Determination of shear rate and viscosity from batch mixer data

Transcription

1 Determination of shear rate and viscosity from batch mixer data M. Bousmina, a) A. Ait-Kadi, and J. B. Faisant b) Laval University, Department of Chemical Engineering, CERSIM, Quebec, G1K 7P4 Canada (Received 20 August 1998; final revision received 16 December 1998) Synopsis A general analysis allowing the determination of shear rate and viscosity from batch mixer rotor speed and torque data is presented. The batch mixer was represented by two effective adjacent sets of concentric cylinders exerting the same torque as that obtained from the batch mixer. The effective internal radius was determined through a general procedure for calibration using non-newtonian fluid. The effective equivalent internal radius, R i, was determined for different polymers and processing conditions. The results revealed that R i is a universal quantity practically insensitive to the nature and to the rheological behavior of the fluid under mixing. In the case of small gaps, it was found that there is a special position in the gap where the effective internal radius, the shear rate and viscosity are independent of rheological characteristics of the fluid under mixing. This validates the Newtonian approximation previously used by Goodrich and Porter to extract the shear rate-viscosity dependence from batch mixer data. The technique was tested on seven different amorphous and semicrystalline polymers and the results were found to be in reasonable agreement with the data obtained independently with cone-and-plate and capillary rheometers. Contributions of both shear stress between the two cylinders and the stress generated at the wall were evaluated. The latter was found predominant The Society of Rheology. S I. INTRODUCTION The batch mixer, illustrated in Fig. 1, is one of the essential instruments widely used in most laboratories working on high polymers. It requires only small quantities of materials and is usually used as a preliminary testing step before the actual processing for verifying for instance i the quality of mixing, ii the feasibility of polymer polymer inter-reaction, iii polymer crosslinking and polymer degradation, etc. The polymer in form of granules or powder is fed in the heated mixing chamber, fused and milled by one or two rotating blades at a fixed rotor speed, while the torque is recorded as a function of time. The typical curve of the torque as a function of time is shown in Fig. 2. The curve illustrates the thermomechanical history experienced by the polymer under mixing. When the polymer is introduced in the mixing chamber, the solid granules or powder offer a certain resistance to the free rotation of the blades and therefore the torque increases. When this resistance is overcome, the torque required to rotate the blades at the fixed speed decreases and reaches for a more or less short time a steady state. The torque increases again due to the melting of the surface of the granules a Author to whom all correspondence should be addressed. Electronic mail: bousmina@gch.ulaval.ca b Deceased on November 27, by The Society of Rheology, Inc. J. Rheol. 43 2, March/April /99/43 2 /415/19/$

2 416 BOUSMINA, AIT-KADI, AND FAISANT FIG. 1. Schematic illustration of the twin rotor batch mixer. that coalesce giving specks of particles. When the heat transfer is sufficient to completely melt the core of the particles, one obtains a macroscopic continuum easier to mix. Consequently, the torque decreases and reaches again a steady state regime and then increases or decreases depending whether crosslinking or degradation phenomena take place. In practice, the time corresponding to the first maximum is very short and is seldom ob- FIG. 2. Typical variation of the measured torque as a function of time.

3 VISCOSITY FROM BATCH MIXER DATA 417 served. The region considered in this work corresponds to the steady state region indicated in the figure. The time required to reach the steady state typically 3 15 min depends on the material and on the processing conditions temperature and rotor speed. Batch mixers are usually fitted with a torquemeter. Due to the small dimensions of the mixing chamber, torque data measured by the torquemeter are reasonably representative of the actual torque exerted on the polymer melt. Since the transfer of momentum in the mixing chamber is a function of the transport properties of the material, it is then desirable to relate the input variable, rotor speed, and the output variable, torque, to the viscosity-shear rate dependence of the material. Many research groups have used batch mixer data to get qualitative indications on melt viscosity Xie et al ; Scott and Young 1995 ; Goodrich and Porter 1967 ; Cheng and Manas-Zloczower 1989 ; Serpe et al ; Yang et al ; Scott and Macosko 1995, degradation or crosslinking Marechal et al ; Kim and Lee In the case of a twin rotor batch mixer, some work has been done on the quantitative determination of viscosity and shear rate from the batch mixer torque data. Goodrich and Porter 1967 converted torque-rotor speed data into viscosity and shear rate using correlations based on the instrument dimensions. They approximated the fluid flow in the batch mixer by an equivalent flow generated between two effective cylinders rotating at constant speed and exerting identical torque as that exerted by the batch mixer on the fluid under mixing. Shear rate and viscosity were calculated from the effective rotor dimensions assuming a Newtonian behavior for the polymer melt. This procedure has been applied to polymers such as polypropylene PP, high-density and low-density polyethylene HDPE and LDPE, polystyrene PS by Goodrich and Porter 1967, and to medium density polyethylene MDPE, and polyamide PA by Serpe et al Both groups found a good agreement between the viscosity estimated from the batch mixer data and the viscosity measured with a capillary rheometer. However, they had to use a calibration procedure with a Newtonian liquid to obtain the effective internal radius from the rotor dimensions. They assumed, without justification, that the same internal radius can be used to determine the viscosity of non-newtonian fluids through empirical correlations. Using the model developed by Tadmor and Gogos 1979, Yang et al estimated empirically the shear rate in the batch mixer from the rotor speed. The rotor was represented by a cylinder with three short low clearance sections. They considered that the viscosity in the clearance section is different from the viscosity within the large gap section. They found two shear rate values, one for the clearance section and one for the large gap section. The maximum shear rate was 1.5 time larger than the average shear rate calculated by the Goodrich and Porter s approach 1967, while the minimum shear rate was three times smaller. Unlike the work of Goodrich and Porter 1967, no estimation of the viscosity from torque data was given. Carreau et al calculated the shear rate and the power consumption of a helicoidal ribbon agitator in the case of non-newtonian fluids obeying a power law behavior. The helicoidal ribbon was represented by a cylinder as in the Goodrich and Porter s work Carreau et al work was however focused on the power consumption analyses and was limited to the mixing of low viscosity liquids at room temperature, and no estimation of the viscosity from torque data was considered. Detailed reviews and excellent work on power consumption in different mixing devices may be found in Oldshue 1983, Ulbrecht and Carreau 1985, Patterson et al. 1979, and Tanguy et al The main objective of this work is to validate the approach of Goodrich and Porter by proposing a general model and experimental procedure allowing for a direct estimation of shear rate and viscosity from batch mixer rotor speed and torque data. Model predictions

4 418 BOUSMINA, AIT-KADI, AND FAISANT are compared to experimental data obtained with cone-and-plate and capillary rheometers for different polymers. II. THEORETICAL BACKGROUND A general schematic representation of the twin rotor internal mixer used in this study is illustrated in Fig. 1. As in the previous works Goodrich and Porter 1967 ; Carreau et al we approximate the fluid motion due to the rotation of the two rotor blades by an overall macroscopic equivalent flow generated by two-adjacent cylinders rotating in a stationary cylindrical chamber. This geometry will be referred to as dual-couette type geometry. Therefore, the rotor blades are replaced by two effective cylinders exerting identical torque as that exerted by the rotors. In one set of cylinders, the inner cylinder rotates at a speed N 1, expressed in terms of number of revolutions per unit time, and in the other set, the inner cylinder rotates at a speed N 2. Let us call 1 the torque exerted on the first set of cylinders, 2 the torque exerted on the second set, and g N 2 /N 1 the gear ratio between the second rotor and the first rotor. The first rotor is directly fitted on the rotor shaft. The rotor speed N 1 is therefore identical to the rotor speed N of the mixer (N 1 N). The second rotor is geared to the shaft. To work out expressions for shear rate-viscosity dependence, we consider simple Couette analogy for each cylinder and then calculate the overall torque exerted by the two cylinders on the mixed fluid. The influence of the aperture that exists between the two sets of cylinders is neglected and, as it is shown in Appendix A, this assumption is reasonable and the maximum error made under usual and practical conditions is less than 4.5%. Each set of cylinders is considered as a simple Couette with a bob rotor having a radius R i and a cup wall having a radius R e. The cup is supposed to be stationary and the bob is rotating at an angular velocity 2 N. We assume laminar steady shear flow under isothermal conditions and perfect adherence of the fluid to the surface of the cylinders. For negligible end effects, we assume that V r V z 0 and V V (r, ) which reduces to V V (r) for an incompressible fluid. For simple fluids, the condition ( ) translates under the above assumptions to rz zr z z 0, and ij ij (r) otherwise with i, j r,,z. The component of the momentum transfer equation reduces then to 1 r 2 r r 2 0. r 1 Equation 1 indicates that under the above assumptions, the torque transferred from one wall to the other is constant throughout the tested fluid irrespective of its rheological characteristics (r 2 r /2 L, being the torque and L the length of the cylinders. Before deriving the equations for an unspecified rheological behavior of the fluid under mixing, let us first examine the simple case of a power law model. In this case, the shear stress is given by r M n M r r v n, 2 r where M is the polymer melt consistency, n the power law index, and (1/2 : ) 1/2 the amplitude of the strain rate tensor V ( V) T, where is the del operator, V the velocity vector, and the superscript T denotes the transpose of the quantity between parentheses. Substituting Eq. 2 into Eq. 1 and integrating the resulting equation with

5 VISCOSITY FROM BATCH MIXER DATA 419 respect to r with the boundary conditions, V R i at r R i and V 0 at r R e, gives the following expression for the shear rate at the position r: r NK r 3 where K r is given by K r 4 n R e R e 2/n r 2/n R i 1. 4 Replacing r by R i or R e in Eq. 4 gives the shear rate at the inner or the outer cylinders respectively. A more interesting position that will be discussed in more details later is r R 1/2 (R i R e )/2. The shear rate at this position is is given by 1/ /n N n 2/n 1 2/n 2/n 1, being the ratio R e /R i. The torque acting on the lateral surface of a cylinder of radius r and length L is given by 5 2 rl r r. 6 Equations 1 6 apply for each set of cylinders with N 1 N for the first set and N 2 gn 1 for the second one; g being the gear ratio. Equation 5 shows that the shear rate, 2, in the second set of cylinders is related to the shear rate 1 in the first set of cylinders by 2 g. The total torque measured by the instrument can be calculated from the total mechanical power by Dividing Eq. 8 by 1 leads to 1 g 2. Then combining Eqs. 2, 3, 6, and 9 gives 2 Lr 2 M K r N n g gk r N n Replacing K r by its expression given by Eq. 4 leads to 2 2n 1 n 1 2 MLR Nn 1 g n 1 e n 2/n 1 n. 11 Equation 11 indicates that the power law index can be readily obtained from the slope of the curve f (N) n d ln d ln N. 12

6 420 BOUSMINA, AIT-KADI, AND FAISANT The shear stress at r R 1/2 is obtained from Eq. 6 r R 1/2 2 L R e R i 2 1 g n From the definition given by Eq. 2, we have finally the viscosity as a function of shear rate 2 L R e R i 2 1 g n 1, 14 where is the shear rate given by Eq. 5. From Eqs. 5 and 14 it is therefore possible to calculate the shear rate and the viscosity from the torque-rotor speed data for a fluid obeying the power law model. The power law index, n, is obtained from Eq. 12 and the consistency, M, is extracted from (, ) data see definition in Eq. 2. The calculations are however possible provided that the mixer dimensions, R i and R e, are known. In the case of the batch mixer, R e is the radius of the mixing chamber but the effective internal radius, R i, is unknown. A possible procedure to overcome this difficulty is to use a polymer with known viscosity-shear rate dependence determined independently from steady shear rheometry and determine the effective radius, R i, using the following expression, obtained from the torque expression given by Eq. 11 : R i 1 4 N n R e 2 MLR 2 1 g n 1 e 1/n n/2. 15 Unlike the work of Goodrich and Porter 1967, it is here possible to use for the batch mixer calibration any power law fluid including Newtonian fluids. However, this supposes that the internal radius is independent of the nature of the fluid under mixing and of the processing conditions, rotation speed and temperature i.e., the denominator in Eq. 15 is constant. These different assumptions will be discussed in more details later. Note also that for simple mixers with a given mixing element rotating inside a cylindrical vessel with radius R e, the expressions developed here can be used to estimate the viscosity-shear rate dependence by setting the gear ratio, g 0, and using the indicated procedure for calibration. The analysis conducted above is based on the calculation of the shear rate at position r R 1/2 (R i R e )/2. This is because in the case of a narrow gap (R e R i )/R i 1, it can be shown see Appendix B that the shear rate evaluated at r R 1/2 (R i R e )/2 is independent of the index n. Therefore 1/2 can be approximated in this case by its Newtonian equivalent expression 2 1/ N N/ln The corresponding viscosity is then given by 2 1 N LR e 1 g This interesting geo-stationary -like position allows the determination of shear rate and viscosity by using the simple Newtonian expressions given by Eqs. 16 and 17.

7 VISCOSITY FROM BATCH MIXER DATA 421 In the previous section, the rheological behavior of the fluid under mixing was supposed to be of power law type. The treatment will now be generalised by considering that the fluid under consideration is of unknown rheological behavior without apparent yield stress. The shear rate-viscosity curve of any type of fluid can be locally approximated in a narrow shear-rate interval log( ),log( ) d log( ) by a power law model: ( ) M( ) (s 1), where s is the local slope of ( )Vs. In the case of a narrow gap, the local index can be assigned to the local slope of the curve ln f(ln N) at the considered rotational speed. Under the condition of narrow gap approximation ((R e R i )/R i 1), the shear rate, r, at a position r is given by Eqs. 3 and 4, where n is replaced by s. Using Eqs. 3 and 4, the shear rate can be expanded in Laurent s series of ln ln R e /R i at different positions in the gap: At r R e, At r R i, e 2 N 2 ln ln 1 ln s 3s 2 O ln i 2 N 2 ln ln 1 ln s 3s 2 O ln The expression for i is similar to that obtained, via a different route, by Krieger and co-workers for Couette-type flow Krieger and Elrod 1953, Krieger and Maron 1954, Krieger 1968, and Yang and Krieger Similarly, at r R 1/2, 1/2 (R 1/2 ) reads 1/2 2 N 1 3s 2 ln 3s 2 ln 2 O ln The corresponding viscosity is given by 1/2 N ln LR e 1 g s 1 1 ln 9s 2 3s 2 12s 2 ln 2 O ln Equations 20 and 21 are the general expressions allowing the calculation of shear rate and viscosity for a material with unspecified rheological behavior. s is the local power law index determined locally by Eq. 12. It is worth mentioning that the dependence of r on s appears in first order of ln at r R i and r R e, whereas at r R 1/2, it appears only in second order of ln. This means that r is a weaker function of the local power law index at r R 1/2 than it is at r R i or r R e. The calculations presented in Appendix B justify, in fact, that the shear rate at r R 1/2 is independent of the power law index. Note also that in the case of narrow gap, the general Eqs. 20 and 21 reduce to the shear rate and the viscosity determined in the special case of power law model Eqs Consequently and for clarity purposes we now confound s and n. Note that the asymptotic terms the first terms of the Laurent s series of r and ( r) of the shear rate and the viscosity coincide with the Newtonian expressions given by Eqs. 16 and 17, respectively. This means that for narrow gap approximation, the shear rate and the viscosity can be determined from Newtonian expressions given by the asymptotic

8 422 BOUSMINA, AIT-KADI, AND FAISANT FIG. 3. Variations of r / asy with respect to R i /R e and n for r R i,(r i R e )/2 and R e. terms ln( ) 0 of Eqs. 20 and 21. This is illustrated in Fig. 3 showing the variation of r / Newtonian as a function of R i /R e at positions r R i, r R e and r R 1/2 for different values of n. r / Newtonian is the ratio of the actual shear rate given by Eq. 5 to the asymptotic Newtonian shear rate r 2 Newtonian R 2/n e ln n r 2/n Clearly, the shear rate evaluated at r R i and r R e shows a strong dependence on n except of course at R i /R e 1. In contrast, at r R 1/2, the dependence of the shear rate on n remains much smaller. For R i /R e 0.85 and n 0.2, which are the conditions generally encountered in practice, the relation r Newtonian at r R 1/2 is valid within 5% which introduces a maximum error on viscosity for g 2/3) of about 10%. Since for narrow gap, the shear rate and the viscosity evaluated at r R 1/2 become independent of n, the shear rate and the viscosity and thus the effective internal radius can be calculated with reasonable approximation by setting n 1. This theoretically legitimates the Newtonian approximation that has been empirically used by Goodrich and Porter Moreover, it should pointed out that the validity of Newtonian approxi-

9 VISCOSITY FROM BATCH MIXER DATA 423 mation we have justified does not mean that a Newtonian fluid should be used for calibration. It indicates rather that the calibration can be used with any power law fluid including Newtonian fluids and molten polymers to obtain the effective internal radius R i from Eq. 15 with n 1. To sum up, the above reasoning shows that when the narrow gap approximation applies, the shear rate and the viscosity can be reasonably approximated by using Eqs. 16 and 17, respectively. These two equations do not require, a priori, the knowledge of the rheological parameters of the fluid under mixing. These can be determined afterward when the data points ( 1/2, ) are collected. The major problem in the analysis of Couette-type flow of non-newtonian fluids is related to the determination of the shear rate. The above development shows that this problem can be overcome by evaluating the shear rate away from the walls and more precisely at R 1/2 (R i R e )/2 when the narrow gap approximation applies. This is one of the key results that will be experimentally assessed in the next section. III. EXPERIMENT A. Materials To validate the approach developed in this work, seven different amorphous and semicrystalline polymers were tested: polystyrene PS, low-density polyethylene LDPE, high-density polyethylene HDPE, poly styrene-acrylonitrile SAN, poly styrene-maleic anhydride SMA and two grades of polypropylene PP. PS with M w and M w /M n 1.63 was obtained from Dow Chemical Inc. SAN was obtained from Monsanto Lustran 31. IthasM w and M w /M n SMA was obtained from Arco Chemical Company with a M w and M w /M n The two polyethylene resins were supplied by Novacor: HDPE, Novapol HB-L455 A/S (MFI 0.4 dg/min at 190 C under 2.15 kg and LDPE, Novapol LF-0219-D (MFI 2.3 dg/min at 190 C under 2.15 kg. The two grades of polypropylene were supplied by Himont Canada: Profax SR-256M and Profax SD-613. Profax SR-256M is a random copolymer with an MFI 230 C/2.16 kg of 2 dg/min. It will be referred to as PPlv in the following sections lv standing for low viscosity. Profax SD-613 is a high melt strength, impact copolymer resin with a MFI 230 C/2.16 kg of 0.3 dg/min. It will be referred to as PPhv in the following sections hv standing for high viscosity. B. Mixing The polymers were processed in the Haake-Büchler System 40-batch mixer with two counter rotating rotors Fig. 1. The two adjacent rotor blades are driven from a common shaft but geared so that the rotor speeds are not the same the gear ratio is 2/3. The measured torque is the total torque exerted by the two adjacent rotor blades on the fluid under mixing. The mixing chamber has a maximum capacity of 60 ml. Depending on the polymer, processing temperatures ranged from 160 to 230 C. Eight rotor speeds ranging from 30 to 150 rpm were tested and the corresponding torques for the different polymers were monitored. The values corresponding to the steady state conditions were then collected for further data analysis. C. Rheological measurements For comparison purposes, steady shear viscosities were measured as a function of shear rate with both cone-and-plate and capillary rheometers. Cone-and-plate measure-

10 424 BOUSMINA, AIT-KADI, AND FAISANT TABLE I. Rheological parameters for PS, LDPE, and PPhv determined from steady shear rheometry data. Sample T C M (Pa s n ) n PS LDPE PPhv ments were performed on the Bohlin-CVO constant stress rheometer. The plate diameter was 12.5 mm, the cone angle was 4 and the truncation gap was 75 m. Depending on the polymer, the temperature was varied from 160 to 230 C. Due to the secondary flow and the effect of normal stresses at high shear rates cone-and-plate measurements were only performed in the shear rate interval 0.01 to 0.5 s 1. Steady shear flow experiments at high shear rates were carried out on the Goetffert Rheotester-1000 capillary rheometer. Dynamic measurements in parallel plate geometry were also carried out and Cox-Merz rule was found to be valid for the different polymers used in this work. IV. RESULTS AND DISCUSSION To determine R i and to verify its sensitivity to the polymer nature and the processing conditions, PS, LDPE, and PPhv were used. The rheological parameters involved in Eq. 15 were extracted from the viscosity-shear rate data obtained from the rheometry measurements. The obtained value of R i was then used to determine the viscosity-shear rate curves for HDPE, PPlv, SMA, and SAN from rotor speed-torque batch mixer data. The results were compared to those obtained from steady shear rheometry. A. Internal radius The rheological parameters of PS, LDPE, and PPhv given in Table I were obtained from the rheometry data. The geometrical parameters of the mixing chamber required to determine R i from Eq. 15 with n 1 are: R e 20 mm, L 47.6 mm, and g 2/3. The calculated values of R i under different processing conditions for the three different polymers are listed in Table II. The results show that R i varies between and mm. These values lie well between the minimum 11 mm and the maximum TABLE II. Internal radius R i determined at different processing conditions using Eq. 15 with n 1. Rotor speed rpm PS 200 C PS 230 C LDPE 160 C LDPE 180 C PPhv 175 C PPhv 200 C The total mean value of R i Mean value of R i

11 VISCOSITY FROM BATCH MIXER DATA 425 TABLE III. Comparison between the values of R i determined with Newtonian approximation (n 1) and the values of R i using the power law model. Sample T C N R i in mm with n 1) R i in mm with actual n PS LDPE PPhv mm measured radius for the rotor of our mixer. For all polymers and processing conditions, R i is almost constant, with a maximum variation of about 1%. Note also that the obtained value for R i shows that the small gap approximation is satisfied for our batch mixer (R i /R e 0.88). To verify the error made in the calculation of R i using Newtonian approximation, R i was also calculated for a power law model using Eq. 15, where the power law index n and the material constant M were determined from the rheometry data see Table I. The FIG. 4. Comparison between the viscosity measured by rheometry and the viscosity determined from batch mixer data for HDPE at 160 and 180 C.

12 426 BOUSMINA, AIT-KADI, AND FAISANT FIG. 5. Comparison between the viscosity measured by rheometry and the viscosity determined from batch mixer data for PPlv at 175 and 200 C. obtained mean values of R i are reported in Table III. The maximum error made for the determination of R i with n 1 instead of considering the actual power law index is less than 2%. Using empirical correlations based on Newtonian approximation and assuming that R i does not depend upon the nature of the fluid under mixing, Goodrich and Porter 1967 calibrated the batch mixer with a Newtonian fluid with known viscosity and got a value of 17.3 mm for R i. The batch mixer geometry they used was similar to the one used in this work. Unlike the Goodrich and Porter s approach, our procedure does not require the Newtonian behavior restriction to calibrate the batch mixer. The calibration can be performed with any fluid with known viscosity-shear rate dependence. Moreover, as has been shown previously, when the narrow gap approximation is satisfied condition R i /R e 0.85 is required, R i is a weak function of n, and thus it can be determined by Eq. 15 with n 1. This value of R i can then be used to determine the shear rate and the viscosity for any given fluid by using Eqs. 16 and 17. In principle, only one polymer is needed to calibrate the batch mixer and determine the value of the effective internal radius R i. This however supposes that R i does not depend on the nature of the fluid nor on the rotor

13 VISCOSITY FROM BATCH MIXER DATA 427 FIG. 6. Comparison between the viscosity measured by rheometry and the viscosity determined from batch mixer data for SMA at 200 C. speed N and the temperature. The data of Table II clearly indicate that this is the case for the batch mixer used in this work. Since R i is a weak function of n,n, temperature and the nature of the polymer under mixing, a universal internal radius may be introduced for the internal mixer we used. For any other mixer, R i is expected to depend only on the geometrical dimensions of the mixing chamber and the gear ratio. For our batch mixer, the data obtained with PS, LDPE, and PPhv under different processing conditions gave a mean value of R i 17.6 mm, which is very close to the value, R i 17.3 mm only 2% difference, found by Goodrich and Porter 1967 using Newtonian fluid for calibration. We then used this value to determine the shear rate and the viscosity from the batch mixer torque and rotor speed data for HDPE, PPlv, SMA, and SAN and then compared the results with the data obtained independently from rheometry. B. Shear rate and viscosity Using the universal value of R i 17.6 mm and the conversion N(sec 1 ) N(rpm)/60, the shear rate and the viscosity were calculated by Eqs. 16 and 17, respectively. Viscosities obtained from steady shear rheometry and viscosities determined from batch mixer torque data are compared in Figs. 4 7 for HDPE, PPlv, SMA, and SAN,

14 428 BOUSMINA, AIT-KADI, AND FAISANT FIG. 7. Comparison between the viscosity measured by rheometry and the viscosity determined from batch mixer data for SAN at 190 C. respectively. The agreement with the four sets of data is very good for the different temperatures tested. Slight disagreements are however observed for high values of N especially for SMA and SAN, which can be attributed to the viscous dissipation that has not been taken into account in the present analysis. The effect seems to be more important for SMA and SAN than for HDPE and PPlv. This can be rationalized by examining the activation energy of the different polymers E R d ln d 1/T, 23 where R is the gas constant, and T the absolute temperature. The activation energy at constant shear rate can be calculated from rheometry data or from batch mixer data by tacking the slope of R ln( /N) vs1/t see Eq. 17. Activation energies determined from rheometry data are reported in Table IV. The activation energy for SMA and SAN are higher than the activation energy of HDPE and PPlv. This means that the viscosity of SMA and SAN are more sensitive to temperature variation than the viscosity of HDPE and PPlv. Therefore, the viscous dissipation should have a large effect on the viscosity drop of SMA and SAN, which is in fact consistent with the observed results at high rotor speeds.

15 VISCOSITY FROM BATCH MIXER DATA 429 TABLE IV. Activation energy at constant shear rate. Material Shear rate s 1 Temperature C E kcal/mol HDPE PPlv SMA SAN We also conducted measurements based on the same approach on low-viscosity solutions for which no or negligible viscous dissipation is expected. The results of such measurements gave indeed better agreement, over the whole range of shear rates tested, between the viscosity measured with cone-and-plate rheometer and the viscosity determined from rotation speed-torque data of a variety of mixing devices data are not shown here. V. CONCLUDING REMARKS The rheological behavior of the melt in a twin rotor blades batch mixer has been examined. The rotor blades and the mixing chamber were represented by a dual-couette exerting identical torque as that obtained from the batch mixer. Identifying the mixing rotor to a cylinder with an effective hydrodynamic radius R i, identical R i was found assuming either Newtonian or power law behavior. For the various polymers used in this work, R i was found to be a universal quantity depending only on geometrical dimensions of the mixer and the gear ratio but independent of the nature of the fluid under mixing and the mixing conditions. Using this universal internal radius, a good estimation of the melt viscosity can be obtained without knowing the polymer melt material constants. We have particularly shown that a Newtonian approximation is sufficient for evaluating the viscosity-shear rate dependence from batch mixer data for any rheological behavior in the case of small gap. Comparison of viscosities determined from batch mixer torque-rotor speed data with those measured with cone-and-plate and capillary rheometers revealed good agreement for all polymers tested in this work. The model developed in this work legitimates the previous empirical approach of Goodrich and Porter 1967 allowing the use of Newtonian approximation to calculate the polymer melt viscosity from the batch mixer data. From one single internal radius value, it is possible to accurately compute the melt viscosity of a large family of polymer melts. It should however be pointed out that unlike the work of Goodrich and Porter 1967, the procedure proposed in this work does not require the calibration with a Newtonian fluid to obtain the universal radius and no empirical correlation were used. In our method, it is possible to calibrate the batch mixer with either a Newtonian or non-newtonian fluid and then determine the viscosity for any non-newtonian fluid with unknown rheological behavior. The model and the calibration procedure proposed here can be applied to any type of mixer with complicated mixing elements. ACKNOWLEDGMENTS The authors would like to thank Professor Irvin M. Krieger for the useful discussions on the subject. This work was supported by the NSERC Natural Sciences and Engineering Research Council of Canada and the FCAR Fonds pour la Formation de Chercheurs

16 430 BOUSMINA, AIT-KADI, AND FAISANT et l Aide à la Recherche du Québec. Novacor-Canada, Himont-Canada, EVAL Co of America are gratefully acknowledged for providing polymer samples. APPENDIX A Since the rotors do not have the same speed, a shear stress is generated in the gap between the two rotor blades. This stress has not been taken into account for the calculation of the torque in the case of a dual-couette geometry. This appendix is aimed at getting an estimation of shear rate and torque due to shear between rolls assuming perfect adhesion at the chamber walls and at the rotor blades surfaces. The rotor blades are represented by cylinders of radius R i. An estimation of the shear rate in the gap between rotor 1 rotation speed N 1 N) and rotor 2 rotation speed N 2 gn) is given for small gap by IS V/e. A1 V is the difference of speed between rotors 1 and 2, e being the gap between rotors. V 2 N 1 R i 2 N 2 R i 2 NR i 1 g. A2 For the mixing chamber used in this study, e 2 R e R i. A3 Therefore, IS N 1 g. A4 R e 1 R i If we neglect the viscous dissipation, the total mechanical power generated by the shear stress at the inter-rotor gap is the given by 1 IS IS IS SR e 1 2. A5 IS is the total torque exerted on the rotor shaft. IS and IS are the viscosity and shear rate generated in the inter-rotor gap, S is the surface of the aperture linking the cylindrical chambers containing the rotors 1 and 2 see Fig. 1. S is given by S HL 2R e tg /2 L, A6 where H is the height of the aperture and L the length of the rotor depth of the mixing chamber. For small, tg( /2) /2 in our case, / and tg( /2) /2 within a precision of less than 2% and Eq. A6 can be rewritten as S R e L. A7 Since the gear ratio g is 1 / 2, IS can be expressed as IS IS IS LR e 2 1 g. A8 For a power law model, IS M LR e 2 NKIS n 1 g. A9 The total mechanical power due to mixing is then given by

17 VISCOSITY FROM BATCH MIXER DATA SW1 2 SW2 1 IS. A10 SW1 ( SW2 ) is the torque due to shear stress between the wall and the first second rotor. Shear rate in the gap between the rotor 1 and the wall is given for small gap by 1 2 N R e /R i 1. A11 Shear rate in the gap between rotor 2 and the wall is given by 2 g 1. In the case of a power law model, the total torque is then given by 2 N 2 2 R el n 1 g R e /R i 1 n g n 1 g A The ratio of the components IS due to shear stress in the inter-rotor gap to the total torque is n IS 1 g /2 1 g /2 1 g n 1 1 /2 1 g /2 n 1 g /2. A13 Knowing g 2/3 and 25.8, it is possible to evaluate this ratio for different values of n. In the case of a Newtonian behavior (n 1), IS / 1.5% and for n 0.2, IS / 5.5%. This means that the contribution to the torque of the shear stress in the inter-rotor gap is very small as compared to the wall-rotor component. Neglecting this component when using the dual-couette viscometer analogy to calculate the total torque is therefore reasonable. APPENDIX B In this appendix, we will calculate the value of r for which the velocity ratio K r r /N becomes independent on the intrinsic rheological characteristic of the polymer melt index n of the power law. It is supposed that the gap between the internal and the external cylinder of the Couette cell is small (R e R i )/R i 1. Equation 4 shows that in the general case, K r depends on n and r for a given Couette cell: K r K(r,n). The independence of K r on n can be written as K(r,n 1 ) K(r,n 2 ) with n 1 n 2. Using Eq. 4, this leads to 1 r R e n 2/n 1 R e /R i 1 2 1/n n 2 R e /R i /n 1. B1 2/n Since is small, the following relation holds: Equation B1 then reduces to 1 2/n 1 2 n 1 1 n 1 2. r/r e 1 /2. Since R e R i (1 ), r is given at first order by r R i 1 1 /2 R i 1 /2. Since (R e R i )/R i, the radius at which K r becomes independent of n is r R e R i /2. B2 B3 B4 B5

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