MATHEMATICAL MODELLING AND SIMULATION OF VARIOUS TYPES OF PAPER MILL CALENDERS

Abstract

Calendering is an important operation in many industries such as paper, polymer, leather and textile, mainly with an aim, to increase the smoothness (or to reduce the roughness) o f surface o f sheet o f substrate and also to enhance gloss in some special quality o f substrate (coated paper for example). In the process o f calendering however, the substrate becomes more densified with the unintentional decrease in bulk and thickness. In paper m ill, the parameters involved in calendering process o f paper substrate which effect the quality are: load applied, speed and radius o f rolls, number o f nips, types o f rolls(hardness, elasticity, surface smoothness) and temperature o f the rolls, temperature and moisture content in paper, and many more. The forecast investigation aims at identifying the relative effect o f each parameter on the quality o f paper in any kind o f calenders (machine calender, soft calender, super calender, temperature and moisture gradient calender etc). The parameters are interactive in nature. In dynamic process o f calendering, it is extremely difficult to evaluate some o f the design and process parameters, under local external and internal conditions and their interactions. For complete analysis o f the problem, it thus necessitates the development o f various mathematical and statistical models. Presently the models available constitute models for nip mechanics, steady and unsteady state heat transfer modelling and model related to stress -strain behaviour (called viscoclaslic model). The nip mechanics model is useful to evaluate the nip width, maximum nip pressure, and pressure distribution in nip. Applying higher temperature in the calender, the smoothness and gloss o f the paper w ill be enhanced. Heat is frequently transferred to the calender rolls from the hot Web leaving the dryer section, also generated during compression in the nip or may be supplied externally from hot roll heated by supplying pressurized hot water, or oil ,or steam/hot air showering to the web to raise the temperature o f the paper. In some case inductions heating alone or in combination with other appliances are also in vogue. Variations in the temperature may occur across the face o f the calender roll due to temperature variations across the web. The third model takes care o f strain-stress relationship along with elasticity variation. Though the above models are available in literature, these are not free from restrictions. There are some limitations of the models which fit to a certain situation o f specific calender type. Therefore, the existing models are not enough to cover the entire spectrum o f modelling o f all types o f calenders as mentioned above. In this present investigation all types o f calenders are categorized as function o f temperature, pressure and material o f construction only. The latter parameter in turn reproduces other dependable parameters like modulus o f elasticity, Poisson’s ratio, hardness index and might be coefficient o f friction. An attempt has been made in this investigation to extend the model o f Hertz [57] and that o f Meijers [95] using more than three terms o f the infinite series obtained from integral equation. Further the sensitivity o f additional terms on the response are to be compared. It is also important to check the validity o f the equations proposed by Hertz [57] for soft roll and other types o f calenders. The. effect o f calendering in terms o f average nip pressure (line load/ nip width) has also to be examined. In addition the models are further validated with different values o f Poisson’s ratio applicable for not only calenders o f different types(machine calender, soft calenders, super calenders, temperature gradient calenders and substrata thermal molding) and also for rolls o f different printing processes, specially lithography offset presses. It may. be recalled that the present investigation do not distinguish among different designs and composition o f rolls and other roll properties, rather it has considered all rolls are o f different values o f poisons ratio, elastic modulus, diameters and may be friction coefficient. This is true in mathematical sense. The results are validated with the experimental data put forward by various investigators, with laboratory pilot calenders or full calenders in practice.. Heat transfer problem can be solved in the case o f calender using various kinds o f initial and boundary conditions which in turn depend on design and operational conditions o f various calenders.The problems have been solved analytically using various transform techniques Laplace, Fourier etc. With various initial and boundary conditions including convection boundary conditions. Considering the calender or a paper or both semi-infinite slabs o f various thicknesses models can also be developed. Using a high roll temperature it is possible to produce a temperature gradient in the thickness direction o f the paper at the same time as the paper is compressed in the nip.The temperature distribution in the X direction (thickness) when there is one heated roll can be estimated An .attempt has been made in this present investigation to further extend the modelling o f the calender -paper system o f various types and improve their solution by considering the various limitations o f earlier investigators, such as variable properties, the effect o f contact resistance, and varying boundary and initial conditions as per the need o f machine calender, supercalender, gloss calender, temperature gradient calender and substrata thermal molding prevailing in industrial practice. In this present investigation to apply the above equations to a machine calender problem (to start with) the parameters of.importance are assumed as follows: T is the temperature o f web, t is the dwell time, W,the nip width, L the web calliper, the thickness direction, cp is the heat capacity and p , the density o f the web. The parameters are however, functionally, dependent o f other parameters (temperature, composition, moisture content, humidity etc.)- Attempt w ill be made to use the functional relationship o f the properties o f roll and paper. When paper passes through a calender nip, there is a compressive pressure developed for a short time. After passing through nip there is a time dependent partial recovery in the initial thickness o f the paper. The most commonly used models for calendering process are as under 1) Linear Maxwell model 2) Linear Kelvin (or Voight) model 3) Burgers four element model 4) Non-linear models o f van Hagg From the detailed survey it appears that all the models proposed for calender designs seem / • inadequate in many respects. Therefore, an attempt has been made in this investigation to examine all the above models in the light o f limitations indicated by various investigators. Further attempt has been made to develop an improved model over the available models considering the effect o f temperature, moisture and viscoelastic effects in paper compression which might be linear or non linear. In order to achieve the objectives ,methodology o f systematic investigation for simulation o f calendering models , namely models for nip mechanics , conduction heat transfer and stressstrain behaviour and viscoelasticity has been developed. In order to achieve the objectives,computer program has been-developed and M ATLAB software has been used. Flow charts as well as algorithms are also developed to facilitate the simulation o f models. Lastly prediction o f paper properties are made with the interlinking all the three models. From the detailed analysis o f data obtained from simulation by M ATLAB software and from various graphs following noteworthy conclusions can be drawn: The conclusions are made for a specific calender and the related models. Studies on nip Mechanics model of Machine calender reveals tlie following conclusions: > * Nip width increases with increase in line load and data obtained from m ill matches more closely with the v=0.28 solution as compared with the Hertz solution. Nip width increases with increase in equivalent diameter.The nip width obtained from Hertz solution matches very closely with the 0.28 solution. • With increase in equivalent elastic modulus nip width decreases. Nip width obtained from Hertz model matches closely with the v=0.28 solution.^ • There is no effect o f cover thickness on nip width obtained from Hertz solution as cover thickness is not considered in this case. But v=0.28 solution shows that nip width increases with increase in cover thickness. • Average pressure remains same in case o f Hertz solution, while v=0.28 solution shows that average pressure decreases with increase in cover thickness. Studies of nip mechanics of Softcalender concludes the following points • W ith increase in line load nip width, increases and all the solutions i.e Hertz solution, Deshpande solution and v” 0.4 solution are quite close to each other but v=0.4 solution matches much closer to the m ill data. • With increase in equivalent diameter nip width increases. Nip width obtained from Hertz solution matches very closely with the v=0.4 solution. • W ith increase in elastic modulus nip width increases.The results obtained by all the models are close, but the nip width obtained by Hertz solution and v=0.4 solution are very close to each other. • Cover thickness has no effect on the nip width obtained by Hertz model,while the nip width obtained by . Deshpande and v=0.4 solution increases with increase in cover thickness.The nip width obtained by v=0.4 solution is close to the nip width obtained • from Hertz solution compared to the nip width obtained from Deshpande solution. i • Cover thickness has no effect on the average pressure in case o f Hertz model.But in case o f Deshpande and v=0,4 solution average pressure decreases, and there is a slight difference in values obtained by Deshpande and 0.4 solution. Studies of Nip mechanics of Super calender bring forth the following important points: • with increase in line load nip width increases in both the Hertz solution and v=0.4 solution. The result obtained by both the Hertz and v=0.4 solution are very close to each other. • W ith increase in diameter o f the roll nip width increases and the result obtained by Hertz solution matches closely with the results obtained by v=0.4 solution. • W ith increase in elastic modulus nip width decreases and the Nip width obtained by Hertz solution matches closely with the v=0.4 solution. / • Cover thickness has no effect on nip width obtained from Hertz model,while nip width obtained from v=0.4 solution increases with increase in cover thickness. • With average pressure remains constant in case o f Hertz solution while in case Oi~v=0.4 solution average nip pressure decreases with increase in nip pressure • Differences between the third approximation and second approximation is insignificant for values o f w/h ./ • With increase in linear load penetration increases and penetration calculated from Deshpande and 0.4 solution are close to each other. Studies on Heat Transfer model reveal the following points In Case 1 (Rolls having same temperature) When paper is inside the calender nip having same roll temperature, the middle part o f the paper remains at the initial temperature, only the outer surfaces from both sides are heated and temperature decreases from outer part to middle part. Also with increase in roll temperature paper temperature increases. In case 2 When paper is inside the calender nip having different roll temperature,the middle part o f the paper remains nearly at the same initial temperature, and only the outer surfaces o f the paper from both sides are heated and temperature decreases from outer part to middle part. Also that side o f paper is more heated which is in touch with the roll having greater temperature as compared to other roll. In case 3(Temperature gradient calendering) when paper is inside the calender nip, in which one roll is heated roll as compared to other roll, which is at the room temperature, one side o f the paper is heated and the temperature decreases as moving towards the centre o f the paper. The other side o f the paper which is in contact with the non heated roll, remains at the initial temperature. In Case 3a (Temperature gradient calendering volume change) there is slighter higher in temperature as compared to the result obtained from fig [6.11 b] and [6.12 b],considering the same conditions except effect o f volume change. In Case 4(one side o f paper is in touch with the calender roll and other exposed to air) the side o f the paper which is in touch with the roll gets heated and temperature decreases as depth increases. The other side o f the paper which is in contact with the air remains nearly at the same average temperature. When the length o f the paper which is in contact with the roll is small, then there is a very slight change in the paper temperature. In case o f substrata thermal molding (case 5) with increase in moisture content, glass transition temperature o f the paper decreases.. With increase in dwell time the temperature required for substrata thermal molding decreases for given moisture content o f paper. Also roll temperature required for substrata thermal molding decreases with increase in moisture content o f paper for the same dwell time.The side o f the paper which is in contact with the heated roll as compared with the other roll gets heated and the other side o f the paper remains nearly at the same temperature. In all the above cases with increase in dwell time, diffusivity and roll temperature paper temperature increases, which in turn increases average paper temperature. ' W ith increase in permanent strain, in nip strain also increases linearly. r With increase in line load, the permanent strain and in nip strain ,Maxwell viscosity, Kelvin modulus, Kelvin viscosity increases, while recoverable strain decreases. Also with increase in line load moisture and paper temperature bulk and roughness o f the paper decreases while gloss increases. W ith increase in speed o f paper Maxwell viscosity, Kelvin modulus, Kelvin viscosity decreases for a particular line load. Also with increase in speed bulk and roughness o f paper increases while gloss decreases Studies of Viscoelastic modelling infer the following vii