- Email: [email protected]
A CFD-Based Model for Predicting Shear-Induced DNA Breakage in Mix Tanks
published by Elsevier Science on behalf of IFAC
A CFD-BASED MODEL FOR PREDICTING SHEAR-INDUCED DNA BREAKAGE IN MIX TANKS
*William J. Kelly, Vipin Narang and Anand Ekambaram
*to whom all correspondence should be sent
*Villanova University Department of Chemical Engineering 800 Lancaster A venue Villanova. PA 19085 william. j. [email protected]
Abstract: A new approach to modeling the degradation kinetics of large biological macromolecules (i.e. DNA) in a mixing tank is proposed which utilizes Computation Fluid Dynamic Modeling (CFD). The approach is based on the premise that molecules break only when they pass through a small region of high shear near the impeller. The CFD simulations can be used to estimate the size of this "high-shear" zone for a given impeller (i.e. RPM , Drr, off-bottom distance). The breakage of 35 kilobase (kB) fragments of DNA from salmon sperm was monitored using gel electrophoresis, and found to increase with the agitation rate of a standard Rushton turbine. The experimental data indicates that the CFD models, employing a RNG turbulence model and a dense numerical grid around the impeller, predict reasonably well the size of the "high-shear" zone in mixing tanks . This modeling and experimental approach can be used to assist in the scaleup of batch bioprocesses, where the shear-induced degradation of the bio-product is a concern.
Keywords: Computational Methods, Biotechnology, Process equipment, Fermentation processes, Turbulence, Shear Stress
251
1. INTRODUCTION
conditions. Levinthal and Davidson (1961) studied the degradation of DNA from the T2 bacteriophage under laminar flow conditions in a capillary. The DNA was linear, double-stranded and approximately 40 - 60 urn in length when fully extended. A theoretical model was developed by assuming the DNA molecule to be a cylinder, and integrating the forces on the surface. The model related the force required for breakage to the shear rate in the capillary. The experimental results supported the model, in that the calculated breakage force (8 x 10.4 dynes) was similar in magnitude to the strength of the chemical bonds holding the DNA backbone together. Another researcher (Reese, and Zimm, 1990), used a different modeling approach to model the breakdown of T7 bacteriophage DNA when subjected to extensional flow conditions. The degradation kinetics were found to be first order. The model assumed that the activation energy for the degradation of the DNA was reduced as the degree of shear in the system increased. The experimental results indicated that the reduction in the activation energy was on the order of 5 - 10 kcallmol. The most generic, and perhaps most effective, model for the shear-induced degradation of biomolecules asserts that: Ge = a(XlXo)b (1)
A biological process used to manufacture a natural product (i.e. an antibiotic or an antigenic vaccine particle) consistently uses fermentors, mixing tanks, homogenizers, centrifuges and pumps. The product is typically manufactured by living cells (i.e. bacteria) in a fermentor. Following fermentation, the cells need to be broke open if the product is intracellular. This is commonly done in industry by pumping the cell-containing solution through a high-pressure homogenizer. Following homogenization, the solution of cell debris and product might then be transferred into a holding tank. The holding tank would typically be agitated by an impeller(s), and equipped with cooling coils or a jacket for cooling. Adequate agitation would have to be provided to ensure rapid heating or cooling of the fluid and to ensure suspension of the solids in the tank. During each of these steps, and especially at large-scales, the biological solution can be exposed to high shear levels. It has been demonstrated that large biological entities (cells, molecules or particles) can break and/or loose activity upon exposure to high shear levels (Elias and Joshi , 1998). For example, filamentous haemagglutinin (FHA) is an anti genic protein that is secreted into the medium during cell growth in an agitated fermentor. It has been demonstrated that this molecule (2 nm in diameter and 100 nm in length ), which is particularly sensitive to shear forces because of its filamentous structure, looses activity as the power per unit volume in a stirred reactor increases (Rodriguez and Yantorno, 1993). DNA is a long (at least 40 Ilm) linear molecule that is present in the nucleus of any fermenting organism. If the cell wall is broken to release an intracellular product during bioprocessing, genomic DNA is also released into the extracellular fluid and exposed to the hydrodynamic shearing forces present. The collection tank following cell disruption is commonly a stirred tank. The degradation of the genomic DNA into smaller fragments, which might occur in this mixing tank could impact the final purity of a DNA vaccine if subsequent purification steps rely on a difference in size to separate plasmid DNA (the product) from the genomic DNA. The goal of this research was to develop a model to predict the kinetics of shear-induced biomolecules (i .e. DNA) in a mixing tank, which transcends impeller type, scale, and tank configuration.
where G is the shear rate, e is the time that the molecules are exposed to that shear force , XlXo is the fraction of un sheared molecules remaining, and a and bare experimentally determined constants. This model has effectively described the shear-induced degradation kinetics of primarily proteins under laminar flow conditions. Interestingly, recent results (Levy and Dunhill, 1999) indicate that plasmid DNA is quite sensitive to breakage in a capillary rheometer and a rotating disc shear device. These results also indicate that the rate of breakage of the DNA is related to the average energy dissipation rate and time of exposure to the shear rate in these devices. Clearly, higher shear levels than those studied in these laboratory devices could occur in large-scale bioprocesses. Characterizing shear levels in large bioprocessing equipment is a difficult task, due to the complex equipment geometries and fluid rheologies encountered. Most work that has been done in this area however, has focused on mixing tanks.
3.CHARACTERIZING SHEAR IN MIXING TANKS
In mixing tanks, shear in the laminar regime can be reliably related to the tip speed. Middler and Finn (1966) found in small tanks that the laminar shear rates were insufficient to account for the damage that was done to bioparticles in solution. Tagucci (1975) showed, also in small mixing tanks, that the frequency of passage
2. KINETIC MODELS FOR DNA DEGRADATION Several researchers have studied the degradation of DNA due to shearing forces , and some have attempted to develop a kinetic model based on the hydrodynamic
252
1989). This approach can involve the importing of the equipment geometry (i.e. in the form of an autocad file) into a commercial code (i.e. F1uent™), and the subsequent generation of a numerical grid to segment the regions of the equipment which would contain fluid . In each cell of the grid, the equations of motion (conservation of mass and momentum) are solved by either a finite difference or a finite volume numerical approach (Patankar, 1983). This approach allows for the inspection of all regions of the equipment, not just those regions conducive to measurement. Two numerical approaches (Brucato and Micale, 1998) have been developed, both which allow for the generation of flow in mixing tanks by the rotation of an impeller, the geometry of which is imported into the CFD model. In the first method, commonly referred to as "sliding mesh", two computational grids are employed: one moving with the impeller and the other fixed to the tank. The moving grid is allowed to slide relative to the stationary grid. The second approach, referred to commonly as "MRF' (or multiple reference frame), requires much less computational time because the steady state form of the governing transport equations are solved. These equations are solved in two domains which are fixed to their respective frames of reference; the outer reference frame is stationary and the inner reference frame rotates with the impeller.
through the impeller zone correlated with damage to biopartic1es. Industrial experience shows that the impeller tip speed on it's own, is a poor correlator of shear damage in larger tanks. In fact , maximum local energy dissipation rate (Em) is considered, in fact to be the best correlating parameter for shear in the turbulent regime typically encountered in large industrial mixing tanks. Maximum energy dissipation rate however cannot be measured directly, and has traditionally been estimated based on a relation proposed by Liepe( 1971): Em
=0.5 E (Drrr
3
(2)
Therefore, if the power per unit volume (E) is measured and the impeller diameter (D) and the tank diameter (T) are known, then the maximum energy dissipation rate can be estimated from this equation. The results from a recent study (Spicer and Pratsinis, 1996) showed that the rate of floc formation that occurred in a mixing tank under turbulent flow conditions was a function of the shear rate (G) that occurred around an impeller in a mixing tank and the circulation time le. where: G
=(E/v)o.5
le= (V/(N~D3»
(3) (4)
and v is the kinematic viscosity, Nq is the impeller flow number, V is the tank volume, and N is the agitation rate.
4. NEW KINETIC MODEL FOR MOLECULE BREAKAGE IN MIXING TANKS
With the development of laser doppler velocimeters, measurements of local velocity components can be made in mixing tanks. From these data, the shear rates can be calculated and local energy dissipation rates can be estimated from turbulence models such as K-E, RNG or Reynolds stresses (Wu and Patterson, 1989). The accuracy of the magnitudes of the shear rates is of course dependent upon the spatial resolution of the measurements. The accuracy of the K and E values calculated to a large extent is dictated by the validity of the turbulence model (and the assumptions contained therein, i.e. isotropic turbulence) in the system being measured. Laser doppler measurements, and the parameters calculated from these measurements (shear rates and energy dissipation rates), are also limited in that the technique is not able to measure in the two regions where high energy dissipation rates would be expected to occur: at the impeller bladelfluid interface and in the swept regions between passing blades.
There is a dearth of literature characterizing the compounding effect of shear on polymers, as they repeatedly pass through a region of high shear over an extended period of time. Therefore, it is of practical importance to develop a scaleable and predictive model for biopolymer losses due to shear forces in batch manufacturing equipment. A new model for the shearinduced degradation of a biomolecule in a mixing tank has been developed. This simple kinetic model is based on information obtained from converged CFD simulations. The model assumes that the mixing tank could be modeled as two separate tank regions: the highshear zone (Vj) near the impeller - where large particles (35 kB DNA fragment, in this case) will be broken, and the remainder of the tank (Vr) where none of the biomolecules are broken. A mass balance on the large DNA molecules can be written for the VI region: QCr-QC j - k.iC;Vj= V j (dC/dt)
Within the last 5-10 years, some researchers have started to use Computational Fluid Dynamic (CFD) models of mixing tanks to estimate the spatial (and most recently temporal) variation of velocity, shear rate and energy dissipation rate versus operating conditions and equipment configuration (Weetman and Hutchings,
(5)
and the Vr region: QC j - QCr= Vr(dC/dt)
253
(6)
Where C j and Cr are the concentrations in the two regions Vi and Vr, k.J is a degradation rate constant and and t is the time that the tank has been agitating for. The volumetric flowrate Q into and out of the two regions can be expressed as Vr/tc ; where le is the circulation time (equation 4). Dividing equations 5 and 6 by Vr then yields:
5.
RESULTS
5.1 CFD mode ling tm
Fluent was used to develop a model of a 20 liter mixing tank equipped with four baffles and a single Rushton turbine. This same tank was used in subsequent laboratory mixing experiments. The numerical grid was unstructured, consisting of 150,000 cells. Many of the cells were concentrated in a region extending 5 mm from the surface of the impeller blades. A grid independent solution was observed, as the power number and the impeller discharge velocities, calculated by Fluenttm, did not change as the number of cells around the impeller increased. Generally, 15-25,000 iterations were required to reach the convergence criteria (normalized residuals < 1 x lO·s) . To validate the accuracy of such a CFD model in predicting the distribution of turbulent energy dissipation rate (£) in a mixing tank, the geometry of a mixing tank used experimentally by S. Kresta (Zhou and Kresta, 1996) was incorporated into a Fluent tm model. Kresta measured energy dissipation rates arou'nd a Pitched Blade Turbine (D=TI2, where T is the tank diameter. For a CFD test run, the impeller was located T/4 off-bottom and rotated at 357 rpm to match an experimental condition. Both the K-£ and the RNG turbulence models predicted the maximum £ to be located on the plane 2 mm below the bottom surface of the PBT. The maximum £ value predicted by the RNG model (21 m 2/s 3 ) however was much closer to the measured value (10.1 m2/s 3 ) than that predicted by the 2 3 K-£ model (80 m /s ) . These results indicate that when a RNG turbulence model is used in conjunction with a multiple reference frame solution technique, the location and value of the maximum energy dissipation rate around a single impeller in a mixing tank matches measured values reasonably well.
(7) (8)
), and A = tc(VjN r). Using the Euler method, equations 7 and 8 can be simultaneously solved numerically for Cj and C" for different combinations of ~ and VjN,. In doing this, the two concentrations (C; and Cr) never differ by much. The model (equations 7 and 8) can be used to predict the degradation kinetics at any scale for any impeller at any RPM (greater than the "threshold" RPM required for molecular breakage) , if V;N, and ~ can be estimated accurately . The following procedure is proposed to calculate ~ and V;N r for a given impeller type and RPM . For a given biomolecule in solution: I.Experimentally identify the minimum agitation rate the corresponds to molecular breakage, at a given scale and for a given impeller type. At this threshold agitation rate (RPM t ), an estimate of the maximum energy dissipation rate (EmT ) in the tank can be obtained from equation 2 .
2. Collect C/Co versus time data for a higher agitation rate (RPM 2) . Through iterative post-processing of a converged CFD simulation at this higher agitation rate, the volume (Vi) in which E is ;::: EmT can be determined.
5.2 Bio-assay for DNA Degradation
3. At that same agitation rate (RPM 2), Equations 7 and 8 can be solved (using VjN r from step 2) to find the ~ value that results in the experimentally determined C/Co versus time profile.
A gel electrophoresis procedure was developed to quantitate the amount of shear damage imparted on DNA molecules in a solution of salmon sperm DNA (Sigma Co.). The gels were stained in an ethidium bromide solution, destained , and then photographed (Fluor-Imager 595, Molecular dynamics Inc .) - the image subsequently being analyzed with the ImageQuant (Molecular Dynamics Inc .) software package. This software allows for the calculation of band intensity or the total (integrated) intensity in a defined region in a lane. From a visual inspection of the gels, the largest DNA fragments in the non-sheared salmon sperm "smear" are 33- 38 kilobase (kB). The largest DNA fragments in the marker lanes are 48.5,38.5,33.4 kB. Using the software, the second and third largest marker bands were used to "section-off' the portion of the
4.Assume that the k.J value solved for in step 3 holds as long as the same molecule is exposed to some energy dissipation rate;::: E mT . For any other mixing condition (agitation rate, impeller type, scale), with this same shear-sensitive molecule/particle: run a CFD simulation to determine V;N r and then use the k.J value from step 4 in equations 7 and 8 to estimate C/Co .. The validity of this approach can be tested, by comparing these predicted C/Co values to experimental C/Co values.
254
salmon sperm smears that contain the largest fragments in the sample. A standard curve was developed, and the results indicate that a linear relati"onship exists between intensity and the DNA concentration in that section.
then tested to see how well C/Co would be predicted at 1000 RPM. A converged CFD simulation at 1000 RPM indicated that VjN r was 0.0182. With a k.J value of 0.41 S· I , equations 7 and 8 predicted an XlXo value of 0 .26 after 6 minutes of shearing, which is very close to the experimentally determined value of 0.3. It may be possible to improve upon the model by determining whether the DNA molecules that are breaking in the high shear zone are "nicked" and the unbroken molecules are not. A better estimate of the volumetric flowrate through the high shear region may be possibly by integrating the local velocity across the surface that contains the high shear volume (Vi) within the CFD model.
5.3 DNA Shearing Experiments Twenty liters of a 50 mg/I Salmon sperm DNA solution (water-like viscosity) were sheared in mixing tanks with a Rushton impeller (Dff = 0.28) at 250 - 1000 rpm for up to 8 hours . Samples were carefully removed from the vessel for analysis via gel electrophoresis. It was clear from the gel results (Figure 1) that breakage of the larger DNA molecules in the salmon sperm samples occurred at 750 and 1000 RPM. Furthermore, the samples that were exposed to 1000 rpm clearly contain DNA with a smaller average molecular weight than the samples sheared at 250 rpm.
6.
In this reserarch, the degradation of salmon sperm DNA due to shearing forces in a mixed tank was monitored using a gel electrophoresis technique. The experimental results were then used to substantiate predictive model has been developed , utilizing information from Computational Fluid Dynamic (CFD) Modeling, that describes the observed degradation kinetics reasonably well. The results from ongoing experiments with different impellers will be used to improve upon the predictability of the model.
7.
REFERENCES
Brucato, A. and Micale, G. (1998). Numerical prediction of flow fields in baffled stirred vessels: A comparison of alternative modeling approaches. Chemical Engineering Science, 53 (21), 3653-3684.
Figure 1 DNA size distributions versus impeller RPM Note:
CONCLUSIONS
Lane 1 on bottom., and lane 30 on top
EJias, C. and Joshi, J. (1998) . Role of hydrodynamic shear on activity and structure of proteins Advances in Biochemical Engineering, 59,47-71.
250 RPM samples -lanes 2,3,10,11,18,19 500 RPM samples -lanes 4,5,12,13,20,26,27 750 RPM samples - lane6,7,14,15,22,23,28,29 1000 RPM samples -lanes 8,9,16,17,24,25
Levinthal, C. and Davidson, P. (1961). Degradation of Deoxyribonucleic Acid under Hydrodynamic Shearing Forces. 1. Mol. Bio!., 3,674 - 683
The disappearance of the 33 - 38 kB DNA fragments in the salmon sperm, due to shear, allowed for determination of C/Co. The threshold RPM (RPM t ) was determined for the salmon sperm to be 375 ± 25 RPM. From equation 2, EmT was then calculated to be 2 2.6 m /s 3. After shearing for 6 minutes at 500 RPM, C/Co was found to equal 0.74. From a CFD solution
Levy, M. and Dunhill, P. (1999). Effect of shear on plasmid DNA in solution. Bioprocess Engineering, 20,7-13. Liepe, F. and Winkler H. (1971). Chem. Techn., 23, 231
at 500 RPM, the volume of the tank in which E is ~ EmT corresponded to VjN r =0.002. Using a ~
Middler, M. and Finn, R. (1966). Biotechnology and Bioengineering, 8, 71
value of 0.41 S· I resulted in equations 7 and 8 predicting the correct C/Co (0.74) for 500 RPM. The model was
255
Patankar, S. Numerical Heat Transfer and Fluid Flow_Hampshire Publishing Co.; 1983 Reese, H. and Zimm, B. (1990). Fracture of Polymer chains in Extensional Flow: Experiments with DNA, and a Molecular-Dynamics Simulations J. Chem. Phys, 92(4),2650 - 2661 Rodriguez, M. and Yantorno, M. (1993). Effect of hydro mechanical forces on the production of filamentous haemagglutinin and pertussis toxin of Bordetella pertussis Journal of Industrial Microbiology, 12, 103-108 Spicer, P. and Pratsinis, S. (1996). The Effect of Impeller Type on Floc Size and Structure during Shear-Induced Flocculation J. of Colloid and Interface Science, 184, 112 - 122 Taguchi, H. and Yomita, Y. (1975). J. of Fermentation Technology, 53 (I) Weetman, R. and Hutchings B. (1989). Computation of Flow Fields in Mixing Tanks with Experimental Verification Applications Annual Winter A.I.CH.E Meeting 1989. Wu, H. and Patterson, G. (1989). Laser-Doppler Measurements of Turbulent Flow Parameters in a Stirred Tank.1..Chemical Engineering Science. 144 (10), 2207 - 2221 Zhou, G. and Kresta, S. (1996). Impact of Tank Geometry on the Maximum Turbulence Energy Dissipation Rate for Impellers AIChE Journal, 42 (9), 2476 - 2490
256
A CFD-BASED MODEL FOR PREDICTING SHEAR-INDUCED DNA BREAKAGE IN MIX TANKS
*William J. Kelly, Vipin Narang and Anand Ekambaram
*to whom all correspondence should be sent
*Villanova University Department of Chemical Engineering 800 Lancaster A venue Villanova. PA 19085 william. j. [email protected]
Abstract: A new approach to modeling the degradation kinetics of large biological macromolecules (i.e. DNA) in a mixing tank is proposed which utilizes Computation Fluid Dynamic Modeling (CFD). The approach is based on the premise that molecules break only when they pass through a small region of high shear near the impeller. The CFD simulations can be used to estimate the size of this "high-shear" zone for a given impeller (i.e. RPM , Drr, off-bottom distance). The breakage of 35 kilobase (kB) fragments of DNA from salmon sperm was monitored using gel electrophoresis, and found to increase with the agitation rate of a standard Rushton turbine. The experimental data indicates that the CFD models, employing a RNG turbulence model and a dense numerical grid around the impeller, predict reasonably well the size of the "high-shear" zone in mixing tanks . This modeling and experimental approach can be used to assist in the scaleup of batch bioprocesses, where the shear-induced degradation of the bio-product is a concern.
Keywords: Computational Methods, Biotechnology, Process equipment, Fermentation processes, Turbulence, Shear Stress
251
1. INTRODUCTION
conditions. Levinthal and Davidson (1961) studied the degradation of DNA from the T2 bacteriophage under laminar flow conditions in a capillary. The DNA was linear, double-stranded and approximately 40 - 60 urn in length when fully extended. A theoretical model was developed by assuming the DNA molecule to be a cylinder, and integrating the forces on the surface. The model related the force required for breakage to the shear rate in the capillary. The experimental results supported the model, in that the calculated breakage force (8 x 10.4 dynes) was similar in magnitude to the strength of the chemical bonds holding the DNA backbone together. Another researcher (Reese, and Zimm, 1990), used a different modeling approach to model the breakdown of T7 bacteriophage DNA when subjected to extensional flow conditions. The degradation kinetics were found to be first order. The model assumed that the activation energy for the degradation of the DNA was reduced as the degree of shear in the system increased. The experimental results indicated that the reduction in the activation energy was on the order of 5 - 10 kcallmol. The most generic, and perhaps most effective, model for the shear-induced degradation of biomolecules asserts that: Ge = a(XlXo)b (1)
A biological process used to manufacture a natural product (i.e. an antibiotic or an antigenic vaccine particle) consistently uses fermentors, mixing tanks, homogenizers, centrifuges and pumps. The product is typically manufactured by living cells (i.e. bacteria) in a fermentor. Following fermentation, the cells need to be broke open if the product is intracellular. This is commonly done in industry by pumping the cell-containing solution through a high-pressure homogenizer. Following homogenization, the solution of cell debris and product might then be transferred into a holding tank. The holding tank would typically be agitated by an impeller(s), and equipped with cooling coils or a jacket for cooling. Adequate agitation would have to be provided to ensure rapid heating or cooling of the fluid and to ensure suspension of the solids in the tank. During each of these steps, and especially at large-scales, the biological solution can be exposed to high shear levels. It has been demonstrated that large biological entities (cells, molecules or particles) can break and/or loose activity upon exposure to high shear levels (Elias and Joshi , 1998). For example, filamentous haemagglutinin (FHA) is an anti genic protein that is secreted into the medium during cell growth in an agitated fermentor. It has been demonstrated that this molecule (2 nm in diameter and 100 nm in length ), which is particularly sensitive to shear forces because of its filamentous structure, looses activity as the power per unit volume in a stirred reactor increases (Rodriguez and Yantorno, 1993). DNA is a long (at least 40 Ilm) linear molecule that is present in the nucleus of any fermenting organism. If the cell wall is broken to release an intracellular product during bioprocessing, genomic DNA is also released into the extracellular fluid and exposed to the hydrodynamic shearing forces present. The collection tank following cell disruption is commonly a stirred tank. The degradation of the genomic DNA into smaller fragments, which might occur in this mixing tank could impact the final purity of a DNA vaccine if subsequent purification steps rely on a difference in size to separate plasmid DNA (the product) from the genomic DNA. The goal of this research was to develop a model to predict the kinetics of shear-induced biomolecules (i .e. DNA) in a mixing tank, which transcends impeller type, scale, and tank configuration.
where G is the shear rate, e is the time that the molecules are exposed to that shear force , XlXo is the fraction of un sheared molecules remaining, and a and bare experimentally determined constants. This model has effectively described the shear-induced degradation kinetics of primarily proteins under laminar flow conditions. Interestingly, recent results (Levy and Dunhill, 1999) indicate that plasmid DNA is quite sensitive to breakage in a capillary rheometer and a rotating disc shear device. These results also indicate that the rate of breakage of the DNA is related to the average energy dissipation rate and time of exposure to the shear rate in these devices. Clearly, higher shear levels than those studied in these laboratory devices could occur in large-scale bioprocesses. Characterizing shear levels in large bioprocessing equipment is a difficult task, due to the complex equipment geometries and fluid rheologies encountered. Most work that has been done in this area however, has focused on mixing tanks.
3.CHARACTERIZING SHEAR IN MIXING TANKS
In mixing tanks, shear in the laminar regime can be reliably related to the tip speed. Middler and Finn (1966) found in small tanks that the laminar shear rates were insufficient to account for the damage that was done to bioparticles in solution. Tagucci (1975) showed, also in small mixing tanks, that the frequency of passage
2. KINETIC MODELS FOR DNA DEGRADATION Several researchers have studied the degradation of DNA due to shearing forces , and some have attempted to develop a kinetic model based on the hydrodynamic
252
1989). This approach can involve the importing of the equipment geometry (i.e. in the form of an autocad file) into a commercial code (i.e. F1uent™), and the subsequent generation of a numerical grid to segment the regions of the equipment which would contain fluid . In each cell of the grid, the equations of motion (conservation of mass and momentum) are solved by either a finite difference or a finite volume numerical approach (Patankar, 1983). This approach allows for the inspection of all regions of the equipment, not just those regions conducive to measurement. Two numerical approaches (Brucato and Micale, 1998) have been developed, both which allow for the generation of flow in mixing tanks by the rotation of an impeller, the geometry of which is imported into the CFD model. In the first method, commonly referred to as "sliding mesh", two computational grids are employed: one moving with the impeller and the other fixed to the tank. The moving grid is allowed to slide relative to the stationary grid. The second approach, referred to commonly as "MRF' (or multiple reference frame), requires much less computational time because the steady state form of the governing transport equations are solved. These equations are solved in two domains which are fixed to their respective frames of reference; the outer reference frame is stationary and the inner reference frame rotates with the impeller.
through the impeller zone correlated with damage to biopartic1es. Industrial experience shows that the impeller tip speed on it's own, is a poor correlator of shear damage in larger tanks. In fact , maximum local energy dissipation rate (Em) is considered, in fact to be the best correlating parameter for shear in the turbulent regime typically encountered in large industrial mixing tanks. Maximum energy dissipation rate however cannot be measured directly, and has traditionally been estimated based on a relation proposed by Liepe( 1971): Em
=0.5 E (Drrr
3
(2)
Therefore, if the power per unit volume (E) is measured and the impeller diameter (D) and the tank diameter (T) are known, then the maximum energy dissipation rate can be estimated from this equation. The results from a recent study (Spicer and Pratsinis, 1996) showed that the rate of floc formation that occurred in a mixing tank under turbulent flow conditions was a function of the shear rate (G) that occurred around an impeller in a mixing tank and the circulation time le. where: G
=(E/v)o.5
le= (V/(N~D3»
(3) (4)
and v is the kinematic viscosity, Nq is the impeller flow number, V is the tank volume, and N is the agitation rate.
4. NEW KINETIC MODEL FOR MOLECULE BREAKAGE IN MIXING TANKS
With the development of laser doppler velocimeters, measurements of local velocity components can be made in mixing tanks. From these data, the shear rates can be calculated and local energy dissipation rates can be estimated from turbulence models such as K-E, RNG or Reynolds stresses (Wu and Patterson, 1989). The accuracy of the magnitudes of the shear rates is of course dependent upon the spatial resolution of the measurements. The accuracy of the K and E values calculated to a large extent is dictated by the validity of the turbulence model (and the assumptions contained therein, i.e. isotropic turbulence) in the system being measured. Laser doppler measurements, and the parameters calculated from these measurements (shear rates and energy dissipation rates), are also limited in that the technique is not able to measure in the two regions where high energy dissipation rates would be expected to occur: at the impeller bladelfluid interface and in the swept regions between passing blades.
There is a dearth of literature characterizing the compounding effect of shear on polymers, as they repeatedly pass through a region of high shear over an extended period of time. Therefore, it is of practical importance to develop a scaleable and predictive model for biopolymer losses due to shear forces in batch manufacturing equipment. A new model for the shearinduced degradation of a biomolecule in a mixing tank has been developed. This simple kinetic model is based on information obtained from converged CFD simulations. The model assumes that the mixing tank could be modeled as two separate tank regions: the highshear zone (Vj) near the impeller - where large particles (35 kB DNA fragment, in this case) will be broken, and the remainder of the tank (Vr) where none of the biomolecules are broken. A mass balance on the large DNA molecules can be written for the VI region: QCr-QC j - k.iC;Vj= V j (dC/dt)
Within the last 5-10 years, some researchers have started to use Computational Fluid Dynamic (CFD) models of mixing tanks to estimate the spatial (and most recently temporal) variation of velocity, shear rate and energy dissipation rate versus operating conditions and equipment configuration (Weetman and Hutchings,
(5)
and the Vr region: QC j - QCr= Vr(dC/dt)
253
(6)
Where C j and Cr are the concentrations in the two regions Vi and Vr, k.J is a degradation rate constant and and t is the time that the tank has been agitating for. The volumetric flowrate Q into and out of the two regions can be expressed as Vr/tc ; where le is the circulation time (equation 4). Dividing equations 5 and 6 by Vr then yields:
5.
RESULTS
5.1 CFD mode ling tm
Fluent was used to develop a model of a 20 liter mixing tank equipped with four baffles and a single Rushton turbine. This same tank was used in subsequent laboratory mixing experiments. The numerical grid was unstructured, consisting of 150,000 cells. Many of the cells were concentrated in a region extending 5 mm from the surface of the impeller blades. A grid independent solution was observed, as the power number and the impeller discharge velocities, calculated by Fluenttm, did not change as the number of cells around the impeller increased. Generally, 15-25,000 iterations were required to reach the convergence criteria (normalized residuals < 1 x lO·s) . To validate the accuracy of such a CFD model in predicting the distribution of turbulent energy dissipation rate (£) in a mixing tank, the geometry of a mixing tank used experimentally by S. Kresta (Zhou and Kresta, 1996) was incorporated into a Fluent tm model. Kresta measured energy dissipation rates arou'nd a Pitched Blade Turbine (D=TI2, where T is the tank diameter. For a CFD test run, the impeller was located T/4 off-bottom and rotated at 357 rpm to match an experimental condition. Both the K-£ and the RNG turbulence models predicted the maximum £ to be located on the plane 2 mm below the bottom surface of the PBT. The maximum £ value predicted by the RNG model (21 m 2/s 3 ) however was much closer to the measured value (10.1 m2/s 3 ) than that predicted by the 2 3 K-£ model (80 m /s ) . These results indicate that when a RNG turbulence model is used in conjunction with a multiple reference frame solution technique, the location and value of the maximum energy dissipation rate around a single impeller in a mixing tank matches measured values reasonably well.
(7) (8)
), and A = tc(VjN r). Using the Euler method, equations 7 and 8 can be simultaneously solved numerically for Cj and C" for different combinations of ~ and VjN,. In doing this, the two concentrations (C; and Cr) never differ by much. The model (equations 7 and 8) can be used to predict the degradation kinetics at any scale for any impeller at any RPM (greater than the "threshold" RPM required for molecular breakage) , if V;N, and ~ can be estimated accurately . The following procedure is proposed to calculate ~ and V;N r for a given impeller type and RPM . For a given biomolecule in solution: I.Experimentally identify the minimum agitation rate the corresponds to molecular breakage, at a given scale and for a given impeller type. At this threshold agitation rate (RPM t ), an estimate of the maximum energy dissipation rate (EmT ) in the tank can be obtained from equation 2 .
2. Collect C/Co versus time data for a higher agitation rate (RPM 2) . Through iterative post-processing of a converged CFD simulation at this higher agitation rate, the volume (Vi) in which E is ;::: EmT can be determined.
5.2 Bio-assay for DNA Degradation
3. At that same agitation rate (RPM 2), Equations 7 and 8 can be solved (using VjN r from step 2) to find the ~ value that results in the experimentally determined C/Co versus time profile.
A gel electrophoresis procedure was developed to quantitate the amount of shear damage imparted on DNA molecules in a solution of salmon sperm DNA (Sigma Co.). The gels were stained in an ethidium bromide solution, destained , and then photographed (Fluor-Imager 595, Molecular dynamics Inc .) - the image subsequently being analyzed with the ImageQuant (Molecular Dynamics Inc .) software package. This software allows for the calculation of band intensity or the total (integrated) intensity in a defined region in a lane. From a visual inspection of the gels, the largest DNA fragments in the non-sheared salmon sperm "smear" are 33- 38 kilobase (kB). The largest DNA fragments in the marker lanes are 48.5,38.5,33.4 kB. Using the software, the second and third largest marker bands were used to "section-off' the portion of the
4.Assume that the k.J value solved for in step 3 holds as long as the same molecule is exposed to some energy dissipation rate;::: E mT . For any other mixing condition (agitation rate, impeller type, scale), with this same shear-sensitive molecule/particle: run a CFD simulation to determine V;N r and then use the k.J value from step 4 in equations 7 and 8 to estimate C/Co .. The validity of this approach can be tested, by comparing these predicted C/Co values to experimental C/Co values.
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salmon sperm smears that contain the largest fragments in the sample. A standard curve was developed, and the results indicate that a linear relati"onship exists between intensity and the DNA concentration in that section.
then tested to see how well C/Co would be predicted at 1000 RPM. A converged CFD simulation at 1000 RPM indicated that VjN r was 0.0182. With a k.J value of 0.41 S· I , equations 7 and 8 predicted an XlXo value of 0 .26 after 6 minutes of shearing, which is very close to the experimentally determined value of 0.3. It may be possible to improve upon the model by determining whether the DNA molecules that are breaking in the high shear zone are "nicked" and the unbroken molecules are not. A better estimate of the volumetric flowrate through the high shear region may be possibly by integrating the local velocity across the surface that contains the high shear volume (Vi) within the CFD model.
5.3 DNA Shearing Experiments Twenty liters of a 50 mg/I Salmon sperm DNA solution (water-like viscosity) were sheared in mixing tanks with a Rushton impeller (Dff = 0.28) at 250 - 1000 rpm for up to 8 hours . Samples were carefully removed from the vessel for analysis via gel electrophoresis. It was clear from the gel results (Figure 1) that breakage of the larger DNA molecules in the salmon sperm samples occurred at 750 and 1000 RPM. Furthermore, the samples that were exposed to 1000 rpm clearly contain DNA with a smaller average molecular weight than the samples sheared at 250 rpm.
6.
In this reserarch, the degradation of salmon sperm DNA due to shearing forces in a mixed tank was monitored using a gel electrophoresis technique. The experimental results were then used to substantiate predictive model has been developed , utilizing information from Computational Fluid Dynamic (CFD) Modeling, that describes the observed degradation kinetics reasonably well. The results from ongoing experiments with different impellers will be used to improve upon the predictability of the model.
7.
REFERENCES
Brucato, A. and Micale, G. (1998). Numerical prediction of flow fields in baffled stirred vessels: A comparison of alternative modeling approaches. Chemical Engineering Science, 53 (21), 3653-3684.
Figure 1 DNA size distributions versus impeller RPM Note:
CONCLUSIONS
Lane 1 on bottom., and lane 30 on top
EJias, C. and Joshi, J. (1998) . Role of hydrodynamic shear on activity and structure of proteins Advances in Biochemical Engineering, 59,47-71.
250 RPM samples -lanes 2,3,10,11,18,19 500 RPM samples -lanes 4,5,12,13,20,26,27 750 RPM samples - lane6,7,14,15,22,23,28,29 1000 RPM samples -lanes 8,9,16,17,24,25
Levinthal, C. and Davidson, P. (1961). Degradation of Deoxyribonucleic Acid under Hydrodynamic Shearing Forces. 1. Mol. Bio!., 3,674 - 683
The disappearance of the 33 - 38 kB DNA fragments in the salmon sperm, due to shear, allowed for determination of C/Co. The threshold RPM (RPM t ) was determined for the salmon sperm to be 375 ± 25 RPM. From equation 2, EmT was then calculated to be 2 2.6 m /s 3. After shearing for 6 minutes at 500 RPM, C/Co was found to equal 0.74. From a CFD solution
Levy, M. and Dunhill, P. (1999). Effect of shear on plasmid DNA in solution. Bioprocess Engineering, 20,7-13. Liepe, F. and Winkler H. (1971). Chem. Techn., 23, 231
at 500 RPM, the volume of the tank in which E is ~ EmT corresponded to VjN r =0.002. Using a ~
Middler, M. and Finn, R. (1966). Biotechnology and Bioengineering, 8, 71
value of 0.41 S· I resulted in equations 7 and 8 predicting the correct C/Co (0.74) for 500 RPM. The model was
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Patankar, S. Numerical Heat Transfer and Fluid Flow_Hampshire Publishing Co.; 1983 Reese, H. and Zimm, B. (1990). Fracture of Polymer chains in Extensional Flow: Experiments with DNA, and a Molecular-Dynamics Simulations J. Chem. Phys, 92(4),2650 - 2661 Rodriguez, M. and Yantorno, M. (1993). Effect of hydro mechanical forces on the production of filamentous haemagglutinin and pertussis toxin of Bordetella pertussis Journal of Industrial Microbiology, 12, 103-108 Spicer, P. and Pratsinis, S. (1996). The Effect of Impeller Type on Floc Size and Structure during Shear-Induced Flocculation J. of Colloid and Interface Science, 184, 112 - 122 Taguchi, H. and Yomita, Y. (1975). J. of Fermentation Technology, 53 (I) Weetman, R. and Hutchings B. (1989). Computation of Flow Fields in Mixing Tanks with Experimental Verification Applications Annual Winter A.I.CH.E Meeting 1989. Wu, H. and Patterson, G. (1989). Laser-Doppler Measurements of Turbulent Flow Parameters in a Stirred Tank.1..Chemical Engineering Science. 144 (10), 2207 - 2221 Zhou, G. and Kresta, S. (1996). Impact of Tank Geometry on the Maximum Turbulence Energy Dissipation Rate for Impellers AIChE Journal, 42 (9), 2476 - 2490
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