Rolling of bread dough: Experiments and simulations

Transcription

1 food and bioproducts processing 8 7 ( ) Contents lists available at ScienceDirect Food and Bioproducts Processing journal homepage: Rolling of bread dough: Experiments and simulations Evan Mitsoulis a,, Savvas G. Hatzikiriakos b a School of Mining Engineering & Metallurgy, National Technical University of Athens, Zografou , Athens, Greece b Department of Chemical and Biological Engineering, University of British Columbia, Vancouver, V6T 1Z3 BC, Canada abstract Bread dough (a flour water system) has been rheologically characterized using shear rheometry at room temperature. Its flow curve shows a viscoplastic shear-thinning material, which obeys the Herschel Bulkley model. In addition, substantial wall slip is exhibited by the material, and an appropriate slip law has been formulated. The material has been rolled using the Sentmanat Extensional Rheometer (SER) (Xpansion Instruments) with an aspect ratio of radius to minimum gap of R/H 0 = 4.3. Rolling experiments have been performed for different feed thickness ratios and roll speeds. The exit thickness ratios and the torques to roll the samples have been measured. They have been found to increase substantially with roll speed and initial sample thickness. Two-dimensional finite-element simulations based on the rolling geometry and the rheological data with slip provide a wealth of information regarding yielded/unyielded zones, pressure and stress distributions along the rolls. The results from the simulations are in good agreement with the experimental torque values although underestimate the thickness of the exiting sheets. It is argued that the latter is a consequence of viscoelasticity of the bread dough, which manifests itself in a free-surface flow, where (extrudate) swell becomes significant The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Rolling; Bread dough; Viscoplasticity; Yield stress; Yielded/unyielded regions; Herschel Bulkley fluids; Slip; Sheet thickness; Torque measurements 1. Introduction Rolling (or sheeting) between counter-rotating rolls is used by the baking industry as a dough-forming process for a wide range of products, such as cookies, crackers, pizza, bread and pastry (Levine and Drew, 1990). The rolling process is akin to calendering (Mitsoulis, 2008, and references therein), which is used in many industries, such as the paper, plastics, rubber, and steel industries, for the production of rolled sheets of specific thickness and final appearance. The process is shown schematically in Fig. 1, where a material enters as a sheet of finite thickness 2H f and exits as a sheet of reduced thickness 2H (Middleman, 1977). In contrast to calendering, where the aspect ratio of roll radius to minimum gap R/H 0 > 100, rolling has smaller aspect ratios that may often give R/H 0 < 10. However, it is noted that industrial rolling lines may well exceed the aspect ratio of 10. Rolling has been reported to have an impact on the behaviour of dough in subsequent process steps, particularly proving and baking, as well as on the properties of the final products (see Engmann et al., 2005, and references therein). On the other hand, the rheology of bread dough is known to be complicated, exhibiting many rheological phenomena, such as yield stress, time-dependent properties, and slip at the wall (Sofou et al., 2008). Rheological models for bread dough have been formulated that describe its behaviour in several types of flow under a large number of experimental conditions (Ng et al., 2006; Tanner et al., 2007, 2008; Ng and McKinley, 2008; Sofou et al., 2008). Obviously, the use of these true rheological models in flow simulations of the rolling process is a subject of tremendous interest for industry, and thus worthy of further investigation, since many unanswered questions still persist. It is the purpose of this work to undertake an experimental investigation of rheological properties and of the rolling process for bread dough. Measurements in rolling include the entry and exit thickness of the feed sheet of dough as well as the related torque measurements. Then a two-dimensional (2-D) numerical analysis of the rolling process will be performed for viscoplastic materials using the Corresponding author. Tel.: ; fax: address: mitsouli@metal.ntua.gr (E. Mitsoulis). Received 13 March 2008; Received in revised form 1 July 2008; Accepted 4 July /$ see front matter 2008 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi: /j.fbp

2 food and bioproducts processing 8 7 ( ) Experimental methodology Fig. 1 Schematic representation of the rolling process and definition of variables. The feed is a finite sheet, whose thickness is reduced from H f to H. Note the attachment point x and detachment point. f continuous regularized Herschel Bulkley Papanastasiou model. This model has shown good predictions of yielded/unyielded regions in other flows of viscoplastic materials (Mitsoulis et al., 1993) and it is worth applying here to find out its capabilities and limitations for bread-dough rolling. 2. Experimental 2.1. Sample preparation A commercial brand-name flour, namely Robin Hood All- Purpose Flour, produced by Smucker Foods of Canada Co., was used to prepare the dough samples. A typical composition of the flour is summarized in Table 1. A farinograph was used to mix the flour and water (to a 500 BU consistency) in order to prepare the dough samples for the various tests. No salt was added and all mixing was done at 25 C. The farinograph used in this work was a Brabender mixer (C.W. Brabender Instruments, Inc., South Hackensack, NJ). The unit is equipped with dual-z mixing blades and a 300-g mixing container. All batches were prepared using the same settings. The dough develops as mixing time increases, that is, the gluten protein in flour becomes hydrated and creates a network enclosing the starch granules. As will be discussed later, the different batches of the dough were similar enough for the rheological and rolling measurements to be carried out and the data to be reproducible. The various dough samples were prepared as slightly overmixed (torque slightly past the maximum) as it has been found to improve reproducibility of the produced dough samples and therefore consistency of experimental results (Bagley et al., 1998). It was found that a mixing time of 10 min ensured a slight overmix of the dough samples. The effect of overmixing has been reported (Zheng et al., 2000) to have minor effects on the rheological properties and the consistency of dough as compared to undermixing. In the present study we are concerned only with steady-state shear experiments and do not consider any dynamic or extensional measurements, which have been the subject of other investigations (Sofou et al., 2008; Tanner et al., 2007, 2008). It should be pointed out that the concept of steady-state shear is not fully applicable to bread dough, as the above-mentioned previous studies have shown. For example, in steady shear experiments a true steady-state is not obtained. Instead the shear stress undergoes a slow decrease due to structural changes until the dough fractures at high strains. However, to a first and fair approximation, such small gradual changes can be neglected, and the experimental results can be compared with numerical results obtained with the use of a steady-state viscoplastic model, as will be shown below. The Bohlin C-VOR rheometer was used to measure the yield stress of the dough. The rheometer was operated under the stress-ramp mode. In this method, stress is applied onto the sample and the apparent viscosity (shear stress divided by rate of deformation) of the sample is measured. The level of applied stress is gradually increased (ramp rates range from 2.0 Pa/s to 7.0 Pa/s), and as long as the applied stress is below the yield stress, the viscosity exhibits a gradual increase. At a certain stress, the viscosity starts decreasing, indicating the onset of flow. This value of stress, corresponding to the maximum viscosity value, is reported as the yield stress. The Sentmanat Extensional Rheometer (SER from Xpansion Instruments, Akron, OH, USA) was used to carry out the rolling experiments. This rheometer is designed as a fixture to be fitted into rotational rheometers in order to perform extensional rheological measurements. The particular unit used in this work is specifically designed for the Bohlin VOR rheometer equipped with a 86-g cm torque transducer. Its unique design allows the rotational motion of the Bohlin VOR motor to counter-rotate the two drums which are connected by intermeshing gears (see Fig. 2). More details on the performance of this device as an extensional rheometer can be found elsewhere (Sentmanat, 2003, 2004; Muliawan and Hatzikiriakos, 2007). This same SER was used to perform rolling experiments in this work. The set-up for such an experiment using the SER is shown schematically in Fig. 2. The dough sample is fed into the gap between the drums (which act now as rollers), which are set to rotate at a specified rotational speed, ω. Although the rheometer does not have the capabilities to measure the roll-separating force, it yields another equally important parameter, namely the torque required to roll the sample. A constant-speed Instron capillary rheometer (Instron, Norwood, MA, USA) was used to determine the viscosity of dough at high shear rates. The cross head that drives the plunger is capable of travelling at a maximum speed of 85 mm/s. A 890-N load cell was used to measure the extrusion load. A barrel with a diameter of mm was used. All experiments were performed at room temperature (25 C). Capillary Table 1 Atypical compositional analysis of the flour Component Content (%) Detection limit (%) Method Ash AOAC c Protein-total Modification of AO AC Starch 71.8 Method of Biochemical Analysis and Food Analysis, pp (Boehringer Mannheim kit)

3 126 food and bioproducts processing 8 7 ( ) Fig. 3 Typical result from shear ramp measurements for the determination of the yield stress of bread dough. The various experimental results plotted represent three samples from three different batches (a total of nine samples) at shear stress ramps of 2.0 Pa/s and 4.5 Pa/s. The average yield stress value was calculated as y = 277 ± 84 Pa. and lengths. The apparent shear rate is defined as A 32Q D 3, (1) Fig. 2 Schematic representation of the SER and its use for the rolling experiments. dies of various diameters, entrance angles and lengths were used, necessary to determine all the corrections associated with analysis of capillary rheological data including wall slip effects (Dealy and Wissbrun, 1990) Yield-stress measurements The presence of yield stress in dough has been well recognized and attributed to its semi-solid nature, where its ingredients form a three-dimensional network (Tanner et al., 2007, 2008; Sofou et al., 2008). It is, therefore, of importance to measure the yield stress of flour water dough. The method followed to measure the yield stress was based on the results of the time-sweep experiment as described by Sofou et al. (2008). As the rheology of the material was found to be stable only up to 300 s (5 min), the creep test could not have been chosen, as the steady creep compliance could have taken hours to be determined. Structural changes occur in dough at long periods of time. Five batches were prepared on different days, and at least three samples were tested from each batch, using the parallelplate geometry of 25 mm in diameter. The stress-ramp rates vary from 2.0 Pa/s to 7.0 Pa/s. Typical results are plotted in Fig. 3. The average yield stress, as derived from the data analysis, was found to be 277 Pa, while the coefficient of variation among different batches is in the order of ±19%. where Q is the volumetric flow rate and D is the capillary die diameter. In capillary extrusion of various highly viscous materials including polymer melts and foodstuff, wall slip typically occurs at shear stress above a critical value. To check the occurrence of wall slip, a technique developed by Mooney (1931) can be used. Capillaries of different diameters and constant L/D are used to determine the flow curve of the material under study ( w versus A ). In the absence of slip, the flow curves are independent of the capillary die diameter. Any dependence of the flow curve on the die radius at a given L/D ratio implies the existence of slip phenomena. The Mooney technique to determine the slip velocity does not work in the case of dough, and instead a modified technique is used (Geiger, 1989; Mourniac et al., 1992; Aichholzer and Fritz, 1998). Details of these experiments and how one can determine the slip velocity of dough can be found in Sofou et al. (2008). The Bagley- and slip-corrected flow curves of bread dough for various D values are shown in Fig. 4. All flow curves 2.4. Capillary extrusion experiments The capillary experiments involve capillary extrusion at different apparent shear rates using dies of various diameters Fig. 4 The flow curve of bread dough at 25 C and its representation by the Herschel Bulkley viscoplastic model.

4 food and bioproducts processing 8 7 ( ) Table 2 Rheological parameters and physical constant for bread dough at 25 C (Herschel Bulkley model) Yield stress, y (kpa) Stress growth exponent, m (s) 200 Power-law index, n 0.42 Consistency index, K (kpa s n ) Slip exponent, Capillary slip coefficient, ˇs (kpa s 1 ) Slip coefficient, ˇ (m kpa s 1 ) Density, (kg/m 3 ) 1076 for three different dies of various L/D ratios fall on a single line, defining uniquely the flow curve of the bread dough at 25 C. The experimental data have been fitted with the Herschel Bulkley model of viscoplasticity, which has the form (Bird et al., 1982): = K n 1 ± y, for > y, (2a) = 0, for y, (2b) where is the shear stress, is the shear rate (= du/dy), y is the yield stress, K is the consistency index, and n is the power-law index. The values of the constants from the best fit are given in Table 2. It is seen that the value of y = 299 kpa as determined by extrapolation, compares favourably with the value of y = 277 kpa found by the stress-ramp test, within its margins of error. Nguyen and Boger (1992) have differentiated between static (direct measurement) and dynamic (flow curve extrapolation) yield stresses and their different significance in a process flow. Due to the semi-solid-like nature of bread dough, the usual interpretation of capillary flow data can in principle not be applied. However, this analysis is used here as a first approximation, noting the absence of an alternative better interpretation. The semi-solid-like nature of the dough must lead by necessity to slip and/or extremely high shear deformation near the wall, leading to significant weakening of the dough (see Tanner et al., 2007, 2008). As discussed above a modified technique was used to determine the slip velocity of dough as a function of wall shear stress (Sofou et al., 2008). It was also found experimentally that the slip velocity depends on the capillary radius, R d, and this dependence was built into the model used for the data analysis (Geiger, 1989). Similar dependence has been observed in the extrusion of EPDM rubber mixture by Geiger (1989) and thermoplastic starch materials (Aichholzer and Fritz, 1998). The modified Mooney method proposed by Geiger (1989) is expressed by the following empirical model: V s ( w,r d ) = R d ( w ) [ exp ( ) ] asl ( w ) 1, (3) R d in which V s is the slip velocity, ( w ) and a sl ( w ) are functions of the wall shear stress and can be obtained from the plot of log A against 1/R d. A value of ± m was found for a sl to represent the experimental data quite well. The function, ( w ), can be expressed by: ( w ) = ˇs w, (4) Fig. 5 The slip velocity of bread dough as a function of wall shear stress and various capillary diameters at 25 C. with ˇs = kpa s 1 and = Combining Eqs. (3) and (4), the final slip velocity expression is: V s ( w,r d ) = R dˇs w [ ( asl ) ] exp 1, (5) R d where V s is given in m/s, R d and sl in m, w in kpa, and ˇs in kpa s 1. Eq. (5) indicates that the slip velocity depends on the capillary diameter as shown in Fig. 5 that plots the slip velocity of dough as a function of wall shear stress and capillary diameter. It is emphasized that this geometry-dependence slip condition is empirical and lacks any physical argument. Also, the above slip law does not take into account any effect of normal stress on the friction/slip behaviour, which is tantamount to assuming a perfectly wetted contact area. This assumption may need experimental verification in a future work. Based on the modified Mooney plot, the wall shear rates, A,S corrected for slip can be obtained. These have been used to construct the true flow curve of bread dough (Fig. 4), which includes the Rabinowitsch correction according to w = 3 + b A,S, (6) 4 where b is the Rabinowitsch correction, given by b = d(log A,S )/d(log w ) = 2.4. It is noted that A A,S under noslip. All of the flow curves obtained from the capillary dies of different lengths and diameters now fall into one single line, i.e., the true flow curve of the bread dough (Fig. 4). Based on the slope of the true flow curve, the power-law index can be estimated as n = This illustrates the pseudoplastic nature of the dough and is indicative of its shear-thinning behaviour Rolling experiments The rolling experiments were performed on the SER used as a rolling device (Fig. 2). The dependent variables are the final sheet thickness and the torque, while the roll speed of the SER and the initial thickness of the entering sheet are the independent variables. Table 3 lists the roll dimensions and the different test cases of roll speed (U1 U6) and initial thickness (H1 H4) together with the relevant dimensionless numbers (for more information see the mathematical modelling section). To minimize experimental errors, three measurements by a micrometer were made of the initial and final sheet

5 128 food and bioproducts processing 8 7 ( ) Table 3 Geometric and experimental data in rolling of bread dough Roll dimensions Radius, R (cm) Minimum gap, 2H 0 (cm) Ratio, R/H 0 Roll length, L (cm) Sample width, W (cm) Initial sample thicknesses #Experiment H1 H2 H3 H4 Initial thickness, 2H f (cm) Thickness ratio, H f /H Roll speeds #Experiment U1 U2 U3 U4 U5 U6 Speed, U (cm/s) Characteristic variables Characteristic shear rate, (s 1 ) Characteristic viscosity, (Pa s) Dimensionless numbers Bingham number, Bn = yh n 0 KU n Reynolds number, Re = UH 0 Slip number, B = ˇ( ) a U Fig. 6 Experimental results for the final exit thickness as a function of entry thickness for different roll speeds (hence Bingham numbers, Bn, see Table 3). thickness, as well as three measurements of the sheet width. Furthermore, for each test case, at least six (6) tests were performed. The experimental results for the average value of the final thickness and the torque are shown in Figs. 6 and 7, respectively. Due to the small size of the samples and the stiffness of the rolls, no bending of the rolls was possible and therefore there was no change in the value of H 0. Fig. 6 shows that the exit thickness increases with increase of the entry thickness and decrease of the Bingham number. The biggest increase of the final thickness reaches more than 50% of its initial value for Bn = and H f /H 0 = 2.5. For a given roll speed, i.e., for a given Bn number, the exit thickness increases as the entry thickness increases. Also, for a given initial thickness, the final thickness increases as the roll speed increases. If the roll speed increases 40 times, the relative final thickness increases at most by 14%. Fig. 7 shows that the torque per unit width increases with increasing initial thickness and roll speed (or with decreasing Bn number). It is noted that the increase is more significant at low to medium roll speeds and tends to level off at higher ones. This phenomenon naturally occurs with shear-thinning materials, and is also the result of increasing slip. As shown in Table 3, measurements were made for initial thickness ratios in the range 1.46 < H f /H 0 < 2.50 and roll speeds in the range 0.36 < U (cm/s) < It should be noted that the speed range is on the low end of commercial operations, which usually run between 5 and 50 cm/s. The upper and lower limits were determined mainly by the sensitivity limits of the torque transducer. Going over these limits led to either a weak torque signal or a torque greater than the transducer s measurement capabilities. On the other hand, using samples having a thickness smaller than that of the lower test limit resulted in experimental difficulties and led to inconsistent results. Representative samples from the experiments after rolling are shown in Fig. 8. Fig. 8a shows a smooth rolled sample Fig. 7 Experimental results for the torque as a function of roll speed for different entry thickness ratios H f /H 0.

6 food and bioproducts processing 8 7 ( ) Fig. 8 Bread dough view after its rolling for different cases of initial thickness and roll speed. Cases (a) and (c) show a smooth rolled sample. In case (b), a combination of small initial thickness and high roll speed leads to the appearance of small surface defects. In case (d), a combination of high initial thickness and high roll speed leads to the bending of the sample. for low entry thickness and roll speed. No distortions were observed on the roll samples for higher speeds and entry thicknesses than the ones reported here. However, as shown in Fig. 8b and a combination of small initial thickness and high roll speed leads to the appearance of surface defects. These small defects become smaller for larger initial thicknesses. Fig. 8c shows a smooth rolled sample for intermediate entry thickness and high roll speed. On the other hand, Fig. 8d shows that a combination of a large initial thickness and a high roll speed leads to bending of the rolled sample. Despite the fact that bread dough is a malleable (soft semi-solid) material, and returns to the initial shape readily, bending may be undesirable in industry, if the sample is required to pass through successive steps of rolling, as is often the case. It should be noted that similar defects have been observed in calendering of polymer melts and have been attributed to several factors (Agassant and Espy, 1985). On the other hand, this bending (or curling) is probably not an issue when feeding successive rolls in a commercial operation. The dough is slack enough to flatten out under its own weight. Also in industrial operations, conveyor belts between sets of sheeting rolls alter the tension in the dough in the principal flow direction (similar to paper processing). Fig. 9 shows results for the torque as a function of the roll speed and the initial thickness of the sheets. The shaded area represents the range of stable operation for rolling of this particular bread dough, as obtained from the above-mentioned visual observations of the shapes of the rolled samples. It is noted that the region of stable operation is specific to the aspect ratio R/H 0 = Mathematical modeling 3.1. Governing equations The flow is governed by the conservation equations of mass and momentum under isothermal creeping flow conditions for an incompressible viscous fluid in a 2-D Cartesian coordinate system. These are: ū = 0, (7) 0 = P +, (8) where ū is the velocity vector, P is the pressure, and is the extra stress tensor. There is also a need for a constitutive equation that relates the stresses to the velocity gradients. Recent works by Tanner et al. (2007, 2008), Ng and McKinley (2008), and Sofou et al. (2008) offer different options of increasing complexity for such a constitutive modelling. We follow here as a first and pragmatic approximation the generalized Newtonian fluid (GNF), as proposed by Sofou et al. (2008), which is capable of capturing well the apparent viscosity data of bread dough. In full tensorial form the constitutive equation is written as: =, (9) where is the apparent viscosity given by the Herschel Bulkley Papanastasiou model (Mitsoulis et al., 1993): = K n 1 + y [1 exp( m )]. (10) Fig. 9 Range of experiments in torque measurements during rolling of bread dough. The shaded area corresponds to stable operation, outlined by the limitations of the measuring device and the observed surface defects. In the above Eq. (10), m is the regularization parameter (with units of time), which controls the exponential growth of stress, and is the magnitude of the rate-of-strain tensor, = v + v T, which is given by: = [ 1 1 ] 1/2 2 II = 2 { : }, (11) where II is the second invariant of. Similarly a formula is obtained for the second invariant of the stress tensor II.It

7 130 food and bioproducts processing 8 7 ( ) should be noted that the above regularized model is a convenient way to avoid the viscosity singularity of the original Herschel Bulkley model when the shear rate approaches zero. The significance of m is to smooth the flow curve, and when m approaches infinity, the original Herschel Bulkley model is recovered (Mitsoulis et al., 1993). The criterion to track down yielded/unyielded regions is for the material to flow (yield) when the magnitude of the extra stress tensor exceeds the yield stress y, i.e., yielded : = unyielded : = [ 1 1 ] 1/2 2 II = 2 { : } > y, (12a) [ 1 1 ] 1/2 2 II = 2 { : } y. (12b) Contours of = y separate the yielded from the unyielded regions and are drawn a posteriori from the numerical solution Dimensionless numbers In calendering (and hence in rolling), the following dimensionless parameters are introduced (Middleman, 1977): x x =, y = y, h = h = 1 + x2, 2RH0 H 0 H 0 2RH 0 (a) (b) (c) P = P ( H0 ) n, K U 2 = Q 1, 2UH 0 (13) (d) (e) where is a dimensionless flow rate (or leave-off distance), Q is the flow rate, and the rest of the symbols are defined in Fig. 1. Dimensionless numbers are also introduced because they are necessary in assessing the relative importance of the various terms in the conservation equations. To do this, characteristic variables need to be chosen. Thus, a characteristic length of the process is half the minimum gap between the rolls, H 0. A characteristic speed of the process is the roll speed, U. It then follows that a characteristic shear rate is given by: = U H 0, (14) and a characteristic viscosity is given by: = ( ). (15) For a material that obeys the Herschel Bulkley model, the characteristic viscosity is given by: = K n 1 + y = K ( U H 0 ) n 1 + yh 0 U. (16) With these characteristic variables it is now possible to define the following dimensionless numbers. 1. The relative importance of inertia forces to viscous forces is given by the Reynolds number: Re = UH 0. (17) Usually, in several processing operations of highly viscous materials Re 1, the creeping flow approximation, Re 0, is valid. This is the case here (see Table 3). Therefore, no inertia terms are present in the momentum equation. 2. The relative importance of yield forces to viscous forces is given by the Bingham number: Bn = y K ( H0 ) n. (18) U The range of Bn numbers is from Bn = 0, for purely viscous fluids, to Bn, for purely elastic solids. For bread dough the Bn numbers are small but non-zero, ranging between and (see Table 3). 3. The relative importance of slip at the wall depends on the slip law assumed. When this takes the classic power-law form: V s = ˇ w, (19) where ˇ is the slip coefficient and is the slip exponent, then a slip number can be defined as: B = ˇ U ( ). (20) When B = 0, there is no slip at the wall, while for B 1 there is macroscopic (obvious) slip occurring. In rolling of bread dough, it is necessary to consider the slip law as given by Eq. (5) and modify it accordingly. We can set the capillary radius equal to half the minimum gap (R d = H 0 = m), which is a reasonable assumption in the rolling geometry. Then Eq. (5) simplifies to: V s ( w,h 0 ) = H 0ˇs w [ ( asl ) ] exp 1. (21) H 0 Then the term in the square brackets becomes a constant, and Eq. (21) takes the form of the classic power-law (Eq. (19)) with: [ ( ) ] ˇ = H 0ˇs exp sl 1, (22) H 0 where ˇ is now given in m kpa s 1. With the value of ˇ given in Table 2, we find the dimensionless B values of Table 3. We observe that B > 1, which signifies the importance of slip Lubrication Approximation Theory (LAT) In calendering, where the aspect ratio R/H 0 > 100, it is assumed that the process is dominated by shear flow between the rolls and around the nip, so that only the axial velocity u and its derivative du/dy are considered (Middleman, 1977). Furthermore, the roll curvature is approximated by a parabola (see Eq. (13c), above), which is an excellent approximation for large aspect ratios. The boundary conditions are: P = dp dx = 0, at x =, (23a) P = 0, at x = x f. (23b) Note that Eq. (23a) is referred to as the Swift condition (Zheng and Tanner, 1988) and is instrumental in the solution of the problem. The above are the key assumptions for the solution of the problem according to LAT. The problem can then be solved either for a given (the detachment point) and finding

8 food and bioproducts processing 8 7 ( ) Fig. 11 Flow domain and boundary conditions for the 2-D FEM analysis. 4. Method of solution Fig. 10 Exit thickness, H/H 0, as a function of entry thickness, H f /H 0, in calendering of viscoplastic Bingham materials (Sofou and Mitsoulis, 2004a). The symbol refers to the present range of experiments for bread dough. x f (the attachment point), or the reverse. Once or x f are known, the sheet thicknesses are readily available according to: H H 0 = 1 + 2, (24) H f H 0 = 1 + x 2 f. (25) For Herschel Bulkley fluids, Sofou and Mitsoulis (2004a,b) have given an analytical expression and a numerical solution for full parametric studies of the power-law index, n, and the Bingham number, Bn. They have also studied the effect of slip number, B (Mitsoulis and Sofou, 2006). The results for the sheet thickness of viscoplastic Bingham fluids (n = 1) are reproduced here in Fig. 10. For a given entering sheet thickness H f /H 0 and Bn number, the analysis gives the exiting sheet thickness H/H 0, and vice versa. It is worth noting that for H f /H 0 < 15.85, the Newtonian fluid gives the biggest exit thickness, while the reverse is true for H f /H 0 > At H f /H 0 = 15.85, all fluids have the same exit thickness. For the bread dough at hand, the range of rolling experiments is indicated in Fig. 10 (see the arrows). The solution process includes important operating variables, such as the maximum pressure, the roll-separating force, and the torque exerted by the rolls. The roll-separating force per unit width, F/W(n, Bn), is defined by: F W (n, Bn) = x f while the torque for both rolls, (n, Bn) = 2WR P(x)dx, (26) x f (n, Bn), is defined by: xy dx, (27) y=h(x) where W is the sheet width. Eqs. (26) and (27) imply that extra normal stresses are negligible and that roll curvature can be neglected in the integration. The constitutive equation for the viscoplastic fluids must be solved together with the conservation equations and appropriate boundary conditions. We note that due to the two-dimensional character of rolling, the roll surface is not approximated by a parabola (Eq. (13c)) but by the proper circle equation: h = h H 0 = 1 + R H 0 (1 1 ) ( x ) 2. (28) R Fig. 11 shows the solution domain and boundary conditions for the symmetric problem. Because of symmetry, only one half of the flow domain is considered, as was done previously (Sofou and Mitsoulis, 2004a,b). The boundary conditions are (for flow from left to right): (a) symmetry along the centreline AB (v y = 0, xy = 0); (b) slip along the roll walls from point E to point D (tangential velocity v t = t Ū = B( t n : ), normal velocity v n = 0); (c) along the exit free surface DC, vanishing tangential and normal stresses (( n) t = 0, ( n) n = 0) and no flow through the surface ( v n = 0), where n and t are the normal and tangential vectors to the surface, and = pī + is the total stress; (d) along the outflow boundary BC, vanishing tangential and normal stresses (( n) t = 0, ( n) n = 0); (e) along the inflow boundary AF, vanishing tangential and normal stresses (( n) t = 0, ( n) n = 0); (f) along the entry free surface FE, vanishing tangential and normal stresses (( n) t = 0, ( n) n = 0) and no flow through the surface ( v n = 0). The reference pressure is also set to zero at point B. All lengths are scaled with the minimum gap H 0, all velocities with the roll speed U, and all pressures and stresses with K(U/H 0 ) n. Also the stress-growth exponent m gives rise to the dimensionless stress-growth exponent M = mu/h 0. The value of m = 200 s is set as a material property and is kept constant in all runs. The numerical solution is obtained with the Finite Element Method (FEM), using the program UVPTHSLIP, originally developed for multilayer flows (Hannachi and Mitsoulis, 1993; Mitsoulis, 2005), which employs as primary variables the two velocities, pressure, temperature and free surface location (u v p T h formulation). It uses 9-node Lagrangian quadrilateral elements with biquadratic interpolation for the velocities, temperatures and free surface location, and bilinear interpolation for the pressures. The free surface is found in a coupled way as part of the solution for the primary variables.

9 132 food and bioproducts processing 8 7 ( ) Fig. 12 Finite elements meshes used in the computations. Upper half shows a mesh with 2000 elements (M2), lower half mesh has 500 elements (M1). Detailed data for the rolling geometry are given in Table 3. We have used two meshes, which are shown (put together for brevity) in Fig. 12. Mesh M1 (lower half) has 500 elements, while mesh M2 (upper half) has 2000 elements, and is produced by subdividing each element of M1 into 4 sub-elements. Mesh M2 gives 8241 nodes and 26,643 unknown degrees of freedom (DOF) with 41 points in the transverse direction. The less dense mesh M1 with 500 elements was used primarily for preliminary runs to gain experience with two-dimensional rolling flows. More elements have been concentrated near the attachment and detachment points and near the rolls, where most of the changes are expected. The entering sheet is usually set with a length of 3H f and the exiting sheet with a length of 4H to guarantee a fully developed profile at entry and exit. Knowing that viscoplastic fluids as opposed to viscoelastic fluids do not swell much (Abdali et al., 1992; Mitsoulis, 2007a,b), such lengths are adequate for a full rearrangement of the velocity profile. The adequacy of the entry and exit lengths was also checked for each run by plotting the centreline and surface velocity profiles and observing their levelling off and matching in these regions. Variations of several dense meshes showed that the results were virtually and visually the same and did not warrant going to higher mesh densities. So the results with 2000 elements are deemed mesh-independent. 5. Results and discussion We present results from the two-dimensional, creeping, isothermal analysis of rolling of bread dough. Data for the operating conditions of the rolling process are given in Table 3. The dimensionless numbers have been calculated according to Eqs. (17) (20), and from the rheological and physical properties of bread dough as given in Table 2. As noted above, bread dough exhibits strong slip at the wall, and this becomes obvious in the calculation of the dimensionless slip coefficient, B > 1. Typical results are given for the field variables for the H1U6 case, which can be described as the most difficult, having the highest roll speed U6 and the biggest entry thickness H Contours We begin the presentation of the flow field through contours of the kinematic variables in Fig. 13 and of the dynamic variables in Fig. 14. There are 11 contours drawn with the highest legend value being the maximum and the lowest legend value being the minimum of each variable. Fig. 13a shows the velocity vectors based on the magnitude of the velocity. Not all vectors are shown due to their multitude. The velocity vectors at entry and exit are constant to reflect the plug velocity profiles there due to the existence Fig. 13 Flow variables in rolling of bread dough (H1U6 case). Kinematic variables: (a) velocity vectors with magnitude Ū = u 2 + v 2 (UBAR) (cm/s), (b) contours of u (U), (c) contours of v (V), and (d) streamlines, (STR). of the free surfaces. The magnitude of the velocity at the nip is almost 3 times higher than at the entry (10.3 cm/s versus 3.4 cm/s). The axial velocity component contours are shown in Fig. 13b and the vertical in Fig. 13c. The axial velocity is maximum at the nip and is reduced on either side, while the vertical velocity shows extrema further down from the attachment point on the roll surface. Note that it is not possible to know a priori the behaviour of the velocity due to massive slip at the wall, which depends on the local stress values. Fig. 13d shows the streamlines (contours of the stream function,, which has been made dimensionless and normalized by the exit value C = U C H C ). The streamlines are all open and equally spaced (2-D planar flow), and thus there is no recirculation. The squeezing of the material to pass through the minimum gap is obvious from Fig. 13d. Fig. 14a shows the isobars and Fig. 14b contours of the normal stress xx (= yy due to a 2-D planar flow). Both the primary variable of pressure, P, and the secondary variable of normal stress, xx, show extrema at the attachment and detachment points, which are singular points. Fig. 14c shows the contours of shear stress. As expected, the biggest changes occur in the nip region, but also at the singular points of

10 food and bioproducts processing 8 7 ( ) attachment and detachment of the sheet. This is also confirmed in Fig. 14d, where the shear rate contours are presented. The shear rate values range between 55 s 1 and 226 s 1, which are rather high for rolling, but again these values are due to the singular points at attachment and detachment of the sheet to and from the rolls. A better understanding of the singular behaviour will be given further down, when the axial distributions of the variables along the rolls are presented. In all the above figures of the flow field, the flow domain is deformed to accommodate the upstream and downstream free surfaces. It is seen that upstream there is more deformation, as the material swells to satisfy the stress-free boundary conditions imposed at the free surface. Thus, starting from the attachment point as an anchor with H f /H 0 = 2.503, the sheet swells to at entry, giving a relative change of 13.7%. This is reminiscent of the Newtonian planar extrudate swell values of 18%, which viscoplasticity brings down as the Bn number increases (Mitsoulis, 2007a,b). As shown in Table 3, here the Bn numbers range between and On the other hand, the swelling of the exiting sheet, defined as the thickness ratio between points C and D (see Fig. 11), is negligible, being 1.2%. Therefore, viscoplasticity combined with slip at the wall serves to reduce the admittedly small swelling in rolling, which is also the case for the benchmark extrudate swell problem from long extrusion dies (Mitsoulis, 2007a,b) Yielded/unyielded regions Fig. 14 Flow variables in rolling of bread dough (H1U6 case). Dynamic variables (in MPa) and contours of shear rates (in s 1 ): (a) pressure, P (P), (b) xx (TXX), (c) xy (TXY), and (d) xy (GXY). Because of the yield stress, viscoplastic fluids have the characteristics of both viscous fluids and plastic solids. The yield line, y 0, separates the two regions, called yielded and unyielded, respectively. Several cases of viscoplastic flows have been analysed in the literature and have shown explicitly these regions, e.g., flow through contractions (Abdali et al., 1992). In calendering, such regions have been shown only schematically by Gaskell (1950) and more recently by Sofou and Mitsoulis (2004a,b). As mentioned in the latter works, the interesting yielded/unyielded regions found by LAT are erroneous, so the 2-D analysis here is instrumental in finding out where these regions lie. With regard to Fig. 13, we observe that the entering and exiting sheets, some distance from the attachment and detachment points, have a constant thickness and move with a plug velocity as a solid plastic sheet. Therefore, they constitute unyielded regions (shaded) before coming into contact with the rolls and after leaving them. But apart from these regions, it is not a priori clear where the unyielded regions between the rolls might be. We present here results for these regions for the H1U cases, i.e., the thickest sheet for all roll speeds. As seen in Table 3, roll speeds range between cm/s (= U1) and cm/s (= U6), and correspond to Bn numbers between and 0.008, respectively. It was found that due to this small range of Bn, the detachment points,, were almost constant at It should be mentioned that because of the significant slip at the roll surface, the exit velocity is not equal with the roll speed, which changes according to the local values of the stresses. Therefore, the criterion for finding was the flow rate Q as found from integrating the velocities along the roll, and its comparison with the flow rate based on according to its definition (Eq. (13e)). The results are shown in Fig. 15, where the extent and shape of yielded/unyielded regions become evident as the roll speed U increases (or as Bn decreases). The regions are distinguished into shaded (unyielded) and clear (yielded), making use of the criterion that the separating line is the contour with a value = y = 299 Pa. The unyielded regions decrease with increasing speed. However, they remain small due to the small Bn numbers. Mainly the unyielded regions are found in the exit, and no such regions appear between the rolls. The plug velocity profiles at the exit are responsible for the unyielded regions there. In contrast, at the entry there are no unyielded regions in a distance up to 9H 0, since the free surface still has some curvature, and an even larger flow domain is needed to obtain a plug velocity profile at entry. The regions at the exit are truly unyielded regions (TUR), since the velocity is nonzero, but its derivatives are zero (no deformation). The yielded region is continuous and extended for all roll speeds. Some islands appear near the entry only for the smaller speeds (see H1U1, H1U2, H1U3). These islands are apparently unyielded regions (AUR), and in all probability they are a consequence of using the regularization by Papanastasiou in the Herschel Bulkley model. The results for different entry thicknesses and a given roll speed (U1) are presented in Fig. 16. They show the extent and shape of yielded/unyielded regions as the entry thickness H f decreases. Since now the roll

11 134 food and bioproducts processing 8 7 ( ) Fig. 15 Progressive reduction of the unyielded zones (shaded) in rolling of bread dough, with the Herschel Bulkley Papanastasiou model and slip at the wall. Different roll speeds U and a given entry thickness H1 (see Table 3). speed is constant (hence the Bn number), there are no changes in the unyielded regions at the exit, as expected. However, at entry there is an increase of the unyielded regions as the thickness of the entering sheet decreases, since the free surface has negligible curvature away from the rolls, which results in a plug velocity profile there, hence in unyielded regions Axial distributions To better understand the contours of Figs. 13 and 14, Fig. 17 depicts the axial distributions of the primary variables, i.e., the axial and vertical components of the velocity u x and u y, respectively, and pressure, P, while Fig. 18 presents the distri- Fig. 16 Progressive reduction of the unyielded zones (shaded) in rolling of bread dough, with the Herschel Bulkley Papanastasiou model and slip at the wall. Different entry thicknesses H and a given roll speed U1 (see Table 3).

12 food and bioproducts processing 8 7 ( ) Fig. 17 Axial distributions in rolling of bread dough (H1U6 case): (a) axial velocity, u x /U, (b) vertical velocity, u y /U, and (c) pressure, P. butions of the secondary variables, i.e., the elongational rate ε xx (= ε yy ), the elongational stress, xx (= yy ), the shear rate xy, and the shear stress, xy. The symbols cl, w, fs, refer to symmetry line (centreline), roll surface (wall) and free surface, respectively. Regarding Fig. 17a, the material enters the flow field with a plug velocity profile, as u x (fs) = u x (cl). For this to happen it was necessary to extend the flow field upstream (to a length of 9H 0 ). At the attachment point, x f, where the material first comes into contact with the rolls and because of significant slip, u x (fs) = u x (w) < u x (cl). As the material passes through the nip, it accelerates, and at the nip u x (cl) < u x (w), due to slip. As the material has left the detachment point,, it gradually regains a plug profile. Its speed at the exit is higher than its speed at the entry, because of mass conservation, namely, as the thickness decreases, the speed increases. Fig. 17b presents the axial distribution of the vertical velocity component, u y. Due to symmetry, u y (cl) = 0. At the free surface, u y decreases as the material reaches the attachment point, x f, after which u y shows a minimum because of slip. The velocity vector at the nip (x = 0) is parallel to the x-axis, and so u y (x = 0) = 0. After the nip and up to the detachment point, u y (w) increases, to fall down to 0 along the free surface. Fig. 17c presents the pressure distribution, P, where it is clear that the points of attachment and detachment are singular points with FEM. The pressure distribution in rolling (R/H 0 < 10) is remarkably different from its behaviour in calendering (R/H 0 > 10), where bell-shaped curves are obtained. Here in the region before the attachment point, the pressure on the free surface takes negative values. The same occurs at the centerline in the region around the detachment point. Also, the pressure distributions along the rolls and at the centerline do not coincide, which is another manifestation of the twodimensional character of the flow. However, the maxima occur for x =, as predicted by LAT. If we disregard the extrema at the singular points, then the pressure shows a maximum value of MPa at. Regarding the secondary variables, Fig. 18a and b shows the distributions of the elongational rate, ε xx, and elongational stress, xx, respectively, while Fig. 18c and d shows the distributions of the shear rate, xy, and shear stress, xy, respectively. Referring to Fig. 18a, before the attachment point, and because of the free-surface flow with its stress-free boundary conditions, we observe that at entry the elongational rate is zero. After attachment, it increases up to 20 s 1 both at the roll surfaces and at the centerline, while at the nip it crosses from 0 to continue with negative values, until it comes back up to

13 136 food and bioproducts processing 8 7 ( ) Fig. 18 Axial distributions in rolling of bread dough (H1U6 case): (a) elongational rate, ε xx, and (b) elongational stress, xx, (c) shear rate, xy, and (d) shear stress, xy. 0 towards the exit because of the downstream free surface. Of course, at the singular points of attachment and detachment, the wall values are off scale. But the distributions of the centerline are smooth. Fig. 18b shows that the distribution of the elongational stresses follows that of the elongational rates, as it should. The maxima reach now MPa before the nip (if we ignore the extremely high values at the singularities) and the minima reach MPa after the nip. Reaching zero on the free surfaces takes a longer distance to occur. Fig. 18c shows that the shear rate starts at 0 on the free surface, and after the singularity it ranges between 25 s 1 and +25 s 1, before reaching the detachment point and decreasing to 0 on the downstream free surface. Slip at the roll surface serves to decrease the values of the shear rates and also affects the shape of the curves, which now appear to tend to linear between their extreme values. Fig. 18d shows that the distribution of the shear stresses follows the behaviour of the shear rates, as it should. On the free surface it starts from 0 and decreases up to the singular attachment point. Then it starts from 0.02 MPa and reaches again almost linearly the maximum of MPa at the nip. Then it decreases to 0, which occurs on the downstream free surface. Slip on the roll surface is responsible for the almost linear behaviour between the two extreme values Detachment point As mentioned above, the detachment point,, was found by making use of the flow rate at exit and that which results in from integration of the velocities on the roll surface. Therefore, the detachment criterion does not make use of the Swift condition used in LAT, and the results are therefore different from those predicted by LAT. The values of were found constant at 0.22, which corresponds to H/H 0 = 1.048, therefore they are independent of the Bn number, in this limited range. As shown in Fig. 6, the experimental H/H 0 values range between 1.2 and 1.5, and they depend both on the roll speed and the thickness of the entering sheet. Therefore, other phenomena are responsible for the experimental swell, with most important those of viscoelasticity, as shown to exist in our rheological characterization of bread dough (Sofou et al., 2008). However, viscoelastic effects are beyond the scope of the present work Torque As a first attempt at calculating the torque, Eq. (27) was used according to LAT, which integrates simply the shear stresses, xy, along the roll surface. The results with slip and without slip are given in Fig. 19, together with the experimental results (symbols with error bars). The torque per unit width W is given in N m/m. Qualitatively, all results are similar both in shape

14 food and bioproducts processing 8 7 ( ) Fig. 19 Torque per unit width as a function of roll speed in rolling of bread dough. Comparison between experiments and simulations based on LAT (Eq. (27)) with slip or no slip at the roll surface. Fig. 20 Torque per unit width as a function of roll speed in rolling of bread dough. Comparison between experiments and simulations based on the 2-D FEM (Eq. (29)) with slip at the roll surface. and ordering, so that the torque increases with increasing roll speed and entering sheet thickness. However, without slip the LAT results (broken lines) underestimate the experimental ones by an average of 34%. Slip reduces these values even more (continuous lines). Therefore, it becomes obvious that integrating simply the shear stresses produces torque values at least an order of magnitude lower than the experimental ones. The difficulties of predicting correctly the torque for power-law fluids have also been pointed out by Levine (1985, 1996). The previous results from the 2-D FEM analysis have shown that in rolling with small aspect ratio R/H 0 all stresses have values of the same order of magnitude, and therefore they cannot be neglected in the force and torque calculations. In contrast to LAT and its Eqs. (26) and (27), not only the pressure and the shear stresses have to be numerically integrated along the roll, but also the normal stresses. Therefore the equation for the torque substitutes the shear stress xy with r, in a cylindrical coordinate system, with r being the radial coordinate and the azimuthal coordinate, according to the formula used for calculating the drag force around a cylinder (Mitsoulis, 2004): = WF d R = 2WR 2 1 r R d, (29) where F d is the drag force, r is the tangential stress on the roll, is the local angle at each point on the roll, and the integration takes place from the attachment point with a corresponding angle 1 to the detachment point with angle 2. Substituting into Eq. (29) and taking into account the usual trigonometric equalities, we obtain for the torque: = 2WR xd x E ( yx cos + yy sin)dx + 2WR yd y E ( xx cos + xy sin )dy, (30) where ( x E, y E ) are the coordinates of the attachment point E and correspond to the angle 1, and (x D, y D ) are the coordinates of the detachment point D and correspond to angle 2. Note that the total stresses are: xx = P + xx, yy = P + yy, and because of symmetry of the stress tensor, yx = xy. The results from the above integrations are given together with the experimental data in Fig. 20. In contrast to LAT, there is now excellent agreement between simulations and experiments, which is due to integrating all stresses and pressure, since these have the same order of magnitude, confirming the two-dimensional character of the rolling process. Slip is Fig. 21 Schematic diagram of the sheet thickness at the detachment point (as predicted by the simulations) and the final sheet thickness H/H 0 (as measured experimentally). Swelling is observed, which begins right after detachment. The flow up to detachment is confined, with wall boundary conditions ( shear flow ), while after detachment is a free-surface flow without shear or normal stresses. essential in getting good agreement with experiments, since sample runs with no slip for H f /H 0 = 2.5 (cases H1U) gave higher torques, ranging from 1.6 to 1.8 times the torque values for slip from low to high roll speeds, respectively. The contradictory findings of getting the torque correct but not the final thickness can be explained by the nature of the two types of flow in the rolling process and referring to Fig. 21. Namely, the flow with wall boundary conditions (between attachment and detachment points), which may be called (perhaps improperly) shear or shear-dominated flow, and the flow with free-surface boundary conditions, for the entering and exiting sheet. The latter is known to produce large swelling in a viscoelastic material, due to the relaxation of all stresses (particularly the normal stresses) upon exiting from a shear or shear-dominated flow (Tanner, 2000). The flow between the rolls is constrained normal to the plane of shear whereas the material is free to extend in that direction after leaving the rolls. A similar situation is the case for extrudate swell at the exit of a capillary, where the pressure drop inside the capillary may be well predicted by GNF relations, but extrudate swell far exceeds predictions of a GNF model. Apparently, this is the case here, so that viscoelasticity manifests itself as a sudden recovery of the stressed state of the material, as it passes between the rotating rolls. On the other hand, the stresses in the shear flow are well captured by the model, which is better suited for such types of flow. 6. Conclusions Rheological measurements of bread dough have shown that it possesses a yield stress and a shear-thinning behaviour, which is well described by the Herschel Bulkley model of vis-