Polymer Dispersions by the ‘Masterbatch’ Process

0263±8762/00/$10.00+0.00 q Institution of Chemical Engineers Trans IChemE, Vol 78, Part A, January 2000

THE CHARACTERIZATION OF PIGMENT POWDERS FOR TITANIUM DIOXIDE/POLYMER DISPERSIONS BY THE `MASTERBATCH’ PROCESS J. RICHARDSON, A. J. MATCHETT (MEMBER), J. M. COULTHARD, S. GIBBON*, C. WILSON** AND C. WATSON** University of Teesside, Middlesbrough, UK *ICI Research & Technology, Wilton, Middlesbrough, UK **Tioxide Group Services, Billingham, UK

T

he Masterbatch process of mixing titanium dioxide pigment with low density polyethylene polymer powder has been investigated, with the aim of relating pigment properties to performance of the mixer. Five grades of TiO2 were investigated. A series of lab-scale mixing trials were undertaken, and the peak torque from the mixer was correlated against a number of pigment characteristics, including: shear properties from a shearbox; compressibility from Uncon®ned Compression Tests; and the blade force required to move a blade over a bed of material. The blade force was calculated from a model based upon nip technology and not bulk mixing. The blade force gave a good correlation with peak torque for all 5 grades of pigment. The work indicates that solids properties predominate in the early parts of the mixing cycle, but models need to be developed to describe the whole of the mixing process. Keywords: mixing; pigment; Masterbatch; friction; compressibility; powder

INTRODUCTION

shape in a `dumbbell’ shaped chamber. A feedport is located above the centre of the mixer, which feeds material into the area where the rotor tip loci overlap. Operation of the mixer

Pigment powders consist of micron and sub-micron sized particles that are used to colour a wide range of materials, including plastics, paper and paints. However, their characteristic properties tend to make the virgin pigment very cohesive and prone to the formation of dusts. These effects make them very dif®cult to handle and to process. Titanium dioxide pigment is incorporated into polyethylene and other plastics to make white bottles, plastic bags and other similar products. The pigment is often initially incorporated into the polymer at high loadings (>70% by weight) and then sold to the end-user. This has several advantages: (i) Dusting is eliminated, as the pigment is held in the matrix of the polymer. (ii) The material is easy to handleÐthe polymer mix consists of large particles compared to the individual pigment particlesÐof the order of mm. (iii) The pigment is pre-dispersed in the polymer and pigment aggregates are broken down to individual particles by the premixing process. (iv) The end-user simply extrudes the pre-mix with polymer to give the ®nal productÐthe end-user does not require high-energy mixers to disperse aggregates. (v) These factors represent a high convenience factor to the end-user. One method of premixing the pigment with the polymer is the `Masterbatch’ process1,2. The Masterbatch is made in a high-energy, batch mixer of the Banbury type3ÐFigure 1. The mixer consists of two contra-rotating rotors of complex

Figure 1. A Banbury type mixer used in the Masterbatch process.

39

40

RICHARDSON et al.

Figure 2. Torque versus time curves for the Masterbatch process, using the laboratory Haake-Buchler Rheocord mixer.

consists of loading, pre-compression, mixing and unloading. A batch of material is loaded into the mixing chamber with the rotors in motion. The initial volume of the mix is usually such that it occupies both the chamber and all or part of the feed port. A compression ram in the feedport is then depressed, forcing all of the material into the mixing chamber. The mixer is then allowed to work until it is judged that mixing is complete, usually by speci®cation of an empirically-determined cycle time. At this stage, the mixer can be unloaded. Large machines often have `bombdoor’ arrangements for unloading. For a comprehensive review of the Banbury mixer and other internal mixers see White4. The process has characteristic curves of torque and temperature versus time ÐFigures 2 and 3. After a short induction period, the torque rises sharply, passes through a peak, and then steadily declines towards a steady state valueÐFigure 2. There is often a second, smaller peak in the torque pro®le, associated with fusion of the polymer. Similarly, after a small initial fall, the temperature rises steadily to a limiting value. The melting point of the polymer is approx. 968 C. Therefore, at some point in the process, the polymer undergoes transition from solid particle into a non-Newtonian liquid. The Masterbatch process therefore commences as a mixing process of two solids components. As the temperature rises, due to the

input of mechanical energy from the rotors, the pigment particles are impressed into the softening polymer. This is seen visually by a reduction in solids volume and the appearance of voids within the mixing chamber. As mixing proceeds, the material undergoes a transformation from a solids system to a paste. Therefore, the behaviour of the process is complex, including solids particle dynamics, phase changes, mixing, aggregate breakdown and dispersion. Numerous authors have modelled the behaviour of material in an internal mixer5±12, but have tended to neglect the initial powder condition associated with this process. The total amount of energy added to the batch is related to the size and location of the peak torque in Figure 2, and is an important process parameter. The quality of the batch is judged upon the number of aggregates in the batch which exceed a certain size. If a predetermined value is exceeded then the batch is of unacceptable quality. The breakdown of aggregates is strongly related to the energy input and hence the peak torque1,2. Pigment manufacturers can modify the properties of their pigments by several means, such as coatings, which are usually of great commercial con®dentiality, and there is a drive to incorporate higher pigment loadings into the polymer dispersion by manipulation of these pigment properties13±15. Unfortunately, there is only a limited understanding of how the pigment properties in¯uence the behaviour of the Masterbatch process.

.

Figure 3. Temperature versus time curves for the Masterbatch process, using the laboratory Haake-Buchler Rheocord mixer.

Trans IChemE, Vol 78, Part A, January 2000

CHARACTERIZATION OF PIGMENT POWDERS FOR TiO2/POLYMER DISPERSIONS Table 1. Properties of the pigments used in the Masterbatch trials.

Pigment

Untapped bulk density

Tapped bulk density

Hausner Ratio tapped/ untapped

Angle of internal friction

Cohesion

Code A B C D E

kg m ±3 870 779 842 1031 1111

kg m ±3 1053 964 1074 1250 1220

1.21 1.24 1.28 1.21 1.10

degrees 27.0 25.3 24.0 25.9 20.5

kPa 4.11 6.29 8.04 11.77 9.14

This paper is part of a larger investigation, to determine the mechanisms and behaviour of the Masterbatch system, with a view to process and product improvement. It presents an experimental investigation into possible relationships between solids properties and behaviour in the initial stages of mixing, up to and including the peak torque. The mechanical properties of ®ve grades of TiO2 pigment were measured, Table 1, and their relation to behaviour in the Masterbatch process was investigated, with an emphasis upon prediction of the peak torque. The following properties of the pigment were measured or calculated: (a) Peak torque of pigment/polymer mixtures in a model mixer (b) The angle of internal friction and cohesion from underconsolidated shearbox tests (c) The peak axial stress, rate of dilation and compressibility from Uncon®ned Compression Tests (d) The force required to move a blade over a bed of material undergoing compression and shear. The peak torque was then correlated with the pigment properties and models sought to understand the mixing behaviour. MATERIALS AND MIXING EQUIPMENT Tests were performed upon 5 grades of titanium dioxide pigment. The properties are given in Table 1. The grades were prepared by different techniques, from several suppliers, and some of them were experimental grades. Each grade was assigned a letter for identi®cation purposes. The data in Table 1 indicate that the properties of the grades were quite different, although they were all nominally TiO2. Details of the source, methods of preparation, crystal form and coatings were considered to be a matter of commercial con®dence by the project sponsors, and it was beyond the scope of the present project to relate bulk properties to those at the particle and micro-scale. Micropol L7 low density polyethylene powder was used in the mixing trials, with a Melt Flow Index of 7. The properties of TiO2 are very sensitive to atmospheric conditions. Therefore, the pigment was conditioned prior to testing. This was done by storing each grade of pigment, for 24 hours, in a dessiccator with a saturated solution of CaCl2, prior to use. This gave a standard relative humidity of 30% RH. Conditioning to a standard RH was chosen rather than a constant moisture content due to the different grades of pigment used. Actual moisture contents ranged from 0.25± 0.5% total weight basis for unconditioned pigment and Trans IChemE, Vol 78, Part A, January 2000

41

0.2±0.4% total weight basis for the conditioned pigment, dependent upon grade. The material was then tested immediately from the dessiccator. MASTERBATCH MIXER TRIALS Laboratory-scale dispersion trials were conducted in a Haake-Buchler Rheocord 750 laboratory mixer with a glass end panel to allow observation. This lab-scale machine has been shown to simulate the behaviour of larger machines in mixing tests7,8. There was a heating system to ensure constant starting temperature. The heaters were used to preheat the mixer to 638 C at the start of each test. The temperature probe was located centrally, in the wall at the base of the mixing chamber. The mixer had a capacity of 70 cm3(7 ´ 10±5m3). The rotor speed was 125 rpm and the ratio of rotor speeds was 1:1.13, to promote increased shear strain in polymer mixing. The blades were of the Banbury type ± Figure 1. Mixing trials were conducted on mixtures of TiO2 and LDPE (low density polyethylene) in proportions shown in Table 2. The LDPE was in the form of particles approx. 400 mm in size. Full-scale Banbury mixers usually employ LDPE pellets, of the order of millimetres in size. These were very large compared to the scale of the lab mixer. Therefore, the smaller particles were used as a standard. Each run was conducted by the following procedure. The rotors of the mixers were started and the system was run until a temperature of 638 C was obtained. The datalogging system was then started, which recorded temperature and torque as time series. The mixture of TiO2 and LDPE was loaded into the mixing chamber by a chute. The ram was placed into position and it was loaded into its ®nal position at constant stress of 245 kPa. The mixing process was continuously observed during the loading and mixing. A test was stopped when the mixture had reached uniformity, at a temperature of approx. 1038 C. Runs took 120 seconds. Typical data are shown in Figure 2 and 3, and a summary of peak torque data is given in Table 2 for runs at constant charge weight and runs at constant composition with variable charge weight. The effects of charge weight are shown in Figures 4,5,6 and 7 for two of the grades. Clearly, peak torque was dependent upon both grade of pigment and charge weight, as shown in Figure 8. Table 2. Experimental conditions for the mixing trials. Grade of TiO2 A B C D E A A A A B B B B

Charge weight g

Initial bulk Density kg m ±3

Weight % TiO2

Peak Torque N-m

98 98 98 98 98 96 94 92 90 96 94 92 90

1400 1400 1400 1400 1400 1371 1343 1314 1285 1371 1343 1314 1285

70 70 70 70 70 70 70 70 70 70 70 70 70

55.9 89.8 83.4 73.3 30.5 52.8 50.4 47.7 41.6 73.2 63.9 53.0 45.7

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42

Figure 4. The effects of charge weight upon torque for TiO2 grade A. 70% wt TiO2.

Figure 5. The effects of charge weight upon temperature for TiO2 grade A. 70% wt TiO2.

The purpose of our investigation was to identify and measure properties of the pigments, which correlate with the peak torque, as shown in Figures 2±7. A further aim was to identify mechanisms contributing to the mixing process. SHEAR CELL TESTS Shear tests were conducted on the grades of TiO2 in a standard soil mechanics shearbox, 100 mm ´ 100 mm square, at several normal stresses up to 245 kPaÐFigure 9. This was undertaken to measure shear and cohesion properties at stresses likely to be experienced in the

mixer. Underconsolidated samples were tested in an attempt to recreate the compression regime within the mixer. The TiO2 was preconsolidated in the shearbox, by tamping down with a stress of 20% of the maximum normal stress. The maximum shear stress was recorded for each test. Data are shown in Figure 10 in terms of shear stress versus normal stress. For each set of conditions, the gradient was measured as the angle of internal friction, and the intercept on the shear stress axis as the cohesion. A summary of friction and cohesion data is given in Table 1. The complete data are shown in Figure 10, including data for all the grades tested.

Figure 6. The effects of charge weight upon torque for TiO2 grade B. 70% wt TiO2.

Trans IChemE, Vol 78, Part A, January 2000

CHARACTERIZATION OF PIGMENT POWDERS FOR TiO2/POLYMER DISPERSIONS

43

Figure 7. The effects of charge weight upon temperature for TiO2 grade B. 70% wt TiO2.

Statistical analysis of variance on the data indicated that normal stress and the grade of pigment were signi®cant factors upon the shear stress. However, subsequent analysis revealed that only Grade E was signi®cantly different (at the 0.05 level of signi®cance) from the other grades. There was no signi®cant difference between Grades A-D at the 0.05 level. Therefore, the shearbox was not an appropriate system to differentiate between mixing properties of pigments in the Masterbatch system. UNCONFINED COMPRESSION TESTS(UCT)

sample was not self-supporting and collapsed prior to the test. The maximum bulk density was limited by the applied force. The cylindrical sample was then removed from the mould, placed upon the base of the VUCT, surmounted by the top cap mass, and compression commenced. The system measured axial force, axial displacement, base displacement and radial displacement16,17ÐFigure 9. Typical data from the UCT tests are shown in Figure 11. Three parameters were extracted from the data. The peak stress is the maximum axial stress experienced by the system. The strain ratio, a , is the gradient of radial strain

A series of uncon®ned compression tests were performed upon the TiO2 grades. The equipment used was the Teesside Vibrational Uncon®ned Compression Tester(VUCT)16±17Ð Figure 9. The equipment was used to compress cylindrical samples 76 mm ´ 38 mm diameter in uncon®ned compression without applied vibration. Tests were also performed to investigate the effects of vibration upon uncon®ned compression, but these will be reported at a later date. Details of the VUCT can be found elsewhere16,17, but for the purposes of this paper, the VUCT was able to measure axial stress, axial strain and radial strain. Samples were prepared in a standard soil mechanics mould 76 mm in length and 38 mm diameter. A known weight of preconditioned pigment was placed into the mould, which was then axially compressed, from both ends. Approximately constant compression force was used, which necessitated the use of different bulk densities for the different grades of TiO2. In practice, a limited range of bulk densities was possible. When the bulk density was too low then the

Figure 8. The effects of charge weight on peak torque: Grades A and B.

Trans IChemE, Vol 78, Part A, January 2000

Figure 9. The soil mechanics shearbox and Uncon®ned Compression Test(UCT) systems.

RICHARDSON et al.

44

Figure 10. Shear stress versus normal stress for TiO2: all grades shown.

Figure 11. Uncon®ned Compression Test for TiO2 grade B; Bulk density 1250 kg m ±3; Axial strain compression rate 0.5mm s ±1.

versus axial strain in the pre-peak stress region. Compressibility, Kplane, is the ratio of effective compressive stress increment in plane strain divided by volumetric strain increment. If the cohesive stress is assumed constant during strain, then the major principal stress is equal to the axial stress and the remaining principal stress is equal to zero: (s01 + s03 ) 2 s1 = sl : s3 = 0

p0plane =

Kplane =

(1)

The peak torque is plotted as a function of compressibility in Figure 12. There was a modest correlation, which was statistically signi®cant at the 0.1 level. Grade E pigment was considerably `out of line’ with the remaining 4 grades. However, if Grade E was ignored, there was a very strong linear relationship between the 4 grades, A-D, signi®cant at the 0.01 level. Therefore, compressibility bears some relation to Masterbatch peak torque. However, the insensitivity of compressibility to Grade E indicates severe limitations for this parameter.

0 plane

dp dsl = du 2del (1 + 2a)

MODELS OF THE MIXING PROCESS Two models of the mixing process have been considered. A ¯ooded chamber model is presented in Appendix 1. This

These data are summarized in Table 3.

Table 3. Uncon®ned compression test data. Grade of TiO2 A B C D E

Bulk density of UCT test

Peak axial compressive stress Pa

Compressibility Pa

Strain Ratio

1250 1250 1250 1400 1400

2.73E+04 2.13E+04 1.86E+04 5.28E+04 2.57E+04

811 1197 1133 951 899

0.922 0.434 ±0.166 1.050 0.188

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CHARACTERIZATION OF PIGMENT POWDERS FOR TiO2/POLYMER DISPERSIONS

45

Figure 12. Peak torque from Haake Rheocord tests versus compressibility for the 5 grades of TiO2. Constant charge weight data for the Masterbatch mixing, 98 gÐFigures 2 and 3.

assumes that the mixing chamber is ®lled with material and subject to stress from the feedport ram. This gives an equation for torque, Tor, as:Tor = FR = (s tan w + C)AR

(2)

In this model, all the materials are subject to the same stress, as the feedport is loaded at constant weight. This makes the torque dependent only upon the friction and cohesion within the material. The shearbox tests showed no signi®cant difference between grades A-D. Therefore, this approach cannot explain the variations in peak torque shown in Figure 2. A development of this approach is to assume that the mixer is operating at constant volume, not constant stress. This would occur if the stresses exerted by the feed-port ram were in excess of that required to compress the material into the mixing chamber. The stresses generated would then be approximately proportional to the compressibility. Figure 12 indicates limited success with this approach, but it cannot explain all 5 grades of pigmentÐGrade E is an anomaly. An alternative model is derived in Appendix 2 and is a nip approach. This assumes that most of the work put into the mixer takes place in the small nip between the mixer wall and the blade. It is exerted in compressing the material into the nip and causing plastic failure. The geometry of the nip and compressibility give the effective compressive stress, p0 , at a point, y, between the blade and wall: p0

p00 = Kplane ln

ê

h1 y tan q

h1

Wall friction is limiting, hence the shear stress, t , is found: t = mw s (5) The shear and normal stresses are then integrated over the length of the blade, L, to give the force required to move the bladeÐthe Blade Force, F: …L F = (t cos q + s sin q)dy (6) 0

Equations (3)±(6) (Appendix 2) have been used to calculate the force required in order to move the blade over the material, using conditions approximating to the mixer, as shown in Table 4. Figure 13 indicates a strong linear relationship between the data for all 5 grades of pigment, signi®cant at the 0.01 level. Therefore, the blade force is the best predictor of peak torque available at the present time. DISCUSSION Figures 8 and 13 provide an empirical method for prediction of peak torque, as a basis for characterization of the material in terms of the mixing process. They enable mixer performance(in terms of peak torque and charge weight) to be predicted from a knowledge of the physical properties of the pigment. The data also indicate certain

(3)

ê Manipulation of relevant Mohr circles gives a quadratic in effective normal stress, s0 , which can be solved for s0 : 2

mw (s0

ê

T )2 + (s0 ê

p 0 )2 ê

q02 = 0

(4)

Table 4. Conditions used for calculation of blade force. Initial blade gap h1m 0.034

Blade angle u degrees

Blade width m

Coef®cient of wall friction

30

0.05

As internal friction

Trans IChemE, Vol 78, Part A, January 2000

Figure 13. Blade force (equation A13) versus peak torque from Haake Rheocord tests for 5 grades of pigment.

RICHARDSON et al.

46

dominating features of the mixing process. The blade force data suggest that the early stages of the Masterbatch mixer operation are best described through a nip technology process, with material being compressed and sheared in the small volume between the blade and the wall. Hence, process improvements should be sought from studies of the blade-wall nip, rather than bulk processes in the mixer. Furthermore, the Blade Force was calculated based upon pigment properties. No account was taken of the included polymer. In spite of this, it gave a good correlation with peak torque in pigment polymer systems with 30% weight polymer. This suggests that pigment solids properties dominateÐat least up to the time of peak stress. A simple explanation of these observations is that the softening polymer particles become enrobed by pigment, such that their properties tend to those of the adhered pigment particles, rather than the underlying polymer. The nip model (Appendix 2) is also able to explain the effects of charge weight, Figure 8, in a satisfactory manner. An increase in charge weight increases the initial bulk density of material, and this will cause an increase in initial stresses: po0 in equation (3). There are some limitations in the experimental work and the models that must be considered. The data collected on the pigments was for pigment only. No account was taken of the presence of the polymer. Furthermore, all tests were done at ambient temperature. These remain factors for further investigation. One reason for this omission is that the Uncon®ned Compression Tests were only possible with free-standing samples. All the pigments were capable of formation into such samples. However, inclusion of polymer and elevation of temperature produced samples that were not capable of standing on the VUCT equipmentÐthey were too fragile. The authors are presently developing a system of con®ned compression tests, which will allow inclusion of polymer and elevated temperatures. The blade force model of Appendix 2 assumes simultaneous failure, both internally and at the blade interface. Geometry has been simpli®ed to a wedge in plane strain. The model obviously gives a good correlation with peak torque, but it is clear that more sophisticated models are possible, including DEM (Distinct Element method), FEM (Finite Element method) and CFD (Computational Fluid Dynamics) approaches. Later stages of the mixing process, including polymer softening, melting and subsequent paste formation require further study to provide a comprehensive description of the Masterbatch process. CONCLUSIONS The Masterbatch process of mixing titanium dioxide pigment with low density polyethylene polymer has been investigated, with the aim of relating pigment properties to performance of the mixer. Five grades of TiO2 were investigated. A series of lab-scale mixing trials was undertaken, and the peak torque from the mixer was correlated against a number of pigment characteristics, including: shear properties from a shearbox; compressibility from Uncon®ned Compression tests; and the blade force required to move a blade over a bed of material. The blade force was calculated from a model based upon nip technology and not bulk mixing. The blade force gave a

good correlation with peak torque for all 5 grades of pigment. APPENDIX 1 FLOODED MIXERÐMIXING PROCESS A mixer that is full of material (¯ooded) will be subject to the stresses imposed by the loading ram (s, with a value of 245 kPa in this case). The torque required to move the rotors may be estimated in these circumstances. The motion of the rotors induces a shearzone of surface area A (m 2) within the mixer. The force, F (N), required to cause displacement along the shear area may be calculated from the shear properties of the material, as determined in the shearbox tests: F = (s tan w + C)A

(A1)

Equation (A1) assumes solids friction behaviour, which is valid in the early stages of the mixing process. Therefore, the torque, Tor (N-m), over radius R of the rotors is given by: Tor = FR = (s tan w + C)AR

(A2)

Equation (A2) suggests that torque, and thus peak torque, should be related to the shear properties of the material. APPENDIX 2 A `NIP’ MODEL FOR TORQUE The voids fraction in the mixer increases rapidly during the mixing process, due to the impression of TiO2 particles into the softer polymer particles. Once there are suf®cient voids in the mixer then the feed ram will exert no stresses upon the materialÐ there is no particulate structure to carry such stresses. Therefore, the stress applied to the feed ram will bear very little in¯uence upon the mix. In these circumstances, the mixer can only exert signi®cant stresses on the material by squeezing it between the rotating blades and the wall. This is a `nip’ phenomenon, and is quite different from the ¯ooded mixer system described in Appendix 1. It is possible to derive a simple nip model, as shown in Figure 14, based upon the following assumptions: 1. The blade is approximated to a simple wedge and the curvature of the mixer wall is ignoredÐco-ordinates are therefore Cartesian, as shown in Figure 14. 2. As the blade proceeds, the material compresses in the x-direction only. 3. The system is in plane strain in the x-y plane. 4. The bed of material, trapped beneath the blade, is at failure and is on the yield locus. 5. Stresses at the blade wall interface are s and t , as shown in Figure 14. 6. The properties of the material may be described by the material and wall yield loci determined from solids properties. The volumetric strain of an incremental element of height h, du, by displacement of the blade by an amount dy: dy tan q du = (A3) h Let Kplane be the plane strain compressibility dy dp0plane = Kplane du = Kplane tan q (A4) h Trans IChemE, Vol 78, Part A, January 2000

CHARACTERIZATION OF PIGMENT POWDERS FOR TiO2/POLYMER DISPERSIONS

Thus, at the outlet to the nip: p0

0 2

h1

h1 L tan q

s0 = s + T T=

(A5)

C tan w

Point(t ,s) is on the Mohr circle tangent with the material yield locusÐFigure 15: t = q0 sin 2c

s0 = p0 + q0 cos 2c

s = s1 + T : s = s2 + T : s = s3 + T 0

p0o = Kplane ln

(A8)

ê Knowing p0 at point y, it is necessary to ®nd t and s in order to calculate the force required to move the blade. Assume that the material is at failure, both internally and at the blade-material interface. This situation can be interpreted in terms of Mohr circles, as shown in Figure 15. The system is at failure with principal stresses s1 and s3. Stress state (t ,s) is on the blade yield locusÐFigure 15Ðat angle c to the major principal stress. Consider the system in effective stress space:

Figure 14. Nip model for blade passing over a bed of material at the mixer wall.

p0 is the effective compressive strain in plane strain (s0 + s03 ) p0 = 1 2 or in 3-D (s0 + s02 + s03 ) p0 = 1 3

ê

47

0 3

s01 + s03 2

p0 =

where T = C/ tan q

(A9)

In plane strain:

T is the tensile stress of the material The use of effective stresses simpli®es analysis. A linear yield locus has the form: t = s tan w + C = tan w (s + T )

q0 = q =

s01 ê

s03 2

The system is at failure: q0 = sin w p0

The yield locus cuts the s axis at ( T,0). Hence, the ê effective stress tranposes the stress system such that the yield locus passes through its origin. The height of an element can be expressed in terms of distance along the blade, y

The system is also at failure at the blade interface:

y tan q ê From equations (A4) and (A6):

(A6)

Equations (A9)±(A11) can be manipulated to give a quadratic equation in s0 :

(A7) ê Equation (A7) can be integrated across the length of the blade to give the force necessary to move the blade. If the compressibility, Kplane, is taken as a constant, the effective compressive stress can be found at any point, y:

T )2 + (s0 p0 )2 q02 = 0 (A12) ê ê ê 0 Equation (A8) is used to ®nd p at point y. The force required to move a blade of unit depth is given by: …L F = (t cos q + s sin q)dy (A13)

h = h1

dp

p0 ê

0 plane

= Kplane tan q

p0o = Kplane ln

h1

h1

ê

t = mw s

m2w (s0

dy y tan q

h1 y tan q u

0

is the angle of the nipÐFigure 15

Figure 15. Mohr circle for material showing internal and blade yield loci.

Trans IChemE, Vol 78, Part A, January 2000

(A10)

(A11)

RICHARDSON et al.

48

Therefore, F can be calculated from the following information: · Failure characteristics of the materialÐfrom the shearbox tests · Coef®cient of wall friction Ðfrom shearbox tests · Plane strain compressibility Ðfrom UCT tests · Geometry of the bladeÐfrom the mixer NOMENCLATURE A C del dp0 du L h h1 Kplane p p0 po0 q R T Tor u x y

shear zone area, m2 cohesion, Pa axial strain increment, ± compressive stress increment, Pa volumetric strain increment, ± width of blade, m height of element, m initial height of element, m plane strain compressibility, Pa compressive stress, Pa effective compressive stress, Pa initial effective compressive stress, Pa deviatoric stress, Pa radius of rotor, m tensile strength of material, Pa torque, N-m volumetric strain, ± Cartesian co-ordinate Cartesian co-ordinate

Greek letters w angle of friction of material, degrees c angle between plane and major principle stress plane, degrees mw coef®cient of wall friction, ± s normal stress, Pa s1, s3 major and minor principle stress respectively, Pa sl axial stress (UCT tests), Pa t shear stress, Pa u angle of blade nip, degrees

REFERENCES 1. Haber, A. and Audet, R., 1990, Factors affecting titanium dioxide dispersion in color concentrates, Proc ANTEC, 20±24. 2. Watson, C. L. and Decelles, G., 1994, The performance of titanium dioxide pigments in Masterbatch applications, Proc ANTEC, 2514± 2517. 3. Bergen, J. T., 1960, in Processing of Thermoplastic Materials, E. C. Bernhardt (ed.), (Reinhold, New York) Ch. 7. 4. White, J. L., 1993, Development of internal mixer technology for the rubber industry, Rubber Chem Tech, 65: 527±579. 5. Freakley, P. K. and Wan Idris, W. Y., 1979, Visualisation of ¯ow

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ACKNOWLEDGEMENTS This project was jointly supported by ICI, Tioxide and University of Teesside. The pigment characterization tests were performed in the Particulate Technology Research Laboratories of University of Teesside, and the mixing trials were performed in the Tioxide Group Services laboratories.

ADDRESS Correspondence concerning this paper should be addressed to Professor A. J. Matchett, Department of Chemical Engineering, University of Teesside, Middlesborough TS1 3BA, UK. (E-mail: a.j.matchett@tees. ac.uk). The manuscript was received 30 April 1999 and accepted for publication after revision 28 September 1999.

Trans IChemE, Vol 78, Part A, January 2000