Investigating the effect of sorption time on coalbed methane recovery through numerical simulation

The objective of this work is to study the effect of non-equilibrium sorption time on the gas production rate in coalbed methane (CBM) reservoirs. Numerical simulation is employed to investigate this phenomenon in coal seams with single-phase flow of methane and two-phase flow of methane and water. Radial and rectangular models with vertical and horizontal wells are considered. A multi-layered model is also generated with properties similar to the Horseshoe Canyon (HSCN) formation in Alberta. The results indicate that the sorption time affects the production rate in the early production phase, namely a few months to a few years depending on how slow the desorption/diffusion process is, but this depends on the magnitude of the sorption time. This is valid both for dry and initially water saturated coalbed methane reservoirs. However, in the latter case, the effect lasts longer since the dewatering must occur first for desorption/diffusion process to start. The type of wellbore also influences the dynamics of sorption/diffusion effects. For smaller diffusion coefficients (larger sorption times), the gas decline rate in horizontal wells is larger relative to vertical wells. The results of the multi-layer study indicate that when sorption time is smaller than 10 days, the effect of sorption/diffusion phenomena on total commingled production rate is negligible. In general, we recommend non-equilibrium models for early-time production when diffusion flow from matrix to fracture is still in transient state. For late-time production, when steady-state diffusion flow has been established between matrix and fracture, equilibrium models can be used.     

Effect of backwashing on activated carbon adsorption using plug flow pore surface diffusion model

A fixed bed is gradually exhausted from top to bottom without backwashing; however, backwashing can rearrange the concentration gradient in the bed. After backwashing, saturated particles which are located at the top of the bed are homogeneously distributed in the bed. The used model to predict adsorption and backwashing effect of organic component is the plug flow pore surface diffusion model (PFPSDM). A sensitivity analysis was performed to determine which parameters have the greatest impact on the model results for components which can represent various organics. In addition, the effects of backwashing were examined by rearranging concentration gradient. For single component sensitivity analysis, the molecular weight was an important parameter. The breakthrough of the smaller molecular weight component was impacted more by backwashing. The SPDFR showed a significant impact on the breakthrough pattern. When surface diffusion was the dominant mechanism, high SPDFR, the breakthrough profile was sharper than when pore diffusion was dominant, low SPDFR. The adsorbability was an important parameter in determining the breakthrough pattern. As expected, the strongly adsorbable component showed the later breakthrough. Backwashing yielded earlier breakthrough for all single components and multi-components examined.     

Estimation of organic biocide leaching rate using a modified cavity jump diffusion model

Estimation of biocide lifetime in marine antifouling coatings is of great use to improve and develop technologies. An existing model simulating the diffusion of molecules in polymer networks below glass transition temperature was employed to estimate leaching. This model was modified to allow for swelling due to water uptake and to permit evaluation of copolymer binders as well as homopolymers. This enabled prediction of biocide diffusion coefficients in polymeric coatings of various binder types, including pMMA, a pMMA/butylacrylate binder containing rosin, and a trityl copolymer, using usnic acid as a ‘model’ biocide. For comparison with modelling results, coatings fomulated using each binder type were also submitted to static and dynamic seawater immersion. Fluorescence microscopy techniques were used to quantify biocide leaching from these coatings relative to unimmersed coatings. Agreement of the modified diffusion model with experimental data was good for pMMA, reasonable for the rosin-based binder, and poor for the trityl binder. Comparison of predicted and experimental biocide profiles in the binder demonstrated deviation from the expected Fickian mechanism for the pMMA binder, despite the accurate rate prediction. This work demonstrates a first approach to predicting organic biocide diffusion, and highlights the areas for future attention.     

Solution to the homogeneous surface diffusion model for batch adsorption systems using orthogonal collocation

A solution to the homogeneous surface diffusion model has been developed and incorporated into a batch adsorption model based on external boundary layer mass transport and homogeneous diffusion. The model has been extensively tested using three experimental adsorption systems, namely, phenol on carbon, basic yellow dye on carbon and basic blue dye on silica. The effect of initial solute concentration and adsorbent mass has been studied in 23 batch experiments, which have been modelled using the collocation solution method to solve the homogeneous surface diffusion equation. The theoretical concentration decay curves show a high degree of correlation with experimental data.     

Studies on batch adsorptive removal of cadmium and nickel from synthetic waste water using silty clay originated from Balochistan–Pakistan

The potentials of silty clay(SC), acquired from Chaman, Balochistan, were investigated as adsorbent for Ni(Ⅱ)and Cd(Ⅱ) removal from contaminated media. The influence of different operating factors like dose, pH, temperature, and time of contact was explored, and optimum values were noted under batch adsorption method. Isothermal study was conducted with varying concentrations of solutions on optimized conditions and different adsorption models i.e., Langmuir, Freundlich, Temkin and Dubinin–Radushkevich(D–R) isotherm, which were employed to interpret the process. The isothermal data of both Ni(Ⅱ) and Cd(Ⅱ) were well fitted to Langmuir isotherm suggesting the formation of monolayer of metal ions on silty clay. The values of adsorption capacity noted for Ni(Ⅱ) and Cd(Ⅱ) were 3.603 mg·g~(-1) and 5.480 mg·g~(-)1, respectively. Kinetic studies affirmed that pseudo second order(PSO) kinetics was being obeyed by the removal of Ni(Ⅱ) and Cd(Ⅱ). Thermodynamic variables like free energy change(ΔG°), enthalpy change(ΔH°) and entropy change(ΔS°) were calculated. The negative value of ΔG° and the positive values of ΔH° and ΔS° unfolded that the removal process of both metal ions of by SC was spontaneous, endothermic and feasible.     

Open-loop optimization for batch reaction systems: a case study

A comparison of the results obtained from deterministic and stochastic model-based optimization approaches for the determination of the optimal open-loop operating policy for a semi-batch reaction system are presented. The commercial synthesis of an intermediate in the production of the hypertension agent Eprosartan is considered in this case study. Identification of the reaction mechanism and estimation of the kinetic model parameters from experimental data are carried out as part of the development of a detailed semi-batch reaction system model that is used with the optimization approaches. The results of this study indicate that conversion optimization of the operating conditions can provide a good approximation to the economic optimum in the case of high conversion operation. This study also indicates that the optimal operating policy for the catalyst and reactant are batch, with the catalyst addition occurring prior to the reactant, and that a reduction in the amount of reactant from current practice is possible.     

Scheduling of displacement batch digesters using discrete time formulation

This paper provides mathematical programming based optimization model and computational results for short-term scheduling of displacement batch digesters in a pulp industry. The scheduling problem involves development of an optimal solution that yields the best sequence of operations in each of the parallel batch digesters sharing common resources. The constraints are imposed on meeting the demand of pulp of different qualities within a specified time horizon. The problem comprises of both fixed-time and variable time durations of the tasks, different storage policies, zero-wait and finite wait times, and handling of shared resources. The scheduling problem is formulated using a state-task-network (STN) representation of production recipes, based on discrete time representation resulting in a mixed-integer linear programming (MILP) problem which is solved using GAMS software. The basic framework is adapted from the discrete-time model of Kondili et al. (Comput. Chem. Eng., 1993, 17, 211–227). Different case studies involving parallel digesters in multiple production lines are considered to demonstrate the effectiveness of the proposed formulation using two different objective functions.     

Modeling separation of proteins by inert core adsorbent in a batch adsorber

Adsorption/desorption kinetics of protein on the binding ligand of inert core adsorbent in a batch adsorber is analyzed theoretically for Langmuir isotherm coupled with the intraparticle diffusion and film mass transfer resistances. For the two limiting cases of Langmuir isotherm, there are analytical solutions. New analytical solutions are derived for Henry isotherm, and the analytical solution of shrinking core model is recommended for rectangular isotherm. The effects of the inert core radius, equilibrium constant, intraparticle diffusion and film mass transfer resistances on the time evolution of bulk concentration and particle radial profiles were investigated. The applicable range of the analytical solution with rectangular isotherm is given. A new method to estimate both film mass transfer coefficient, k_f, and effective pore diffusivity, D_pe, from a single bulk concentration-time curve in batch adsorber is given and tested with literature data for the adsorption of BSA on CB-6AS inert core adsorbent.     

Use of stirred batch reactors for the assessment of adsorption constants in porous solid catalysts with simultaneous diffusion and reaction. Theoretical analysis

A simple, pseudo-equilibrium model was derived for a catalytic system with a first order chemical reaction and simultaneous diffusive and adsorptive processes, in order to assess the corresponding kinetics and Henry law's-type adsorption parameters. Solutions from this model were compared to exact solutions from a more detailed, general model. It was shown that under most of the experimental conditions used in stirred batch reactors and the usual model considerations, it is only possible to assess apparent adsorption parameters. Also, we observed that a stable relationship between the concentrations in the gas and solid phases is reached. The error produced in assuming that the apparent adsorption constant is the real one was calculated to be very important. The value of the apparent adsorption constant depends on various system properties and experimental conditions, such as the Thiele modulus, the amount of catalyst and the contact time. The ratio between the apparent and real adsorption constants was shown to be the transient effectiveness factor at any moment. This ratio reaches a maximum value for the pseudo-equilibrium state, that is always larger than the steady-state effectiveness factor, becoming closer as long as the system's adsorption capacity decreases. The analysis determines the operative conditions to reduce the parametric correlation. Also a criterion for the applicability of usual approximations in the assessment of kinetics and equilibrium adsorption parameters in porous solid catalysts by means of pulse injection methods is established.     

Diffusion in a mesoporous silica membrane: Validity of the Knudsen diffusion model

Experimental permeance data for several light gases in a mesoporous silica membrane are analyzed in detail and shown to conform closely to the Knudsen diffusion model. The results of this study do not support the conclusions drawn from recent molecular simulations concerning the inadequacy of the Knudsen model.     

A modified iterative IOM approach for optimization of biochemical systems

The presented previously indirect optimization method (IOM) developed within biochemical systems theory (BST) provides a versatile and mathematically tractable optimization strategy for biochemical systems. However, due to the local approximations nature of the BST formalism, the iterative version of this technique possibly does not yield the true optimum solution. In this work, an algorithm is proposed to obtain the correct and consistent optimum steady-state operating point of biochemical systems. The existing linear optimization problem of the direct IOM approach is modified by adding an equality constraint of describing the consistency of solutions between the S-system and the original model. Lagrangian analysis is employed to derive the first order necessary optimality conditions for the above modified optimization problem. This leads to a procedure that may be regarded as a modified iterative IOM approach in which the optimization objective function includes an extra linear term. The extra term contains a comparison of metabolite concentration derivatives with respect to the enzyme activities between the S-system and the original model and ensures that the new algorithm is still carried out within linear programming techniques. The presented framework is applied to several biochemical systems and shown to the tractability and effectiveness of the method. The simulation is also studied to investigate the convergence properties of the algorithm and to give a performance comparison of standard and modified iterative IOM approach.     

Freeze drying process: real time model and optimization

Freeze drying is a separation process based on the sublimation phenomenon. This process has the following advantages compared to the conventional drying process: the material structure is maintained, moisture is removed at low temperature (reduced transport rates), product stability during the storage is increased, the fast transition of the moisturized product to be dehydrated minimizes several degradation reactions. Freeze drying process has not been studied well enough. In order to put it to practice, a mathematical model based on fundamental mass and energy balance equations has been developed, based on a deterministic mathematical model proposed by Liapis and Sadikoglu [Drying Technol. 15 (3–4) (1997) 791], and used to calculate the amount of removed water and amount of residual water. The proposed model contains the freeze drying equations, which are solved by the orthogonal collocation and polynomial approximation—Jacobi method. The results show that the dynamic mathematical model represents well the process and is especially well suited for real time optimization. As a case study to illustrate the model utilization in a real time optimization procedure, the freeze drying process was optimized by the method of Successive Quadratic Programming (SQP) used for solution of non-linear equations, for skimmed milk and soluble coffee. The optimization procedure showed to be an important tool to improve the process performance since lower energy consumption and hence lower cost has been achieved to obtain the product with the same quality.     

Asymptotically exact driving force approximation for intraparticle mass transfer rate in diffusion and adsorption processes: nonlinear isotherm systems with macropore diffusion control

An asymptotically exact nonlinear driving force model of intra-particle mass-transfer rate for nonlinear isotherm systems with macropore diffusion control is presented. The obtained expression is compared with the solutions of the Fickian diffusion and adsorption model and excellent accuracy over the entire time (fractional uptake) domain is demonstrated. In the case of an irreversible isotherm the model reduces to the equations resulting from the shrinking core model a fact that guarantees its accuracy for highly nonlinear systems. The high accuracy of the model is further demonstrated by comparison with experimental data under various operational conditions.     

Experimental design and batch experiments for optimization of Cr(VI) removal from aqueous solutions by hydrous cerium oxide nanoparticles

Hydrous cerium oxide (HCO) was synthesized by intercalation of solutions of cerium(III) nitrate and sodium hydroxide and evaluated as an adsorbent for the removal of hexavalent chromium from aqueous solutions. Simple batch experiments and a 2⁵ factorial experimental design were employed to screen the variables affecting Cr(VI) removal efficiency. The effects of the process variables; solution pH, initial Cr(VI) concentration, temperature, adsorbent dose and ionic strength were examined. Using the experimental results, a linear mathematical model representing the influence of the different variables and their interactions was obtained. Analysis of variance (ANOVA) demonstrated that Cr(VI) adsorption significantly increases with decreased solution pH, initial concentration and amount of adsorbent used (dose), but slightly decreased with an increase in temperature and ionic strength. The optimization study indicates 99% as the maximum removal at pH 2, 20 °C, 1.923 mM of metal concentration and a sorbent dose of 4 g/dm³. At these optimal conditions, Langmuir, Freundlich and Redlich–Peterson isotherm models were obtained. The maximum adsorption capacity of Cr(VI) adsorbed by HCO was 0.828 mmol/g, calculated by the Langmuir isotherm model. Desorption of chromium indicated that the HCO adsorbent can be regenerated using NaOH solution 0.1 M (up to 85%). The adsorption interactions between the surface sites of HCO and the Cr(VI) ions were found to be a combined effect of both anion exchange and surface complexation with the formation of an inner-sphere complex.     

A pore network model for diffusion in nanoporous carbons: Validation by molecular dynamics simulation

A hybrid molecular dynamics simulation/pore network model (MD/PNM) approach is developed for predicting diffusion in nanoporous carbons. This approach is computationally fast, and related to the structure of the real material. The PNM takes into account both the geometrical (a distribution of pore sizes) and topological (the pore network connectivity) characteristics of nanoporous carbons, which are obtained by analysing adsorption data. The effective diffusion coefficient is calculated by taking the transport diffusion coefficients in single slit-shaped model pores from MD simulation and then computing the effective value over the PNM. The reliability of this approach is evaluated by comparing the results of the PNM analysis with a more rigorous, but much slower, simulation applied to a realistic model material, the virtual porous carbon (VPC). We obtain good agreement between the diffusion coefficients for the PNM and the VPC, indicating the reliability of the hybrid MD/PNM method and it can be used in industry for materials design.     

Some pitfalls in the use of the Knudsen equation in modelling diffusion in nanoporous materials

The Knudsen model of diffusion in small pores, originally verified in macropores, is widely applied at the mesopore scale with adsorption effects neglected, largely based on linearity of the correlation. Here, we show that this approach is misleading, and that the correlation masks inconsistencies arising from neglect of van der Waals forces in the Knudsen model. We examine the tortuosity for diffusion of light gases in nanoporous carbons using the Oscillator model of low pressure transport developed in the first author’s laboratory, which incorporates van der Waals interactions. Pore network effects are considered through a hybrid correlated random walk-effective medium theory approach. It is shown that in the presence of a pore size distribution the apparent tortuosity is not a porous medium property alone, but depends also on the temperature and on the diffusing molecule, because of the temperature and the gas-dependent short circuiting effects associated with pores that have high conductance. This short circuiting effect leads to a complex and rich variety of behaviour with respect to pore size, temperature and diffusing gas, which is consistent with experimental evidence, but is absent when the Knudsen model is used with adsorption effects neglected. It is shown that when effects of adsorption on equilibrium and transport are overlooked the commonly used correlation of diffusivity with is deceptive, as the product of the adsorption equilibrium constant and diffusivity (or pore conductance) also approximately scales linearly with the Knudsen diffusivity (i.e. with ). Such behaviour is found for diffusion in mesoporous carbons and silica as well as in silicon. Consequently, claims of validity of the Knudsen model based on such a correlation may be misconceived.     

Statistically modeled adsorption isotherms for the multilayer gas molecules adsorbed on non-porous solid adsorbents of two and three groups sites

The adsorption isotherms with each saturation vapor pressure factor (c_s1, c_s2 or c_s3) for two groups of sites in two cases of the multilayer and for three groups of sites in one case of the multilayer are derived statistically in heterogeneous non-porous solid adsorbents without interactions among the adsorbed molecules. When some sites of BET isotherm are substituted by less energetic sites, the two-group isotherm obtained by the substitution shows less adsorption over the whole range of relative pressure than the BET isotherm prior to the substitution, at any combined values of f₁ with M₁ of the two-group isotherm with the same saturation vapor pressure factor. A method to get the monolayer sites (v_m) from the ratios of the experimental isotherm to the theoretical isotherm at the whole relative vapor pressure minimizing the standard error is suggested. Our two- or three-group isotherms calculated through many experimental adsorption isotherm data selected appropriately provide larger values of v_m than those obtained from BET isotherms. Differential heat vs. v/v_m and Bose-Condensation heat are mentioned.     

Adsorption of methylene blue from aqueous solutions by modified expanded graphite powder

The modified expanded graphite (MEG) powder was used as a porous adsorbent for the removal of the cationic dye, methylene blue (MB), from aqueous solutions. The dye adsorption experiments were carried out with the bath procedure. Experimental results showed that the basic pH, increasing initial dye concentration and high temperature favored the adsorption. The dye adsorption equilibrium was attained rapidly after 5 min of contact time. Experimental data related to the adsorption of MB on the MEG under different conditions were applied to the pseudo-first-order equation, the pseudo-second-order equation and the intraparticle diffusion equation, and the rate constants of first-order adsorption (k₁), the rate constants of second-order adsorption (k₂) and intraparticle diffusion rate constants (k_int) were calculated, respectively. The experimental data fitted very well in the pseudo-second-order kinetic model. The thermodynamic parameters of activation such as Gibbs free energy, enthalpy, and entropy were also evaluated. The results indicated that the MEG powder could be employed as an efficient adsorbent for the removal of textile dyes from effluents.     

Atomically detailed models of gas mixture diffusion through CuBTC membranes

Metal–organic frameworks are intriguing crystalline nanoporous materials that have potential applications in adsorption-based and membrane-based gas separations. We describe atomically detailed simulations of gas adsorption and diffusion in CuBTC that have been used to predict the performance of CuBTC membranes for separation of H₂/CH₄, CO₂/CH₄ and CO₂/H₂ mixtures. CuBTC membranes are predicted to have higher selectivities for all three mixtures than MOF-5 membranes, the only other metal–organic framework material for which detailed predictions of membrane selectivities have been made. Our results give insight into the physical properties that will be desirable in tuning the pore structure of MOFs for specific membrane-based separations.     

Study of the mechanism of pore diffusion in batch adsorption systems

In this study the film-pore diffusion model was applied to describe system transport kinetics of three basic dye-carbon systems, namely Basic Blue 69, Basic Red 22 and Basic Yellow 21. The mass transfer parameters evaluated were the external mass transfer coefficient k_f (cm s^?1) and the effective diffusivity D_eff (cm² s^?1). A single k_f value was sufficient to describe each dye system: these were 0.15 × 10^?2, 0.20 × 10^?2 and 0.50 × 10^?2 cm s^?1 for BB69, BR22 and BY21, respectively. The effective diffusivity was found to have values much larger than those of pore diffusivities calculated from liquid diffusivities and its value decreased with increasing initial dye concentration. This was attributed to the effect of surface diffusion, hence pore diffusivity was exchanged by the effective pore diffusivity in the model. The present model was solved by the exponential curve fit technique; results were expressed in the form of experimental and theoretical Sherwood Numbers compared in terms of the residual.