axial distance variable. Equations (1) through (5) are solved by using the initial condition that CA =:O when t = 0 for X 2 X0’ along with the boundary conditions _ _ > at X — XO , CA - CA0“) for t __ O . (6) - _ + DEA BCA + Db‘bAb aCA - - + and cA(x1) - W 8X (x1) = CA(X1>'—‘w—— Tag-(x1) (8) at X = 00 , CA(t, 00) = finite for t Z O (9) . A restriction on CA0(t) is that it must be a "hump" function of time. This represents a pulse input. Aria5 has given the mathematical restrictions for CAo(t). Boundary conditions (7) and (8) state that both the concentration and the flux are continuous at X a X1. The complete solution of equations (1) through (9) for CA’ CA3: and 9A1 in closed form, at the present time, is not possible, but by utilizing some formal mathematical properties of the Laplace transform it will be shown that it is not necessary to know the complete solution. For convenience CA, CA1, and CAS will be normalized such that _E_4_ <3 = m (10) of CA dt 0. (:1: Te}... (11) C c,s = on As (12) of CA dt where 00 Q and Q = the total amount of A in the pulse W volumetric flow rate Since c = w CA/Q, thus [0°C dt= l (14). 0 Thus, the subscript A may be dropped from equations (1) through (9) and this set of boundary value equations is solved using the Laplace transform method. Since we are interested in that portion of the bed in which the adsorption process takes place, we will solve for the Laplace transform concentration 5 when X f X1, that is the section of the bed containing both measuring points. If the Laplace transform of the concentration (5) is known, then the nth moment of the concentration is defined by n... 3.. =(-1)" [EL—{l (15) 3p p=0 where p is the transform variable. If all n moments of a distribution, in this case the concentration, are known, the distribution is completely described. Later, it will be shown that the mean and the variance, the first two moments of the concentration, give sufficient infomtion to calculate the dispersion coefficient from pulse input data. This is very important since higher moments become very laborious to calculate. The Laplace transform of C with respect to t, E", is E = f°°exp (-pt) C(t,x) dt (16). 0 Then, equation (5) is substituted into equation (3), and we take the Laplace transform of equation (1), (2), and (3). The transformed equations are: in the liquid phase 7'52”? ‘32" De (E’Ei)=1%pE (I'A) 5 (Z-A) and on the solid for X5 X1 (N I 5‘ 'U 0 1(Xa. __ I-e (C ' i ' i (3-A) for XZXl c“: = o (4-A) This set of equations now have been reduced from a set of partial differential equations in C to a set of ordinary differential equations in 5: 5; is substituted from equation (3) into equation (1) in terms of 5: lO BEE-W f_Kxa [l+€p .. 1 ]f‘—O 2 56A 3—K- DE K a. mp(1—€) I 3X x 1+ K a x forX_<_ X1 2. 8C W 8C p — > - - —— C = O for X_.X 3X2 fibebA'b 53(— Db 1 The transformed boundary conditions are: at X:X09 C : C0(p) — — — + at x=x1. C(xl) = C and — De Ab - — - DeA 8C — _ — + bb BC + C(Xl) ' _‘W- 37 (X1) - C0(1)" —W_—' er‘xl) and atX=co , 5 = finite (l-B) (Z-B) (6-A) (7-A) (8-A) (9-A). Equations (1) and (2) are solved simultaneously with the transformed boundary conditions to obtain (.2- for X 5 X1. amp) = Eon») F(X.p> where, for X _<_ X1 Aexp [-3— L] [G — Hexp [54]] G - H expT s(§,+L)] F(X.p) = A: exp [—E-L] L=X -X m o L: Xl-Xm b: W (17) (18) ll +4f _ d p f—e[l-E+—d.+I—)_é] mKa d: x I-e I( a e= x 136 I (3:: I{-%}- s R2"fb'?‘+4-3- Db b'= W Dbebfi I 13 = R.+ E—- 5 Expanding the functions 5, 50, and F in a Maclaurin series in powers of the Iaplace transform variable p, there results: _ _ a) "' n C(X.p) = C(X.0) + z —1— 19 P (19) n=l n! apn O p: c — 'c': + w ant—3° Pn 20 0(P)- 0(0) n=1 "'— 8 n ( ) n. p p=0 co 1'1 n F(x.p)=F(X.0) + 231 3—— 3—F p (21) n— n! 8'p P=0 From equation (16) it is readily seen that both 5(X,o) and 50(0) are equal to ()wa (it which is 1. By substituting the expansiona,(l9), (20) and (21) into equation (17) and equating coefficients of p, it is easily shown that l2 F(x, 0) = 1 (22) —: at: BF‘ 8C: 0 = , - (23) [51>] p_—_o [51). p___0 8p pzo [— z— - 2 8217‘ 8F 2 _ 326 8‘6: 2 a C0 8C0 24 —Z ' p ' a ' p ' a ' ‘p ( )' 8p p:0 pzo p pzo p:0 p p:0 pzo — The terms on the left hand side of equations (23) and (2h) can be evaluated by differentiation of equation (l8). The terms on the right hand side of equations (23) and (2h) are related to well known pr0perties of the concentration distribution at Km and X0. The definition of the Laplace transform C = foo exp (-pt) C dt 0 and the first and second derivatives of E-as p equals zero will be considered. The first derivative with p equal zero is [3;] O = -Of°°tht (25). p: The integral is recognized as the mean time, pin, of the concentration at xm. Therefore, a— : - 2 [155—] P=0 Hm ( 6) l3 and similarly for 56 8323 3—— = w: (27) P p=0 0 where ”o is the mean time of the concentration at XO Then equation (23) is rewritten BF‘ = '(P- _ H ) (28). [53]?” m ° The second derivative of 5 with p equal zero is z. Lg = f°°t2cat (29> 3p 13:0 0 and 7'2— —— Z ’ 8;] - Egg] = we..- .1: (3.). apJ __ :0 O a p—O p Equation (30) is rewritten as 2 3233 8C? m 2 _z _ .5— : f (t-um) Cdt (31) 8p p=0 p p=0 O and the integral is recognized as the second moment about the mean, or the time variance O’rzn at X , Z—- - 2 (I:a == 8 C: - %§% (32). 81) pzo p20 Similarly for 6; z. ._ 8 C 8C w 2 O - “539- : f (t - “0) C dt (33) a? :0 p=0 and 326 — z 2 0 BC) 0’0 = —z— - 5— (34) BP p=o p on lh where as: is the time variance of the concentration at X0. With these definitions equation (2h) is written 2 32F _ 8F _ 2 _ 2 (3:) g; 713—- — am 0.0 4 ' p=0 _Thus, from experimental data of concentration versus time at two 2 Z and U - 0 positions in a bed, one can calculated ani' HO n3 0 as numbers. From this data the unknown parameters of the bed can be evaluated by equating these numbers to the respective equation, either (28) or (35). Differentiating equation (18) with respect to the transform variable p, we obtain, e l (a +5) o1r* ' [ 5—3;: - (3’ + 119-11374 exp (-b§)[exp (-bL) —l] (36). b The second derivative, minus the square of the first derivative, is 2 2 2 1 e 1 2 Um‘O'O 1' 2L[EZ-g+b—3 (-d-+T))] l e l 2 1 l e + 4[ (_+—) - — ][exp(-bL)-1] exp(—b§) 17 d D Déb'l Z 13de 6+1) 2 3' I) 1 e 1 -4(§+L)[—T— -—( +-) ]eXP[-b(L+é)] b' bDb b3 a D 1 (E +—) 2 d D 1 e 1 +4Ll—T— - 713+Euexp1-b41 'b' bIDb b 1 1 e 1 7- D I b 15 These are the desired but cumbersome results. There are three limiting cases for which the solutions to this problem are of interest. 2 2 The results for u. -}l and c" - U will be tested for these limiting 1 n1 0 n1 0 (38888 . a). Axial Dispersion with No Adsorption E. = o and d e = 0 27' I21: 131. =13 and p _ sz b' 13b equation (36) becomes um - uo = 5%[1' LEI—Du] [exp (-p)-l] exp-(:EQ- (38). This is the difference of mean times which Bischoff8 obtained for no adsorption. With the same values inserted into equation (37), the difference of variances is obtained. 2 . a; - (r: {($13 Ep + $2 (~1- pZHEXP('P)-1)exp—rg—(-p) Egg“). ((3 - 1) exp(:1¥ “+14“ $3; ((3 " 1) eXptpEQ‘ + {g (p _ 1),2(exp(-ZP)-l) exp Sig-1‘1] (39). This equation also checks with the result of Bischoff. :16 As the bed goes to infinity, i.e. Q -—> m (P-m - ”0)... —> 6%)“ (40) 2 2 eAL Z 2 (Um " 0.0)(1) -—> ( W ) fi (41). b). Adsorption in the bed with no axial dispersion For the case of adsorption, with no axial dispersion D=0 b:co e _ f: (1-6) Fig-W m __1_-sf: bD-W With these substitutions equation (36) reduces to (rm-no: %% [1+€(m-1)] (42). By inserting D = 0 into equation (37), it is seen that all the exponential terms go to zero because b goes to infinity. Also the 2 . :7; (sea + %) term is zero when D = 0. Therefore, only the first term 2 2 for a- - 0' remains: n1 0 2 2 2 LA (1 - e)2 o_rn - 60 = 2 We (43). W m X c). Infinitely long bed with both axial diapersion and adsorptigp_ For an infinite bed, L = m is inserted into equations (36) and (37), they reduce to m. - w. = 5%- + ‘1 $3.“ <44> 17 and the exponential terms in equation (37) are zero, hence Z 2 2 2 ZAL (l-e) 2L 3 (1-6) 1 (45). (0' -5) = +——3(DeA)[ +—] m 0 (I) mZKXa- W W EmD D Let 2 2 (er - or ) 11 = m °§° 146) (pm - #0))” where w is a measure of the total spread of a pulse. Then let (62 - G2) (JD = m 0; (47) (Pm - so) a, 2 2 (cm - (foxJo (11m - “0’00 where 433 is the spread due to axial dispersion for an infinite tube, evaluated in case (a), and 43* is the spread due to adsorption, evaluated in case (b). An interesting result of this case (c), is 2 q” " WE ALKXa 1+e(m-1) ‘ If we examine (a) for the infinite case we find that _ZeAD LPD “ "Tw (50) and if we examine (b) for an infinite bed we find that 2 _ 2w 1 -6 “PA- Xe. m] (51)- 18 Therefore, for the infinite tube case with both processes, the total spread of a pulse input is the sum of the axial dispersion and adsorption spreads. Therefore, we may write ¢=¢D+¢A (52). van Deemter, Zuiderweg, and Klinken'bergl+5 have shown that, if the concentration-time distribution were Gaussian, the dimensionless variance at a position in the bed is the sum of the dimensionless variances due to axial diSpersion and adsorption. In this work, it has been shown that the difference in dimensionless variances between two planes in an infinitely long bed is the sum of the differences of dimensionless variances due to the axial dispersion and adsorption for any hump like input. That is, the restriction of a Gaussian distribution is removed. There is a great experimental advantage in removing this restriction since it is impossible to introduce a perfect pulse input. Previous work16:20 has required that the concentration time curve also be Gaussian at the measuring point. It is shown in Appendix II that any pulse input approaches a Gaussian time distribution for long columns. All previous investigators have made this a criterion for the design of their apparatus. Thus, both a long column and a perfect injection system were needed. The present work allows one to analyze data from short columns with no particular care taken to introduce the pulse into the column. The only requirement of the pulse is that it be "hump-like". 19 In summary, the relationship between the difference in variances and means between two points in a finite packed bed and the parameters of the bed is derived with both axial dispersion and adsorption occurring simultaneously. Three specific cases of this general solution are obtained and case (a) is compared with a previously derived result, first by Aria2 and later, corrected by Bischoffa. In summary these three specific cases are: (a) axial dispersion with uniform flow in both portions of the bed. (b) adsorption process occurring in the first section of the bed with axial dispersion in the second section along with uniform flow. (c) infinitely long bed with both adsorption and axial dispersion occurring simultaneously and individually with uniform flow. As a result of considering case (c) the term 2 2 Urn.' oc> 4’: Z (Hm'Ho) from axial dispersion may be added to the corresponding term for adsorption to obtain the result for both processes occurring simultaneously. This is the mathematical Justification for separating the two phenomena,axial dispersion and adsorption. AXIAL DISPERSION IN PACKED BEDS This discussion will present background material concerning dispersion, primarily in packed beds. The three common approaches to dispersion are: (A) the diffusion model; (B) the mixing cell model; and, (C) the statistical approach. All three of these models have been analyzed theoretically and experimentally by many investigators. Diffusion Model The most widely used model for axial diapersion is the diffusion model, which assumes that the equations which apply to mass transfer of species are of the same form as those describing a molecular diffusion process. The conservation equation is 1332c uac_ac 53 EZ- ex-s? (’- It is shown in Appendix I that D, the dispersion coefficient, is the sum of the molecular and eddy diffusivities. The use of the Fick's law equation for turbulent transfer has been discussed by Prausnitz35. He considers an observer traveling with the mean flow velocity, U6. Two packets of fluid, A and B of Figure 3, interchange by the following mechanism: 1. Fluid element A, coordinate (r0, X-l) and traveling at U0, encounters a solid particle and, in order to pass it, side-steps to position r1, where its velocity is changed to U1. 20' 21 Cl X .‘* B ._._ l I ‘ Ull I observer traveling l at U0 l I X0 1 l 1 U1 l I X.1 L — q...L____. -l r“l r0 r1 Figure 3. Side-stepping Model of Dispersion. 2. Fluid element B, coordinate (r0, X1) and traveling at Uo, encounters a solid particle and, in order to pass it, side-steps to position r_1 with a new velocity U_1. In order that A and B interchange positions in the stream, (01 - Uh) := (U5 - U-l)' The flowing fluid is a solution of a solute concentration Cl at X1 and C-1 at X_1. Thus, there is a net flux of solute across the plane which is moving at velocity U5 and is given by: m FLUX = (U1 - U0) c_l + (U_1 - U0) c1 (54). The velocity increments in excess of U6 within a pore are given by U1 ' Uo (U-1’ U0) 8U “Tr—— = ' Tr = a; (55) where 11. is the radial scale of mixing similar to the Prandtl mixing length for turbulent flow. It represents an average side stepping 22 distance. Similarily, C1"C.1 ac T = a ‘56) where 1X is the axial scale of mixing. By inserting equations (55) and (56) back into equation (53), the net flux is given by: _ 811 BC: Net Flux— -1X1r —a—r- 8—)? (57). This simplified model shows how one can deduce an expression for the net flux in terms of the axial concentration gradient, the radial velocity gradient, and the axial and radial side-stepping lengths. By comparison with Fick's law for molecuhmrdiffusion the mixing or eddy diffusivity, De, is given by: _ aU De _ 1X 1r 5? (58) and the net flux due to mixing is 8C3 Net Flux = - De 6—): (59). This development Justifies the use of a Fick's law diffusion model to describe eddy mass transfer. waever, it is emphasized that the eddy diffusivity depends upon the flow characteristics of the flow region; whereas, the molecular diffusivity depends on the intrinsic prOperties of the fluid. Prausnitz and Wilhelm36 have obtained concentration fluctuations at high interstitial velocities which indicate orders of magnitudes and approximate values for the parameters in the eddy diffusivity. These 23 approximate values are 11. = l/h dP where dP is the particle diameter and l: = 7 lr. From dimensional considerations a reasonable estimate for the order of magnitude of dU/dr is U/dp. Thus the value of the Peclet number (P8 = ESEB) is approximately two. This agrees, probably fortuitously, with exgerimental data, at turbulent flow conditions. ‘With this type of model one can.predict qualitatively the effect of void fraction on the eddy diffusivity. As a first approximtion the tendency for 1r to increase with void fraction will be compensated for‘by a tendency for dU/dr to decrease; however, 1; would'be expected to increase. Thus one would expect an increase in the eddy diffusivity and.a decrease in the Peclet number with an increase of void fraction. This has'been experimentally verified‘by Jacques25. The effect of particle diameter is similarly analyzed, and it is expected that the eddy diffusivity would.increase proportionally with the particle diameter and as a result one would expect no change in the Peclet lumber. This has been experimentally verified by Ebach and white20. The theory previously discussed is a generalization of the model proposed by Taylor“? for flow in pipes . Taylor derived an equation to describe axial dispersion for turbulent flow in pipes. De = 3. 7 <51t 11 NH (60) where dt a diameter of the pipe u a ‘mean.flow velocity 2k and y is defined by To 2%ypu2 (61) where To = shear stress at wall of pipe P = the density of the fluid. Y is interpreted as a resistance coefficient for the flowing fluid. In another paper, Taylor”3 derived the expression for the axial diapersion coefficient for laminar flow in a tube dtz‘u2 Daxial = Wit; (67-) where Dv is the molecular diffusivity of the solute. Taylor's deve10pment shows that the dispersion is due to radial gradients. Fluid flowing slowly in a packed bed will have dispersion characteristics which are a combination of the characteristics of a fluid flowing in a tube in both laminar and turbulent flow. If the fluid is actually in laminar flow in the interstices of a packed bed, it should exhibit turbulent-like dispersion characteristics, since the packing restricts the flow such that a laminar profile is not set up in the bed. The interstices are interconnected such that the boundary conditions and geometry of each cannot be represented mathematically as simply as those of a tube. Therefore, it is realistic to qualitatively describe dispersion in a packed bed of a slow moving fluid by the side-stepping model developed by Prausnitz37. 25 Solutions to the Diffusion Equation There are four types of boundary conditions which have been applied to equation (53) for dispersion in packed beds from which a description of the concentration profile is obtained. From these solutions and appropriate experimental data, one may obtain the value of the axial dispersion coefficient D. Three of these sets of conditions are for the unsteady state and the other is for the steady state. The pulse input - This method is an unsteady state case of equation (53) solved with the following boundary conditions: att=0, c=o forX>O (63) atX 00, C =finite for t > 0 (6’4) and at X = 0 there is an infinitely high pulse of zero thickness introduced. The solution is approximately (Appendix III) o "1 '%%’ C(X,t)= VNf expl: 4D ] (65) 411’ D/uX '11—)? where CI) Q = W f Cdt (66) 0 which is the total solute injected. The maximum value of C appears at 2 1/2 -I) }(tl) . tm = —Z+ 2-2- [1+(T )] (67) and can be approximated by x D XU 2 tm = v - —-—2 when (—D—) >> 1 (68). 'U 26 Equation (65) is in the form of a normal distribution with the variance given by 2 2D U z-IJT (69). The variance of the concentration-time distribution at X ==L is determined experimentally, and D is calculated from equation (69). The step_input - Equation (53) is solved for a finite bed of length L with the following conditions: at X=L 8C=0 for t>0 (70) 535 and at X = O and t >'O 8C3 _ UC'D‘JY‘UCO (71) where Co is the concentration of the input, and at t = O C = O for X > O (72). The set of equations to be solved in dimensionless form is ac»: + ac* _ 1 32m (73) 56 5y _4P' ayZ 8CD“ . 4P'C* - =0 at =Oior6>0 74 537' Y ( ) 8CDk 67-:0 at y=1for 6>0 (75) C*=1 at 6:0 for 00 (79) 337 aT+g1=NT=O aty=1for6>0 (80) 337 T(y,0) = exp(-2P'y) at 9:0 (81). These boundary conditions correspond to convection from the ends of a slab into a media at zero temperature. Carslaw and J'aeger17 have given the general solution of this arbitrary initial condition. The solution to these equations is C*(y.9) = 2 exp[P(Zy - 9)] x 15:1? ‘TK/(‘Té't P'2+ p!) (7}: + P'z) x [7K cos (ZTKY) + P' sin (ZTKY)] xexfi-Té e/P') (82) where the 7K(K=1, Z. .) are the positive roots, taken in order of increasing magnitude, of the transcendental equation I Tan 27: 27? , and'rf’o (83). 72 _ P,2 28 Equation (82) converges too slowly to be of much use when P' is large and/or t is small. Brenner12 develOped an asymptotic solution by the Laplace transformation and has tabulated values of 0* as a function of t for large values of P'. These can be used to calculate P' from experimental data,C-t for a given column. The step input is also solved for an infinite column extending from -m to +c: . Initially, the section from X = —a>'to X = O is filled with solute of concentration C ==Co, and for X greater than 0, 0 == 0. Thus the boundary conditions are: at X = 0, C = Co for t > O (84) at X=00, C=0 for t>0 (85) andat C=C0 for XO The solution to equation (53) with these conditions is given by Danckwertsl9 C l (X - ut) = - [l - erf ] (87) t; 2 (4 Dt)1; z where erf (Z) = VET-r- ofz exp (yz) dy (88). By setting X = 1.and differentiating equation (86) with respect to ut/I.and evaluating when ut = L, one obtains C (134 8 o = l vJ—T—uL “D (89). ut 2 (1:) if}: 1 29 From data of C/Co versus ut/L, the slope at ut/L ==l gives the value of D. Frequency response technique - This is a quasi-steady state method for determining the parameters of the bed in which the solute concentration input to the bed is a sinusoidal function of time. The amplitude and the phase angle lag at the output section of the bed are related to the bed parameters. The boundary conditions are represented by: C=Cm+A(X,0) cos (w t) atX=0 (90) c=cm or A=O atX=00 (91). Ebach and White20 have shown the solution to equation (53) with these conditons is: C (X, t) = Cm + A(O) exp (-B) cos ((0 t - 4)) (92) where }( - 1 2 B = - 7—3- 1 -('\f1f4+(ZDw/ul)‘ +1/Z) /] (93) and q; = $2517 'xll/r + (.2Dm/u‘Z)Ar -1/2]1/2 (94) and A(o) = input amplitude Cm = mean composition about which concentration oscillates w = angular frequency of periodic wave, rad/sec. 30 B and ¢> represent the decrease in amplitude and the phase shift of the output wave, respectively. The ratio of the inlet amplitude to the outlet amplitude is given by Ai/Ae = B (95). Equation (93) is compared with (95) for eXperimental data and D is then calculated. Steady state solution - Equation (53) is solved with the boundary conditions: C=CO at X=L (96) and 3C: uC=D-57{ at X30 (97). Since the steady state is assumed dC/dt = o (98) and the solution to this set of conditions is C=Coexp[%(X-L)] (99). These conditions correspond to a tracer being continuously injected downstream from the inlet of the column and the solution is for the concentration profile between the inlet to the bed and the tracer injection. A plot of ln C versus X yields a slepe of u/D from which D can be determined. The four solutions to the diffusion equation (the pulse input, the step input, the frequency response technique, and the steady 31 state method) all represent different experimental techniques to calculate the axial dispersion coefficient D of equation (53). These solutions differ only in the boundary conditions which are applied. Consequently, D should have the same value under the same bed conditions for each technique. The pulse input as given here requires the pulse to be of zero thickness and of infinite height. This is experimentally impossible, and in order to approximate this type of input, considerable care must be taken in the design of the injection system. 2 and later corrected by Bischoff8 The method derived by Aria (see Case (a) in Statement of Problem), allows one to calculate the axial dispersion coefficient with a minimum of assumptions. The only restrictions on the input are those given by Aris5 which essentially state that the pulse input be a "hump like" function of time. This method requires two points of measurement rather than one. However, the column does not have to be long as in the case of the approximation of a normal distribution. The step input method is a popular method of analysis and requires concentration versus time measurements at one position in the column, however, it is experimentally difficult to obtain a sharp step input. The frequency response technique is probably the most difficult method experimentally. It takes a great deal of elaborate equipment to generate a sinusoidal concentration input. In addition, a fast response measuring technique is required. 32 The steady state method, in principle, is the simplest method for the determination of the axial dispersion coefficient. waever, u/D must have an order of magnitude of about 10 ft“1 to be practical. This is considerably smaller than the approximate value of 200 ft'1 for this work. The Mixing Cell Mbdel. The mixing cell model was first discussed by Kramers and Alberda3O and later by other investigators.6’15’l6’35 They assumed the voids between the particles of packing may be considered as cells of complete mixing. Equations (100, (101), and (102) describe a series of cells numbered from O to N, with the zeroth cell being fed a solution of concentration Cf(t). It is assumed that the cells are initially free of solute, and their volumes and the flow rate are V and V, respectively. dCI‘I ch_l - ch = v d‘t (100) = = 101 Cit 0 at o ( ) dCcD - = ____ 02 . WCf WCo V at for t>0 (1 ) Aris and Amundson6 have applied the Laplace transformation to equations (100), (101), and (102) and found 6k, the Laplace transform of 0N, rq+1 c: _. vv f (3 = __ (103) N [V] (P + W/V)N+l where p is the transform variable. 33 The inverse transform by means of the convolution integral is Iq+l CN = filT [VTVI'] 0ft Cf(s) (t-s)N exp [- g— (t-s)] ds (104) which gives the concentration at the Nth mixing cell as a function of t and the input Cf(8). For a delta pulse function input, equation (10h) has been integrated (see Appendix III) with the result N+1 WCN [3-379 w T W? “P (' v“ ”05’ where Q is the amount of material injected. Equation (105) is a relationship for the concentration of the Nth cell as a function of time for a pulse input. For a step-function feed of tracer of concentration Ci at the inlet to a sequence of mixing cells,the concentration at cell N is given as a function? time by: _jfl” ELI“ g— =1- V V ...+1 exp(-‘%t) (106). i N' +(PJ-1) ' The Random4Walk Medal. Jacques and Vermeulen25 have develOped a randomawalk model for the transport of a tracer molecule through a packed bed. The randomdwalk of a tracer molecule through a packed bed is characterized by the advancement of the molecule from point to point in a some what erratic way, always moving in the direction of flow. The path followed by the molecule is made up of a succession of random steps involving different distances and occurring at different times. 3h Scheideggerl‘O has pointed out that the fluid flow through porous media is governed by the Navier-Stoke's equations. It is the random- ness of the media itself which leads to a stochastic treatment of the flow. The stochastic approach to the problem is based on the "ergodic hypothesis" which states the motion of a particle at any point in the media is independent of the particles at other points in the media. Jacques and Vermeulen have derived a normalized probability density function PT(N) for this case such that PT (N) dT = exp [ -(N+T)] Io (ZN/_NT ) dT (107) where N == L/h (number of steps) dimensionless length T = ut/h dimensionless time and L = length of bed 11 = average length of steps 11 = a characteristic velocity t = time. I0 is the zero-order Bessel function of the first kind with an imaginary argument. PT(N) dT is the prdbability of finding a packet of fluid N mixing lengths from the inlet between time T and T 4- dT. For a step function input of concentration Co the equation for the concentration at plane N is a; e ofT exp [ -(N+T)] Io (2 NI—NT ) dT (108). 35 Hennico, Jacques and Vermeulen23 have tabulated values of C/Co versus T for a step input for values of the parameter N. Their values are based on an approximation of equation (108) which is suggested by Klinkenberg28. The Klinkenberg approximation simplified equation (108) to g— : %[l+erf[Nr(N+l)6-l/4 -'\/N+174fl (109) where 'T IQ+1 ' 9 = -——— For a pulse input of amount Q and a column having a void volume of V the concentration C is given by: EQY = exp [-(N+T)] Io (le NT) (110), Comparison of Models. Several investigators 6:30:35 have shown that the cell size in the mixing-cell model corresponds to twice the mixing length in the diffusion model. The diffusion equation is 2 so so _ ac D552 ”1‘5: " a? (“1" The finite-difference form of the diffusion equation is D [Cx+Ax - ZCX + Cx-Ax] _ 2 [Cx+Ax - Cx-Ax] _ AC (112) sz 2 AX ‘ At ' Then, if one lets D = 1 u, Ax = 21. and AC/At = 8C/8t where l is the mixing cell length, equation (112) can be rearranged to give 36 (113) where N=x/21. Equation (113) is the same as equation (100) for the Nth mixing cell with the mean residence time equal to 2l/u. The finite difference approximation to equation (111) requires that the increment AX be much less than the total length of the bed. Hence, this comparison is valid only for large N. Physically it seems realistic to use the mixing cell model for high flows, but for low flows it is not realistic to assume perfect mixing in each cell. For this reason one should use either the diffusion or the randomawalk model when the flow rate is small. The randomawalk and diffusion models are compared by defining D: u 1 (114) and P6: 9}}— : 915% (115). Then it is consistent to write 11 __ L1u N- 1— — —D— (116). A further comparison of the diffusion and the randomawalk model is made by comparing the C/Co versus 0 curves of each model for a step function input. This comparison for N = 2 and N = 80 is shown in Figure h. It is seen here that the shape and values of . mHmdoz 433180.?“de pom QOflmSMHB was. .Ho modfiwofim nowapmhpcoonoo 05. mo nOmHHmmEoo .: ohsmflh O. N m 01—” OOH m e O _ _ . 0.0 Hopoa aznaopcmn I I I \ ownz specs gametes BEE \ hem \ II N. O \. \ [$4.0 O NHZ .\ W \ O \ \ \ .11 w o \ \ \ \ \ . \ \ \ .lwoo \ \ 2“ \ o.._.. 37 38 the curve for each model are approximately the same. Since neither the diffusion model nor the random-walk model is completely rigorous, it makes no difference which model is used to correlate experimental data. The only criterion for choosing one model over the other is the simplicity of the calculations. THE EXPERIMENTAL PROGRAM It has been shown that the total spread number, 1L, of a pulse input as it passes through a column packed with an adsorbing material, is the sum of the spreads due to axial dispersion and adsorption. It is seen in equation (51) that the adsorption spread is a function of the velocity, the void fraction, the mass transfer coefficient, the equilibrium constant, and the distance between the measuring points. If this part of the process is scaled-up such that the length, void fraction, and the velocity are unchanged; then for an increase in the column cross sectional area there is no reason to expect a change in the mass transfer coefficient or the equilibrium constant. There- fore, it is expected that the spread of the pulse due to adsorption would not be a function of the column diameter. Thus, if the total spread of a pulse is a function of the column diameter this effect is a result of axial dispersion and not adsorption. Therefore, an eXperimental investigation of flow in a non-adsorbing packed bed was made in order to establish the effect of column diameter on the spread of pulse input. At the present time, it is not possible to predict a variation of the axial dispersion coefficient with column diameter. Axial dispersion has been shown to be a result of radial velocity gradients for laminar flow in a tube while for turbulent flow the dispersion results from eddy's or fluctuations. Since flow in a packed bed exhibits character- istics of both laminar and turbulent flow, it is expected that dispersion is caused by both the radial velocity gradients and fluctuations. 39 hO It is not known if these fluctuations and gradients change with the column cross section; therefore, it is necessary to determine experimentally the relationship between the axial dispersion coefficient and column diameter of a packed bed. 11,18. Viscous Fingering_ In chromatography applications it is common for the pulse of material to be separated to have a viscosity much higher than that of the solvent. It may be possible that this factor may have an over-all effect of changing the axial dispersion coefficient with a change in column cross section. This effect may be explained by the phenomenon known as viscous fingering. Viscous fingering is a result of a difference in viscosity between the pulse of material injected and the continuous solvent phase. When a fluid of low viscosity is used to displace one of high viscosity in a packed bed, small irregularities in the velocity profile tend to become magnified. As a result "fingers" of the low viscosity fluid extend into the viscous phase, and grow exponentially with time. For a high viscosity material displacing one of low viscosity the boundary is stable, fingers do not form. No studies have been made, however, on a pulse with a different viscosity than that of the solvent, to determine the effect on the dispersion. A pulse of high viscosity material might be expected to have a stable front boundary and an unstable rear boundary. 0n the other hand, it is reasonable to expect the combination of the two boundaries of a pulse to be stable. This problem is investigated in this work. #1 End Effects. End effects are a result of conditions at the ends of the column which disrupt the flow pattern. This disturbance in flow pattern may increase with the column cross section, possibly making the axial dispersion coefficient a function of diameter. Objectives. The experimental objectives of this research were to determine; (a) the effect of column diameter and (b) the effect of a high viscosity pulse on dispersion in a packed bed. In order to fulfill these Objectives, it was decided to use the pulse tracer input method with two points of measurement for the determination of the axial dispersion coefficient. The two major advantages of this method are that this input is similar to that used in chromatographic separations and requires only that the pulse be "hump like". In order to eliminate complicating adsorption effects a non-adsorbing packing was used. values of the axial dispersion coefficient for a pulse viscosity equal to that of the solvent were obtained in a six inch column and a one-half inch diameter column with the same packing material of two sizes over a wide flow rate range. Particular attention was given to low flow rates where data are scarce. Additional data were obtained in the sixpinch column with a pulse of higher viscosity than the viscosity of the solvent. APPARATUS AND PROCEDURE A schematic diagram of the flow system used in the measurement of tracer concentration versus time in a nonaadsorbing packed bed is shown in Figure 5. A one-half inch and a six inch diameter column were used with distances between the measuring points of six, twelve, and eighteen inches. The Flow System. Each column consisted of various lengths of flanged pyrex pipe. The water and tracer tanks were connected to the column with one quarter inch copper tubing. The tracer and water containers were plastic lined galvanized steel pressure tanks. Nitrogen was used to drive the fluid from the tanks through the column. The outlet line had a micrometer valve for fine adjustment of the flow rate. The Tracer. It was decided to use a dilute aqueous sodium chloride solution as a tracer. Sodium chloride concentrations are easily measured by electrolytic conductance and the prOperties of the system are well known. The concentration of the sodium chloride in water was 0.5 g/l, approximately 500 parts per million. It is shown in Figure 6 that the specific conductance of sodium chloride is proportional to its concentration in the range of O to 700 parts per million. The value of the molecular diffusivity 1)v at 30 °c is 1.811 cm2/sec x 10'5 at 0.5 mol/liter concentration2h. A2 nitrogen pressure tracer water ._L nitrogen T Figure 5. Schematic of Flow Equipment. . h3 pressure cell I test section cell II micrometer valve 11100 g 1200 'g .H 1000 E El a, 800 C) i 0 600 8 O #00 H 1'11 8 g. 200 (0 Figure ‘T I 200 1100 600 parts per million Specific Conductance of Sodium Chloride at 25° C. #5 A 65% sucrose in water with 0.035% sodium chloride was used as a tracer for the high viscosity pulse data. At room temperature this solution had a viscosity of 100 C.P. Experimental data of conductance of the sugar-NaCl solution were obtained in the laboratory since they were not available in the literature. This data is shown in Figure 7. Packing_Material. Spherically shaped glass beads of 0.0071 and 0.00h7 inches were used as the packing material for this investigation. It was experimentally verified that these beads did not adsorb sodium chloride. The conductivity of the sodium chloride solution remained unchanged after being contacted with the glass beads. The average void fraction of these beads of O.h was determined by the displacement of water. Conductance Cells. The conductivity cells for the one-half inch and the six inch columns are shown on Figures 8 and 9, respectively. Considerable difficulty was encountered in the design and construction of reliable cells. From the standpoint of true measurements of concentration at a cross sectional plane in the bed, it would be desirable to have a sensing device which would be a part of the packed bed, or at least not change the characteristics of the bed in the region of the measuring element. Preliminary studies of various conductivity cell designs were carried out in a one inch column. Cells similar to the one shown in Figure 9 were tested in the packed bed, the cell being inserted in the packing material itself. Data on this arrangement were not reproduceable and the design was abandoned. 0.80 0070 — 13 O '8 0.60- .fl F1 ii 0.50 - Q) 0 g Geno _ 0 3 8 O 0030 _‘ O H :1 O 0020 - Q) 94 U) 0.10 - 0'0 ' I I I* I I I I I 0 200 1100 600 800 sugar concentration (grms/liter solution) Figure 7. Specific Conductance of Sucrose with a Trace Amount of NaCl at 25° c. h6 top view platinum wire i I -——---II- T--—-- . 1'6" 1/8 u __.1 ( f edge view Figure 8. Plexiglas Conductivity Cell for One-half Inch Column. 1&7 terminals for {P platinum'wire 'vvv —-| I‘- 1/8" t0p view edge view Figure 9. SiXeinch Conductivity Cell. 1+8 1+9 It was then decided that the conductance cell had to be separated from the packed bed. After several other designs had been tested, the one shown in Figure 8 was tried and found to be reliable. It was inserted in the column with a piece of canvas over it to keep the packing out of the sensing area. The cell was made as thin as possible so its contribution to dispersion would be small. The amount of dispersion with this design was undetectable in the one- half inch column. However, a similar design for the six inch column proved to be unsatisfactory, in that it contributed the equivalent of fourteen inches of packed bed to the dispersion coefficient. The improved design, shown in Figure 9, made this effect negligible. The Measuring Circuit. The electrical circuit for measuring and recording concentration as a function of time is shown in Figure 10. The circuit is designed so that the voltage drop across the resistor R is pr0portiona1 to the reciprocal of the cell resistance Re. The voltage drop across R is E == IR where I _ V _ Ike-+11 ’ As long as Rc is much greater than R, then \flt E: F— c Since both V and R are constant, then E is pr0portional to l/Rc R is fixed a 1 ohm and Re is never lower than 1000 ohms. There- fore the maximum error due to this approximation is 0.1%. .mpooacHSmmmz moomposcooo mow pfldoawo Hmoflhpooam .OH madmwm Hovmfldfiomo Home hophooca ow o. Jargtioex pus mundane 51 A variable frequency audio oscillator was used as the voltage source. The A.C. voltage across R was amplified, rectified, and measured with a Sargent multi-range recorder. A schematic diagram of the rectifying-amplifying circuit is shown in Figure 11. Packing_Procedure. It was important that considerable care be taken in packing the columns. Trapped air or unpacked channels would cause the column to operate ineffectively. The bottom section of the column containing the packing support was filled with water. The beads were mixed as a thin slurry to remove the air from their surface and were poured into this section until it was full. During this time the column was vibrated to settle the packing. A conductivity cell was clamped between the top of the packed section and the bottom of the next. A packing support was placed on top of the conductance cell and the packing procedure was repeated for a new section. The column was packed to three inches above the top conductance cell and a liquid distributor with approximately the same void fraction as the packing was placed on t0p of the bed. Run Procedure. While water was flowing through the bed each cell was monitored until a constant zero reading was obtained. The liquid level was allowed to dr0p to the top of the bed at which time the flow was stOpped. .A pulse of tracer was injected uniformly across .pssosao msaamaams< msaaaapoom mo oasesoaom .HH enemas Havoc: b mama o>m.m VI) I ‘l m a III I— .H..I l I I _ 15W Ho. _ I Hopaoooa H _ m on “Mm/men .Eom + ncpaooma M , 2..” \1 ad NH «\H \(z) SJ,“ _ @ roam s a ma «\H EH _ _ I- .1. .11.. _ 1 9% Ho. 9m: Ho. . aw _ >oom+ 52 53 the top of the bed and the flow rate was gently adjusted to allow the pulse to enter the bad, after which the water flow was resumed at the desired rate. Conductance versus time data was obtained at each cell position in the bed. After each run resistance versus recorder scale readings were obtained to insure that the conductance was proportional to the scale reading. Calculation of the Dispersion Coefficient from the Data. The dispersion coefficient D is determined by the relationship 0' 2 0" 2 ' Z Z 1.2 = ‘u‘L— (5°) (“-2 ' P1) and the 2 represents cell number 2 down stream from cell number 1 where u = fmCt at/ [”0 8t (27) n 0 0 and z - “’c (t - )2 8t/ “’0 3t (31) on — of (Ln of . These integrals are evaluated numerically with a Control Data Corporation 3600 computer from the experimental data using Simpson's. rule. Scale readings are obtained from the recorder chart at equally spaced time intervals . For the aqueous sodium chloride tracer the zero reading is subtracted from each point, these differences being 5’4 proportional to the concentration. Therefore, in place of C, in equations (27) and (31), it is necessary to insert only the corrected scale readings since the proportionality constant cancels. For the sucrose sodium chloride tracer, the scale readings are corrected by subtracting the zero reading from each value and then obtaining the values of the concentration from Figure 7. It can be easily verified that on2 is independent of the chosen reference time. Thus, for computational purposes, the time is taken as zero at the beginning of each cell. The time between the beginning of the two cells is noted and is added to pa for the final calculation in equation (50). The value of the interstitial velocity u is calculated from the data according to the relationship Uz; (117). “2 ‘ “1 Figure 12 is a plot of the conductance versus time measurements made at two positions in the column. .gm .m Ho.“ ”v0.3 059 manned? ooqmpodvqoo .NH mcflrwwm Aommv 95$ 0% com 8H 8H ow 3 o .y _. _ _ _ _ _ _ _ _ _ p a o _ _ .. a A.~¥.Vl 0: l om uem N .mvm n Nb 00m w .3. n Hb N N N N .l ow N - :3 I 09” i _f N1 L _ I 8H J - N1 e. him e 33 .o u a 2}“ m .2 "a: I 03 3:2: 2.8 .o u s a :8 .3 cm .o n A - CESHOU tome £0GTN\H C .l. owH cautionpu‘oo 55 RESULTS AND DISCUSSION It is seen from equation (60) D=3.7dtu~/'y (60) the relationship for the eddy diffusivity for turbulent flow in a pipe, and equation (62) D = dt2 112/192 Dv . (62) the relationship for the axial dispersion coefficient for laminar flow in a pipe, that a log-log plot of D/Dv versus dtu/Dv should be a straight line. In the turbulent region the line should have a slape of l and in the laminar region the sIOpe should be 2. It is expected, therefore, that the same type of plot would correlate data for packed beds, although, a straight line would not necessarily result since it may be difficult to determine the transition between turbulent and laminar flow in a packed bed. For correlation purposes, the tube diameter dt is replaced by four times the hydraulic radius, hM. For a bed packed with spheres, 2 — ._ ._£_ 4M- 3 1-6 dp e The dispersion coefficient is divided by the molecular diffusivity to illustrate that the axial dispersion coefficient should approach the magnitude of the molecular diffusivity at a low value of the velocity. It has been pointed out that the axial dispersion coefficient should go to approximately 2/3 By and not Dv, because the path for 56 57 the solute to diffuse is actually longer than the length of the packed bedll. A plot of D/Dv versus h Mu/Dv is shown on Figure 13 for the data of this work. Column Diameter. The effect of the column diameter was studied experimentally with six inch and one-half inch columns. It is seen from Figure 13 that there is no detectable effect of column diameter on the axial dispersion coefficient. This data covers a range of velocities from 0.1 ft/hr to 20 ft/hr in a packed bed of spheres with a void fraction of 0.h. Pulse Viscosity. 100 ml of 100 centipoise sugar solutions with a small amount of sodium chloride was used as a tracer. Values of the dispersion coefficient with this pulse input were obtained in the six-inch column packed with 0.0071 inch glass beads and a void fraction of 0.h. These values are compared in Figure It with those values obtained for an aqueous sodium chloride tracer solution where the viscosity of the pulse was the same as that of the solvent. From this figure it can be concluded that there is no detectable effect of viscous fingering with column diameter since the values of the dispersion coefficient for a high viscosity pulse fall on the same line as those for pulse viscosity equal to that of water. Particle Diameter. The effect of the particle diameter was studied in the one-half inch column and it is seen from Figure 15 that the dispersion coefficient effect increases with the particle diameter. lOOO mégffigD Eff) 100'” C) Q d Q90 10 ‘i O > [F] Key Q o E El 6-inch column B .0071" glass beads C l/2-inch column .0071" glass beads 1.0 T I 0.1 1.0 10 100 km u D v Figure 13. D/DV versus l’rMu/DV for Column Diameter Comparison. 58 .l [s .01 — get A B E] a: Key ‘H 6-inch column with .007 " g .001 — glass beads d3 D pulse to solvent viscosity ratio of l E] A pulse to solvent viscosity ratio of 100 .0001 - I I 0.1 1.0 10 100 u ft/hr Figure 11+. D versus 1.1 for Pulse Viscosity Comparison. 59 .1 .01 - a .m \\ 0t» CH 0001— n / Key 0 / l/2-inch column 0 .OOLIT'glass beads C .007l"g1ass beads .0001 I 0.1 1.0 10 100 u ft/hr Figure 15. D versus u for Particle Diameter Comparison. 61 The fact that D is proportional to dp is apparent from the correlation of Figure 16. This conclusion is in agreement with the data of Ebach 4 Mup and White20 in their experimental range of from 0.1 to 1+0. 16,20,38 Column Length. Previous investigations have shown experimentally that the axial dispersion coefficient is independent of the bed length. The data obtained in both the six inch and the one-half inch columns verify their conclusion for w investigation. Values of the dispersion coefficient for these lengths are listed in Tables I through VI in Appendix IV. These values are shown on Figure 13 from which it is seen that there is no effect on the dispersion coefficient with column length. Minn with Other Work. Figure 17 is a plot of the packed bed Peclet Number (Pa = 1:52 ) versus a modified Reynolds Ember (Re = 3%33) shows a transition range for Be between 10 and 1000. This region corresponds to the transition range between laminar and turbulent flow as illustrated by Ergun ' a plot7 of the friction factor versus the Reynolds Number for packed beds . The upper limit of the curve is described by a Pe of about 2.0 which is in excellent agreement with that predicted by the side- stepping model developed by Prausnitz37 and, with gas phase dispersion data35. The lower limit of Fe is 0.22 for liquids in the experimental range. As suggested earlier, as the velocity approaches zero the axial dispersion coefficient should approach the magnitude of the lOOO 100—' 6 Q 0 C890 0 Q> 10 -J E CD Key l/2-inch column e .ooLIT' glass beads O .0071" glass beads 1 I I 0.1 1.0 10 100 AM u D Figure 16. D/DV versus LIMu/Dv for Particle Diameter Comparison. 62 .xmok Hmnpo 39H? mOmHHmmEoo 90% 1\Ansz : m5m~o> Q\mdd .NH mHSmHm 1 use OH mOH mOH OH O H HuOH muOH muOH I - OH _ _ _ _ H a _ H- mHHmoMmmoO com mmmapm .m moHdmsuo> com nonvomh «OUquem .: 1| movemam 6mm maamnamo .m I. opan com nomnm .m H moHpmepmo>mH mHgB .H n d/ [I I G I: m II Isl-I. Con—u. _ _ p. O.m 63 6% molecular diffusivity. The data obtained in this investigation at Re of 10"3 is in the order of magnitude of the molecular diffusivity. Consequently, one would expect Pe to go to zero with the velocity for Re less than 10‘3. Ebach and White20 used both a sinusoidal and a pulse injection of a dye into a 2.0 inch diameter column 5 feet high, packed with spheres of 0.0083, 0.0h, 0.13 inches in diameter. They showed that the axial dispersion coefficient varied in prOportion to the particle diameter. Carberry and Brettonl6 used a pulse of dye into a 1.5 inch diameter column at various lengths, packed with spheres ranging from 0.02 to 0.2 inches in diameter. Hennico, Jacques, and Vermeulen23 used the step injection technique for spheres of 0.75 inches and 0.38 inches in diameter. They used a column diameter of 6 inches and lengths of 12 and 2h inches. Strang and Geankopolis2+1 used a sinusoidal input of dye, for 0.23 inch glass spheres in a 1.65 inch diameter and 23 inch length column. In order to compare these experiments, one must keep many sources of error in mind, such as instrument lags, injection effects, column wall effects, and adsorption of the tracer on the particle surface. It is quite possible that the correct plotting coordinates have not been found. In this study, the injection problem was eliminated by the use of two measuring positions. A tracer was chosen such that it 65 did not adsorb on the glass spheres. Since the axial dispersion coefficient was independent of column diameter, there were probably no wall effects. CONCLUSIONS The spread of a pulse input, as it travels through an adsorbing section of a packed bed as in chromatography, is the sum.of the spreads due to axial dispersion and to adsorption. The two restrictions on this are that the input be "hump-like", and that the mass transfer process be described by a mass transfer coefficient. There is no effect of column diameter on the axial dispersion coefficient in the range of LImu/Dv from 0.5 to 80. In this range, the axial dispersion coefficient is proportional to the particle diameter. The results of this investigation are described by a constant modified Peclet Number (PE = 0.22). The particle diameter effect substantiates the findings of Ebach and Whiteeo. The ratio of the viscosity of the pulse to that of the solvent can be as high as 100 and not affect the axial dispersion coefficient in a six inch column. It is concluded that viscous fingering has no effect on the dispersion coefficient for column diameters up to six inches, and for flow velocitieS'within the range of 0.1 to 10 ft/hr. The column length for short columns has no effect on the axial dispersion coefficient. This is a prime requirement for using the dispersion model, and has been verified by otherinvestigatorsl6’20’38 for long columns. The dispersion coefficient is independent of column length for short beds as used in this investigation, because the experimental technique of two measuring points removed the criterion that the pulse was required to fit a normal curve as is the case for one measuring point. 66 .APPENDIX I The Reynolds Method of Treating_Turbulence or Eddies7. Consider the case of one dimensional fluid flow through a packed bed. The equation of continuity of species A is: 2 8 C3 8 01(3 ) 8C1 D A _ A _ A (I-l) v 35,—2— T— I 7)?— In turbulent motion CA and u are rapidly oscillating functions of time. Let a bar over a quantity represent its average value. The concentration at any time is given by: CA:EA+CA (I-Z) where CA is a time dependent concentration fluctuation. Thus, EA :0 Similarly, u=E+u' (I-3) and If = 0 where u = the velocity 6' = time average velocity u' = velocity fluctuation. Inserting the values for CA and u into equation CDéD, one obtains: 82(EA+CA) a [(6A +91.) (E + u')] V - : ax2 OK I) (I-4) 8(EA+CA) 6t 67 68 By time averaging this equation, one is left with: 2— — ' , — D acA 8(CAu) 8(CAu) _ acA (15) v 79;?— " T)?— ' —5'X—— " ‘5?— ' Then, CA u' is the turbulent flux and it can be seen by analogy with Fick's law of diffusion that: BCJA and _I BE D(eddy) = - c u'/ A (1-7) Thus, the equation of conservation may be rewritten with the previous analogy: z_ _"_ 8 CA 8(u CA BCA [Dv + D(eddy)] _TBX - T = T—t (I-8) and the total axial diffusivity is made up of two terms, the molecular diffusivity, and the eddy diffusivity. APPENDIX II gplanation of Pulse Skewness. The solution to the differential equation31: D 32c : ac (m1) 3;? 8t— where Z = X - u t (II-Z) for an infinite bed with a tracer injected instantaneously at X = 0 is: ‘ CV exp [-(1 - goal/4 (2,1 ”(5%) T : (II-'3) 2 «L: (36. t) D/uL where = the volume of the bed considered a amount of tracer injected volumetric flow rate of solvent :- the value of X where curve is obtained dh£o< n time . Figure 18 is a plot of %’ versus th for several values of D/uL. 69 2.0 3 1.0- Figure 18. Plot of CV/Q versus F. t to Show Skewness. V 70 71 The concentration approaches a Gaussian distribution for low values of D/uL, but for high values, the curves become skewed. For small values of D/uL, equation (3) can be approximated by: VVt 2 CV _ 1 exp[-(l-T',—) /4D/uL T — 2 V17 D/uL (II-4) which is a normal distribution. By differentiation of equation (3) with respect to t, and setting the derivative equal to zero, the time when the maximum concentration appears is VV _ = ‘ Z - _ V tm NHD/uL) +1 D/uL (II 5) ' VVt Thus, the maximum concentration always occurs between values of 17- of 0 and 1. For small values of D/uL, Y%; is about 1, while for larger values, it is significantly less than 1. APPENDIX III The solution to equation (101+). IQ+1 XIV) Of‘cf (t-s)N exp [- 2"]! (t-s)] ds (104) CN: NT(— The solution for a delta pulse function input, that is with the condition: (Si 0 < t ‘< 6 f: =0 t<0andt>8 C3 C3 is a tabulated integral. The solution to this equation is VV Tq-l ((33—11: exp[-vTv/.(t-6)] [¥(tfi?)]N {Pig-:16; +...+l i O - 0 - exp[-¥t] [kgfiN/N! + +3 (III-l) This is the same result obtained if one superimposed two step inputs 3 the first step followed after time 5 by a solvent step. Equation (1) is simplified by expanding the (t-6)N terms according to the binomial expansion along with expanding the exp[%.-v 6 ] term, and then dropping all second and higher order terms of 6. With this approximtion , there results CN N+l t—= [(vvvt) fig] {5— exp[-¥t1 (III-2) 1 72 73 and 01 6 is replaced by oh: and then N T- = (Vivi) N1,- exp[--¥,‘—’t1 (III-3) 7h .mIHQ can .mnNQ .NIHQ mo 25.?» ommaodwm on? n >mQ N s" (39$?on on .230 wait was «new 93 was 6.35 can Omooom .mcooom pan mum on: smegma. pseudo on”. 80.3 woumHsono can mIHQ mam .MINQ .NIHQ H "w awooo.o MPH .O M.© In I: wVVOOOAV mw¢.0 MMH .O OVO0.0 mwm.o N.N~ II II HOOO0.0 ©~.N mwm.o ovoo.o MON .0 ¢.®N ~NO0.0 ~NO0.0 ~NO0.0 ©©.N NP.O ¢moo.o Nod .o ©.OV In In ®NO0.0 ¢®.N m0>.0 NH0.0 MNN.O méwo In In ©¢O0.0 ¢.© fish; bao.o NNN .o N (In. mmoo.o mmoo.o Vmoo.o m.® omd NN0.0 Owfi .O 06*; mo~o.o NOH0.0 ~®O0.0 N.NH amim mvo.o ofiN.O OJJVN mufifioéu ®QH0.0 ®NH0.0 m.MN mméw «3.0.0 ¢~N.O O.HwN OON0.0 ~®H0.0 OHN0.0 QAwN ma .N. 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Gas—Hoe goawuo H3 3mm > cash. 78 h000.0 KEN .0 ~N00.0 00.0 m-o.o oo~.o vmoo.o mw.~ fiwdo.c ~m~.o mooo.o po.~ oo~o.o me.o ~ooo.o do.m oomo.o mo~.o ohoo.o do.m wo~o.o mv~.o oooo.o mo.m vpuo.o ~w~.o ammo.c ¢.~H 123.4 v min? him: Q hit : imam a m 4 330 N mwmmfi mmmfim £33 :00. £8530 gunwuo mmgnw mm H062 mo oumuH .m :33 cofidfiom Hmmnm ommomfldmu 00m new mama H> 03MB 79 NOMENCLATURE Cross-sectional area of tube. Area of solids/volume. Nbrmalized concentration (defined by equation 10). Concentration of component A mol/vol. Concentration of A at the solid interface. Concentration of A on the solid (mols/vol. solid). Axial dispersion coefficient in adsorbing section. Axial dispersion coefficient in non-adsorbing section. Molecular diffusivity of solute. Particle diameter. Tube diameter. Distance between measuring points. Mass transfer coefficient (mols ft/time Conc. driving force). Equilibrium constant between solid and liquid (defined by equation 5). Hydraulic radius for packed bed Laplace transform variable. Total amount of solute injected. Time. Interstitial velocity. Volumetric flow rate. Distance variable. Incremental time for pulse to be introduced. 80 81 Void fraction. Distance from second measuring point to end of adsorbing sector. The mean time of C-t at Xm. The mean time of C-t at X0. Viscosity of solvent. Spread of pulse (defined by equation #6). Spread of pulse due to dispersion (defined by equation #7). Spread of pulse due to adsorption (defined by equation #8). Density of solvent. The variance of C-t at Xm. The variance of C-t at X0. 10. ll. l2. 13. 1h. 15. 16. BIBLIOGRAPHY Aria, R., Chem. Eng. Sci., 11, 19h (1959). Aria, R., Chem. Eng. Sci., 9, 266 (1959). Aria, R., Chem. Eng. Sci., ll, 19h (1959). Aria, R., Proc. Roy. Soc.,.gggg, 63 (1956). Aria, R., Proc. Roy. Soc., @232, 268 (1958). Aria, R. and Amundson, N. R., A.I.Ch.E. Journal, 12 280 (1957). Bird, R. B., Stewart, W. 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