DOI: 10.3303/CET24109016 Paper Received: 26 November 2023; Revised: 4 February 2024; Accepted: 14 April 2024 Please cite this article as: Trindade Júnior V.N., Guirardello R., 2024, Numerical Simulation of Gamma-valerolactone Adsorption Into Porous Spheres in a Batch Process, Chemical Engineering Transactions, 109, 91-96 DOI:10.3303/CET24109016 CHEMICAL ENGINEERING TRANSACTIONS VOL. 109, 2024 A publication of The Italian Association of Chemical Engineering Online at www.cetjournal.it Guest Editors: Leonardo Tognotti, Rubens Maciel Filho, Viatcheslav Kafarov Copyright © 2024, AIDIC Servizi S.r.l. ISBN 979-12-81206-09-0; ISSN 2283-9216 Numerical Simulation of -Valerolactone Adsorption into Porous Spheres in a Batch Process Valter Nunes Trindade Júnior, Reginaldo Guirardello School of Chemical Engineering, University of Campinas, Av. Albert Einstein 500, 13083-852, Campinas-SP, Brazil guira@unicamp.br The production of ethanol from lignocellulosic biomass is more complex in relation to ethanol produced from feedstock containing sugars or starch, since it requires the conversion of polysaccharides, cellulose or hemicellulose, into fermentable sugars. An alternative to improve this conversion of biomass polysaccharides into sugars is the use of -valerolactone (GVL) as a solvent. However, the GVL must be removed before the fermentation, since its presence is harmful to the microorganisms that metabolize glucose to produce ethanol. One extraction technique is the use of subcritical CO2, which is able to separate large amounts of GVL, but a complementary method is still required to separate small quantities of GVL remaining in the sugar solution, which can be done efficiently with adsorption. In this work, the adsorption of -valerolactone by porous spheres made of a resin in batch mode is studied to remove the GVL from the sugar solution containing glucose. This work is computational and the modelling considers the mass transfer from the solution to the surface of the spheres, the diffusion inside the pores of the spheres, and then the adsorption of GVL. The numerical resolution is done using two numerical methods, the finite differences and the variational method. The results are compared using different numbers of radial intervals in the spherical particles. Keywords: biomass, ethanol, glucose, -valerolactone, adsorption, finite differences, variational formulation. 1. Introduction Ethanol can be produced from lignocellulosic biomass by the hydrolysis of cellulose and hemicellulose into monosaccharides (sugars). Lignocellulosic sources available at industrial scale are corn crop residues and sugarcane bagasse, for example, but there are many others. However, the hydrolysis process of these materials is complex and expensive, so improvements are required to make this process economically viable. It has been shown that -valerolactone (GVL) as solvent in the conversion of the lignocellulosic biomass into sugars, in the presence of low concentration acid, has advantages over water alone (Mellmer et al., 2014), with increased catalytic activity (Alonso et al., 2013), and higher selectivity and efficiency of the hydrolysis reaction, minimizing sugar degradation (Luterbacher et al., 2014). After the hydrolysis, GVL must be separated from the sugar solution, because a high concentration of this solvent is harmful in the fermentation stage, due to its toxicity to the microorganisms that metabolize glucose to produce ethanol. It was found that an extraction with subcritical CO2 was able to decrease the concentration of GVL, but a further separation was still required in order to reduce the GVL to acceptable levels (Luterbacher et al. 2014). Adsorption has been found to have high efficiency as a complementary separation step to selectively remove GVL from the solution (Trindade, 2015a; Trindade et al, 2015b). Trindade (2015a) evaluated the use of four adsorbents in the removal of GVL from the glucose solution, and found that the resin Sepabeads SP850, which is a highly porous styrenic adsorbent, was the most selective to adsorb GVL compared to glucose. The objective of this work was to model the adsorption of GVL into spheres of SP850 in a batch process, in order to reproduce the batch experimental results obtained by Trindade (2015a), and thus help improve this separation step. The model considers both external and internal mass transfer, using a Langmuir type isotherm for adsorption in the spheres, and was solved with two numerical methods, finite differences and variational method. The results were then compared with the experimental data available. 91 2. Mathematical Model Adsorption in a porous medium is a very important separation process, for a variety of substances. For a batch system containing a volume of solution (𝑉) and a volume of porous spheres (𝑣), the modeling of the solute concentration in the solution, 𝐶(𝑡), and in the spheres, 𝑐(𝑟, 𝑡), can be written as a set of differential equations (Horstmann and Chase, 1989). In this work, a Langmuir type isotherm is considered, given as: 𝑞 = 𝑞𝑚 ∙ 𝑏 ∙ 𝑐 (1 + 𝑏 ∙ 𝑐) (1) where 𝑞 is the amount of solute adsorbed in the spheres, and 𝑞𝑚 is the maximum amount of solute that can be adsorbed by the spheres. The model can then be made dimensionless by defining 𝑢 = 𝑐 𝐶0⁄ and 𝑤 = 𝐶 𝐶0⁄ (Guirardello, 2013), so that: 𝑓′(𝑢) ∙ 𝜕𝑢 𝜕𝜏 = 1 𝑥2 ∙ 𝜕 𝜕𝑥 [𝑥2 ∙ 𝜕𝑢 𝜕𝑥 ] (2) 𝑑𝑤 𝑑𝜏 = −𝛼 ∙ [𝑤 − 𝑢(1, 𝜏)] (3) with boundary and initial conditions given by: 𝜕𝑢 𝜕𝑥 = 𝛽 ∙ (𝑤 − 𝑢) at 𝑥 = 1 𝜏 > 0 (4) 𝜕𝑢 𝜕𝑥 = 0 at 𝑥 = 0 𝜏 > 0 (5) 𝑢 = 0 at 𝜏 = 0 0 ≤ 𝑥 ≤ 1 (6) 𝑤 = 1 at 𝜏 = 0 (7) where 𝑥 = 𝑟 𝑅⁄ , 𝜏 = 𝑡 ∙ 𝐷 𝑅2⁄ , and: 𝛼 = 3 ∙ 𝑣 𝑉 ∙ ( 𝑅 ∙ 𝑘𝑓 𝐷 ) (8) 𝛽 = ( 𝑅 ∙ 𝑘𝑓 𝜀 ∙ 𝐷 ) (9) 𝑓′(𝑢) = 1 + (1 − 𝜀) 𝜀 ∙ 𝑞𝑚 ∙ 𝑏 (1 + 𝑏 ∙ 𝐶0 ∙ 𝑢)2 (10) 3. Methodology Two numerical approaches were used in this work to make the discretization in the 𝑥 variable: finite differences and variational calculus. Both approaches are detailed here. 3.1 Finite Differences The finite differences method is well known and is based on the substitution of the exact derivatives by numerical ones. However, here it is used appropriate expressions for the boundary points (𝑥 = 0 and 𝑥 = 1), so that the boundary conditions are naturally inserted in the numerical resolution of the PDE: ( 𝜕2𝑢 𝜕𝑥2)| 𝑖 = 𝑢𝑖 ′′ ≅ −7 ∙ 𝑢𝑖 + 8 ∙ 𝑢𝑖+1 − 𝑢𝑖+2 − 6 ∙ ∆𝑥 ∙ 𝑢𝑖 ′ 2 ∙ ∆𝑥2 𝑖 = 0 or 𝑥 = 0 (11) ( 𝜕2𝑢 𝜕𝑥2 )| 𝑖 = 𝑢𝑖 ′′ ≅ −7 ∙ 𝑢𝑖 + 8 ∙ 𝑢𝑖−1 − 𝑢𝑖−2 + 6 ∙ ∆𝑥 ∙ 𝑢𝑖 ′ 2 ∙ ∆𝑥2 𝑖 = 𝑁 or 𝑥 = 1 (12) Both expressions are second order in the interval (error ~ ∆𝑥2) and make use of the known first derivative, so that the boundary conditions can be inserted directly. There is no need for similar expressions for the first derivatives, since they are given by the boundary conditions at these two points: 92 ( 𝜕𝑢 𝜕𝑥 )| 𝑖 = 𝑢𝑖 ′ = 0 𝑖 = 0 or 𝑥 = 0 (13) ( 𝜕𝑢 𝜕𝑥 )| 𝑖 = 𝑢𝑖 ′ = 𝛽 ∙ (𝑤 − 𝑢𝑖) 𝑖 = 𝑁 or 𝑥 = 1 (14) However, at 𝑥 = 0 the L'Hospital theorem must be applied in Eq (2) in order to avoid getting 0 0⁄ , since the first derivative is divided by 𝑥 after rearrangement of the right hand side of that equation. For the interior points (𝑥𝑖, 𝑖 = 1, ⋯ , 𝑁 − 1), the usual second order expressions (error ~ ∆𝑥2) are used: ( 𝜕2𝑢 𝜕𝑥2)| 𝑖 = 𝑢𝑖 ′′ ≅ 𝑢𝑖+1 − 2 ∙ 𝑢𝑖 + 𝑢𝑖−1 ∆𝑥2 (15) ( 𝜕𝑢 𝜕𝑥 )| 𝑖 = 𝑢𝑖 ′ ≅ 𝑢𝑖+1 − 𝑢𝑖−1 2 ∙ ∆𝑥 (16) 3.2 Variational Formulation It is not possible to arrive directly at the diffusion equation using a variational formulation, since the first derivative vanishes when the Euler-Lagrange equation is applied to the functional. However, it is possible to asymptotically arrive to the diffusion equation by using a procedure starting with a similar equation and then making a quantity going to infinity. The following functional is then proposed: 𝐼 = ∫ [∫ 𝑒−𝜆∙𝜏 ∙ (λ∙ 𝑓′(𝑢) ∙ 𝑢𝜏 2 + 𝜆2 ∙ 𝑢𝑥 2) ∙ 𝑑𝜏 − 𝜆 ∙ 𝑢𝑥 2(𝑥, 0) ∞ 0 ] ∙ 𝑥2 ∙ 𝑑𝑥 1 0 + ∫ 𝛽 ∙ 𝑒−𝜆∙𝜏 ∙ [𝜆2 ∙ (𝑤 − 𝑢(1, 𝜏)) 2 + 𝜆 ∙ 1 𝛼 ∙ 𝑤𝜏 2] ∙ 𝑑𝜏 ∞ 0 (17) where 𝑢𝑥 is the first derivative of 𝑢 with respect to 𝑥, 𝑢𝜏 is the first derivative of 𝑢 with respect to 𝜏, and 𝑤𝜏 is the first derivative of 𝑤 with respect to 𝜏. Some asymptotic relations can be obtained for large values of 𝜆. From the theory of Laplace transforms, the following equation is valid (Spiegel, 1965): ∫ 𝑒−𝜆∙𝜏 ∙ 𝑓(𝜏) ∙ 𝑑𝜏 = ∑ 1 𝜆𝑘+1 ∞ 𝑘=0 ∞ 0 ∙ 𝑓(𝑘)(0) (18) so that Eq (17) converges to a finite value when 𝜆 → ∞. The stationary condition of a functional is given by the Euler-Lagrange equation. For an integrand (𝐹) that depends only on 𝑢, 𝑢𝑥, 𝑢𝜏, the stationary condition is: − 𝜕𝐹 𝜕𝑢 + 𝜕 𝜕𝜏 ( 𝜕𝐹 𝜕𝑢𝜏 ) + 𝜕 𝜕𝑥 ( 𝜕𝐹 𝜕𝑢𝑥 ) = 0 (19) Applying this stationary condition to the proposed functional with respect to the profile 𝑢(𝑥, 𝜏), after some rearrangements, the results is: [−𝑓′(𝑢) ∙ 𝑢𝜏 + 1 𝑥2 ∙ 𝜕 𝜕𝑥 (𝑥2 ∙ 𝑢𝑥)] + 1 2 ∙ 𝜆 ∙ [𝑓′′(𝑢) ∙ 𝑢𝜏 2 + 2 ∙ 𝑓′(𝑢) ∙ 𝑢𝜏𝜏] = 0 (20) This is not Eq (2), but as 𝜆 → ∞ this equation asymptotically tends to Eq (2). Boundary condition Eq (5) is naturally satisfied, while boundary condition Eq (4) comes from the stationary condition for the term 𝑢(1, 𝜏). Initial condition Eq (6) is given. The stationary condition for the profile 𝑤(𝜏) is given by: − 𝜕𝐹 𝜕𝑤 + 𝑑 𝑑𝜏 ( 𝜕𝐹 𝜕𝑤𝜏 ) = 0 (21) which, after rearrangements, leads to: 93 [𝑤 − 𝑢(1, 𝜏)] + 1 𝛼 ∙ 𝑤𝜏 − 1 𝜆 ∙ 1 𝛼 ∙ 𝑤𝜏𝜏 = 0 (22) This is not Eq (3), but as 𝜆 → ∞ this equation asymptotically tends to Eq (3). Initial condition Eq (7) is given. 3.2.1 The Ritz Method - Solution with an Interpolating Polynomial One particular application of the variational method is finding approximated solutions to differential equations, also known as the Ritz method. It consists in finding a corresponding functional that results in the differential equation, then using trial functions in the functional, and then applying the stationary conditions with respect to the parameters in the trial functions. Using Lagrangian interpolation, the following trial function is considered: 𝑢(𝑥, 𝜏) = ∑ 𝑙𝑗(𝑥) ∙ 𝑢(𝑥𝑗 , 𝜏) 𝑀 𝑗=1 (23) where 𝑙𝑗(𝑥) are the Lagrangian polynomials at some given points 𝑥1, ⋯ , 𝑥𝑀, so that 𝑙𝑗(𝑥𝑘) = 0 for 𝑘 ≠ 𝑗 and 1 otherwise. The trial function is then substituted into Eq (17). A Gaussian quadrature is applied, and since the integration is from 𝑥 = 0 to 𝑥 = 1, with weight function 𝑥2, the Gaussian integration of moments with 𝑘 = 2 is used (Abramowitz and Stegun, 1972). The following results will then be used: ∫ 𝑓′(𝑢) ∙ 𝑢𝜏 2 ∙ 𝑥2 ∙ 𝑑𝑥 1 0 = ∑ [𝑊𝑗 ∙ 𝑓′(𝑢𝑗) ∙ ( 𝑑𝑢𝑗 𝑑𝜏 ) 2 ] 𝑀 𝑗=1 (24) ∫ 𝑢𝑥 2 ∙ 𝑥2 ∙ 𝑑𝑥 1 0 = ∑ [𝑊𝑗 ∙ (∑ 𝑙𝑘 ′ (𝑥𝑗) ∙ 𝑢(𝑥𝑘 , 𝜏) 𝑀 𝑘=1 ) 2 ] 𝑀 𝑗=1 (25) where 𝑢𝑗 = 𝑢(𝑥𝑗 , 𝜏) is a function of 𝜏 at each given point 𝑥𝑗, and where the points 𝑥𝑗 and the weights 𝑊𝑗 can be found in Abramowitz and Stegun (1972) for Gaussian integration of moments with 𝑘 = 2 (the weight function 𝑥2 is already included in the weights 𝑊𝑗). The functional 𝐼 is then given by Eq (26), where 𝑢𝑗 = 𝑢(𝑥𝑗 , 𝜏) and 𝑢𝑗,𝜏 = 𝑑𝑢𝑗 𝑑𝜏⁄ : 𝐼 = ∫ 𝑒−𝜆∙𝜏 ∙ ∞ 0 [𝜆 ∙ ∑ 𝑊𝑗 ∙ 𝑓′(𝑢𝑗) ∙ ( 𝑑𝑢𝑗 𝑑𝜏 ) 2 + 𝜆2 ∙ ∑ 𝑊𝑗 ∙ (∑ 𝑙𝑘 ′ (𝑥𝑗) ∙ 𝑢(𝑥𝑘 , 𝜏) 𝑀 𝑘=1 ) 2𝑀 𝑗=1 𝑀 𝑗=1 ] ∙ 𝑑𝜏 −𝜆 ∙ ∑ 𝑊𝑗 ∙ (∑ 𝑙𝑘 ′ (𝑥𝑗) ∙ 𝑢(𝑥𝑘 , 0) 𝑀 𝑘=1 ) 2𝑀 𝑗=1 + ∫ 𝛽 ∙ 𝑒−𝜆∙𝜏 ∙ [𝜆2 ∙ (𝑤 − 𝑢(1, 𝜏)) 2 + 𝜆 ∙ 1 𝛼 ∙ 𝑤𝜏 2] ∙ 𝑑𝜏 ∞ 0 (26) where the value of 𝑢(1, 𝜏) is calculated by substituting 𝑥 = 1 in Eq (23), since all 𝑥𝑗 ≠ 1. Also, using the initial conditions, all 𝑢(𝑥𝑘 , 0) = 0, so that the corresponding term is zero. The stationary condition for 𝐼 with respect to the profiles 𝑢(𝑥𝑗 , 𝜏), for 𝑗 = 1, ⋯ , 𝑀, is given by: − 𝜕𝐹 𝜕𝑢𝑗 + 𝑑 𝑑𝜏 ( 𝜕𝐹 𝜕𝑢𝑗,𝜏 ) = 0 (27) Applying this stationary condition for 𝐼, for all 𝑗, and then making the limit 𝜆 → ∞, the following equation is obtained, after rearrangement: 𝑊𝑗 ∙ 𝑓′(𝑢(𝑥𝑗 , 𝜏)) ∙ 𝑑𝑢 𝑑𝜏 (𝑥𝑗 , 𝜏) = − ∑ 𝐴𝑗𝑘 ∙ 𝑢(𝑥𝑘 , 𝜏) 𝑀 𝑘=1 + 𝛽 ∙ (𝑤 − 𝑢(1, 𝜏)) ∙ 𝑙𝑗(1) (28) This same equation was obtained by Guirardello (2013), using a different approach, where 𝐴𝑗𝑘 is given by: 𝐴𝑗𝑘 = ∑ 𝑊𝑖 ∙ 𝑀 𝑖=1 𝑙𝑗 ′(𝑥𝑖) ∙ 𝑙𝑘 ′ (𝑥𝑖) (29) 94 Repeating the same procedure for 𝑤 results in Eq (3). Finally, the set of ordinary differential equations (ODE), given by Eq (3) and Eq (28), is solved, using the initial conditions 𝑢(𝑥𝑗 , 0) = 0 and 𝑤(0) = 1. 4. Experimental Data The experimental values used in this work were taken from Trindade (2015a), who conducted batch experiments with several adsorbents, one of them the synthetic resin SepabeadsTM SP850. This is a hydrophobic resin which showed a high selectivity for -valerolactone, while adsorbing very little glucose. It is also very porous, so that the aqueous solution can enter the macro-porous, but without the resin adsorbing the water. Therefore, this resin was found suitable for the separation of -valerolactone from the glucose solution. The results are presented in Tables 1 and 2. These batch tests were conducted at 25 ºC, with an acid pH. In the tests, the diameter of the particles was 500 m, and they were carried out by mixing the dry resin with the solution in the proportion of 𝑚res 𝑉⁄ = 0.01 g/mL. Table 1 presents the results for selectivity in the adsorption, for initial concentrations of 27.2015 g L⁄ for -valerolactone (GVL) and 66.4235 g L⁄ for glucose (GL). It can be seen from Table 1 that adsorption of glucose and -valerolactone are not competitive for this resin. Table 2 presents the concentration in the solution. From other tests, it was found that for -valerolactone the fitted parameters for a Langmuir type isotherm were 𝑞𝑚 = 2.6366 gGVL gres⁄ and 𝑏 = 0.0044 L/g (Trindade, 2015a). However, in the model 𝑞𝑚 has units of [mass solute volume resin⁄ ], so it is necessary to convert it using the density of the resin. Table 1: Adsorbed quantities for glucose and -valerolactone by SP850. time gGL gres⁄ gGVL gres⁄ 15 min − 0.2056 48 h 0.0060 0.2528 '−' means too low to measure by HPLC Table 2: Batch trials for -valerolactone adsorption by SP850. time 𝐶 𝐶0⁄ 0 1.0000 15 min 0.9168 48 h 0.9069 5. Results and Discussion 5.1 Case Study The model was solved for a case study considering porous particles of SP850 with the following adsorption parameters: 𝑞𝑚 = 2663.0 g/Lres and 𝑏 = 0.0044 L/g. The conditions for the solution and resin in the stirred tank were: 𝑉 = 10 mL, 𝑣 = 0.209 mL, and 𝐶0 = 27.2015 g/L. The porosity of the particles was 𝜀 = 0.526. Also, the spheres radius was 𝑅 = 250 μm (𝑑𝑝 = 500 μm). These values were used to reproduce the conditions of the experimental tests (𝑚res = 𝜌res ∙ 𝑣 ∙ (1 − 𝜀) = 0.1 g and 𝑞𝑚[by volume resin] = 𝜌res ∙ 𝑞𝑚[by mass resin]). The value of 𝑘𝑓 = 1.7 ∙ 10−5 m s⁄ was estimated from the correlation of Geankoplis for particles in stirred tanks (Geankoplis, 1983, apud Skidmore et al, 1990), using the molecular diffusion coefficient of GVL in water which was estimated as 𝐷𝐴𝐵 = 1.03 ∙ 10−9 m2 s⁄ with the correlation of Othmer and Thakar for dilute aqueous systems (Sherwood et al, 1975). 5.2 Results The variational method was solved for M=3 and M=5 (3 and 5 points, respectively), and the resulting ODE system was solved with a 4th order Runge-Kutta method, with ∆𝑡 = 0.01 min. The finite difference method was solved with 5 and 10 intervals (6 and 11 points, respectively), and the resulting ODE system was also solved with a 4th order Runge-Kutta method, with ∆𝑡 = 0.01 min. The results for the dimensionless concentration in the solution (𝑤) are presented in Table 3, where it can be seen a good agreement between the two methods. It is important to point out that some care must be taken with respect to the time interval ∆𝑡. Parabolic PDEs as Eq (2) may result in a stiff ODE system when a discretization is applied in the 𝑥 domain. 95 Table 3: Results for the dimensionless concentration in the solution (𝑤) for the case study. Time (min)  Variational (M=3) Variational (M=5) Finite Diff. (∆𝑥 = 0.2) Finite Diff. (∆𝑥 = 0.1) 0.00 0.0000 1.0000 1.0000 1.0000 1.0000 5.00 0.3600 0.9446 0.9434 0.9451 0.9439 10.00 0.7200 0.9274 0.9268 0.9280 0.9271 15.00 1.0800 0.9172 0.9169 0.9178 0.9171 20.00 1.4400 0.9105 0.9104 0.9111 0.9105 25.00 1.8000 0.9059 0.9059 0.9065 0.9060 30.00 2.1600 0.9028 0.9028 0.9033 0.9030   0.8962 0.8962 0.8962 0.8962 The value of 𝐷 = 7.5 ∙ 10−11 m2 s⁄ was fitted to reproduce the experimental values in Table 2. However, there is a difference at equilibrium, 𝑤∞ = 0.8962, which is due to the experiments had used dry resin at 𝑡 = 0, while the model implicitly assumes a wet resin at 𝑡 = 0 (𝑐 = 0 means no solute, but the solvent is inside the pores in the model). This affects the mass balance. The calculated value at equilibrium, considering dry resin at 𝑡 = 0, is 𝑤∞ = 0.9052, which is closer to the experimental value in Table 2. One way to solve this is to use 𝑉∗, such that 𝑉∗ = 𝑉 − 𝑣 ∙ 𝜀, and 𝑐 ≠ 0 at 𝑡 = 0 (but 𝑐0 ≠ 𝐶0, due to 𝑞0), to compare experimental and simulated values. 6. Conclusions The numerical methods used were able to give reliable results to represent the adsorption of GVL into spheres of SP850 resin. This resin is very selective to separate GVL from a solution of glucose. Therefore, the model and the numerical methods studied here can be used to design a better separation process using adsorption. The fitted diffusion coefficient inside the particles, 𝐷, lead to the dimensionless number 𝑘𝑓 ∙ 𝑅 𝐷⁄ = 56.67, indicating that for this case study the internal resistance for mass transfer is much higher than the external resistance. The value of 𝐷 can then be used in other systems, since it is a parameter inside the particles, for example a fixed bed column, but then the value of 𝑘𝑓 may be different from the one in a stirred tank. References Abramowitz M., Stegun I.A., 1972, Handbook of Mathematical Functions, National Bureau of Standards, Applied Mathematics Series 55, Tenth Printing. Alonso D.M., Gallo J.M.R., Mellmer M.A., Wettsteinab S.G., Dumesic J.A., 2013, Direct conversion of cellulose to levulinic acid and gamma-valerolactone using solid acid catalysts, Catalysis Science & Technology, 3, 927-931. Geankoplis C.J., 1983, Transport Processes and Unit Operations, Allyn and Bacon, Boston, MA, 2nd ed. Guirardello R., 2013, Numerical solution for adsorption in porous spheres using a variational formulation, 11th International Conference of Numerical Analysis and Applied Mathematics 2013 - ICNAAM 2013, Rhodes, v. 1558. p. 2018-2021. Horstmann B.J., Chase H.A., 1989, Modelling the affinity adsorption of immunoglobulin G to protein A immobilised to agarose matrices, Chemical Engineering Research and Design 67, pp. 243-254. Luterbacher J.S., Rand J.M., Alonso M.D., Han J., Youngquist J.T., Maravelias C.T., Pfleger B.F., Dumesic J.A., 2014, Nonenzymatic sugar production from biomass using biomass-derived γ-valerolactone. Science, American Association for the Advancement of Science, 343, 277-280. Mellmer M.A., Alonso D.M., Luterbacher J.S., Gallo J.M., Dumesic J.A., 2014, Effects of γ-valerolactone in hydrolysis of lignocellulosic biomass to monosaccharides, Green Chemistry, Royal Society of Chemistry, 16, 4659-4662. Sherwood T.K., Pigford R.L., Wilke C.R., 1975, Mass Transfer, McGraw Hill Inc. Skidmore G.L., Horstmann B.J., Chase H.A., 1990, Modelling single-component protein adsorption to the cation exchanger S Sepharose® FF, Journal of Chromatography, 498, 113-128. Spiegel M.R., 1965, Theory and Problems of Laplace Transforms, Schaum’s Outline Series in Mathematics, McGraw-Hill Book Company, New York, reprinted 1986. Trindade Junior V.N., 2015a, Análise dos Processos de Separação na Produção de Etanol Celulósico a partir de Biomassa por Rota não Enzimática, doctoral thesis, University of Campinas, Brazil. Trindade Junior V.N., Luterbacher J.S., Dumesic J.A., Silva E.A., Guirardello R., 2015b, Adsorption of water/glucose mixture onto amberlite resin, Chemical Engineering Transactions, 43, 607-612. 96 7guirardello.pdf Numerical Simulation of -Valerolactone Adsorption into Porous Spheres in a Batch Process