MATHEMATICAL MODELING OF PULP WASHING SYSTEMS AND SOLUTIONS

Abstract

Pulp and paper industry is highly capital intensive industry. It is intensive in terms of raw materials, water, chemicals, thermal and electrical energy, labour and pollution loads. Approximately 2.5-3.0 t of raw materials, 200-350m3 of water, 9 -20 t of steam, 1000-1700 kWh of electric energy are required for one tonne of pulp. This leads to generation of pollution loads to the extent of 24-45 kg of BOD5, 80-150 kg of COD, 2.5-8 kg of AOX in the effluents. There are all round efforts globally to reduce the above intakes and outflows from a paper mill keeping the quality for its acceptance in the international market. Concerted efforts are also being made in Indian paper mills to address these issues for mere survival point of view and also to keep pace with sustainable production to meet the demands of paper. Some of the measures taken into consideration by mills are: seeking of optimum design and operational parameters, adaptation of modern control measures and use of new equipment and process technology. Mathematical model can help significantly to fulfill the above objective. For research work a model can be highly complicated to obtain higher accuracy but in actual practice there is a compromise between the accuracy and the complexity of the model. A mathematical model can be macroscopic, microscopic or semi quantitative in nature. Macroscopic / empirical / black box / white box models give a general outside description (in terms of material balance) of a brown stock washer. As a whole microscopic / physical / gray box models gives a deep inside description (in terms of fundamental parameters) of a brown stock washer. Semi quantitative models are intermediate between macroscopic and microscopic models. Higher is the order of the model, higher is the degree of difficulty expected to solve the problem. iv In this present study up-to date review of mathematical models relating to displacement washing or similar systems with or without adsorption or dispersion is made. The models developed very recently for some other systems (quite similar to pulp fibers) are also presented. It is found from the literature that there is abundant of information regarding steady state models mainly required for the material balances for the macroscopic evaluation of the design estimates. However, there is little information about the interaction of various time dependent operational parameters for control purposes. Very few investigators have attempted to throw light on these aspects. Majority of the investigators have concentrated on displacement washing studies though in normal practice the process is related to dilution and extraction and also with displacement. Many investigators did not take into account the parameters related to diffusion, dispersion, adsorption, desorption, multiporosity values for inter particle and intra particle voids. Although some mathematical models are available in literature limited studies are carried out for pulp washing under the influence of longitudinal dispersion coefficient, mass transfer coefficients and solute accumulation capacity of pulp fibers. Many works are also applicable to only nonporous solids. Though these are fantastic in their approach and the corresponding solution techniques, these are not truly applicable to porous adsorptive beds like pulp mat. Besides, there are significant variations noted among the models of many investigators in their adsorption-desorption isotherm equations. The solution techniques (either analytical or numerical or quasi- analytical or statistical) are also remarkably different. No investigator has ever compared the results evolved out by assuming different adsorption isotherms that too for different boundary conditions with mill practice values. Very few workers applied the results of simulation with displacement washing data to brown stock washer operation in rotary vacuum filter equipment in a very concise form but it lacks clarity. From design point of view these are of little relevance. The solution of partial differential equation can also be possible with numerical techniques which is not very complex yet preserving the required accuracy for evaluating an industrial system. The system equation must address the issues of practical significance. It therefore demands a rational approach for interlinking the output data from a mathematical problem and the parameters normally known to the practicing engineers of a pulp and paper industry. In this investigation mathematical models are derived for displacement washing in the washing zone of a rotary vacuum washer from basic equation of continuity in one dimensional form for flow through porous media, Fick's law of diffusion and dispersion with adsorption and desorption isotherm equations for two species namely Na+ and lignin. The equations are coupled with many other fluid mechanical parameters and thus general in nature. The models resulted in the form of a linear parabolic partial differential equation. For solving the models varying initial and boundary conditions applicable to rotary vacuum brown stock washer are proposed. The developed models have been reduced to various earlier proposed models using simplifying assumptions and neglecting certain terms. . As the developed models are mathematically complex to solve, these are simplified for further investigation in order to compare with some of the models proposed by Grah [24], Brenner [6], Sherman [88], and Pellett [71] but with different numerical techniques. Four set of models differing in adsorption isotherm with same boundary conditions are proposed to solve through finite difference method. In fact earlier investigators did not attempt to use this simple technique. En order to solve the proposed models, various equations for parameters like porosity, permeability, specific cake resistance, filter medium resistance, compressibility constants, design equations for filtration are reviewed in detail and selected for our investigation. For the solution of the models developed above. MATLAB software is employed after developing some specially designed C++ programmes. vi The present models (four models) are validated with those of Grah [24] based on numerical technique (Orthogonal Collocation) and Brenner 161 with analytical methods with same species and Pe numbers estimated by Grah [23]. In the present investigation the sorption values are also calculated using non-linear adsorption isotherm which is of Langmuir type for estimation of Na/ soda loss and lignin. At practical values of kappa number for bleachable grade kraft pulp, the adsorption of Na+ and soda loss will be on the order of 1.5-1.75 kg / t and 4.8-5.5 kg / t respectively. At kappa number admissible for bleachable grade pulp the amount of lignin sorption will be on the order of 3 kg / t to 5.5 kg / t. Expressions are given for dimensionless concentration of solute in liquid phase C at any dimensionless pseudo time T, and dimensionless concentration of solute in solid phase, N as a function of T and also as a function of dimensionless distance (bed depth). The C-T profiles are in excellent agreement with those of Grah [24], Poirier et. al.[79] as well as Sherman [88], and Brenner [6]. The C-T profile closely agrees also with Kuo and Barrett [43] and Kukreja [40]. However, there is no work available for pulp washing system with N-T profiles though Kuo [42], Kuo and Barrette [43] and later Kukreja [40] have attempted to show the profile for stagnant liquor. The N-T profile for the present investigation has been in close agreement with many other investigators [17,45,51,92] working with adsorption in solid phase for allied type of systems. Therefore this investigation with N-T profile can claim for the first time a new dimension for solving adsorption related issues in pulp washing situation using the concentration terms in solid phase. The N-T profile in this present study gives an opposite trend with C-T profile of solute in liquor phase or solute in stagnant liquor phase as advocated by Kuo [42] and Kukreja [40]. The four different models are used to predict for two species, namely Naf and lignin to examine their washing behavior. Parameters like bed thickness. Peclet number, liquor velocity through cake vii pores, dispersion coefficient, porosity and mass transfer coefficients are varied in the C-T and N-T profiles of both Na} and Lignin species to examine their influences. From the displacement washing study for which our models are based the following important conclusions can be drawn. Investigations of the simultaneous displacement washing of Na+ and Lignin in sulphate pulp have shown great difference between the two substances in their behavior during washing. At the same time interval the AT values (proportional to solute removal) of lignin are equal to or higher than those of Na+ for the models 1 and 2 but it is reversed in case models 3 and 4 (i.e. Na+ is higher than lignin) though the differences are very marginal. Poirier et. al.[77] however found the similar or slightly higher values in case of Nat It is probable that Poirier et. al.[77] did not consider the dispersion effects. The model predicted data can be used reliably for brown stock washer simulation. The simulated data on DR from displacement washing are in close agreement with mill data. The washing efficiency WE based on total solute removal in a brown stock washer closely tally with data calculated from the breakthrough curve obtained from the model. The difference arises due to different conditions employed by different investigators for brown stock washer. It however does not consider the removal efficiency of nature of solutes namely, Na+ and lignin species. The DR and WE values increases with time, valid for both Na+ and lignin. This is in excellent agreement with those of Grah [25]. The DR and WE data for sodium are slightly higher than those for lignin for model I and model 2. However, for the case of model 3 and model 4 they yield identical values. The model 1 and model 2 give the same value whereas model 3 and model 4 yield identical value for both Na and lignin species. The latter models (models 3 and 4) however provide higher values of C than those from viii the former (models I and 2). This is caused as earlier indicated by the dispersion which hampers adsorption. All the above experimental and theoretical findings suggest that the present model agrees quite well with the experimental data and takes care of many of the important aspects of diffusion. dispersion, adsorption-desorption and different porosity values in the system. This verifies that numerical solution of resultant non-linear parabolic partial differential equation with finite difference method can be used though it is simpler and approximate compared to Orthogonal Collocation method employed by many authors. Present investigation also attempted to correlate the DR and WE obtained from microscopic models (mathematically complex) with those from industrial system of brown stock washer after modifying certain parameter as suggested by Grah [24] using one or two parameter models. The present study also designed a systematic procedure to examine whether a displacement washing data can be used as a guideline to simulate a BSW or not. It is found that the present computer simulation can be employed to find optimum operating conditions for practical washers, however, it is limited to only washing zone of BSW and also for a single stage equipment. The present mathematical model is suitable for the simulation of displacement washing in other existing equipment and in equipment under design. This present analysis can also be extended for the simulation of the washing operation on a multistage rotary washing filter for pulp mill.