PLASTIC DEFORMATION OF POROUS MATERIALS DURING CONSOLIDATION PROCESS

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1 Copyright 009 by ABCM January 04-08, 010, Foz do Iguaçu, P, Brazil PLASTIC DEFOMATION OF POOUS MATEIALS DUING CONSOLIDATION POCESS Ireneusz Malujda, Poznań University of Technology, Chair of Machine Design Fundamentals, No.3, Piotrowo St, Poznań, Poland Abstract. Designing of machines for densification and compression of structural and waste materials plays a major role in development of new processing techniques. The processes of densification and compression are utilized in the production of certain kinds of biomasses as an environment friendly and renewable source of energy. Moreover, specific physical properties may be obtained in the superficial layer of material, an example of which could be improving the quality of wood surface by hot rolling. For modeling of compression and plasticization of loose materials and materials with porous and anisotropic characteristics the primary parameter is the critical strain at which plastic flow commences. The critical strain value, which depends on the thermo-mechanical properties of the material and the key parameters of the process, is critical for effective plasticization. The strength of certain materials such as wood and sawdust significantly decreases with temperature. Therefore, temperature distribution, especially in the thin superficial layer, is one of the main characteristics taken into account in formulating constitutive equations concerning thermal conductivity and plasticity. Keywords: yield condition, critical load, temperature, porosity, loose material 1. INTODUCTION This paper attempts to formulate the strength criterion of particulate materials. This criterion is the main element of a mathematical model describing the said plasticization processes of a bounded layer of such materials taking into account the influence of heat. This is the basis for determining actual parameters of forming the material and deciding on the design characteristics of machinery used for plasticization (Malczewski, J., 1994.). esearch in this area was started due to the needs of engineering practice, as a result of incomplete description of plastic forming, strength and resistance in such materials. This plasticization and compression mechanism requires designing of appropriate geometrical parameters of the briquetting system. Moreover, it is necessary to relate the compression resistance the generated heat in order to stimulate the required critical stresses in the structure of formed briquette. The experience gained in the design, assembly and implementation of machines for briquetting of wood processing waste has been used to formulate consecutive equations describing the process of compression of particulate materials, allowing for the effect of temperature. The critical stress condition has been defined by a relationship combining the constitutive equations describing the process. Plastic flow in a thin layer of wood was analyzed on the basis of the hypothesis of limit energy of shape deformation formulated by Huber Mises Hencky. This condition is the basis for defining the effective parameters of shaping the material and design parameters of the machine in which the plasticization process is effected. Solution to this problem has been presented in the referenced publications (Dudziak M., Malujda I., Meler I., 199). The yield point defining the critical stress has been determined empirically from the compressive strength test as the strength criterion for mathematical model has the process under analysis (Mises, 198 and Szczepiński, 1974).. STENGTH CITEION The effects of friction related to pushing of sawdust through the main chamber and through the forming sleeve (Fig.1a, b) results in an increase of temperature in the outer layer of briquette to over 100 C, which promotes plasticization of wood lignin. In this way, a very thin crust is formed on the briquette, which after cooling down to the ambient temperature provides a uniform and smooth consolidated structure (Malujda, I., 006). For effective plasticization and compression the briquetting system must be designed with appropriate geometrical characteristics. Moreover, it is necessary to relate the compression resistance and the generated heat in order to stimulate the required critical stresses in the structure of formed briquette. The experiences gained in the design, assembly and implementation of machines for refining of furniture and briquetting of wood processing waste have been used in the present attempt to formulate consecutive equations describing the process of compression of particulate materials, allowing for the effect of temperature and humidity. Hence, a mathematical model based on a generalized model of a perfectly plastic body has been developed. This model describes the plasticization of loose material in briquetting process.the intent here was to determine the force defining the stress causing plastic flow of the compressed material. It was assumed that area ABCD defining the actual pressing work (Fig. 1) could be replaced with an equivalent rectangular work area. The height h of parallelogram has the length

2 Copyright 009 by ABCM January 04-08, 010, Foz do Iguaçu, P, Brazil equal to 3 k T,W corresponding to the equivalent yield point and value of the second largest deformation ε is based on the actual material curve. The k T, W stands for the average stress of plastic flow and depends on the influence of heat and humidity. It is equivalent to the yield point on shearing of the material in briquettes (Piwnik, J., 1987). σ D E C h A ε B F F 1 s 1 3 F Figure 1. Change of briquetting pressure σ as a function of piston travel s during uniaxial compression of briquette in an extruder barrel with adjustable taper In the above figure positions 1 and denote the start and the end of piston travel, 3 means a pressure relief phase (withdrawal of piston) resulting in unwelcome decompression of briquette, ε is relativity strain, F 1 defines a main pressing force, F defines a pressing resistance adjustment force. Curve DC illustrates pushing of briquette with characteristic drop of pressure due to friction-generated temperature. Subsequently, the briquette pressing work determined from the actual diagram is compared with the work defined by the area of the rectangle whereσ is temperature and humidity dependent yield point determined from the actual material curve during compression while a is field of this parallelogram. 3 k T W, ε = σε dσdε (1) a The relation Eq. (1) is used to determine the value of k T,W the average plastic flow stress which is equivalent to the yield point of the compressed material. Therefore, the strength criterion of the compressed material becomes the yield point which depends on temperature and has to be determined empirically. By calculating the temperature distribution in the layers of briquette it is possible to determine the value of pressing force which depends on the temperature and humidity, where 1 is a radius of the opening on entry into the main chamber, is a radius of the opening on exit from the main chamber, s is a radius as a function of the main chamber s length, l is a length of the forming sleeve, α determines half of main chamber s taper angle and b defines the length of the main chamber. α y s dx d s (x) ν 1 1 x b Figure. Geometrical characteristics of the pressing channel In order to determine the critical pressing force F 1 it is necessary to predetermine the components of the tensor of displacement velocities and components of the tensor of deformation velocities. Pressing force F 1 as in Eq. () can be

3 Copyright 009 by ABCM January 04-08, 010, Foz do Iguaçu, P, Brazil determined from the power balance of the external forces P z which are used for the power of dissipation in the plasticization area P d and the power of overcoming frictional resistance P f P z = F 1 ϑ 1, P z = P d + P f () where ϑ 1 is en initial velocity at the entry into the axially tapered main chamber (Fig.). In order to determine the power of external forces it is necessary to predetermine the components of the tensor of displacement velocities and components of the tensor of deformation velocities The dissipation power of plastic deformations in the conical channel is described by the following equation P d = συ dv (3) where υ means tensor of deformation velocities and finally, after transformation, power of dissipation has following form N d 1 = 3 3πkT,Wϑ1 1 ln (4) Power of friction is described by the following equation = k fϑ da (5) Pf TW 1 A were A is a surface area of conical channel and after transformation, power of friction in the axially tapered conical opening and the forming channel has the following form N t ds( x ) 1+ dx 1 = πkt,w 1 ϑ1( f ln ) (6) cosα where f means a roughness ratio of the main chamber. Finally, after adding Eq. (4) and Eq. (6) and comparing with Eq. () the equation for calculating of the critical pressing force F 1 in the conical main chamber is obtained 1 1+ ( tgα ) 1 F1 = 3πkT,W 1 [ 3 ln + ( f ln )] (7) 3 cosα The value of critical force drops with the increase of temperature and depends on humidity. Because a direct measurement of temperature, especially during machine s operation, is very difficult, an attempt was made to calculate the distribution of temperature as a function of time in a layer of briquette. 3. MATHEMATICAL MODEL OF HEAT TANSFE The temperature is a key parameter in formulation of the constitutive equations describing the analysed process. There are various ways in which heat penetrates a body. Here, two methods will be briefly described, relevant to the process under analysis and resulting from the actual transfer of heat between the hot wall of the chamber and closely abutting side of briquette. Unsteady heat conduction is described by the second Fourier law T = a T t (8)

4 Copyright 009 by ABCM January 04-08, 010, Foz do Iguaçu, P, Brazil λ where a = describes a thermal diffusivity, T is a temperature, t is a time, ρ means a specific density, c p is a specific c p ρ heat and λ means a thermal conductivity. The thermal diffusivity is usually determined experimentally, taking into consideration influence of temperature and humidity. Highly relevant here is the convective transfer of heat expressed in the following equation Q= α (T p T 0 ), B i αd = < 1 λ (9) where T p it is a surface temperature and T 0 is a temperature of the boundary layer, α defines a heat transfer coefficient, B i describes a Biot s number and d is a layer thickness. Convection is more appropriate than conduction in describing the heat transfer when the Biot number is less than one, and for B i > 1 heat is transferred by conduction ( Hobler, T., 1968) Transfer of heat by unsteady conduction is the closest approximation of the actual transfer of heat between the hot wall of forming sleeve and the briquette. The hot wall of the forming sleeve is in contact with the surface of the formed briquette, and thus it can be assumed that the heat transfer coefficient α tends to infinity, and consequently the Biot number (7) is greater than 1. Thus, the criterion for conductive heat transfer is met. Therefore, the analysis of the constitutive relations describing the heat transfer between the wall of forming sleeve and the briquette will be related to conduction only. Therefore, the role of convection and radiation has been considered insignificant and ignored in modelling the transfer of heat between the surface of hot roller and the plasticized wood surface. Conduction was assumed to be the only way in which heat was transferred in the process (Malujda, I., 006). The above term is known as Fourier s law describing the change of temperature in time and as a function of temperature gradient variation in 3D. With correct boundary and initial conditions it allows for determination of time dependent temperature at any point throughout the analysed layer of material. The transformed equation (8) draws our attention to the importance of the temperature conduction coefficient a (and, specifically, to the three parameters describing it, namely: c p, ρ, λ). For solving the problem of heat conduction in the layer of briquette the approximate solutions method is used. Following the spatial digitisation of Eq. (8) selection of appropriate initial and boundary conditions and approximation of the temperature field with the finite element shape function a system of simple differential equations is obtained as a function of node temperatures and their time derivatives. The system of equations may be expressed with the following differential equation: d [C] {T}+[K]{T}={F} (10) dt where [C] is a heat capacity matrix, [K] is a conductivity matrix, {F} means thermal force vector, {T} is a nodal temperature vector. Software program I-DEAS equipped with TMG Thermal Analysis module was used to prepare a 3-dimensional physical model of a layer of briquette on the basis of geometrical parameters (Fig. ). Spatial analysis was based on eight-nodal spatial elements of rectangular prism shape. As a result, a linear interpolation of the analysed function between the nodes is obtained. The calculations were based on the properties of the material and dimensions of briquettes produced by industrial machines, namely 60 mm diameter and 50 mm length. Two minute duration of briquette formation in the sleeve has been assumed to account for production capacity limitations. Although sawdust is classified as a composite particulate material with anisotropic and non-linearly elastic behaviour, for the purpose of this study the pressed material is regarded as a material continuum. This allows for determination of certain thermo-mechanical properties as the average values. Such model is fitted with equivalent coefficients such as equivalent heat conductivity and average density. In the analysed model the so-called Dirichlet s condition has been chosen from among the four basic types of boundary conditions. This condition assumes known temperature on the surface of briquette closely abutting the side of forming channel. This known temperature is ca C, as this level is crucial for effectiveness of the compression process and cohesion of the produced briquette. The initial-value conditions (a.k.a Cauchy conditions) are the values of body temperature at the initial moment t 0 = 0 s. Hence T(x,0) = T 0 (x) = 0 0 C.

5 Copyright 009 by ABCM January 04-08, 010, Foz do Iguaçu, P, Brazil 3.1. Calculation results The above described mathematical model of spatial heat conduction is projected using the I-DEAS software. In order to increase the accuracy of temperature distribution in the thin superficial layer of briquette the density of the finite element grid (Fig. 3a) is increased in the contact zone between the hot forming sleeve and the surface of plasticised material. The resulting spatial distribution of temperature after minutes is presented in (Fig. 3b). Distribution of temperature on a chosen XY plane is presented in Fig. 3b. As the layered temperature distribution profile (Fig. 3a) does not allow for qualitative evaluation of the calculation results, they are presented as curves (Fig.3b) representing the increase of temperature as a function of time in the subsequent temperature layers (Malujda, I., 006). a b TEMPEATUE [ C ] TIME [s] Figure 3. Calculation results: a) a fragment of grid showing temperature layers, b) curves representing the change of temperature as a function of time and depth of the respective layers, where the temperatures at the respective depths are: mm, 1 mm, 3 1.5mm, 4 mm. 3.. Final remarks Critical pressing force obtained in analytical form Eq. (7) can be easily applied in practical calculations of the geometrical parameters of briquette and in designing the machines utilizing this process. According to the obtained temperature distribution in the direction inwards the briquette and for the assumed input data the temperature increases to ca C (at mm depth) and to ca C (at mm depth). This supports the conclusion that plasticization of the sawdust layer reaches down to ca. 1 mm depth, accompanied overheating to the temperature at which plastic flow of lignin occurs. This provides the manufactured briquette with a consolidated and robust crust promoting integrity of the material. 4. EFEENCES Dudziak M., Malujda I., Meler I., 199. Modeling plastic flow loose continuous medium in convergent channels PUT Press, Nr 37, Poznan, Poland, pp Hobler, T., Heat transfer and heat exchangers. WNT Press, Warszawa. Malczewski, J., Mechanics of loose materials WUT Press, Warszawa, Poland. Malujda, I., 006, Plasticization of a bounded layer of anisotropic and loose material, Machine Dynamics Problems,Vol.30, Nr 4, pp Mises,., 198. Mechanik der plastischen Formänderung von Kristallen, ZAMM,8,3, pp Piwnik, J., Analysis of symmetrical-axial extrusion process, Engineering Transaction, 3,. Szczepiński, W.,1974. Ultimate limit states and kinematics of loose medium, PWN Press, Warszawa, Poland. 5. ESPONSIBILITY NOTICE The author is the only responsible for the printed material included in this paper.