A finite-difference method for the prediction of velocity field in extrusion process

Transcription

1 Journal of Food Engineering 83 (2007) A finite-difference method for the prediction of velocit field in etrusion process Gino Ferretti, Roberto Montanari * Dipartimento di Ingegneria Industriale, Università degli Studi di Parma, Viale delle Sciene, Parma, Ital Received 26 September 2006; received in revised form 30 December 2006; accepted 3 Januar 2007 Available online 18 Januar 2007 Abstract Etrusion defects and flow instabilities are an important limitation in most product processing operations. Investigation of the fluid kinematics to deduce the dnamical response of the fluid can be ver useful to characterie these instabilities and understand the mechanisms involved in their triggering and enhancement. To do so, the flow of a Newtonian fluid through a screw press has been studied in order to predict the downstream velocit field in etrusion process. Flow in the screw press is modeled b adapting the classical theor of single screw etrusion. Some publications have addressed this subject matter, especiall in the nineties. However, most of these publications develop models that are not so eas to appl. To complement those earlier works, an approimate model base on finite-difference method is developed. The models in this paper ma be considered as of pedagogical value because of the ease of their application; nonetheless, results are obtained basing on the real etruding conditions, such as a general screw geometr. In order to achieve this pedagogical purpose, three important properties pertaining to model building are considered in this article. These properties are applicabilit, simplicit, and level of technique applied. Moreover, the models introduced in this paper are straightforward application b both practitioners and students. The ease of the applicabilit is confirmed b the suitabilit to adopt a widel emploed software, such as Microsoft Ecel, to implement the model developed. Ó 2007 Elsevier Ltd. All rights reserved. Kewords: Etrusion; Finite-difference; Downstream velocit field 1. Introduction At the beginning, single screw etruders have been widel used in the polmer industr, but in the last decades the have been etensivel applied in the ceramic, food and pharmaceutical industries, due to the fact that the are cheap, robust and continuous. Etruders geometr is an important parameter of the etrusion process, since it affects shear rate and residence time values, involving direct outcomes on the qualit of the final product. Defects and flow instabilities are important limitations in processing operations of most etruded products. Investigating fluid kinematics to deduce the * Corresponding author. Tel.: ; fa: address: [email protected] (R. Montanari). dnamical response of the fluid can be ver useful to characterie those instabilities and understand the mechanisms involved in their triggering and enhancement. In addition, better control on the downstream velocit field in etrusion process can be a significant source of qualit differentiation between food products. In other words, a more regular profile and lower values of the mean shear rate along the etruder determine a reduced gelatiniation of the compound. Conversel, etruders characteried b high values of shear rate can involve several issues in controlling the final qualit of the products, due to the comple behavior of the shear rate profile along the screw ais. On the ground of these premises, models for predicting the etrusion process variables have been receiving ever increasing attention b the researchers. Until the 1970s, food etrusion was regarded as an art rather than a science; thus, onl trained and eperienced personnel could handle /$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi: /j.jfoodeng

2 G. Ferretti, R. Montanari / Journal of Food Engineering 83 (2007) Nomenclature D b R s W H h h b l V s internal barrel diameter screw root radius rotation sped of screw width of channel at the internal radius of barrel channel depth heli angle of screw heli angle of screw at internal barrel radius that is R s + H viscosit of the Newtonian fluid screw root velocit down channel coordinate channel depth coordinate channel width coordinate V, V, V velocit component of, and directions pressure gradient in down channel direction o o o n i, j pressure gradient in channel depth direction pressure gradient in channel width direction generic node (i,j) inanw * nh grid mesh nw, nh number of nodes along and direction h, h distance between two adjacent nodes along and direction Q leakage volumetric flow rate of leakage flow such equipment. However, over the past 30 ears and especiall in the nineties, several models have been developed to predict process throughput, energ requirements, miing patterns and residence time distribution of etruders. One of the earliest models for single screw etrusion was presented b Darnell and Mol (1956). The model developed b the authors described plug flow of a solid feed material along a screw of constant pitch and depth. The original approach proposed b Darnell and Mol (1956) has been subsequentl developed b Burbidge and Bridgwater (1995), Englander, Burbidge, and Blackburn (2000), b adding the rheological parameters characteriing past feed materials. Following a similar approach, further authors focused on etrusion process for food materials. Among recent works, Vergnes, Le Rou, Chaurand, and Abécassis (1995), developed and eperimentall validated a thermomechanical model for pasta etrusion. More specificall, the flow along the screw channel was computed using finite element simulation and described b means of an approimate model, which has been derived starting from wellknown analses of plastic etrusion. Karwe, Chiruvella, and Jaluria (1996), studied the behavior of starch-based materials under high temperature environment, b a numerical simulation of flow inside a single screw etruder. Else, Riepenhausen, Mcka, Barton, and Willis (1997), developed a fleible dnamic model of the etrusion process in order to predict and control realistic qualit variables, such as product gelatiniation fraction. Along the same line, the earlier work of Blackburn, Bottena, and Burbidgea (2003), presented a general one-dimensional model for the prediction of the pressure field in single screw etrusion including corrections for non-constant channel depth in the screw. As stated above, several models have been developed and presented on etrusion processes. Nevertheless, the analsis of the scientific literature has pointed out that the issue of accurate predicting the velocit distributions has been somehow neglected. In particular, issues such as how to predict the downstream velocit field in etrusion processes have not received the required attention in technical literature and onl few authors faced this challenge. Both mathematical and eperimental approaches have been emploed to achieve the required accurac in order to describe and solve more realistic generalied problems. Choo, Neelakantan, and Pittman (1980), developed a computer model for flow in a single-screw etruder, pumping a Newtonian liquid under isothermal and developed flow conditions. Flow rate, screw speed, and pressure gradient characteristics were measured, and a tracer particle technique was emploed to determine channel velocit profiles. The data were required for the testing the model which takes into consideration channel curvature. Using the resulting eperimental data, Li and Hsieh (1996), developed and tested an analtical solution of an isothermal, Newtonian fluid in a single screw etruder with finite channel. Following this analtical approach, Yu and Gunasekaran (2003), proposed and tested a numerical approach to anale flow field and heat transfer developed inside a single-screw channel. Finite-difference and finite element methods were used to obtain numerical solutions for flow and energ equations also in the case of non-newtonian fluids. Along the same line, but more focused on eperimental approaches, Bakalis, Co, Russell, Parker, and Frer (2006), analed velocit distributions in single-screw etruder a speciall designed and manufactured, b using positron emitting particle tracking. In particular, velocit profiles in the etruder, that could be predicted b computational fluid dnamics were generated, measured and compared to the eperimental data b setting process parameters, flow rate and rotational speed in the range used in commercial processes. On the ground of literature analsis, it can be right stated that most of the papers eamined present approaches that are able to accuratel predict the downstream velocit field in etrusion process. However, it is also self-evident that the wa approaches were developed has can significantl affect both their practical applicabilit and their degree of current adoption. Generall speaking, the higher the accurac of the approach proposed, the higher the resulting compleit of the model; consequentl, higher difficulties ma be involved in prac-

3 86 G. Ferretti, R. Montanari / Journal of Food Engineering 83 (2007) tical implementation of the model itself. As a result, a necessar trade-offs between model compleit and practical applicabilit rise. Moreover, in some cases a detailed eplanation on how to appl well-know techniques, such as finite-difference methods, eploiting widel emploed software (e.g. MS Ecel) should be considered as essential for the applicabilit of the models developed. The analsis of scientific literature related to the etrusion process, and in particular to the downstream velocit field, has pointed out that the issue of how readers could appl the proposed models has been somehow neglected. Based on these premises, the aim of this paper is twofold. First, we propose a finite-difference approach for predicting the downstream velocit field in etrusion process. Fluid flow in the screw press is modeled b adopting the classical theor of single screw etrusion for Newtonian fluids in a general screw geometr. In addition, we present a detailed numerical case stud, in order to provide the required knowledge for a quickl propagation of the model without epensive and obscure software. Usuall, these tools act as black boes where onl few people know how the work and consequentl onl few people have a full control on them. To overcome this problem the numerical case stud is developed step b step b user-friendl and widel emploed software as MS Ecel. The model proposed in this paper ma be considered as of pedagogical value because of its applicabilit, simplicit, and good performance in predicting the velocit field values. Moreover, the model introduced in this paper is of straightforward application b both practitioners and students. The ease of the applicabilit is confirmed b the suitabilit to adopt a widel emploed software, such as Microsoft Ecel, to implement the model developed. The good performance of the model is confirmed b the comparison between numerical results and eperimental data from published literature. This stud is part of a wider project aiming to control microbiological parameters of etruded products. The whole project is founded b Regione Emilia-Romagna inside the PRRITT (Programma Regionale per la Ricerca Industriale, l Innovaione e il Trasferimento Tecnologico). 2. Safet emphasis This stud is part of a wider project whose aim is to define process operative leverages in order to control microbiological polluters in food etruded products. In particular, a mathematical model, and subsequentl a simulation software, will be developed in order to predict phsical properties affecting the proliferation of microbiological polluters in products during the whole process. Thanks to the software developed, proper conditions could be designed to optimie the safet of the etrusion process. 3. Mathematical formulation The material fed to the screw etruder passes through a relativel long, shallow helical channel formed b the flight of the screw. The channel bounded b the inner surface of the barrel, the root of the screw and the flight. In order to simplif the problem man approaches hpothesie stationar screws and rotating barrels (in opposite direction). This does not alter the analsis when the internal barrel diameter is much larger then the channel depth. Nevertheless, Li and Hsieh (1996), underlined the source of error and the proposed a new analtical solution for flow in etruder with a rotating screw and stationar barrel. In accordance with Li and Hsieh (1996), and in order to provide better and more realistic results, the model developed eploits a rotating screw and stationar barrel approach. The channel and the screw geometr studied is qualitativel shown in Fig. 1; in the scheme,, and represent channel down stream, channel height and channel width directions, respectivel. Since in man practical etruders the height of the screw channel is much less than the barrel diameter, the screw curvature is assumed to be ver small. As a consequence, the barrel surface and the screw channels can be mathematicall modelled as unwrapped. In other words, from a geometrical point of view, the higher the ratio of the barrel diameter to the height of screw channel, the lower screw curvature should be and the screw channel tends to become flat plate. This does not alter the analsis, since centrifugal force are negligible. In order to further simplif the problem, the following assumptions are commonl used without losses in accurac: fluids flow is assumed as laminar, since the ver high viscosit of the common etruded fluid (>10 3 Pa s) leads to Renolds numbers much less than unit; the inertia terms in the Navier Stokes equation can be neglected in comparison with viscous and pressure terms; gravitational force is negligible; the fluid is incompressible and Newtonian; the flow is full developed; no slips at the walls occur. Under these hpotheses, equations describing fluid motion for this problem in rectangular coordinates can be epressed as follows: o ¼ l o2 V o 2 þ o2 V o 2 o ¼ l o2 V o þ o2 V 2 o 2 o ¼ l o2 V o þ o2 V 2 o 2 component component component Assuming that V 0, which is a reasonable assumption for screws with a small channel depth to width ratio, ð1þ

4 G. Ferretti, R. Montanari / Journal of Food Engineering 83 (2007) Barrel H D b ω W R s θ Lead Barrel H V V s θ V Screw W Fig. 1. Real and unwrapped geometr of a single screw etruder. H/W, and using the continuit equation; then, ov /o 0, ov /o 0, and ov /o 0. Thus Eq. (1) becomes: o ¼ l o2 V o þ o2 V 2 o 2 component o 0 component ð2þ o l o2 V o 2 component The above simplified equations of motion can be found in Li and Hsieh (1996). Since the proposed model utilies a rotating screw and stationar barrel approach, boundar conditions should take into account the screw velocit. In particular, the screw flight velocities are not constant but variable as a function of the radius; from the mathematical point of view, in the case of the -component the function is epressed in the following boundar conditions: Barrel Screw root Screw flights V ð; HÞ ¼0 V ð0; Þ ¼R s cosðhþ V ð0; Þ ¼ðR s þ Þ cosðhþ V ðw ; Þ ¼ðR s þ Þ cosðhþ ð3þ As can be seen from the above equations, the -component of Eq. (2) is a non-homogeneous, second order partial differential equation with non-homogeneous boundar conditions. Li and Hsieh (1996), proposed an analtical solution of this problem for finite rectangular channel. In order to etend the solution to a general duct geometr, a finite-difference method is applied to solve the differential -component equation. To this etent, the domain is discretied b imposing a grid defined for a set of nodes n i, j (see Fig. 2). On the ground of these premises, the -component of the above equation can be written in discrete form as follows (the mesh is an nw * nh grid along and direction, respectivel): o l V ði þ 1; jþ 2V ði; jþþv ði 1; jþ h 2 þ V ði; j þ 1Þ 2V ði; jþþv ði; j 1Þ ð4þ h 2 Hence, for an estimated pressure gradient in the down channel direction, the component of velocit V (i,j) for the (i,j) node in direction becomes:!

5 88 G. Ferretti, R. Montanari / Journal of Food Engineering 83 (2007) n 1,nH n i+1,j n i,j-1 n i,j n i,j+1 n i-1,j n 3,1 n 2,1 n 1,1 n 1,2 n 1,3 n 1,nW Fig. 2. Discretied domain. " V ði; jþ h2 h2 V ði þ 1; jþþv ði 1; jþ 2ðh 2 þ h2 Þ h 2 # þ V ði; j þ 1ÞþV ði; j 1Þ 1 ð5þ h 2 l o It is noteworth that the velocit value in the generic node (i,j) is a function of the velocit values close to the node itself, and, in particular, these are the velocit value at the up node V (i +1,j), the velocit value at the down node V (i 1,j), the velocit value on the right side node V (i,j + 1), and the velocit value on the left side node V (i,j 1) as shown in Figs. 2 and 4. These velocit values have been derived in the previous iteration and are assumed as constants for the current iteration. Since boundar conditions are constant along the procedure, Eq. (5) can be adopted iterativel for each node in the mesh (ecept the boundar nodes) until a required level of accurac is achieved. Based on the same rationale, slanted walls of the channel can be taken into account b a suitable mesh as shown in Fig. 3. As an oblique line can be shown in a PC monitor b piels, slanted walls of the channel can be modeled b rectangular grid and, as can be seen in Fig. 3, the higher the number of nodes, the higher the qualit of the model should be. As shown in Li and Hsieh (1996), for the cross channel flow, component of Eq. (2) can be integrated twice to ield: V ¼ R s sin h b ðh Þ H 4. Software implementation h 3 iþ H 1 6Q leakage ½H 2 Š Nowadas, commercial solutions are available to solve comple fluid dnamic problem, although these solutions are often epensive, unknown, and not user friendl if compared to spreadsheet programs which are H 3 available in widespread, cheap and user friendl software packages as MS Office, Star Office or Open Office. Besides, as stated above, the commercial solutions for solving comple fluid dnamic problem, in several cases, act as black boes and onl few people know how the eactl work; consequentl, few people have a full control on them. This means that few people can utilie and full understand the outcomes from the software. Conversel, when adopting user friendl and widel emploed softwares, a general model can become a powerful tool for a large number of users. To this etent, and due to its wide emploment, a MS Ecel implementation of the presented approach has been developed. Due to its framework, MS Ecel is particularl suitable to solve the finite different approach. As well known, it is structured b cells can contain labels, numerical values and mathematical relationships between cells. Due to its similarit to the finite different approach, this structure is particularl appropriate to describe the mesh and the mathematical relationships between each node. In other words, each cell in MS Ecel is approached as a node, where both numerical value and analtical formula represent the down stream velocit and the mathematical relationship to solve the problem, respectivel. Moreover, thank to the large number of available cells, MS Ecel is particularl suitable to be adopted to implement the iterative approach required to achieve a defined level of accurac. Once the number of nodes is defined, the cells in MS Ecel to be implemented, can be identified. Clearl, the higher the number of nodes of the mesh, and consequentl the number of cells involved, the higher the resulting accurac of the model. Following the boundar conditions of Eqs. (3), the boundar cells have to be implemented with the respective formulation, whereas the others cells have to be initiall set to 0. Fig. 4 shows the mesh (the boundar cells are in gre) at the iterative step 0, the related data and the screw flights velocit formulation in a boundar cell on the left side.

6 G. Ferretti, R. Montanari / Journal of Food Engineering 83 (2007) Fig. 3. Discretied domain with slanted walls. Fig. 4. MS Ecel application at the iterative step 0. In order to develop the iterative procedure, a set of tables have to be computed. In other words, Eq. (5) has to be adopted for the inside cells where the velocit values close to the specific cell (i.e. V (i,j + 1), V (i,j 1), V (i +1,j) and V (i 1,j)) are obtained in the previous table and are treated as constants values

7 90 G. Ferretti, R. Montanari / Journal of Food Engineering 83 (2007) Fig. 5. Iterative MS Ecel application. for the current table (see Fig. 5). To this etent, to implementing the iterative procedure the user can quickl cop the second table as shown in Fig. 5. The higher the number of iterations, and consequentl the number of tables computed, the higher the accurac of the model. It is noteworth that the above implementation developed for a rectangular channel can be easil etended to applications for general channel geometr: different geometries can be built b simpl adding and/or removing opportune cells in MS Ecel. Table 2 Down channel velocit =15mm V (mm) Numerical (mm/s) Analtical (mm/s) Comparison (%) Results To assess validit and accurac of the proposed approach, results provided b the finite-difference methodolog are compared with those presented in published papers. Since eperimental and/or analtical data are currentl available onl for rectangular channel, the application is developed with a rectangular dept. The Table 1 Pertinent data H 10 mm W 30 mm h 1 mm h 1 mm nh 11 nw 31 R s 9 mm l 10,000 Pa s / b Rad /s V mm/s /o 5000 Pa/mm predicted down channel velocit is compared with the analtical solution proposed b Li and Hsieh (1996), whose approach was validated b eperimental results measured b Choo et al. (1980). Table 1 shows the pertinent data, while Table 2 shows the down channel velocit computed and the comparison results in the middle of the channel. As can be observed from Table 2, the difference is less than 1% even when onl a mesh is used. These results are obtained after onl 79 iterations. Clearl, the higher the number of iterations and the higher the number of nodes in the mesh, the higher the accurac should be. It is noteworth that the user can suitabl set these two parameters in order to achieve a desired level of accurac. In this case stud, the maimum difference less than 1% from the analtical solution proposed b Li and Hsieh (1996), is the criteria utilied for deciding when the solution had converged. In Fig. 6, the three dimensional down channel velocit representation is shown b MS Ecel surface graph.

8 G. Ferretti, R. Montanari / Journal of Food Engineering 83 (2007) Fig. 6. A three dimensional down channel velocit representation. 6. Conclusions The paper proposes a finite-difference approach for solving the down channel velocit in a single screw etruder of a Newtonian fluid. The approach completes those eisting in toda s literature b adding an efficient as well as eas to appl procedure to obtain the velocit field. Moreover, the model ma be considered as of pedagogical value because of its applicabilit, simplicit, and level of technique applied. Moreover, the model introduced in this paper is of straightforward application b both practitioners and students. Since of a user friendl, widel emploed, and cheap software as MS Ecel is utilied for implementing the model, the proposed approach becomes an interesting and practical tool aiming at a quickl propagation. In other words, this stud has presented a tool that can be adopted b ever people for constructing the screw characteristics and the analing the etruder performance. The predicted down channel velocit data are compared with the analtical data of Li and Hsieh (1996), that were validated with the eperimental data of Choo et al. (1980). Testing the proposed approach for a small mash (10 30 nodes), has ielded less than 1% difference, providing good findings of the methodolog developed. The full developed velocit profile reveals important information about the effect of the screw flight. Besides, it represents the starting point for the fluid dnamic and thermodnamic approach for the purposes of screw design and etrusion simulation.

9 92 G. Ferretti, R. Montanari / Journal of Food Engineering 83 (2007) Acknowledgement The authors appreciate the efforts and the constructive comments of the anonmous reviewers that have helped in sharpening the manuscript. References Bakalis, S., Co, P. W., Russell, A. B., Parker, D. J., & Frer, P. J. (2006). Development and use of positron emitting particle tracking (PEPT) for velocit measurements in viscous fluids in pilot scale equipment. Chemical Engineering Science, 61, Blackburn, S., Bottena, J., & Burbidgea, A. S. (2003). A model to predict the pressure development in single screw etrusion. Journal of Materials Processing Technolog, 135(2 3), Burbidge, A. S., & Bridgwater, J. (1995). The single screw etrusion of pastes. Chemical Engineering Science, 50, Choo, K. P., Neelakantan, N. R., & Pittman, J. F. T. (1980). Eperimental deep-channel velocit profiles and operating characteristics for a single-screw etruder. Polmer Engineering and Science, 20, Darnell, W. H., & Mol, E. A. J. (1956). Solids conveing in screw etruders. Social Phsics Engineering Journal, Else, J., Riepenhausen, J., Mcka, B., Barton, B., & Willis, M. (1997). Modeling and control of a food etrusion process. Computers and Chemical Engineering, 21, Englander, A., Burbidge, A. S., & Blackburn, S. (2000). A preliminar evaluation of single screw paste etrusion. Chemical Engineering Research and Design, 78, Karwe, M. V., Chiruvella, R. V., & Jaluria, Y. (1996). Numerical simulation of the etrusion porcess for food materials in a single-screw etruder. Journal of Food Engineering, 30, Li, Y., & Hsieh, F. (1996). Modelling of flow in a single screw etruder. Journal of Food Engineering, 27, Vergnes, B., Le Rou, D., Chaurand, M., & Abécassis, J. (1995). A thermo-mechanical approach to pasta etrusion. Journal of Food Engineering, 26, Yu, C., & Gunasekaran, S. (2003). Modeling of melt conveing in a deepchannel single-screw cheese stretcher. Journal of Food Engineering, 61(2),