Roller Extrusion of a Ceramic Paste - Industrial & Engineering

Ind. Eng. Chem. Res. 2005, 44, 4099-4111

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Roller Extrusion of a Ceramic Paste M. C. Peck, S. L. Rough, and D. I. Wilson* Department of Chemical Engineering, University of Cambridge, New Museums Site, Pembroke Street, Cambridge CB2 3RA, U.K.

The roller extrusion of a stiff model ceramic paste has been studied experimentally using a fully instrumented roller apparatus that allows simultaneous measurement of roller separation, separating force, torque, and roll surface pressure. The paste exhibited no noticeable wall slip and little expansion, detaching from the rollers shortly downstream of the nip. Data collected over reduction ratios of 0.92-0.15 and speeds of 1.5-30 rpm (7.9-160 mm s-1) showed consistent trends when plotted against reduction ratio. A small strain rate dependence was observed, which was consistent with the rheological parameters obtained from the Benbow-Bridgwater characterization approach that models the material as a yield strength material. The data were also compared with the results from three models for rolling based on the plasticity approach: an upper bound model, a lower bound model, and a hot metal rolling model originally presented by Orowan. The experimental and predicted results showed good agreement for the latter model, and it was then extended to incorporate a simple strain rate dependency based on the BenbowBridgwater characterization approach. These results indicate that this characterization approach can be used to predict or gauge the rolling performance of such soft-solid materials. 1. Introduction 1.1. Pastes. Pastes are soft-solid materials comprising a high volume fraction of particles that can be readily shaped by extrusion to give macro-structured products such as pellets or catalyst supports. Several paste-forming processes employ rolling, where counterrotating cylinders force the material through a constriction or nip, or rotary moulding, where material is forced into moulds or through dies located in the roller surface. The formulation of materials and design of machines to perform these operations is problematic because the mechanics of paste flow in such rotating geometries has not been studied extensively, principally due to the complex rheology exhibited by pastes and related softsolids. In contrast, rolling is well-understood in applications such as metal sheeting where the material is treated as a plastic1-4 and in the calendering of nonNewtonian polymer melts.5-7 Pastes belong to the class of materials often described as soft-solids as they possess an apparent yield stress (typically of order 10-200 kPa) but are sufficiently soft enough to be shaped readily. Examples include ceramic catalyst supports, cold margarine, soap, chocolate, and detergents. Soft-solids frequently exhibit complex flow behavior, including strain and strain rate dependent responses as well as a yield stress. Their multiphase nature results not only in complex bulk properties but also in a tendency for surface fracture and a high sensitivity to small changes in composition. In response to the problems encountered when characterizing these materials using traditional techniques, Benbow and Bridgwater8 developed an approximate method of material characterization based on simple ram extrusion through capillaries, in which pastes are treated as yield stress materials. The parameters thus obtained can be used to elucidate formulation studies and predict the * To whom correspondence should be addressed. Telephone: +44-1223-334777. Fax: +44-1223-334796. E-mail: [email protected].

performance of different die designs9 and extruder and flow geometries.10,11 The reliability of the BenbowBridgwater approach has been discussed previously.12,13 One of the aims of this work was to establish whether this methodology can be used to translate rheological parameters obtained using ram extrusion to a significantly different extrusion geometry, namely rolling. In the Benbow-Bridgwater approach, the total pressure drop (P) required to extrude a paste from a circular barrel of diameter (D0) through a concentric circular die of diameter D and length L is given by

( )

P ) 2(σ0 + RV) ln

()

D0 L + 4(τ0 + βV) D D

(1)

where V is the mean velocity of the paste in the dieland (i.e., assuming the flow is wall-slip dominated). The first term on the right-hand side represents the work due to the paste undergoing quasi-plastic deformation associated with the contraction; σ0 is the die-entry bulk yield stress of the paste. The second term represents the work due to the shear effects within the die-land; τ0 is the paste-die wall shear yield stress. The parameters R and β are used with V to introduce a strain rate dependence of the yield and shear stresses, respectively. For materials with a strong strain rate dependency, several workers have used the Herschel-Bulkley fluid constitutive model to describe the paste rheology (e.g., Adams et al.14). In this context, we use the term “stiff” to describe pastes where the rate-invariant term (σ0 in eq 1) exceeds the strain rate contribution (RV) for the operating regime (i.e., the assumption that the material may be treated as an engineering plastic is reasonable). 1.2. Rolling. A schematic of the process under consideration is shown in Figure 1: a single pair of equal-sized rollers counter-rotate at the same speed with a feed sheet of finite thickness (Hf), which is reduced to the nip thickness (H0) before some recovery to its final exit thickness (He). The sheet is wide and thin so that it may be assumed to be undergoing plane

10.1021/ie040270g CCC: $30.25 © 2005 American Chemical Society Published on Web 04/29/2005

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Ind. Eng. Chem. Res., Vol. 44, No. 11, 2005 Table 1. Constituents of the Ceramic Paste Mix25

component

supplier

R-alumina 1500 Universal Abrasives Ltd, R-alumina 600 Stafford, UK R-alumina 80

strain. Some details of the rolling process depend on the nature of the material and the relative sizes of the sheet and rollers. However, there are essential features common to all rolling of wide sheets in the absence of forward or backward tensions, as follows.2 During rolling the sheet is compressed vertically and expands longitudinally in the direction of rolling. In the region near the initial contact between sheet and roller, the surfaces of the sheet are being drawn toward what is called the “roll nip”. Continuity demands that the roll surfaces must be moving faster than the average sheet velocity, so there is a shear force acting as indicated in Figure 1 causing the pressure in the sheet to increase toward the right. Due to its longitudinal extension, the material is being squeezed out from the roll gap. In the region toward the sheet exit the shear forces at the roller surfaces are therefore opposing the resulting flow forward, and pressure is decreasing toward the right in Figure 1. Somewhere near the middle of the arc of contact there is a “neutral” plane, or “point”, where the pressure is at a maximum and the rollers are neither pulling the sheet toward the nip nor impeding its exits the material at the neutral plane is moving at the velocity of the roll surfaces. The challenge in modeling the rolling process is in quantifying the associated shear forces and deformation behavior with an appropriate model for the material rheology. Two detailed studies have been reported investigating the plane strain rolling of pastes or soft-solid materials. Studies of Plasticine modeling clay reported by Adams et al.,15 summarizing the work by Sinha,16 compared models based on two types of constitutive equation, namely, (i) a modified plasticity model of the form:

σy ) B˘ ab

(2)

where σy is a uniaxial yield stress; ˘ is extensional strain rate; is extensional strain; and B, a, and b are parameters characteristic for a given material; and (ii) a power law or Herschel-Bulkley fluid. Values for σy were obtained using approximations of Orowan’s hot metal rolling model equations.17 An average effective strain rate was calculated on the assumption of a von Ka´rmań uniform flow field,18 and the incremental strain was defined as the natural logarithm of the ratio of feed sheet thickness to current sheet thickness. Sinha16

26.3 26.3 26.3

potato starch

BDH Laboratory Supplies, Poole, UK

3

bentonite clay

Steetley Bentonite and Absorbents Limited, Retford, UK

3

water (reverse osmosis)

Figure 1. Cross-section of the rolling process with descriptive terms used.

wt fraction (%)

15

concluded that this plasticity model may not have provided an adequate description for the rolling of this material: the values for σy obtained from the force and torque readings differed by 40-50%, and the material appeared to have a high degree of strain dependence (b ) 0.3). The data correlated extremely well with a (strain independent) power law model for calendering. However, despite a good level of correlation with the rolling data, the characteristic parameters obtained by the calendering analysis were 6-10 times lower than comparable material parameters obtained via other techniques, highlighting the difficulty in quantifying the paste rheology. Cheyne and Whitlock19 studied the rolling of two different R-alumina-based ceramic pastes and Plasticine modeling clay. They did not observe a significant increase in roll torque with speed and found that a lower bound plasticity model showed greater promise than calendering approaches for their stiffer pastes. However, they reported poor data reproducibility owing to shortcomings in their equipment. A similar apparatus, extensively modified, has been employed in the current work. The aims of this study are 3-fold. First, to observe and record the behavior of a well-behaved ceramic paste (i.e., free from flow phenomena such as liquid-phase migration20) undergoing rolling. Second, to investigate the application of plasticity-based rolling models to describe this system. This approach requires independent estimations of the material yield stress, and the third aim is therefore to explore the application of the BenbowBridgwater characterization method to provide quantitative parameters appropriate to this geometry. 2. Materials and Experimental Apparatus 2.1. Paste Preparation and Characterization. The ceramic paste used is composed of white fused R-alumina powders of various size fractions, water, and liquid-phase modifiers in the form of potato starch and bentonite clay (Table 1). The material is referred to as “Mix25” and is a model paste of long-standing interest.10,21,22 The material is readily moulded by extrusion at room temperature, exhibits low strain rate dependency, and is not prone to liquid-phase migration during extrusion. The 2.5 kg batches of Mix25 were prepared as follows. The dry components of each batch were weighed into a planetary mixer (AE 200, Hobart Manufacturing Company Ltd., London, UK) and mixed at 40 rpm for 10 min. The water was added, and the material was wet mixed for 5 min until the water was satisfactorily dispersed. The material was then passed twice through a high shear pugging attachment in order to break up agglomerates and yield a uniform and

Ind. Eng. Chem. Res., Vol. 44, No. 11, 2005 4101 Table 2. Benbow-Bridgwater Characterization Parameters Obtained by Amarasinghe21 for Mix25, as Shown in Eq 1 parameter

value

σ0 [MPa] R [MPa‚s‚m-1] τ0 [MPa] β [MPa‚s‚m-1]

0.11 0.15 0.01 0.16

cohesive paste. The paste was stored in a sealed bag for 2 h to equilibrate to room temperature. Characterization parameters are generally determined by loading a circular steel barrel of 25 mm i.d. with paste and ram extruding the material through concentric square entry circular dies of diameter 3 mm and lengths 6, 12, 24, 36, and 48 mm. In each test with a given die, the ram is stepped through eight different speeds using a Dartec SA100 loading frame. Assuming the paste to be incompressible, the ram speeds of between 0.016 and 0.833 mm s-1 produce extrudate velocities (V) between 1.1 and 57.8 mm s-1 and apparent die-land shear rates between 3 and 154 s-1. The characterization parameters used in this study were those determined by Amarasinghe21 and are listed in Table 2. Details of this choice are discussed by Peck.23 The values indicate that σ0 . RV over the range of V employed, which indicates that the strain rate dependence of the overall bulk yield stress term is slight. 2.2. Paste Feed Sheet Preparation. A number of rectangular paste feed sheets of uniform thickness were prepared for feeding one by one through the rolling apparatus. Appropriate quantities of the paste were rolled by hand with a long brass rolling pin on a flat surface between brass strips of a range of known and uniform cross-sections. The flat preparation surface was initially covered securely with a smooth sheet of aluminum foil, which allowed the sheets to be lifted with minimum disturbance since the foil could subsequently be peeled easily from the sheets. Six brass strip heights were employed, namely, 2.34, 3.20, 4.80, 6.40, 8.10, and 9.60 mm. Narrow sheets of feed material measuring at least 100 × 350 mm were prepared between two of these strips of the chosen size and then trimmed with a scalpel to a width of 100 mm. This was the maximum practical width given the length of the rollers and was employed to minimize the lateral spreading of the sheets during rolling, thereby approximating planar flow. The strips of sheet were cut into a number of separate feed sheets of length ranging from 40 to 70 mm and placed in a sealed plastic bag where they were stacked three or four sheets high and interleaved with aluminum foil so as to minimize moisture loss from the sheet surface. In general, experiments were designed so that the output sheets were 80-100 mm long. These lengths allowed a steady state to be reached during the rolling experiments. No feed sheet was used more than once. 2.3. Rolling Apparatus. A schematic plan of the experimental apparatus is shown in Figure 2. The apparatus comprises an electric motor driving two counter-rotating shafts connected by directly intermeshing cog-wheels. Each shaft is fitted with a brass roller of diameter 100 mm and length 110 mm. The apparatus allows the distance between the rollers to be altered up to a maximum of 40 mm ((0.1 mm) and enables the measurement of several key rolling parameters. Each drive shaft is fitted with commercial torque transducers: a 20 Nm unit (type TT24AD, Experimental and Electronic Laboratories, East Cowes, Isle of Wight,

Figure 2. Schematic plan view of roller extrusion apparatus.

UK) measures the torque driving roller 1; a similar unit (type E300RWT, Sensor Technology Ltd, Banbury, Oxon, UK) measures the torque driving roller 2. Roller 2 is mounted on framework held in place with a commercial load cell (5 kN type D96 load transducer, Sangamo Schlumberger), and roller 1 is mounted on framework, the position of which may be altered. At the end of the roller 1 shaft is a cellophane disk with 180 opaque segments and a carefully aligned optical sensor. This disk, in conjunction with a computer program, permits precise calculation of the orientation of the rollers ((0.004 rad) relative to the roll nip. The surface of roller 2 is fitted with a 100 kPa piezoelectric pressure sensor (type XTM-190-100, Kulite Semiconductor Products Inc., Leonia, NJ), and the drive shaft includes a slip ring providing electrical connection to an amplifier. The diameter of the circular flat surface of the pressure sensor is 3.75 mm. It is slightly recessed from the curved roller surface by a maximum of 0.1 mm. Roller 2 is powered directly by a 750 W variable speed electric motor (type 90L 4GMVFOTG, Anyspeed, Halesfield, Telford, UK), and roller 1 is powered by means of the intermeshing cog-wheels. Due to the compliance of the apparatus during operation, laser displacement sensors were used to obtain real time measurement of the separation of the rollers. Both units were made by Baumer Electric, model OADM 2014460/S14C, with a resolution of 0.06 mm. The torque and force transducers were calibrated in situ by loading with weights via a pulley and string. The static calibration was tested dynamically by applying tangential loads to the rollers rotating at various speeds. The pressure transducer was calibrated with a dedicated Druck DPI601 pressure indicator prior to installation. The experimental uncertainty of the roll torque measurement was estimated as (0.2 Nm, and that of the roll separating force was estimated as ( 5 N. 2.4. Experimental Procedure. In each experiment the gap between the rollers was adjusted, and the rollers were started and set to the desired speed. The feed sheet was held just above the roll nip and then releasedsthe sheet was allowed to pass between the rollers under its own weight. Care was taken to release the sheets so that the pressure transducer made contact with the center of each. While being rolled the sheet was supported laterally as required, depending on its length and rigidity. The rollers were cleaned with soapy water and dried thoroughly after each test. Experiments were conducted with the range of feed sheet sizes described in section 2.2, which were reduced in thickness by 8-85%, at three rotation speeds of

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1.5-2, 6, and 20 rpm ((2%). Further experiments were conducted in which only the roller speed was varied, by increments between 1.5 and 30 rpm. Sheet width was measured with a ruler to the nearest millimeter. Feed sheet thickness ((5%) was measured by placing the sheet between two rigid Perspex plates of known thickness and measuring the total width with callipers at a number of positions. In cases where the output sheet did not stick to a roller surface, this method was also used to measure the exit sheet thickness ((0.2 mm). However, in the majority of cases where the sheet adhered to one or both rollers, the depth gauge of the callipers was used to measure the thickness of the sheet in situ. Readings from the various transducers were recorded digitally 4600-5000 times per roller revolution (a maximum of 17 500 readings per second). Oscillations originating from misalignment of the torque transducers were removed by fitting a Fourier series curve to the signal recorded before the sheet was fed to the rollers, extrapolating this trend, and deducting the predictions in order to estimate the actual torques being imposed on the rollers at the end of each drive shaft. As well as correcting for the poor performance of the transducers, this procedure also compensated for the rolling resistance of the various bearings in the drive shafts. 3. Experimental Results 3.1. General Observations. One hundred and fifteen experiments were completed with sheets of Mix25. The paste was fairly stiff and cohesive, so there was seldom any problem with sheets crumbling or distorting during preparation or feeding to the machine. The lateral spreading of the thickest sheets undergoing large deformations was less than 6%; hence, one can assume plane strain conditions.7 The rolling of the paste was characterized by a high level of adhesion between the sheet and the roller. At the initial contact, the level of friction between the paste and the rollers was always sufficiently high for the roller to draw the leading edge of the sheet toward the nip, even for the largest sheet reductions. At the end of the roll bite, the sheets generally adhered to one (or sometimes both) of the roller surfaces. This effectively eliminated the possibility of slip after the neutral point. In only 11 of the 115 experiments did the sheet remain straight and pass through the roll gap without sticking to either roller. The tendency for such adhesive contact contrasts with the rolling of metals and is believed to be due to greater cohesion between the paste and the roller, the larger reductions possible, and the relatively flexible nature of the Mix25 sheets. A sheet was less likely to stick to a roller if it were thick (perhaps on account of the greater rigidity of the feed and exit sheets) and if it were reduced in thickness only slightly. All the nonsticking sheets were reduced in thickness by less than 23%. Furthermore, when the sheets did stick to a roller, the apparent strength of adhesion increased as the percent reduction increased. For very large reductions the sheet became stuck to one or both rollers so firmly that it had to be scraped off. The degree of adhesion appears to be related to the pressure generated during the deformation, but a reliable method of measuring the adhesive force would be required to take investigations further. The flow of material at the entry to the roll bite was also notably asymmetric. Upon being fed to the rolls, the sheet would frequently “tilt” or be dragged toward one of the rollers and remain tilted for the duration of

Figure 3. (a) Roll torque and (b) roll separating force data as a function of time for the rolling of Mix25 at 1.7 rpm. Feed sheet thickness Hf ) 9.6 mm, exit sheet thickness He ) 3.1 mm.

the experiment. Attempts to keep the feed sheet straight involved the application of a significant force against the appropriate face of the sheet, a force that the sheet would continue to resist, often curling against it and still entering the roll bite at an angle. It was noted that if the feed entered asymmetrically, the product was more likely to adhere to the opposite roller, away from which it was tilting. However, this asymmetry had little effect on the average roll torque and roll separating force, as now discussed. 3.2. Roll Torque and Roll Separating Force. Figure 3 shows some typical roll torque and separating force data for the Mix25 sheets. The average roll torque and roll separating force show similar trends. Most tests featured periods of steady state, with transient responses at the beginning and particularly at the end of the runs. The average fluctuation in the average roll torque and roll separating force about the steady-state value was 12% and 8%, respectively. Figure 4 demonstrates an example of asymmetry in the torque readings. The difference between the maximum and minimum torque is ∼1 Nm; the average of this quantity over all the Mix25 experiments was 1.32 Nm and is equal to 90% of the mean of the average torque. However, it can be seen that such asymmetry does not have a significant effect on the average torque or the roll separating force. The work input to deform the sheet, a function of the sum of the two torque values, is reasonably steady. The average roll torques and roll separating forces at a nominal speed of 6 rpm for a range of feed sheet thickness are shown in Figure 5 as functions of exit sheet thickness. The trends in the experimental data are generally linear, and similar trends were obtained at speeds of 1.5 and 20 rpm (data not reported). When plotted against a reduction ratio (defined as sheet thickness/feed sheet thickness), the roll separating forces follow a straight line, as illustrated in Figure 6. Since the sheet was found to expand beyond the roll nip, there are two distinct ratios that could be used: Figure

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Figure 4. (a) Roll torque and (b) roll separating force data as a function of time for the rolling of Mix25 at 3 rpm. Feed sheet thickness Hf ) 11.0 mm, exit sheet thickness He ) 5.4 mm.

Figure 6. Steady-state roll separating force data as a function of sheet reduction ratio for the rolling of Mix25 at 6 rpm, for feed sheet thickness Hf ranging from 2.7 to 10.7 mm as indicated in the legends. Reduction ratio defined in terms of (a) exit sheet thickness He and (b) nip height H0. Lines show linearly regressed trend with correlation coefficient R2.

Figure 7. Average roll torque as a function of sheet reduction ratio He/Hf for the rolling of Mix25 at 6 rpm, for feed sheet thickness Hf ranging from 2.7 to 10.7 mm as indicated in the legend.

Figure 5. (a) Average roll torque and (b) roll separating force data as a function of exit sheet thickness He for the rolling of Mix25 at 6 rpm, for feed sheet thickness Hf ranging from 2.7 to 10.7 mm as indicated in the legends. Error bars indicate fluctuation in parameter about steady-state mean value.

6a employs the ratio of exit thickness to feed thickness (He/Hf), while Figure 6b employs the ratio of nip height to feed thickness (H0/Hf). It can be seen that the former gives a marginally better match to a best fit linear trendline. Also, Figure 6a shows the trendline crossing the abscissa closer to the expected value of He/Hf ) 1. The corresponding plot of the average roll torques, shown in Figure 7, does not exhibit a similar correlation. As well as not following a linear trend, the data series

are curved such that they appear to intersect the abscissa at a reduction ratio of 0.9 or less, rather than 1 as might be expected. The paste exhibited a slight dependence on roller speed. Figure 8 illustrates that the average roll torque and roll separating force both increase by about 25% as the roll speed is increased from 1.5 to 30 rpm. The rise is approximately linear although there is a noticeable degree of scatter. 3.3. Pressure Profiles. Figure 9a shows the roll surface pressure profiles recorded for a set of experiments at 6 rpm. The theoretical contact and exit angles of the sheet may be calculated from the feed and exit sheet thickness and the roll nip dimension. These theoretical angles are shown on the graph as vertical solid lines. There is reasonable agreement between these and the pressure profile, but there is clearly some

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adhesion as the sheets detached from the roller is indicated by the negative pressure detected by the transducer. The adhesion between the surface of the transducer and the sheet may not be entirely representative though, since the surface texture and material of construction is not the same. Finally, the one sheet that did not stick to either roller, the 6.8 mm exit sheet in Figure 9b, did not exert any force on the transducer beyond the roll nip. 4. Modeling Figure 8. Influence of roller speed on average roll torque (b) and roll separating force (0) for Mix25. Hf ) 7.5 mm, He ) 3.5 mm. Error bars indicate fluctuation in parameter about steady-state mean value.

Figure 9. Roll surface pressure profiles for Mix25 as a function of angular position from nip (positive toward roll exit): (a) roll speed ) 6 rpm, Hf ) 3.7 mm, He ranging from 3.1 to 1.3 mm as labeled; (b) roll speed ) 20 rpm, Hf ) 7.1 mm, He ranging from 6.8 to 1.5 mm as labeled. Predicted entry and exit angles marked as solid vertical lines.

fluctuation in the feed angle. The agreement with the theoretical exit angle is generally good. A second set of pressure profiles recorded at 20 rpm is shown in Figure 9b. Three of the profiles show a prominent step or hump at 50 kPa at the entry to the roll bite. This hump was observed in many of the tests. In the experiments featured in the above plots, the exit sheet adhered to roller 2 (the roller with the transducer) on three occasions, for all of which the transducer recorded a negative pressure long after the sheet detached from roller 1. Furthermore, in all but one of the other cases the sheet stuck to roller 1 and the pressure profiles confirm that the exit sheet is thicker than the roll nip. The data show that the pressure does not return to zero until approximately 0.05 rad downstream of the nip, equating to a distance of about 2.5 mm. This corresponds to an exit sheet being about 0.12 mm thicker than the nip. The degree of

A perfect plastic material is one that exhibits a yield stress, above which it is observed to flow with no significant increase in shear stress with shear rate. Plasticity theory and the concept of ideal work underpin the die-entry analysis of the Benbow-Bridgwater model for paste flow in a ram extruder. This approach is wellsuited to the extrusion of the Mix25 ceramic paste, which exhibits only limited strain rate dependence. Plasticity theory is therefore expected to yield a useful description of the rolling data. The perfect plastic material model is a popular and convenient approximation in the study of metal-forming processes, even though it fails to accommodate the elastic properties, anisotropy, or strain hardening effects of the metal. A comprehensive review of plasticity theory and the perfect plastic model is given by Chakrabarty,18 but briefly the three elements required to define completely the response of a perfect plastic material are (i) a statement of force equilibrium; (ii) a yield criterion, describing the stress state at the onset of plastic flow; and (iii) a flow rule, describing how the material deforms in response to the critical stress state. Since sheet rolling is predominantly a two-dimensional process, it is usual to present the subsequent equations as such. Even when simplified in this manner, however, it is not clear how the equations may be used to solve practical problems. Hence three analytical methods commonly used to solve problems involving perfect plastic flow are used to investigate the rolling behavior of Mix25 paste: an upper bound theorem; a lower bound theorem; and a hot metal rolling model. An extension of the analogy between die extrusion and rolling is also considered by attempting to modify the plastic shear yield stress with a simple strain rate dependence derived from the R Benbow-Bridgwater characterization parameter. 4.1. Upper Bound Theorem. The upper bound theorem will generally yield an overestimate of the forces required for a particular plastic deformation. To apply the theorem, a collapse pattern is first postulated, which consists of a compatible set of displacements and strain increments within the material by which it could feasibly deform. The mechanism may be continuous or involve rigid body motions with sliding contact between them. The internal energy dissipation that would occur for that mechanism is then calculated and equated via the principal of virtual work18 to the rate of work being done by the loads on the system. The convenience of the upper bound theorem stems from the fact that, in the case of two blocks separated by a plastically shearing layer, for example, the total rate of energy dissipation is independent of the thickness of the layer. In many upper bound solutions the deformation is thus proposed to occur within infinitesimally thin planes of material. In this case the rate of energy dissipation per unit area of the slip plane is given very simply by the shear yield


Figure 11. Upper bound model geometry angle φc as a function of reduction ratio He/Hf for feed sheet thickness Hf ) 3, 5, 7, 9, and 11 mm, shown in order as indicated by arrow.

Figure 10. Upper bound solution for rolling with a wide range of geometries. The sketches show different configurations which arise if φc is permitted to vary: (a) φc ) 45° as proposed by Johnson and Mellor,24 where φn is the angular position of the maximum roll surface pressure from the nip; (b) thick sheets with small reductions, 45° < φc < 90°; and (c) thin sheets with large reductions, 0° < φc < 45°.

stress k × the slip velocity.24 Since upper bound models are based on a proposed velocity field, they can also provide an insight into the deformation process. Various authors have proposed upper bound solutions for the sheet rolling of metal. The simplest is that by Johnson and Mellor,24 in which the metal is sectioned into curvilinear triangles that rotate with the roller, thereby modeling the sticking friction of hot rolling. It is this analysis that has been used as a basis for calculation of an upper bound value for the observed rolling torque, because of the ease of application to different sheet sizes. Figure 10a shows Johnson and Mellor’s suggested velocity field, in which they specify a 45° angle between the second discontinuity and the center line. However, it was found that in so doing they have imposed an unnecessary constraint on the solution. Lower, and therefore more accurate, values for torque may be achieved by permitting the angle φc in Figure 10 to take an optimal value (defined as that which minimizes the predicted roll torque) between zero and π. The optimum angle in the proposed geometry was calculated using Mathematica (Wolfram Research Inc., Champaign, IL), and predictions of the roll torque were made for comparison with other models and experiment. The optimal values calculated for φc are shown in Figure 11 as a function of reduction ratio. They lie across the full range of permitted values (0 - π) and are sensitive to both He/Hf and He, particularly for reduction ratios >0.5, the main region of interest. 4.2. Lower Bound Theorem. One way of obtaining a lower bound solution for plastic deformation is to equate the rate of work done by external forces on the system with the rate of internal energy dissipation

during homogeneous deformation between initial and final geometries. Homogeneous deformation is the most efficient method of achieving the desired transformation24 and is termed the “ideal work”. It does not include any of the redundant deformations (distortions) that occur in real processes, which are included in upper bound solutions. Such calculations are usually straightforward and form the basis of the Benbow-Bridgwater analysis of paste extrusion. Thus a lower bound solution, together with an upper bound prediction, should encompass the experimental data. In the rolling of pastes, the work is done by the rotating rollers. To generate a lower bound prediction, the power supplied by the rollers and the work done on the paste during this deformation are equated. Inertial terms are ignored, and it is assumed that the whole exit sheet leaves the bite at the roll surface velocity, which is consistent with the majority of experimental observations. For rolling, continuity of an incompressible material gives

Hfl0 ) Hel

(3)

where l0 and l are the lengths of paste sheet being consumed and produced per unit time, respectively. Since the exit sheet moves with the rollers at the tangential velocity and neglecting the curvature of the roll surface, then

l ) ωR

(4)

where R is the radius of the rollers and ω is their angular velocity. The ideal plastic work per second associated with the deformation is thus given by

∫HH f

e

-

( )

σyHfW0l0 Hf dH ) σyW0RωHe ln H He

(5)

where W0 is the width of the sheet. The power supplied by the two rollers is 2Tω, where T is the average torque. Equating these expressions, and using a von Mises yield criterion to relate the uniaxial yield stress σy to the shear yield stress k, yields a prediction for the torque:

T)

( )

Hf x3 kW0RHe ln 2 He

(6)

Since the ideal work calculation does not concern itself with the stress field within the material, it cannot offer any predictions of roll separating force or surface pressure profile.

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average longitudinal pressure (p) and produces the following relation between q and p:

π q-p) k 2

(7)

This is the relationship between the roll normal stress and the average horizontal normal stress in the material, on the assumption that it is deforming throughout and exhibits the proposed stress distribution. The final part of the analysis equates the change in p across the arc to the hoop stress within it and thus to q. This is achieved via a simple force balance over the arc of material and the differentiating of eq 7:

Figure 12. Geometry of hot metal rolling analysis, with detail of shaded elemental section of arc.

4.3. Hot Metal Rolling Model. Problems involving material flow must consider the conditions at the boundary of the deforming region. In plasticity theory, contact with a containing wall is often assumed to be either smooth (no friction with perfect slip) or rough (sticking friction). However, there can be slipping at the boundary when the wall friction is insufficient to cause the material to yield internally. Metal-forming processes are categorized as cold forming or hot forming, the main difference lying in the relative magnitudes of the bulk yield and the surface frictional stresses. In hot rolling the bulk yield stress is sufficiently low that the sheet and rollers tend to be in sticking contact. In cold rolling, slippage against the roller surface is more common, and one of the simplest and earliest analyses, by von Ka´rmań,25 considers this scenario. The experimental observations reported in section 3 indicate that there is a predominately nonslip interface between the paste sheet and the rollers, which suggests that descriptions of hot metal forming will have more relevance to paste processes. The following analysis of hot rolling is by Orowan2 and is included in the text by Chakrabarty.18 While the von Ka´rmań analysis is used for small sheet reductions found in cold metal sheet rolling, hot rolling of metals allows for a slightly larger range of sheet reductions. The sheet is therefore in contact with the roller surfaces through a larger angle, and the one-dimensional analysis of von Ka´rmań is no longer sufficient. The hot rolling model of Orowan extends the cold rolling principles to a curved geometry and changes the boundary condition at the contact with the roll surfaces from slipping to sticking friction. The model considers an arc of material running from the roller to the center line, as shown in Figure 12. The analysis begins by assuming a stress distribution in this arc of material. The first assumption is that the entire arc of material between the rollers is at the yield point. Bounded by the roller, the material touching the roller surface is confined to pure shear, so the assumption defines the shear stress τFθ as being equal to the shear yield stress k at this point. The shear stress is also well defined at the center line where, due to symmetry, τFθ is 0. Between these two points, the analysis proposes that τFθ increases linearly with distance from the center line and also assumes that the normal stress component σθ has a constant value within the arc. The analysis proceeds by relating the stresses σF and σθ to the normal stress against the surface of the roller (q) and the

d d π (ph) ) q - k h ) 2R(q sin φ ( k cos φ) dφ dφ 2

[(

)]

(8)

where h is the vertical roll gap at position x. The direction of the tangential force at the roller surface must change at some point along the arc of contact for the pressure within the sheet to be zero at both ends. The positive sign holds on the exit side and the negative sign on the entry side of the roll bite. The position at which the solutions meet is the prediction of the neutral point. The differential equations for the entry and exit side in eq 8 of the neutral point were solved numerically with the aid of Mathematica by setting p to 0 (and thus q to πk/2) at φ ) 0 (the roll nip) and at φ ) φf, the point of first contact of the sheet with the rollers. This choice of boundary conditions implies that in hot metal rolling it is appropriate to treat the sheet as leaving the rollers at the nip. Unlike the calendering of polymers, the sheet does not expand significantly beyond it. The position of the neutral point (at φ ) φn) determines the roll torque because of the constant tangential force at the roller surface:

T ) W0kR2(φf - 2φn)

(9)

The roll separating force (F) is given by

F ) W0 R

∫0φ q cos φ dφ + φ φ W0kR(∫φ sin φ dφ - ∫0 sin φ dφ) f

f

n

n

(10)

Approximate algebraic solutions to these equations for small reductions were published by Sims26 and tabulated by Chakrabarty.18 4.4. Comparison of Models. The predictions of Orowan’s model (eq 9) are illustrated in Figure 13 alongside those of the upper bound and lower bound (ideal work; eq 6) models, for the experimental configuration of R ) 50 mm; 3 mm < Hf < 11 mm. The ideal work model diverges significantly from the other two predictions as the exit thickness decreases. This discrepancy occurs since in practice large reductions in sheet thickness involve a greater proportion of redundant work. The maximum torque predicted by the ideal work model can be found by differentiating eq 6 and occurs when He ) 0.368Hf. This maximum occurs because, for a given reduction, the ideal work is proportional to volumetric flow rate, and for a small exit thickness this relation dominates. The ideal work calculation is the analogy of the Benbow-Bridgwater analysis of ram extrusion. However, a maximum in the rolling torque profile is physically implausible and


Figure 13. Roll torque (T) predictions as a function of the sheet reduction ratio He/Hf using the upper bound, ideal work (eq 6) and Orowan (eq 9) models, expressed in terms of T/W0k, where W0 is the feed sheet width and k is the material shear yield stress. Roll radius R ) 50 mm; Hf ) 3, 7, and 11 mm, shown in order as labeled for each model.

reveals a fundamental difference between the two processes: ram extrusion, unlike rolling, would still be possible if the equipment surfaces that the paste came into contact with were all smooth. The ideal work model therefore cannot be expected to provide an accurate prediction for the rolling process, even if used in combination with the Benbow-Bridgwater parameters from the analogous ram extrusion analysis. The upper bound and ideal work predictions are fairly close for reduction ratios > 0.8. However, the nonzero intercept of the upper bound model for no reduction in sheet thickness would not be expected in experimental data. This peculiarity occurs because the model involves shearing the material along the first slip surface, only to shear it immediately back to the original configuration at the second. The speed of rolling is not predicted to affect the torque on the rollers. This result follows directly from the assumption of perfect plasticity whereby the stress within the material, and thus the shear stress at the roller surfaces, is limited to the material yield stress and is independent of the rate or extent of deformation. The hot metal rolling model is thus observed to combine the plausible features of the upper bound and ideal work models. The unlikely prediction of a nonzero value for torque by the upper bound model for the case of no reduction in sheet thickness has been eliminated, as has the implausible decrease in torque for very large sheet reductions predicted by the ideal work model. Since the prediction for the torque is derived from a model of the stress between the rollers, estimates for the roll separating force and the roll surface pressure profile can also be calculated. The upper bound and ideal work models do not allow for the prediction of these quantities. Orowan’s model will therefore be used for the following comparisons with rolling test data. The model is typical in assuming that the nip height is the same as the thickness of the exit sheet, although experimental results indicate that for pastes this difference can sometimes be significant. However, to assess the model, it will be assumed that nip height equals the exit sheet thickness. 4.5. Comparison with Experiment. To compare the plasticity theory predictions with experiment, a value for the shear yield stress (k) is required. Using the von Mises yield criterion together with a result by Horrobin27 that relates the uniaxial yield stress (σy) to the die-entry Benbow-Bridgwater characterization term

Figure 14. Comparison of Mix25 experimental results (points) and model predictions (lines) for roll torque (T) as a function of the sheet reduction ratio He/Hf, expressed in terms of T/W0k, where W0 is the feed sheet width and k is the material shear yield stress, for a roll speed of 1.5 rpm: (a) Hf ) 3.8 mm, Orowan model, and ideal work predictions; (b) Hf ranging from 2.9 to 11.1 mm as indicated in the legend, with corresponding Orowan model predictions.

(σ0) for extrusion of a plastic material through smooth dies gives

k)

0.82σ0

x3

(11)

The estimate of k thereby obtained from the Mix25 characterization parameters (Table 2) is 0.05 MPa. Since Mix25 is relatively insensitive to strain rate and approximates a perfect plastic material, the velocity terms of the characterization may be neglected, and this value of k is used alone. However, to minimize the error in the approximation, the experimental data from the slowest tests have been used for comparison with Orowan’s models. Figure 14a shows the experimental and theoretical roll torques for a feed sheet thickness of 3.8 mm as a function of reduction ratio. The experimental error and fluctuation data have been used to provide confidence limits on both the experimental results and the model. The ideal work solution is also marked. Within the bounds of error the agreement with the Orowan model is good. Figure 14b shows the empirical and theoretical roll torques for all the data collected at 1.5 rpm. Figure 7 showed a marked spread in the roll torques for Mix25 at 6 rpm when plotted in this way. The experimental results at 1.5 rpm are spread similarly. Qualitatively the model predicts the spread with increasing feed thickness correctly, although there is some discrepancy in the form of the trendlines. Confidence limits have been omitted for clarity. Figure 15 shows the corresponding plots for roll separating force. The agreement is again reasonably

4108


Figure 16. Comparison of Mix25 experimental (1.5 rpm) roll surface pressure profiles and Orowan model predictions as a function of angular position from nip (positive toward roll exit), expressed in terms of the material shear yield stress, k. Hf ) 3.8 mm; He ) 3.2, 2.8, 2.7, 2.1, 1.7, and 1.3 mm, presented in order from bottom to top.

Figure 15. Comparison of Mix25 experimental results (points) and model predictions (lines) for roll separating force (F) as a function of the sheet reduction ratio He/Hf, expressed in terms of F/W0k, where W0 is the feed sheet width and k is the material shear yield stress, for a roll speed of 1.5 rpm: (a) Hf ) 3.8 mm, Orowan model prediction; (b) Hf ranging from 2.9 to 11.1 mm as indicated in the legend, with corresponding Orowan model predictions.

good within the bounds of experimental uncertainty. The model correctly predicts a smaller spread than it did for the roll torques, although it does not predict a single straight line that was obtained experimentally at 6 rpm (Figure 6). The spread in the results at 1.5 rpm is greater but does not match the trend predicted by the model. While there is good overall agreement, the trends displayed by the model are not entirely satisfactory. In particular, the curved trendlines at small reductions do not resemble those observed experimentally. Some insight into the cause of the discrepancy is provided by the predicted pressure profiles. The experimental and theoretical profiles for the 3.8 mm feed sheets are shown in Figure 16. The two most notable features of the model are the pressure step up at the initial contact between sheet and roller (and step down at the exit) and the discrepancy between the exit angles. The steps in pressure of height πk/2 (see eq 11) are due to the yield stress; stresses within the sheet have to be sufficient to overcome the yield stress when the sheet enters the rollers. They bear some resemblance to the humps observed experimentally (Figure 9) although they tend to be higher, by 50-100%. The size of this step is derived from the assumptions that the entire arc of material between the rollers (Figure 12) is undergoing deformation and that the average longitudinal pressure in the sheet is zero. The geometric constructions of Figure 12 have no purpose other than determining the entry and exit boundary conditions. The shape of the pressure profile in the roll gap is determined by a simple force balance (eq 8). However, the boundary conditions

do not seem appropriate in this case. In practice, even hot rolling of metals features a mixture of slipping and sticking friction.2 The assumption of complete sticking friction is an approximation. The velocity discontinuity at the sheet surface that is implied in these analyses is eliminated by the slipping of the sheet against the roller before it reaches the nip. Orowan2 discussed this briefly by examining the rolling of a laminated Plasticine bar. The original purpose of the experiment was to determine the mechanism by which sticking friction can exist over the whole contact area. The results made it clear, however, that the deformation region does not extend across the whole bar of material at the roll entry. Some regions are deforming, which are observed to grow toward the center of the sheet as rolling proceeds. The boundary conditions are therefore not exact: a more detailed elasto-plastic analysis would be required to determine the true stress profile at the inlet. Unlike metals, paste sheets do not slip against the rollers as they approach the nip. They tend to adhere to the rollers, and the pressure profile results indicate a significant pressure between paste and roller beyond the nip. It may be that the elastic properties of Mix25 contribute to swell of the exit sheet before detaching from the rollers. The discrepancy in the point of the sheet exit is a direct result of the model’s assumptions. The model is overestimating the pressure for low reductions and underestimating it for high reductions, which correlates with the roll separating force results in Figure 15b. Although the positions of the neutral points (where the pressure reaches a maximum) do not agree in Figure 16, the trends of the positions with changing sheet sizes are similar. One further discrepancy is the rounded appearance of the observed pressure peaks, which contrast with the angular appearance of the predictions. Orowan2 explains that this is due to a triangle of undeforming material that will often form against the roller surface at the neutral point. The model is very sensitive to the location of this neutral point, but the predictions remain reasonably accurate because this triangle will tend to be symmetrical and thus does not alter its position. To some extent it is therefore by good fortune that the model makes reasonable predictions of roll torque and separating force. It has been discovered that the hump in the pressure profiles is related to the yield stress of the paste. In other respects, however, the model does not describe the rolling process of pastes


quite as expected from roll torque and separating force predictions. 4.6. Strain Rate Dependence. None of the plasticity models in their form above can describe the slight increase in roll torque and separating force that are observed for Mix25 as the roller speed is increased. To reflect the influence of roller speed, they must be modified to include a strain rate dependence for the material. The rolling model of Orowan is formulated so that strain or strain rate dependence of the shear yield stress can be incorporated. In principle it would be possible to include a strain rate dependent yield stress for pastes. However, such calculations would require a reliable measure of the strain rate dependency of these materials, detailed information of which is notoriously difficult to obtain. In the Benbow-Bridgwater characterization of strain rate dependent materials, the mean extrudate velocity in the die-land is used as an indicator of the strain rate in the die-entry region. However, since the Benbow-Bridgwater R velocity factor is too specific to ram extrusion to be applied directly to rolling, a mean strain rate for the die-entry region deformation has been estimated. The mean extensional strain rate may be expressed as28

jε˘ ) 1 Ls

∫0L ε˘ dx ≈ L1s∫0L s

s

( )

∂ux dx ∂x

(

)

(13)

where Q is the volumetric flow rate of the paste. Evaluated for the extrusion geometry used for the characterizations and the estimated deformation zone length of (25-3)/2 ) 11 mm, this may be expressed as

jε˘ ) 0.27 V D

(14)

This is equivalent to mean strain rates of up to about 5 s-1. To insert this expression into the BenbowBridgwater equation for the entry region, the die-entry pressure drop term is transformed to the following:

(

P0 ) 2 σ0 + R′

[DV]) ln( D ) D0

(15)

where R′ ) RD. Combining eqs 14 and 15 gives an expression for the shear rate adjusted die-entry uniaxial yield stress (σ′0), which can be applied to rolling:

[ ] jε˘ 0.27

(16)

To replace the yield stress used in the plasticity models by this expression, an estimate of the mean strain rate during rolling must be obtained. In the plane strain of rolling, an equivalent statement of the strain rate is the rate at which the thickness of the rolled sheet decreases in relation to its current thickness.18 This may be expressed as

dH dφ dx dφ dx dt ε˘ ) H

(17)

where a plane contacting the rollers at an angle φ from the nip is being considered, and defining x as the distance upstream from the nip. Geometry gives the sheet thickness as

H ) H0 + 2R(1 - cos φ)

(18)

, and the sheet average velocity can be expressed in terms of the volumetric flow rate (Q) and the sheet width (W0):

(12)

where Ls is the length of the deformation zone and ux is the paste velocity in the direction of extensional flow. When ux is taken as the plug flow velocity, the mean strain rate is found to be dependent only on the length and not on the shape of the deformation profile. Mix25 was seen to produce static zones during ram extrusion at the edges of the deformation zone. On the basis of this evidence, the deformation zone is assumed to have the length of a conical die-entry region with an entry angle of 45°, and thus the diameter of the deformation zone decreases linearly in the axial direction of paste flow. When evaluated for a barrel diameter of D0 and a die-land diameter of D, the mean strain rate is therefore given by

jε˘ ) 1 4Q 1 - 1 Ls π D2 D 2 0

σ′0 ) σ0 + R′

dx Q ) dt HW0

(19)

Differentiating eq 18 and combining with eqs 17 and 19 yields

ε˘ )

2Q sin φ W0H2 cos φ

(20)

Assuming that the sheet exits at the roll surface speed, this can be written as

ε˘ )

2H0ωR sin φ H2 cos φ

(21)

The mean strain rate for rolling is then obtained from

jε˘ )

1 φf - φe

∫φφ ε˘ dφ f

e

(22)

where φf and φe are the feed and exit angles, respectively. When evaluated, this indicates that the rolling experiments featured mean strain rates of up to 6 s-1. In the forthcoming comparisons this expression is evaluated for each test, and the yield shear stress (k) as given by eq 11 is modified for strain rate effects via eq 16. Corrections for the Mix25 velocity terms are small. The above procedure therefore has a small effect on the model predictions. The best demonstration is to examine the increases in roll torque and separating force that have been observed when the roller speed is increased. Figure 17 shows experimental results and model predictions for feed sheets of 7.5 and 3.6 mm being reduced to 3.5 and 2.1 mm, respectively. The small increases with roller speed that were observed in Figure 8 are reproduced by the model. The predicted increases are smaller but of the correct order of magnitude. No pretence will be claimed about the accuracy of this approach. In ram extrusion, as well as in rolling, the strain rate varies throughout the deformation zone. This

4110


Figure 17. Influence of roller speed on (a) average roll torque per unit width and (b) roll separating force per unit width for Mix25. Experimental results (points) and model including strain rate dependence (solid lines) for feed sheet to exit sheet thickness of 7.5 to 3.5 mm (0) and 3.6 to 2.1 mm (b).

has not been taken into account, either in the modification of the R term or in the application of the new parameters to the rolling analysis. If the strain rate dependence of materials is significantly nonlinear, then using an average value for the strain rate in calculations can only approximate the stresses in the system, at best. However, there are some qualitative similarities between the ram extrusion and rolling processes. Both feature parallel feed streamlines, a region of convergence with extensional strain and some shear strain energy losses and parallel product streamlines. In general terms, if the pragmatic Benbow-Bridgwater approach is to be of practical use in other geometries, then these similarities will be sufficient to cancel some of the errors associated with using these mean values. 5. Conclusions A description of the behavior of a model ceramic paste (Mix25) during roller extrusion has been obtained, and the application of existing models to the rolling process has been explored. The paste yielded reliable and consistent rolling results. The effect of sheet adhesion to the rollers was captured by the roll surface pressure transducer. Small negative pressure peaks were sometimes observed as the sheet detached from the rollers: such features are not commonly observed in the production of metals or polymers. Significant asymmetry was observed in the behavior of feed and exit sheets and in many of the torque results. Reasons for this behavior are unclear. It is conceivable that the compliance of the torque transducers could have some influence, but it seems unlikely that this could be the sole cause of such a pronounced effect. A plasticity-based model drawn from the hot rolling of metals was used with a shear yield stress value

derived from the Benbow-Bridgwater method of paste characterization. The choice of the hot rolling model was based on the observation that sticking friction dominated at the roller surface. The agreement within the bounds of experimental or prediction error was good for both the roll torque and separating force. However, some discrepancies became evident when the theoretical and empirical pressure profiles were compared. The model is not able to predict the thickness of the exit sheet, which unlike rolled metal sheets, is larger than the roll nip. The experimental pressure profiles displayed a step at the start of the roll bite which was predicted, but the predicted step was too large. By considering estimates of the mean strain rates in rolling and ram extrusion, the velocity dependency observed in the Benbow-Bridgwater characterization was transferred to the rolling analysis. The BenbowBridgwater equation was first reformulated so that it described the shear rate dependence in terms of the shear strain ratio V/D rather than the dimensionally inconsistent extrudate velocity, as described by Martin et al.13 The strain rate dependence of the paste was reflected well by this modified plasticity model. This is likely to be due in part to its strain rate dependence being slight. It was convenient that the estimated mean strain rates in the ram extrusion tests, and the rolling experiments were of the same order of magnitude. The study was moderately successful in the prediction of roll torque and separating force. The analogy between extrusion and rolling was promising and should be extended to other “stiff” paste materials. In the modeling work, flow patterns have been assumed and not verified in experiments. Further modeling should be accompanied by flow verification studies, either via interrupted experiments or tomographic techniques. A promising example of the latter is the recent positron emission particle tracking (PEPT) study of powder roll compaction by Perera.29 The sticking phenomenon underpins a range of industrially important processes such as rotary moulding. The apparatus in its current configuration allows the stresses developed during rolling to be measured reliably. Modifications could be commissioned to study the rotary moulding processs in the first instance the adhesion of the material would need to quantified. Acknowledgment A studentship for M.C.P. from the EPSRC and support from United Biscuits Group R&D (High Wycombe, UK) are gratefully acknowledged. Nomenclature a ) modified plasticity strain rate exponent in eq 2 [-] b ) modified plasticity strain exponent in eq 2 [-] B ) modified plasticity factor in eq 2 [N m-2] D ) die-land diameter [m] D0 ) extrusion barrel diameter [m] F ) roll separating force [N] h ) vertical roll gap at position x [m] H ) sheet thickness [m] He ) exit sheet thickness [m] Hf ) feed sheet thickness [m] H0 ) roll nip dimension [m] k ) shear yield stress [N m-2] l0 ) length of paste sheet being consumed per unit time [m s-1]

Ind. Eng. Chem. Res., Vol. 44, No. 11, 2005 4111 l ) length of paste sheet being produced per unit time [m s-1] L ) die-land length [m] Ls ) length of the deforming region in the die-entry zone [m] p ) principal stress within sheet oriented along length [N m-2] P ) total extrusion pressure [Pa] P0 ) pressure drop across the die-entry region [Pa] Q ) volumetric flow rate [m3 s-1] q ) principal stress within sheet perpendicular to plane of symmetry [N m-2] R ) roll radius [m] R2 ) Pearson coefficient of correlation [-] t ) time [s] T ) roll torque [N m] ux ) velocity of paste in extensional flow direction [m s-1] V ) mean extrudate velocity [m s-1] W0 ) feed sheet width [m] x ) distance upstream along plane of symmetry from nip; extensional paste flow coordinate [m] Greek Letters R ) Benbow-Bridgwater yield stress velocity factor in eq 1 [Pa s m-1] R′ ) Benbow-Bridgwater modified yield stress strain rate factor [Pa s] β ) Benbow-Bridgwater wall shear stress velocity factor in eq 1 [Pa s m-1] ) extensional strain [-] ˘ ) extensional strain rate [s-1] εj˘ ) mean extensional strain rate [s-1] φ ) angle from nip relative to roll center [-] φc ) center line/second slip line angle, defining the upper bound solution [-] φe ) angle subtended at roll center of exit sheet contact with roll surface [-] φf ) angle subtended at roll center of feed sheet contact with roll surface [-] φn ) angular position of neutral point from nip [-] θ ) angle of arc [-] σF ) normal stress component perpendicular to arc in Orowan’s theory [N m-2] σθ ) normal stress component within arc in Orowan’s theory [N m-2] σ0 ) Benbow-Bridgwater die-entry yield stress [N m-2] σ′0 ) shear rate adjusted die-entry yield stress [N m-2] σy ) uniaxial yield stress [N m-2] τFθ ) shear stress component acting on the plane perpendicular to arc in Orowan’s theory [N m-2] τ0 ) Benbow-Bridgwater paste-die wall shear yield stress [N m-2] ω ) angular velocity of rollers [s-1] F ) radius of arc

Literature Cited (1) Siebel, E. Kra¨fte und Materialfluss bie der Bildsamen Forma¨nderung. Stahl Eisen 1925, 45, 1563. (2) Orowan, E. The calculation of roll pressure in hot and cold flat rolling. Proc. Inst. Mech. Eng. 1943, 150, 140. (3) Helmi, A.; Alexander, J. M. Geometric factors affecting spread in hot flat rolling of steel. J. Iron Steel Inst. 1968, 23, 1110. (4) Lippmann, H. Elementary theory of metal forming. In Engineering Plasticity: Theory of Metal Forming Processes; Lippmann, H., Ed.; Springer-Verlag: New York, 1977. (5) Middleman, S. Fundamentals of Polymer Processing; McGraw-Hill Book Company: New York, 1977.

(6) Kiparissides, C.; Vlachopoulos, J. The study of viscous dissipation in the calendering of power law fluids. Polym. Eng. Sci. 1978, 18, 210. (7) Levine, L.; Corvalan, C. M.; Campanella, O. H.; Okos, M. R. A model describing the two-dimensional calendering of finite width sheets. Chem. Eng. Sci. 2002, 57, 643. (8) Benbow, J. J.; Bridgwater, J. Paste Flow and Extrusion; Clarendon Press: Oxford, UK, 1993. (9) Benbow, J. J.; Jazayeri, S. H.; Bridgwater, J. The flow of pastes through dies of complicated geometry. Powder Technol. 1991, 65, 393. (10) Barrett, A.; Bridgwater, J.; Burbidge, A. S.; Hargreaves, M. Flow of pastes in bent tubes. J. Eur. Ceram. Soc. 1997, 17, 233. (11) Englander, A.; Burbidge, A. S.; Blackburn, S. A preliminary evaluation of single screw paste extrusion. Trans. IChemE. 2000, 78, 790. (12) Blackburn, S.; Burbidge, A. S.; Mills, H. A critical assessment of the Benbow approach to describing the extrusion of highly concentrated particulate suspensions and pastes. Proceedings of the XIIIth International Conference on Rheology, Cambridge, UK, 2000. (13) Martin, P. J.; Wilson, D. I.; Bonnett, P. E. Rheological study of a talc-based paste for extrusion-granulation. J. Eur. Ceram. Soc. 2004, 24, 3155. (14) Adams, M. J.; Biswas, S. K.; Briscoe, B. J.; Kamyab, M. The effect of interface constraints on the deformation of pastes. Powder Technol. 1991, 65, 381. (15) Adams, M. J.; Biswas, S. K.; Briscoe, B. J.; Sinha, S. K. A two roll mill as a rheometer for pastes. Mater. Res. Soc. Symp. Proc. 1993, 289, 245. (16) Sinha, S. K. Interface and Bulk Rheologies of Soft Solid Pastes. Ph.D. Thesis, Imperial College, University of London, UK, 1994. (17) Johnson, K. L. Contact Mechanics; Cambridge University Press: Cambridge, UK, 1985. (18) Chakrabarty, J. The Theory of Plasticity; McGraw-Hill Book Company: New York, 1987. (19) Cheyne, A.; Whitlock, B. Roller Extrusion of Paste, Part II; Project Report; University of Cambridge: Cambridge, UK, 1996. (20) Rough, S. L.; Bridgwater, J.; Wilson, D. I. Effects of liquid phase migration on extrusion of microcrystalline cellulose pastes. Int. J. Pharm. 2000, 204, 117. (21) Amarasinghe, A. D. U.S. Interpretation of Paste Extrusion Data. Ph.D. Thesis, University of Cambridge, Cambridge, UK, 1998. (22) Russell, B. D.; Lasenby, J.; Blackburn, S.; Wilson, D. I. Characterising paste extrusion behaviour by signal processing of pressure sensor data. Powder Technol. 2003, 132, 233. (23) Peck, M. C. Roller Extrusion of Pastes. Ph.D. Thesis, University of Cambridge, Cambridge, UK, 2002. (24) Johnson, W.; Mellor, P. B. Engineering Plasticity; Van Nostrand Reinhold Company Ltd: Wokingham, Berkshire, UK, 1973. (25) von Ka´rmań, T. Beitrag zur theorie des walzvorganges. Z. Angew., Math. Mech. 1925, 5, 139. (26) Sims, R. B. The calculation of roll force and torque in hot rolling mills. Proc. Inst. Mech. Eng. 1954, 168, 191. (27) Horrobin, D. J. Theoretical Aspects of Paste Extrusion. Ph.D. Thesis, University of Cambridge, Cambridge, UK, 1999. (28) Steffe, J. F. Rheological Methods in Food Process Engineering; Freeman Press: East Lansing, MI, 1992. (29) Perera, L. N. F. J. An experimental validation of Johanson’s theory of roll compaction using PEPT. IChemE Particle Technology Subject Group Meeting, September 18, 2002, Birmingham, UK.

Received for review November 1, 2004 Revised manuscript received March 22, 2005 Accepted April 4, 2005 IE040270G

Roller Extrusion of a Ceramic Paste - Industrial & Engineering

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