mi 129 masts I m» . gar MiCkI‘Em’m mate % University This is to certify that the thesis entitled MODELING OF ENZYME DEGRADATION IN STIRRED TANKS presented by Michae] Douglas Waite has been accepted towards fulfillment of the requirements for M. S. degree inChemi cal Engineering dang/a W Major professor Date May 17, 1984 0-7639 MS U is an Affirmative Action/Equal Opportunity Institution IVISSI_} RETURNING MATERIALS: Piace in book drop to mummies remove this checkout from “ your record. FINES win be charged if book is returned after the date stamped below. MODELING OF ENZYME DEGRADATION IN STIRRED TANKS By Michael Douglas Waite A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Chemical Engineering 1981 ABSTRACT MODELING OF ENZYME DEGRADATION IN STIRRED TANKS by Michael Douglas Waite Enzymes are known to lose activity when exposed to shear forces in turbulent flow systems. This thesis is an attempt to predict the deactivation of catalase in a stirred tank. The method requires that the shear dependence of the degrada- tion kinetics be determined. The enzyme is subjected to the pure shear flow generated in a couette viscometer. Spectro- photometry is used to follow the activity loss and determine the shear degradation kinetics. The extrapolation of the knowledge of shear inactivation in pure shear flow to a stirred tank is hindered by the lack of basic understanding of turbulence in stirred tanks. Various hypothetical models for a shear probability distribution func- tion are proposed. Using these distribution functions and the shear degradation kinetics, rate expressions for the degrada- tion in stirred tanks are developed. The predictions are very sensitive to the choice of shear distribution function. A successful fit is achieved using this method with a distribution function as follows: {-1n( 5 )}2 3 s F _ max 3 s (S) S 2 S max {1n(—§F—)}3 3 So ln( :ax) o SofiS§Smax US$580 To my wife Retha ii ACKNOWLEDGMENTS Sincere appreciation is extended to Professor Donald Anderson, who suggested this area of research, and for his support throughout the course of this work. I would like to acknowledge Dr. Carl Beck, whose enzyme studies preceded this and laid the groundwork for my research. Gratitude is also expressed to Professor Charles Petty, whose interest and suggestions were of great value. iii TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . . . . LIST OF FIGURES. . . . . . . . . . . . . . NOMENCLATURE . . . . . . . . . . . . . . . . I. INTRODUCTION . . . . . . . . . . . . . . II. BACKGROUND. . . . . . . . . . . . . . . III. EXPERIMENTAL EQUIPMENT . . . . . . . . A. B. Viscometer . . . . . . . . . . . Stirred Tank . . . . . . . . . . . IV. EXPERIMENTAL METHOD . . . . . . . . . . A. B. Assay Method . . . . . . . . . . . Procedure. . . . . . . . . . . . . V. EXPERIMENTAL RESULTS . . . . . . . . . . A. B. C. Viscometer O O O O O O O O O O O O Stirred Tank . . . . . . . . . . . Improved Activity at Short Times . VI. THEORETICAL ANALYSIS. . . . . . . . . . A. DeveIOpment of Stirred Tank Rate Expressions. . . . . . . . . . . . Several Distribution Functions and Expressions. . . . . . . . . . . . iv Page . vi viii 11 U: 18 22 22 26 29 32 32 33 VII. A. Comparison of Mathematical Models and DISCUSSION . . . . . . Experimental Results . B. Possible Degradation Mechanisms. C. Suggestions for Further Work . VIII. APPENDIX A: APPENDIX B: INELRXfiFPHY. CONCLUSIONS . . . . . Viscometer and Tank Dimensions. Tabulated Data. 47 47 49 50 52 55 57 72 LIST OF TABLES Molecular Activity of Some Enzymes Commercial Uses of Enzymes . . . . Kv Versus Shear Rate . . . . . . . Kt Versus (P/V)% . . . . . . . . . Summary of the Mathematical Models vi Page 25 25 45 “I I 0 8a. 8b. LIST OF FIGURES Schematic of the Coaxial Cylinder Couette Viscometer . . . Schematic of the Stirred Tank. -ln ([S]/[Si]) Versus Time for the Breakdown of Hydrogen Peroxide . -ln (k/ki) Versus Exposure Time in the Viscometer . . . . . ln (K - K ) Versus ln(s) in the s so Viscometer . . . . . . . t Measured Torque Versus Impeller Speed The Linear Distribution Function for B = O, 1, 2 . . . . . . A Schematic Representation of Models 2 Through 5 . . . . . . t and the Stirred Tank Data vii 1/ K Versus (P/V)2 . . . . . . 1 1 K Versus (.‘7./1.i/"')(P/V)/2 for Each of the Models Page 10 12 17 24 27 28 35 38 48 [e] F'F(s) k! k! K V0 NOTATION enzyme activity constant constant constants constant enzyme concentration shear probability density function constant H202 decay reaction rate constant H202 decay reaction rate constant with enZyme concentration initial rate constant for H202 decay enzyme degradation rate constant due only to shear enZyme degradation rate constant in the stirred tank enZyme degradation rate constant in the viscometer initial viscometer enzyme degradation rate constant height of control volume for Model 5 constant order of enzyme degradation reaction with respect to shear rate speed of outer cylinder viii units/cm3 sec sec2 ug/ml SEC ml'l pg mM sec2 mM sec2 mM sec2 min CID min P power input to tank erg/sec Q geometrical constant r radius of control volume for Model 5 cm rg radial position within the visco- cm meter gap RO inner radius of outer cylinder cm RS rate of enzyme degradation due only actiyity cm3 sec to shear . . @1113]. Rt rate of enZyme degradation in tank cm3 sec Rv rate of enzyme degradation in the activit viscometer cm3 sec R initial viscometer degradation rate 22%AZLLX' vo cm sec 5 shear rate sec--l [S] substrate concentration mM [Si] initial substrate concentration mM mean shear rate sec-1 <32) variance in shear rate sec."2 5 maximum shear rate sec-1 max 50 intermediate shear rate sec—l t time of assay sec Tq torque erg v control volume for Model 5 cm3 V tank volume cm3 Z t' l't con tant sec prOpor iona l y S TIE—n- GREEK SYMBOLS 8 shear eXposure time viscosity gamma function angular velocity min poise radians/sec INTRODUCTION Enzymes are high molecular weight proteins that exhibit bioloqical catalytic activity. As catalysts they are remarkable, often accelerating reactions by a factor of 108 to 101°. The extremely complex three dimensional structure of enzymes is necessary for their effectiveness and specificity. The conformation of enzymes is unique to each species and depends upon the peptide bond "backbone" and the interaction of hydrogen bonds and van der Waal's forces act- ing as cross members. Enzyme structure can be effected by pH, temperature, chemical inactivators, microbial contami- nants or mechanical forces induced by shear or possibly elongational flow fields. The peptide bonds are resistant to cleavage, and are prevented from rotating by the hydrogen bonds and van der Waal's forces. These weaker bonds can dissociate more easily resulting in a loss of catalytic activity or even denaturation (unraveling). Although enzymes are fragile, renaturation and a renewal of activity is possible and often rapid so that only those enzymes which have been irreparably damaged contribute to the loss of activity. A number of industrial applications have been found for enzymes but permanent degradation of the active site pre- vents exploitation of many potentially beneficial enzymes. The deactivation of enzymes in a shear field would be un— avoidable in any large-scale chemical process since all mixing or pumping operations involve shear forces. The problem has been studied by a number of investiga- tors. Much of this work concerned immobilized enzymes used in fluidized or fixed bed reactors (6, 14, 15). There are many commercial applications that might require the use of free enzymes in solution such as in the conversion of cellu- lose to glucose, production of some pharmaceuticals, or in the removal of hydrogen peroxide from milk during cheese processing (16). One of the first investigations of shear deactivation of free enzymes was performed by Charm and Wong (7). The authors reported activity losses for the enzymes rennet, catalase, and carboxy peptidase when exposed to known shear rates in a couette viscometer. Additional work was com- pleted by this group in which different flow systems were examined and activity loss was predicted based on the known response to the pure shear field (9). Studies by Tirrell (19, 20, 21) modeled the degradation of urease and lactic dehydrogenase using the shear stress rather than shear rate. Work by Beck (3) attempted to pre- dict the response of the enzyme catalase to the turbulent 3 shear eXperienced in a standard stirred tank. Couette viscometer data provided information about the rate of degradation in a pure shear field. This study will attempt to validate the preposed rate expression, where, K = an 2 RS is the rate of enzyme inactivation in a shear field, K8 is the enzyme degradation reaction rate constant, 5 is the shear rate, and n is an unknown constant. These expressions imply that an enzyme solution exposed to a constant shear field will lose activity at a rate proportional to the con- centration of active enzyme. The rate constant is a function of the shear rate and a plot of ln(Ks) vs. ln(s) gives the order on the shear rate n, as well as the constant of pro- portionality 2. It would be useful to be able to extrapolate knowledge about shear inactivation in a couette apparatus to stirred tanks or other equipment with unknown shear structures. Various hypothetical models for the distribution of shear rate magnitudes are prOposed here for stirred tanks and are tested against eXperimentally determined deactivation rates. For an arbitrary shear distribution function the rate of degradation in a tank is given by Rt = fRsF(s)ds 3 where F(s) is the shear probability density function. The integration is performed over the domain of F(s)' and results in an enzyme degradation rate eXpression. It is proposed that one (unknown) parameter can be evaluated in terms of the rate of energy dissipation for shear flow given by dP_ 2 aV'US 4 where u is the viscosity of the fluid and P is the power input. The final rate eXpression is of the form where G is a dimensionless constant and the enzyme degrada- tion rate constant in the tank is given by - 5 P15 u Each of the proposed rate expressions are checked against degradation rate data taken from the stirred tank. BACKGROUND EnZymes are polymers of the a-amino acids. They demon- strate biological catalytic activity and are essential to all life forms. The most common unit of enzyme activity is defined as the amount of enzyme which causes transformation of one micromole of substrate per minute at 25°C under Optimal conditions of measurement. The catalytic ability of some enzymes is given in Table 1 (11). Table 1. Molecular activity given in moles of substrate transformed per mole of enzyme per minute under Optimal conditions EnZyme Activity; Carbonic Anhydrase C 36,000,000 Catalase 5,600,000 8 Amylase 1,100,000 8 Galactosidase 12,500 Phosphoglucomutase 1,240 Succinate Dehydrogenase 1,150 The great activity of enzymes is the result of a highly refined structure which has four major levels. Primary structure refers to the covalent linkage and sequence of the amino acid backbone. The amino acid residues are bound head to tail by peptide bonds. Secondary structure refers to the recurring arrangement in space of the polypeptide 5 chain along one dimension. These chains may have a longitu- dinally coiled or extended conformation. Tertiary struc- ture refers to the folding or bending of the polypeptide chain in three dimensions to form the compact structure of the globular protein. Quarternary structure refers to the arrangement of chains in relation to one another. Typical large proteins contain two or more polypeptide chains or subunits which are not covalently linked. In addition to the covalently bound sequence of amino acids, the effects of hydroqen bonding, van der Waal's forces, electrostatic forces, hydrOphobic forces, and intra- molecular covalent crosslinks also play an important role in the native conformation of a protein molecule. This structure is almost always identical in every molecule of that protein. Enzymes or other proteins whose function involves binding small molecules, have as an essential part of their structure an "active site". The ultimate purpose of the three dimensional form of en2ymes is to generate these active sites so that those groups responsible for binding and catalysis are apprOpiately positioned. Obviously, there is considerable interest in eXploiting the catalytic prOperties of enzymes. Many commercial pro- cesses currently use enzymes and many more processes which require moderate to extreme conditions of temperature, pressure, pH, etc. could become increasingly economical. Several notable examples of the practical uses of enzymes are listed in Table 2 (16). Table 2. Commercial Uses of Enzymes Enzyme Typical Uses a-Amylase Textile desizing; starch liquefaction; glucose production. Invertase Production of confections such as soft- Pectic enzymes Celluloses Bromelain Papain Trypsin Rennins Lipases Pancreatin Glucose oxidase Catalase Glucose isomerase center candies. Clarification of fruit juices and wines. Digestive aid; reduction of viscosity of vegetable gums such as those in coffee. Digestive aid; anti-inflammatory prepa- rations; meat tenderizer. Meat tenderizer; chill proofing beer. Digestive aid; leather bating. Curdle milk in cheese formation. Digestive aid; waste disposal; alter flavors by modifying milk fats. Digestive aid. Removal of oxygen from food products; desugars eggs; diagnostic aid (glucose in diabetes). Removal of hydrogen peroxide when used for sterilization, especially in milk. Production of high-fructose corn syrups. The enzyme used in this study is bovine liver catalase. It is a globular protein classified as a metaloenzyme with a molecular weight of approximately 250,000. Catalase is high in sulfhydryl groups and has extensive areas of hydro- phobic random coil. Its high molecular weight and chemical prOperties are believed to be responsible for the relative sensitivity to shear modification. Catalase has a distinctive absorption spectrum due to the presence of an iron-porphyrin prosthetic group as part of the active site. Upon mixing with the substrate hydro- gen peroxide, catalase exhibits a transient change in absorption characteristics reflecting the formation and decomposition of an enzyme-substrate complex (2). It is believed that the free iron ligand bonds the hydrogen peroxide directly as a prelude to catalysis. EQUIPMENT A. Viscometer The shear field generator used in this study was con- structed by Beck (3), Figure 1. A couette viscometer is capable of shearing a relatively large amount of fluid and is readily modeled. The outer cylinder is rotated at a constant rate while the inner cylinder is stationary and acts as a heat sink for the sheared fluid. The shear rate experienced by any differential volume of fluid in the viscometer is a function of radial position rg, and turning rate of the outer cylinder in revolutions per minute N. Shear is a maximum at the outer wall and de- creases to a minimum at the inner wall. Since the gap is very narrow, the shear rate is nearly constant throughout the fluid. An average shear rate E can be found by (N)’ integrating the shear for any differential volume over the radial position and dividing by the average radius. In cylindrical coordinants this becomes - fs(r'N)rgdr S 7 (N fr dr ) g 9 An expression for 5(r N) can be obtained from the equation I of motion (4). _ 1m R02_Qf_ 3mm “ 2 332—21422 8 9 10 in h tap water 20°C sampling J port drive L,J.shaff Schematic of the Coaxial Cylinder Couette FIGURE 1 . Viscometer. Scale is %, Q=0.9865 11 where the quantity (UN/30) is the angular velocity in reciprocal seconds, R0 is the inner diameter of the outer cylinder, and Q is the ratio of inner gap wall radius to outer gap wall radius. The average shear rate can be eXpressed as R0 NN R02 Q2 f 2 '36 “a" 1-92 " QRO g 9 50:) R0 f r dr QRo g 9 Integration gives, - __ '1 8(N) — 6.859 N (sec ) Thus the average shear rate within the viscometer can be determined directly from the known Speed of the outer cylinder. B. Stirred Tank A schematic diagram of the stirred tank appears in Figure 2. The tank is made of stainless steel and is industrially standardized with geometrical relationships as follows (10). Four baffles are used, one twelfth the tank diameter in width . Fluid depth equals tank diameter . A radially discharging six blade turbine is used 12 -4 A‘_‘-‘_A‘AMM~ MW- .— A 7 Av v— - ‘ ‘va-hV ;' FIGURE 2. Schematic of the Stirred Tank Dimensions are specified in the Appendix. l3 . Impeller diameter is one third the tank diameter . Impeller is located one third of the tank diameter from the bottom . Impeller blade width is one fifth of the impeller diameter . Impeller height is one fourth of impeller diameter The temperature of the apparatus was maintained at twenty degrees centigrade with a constant temperature room. At high impeller speeds, the heat generated by viscous dis- sipation required the use of an external constant tempera- ture bath. Due to the turbulence within the tank, a drive system with good speed stability characteristics was required. A Master Controller Servodyne Drive System was used for this purpose. The motor-generator provides the impeller torque and a feedback signal to the controller which is compared to a reference. The controller then adjusts the power to the motor windings to maintain a constant output of speed. This system provides a check on the torque measurements as well as a constant impeller speed. A dynamometer was used to measure the power input. The product of the torque arm radius and the force measured with a known mass is the torque transferred to the wall of the tank through the fluid by the impeller. The power input per unit volume is 14 = TqQ v 10 m eXpI 2 '5 } 39 2 where <52> is the second moment or variance of 5. Notice that no constants must be evaluated and that the domain of F is from negitive infinity to positive infinity, Figure 8b. This function can be substituted into equation 3 to give -52 _ m exp{—————} Rt - f -s Zas «asT; /7r 2 55 4° Removing constant terms from the integral and solving re- sults in a rate expression in terms of the second moment R = —3— 7 Za 41 t Vin Since by definition (52> = fstds 42 38 5 5 Model 2 Model 3 S o 3 Model 4 Model 5 FIGURE 8b. A Schematic Representation of Models 2 Through 5. 39 one can show by equation 32 that for any arbitrary distri- bution function 2._.P_ <5 > V 43 T: This result is substituted into equation 41 to give the rate eXpression 2 Z P k R = (—) a 44 t ,7? “g V so that 2 Z P 8 K = - (-) 45 t ,7; us V Since there is no artificial end point for this distribution (e.g. S ), then the resulting rate eXpression has no ad- max justable parameters. 3. Exponential Form A simple and intuitive distribution function predicts that low shears will be more abundant, and that the proba- bility distribution decays exponentially with the magnitude of s. This model is shown in Figure 8b and can be written = -cs F(s) ce 46 The constant can be evaluated by finding the first moment (or mean) of shear rate. = fjstds 47 40 By substituting equation 46 the mean Shear can be found in terms of c. 1 <5) — E 48 The distribution function then becomes _ l s P(S) — 2'5 €Xp( ) 49 Substituting F into equation 3, the resulting rate eXpres- sion is given in terms of the mean shear rate. Rt = I(2)Za 50 Once again, the energy dissipation eXpression can be used to determine in terms of P/V. _==l52]=‘ . . d 1 11V (5) S 5 Solving the integral gives (52> = 2(5)2 52 By substituting equation 52 into the rate eXpression from equation 50, one can show that Z P % R = (—) a 53 Q V u 4. Loq-Normal Form The flow pattern in a stirred tank has been fairly well established (10). While basically radial in nature, the 41 presence of the baffles often causes regions of relatively stagnant fluid at very low shear rates. The fraction of fluid is normally low in these stagnant pockets suggesting that the distribution function will reach a maximum at some moderate rate of shear. The following model shown in Figure 8b provides for this type of behavior. =t _l2 F(S) C 5 exp( C s ) 54 The constant can readily be determined in terms of the second moment to give a distribution function of the form F ) 55 eXp(- 2 2 (S)= Substitution into equation 3 and integration results in a rate expression given by E_ 8 “l3 P t (v) a 56 T: 5. Composite Form According to Holland and Chapman (10), the shear rate can be modeled as decaying exponentially from the center of the tank. This model can be eXpressed as U) II r smax exp(-mi) 57 where r is the distance from the tank axis and R is the tank radius. The volume contained in some radius r is, = 22 58 V(r) Tl'r 42 where i is 2r. The tank volume is, V = nR22R 59 The ratio of r/R can now be exPressed as a function of v/V. r/R = 9v7V 60 The constant m can now be evaluated for the tank wall condition where v/V = l and s is equal to some wall shear rate so. Solving for m gives, 80 m = —ln(s ) 61 max At a specific shear rate (or radius) v is given by s -ln(smax) v = v 62 (r) _ln(SSO ) max Using equation 33, one can write _ l dv where the differential change in volume with respect to shear rate is, 1n( 3 ) dv d 5max 3'? = 'a's—‘I-WIV 64 1n( s ) 0 or dv {1 ( maxl}2 -s 8max -—-— = 7,". ds .1 ( 2 ) 65 43 so the distribution function becomes, 5 2 F = 66 (S) s {REE-ii)” S So VET—I 2 0.799 s 3 exp{-} 0.707 s 82 4 §?§3;-GXPI’EZ§3;1 0.627 3 {-ln(s : )}-’- max 8 {ln( max)}3 s so So 72 Recall that for any distribution function < 2> = E_ 5 UV 43 so that the rate eXpression becomes, _ ZP Rt - UV a 73 Hence, for a second order dependence on the shear, the predicted rate of degradation in the tank is not dependent upon the form of the shear distribution function. This would seem to imply that the enzyme degradation kinetics as determined by the couette experiments can be extra- polated to any turbulent flow system. Only the rate of energy dissipation and viscosity of the system would be required to characterize the rate of enzyme activity loss. DISCUSSION A. Comparison of Mathematical Models and Experimental Results Each of the models presented above results in an over- all enzyme degradation reaction rate eXpression for the stirred tank which has the form, where (3)3" 75 The models vary only in the dimensionless constant of pro- portionality G. Recall that the viscosity u is known, the value of Z is determined eXperimentally, and the power input is ob- tained from torque data. The value of G can then be calcu- lated from the eXperimental results by plotting Kt vs. the quanity £8 (5-7. Figure 9 Shows this plot for the experi- mental rgsults and each of the prOposed models. In general the predicted rates of activity loss are somewhat high. As expected, those models which are more carefully develOped result in a progressively better fit to the data. Model 5, which contains an adjustable parameter, fits the data well when Smax/So is 37.33. The implication is that a shear rate of roughly 2.7 percent of the maximum will be most commonly found. An important result of this study is 47 48 0.0/8“ B=0 =l B=2 ’ 2 3 0.0/2l .4 K. (min") 1% 0.0063‘ . 5 o b O o () —¢* ; :4 +*:. .: sis ~¢s if t 0 t .-: . l (Z/uMP/vfilmm) 00" FIGURE 9. Kt vs. (Z/ukHP/V)15 for the models and for the stirred tank data. 49 that predicted reaction rates using couette data are indeed sensitive to the distribution of shear rates in the tank. B. Possible Degradation Mechanisms A number of scenarios have been developed to describe the modification of protein structure or function at the molecular level. Much of the work is isolated and provides no common basis from which a generally accepted mechanism of shear modification can be developed. Early research by Joly and Barbu (l) examined low con- centrations of horse serum albumin and tobacco mosaic virus. They found that for low shears in a couette viscometer, there was an increase in the effective particle length as measured by flow birefringence. According to the authors, at the molecular level shearing increases the collision frequency of the particles, promoting aggregation and in- creasing apparent length. This phenomena was explained by the collision coagulation theory of Smoluchowski (l7) modi- fied to account for varying interaction strengths between particles. One may infer that the loss of enzyme activity could be caused by this shear-induced aggregation effect. At higher shears (above 2000 sec-1), the aggregates were ruptured but the authors did not investigate the further action of shear on the particles. There is evidence in the medical literature for plasma protein denaturation in flow through extracorporeal devices. 50 The strong intermolecular forces at blood-gas interfaces have been shown to cause protein denaturation (22). This group claims that the presence of blood-solid interfaces may also cause damage to blood proteins. Studies of flow induced erythrocyte damage Show that near solid boundaries, shear stresses of one Pascal or less can cause erythrocyte lysis (12). In the absence of walls, the critical shear stress for lysis has been estimated to be 6000 Pascals in liquid-into-liquid jet experiments (5). The validity of this comparison must be questioned in light of the great difference in the method of applied stress. Within the last ten years, considerable work has been done with specific enzymes which have been fairly well characterized. The results of much of this work imply that hydrodynamic forces are not directly causative of protein modification but act to reduce somewhat the activation energy for the degradation reaction (20). The energy of the turbulent shear causes tension in some bonds making them more labile and resulting in a less stable protein structure. C. Suggestions for Further Work The structure of the turbulence within a stirred tank is not well known. A better understanding of this structure would provide a valuable tool for the modeling of hydro- dynamically related phenomena. The structural modification of biological compounds in a stirred tank is ultimately 51 caused by the dissipation of energy. It is known that most of the kinetic energy of an agitated fluid is dissipated at the smallest scales of turbulence, and that extensional flows may become an important or even the predominant mechanism of energy dissipation. In order to more accurately predict activity losses and describe protein deactivation at the molecular level, it may become necessary to understand the sensitivity of enzymes to extensional flow as well as Shear flow. Although devices have been designed which produce a pure elongational flow (4), they may be difficult to model as a continuous flow system. The couette apparatus can be used to generate Tay- lor vortices which contain both shear and extensional flows but are readily modeled. CONCLUSION The primary goal of this thesis is to provide a relia- ble method for the prediction of the rates of enzyme acti- vity loss in a turbulent system of practical value, e.g. a stirred tank. First the response of enZyme activity to a pure shear field is determined experimentally in a couette viscometer. Then, if the stirred tank turbulence is modeled as a collection of differential packets each experiencing a distinct shear, the couette kinetic data can be used to pre- dict the total rate of activity loss. Turbulence research has as yet not established the distribution of shears in a complicated system like a stirred tank. Consequently, various hypothetical functions are prOposed to characterize the shear distribution. In each case, the form of the predicted rate expression re- duces to, R = G E (:37)!5 a 76 u t 5 where the rate constant is given by (3) ’5 77 K = G V t t [N \ The models differ only in the form and value of the con- stant G. In reviewing the forms of the distribution functions diagrammed in Figure 8, one may conclude that the shear 52 53 probability function reaches a maximum at some intermediate shear rate. For the tank used here, a shear rate equal to 2.7 percent of the maximum was predicted to be most common. Although this model seems to provide an adequate descrip- tion of shear for enzyme degradation, it would be unwise to assume an a priori knowledge of the shear distribution for use in other shear sensitive operations (e.g., mixing). Rate data were taken in two geometrically similar tanks. The resultant rate expressions are functions of enzyme sen- sitivity to shear, viscosity, and power per volume. Shear sensitivity varies with temperature and between species. These results seem to indicate that a useful scale-up cri— terion can be based on the power input. This would elimi— nate the necessity of first predetermining the degradation characteristics in a coaxial device for each tank volune change. A transient enhancement of activity was noticed for both the viscometer and the stirred tank. It is believed that aggregates of enzyme molecules are broken up by the shear force resulting in an increase in the initial activity. Increases of two to ten percent were found. Data taken in a smaller, geometrically similar tank, support the findings in this work. In both tanks, the rate of enzyme degradation has been shown to be proportional to the square root of power per volume. It is suggested that this observation can be used as a criterion for scaling to other sizes of stirred tanks. APPENDICES 54 APPENDIX A Viscometer and Tank Dimensions 55 APPENDIX A Viscometer Dimensions (cm) Inner diameter of outer cylinder . . . . . . . . . 11.8491 Outer diameter of inner cylinder . . . . . . . . . 11.6891 Outer cylinder height . . . . . . . . . . . . . . 20.10 Inner cylinder height . . . . . . . . . . . . . . 25.6 Gap width . . . . . . . . . . . . . . . . . . . . 0.08 Gap volume . . . . . . . . . . . . . . . . . . . . 70 cm Stirred Tank Dimensions (cm) Tank diameter . . . . . . . . . . . . . . . . . . 21.0 Fluid depth . . . . . . . . . . . . . . . . . . . 21.0 Baffle width . . . . . . . . . . . . . . . . . . . 1.75 Impeller width . . . . . . . . . . . . . . . . . . 7.00 Blade width . . . . . . . . . . . . . . . . . . . 1.40 Blade height . . . . . . . . . . . . . . . . . . . 1.12 Impeller height from bottom . . . . . . . . . . . 7.00 Shaft diameter . . . . . . . . . . . . . . . . . . 1.61 Tank volume . . . . . . . . . . . . . . . . . . 6670 cm 56 APPENDIX B Tabulated Data Tabulated Data APPENDIX B Torque vs. 57 Impeller Speed Mass (gm) N Torque X10"7 erg 0 125 -0- 10 175 0.1 20 225 0.2 30 270 0.3 50 325 0.5 80 440 0.8 100 495 1.0 120 545 1.2 170 655 1.7 200 700 2.0 220 750 2.2 270 830 2.7 300 900 3.0 320 925 3.2 350 975 3.5 370 1020 3.7 395 1070 3.9 455 1150 4.6 475 1190 4.7 500 1210 5.0 550 1315 5.5 600 1430 6.0 650 1500 6.5 700 1590 7.0 750 1650 7.5 800 1725 8.0 850 1790 8.5 895 1900 8.9 925 1920 9.2 975 2020 9 7 58 Viscometer N = 0 rpm 8 = 0 sec".1 Kv = .001027 min"1 Assay t 6 k' 1 12:15 0 .019263 2 12:19 4 .02129 3 12:23 8 .02085 4 12:27 12 .02222 5 12:30 15 .02091 6 12:33 18 .02108 7 1:15 60 .01984 8 1:25 70 .02001 9 1:35 80 .01948 10 2:40 145 .01833 11 2:43 148 .01659 12 2:46 151 .01775 13 2:49 154 .01770 14 3:35 200 .01698 15 3:38 203 .01782 16 3:42 207 .01781 17 3:45 210 .01679 59 Viscometer N = 833 rpm 3 = 643.0 sec"l KV = .00131 min"1 Assay t 9 k' 1 1:37 -13 .022546 2 1:41 - 9 .019823 3 1:44 - 6 .020030 4 1:50 0 .019360 5 1:55 5 .019904 6 2:00 10 .019945 7 2:10 20 .018998 8 2:20 30 .018337 9 2:50 60 .017362 10 3:20 90 - 11 3:50 120 .016387 12 4:20 150 .015314 13 4:55 185 .015316 14 5:20 210 .014877 15 5:25 215 .014950 16 5:30 220 .015124 17 5:35 225 .014760 60 Viscometer N = 166.7 rpm 5 = 1286 sec"1 Kv = .00139 min-1 Assay t 8 k' 1 9:50 -15 .01629 2 9:54 -11 .01690 3 9:58 - 7 .01661 4 10:05 0 .01669 5 10:10 5 .01827 6 10:15 10 .01739 7 10:20 15 .01679 8 10:25 20 .01646 9 10:35 30 .01606 10 11:05 60 .01469 11 11:35 90 .01451 12 12:05 120 .01494 13 12:35 150 .01222 14 1:05 180 .01460 15 1:35 210 .01235 16 2:05 240 .01324 17 2:35 270 .01145 61 Viscometer N = 250 rpm 5 = 1929 sec-1 Kv = .001628 min-l Assay t 8 k' 1 11:10 -10 .02200 2 11:14 - 6 .02073 3 11:17 - 3 .02268 4 11:20 0 .01911 5 11:25 5 .01918 6 11:30 10 .02068 7 11:40 20 .01524 8 11:50 30 .01664 9 12:20 60 .01600 10 12:50 90 .01540 11 1:20 120 .01481 12 1:50 150 .01442 13 2:20 180 .01398 14 2:50 210 .01336 15 3:00 220 .01306 16 3:10 230 .01370 17 3:20 240 .01232 62 Viscometer N = 333.3 rpm 5 = 2572 sec"l Kv = .001860 min-1 Assay t 8 k' 1 11:31 - - 2 11:40 -22 .01876 3 11:44 -18 .01776 4 11:50 -12 .01762 5 11:57 - 5 .01770 6 12:02 0 .01757 7 12:07 5 .01881 8 12:12 10 .01718 9 12:17 15 .01669 10 12:22 20 .01585 11 12:32 30 - 12 12:47 45 .01638 13 1:02 60 .01556 14 1:17 75 .01446 15 1:36 94 .01449 16 1:47 105 .01486 17 2:02 120 - 18 2:17 135 .01497 19 2:32 150 .01452 20 2:42 160 .01315 21 2:52 170 .01260 22 2:57 175 .01233 23 3:02 180 .01194 24 3:07 185 .01310 63 Viscometer N = 500 rpm 5 = 3858 sec--1 Kv = .002227 miti-l Assay t 0 k' 1 9:03 -22 .02022 2 9:10 -15 .01849 3 9:13 —12 .01975 4 9:17 - 8 .02095 5 9:20 - 5 .02071 6 9:25 0 .01959 7 9:30 5 .01804 8 9:35 10 .01759 9 9:45 20 .01734 10 9:55 30 .01697 11 10:25 60 .01536 12 10:55 90 .01505 13 11:25 120 .01370 14 11:55 150 .01320 15 12 25 180 - 16 12 55 210 .01130 17 1:05 220 .01028 18 1:15 230 .01117 19 1:25 240 .01098 64 Stirred Tank N = 500 rpm Kt = .000947 min-1 Assay t 8 k' 1 8:53 -17 .02014 2 8:59 -11 .02000 3 9:03 - 7 .01937 4 9:10 0 .02093 5 9:15 5 .02103 6 9:20 10 .02062 7 9:25 15 .02091 8 9:30 20 .02064 9 9:40 30 .02054 10 10:10 60 .01937 11 10:40 90 .01944 12 10:50 100 .01919 13 11:00 110 .01885 14 11:10 120 .01885 15 11:40 150 .01805 16 11:55 165 .01806 17 12:00 170 .01808 18 12:05 175 .01771 19 12:10 180 .01770 65 Stirred Tank N = 1000 rpm Kt = .00141 min-l Assay Time a k' 1 8:43 -17 .00858 2 8:50 -10 .00826 3 8:56 - 4 .00779 4 9:00 0 .00880 5 9:05 5 .00881 6 9:10 10 - 7 9:15 15 .00872 8 9:30 30 .00847 9 10:00 60 .00807 10 10:30 90 .00822 11 11:00 120 .00788 12 11:30 150 .00752 13 12:00 180 .00724 14 12:30 210 .00652 15 1:00 240 .00643 16 1:30 270 .00619 17 2:00 300 .00589 18 2:30 330 .00534 19 3:00 360 .00566 66 Stirred Tank N = 1000 rpm Kt = .001633 mifi" Assay t 0 k ' 1 11:10 -20 .01994 2 11:15 -15 .01981 3 11:19 - 9 .01992 4 11:30 0 .01988 5 11:35 5 .02078 6 11:40 10 .01919 7 11:45 15 .01942 8 12:00 30 .01946 9 12:30 60 .01805 10 1:00 90 .01773 11 1:30 120 .01665 12 2:00 150 .01564 13 2:20 170 .01526 14 2:25 175 .01503 15 2:30 180 .01523 67 Stirred Tank N = 1500 rpm Kt = .001983 min-1 Assay t 6 k' 1 9:30 —80 .01736 2 10:40 -10 .01792 3 10:43 - 7 .01779 4 10:50 0 .01791 5 10:55 5 - 6 11:00 10 .02212 7 11:10 20 .01819 8 11:20 30 .01768 9 11:50 60 .01663 10 12:20 90 .01534 11 12:50 120 .01533 12 1:20 150 .01404 13 1:50 180 .01313 14 2:20 210 .01293 15 2:50 240 .01286 Stirred Tank 1500 rpm Kt = .002643 min"1 Assay t 8 k' 1 9:43 -27 .01306 2 9:54 -16 .01293 3 10:00 -10 .01389 4 10:10 0 .01296 5 10:15 5 .01449 6 10:20 10 .01231 7 10:25 15 .01350 8 10:30 20 .01358 9 10:40 30 .01283 10 11:10 60 .01158 11 11:40 90 .01084 12 12:10 120 .00929 13 12:40 150 .00902 14 1:10 180 .00857 15 1:40 210 .00802 16 2:10 240 .00752 69 Stirred Tank N = 1750 rpm Kt = .003097 min-1 Assay t 8 k' 1 12:40 -20 .02018 2 12:45 -15 .01986 3 12:51 — 9 .02003 4 1:00 0 .01994 5 1:05 5 .02117 6 1:10 10 .01935 7 1:15 15 .01837 8 1:30 30 .01815 9 2:00 60 .01597 10 2:30 90 .01513 11 3:00 120 .01456 12 3:30 150 .01260 13 3:45 165 .01167 14 3:50 170 .01177 15 3:55 175 .01200 16 4:00 180 .01141 70 Stirred Tank N = 2000 rpm Kt = .003097 min-l Assay t 8 k' 1 8:50 -20 .01876 2 8:56 -14 .01940 3 9:00 -10 .02004 4 9:03 - 7 .01928 5 9:06 - 4 .01904 6 9:10 0 .01895 7 9:15 5 .02039 8 9:20 10 .01902 9 9:30 20 .01788 10 9:40 30 .01796 11 10:10 60 .01478 12 10:40 90 .01519 13 11:10 120 .01360 14 11:40 150 .01078 15 11:50 160 .00962 16 12:00 170 .00969 17 12:10 180 .01024 71 Stirred Tank N = 2000 rpm Kt = .003439 min"1 Assay t 8 k' 1 1:02 -13 .02138 2 1:05 -10 .02084 3 1:09 - 6 .02098 4 1:15 0 .02073 5 1:20 5 .02250 6 1:25 10 .02141 7 1:35 20 .02010 8 1:45 30 .01981 9 2:15 60 .01780 10 2:45 90 .01737 11 3:15 120 .01477 12 3:45 150 .01206 13 4:15 180 .01163 14 4:25 190 .01076 15 4:35 200 .01159 16 4:45 210 .01166 BIBLIOGRAPHY 10. 11. 12. 13. 14. 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