SIMULATION OF A VIBRATED FLUIDISED BED DRYER FOR SOLIDS CONTAINING A MULTICOMPONENT MOISTURE

Transcription

1 XXII IACChE (CIIQ) 6 / V CAIQ SIMULATION OF A VIBRATED FLUIDISED BED DRYER FOR SOLIDS CONTAINING A MULTICOMPONENT MOISTURE A. Picado 1, * and J. Martínez 1 Faculty of Chemical Engineering, National University of Engineering (UNI) PO Box 5595, Managua, Nicaragua picado@kth.se Dept of Chemical Engineering and Technology, Royal Institute of Technology (KTH) SE-1 44 Stockholm, Sweden jmc@ket.kth.se Abstract. The drying of solids in a continuously worked vibrated fluidised bed dryer is studied by simulations. A model considering the drying of a thin layer of particles wetted with a multicomponent mixture is developed. Particles are assumed well mixed in the direction of the airflow and only the longitudinal changes of liquid content, liquid composition and particle temperature are considered. Interactive diffusion and heat conduction are considered the main mechanisms for mass and heat transfer within the particles. Assuming a constant matrix of effective multicomponent diffusion coefficients and thermal conductivity of the wet particles analytical solutions of the diffusion and conduction equations are obtained. The variation of both the diffusion coefficients and the effective thermal conductivity of the particles along the dryer is taken into account by a stepwise application of the analytical solution in space intervals with averaged coefficients from previous locations in the dryer. The analytical solution gives a good insight into the selectivity of the drying process and can be used to estimate aroma retention during drying. The solution is * To whom all correspondence should be addressed

2 XXII IACChE (CIIQ) 6 / V CAIQ computationally fast; therefore, the experimental verification of this approximate model would introduce an important computational economy since the rigorous treatment of multicomponent drying involves tedious and time-consuming calculations. Keywords: Aroma Retention, Drying Selectivity and Multicomponent Drying. 1. Introduction Continuously worked vibrated fluidised bed dryers (VFBD) have been used to dry a variety of particulate solids such as inorganic salts, fertilizers, foodstuffs, pharmaceuticals, plastics, coated materials, etc. In some industrial processes, the VFBD is the only drying unit responsible for moisture removal, but it is also used as a second stage in two stages drying processes. For instance, the first stage is performed in a spray dryer to concentrate the product and the VFBD is used in a second stage to reduce the moisture content to the value required by the final product. This second stage saves energy and assures better control of the product quality (Cruz et al., 4). Other advantages of VFBD are: good performance, relative low investment cost, low maintenance costs, robustness of the equipment and versatility. Many different types of particulate solids, from chemicals to foodstuffs, usually with large continuous throughputs are treated in this way. In most of the cases, the moisture to be removed consists of water but there are important applications such as the drying of pharmaceuticals, plastics and coated materials where the moisture consists of a multicomponent mixture. The drying of foodstuffs is a special case of multicomponent drying since the moisture usually consists of water and a large number of low concentration volatile compounds (e.g., coffee, cocoa or milk). Considerable work has been devoted to the study of VFBD concerning particle behaviour and its interaction with the gas, wall and effects of vibration, as well as mass and heat transfer during drying (Pan et al., ; Pakowski et al., 1984; Hovmand, 1987). Eccles and Mujumdar (199) carried out an extensive review of work on VFBD. There are numerous incremental models to simulate the drying process in continuous fluidised bed dryers (Keey, 199; Kemp and Oakley, ; Izadifar and Mowla, 3;

3 XXII IACChE (CIIQ) 6 / V CAIQ Daud, 6). Most of the equipment models assume plug flow of the solids but solids non-ideal flow has been also studied. Gas cross flow is modelled in some extent. On the other hand, the material model does not include the drying of solids wetted with a mixture of solvents. These cases are important because of the great influence of the composition of the remaining mixture on product quality. A great deal of multicomponent drying research has been performed by Schlünder and co-workers in Karlsruhe (Schlünder, 198; Thurner and Schlünder, 1986; Riede and Schlünder, 199; Wagner and Schlünder, 1998). The research has been focused on the behaviour of the evaporating mixtures in a rather simple geometry. Depending on the prevailing drying conditions, drying of solids containing multicomponent mixtures can be controlled by transport in the liquid phase, in the gas phase or by equilibrium. Gasphase-controlled drying of a multicomponent liquid film in continuous contact with the gas phase has been studied by Vidaurre and Martínez (1997). Luna and Martínez (1999) showed that a deep understanding of the process can be obtained by a stability analysis of the ordinary differential equations that describe the dynamical system. Liquid-phasecontrolled drying of multicomponent mixtures has been analysed by Pakowski (1994). Gamero et al. (6a) studied the continuous evaporation of a falling liquid film into an inert gas numerically. Recently, Gamero et al. (6b) reported an analytical solution for batch drying of a multicomponent liquid film in non-isothermal conditions assuming constant physical properties. The changes of physical properties during the process were accounted for by a stepwise application of the solution with averaged coefficients from previous steps. The purpose of this study is the development of a model to simulate the drying of particulate solids containing multicomponent liquid mixtures in a vibrated fluidised bed dryer. The model is developed by incorporating a material model for a single spherical particle wetted with a liquid mixture in an incremental equipment model assuming plug flow of the solids. The model would be a useful tool to explore the selectivity of the drying process and choose appropriate drying conditions to control the composition of the final moisture.

4 XXII IACChE (CIIQ) 6 / V CAIQ. Theory A schematic description of the VFBD is shown in Figure 1. In such equipment, effective mixing of the particles takes place and a homogeneous material at a vertical cross section of the dryer is usually obtained. The residence time distribution of the particles measured at the outlet does not differ very much of that calculated for a plug flow model (Strumiłło and Pakowski, 198). Vibration allows for lower gas velocities to achieve a good contact between the gas phase and the wet particles. Fig. 1. A Plug Flow Vibrated Fluidised Bed Dryer..1. Mass and Energy Balances in the Dryer In the analysis of the dryer, it is assumed that the bed of particles is moving forward with a uniform velocity and that the dryer has been operated during sufficient time for steady state conditions be reached. A moisture balance applied to the volume element shown in Figure yields: F s dx dz i = am G i = 1,.... n (1) i gi where n is the number of components in the moisture. Since all the evaporated liquid goes to the gas the changes of air humidity are given by the following balances:

5 XXII IACChE (CIIQ) 6 / V CAIQ F g dy i dx i = Fs H b i = 1,.... n () dz In Equations (1) and () the air humidity, Y i, and the solid liquid content, X i, are in dry basis. F is a mass flow of inert per cross section in the direction of the flow. The subscripts s and g denote solid and gas respectively. M is the molecular weight, a is the specific evaporation area per bed volume, G g,i is the molar evaporation flux of component i, and H b is the bed height. Fig.. Scheme of a differential dryer element. If heat losses in the dryer are neglected the energy balance over the volume element becomes: di g Fs di s = H b (3) F dz g where I is the enthalpy of the phases per unit mass of inert. The bed height is calculated as: H b = S vρ (1 ε )(1 ε )B (4) p p b

6 XXII IACChE (CIIQ) 6 / V CAIQ where S is the flow of dry solids, v is forward bed velocity, ρ is the density, ε is the porosity and B is the dryer wide. The subscripts p and b denotes particle and bed respectively. To integrated Eq. (1) along the dryer, apart from inlet conditions, the evaporation fluxes must be provided. These fluxes depend on the temperature and liquid composition at the surface of the particles. This information can be obtained by analysing what happens with a single particle moving along the dryer... Drying of a Single Particle 3. The drying of a single particle into an inert gas is schematically described in Figure Fig. 3. Schematic drying of a single particle into an inert gas..3. Governing Equations If diffusion inside the particle is the main contribution to mass transfer, the process is described by the diffusion equation: x x = D + z r v r x r (5) If conduction is the only mechanism for heat transfer within the particle the corresponding equation to describe changes of temperature is the conduction equation: T = D z T r v h + T r r (6)

7 XXII IACChE (CIIQ) 6 / V CAIQ where x is a column vector with the molar fractions of the independent diffusing component in the liquid, D is the matrix of multicomponent diffusion coefficients and D h the heat diffusivity. Equations (5) and (6) represent a system of partial differential equations. If evaporation and convection heat occurs only at the surface of the particle and the initial composition as well as temperature of the particles are given functions of r, the initial and boundary conditions are: At z = and r δ, x= { r} ; T { r} x T = (7) x T At r = and z >, = ; = r r (8) x T T At r = δ and z >, C LD = G g, n 1 ; k = h(t T g, ) + λ G g (9) r r where λ is a column vector of heat of vaporisation. The superscript T denotes transposition. The subscript n-1 in the column vector of evaporation fluxes in gas phase indicates that only n-1 of the fluxes are considered to match the dimension of the independent diffusion fluxes within the particle..4. The Matrix of Multicomponent Diffusion Coefficients The matrix of multicomponent diffusion coefficients, D, is of order n-1 n-1. This expresses the fact that the nth component does not diffuse independently. In non-ideal mixtures, the matrix of multicomponent diffusion coefficients is defined as: 1 D= ιb Γ (1) where ι embodies the constriction and tortuosity factors to take into account that the liquid is confined in a porous particle. The matrix B, which can be regarded as a kinetic contribution to the multicomponent diffusion coefficients, has the elements:

8 XXII IACChE (CIIQ) 6 / V CAIQ xi Bii = D in + n x D k k= 1 ik k i ; 1 1 B ij (i j) = xi (11) Dij Din where i, j = 1,, n-1 and D ij are the Maxwell-Stefan diffusion coefficients. The elements of the matrix of thermodynamic factors, Γ, are given by: x lnγ i i Γ ιj = δij + (1) x j lnx j where γ i is the activity coefficient of compound i and δ i,j is the Kronecker delta (δ i,j = 1 for i = j and δ i,j = for i j). For ideal solutions, the matrix of thermodynamic factors reduces to the identity matrix..5. Mass and Heat Transfer Rates If diffusional interactions in gas phase are included evaporation fluxes may be written as: G = K{ y δ y } (13) g Here the matrix K is the matrix product βek in which β embodies an extra relationship between the fluxes to calculate molar fluxes from diffusion fluxes, E is a matrix of correction factors to account for the finite mass transfer rate and k is a matrix of mass transfer coefficients at zero mass transfer rates. The columns vectors y δ and y are the molar fractions of the vapours at the gas-liquid interface and the bulk of the gas respectively. For details see Taylor and Krishna (1993). The convective heat flux can be expressed by: q g, = h (T T ) (14) δ where h is a heat transfer coefficient between the heating medium and the particles.

9 XXII IACChE (CIIQ) 6 / V CAIQ.6. Coupling between Phases If the gas phase is considered to be in equilibrium with the liquid at the interface, then at r = δ: y δ = 1 P t P γx = K n γ x n (15) is obtained, with P t being the total pressure. P and γ are diagonal matrices containing the saturated vapour pressures of the pure liquids, and activity coefficients respectively. The subscript n indicates that the vector x contains the molar fractions of the n components of the liquid mixture..7. Integrating along the Dryer The solution of Eqs. (5) and (6) subjected to inlet and boundary conditions (7) through (9) provides the temperature and liquid composition gradients within the particle. In addition, mass and heat transfer rates at the particle surface are obtained. The analytical solution assuming constant transport coefficients as well as heat and mass transfer rates is shown in details in Appendix A. Since these conditions change along the dryer, the analytical solution is applied to an interval dz, with inlet conditions and averaged transport coefficients corresponding to the outlet conditions of the previous step. As the integration of Eq. (1) proceeds the procedure is repeated. The outlet composition of the gas at each step dz is calculated using Eq. (). Then, the energy balance (3) allows for the calculation of the exhaust gas enthalpy using the particle mean temperature to calculate the outlet enthalpy of the wet solids. Since the gas enthalpy is a function of gas composition and temperature, the outlet gas temperature can be calculated from a non-linear equation that relates gas temperature with enthalpy. Integration proceeds in this way until the exit of the dryer is reached. 3. Results and Discussion Calculations were performed with particles containing two different liquid mixtures: ethanol--propanol-water, and acetone-chloroform-methanol. The evaporation fluxes

10 XXII IACChE (CIIQ) 6 / V CAIQ were calculated according to Eq. (13) using an algorithm reported by Taylor (198) with diffusion through stationary gas as bootstrap relationship. The matrix of correction factors was evaluated using the linearised theory. Mass and heat transfer coefficients at zero-mass transfer rates were computed by correlations of Kunii and Levenspiel (1969) with binary diffusion coefficients in gas phase predicted by the method of Fuller et al. (1966). Physical properties of pure component and mixtures were evaluated using methods described by Poling et al. (). Activity coefficients were calculated according to the Wilson equation with parameters from Gemhling and Onken (198). Antoine method was used for computing the vapour pressure of pure liquids. For determining the Maxwell-Stefan diffusion coefficients in liquid phase the method of Brandowski and Kubaczka (198) with an empirical exponent of.5 was used for both liquid systems. Physical properties of Pyrex were used for the solid. A typical result for a simulation for a solid containing ethanol--propanol-water is shown in Figure 4. In this mixture the volatility of water is much less than ethanol and -propanol. According to the theory, to remove water preferentially and keep the volatiles in the solid, the resistance against mass transfer within the solid must be high. This situation is favoured by an intensive drying regime. The resistance within the solid increase when the ratio between constriction and tortuosity has a low value and the diameter of the particles is large. Drying intensity can be increased by increasing external factors such as gas velocity and temperature. The Tables below show the influence of these parameters on the ratio of retention defined as X / X ) /( X / X ) ( e,i,i e. The results revealed that retention of volatile compounds is favoured by the resistance against mass transfer within the solid. However increasing gas velocity and temperature has a negative effect. The selectivity of the process is not expected to be affected by the external conditions but to induce internal resistance. Clearly, in the conditions examined, the effects of gas velocity and gas temperature on particle temperature and transport coefficients seem to have an opposite effect.

11 XXII IACChE (CIIQ) 6 / V CAIQ Fig. 4. Drying simulations for particles containing ethanol--propanol-water. u g = 1.5 m/s, T g = K, Y = [ 1], S = kg/s, δ = m, v =. m/s. Table 1. Influence of gas velocity on volatile retention. T g = K, S = kg/s, δ = m, v =. m/s. Components Retention ratio u g = 1. m/s u g = 1.5 m/s u g = 1.9 m/s Ethanol propanol Water

12 XXII IACChE (CIIQ) 6 / V CAIQ Table. Influence of the particle diameter on volatile retention. u g = 1.5 m/s, T g = K, S = kg/s, v =. m/s. Components Retention ratio δ =. m δ =.3 m δ =.4 m Ethanol propanol Water Table 3. Influence of the solid structure on volatile retention. u g = 1 m/s, T g = K, S = kg/s, δ = m, v =. m/s. Components Retention ratio ι =.35 ι =.65 ι = 1. Ethanol propanol Water Table 4. Influence of the gas temperature on volatile retention. u g = 1.5 m/s, S = kg/s, δ = m, v =. m/s. Components Retention ratio T g = 6 C T g = 7 C T g = 8 C Ethanol propanol Water Simulation results for the drying of particles wetted with a liquid mixture consisting of the highly volatile components, acetone-chloroform-methanol are shown in Figure 5 and Table 5. It is clear that drying rates are higher than of the mixture containing water and particle temperature decreases much more along the dryer. In the presence of such solvents, the main concern should be to keep the concentration of all or some components in the product below certain limits. The results of the simulations shown in Table 5 evidence the particular features of multicomponent drying that can lead to

13 XXII IACChE (CIIQ) 6 / V CAIQ unexpected results and the application of unconventional measures to fulfil product quality requirements. Slight changes of liquid composition in the feed by adding small amount of the other components to the solid reduce methanol concentration in the product to less than 5 % of that of the first case. Furthermore, the final total liquid content is reduced despite the higher total liquid content of the feed. Fig. 5. Drying simulations for particles containing acetone-chloroform-methanol. u g = 1.5 m/s, T g = K, Y = [ 1], S = kg/s, δ = m, v =. m/s. Table 5. Adding solvents to the solid feed. 1) Acetone, ) Chloroform, 3) Methanol. x (kmol/kmol) X (kg/kg) X e (kg/kg) X 3,e, Methanol (mg/kg) [..] [.1.] [..1]

14 XXII IACChE (CIIQ) 6 / V CAIQ 4. Conclusions The incremental model to simulate drying of particles containing liquid mixtures in a vibrated fluidised bed dryer describes qualitatively well the main features of multicomponent drying established theoretically and experimentally in previous works, particularly, the effects of the solid resistance against mass transfer on the retention of volatile components. Factors intrinsically connected to an increase of solid resistance, such as a more intricate solid structure and larger particle diameters, increase volatile retention. Remarkably, external factors that make drying more intensive and thereby more evident the existence of internal resistance, such as gas velocity and temperature, seems to have an opposite effect on volatile retention. A deeper study using other conditions is necessary to elucidate this behaviour. Simulations with a mixture containing highly volatile components showed that the composition of the remaining liquid in the product can be controlled by adding small amount of the other components to the solid feed. For instance, the concentration of methanol in the product can be kept under a certain limit by adding small amount of chloroform to the solid feed. This unconventional solution in drying practice evidences the complex features of multicomponent drying and the need for suitable tools to predict the entire trajectory of a drying process. To make this model such a useful tool for aiding dryer design requires the experimental verification of the model. Acknowledgments The authors gratefully acknowledge the financial support provided by the Swedish International Development Cooperation Agency (Sida/SAREC) for this work. References Bandrowski, J., Kubaczka, A. (198). On the Prediction of Diffusivities in Multicomponent Liquid Systems. Chem. Eng. Sci. Vol. 37, pp Carslaw, H.S., Jaeger, J.C. (1959). Conduction of Heat in Solids. nd Ed. Oxford University Press. London, Great Britain. Cruz, M.A.A., Passos, M.L., Ferreira, W.R. (4). Final Drying of Milk Powder in Vibrated-Fluidized Beds. In Proceedings of the 14th International Drying Symposium. São Paulo, Brazil. Vol. B, pp Daud, W.R.W. (6). A Cross Flow Model for Continuous Plug Flow Fluidised Bed Cross Flow Dryers. In Proceedings of the 15th International Drying Symposium. Budapest, Hungary. Vol. A, pp

15 XXII IACChE (CIIQ) 6 / V CAIQ Eccles, E.R., Mujumdar, A.S. (199). Cylinder-to-bed Heat Transfer in Aerated Vibrated Bed of Small Particles. Drying Technol., Vol. 1, pp Fuller, E.N., Schettler, P.D., Giddings, J.C. (1966). A New Method for Prediction of Binary Gas-Phase Diffusion Coefficients. Ind. & Engng. Chem., Vol. 58, pp Gamero, R., Luna, F., Martínez, J. (6). Convective Drying of a Multicomponent Falling Film. In Proceedings of the 15th International Drying Symposium. Budapest, Hungary. Vol. A, pp Gamero, R., Picado, A., Luna, F., Martínez, J. (6). An Analytical Solution of the Convective Drying of a Multicomponent Liquid Film. In Proceedings of the 15th International Drying Symposium. Budapest, Hungary. Vol. A, pp Gmehling, J. and U. Onken (198), Vapor-liquid equilibrium data collections. DECHEMA, Chemistry data Series, I/1a, I/a. Hovmand, S. (1987). Fluidized Bed Drying. In Handbook of Industrial Drying. Marcel Dekker, Inc. New York, USA. Izadifar, M., Mowla, D. (3). Simulation of a Cross-Flow Continuous Fluidized Bed Dryer for Paddy Rice. J. Food Eng., Vol. 58, pp Keey, R.B. (199). Drying of Loose and Particulate Material. Hemisphere Publishing Corporation. New York, USA. Kemp, I., Oakley, D.E. (). Modeling of Particulate Drying in Theory and Practice. Drying Technol., Vol., pp Kunii, D., Levenspiel, O. (1969). Fluidization Engineering. John Wiley, New York, USA. Luna, F., Martínez, J. (1999). Stability Analysis in Multicomponent Drying of Homogeneous Liquid Mixtures. Chem. Eng. Sci., Vol. 54, pp Pakowski, Z. (1994). Drying of Solids Containing Multicomponent Mixture: Recent Developments. In Proceedings of the 1th International Drying Symposium. Gold Coast, Australia. Vol. A, pp Pakowski, Z., Mujumdar, A.S., Strumiłło, C. (1984). Theory and Application of Vibrated Beds and Vibrated Fluid Beds for Drying Processes. In Advances in Drying: volume 3. Hemisphere Publishing Corporation. Washington, USA. Pan, Y.K., Li, J.G., Zhao, L.J., Ye, W.H., Mujumdar, A.S., Kudra, T. (). Drying of a Dilute Suspension in a Vibrated Fluidised Bed of Inert Particles. In Proceedings of the 1th International Drying Symposium. Noordwijkerhout, The Netherlands. Paper No. 66. Poling, B., Prausnitz, J., O Connell, J. (). The Properties of Gases & Liquids. 5 th Ed. McGraw-Hill, New York, USA. Riede, T., Schlünder, E.U. (199). Selective Evaporation of a Ternary Mixture containing one Non-Volatile Component with regard to Drying Process. Chem. Eng. & Proc., Vol. 8, pp Schlünder, E.U. (198). Progress towards Understanding the Drying of Materials Wetted by Binary Mixtures. In Proceedings of the 3rd International Drying Symposium. Birmingham, England. Vol., pp Strumiłło, C., Pakowski, Z. (198). Drying of Granular Products in VibroFluidized Beds. In Drying 8 Volume 1: Developments in Drying. Hemisphere Publishing Corporation. Montreal, Canada. Taylor, R., Krishna, R. (1993). Multicomponent Mass Transfer. John Wiley & Sons. New York, USA. Taylor, R. (198), Film Models for Multicomponent Mass Transfer: Computational Method II: The Linearized Theory. Computer & Chem. Engng., Vol. 6, pp

16 XXII IACChE (CIIQ) 6 / V CAIQ Thurner, F., Schlünder, E.U. (1986). Progress towards Understanding the Drying of Porous Materials Wetted with Binary Mixtures. Chem. Eng. Process. Vol., pp Vidaurre, M., Martínez, J. (1997). Continuous Drying of a Solid Wetted with Ternary Mixtures. AICHE J, Vol. 43, pp Wagner, G.R., Schlünder, E.U. (1998). Drying of Polymeric Solvent Coatings. In Proceedings of the 11th International Drying Symposium. Halkidiki, Greece. Vol. C, pp Appendix A: Analytical Solution of the Equations for the Particle Equations (5) and (6) can be made dimensionless by introducing the following dimensionless variables: z τ = ; ζ r Tg T = ; θ = (A1) L δ Tg T The system of partial differential Eqs. (5) and (6) become: x x x θ θ θ = D + d ; τ ζ = κ + (A) τ ζ with LD LD D d = ; κ = h (A3) vδ vδ The inlet and boundary conditions are: At = τ and ζ 1, x x { ζ} ; θ { ζ } = = (A4) θ x At ζ = and τ >, = θ ; = (A5) x θ At ζ = 1 and τ >, = φx + y b ; = aθ + b (A6) with and L { γ } n φ = D d ΚΚ ; b = D d { y } vδc n 1 L vδc L h a= δ ; k a b = h(t T { λ G } g g T L y Κ (A7) ) (A8)

17 XXII IACChE (CIIQ) 6 / V CAIQ The subscript n-1 indicates that the matrix product consists of the first n-1 column and rows of the original matrix product. The same applies to the resulting column vector in Eq. (A7). Equations (A) may be now transformed into ones describing linear flow in one direction by introducing the following new dependent variables u = ζ ( φ x+ y ) ; Θ = ζ ( aθ + b ) (A9) b Equations (A) become: u ~ u = D ; τ with ~ 1 = φd d φ Θ Θ = κ τ (A1) D (A11) The new inlet and boundary conditions are: At = τ and ζ 1, u { ζ }; { ζ } = u Θ = Θ (A1) At ζ = and τ >, u = ; Θ = (A13) u Θ At ζ = 1 and τ >, + ( φ I ) u = ; + ( a 1) Θ = (A14) where I is a diagonal matrix of ones. The composition in Eq. (A1) can be de-coupled through the similarity transformation P DP = Dˆ 1 ~ (A15) 1 uˆ = P u (A16) The matrix P is the modal matrix whose columns are the eigenvectors of D ~ and Dˆ a diagonal matrix of its eigenvalues. The transformation yields: uˆ ˆ uˆ = D τ with initial and boundary conditions: (A17) At τ = and ζ 1, u= ˆ uˆ { ζ } (A18) At ζ = and τ >, u ˆ = (A19) uˆ At ζ = 1 and τ >, = ξ f uˆ (A)

18 XXII IACChE (CIIQ) 6 / V CAIQ where ξ f = P 1 ( φ I)P (A1) Since the solution demand ξ f to be a diagonal matrix a new diagonal matrix ξ is defined so that it satisfies: ξ f uˆ = ξuˆ (A) giving the new boundary conditions: uˆ = ξuˆ (A3) Equation (A17) is not explicitly dependent on temperature and can be solved separately. Under the assumption that the matrix ξ is constant the de-coupled differential equations can be solved by the method of variable separation. The solution reported by Carslaw and Jaeger (1959) is: uˆ = m= 1 e ˆ Dν mτ ν m ν m + ξ + + sin( ν ξ( ξ I ) m ζ ) 1 { uˆ ( ζ ) sin( ν mζ )dζ } (A4) To preserve the formalism of matrix product, the integral in Eq. (A4) is a diagonal matrix that contains the value of the integral. The eigenvalues in Eq. (A4) are defined implicitly by tan ν 1 m = ξ ν m (A5) Finally, by using Eq. (A9) and the inverse of Eq. (A16) û is transformed back to obtain the liquid composition: x 1 Puˆ φ y (A6) ζ = b At the centre of particle, when ζ =, the composition is undetermined and the expression must evaluated as a limit. The limit of the expression is related to the derivative of the transformed composition with respect to the dimensionless space uˆ dˆ u lim = lim (A7) ζ ζ ζ dζ By evaluating the derivative of Eq. (A4) at ζ = :

19 { uˆ ( ζ ) sin( mζ )dζ } XXII IACChE (CIIQ) 6 / V CAIQ τ + d uˆ ˆ ν ν 1 m ξ lim = e D m ν m ζ dζ = + + m 1 ν m ξ( ξ I ) ν (A8) Equation (A6) provides the mole fractions of n-1 components in the liquid. The mole fraction of the nth component is calculated taking advantage of: n 1 x n = 1 x j (A9) = j 1 For the temperature: κν τ ν h,m + ( a - 1 Θ = e h, m ) sin( ν h,mζ ) = ν + m 1 h,m a( a - 1) with eigenvalues defined implicitly by tanν 1 h,m = 1 a ) h,m 1 { Θ ( ζ ) sin( ν h,mζ )dζ } (A3) ( ν (A31) Substitution back to temperature: (Tg T ) Θ T = Tg b (A3) a ζ The values of the centre are calculated using a similar relation between the limits. In this case: Θ dθ lim = lim (A33) ζ ζ ζ dζ Applied to Eq. (A3): dθ lim = ζ dζ m= 1 e κν 1 { Θ ( ζ )sin( ν h, mζ ) d }, ν, + (a -1) h mτ h m ν h, m ζ (A34) ν + h, m a(a -1) Even though the solution is only valid for constant physical properties the variation of coefficients for the whole process can be taken into account by a stepwise application of the analytical solution along the process trajectory. That is, by performing the solution in successive steps where the final conditions of the previous step are used to calculate the coefficients and as initial condition of the next step.