Acta Polytechnica doi:10.14311/AP.2014.54.0028 Acta Polytechnica 54(1):28–34, 2014 © Czech Technical University in Prague, 2014 available online at http://ojs.cvut.cz/ojs/index.php/ap DESIGN THEORY FOR THE PRESSING CHAMBER IN THE SOLID BIOFUEL PRODUCTION PROCESS Monika Kováčová∗, Miloš Matúš, Peter Križan, Juraj Beniak Slovak University of Technology in Bratislava, Faculty of Mechanical Engineering, Námestie slobody 17, 81231 Bratislava, Slovakia ∗ corresponding author: monika.kovacova@stuba.sk Abstract. The quality of a high-grade solid biofuel depends on many factors, which can be divided into three main groups — material, technological and structural. The main focus of this paper is on observing the influence of structural parameters in the biomass densification process. The main goal is to model various options for the geometry of the pressing chamber and the influence of these structural parameters on the quality of the briquettes. We will provide a mathematical description of the whole physical process of densifying a particular material and extruding it through a cylindrical chamber and through a conical chamber. We have used basic mathematical models to represent the pressure process based on the geometry of the chamber. In this paper we try to find the optimized parameters for the geometry of the chamber in order to achieve high briquette quality with minimal energy input. All these mathematical models allow us to optimize the energy input of the process, to control the final quality of the briquettes and to reduce wear to the chamber. The practical results show that reducing the diameter and the length of the chamber, and the angle of the cone, has a strong influence on the compaction process and, consequently, on the quality of the briquettes. The geometric shape of the chamber also has significant influence on its wear. We will try to offer a more precise explanation of the connections between structural parameters, geometrical shapes and the pressing process. The theory described here can help us to understand the whole process and influence every structural parameter in it. Keywords: densification process, numerical optimalization for structural parameters, mathematical model for cone chamber, mathematical model for cylindrical chamber. 1. Introduction Current European legislation set targets for using renewable energy sources, which will result in the gradual replacement of fossil fuels. Biomass is the most promising renewable energy source, and offers the most effective options for energy storing. This leads to a need to carry out research in the area of pro- cessing biomass and transforming it into a high-grade solid biofuel. The compaction process can affect the mechanical quality indicators of biofuels, especially their density and mechanical resistance. The geome- try of the pressing chamber has an enormous impact on the quality of the briquettes and on the required press pressure. It is therefore appropriate to work on optimizing the geometry of the pressing chamber in order to achieve high briquette quality together with minimum energy input. 2. Structural parameters in the densification process The quality of solid high-grade biofuels depends on many factors, which can be divided into three groups: • material parameters, • technological parameters, • structural parameters. The material parameters affecting the quality of briquettes are mostly linked to the characteristics of the starting material (material strength, composition etc.) and some physical constants. The technolog- ical parameters (humidity, size of the compression pressure, pressing temperature, pressing speed, etc.) can dramatically affect the process of compaction and the quality of the briquettes. However, the structural parameters have a special place in the pressing pro- cess, since the successful production of high-quality briquettes involves synergies between all the groups. The main structural parameters affecting the quality of briquettes are: • the diameter of the pressing chamber, • the length of the pressing chamber, • the convexity of the pressing chamber. Only a limited amount of work is currently being done on mathematical descriptions of the biomass briquetting and pelleting process, the influence of the parameters of the process on the final quality of the briquettes, and descriptions of the effects of pressure in the pressing chamber. There are no complete math- ematical models that deal mainly with the impact of structural parameters on the pressing process. It is clear that a detailed study of the impact of all of structural parameters on the pressing process and the resulting quality of briquettes is a very extensive 28 http://dx.doi.org/10.14311/AP.2014.54.0028 http://ojs.cvut.cz/ojs/index.php/ap vol. 54 no. 1/2014 Design Theory for Pressing Chamber Figure 1. Specification of the geometry of the press- ing chambers in the process of pelleting biomass: a) normal, b) deep, c) flat d) well, e) cylindrical, f) coni- cal, g) stepped. Figure 2. Pressing chamber geometry of the screw briquetting press. undertaking, and requires a detailed analysis of this issue. The most significant influence on the pressing process is from the geometric parameters of the press- ing chamber, i.e. the shape and dimensions (diameter, length and convexity of the chamber). The geometry of the pressing chambers currently used for producing solid biofuels is very diverse. It consists of a cylin- drical part, in most cases also with a conical part. There is often a combination of several cylindrical and conical parts (Figure 1, Figure 2). The length of the cylindrical part provides the necessary back pressure by the friction part of the force. It also provides bio- fuels with a high-quality smooth surface. The conical part of the chamber provides spatial movement of the particles and a higher degree of compaction, resulting in higher production quality. When the material is extruded through the conical part of the chamber, the briquettes are given greater density and strength. However, the friction and pressing conditions in the conical chamber greatly increase the required press pressure. The shape and the size of the pressing cham- ber have a direct impact on the production quality and on the size of the required compression pressure. It is therefore necessary to provide a mathematical description of the whole physical process of densify- ing a particular material and extruding it through a cylindrical chamber, a conical chamber and also a combined chamber. The mathematical models de- scribing the pressure conditions that are presented here form the basis of our study of the geometry of the chamber. Our study focuses on optimizing the chamber geometry in order to achieve high briquette quality together with minimum energy input. Figure 3. Forces in cylindrical pressing chamber. 3. Mathematical background — a cylindrical chamber We will use the following notation in this paper, see Figure 3: • dx — height of an infinitesimal cylinder, dx > 0; • d — cylinder diameter; • Sv — area of the bottom of the cylinder; • S — surface area of the cylinder; • pa — axial pressure • pr — radial pressure • dpa — the pressure change between the top and bottom base, dpa < 0. We suppose that F1 > F2. Based on force equilibrium, we can state the following equation: F1 − F2 − F3 = 0. By simple computation we can state that the force acting on the top base is F1 = paSv = paπ (d 2 )2 = pa πd2 4 , and the force acting on the bottom base is F2 = (pa + dpa)Sv = (pa + dpa)π (d 2 )2 = (pa + dpa) πd2 4 . 29 M. Kováčová, M. Matúš, P. Križan, J. Beniak Acta Polytechnica The friction force F3 is a special case. We will use the coefficient of support friction and the axial force to evaluate F3: F3 = µFN = µpeS = µprπddx. We know that the radial pressure and the axial pres- sure should be connected with their horizontal com- pacting ratio λ: λ = σr σa = pr pa , where σr is the radial stress and σa is the axial stress. So we have F3 = µFN = λpaπddx. Based on equilibrium of forces, we can derive the differential equation for pressure changes between the two bases of the cylinder: F1 − F2 − F3 = 0, pa πd2 4 − (pa + dpa) πd2 4 −µλpaπddx = 0. Let us suppose that dx → 0 and dpa → 0, then d 4 dpa dx + µλpa = 0. (1) The axial pressure depends on the place, so we need to locate our cylinder on the axes. Based on this, we are able to rewrite the axial pressure to the function relation d 4 p′a(x) + µλpa(x) = 0. Hence we have a linear differential equation with con- stant coefficients, and we can find its solution in the form pa(x) = c1e−kx, wheree k is a constant given by k = 4µλ d . The result is in accordance with the physical princi- ple that pressure decreases according to distance from the origin of the coordinate. Based on our coordinate system, we have: • x = 0 — the start position of pressing chamber between compactor and material, • x = L — the start position of pressing chamber. Thus we are also able to compute the Cauchy problem with the initial conditions pa(x) = pap, where pap is the constant pressure of the compactor on the material throughout the pressing phase: pa(x) = c1e−kx =⇒ pa(0) = c1e0 = pap, pap = c1, pa(x) = pape− 4µλ d x. (2) Figure 4. Forces in a conical pressing chamber. The outgoing pressure on position L can be computed: pa(L) = pape− 4µλ d L. We are also able to express the incoming pressure pap in terms of the outgoing pressure: pap = pa(L)e+ 4µλ d L. 4. Mathematical background — a truncated cone chamber A truncated cone is a more complicated case than the classical cylinder. Simply speaking, the cylinder is only a special case of the truncated cone, with the elevation angle α = 0. We will use the same ideas and the same notation as for the cylinder — see Figure 4. In the case of a truncated cone, the force equilibrium will change: F1 − F2 − cos α F3 = 0. The direction of friction force F3 contains elevation angle α with the direction of forces F1 and F2. So we can write paS1 − (pa + dpa)S2 − cos α F3 = 0, where S1 is the area of the top case and S2 is the area of the bottom case. The same coordinate system is used as for the cone. By simply computation we can state that the force acting on the top base is F1 = paS1 = paπ (d2 + 2v 2 )2 , where d2 is the diameter of the bottom case and v is the width of ring of the top case. The force acting on the bottom base is F2 = (pa + dpa)S2 = (pa + dpa)π (d2 2 )2 = pa πd22 4 + dpa πd22 4 . 30 vol. 54 no. 1/2014 Design Theory for Pressing Chamber Figure 5. Essential dimensions of elementary trun- cated cone. Then we have paπ (d2 + 2v 2 )2 −pa πd22 4 −dpa πd22 4 = F3 cos α. (3) By simplification we get F3 = π 4 cos α ( pa ( (d2 + 2v)2 −d22 ) −dpa d22 ) = µFN, where FN is the normal force. Now we will try to find a proper evaluation for the friction force. We need first of all to compute the surface area of the cone. We have Sv = π (d1 2 + d2 2 ) s, where d1 and d2 are the diameters of the top and bottom cases of the cone, and s is the length of the lateral surface. Based on Figure 5, we can compute sin α = d1 −d2 s or cos α = dx s , where s = d sin α . Hence for the surface area of the cone we have Sv = π (d1 2 + d2 2 ) s = π (d1 2 + d2 2 ) dx cos α . Then F3 = µFN = µprπ (d1 2 + d2 2 ) dx cos α . We know that the radial pressure and the axial pres- sure should be connected with their horizontal com- pacting ratio λ. In the case of a truncated cone, the situation is slightly different. Radial pressure pr is perpendicular to the lateral surface, so the horizontal ratio must make provision for this: λ = σr σa cos α = pr pa cos α, where σr is radial stress and σa is axial stress. So we have λ = pr pa cos α and pr = λpa cosα . Finally, we have F3 = µFN = µλpa 1 cos α π (d2 + 2v 2 + d2 2 ) dx cos α . (4) Let us go back to the equilibrium state equation. From (3) and (4) we have π 4 cos α ( pa(d2 + 2v)2 −pad22 −dpa d 2 2 ) = µλpa 1 cos α π (d2 + 2v 2 + d2 2 ) dx cos α . By simple computation we have pa ( (d2 + 2v)2 −d22 ) −dpa s22 = 4µλ cos α pa (d2 + 2v 2 + d2 2 ) dx. We can express the ratio tan α = v/dx. It implies v = tan αdx. The left-hand side should be simplified: pa ( (d2 + 2v)2 −d22 ) −dpa d22 = pa(d22 + 2vd2 + 4v 2 −d22) −dpa d 2 2 = pa(2 tan α dxd2 + 4 tan2 αdx2) −dpa d22. The infinitesimal element should be considered as sufficiently small, so we can omit the term 4 tan2 αdx2. Then we have pa(2 tan αdxd2) −dpa d22 = 4µλ cos α pa (d2 + 2v 2 + d2 2 ) dx, −dpa d22 = −pa(2 tan αdxd2) + 4µλ cos α pa (d2 + 2v 2 + d2 2 ) dx. Let us suppose that dx → 0 and dpa → 0, then dpa dx d22 = +pa(2 tan αd2) − 4µλ cos α pa (d2 + 2v 2 + d2 2 ) . The axial pressure depends on the place, so we need to locate our cylinder on the axes. Based on this, we are able to rewrite the axial pressure pa to the function relation dpa(x) dx d22 = +pa(x)(2 tan αd2) − 4µλ cos α pa (d2 + 2v 2 + d2 2 ) . Hence we have a linear differential equation with a constant coefficient: d22p ′ a(x) + ( 4µλ cos α (d2 + 2v 2 + d2 2 ) − 2 tan αd2 ) pa(x) = 0 and we can find its solution in the form pa(x) = c1e−kx, 31 M. Kováčová, M. Matúš, P. Križan, J. Beniak Acta Polytechnica Λ = 0.15 Λ = 0.25 pap = 140 M Pa 10 20 30 40 50 80 100 120 140 Figure 6. Cylindrical chamber. where k is a constant given by k = 2 sec α(2d2λµ + 2vλµ−d2 sin α) d22 . The result is in accordance with the physical principle that the pressure decreases according to the distance from the origin of the coordinate. Based on our coor- dinate system, we have: • x = 0 — start position of pressing chamber, • x = L — start position of pressing chamber. Thus we are also able to compute the Cauchy problem with the initial conditions pa(x) = pap, where pap is the constant pressure of the compactor on the material during the whole pressing phase: pa(x) = c1e−kx =⇒ pa(0) = c1e0 = pap, pap = c1, pa(x) = pape −2 sec α(2d2λµ+2vλµ−d2 sin α) d22 x . (5) The outgoing pressure on position L can be com- puted: pa(L) = pape −2 sec α(2d2λµ+2vλµ−d2 sin α) d22 L . We are also able to express the incoming pressure pap in terms of the outgoing pressure: pap = pa(L)e + 2 sec α(2d2λµ+2vλµ−d2 sin α) d22 L . As was mentioned above, the cylinder is only a special case of the cone, so in the case of angle α = 0, the result of (5) should be the same as in (2): pa(x) = pape −2 sec α(2d2λµ+2vλµ−d2 sin α) d22 x = pape − 2 1cos α (2d2λµ+2vλµ−d2 sin α) d22 x = pape −2(2d2λµ+2vλµ) d22 x = pape −4µλ(d2 +2v) d22 x = pape −4µλd1 d22 x . If diameters d1 and d2 are the same (d1 = d2 = d), we have pa(x) = pape− 4µλd d2 x = pape− 4µλ d x. The outgoing pressure on position L can be computed: pa(L) = pape − 2 1cos α (2d2λµ+2vλµ−d2 sin α) d22 L . We are also able to express the incoming pressure pap in terms of the outgoing pressure: pap = pa(L)e + 2 1cos α (2d2λµ+2vλµ−d2 sin α) d22 L 5. Numerical experiments The exact expressions for a conical chamber and for a cylindrical chamber have been described above. Now we will deal with some simple numerical experiments. As has been shown, a linear differential equation was used in both cases for describing the mathematical model. In the case of a cylindrical chamber, the relation between the outgoing pressures on position L should be computed by the expression pa(L) = pape− 4µλ d L. 32 vol. 54 no. 1/2014 Design Theory for Pressing Chamber Λ = 0.15 Λ = 0.25 pap = 140 MPa 10 20 30 40 50 60 80 100 120 140 Figure 7. Cone case for α = 2°. Λ = 0.15 Λ = 0.25 pap = 140 MPa 10 20 30 40 50 80 100 120 140 Figure 8. Cylindrical shape – solid line, Conical shape for α = 2° – dashed line. In the case of the cone, the outgoing pressures should be computed by a more complicated expression: pa(x) = pape −2 sec α(2d2λµ+2vλµ−d2 sin α) d22 L . Let us take the concrete example of a conical pressing chamber and a cylindrical pressing chamber. In the case of a cylindrical chamber, we will take d = 20 mm, L = 50 mm, µ = 0.35 and λ comes from the range λ ∈ [0.15, 0.25]. We will suppose that pap = 140 MPa. Then the outgoing press can be modeled by the graph in Figure 6. For a conical chamber, the situation is more com- plicated. Let us suppose α = 2° and the length of the pressing chamber is the same L = 50 mm. Then by simple computation we can set v = tan 2° · 50 mm = 1.74604. Let us assume similar conditions as in the previous case d1 = d = 20 mm. Then d2 + 2v = d1 and d2 = 20 − 2 · 1.74604 = 16.5079. The result is in Figure 7. Simply stated, the shape of the pressure curve re- mains the same in both cases, but in the case of the cone the outgoing pressure is smaller for some pa- rameters λ than in the case of a cylindrical pressing chamber. We can compare the two cases in one picture. We can see that for λ = 0.15 the outgoing pressure is greater in a conical shape, but the situation is com- pletely different for λ = 0.25. In that case, it seems that the conical shape will be more effective. The conical shape is drawn with a dashed line in Figure 8. 33 M. Kováčová, M. Matúš, P. Križan, J. Beniak Acta Polytechnica 6. Conclusions This paper has presented mathematical models for de- scribing the cylindrical part and the conical part of a pressing chamber. These models form the basis for our whole study in the field of densifying biomass into a solid biofuel. All these mathematical models allow us to optimize the geometry of the pressing chamber and the energy input of the process, to control the final quality of the briquette, and to wear to the chamber. Practical results show that reducing the diameter and the length of the chamber and the angle of the cone have a direct influence on the compacting mechanism and, as a consequence, on the quality of the briquettes. The geometry of the chamber also has a significant influence on its wear. Until now, the geometry of the chamber has been designed mostly empirically, with- out any research. However, the theory described here can help to understand whole process and influence every structural parameter in the process. The next step in our research leads toward a mathematically optimized chamber geometry together with minimum energy input (minimal pressure). Acknowledgements This paper is an outcome of the project “Development of progressive biomass compacting technology and pro- duction of prototype and high-productive tools” (ITMS project code: 26240220017), which received funding from the European Regional Development Fund’s Research and Development Operational Programme References [1] MATHEWS, J H. – FINK, K D. Numerical Methods Using Matlab. Upper Saddle River: Pearson Prentice Hall, 2004. pp. 680. ISBN 0-13-191178-3. [2] HOFFMAN, J D. Numerical Methods for Engineers and Scientists. New York: McGraw-Hill, 1993. pp. 825. ISBN 0-07-029213-2. [3] HORRIGHS, W. Determining the dimensions of extrusion presses with parallel wall die channel for the compaction and conveying of bulk solids, No. 12/1985 – Aufbereitungs Technik. [4] MATÚŠ, M. – KRIŽAN, P.: Influence of structural parameters in compacting process on quality of biomass pressing. In: Aplimat - Journal of Applied Mathematics. ISSN 1337-6365. Vol. 3, No. 3 (2010), pp. 87-96. [5] KRIŽAN, P. – ŠOOŠ, Ľ. – MATÚŠ, M. – SVÁTEK, M. – VUKELIČ, D.: Evaluation of measured data from research of parameters impact on final briquettes density. In: Aplimat - Journal of Applied Mathematics. ISSN 1337-6365. Vol. 3, No. 3 (2010), pp. 68-76 [6] KRIŽAN, P. – ŠOOŠ, Ľ. – MATÚŠ, M.: Optimisation of briquetting machine pressing chamber geometry. In: Machine Design. ISSN 1821-1259, 2010, s. 19-24 34 Acta Polytechnica 54(1):28–34, 2014 1 Introduction 2 Structural parameters in the densification process 3 Mathematical background — a cylindrical chamber 4 Mathematical background — a truncated cone chamber 5 Numerical experiments 6 Conclusions Acknowledgements References