Linear Driving Force Equation Research Articles

Mathematical Evaluation of Direct and Inverse Problem Applied in Breakthrough Models of Metal Adsorption

The treatment of industrial effluents has great environmental and human health importance. The purification of water from polluting components, such as metals and organic compounds, can be considered one of the main applications in this field, with adsorption being one of the main treatment methods. Therefore, with the objective of describing the dynamics of the process in an adsorption column and estimating the parameters involved, in this work, an algorithm for the Method of Lines (MOL) was used in order to numerically solve the model formed by the mass balance in liquid phase, the linear driving force equation (LDF), and the Langmuir isotherm for equilibrium. In addition, a sensitivity analysis of the phenomenon was carried out in relation to the parameters and a subsequent estimation of these was made through the Monte Carlo technique via the Markov chain (MCMC). The validation algorithm was created using data from actual breakthrough curves found in the literature. The experimental data were obtained from the literature for the adsorption of Cadmium (Cd), Copper (Cu), Nickel (Ni), Zinc (Zn), and Chrome (Cr) ions. Among all the estimates, the one that had the lowest adjustment to the data was that related to zinc metal, which had an R2 equal to 0.8984. For the other metals, the correlation coefficient had a value closer to unity. This demonstrates that, in general, the estimates were good enough to represent the dynamics of adsorption.

A numerical modelling study of SO2 adsorption on activated carbons with new rate equations

Modelling dynamic adsorption of sulfur dioxide (SO2) on activated carbons (ACs) is significant in guiding practical desulphurization processes and making highly efficient use of adsorbents in terms of the adsorption rate which largely depends on particle size. In this work, models derived from the Vermeulen and an improved linear driving force (LDF) rate equation were studied for the first time on SO2 adsorption over AC particles with different sizes. For larger particles (≥3 mm), breakthrough curves predicted by the Vermeulen equation showed good agreement with experimental data, demonstrating that intraparticle diffusion resistance varied with particle size, feed concentration, adsorption time and location. For smaller particles (1 mm), a correction on the volume-averaged adsorption capacity as a function of adsorption time and saturation in the rate equation was developed to avoid the underestimation of adsorption rate due to the inappropriate parabolic concentration profile inherent in the conventional LDF model. By providing a concentration gradient and adsorption rate closer to actual values, the improved LDF equation was confirmed to provide excellent prediction results on 1-mm particles. Different modelling characteristics of the two models indicates varying effects of intraparticle diffusion on adsorption rate with particle size regarding the specificity of SO2 physisorption on ACs.

Analytical solution of the Langmuir-based linear driving force model and its application to the adsorption kinetics of boscalid onto granular activated carbon

The application of intraparticle diffusion models is generally difficult because of complex mathematical structure. The linear driving force (LDF) approach reduces the mathematical effort and provides a reasonable approximation of the intraparticle diffusion theory. In this work, an exact analytical solution of a LDF equation based on the Langmuir equilibrium model (LLDF) was derived. The LLDF model depends on three unknown parameters, namely the adsorbed amount at equilibrium, the maximum adsorbent capacity and the mass transfer coefficient. The LLDF model was used for analyzing the adsorption kinetics of boscalid onto granular activated carbon. The experimental results at equilibrium showed that the maximum adsorption capacity of activated carbon for boscalid and the Langmuir equilibrium constant of the process were 167 mg g−1 and 0.53 L mg−1, respectively. The LLDF equation was successfully applied to the kinetic data, allowing the evaluation of the mass transfer coefficient of boscalid (8.4 × 10−3 h−1). The LLDF model has general validity for describing intraparticle diffusion-adsorption onto porous media and its reliability can be assessed by a simple graphical method.

Equilibrium and kinetics of CO2 adsorption onto activated carbon

Knowledge of adsorption characteristics of adsorbent-adsorbate pairs is essential for designing adsorption beds for adsorption cooling and adsorptive gas capturing applications. We investigated the adsorption isotherms and the adsorption kinetics of CO2 onto microporous activated carbon powder of type Maxsorb III. Measurements were performed with gravimetric apparatus for temperatures from 30 to 70°C and pressures up to 7MPa for adsorption isotherms and up to 4MPa for adsorption kinetics. The gravimetric adsorption data obtained were consistent with previously measured isotherms with volumetric apparatus. Both absolute and excess adsorption data have been fitted precisely with Tóth and Dubinin-Astakhov isotherm equations. The classical linear driving force (LDF) model with a constant mass transfer coefficient failed to correlate the experimental adsorption kinetics data. To overcome this problem, the authors presented a modified LDF equation with a variable mass transfer coefficient which is a function of the equilibrium and instantaneous uptakes. This modified LDF equation led to a better fitting and could be implemented easily in simulation of pressure swing adsorption (PSA), temperature swing adsorption (TSA) and adsorption chiller applications.

Corrected adsorption rate model of activated carbon–ethanol pair by means of CFD simulation

The adsorption rate is an important parameter for accurate performance estimation of adsorbent-refrigerant based adsorption cooling cycles. Here, we have investigated the response of two adsorption kinetics models of activated carbon–ethanol pair by means of CFD simulation. The isothermal assumption used in estimating the diffusion time constant of Fickian diffusion and linear driving force (LDF) models led to divergence and under-estimated adsorption uptakes, respectively. By including the simulated adsorbent temperature profile in fitting of LDF model to experimental data, we assessed the non-isothermal diffusion time constants which were 2.5 to 5 times higher than those evaluated previously with isothermal assumption. The goodness of fitting, evaluated with coefficient of determination (R2), improved and became higher than 0.95 from 0.73 initially. The developed non-isothermal LDF equation allows accurate heat and mass transfer simulations and performance optimization of large scale adsorption/desorption bed employing activated carbon-ethanol pair for adsorption cooling applications.

Performance Evaluation of a Continuous Operation Adsorption Chiller Powered by Solar Energy Using Silica Gel and Water as the Working Pair

In the present study, dynamic analysis and performance evaluation of a solar-powered continuous operation adsorption chiller are introduced. The adsorption chiller uses silica gel and water as the working pair. The developed mathematical model represents the heat and mass transfer within the reactor coupled with the energy balance of the collector plate and the glass cover. Moreover, a non-equilibrium adsorption kinetic model is taken into account by using the linear driving force equation. The variation of solar radiation, wind speed, and atmospheric temperature along a complete cycle are considered for a more realistic simulation. Based on the case studied and the baseline parameters, the chiller is found to acquire a coefficient of performance of 0.402. The average thermal efficiency of the solar collector is estimated to be 62.96% and the average total efficiency approaches a value of 50.91%. Other performance parameters obtained are 363.8 W and 1.82 W/kg for the cooling capacity and the specific cooling power of the chiller, respectively. Furthermore, every 1 kg of silica gel inside the adsorption reactor produces a daily chilled water mass of 3 kg at a temperature of 10 ◦C. In addition, the cooling system harnesses 25.35% of the total available solar radiation and converts it to a cooling effect.

Methane adsorption behavior on coal having different pore structures

The adsorption of methane onto five dry coal samples was measured at 298K over the pressure range from 0 to 3.5MPa using a volumetric method. The isotherm data were fitted to the Langmuir and the Freundlich equations. The kinetic data were fitted to a pseudo second order equation, the linear driving force equation (LDF), and an intra-particle diffusion model. These results showed that higher methane adsorption is correlated with larger micro-pore volumes and specific surface areas. The adsorption was related to the narrow micro-pore size distribution when the previous two parameters are large. The kinetics study showed that the kinetics of methane adsorption onto these five dry coal samples followed a pseudo second order model very well. Methane adsorption rates are controlled by intra-particle diffusion. The faster the intra-particle diffusion, the faster the methane adsorption rate will be.

A Comparison of the Wheeler-Jonas Model and the Linear Driving Force at Constant-Pattern Model for the Prediction of the Service Time of Activated Carbon Cartridges

The linear driving force (LDF) model is applied to predict the service life of activated carbon cartridges. It is compared with the currently used Wheeler-Jonas equation, which results from a model of chemical reaction kinetics. The LDF model is based on a mass transfer model of adsorbate into the particle. The two models are studied in constant-pattern conditions. The properties of the two models are first clarified and then compared. It is shown that the Wheeler-Jonas equation leads to symmetrical breakthrough curves, whereas the constant-pattern LDF equation results in asymmetrical curves. Thus, the curvature of the isotherm has no influence on the shape of the Wheeler-Jonas curve. For the LDF breakthrough curve, it is shown that the asymmetry increases with the curvature of the isotherm. Wheeler-Jonas can be used with a Dubinin-Raduskevitch isotherm, whereas the LDF model analytical solution is valid for a Langmuir isotherm only. The LDF model can be used with the DR isotherm, but a numerical solution is required. At very low concentrations where the isotherm is linear, the constant pattern no longer exists and both models fail. The Dubinin-Raduskevitch isotherm must be fitted with a Langmuir isotherm to use the analytical solution of the LDF model.

On the linear driving force approximation for adsorption cooling applications

This paper aims to address the effect of using classical Linear Driving Force (LDF) model on the performance of adsorption chillers. A comparative study between the well known LDF equation and the Fickian diffusion (FD) model has been conducted. Two adsorbent/refrigerant pairs namely, RD silica gel/water and “CaCl 2 confined to KSK silica gel”/water pairs have been used. Relative and absolute errors between adsorption uptakes estimated by both models have been estimated. It is found that the LDF model has numerous errors at relatively shorter adsorption times which imply that using LDF equation may lead to incorrect evaluation of the performance of adsorption chiller especially at short cycle times. This study also presents an improved form of LDF model considering the dependency of the particle mass transfer coefficient on the dimensionless time. Theoretical calculations show good agreement between the improved LDF approximation and the FD model.

Hydrogeochemical modeling of enhanced benzene, toluene, ethylbenzene, xylene (BTEX) remediation with nitrate

During a 5‐month field test, active remediation of a benzene, toluene, ethylbenzene, xylene (BTEX)‐contaminated aquifer was initiated by injecting water with varying amounts of KNO3. The experiment was performed prior to selecting bioremediation for full‐scale cleanup, particularly to evaluate the competing reaction of nitrate with hydrocarbons and reduced sulfur components. The nitrate oxidized sulfides that had precipitated earlier as a result of the natural degradation of BTEX with SO42− from groundwater. When the sulfides were exhausted, BTEX degradation was enhanced by nitrate. A hydrogeochemical model with kinetic oxidation reactions for Fe(II), FeS and BTEX by nitrate was developed to calculate the observed concentration patterns along a flow line in the aquifer. The rates for the kinetic model were based on published kinetic reaction equations for oxidation with oxygen. Nitrate was introduced in the equations in the same form as oxygen, with a premultiplier added to fit the observed concentration changes in the aquifer. The oxidation of Fe(II) with nitrate in the aquifer was 4 times slower than the abiological oxidation reaction with oxygen in water. Similar rates were found for oxidation of FeS with nitrate as for FeS2 with oxygen, but the specific surface area of FeS in the aquifer was larger. The reaction rate for degradation of BTEX compounds was about 107 times faster than for natural organic matter. BTEX release from pools in the aquifer was modeled with a linear driving force equation in which the pollutant/water interfacial area was linked to the mass of BTEX. The release rate and the denitrification rate were used to calculate the initial amounts of BTEX at the start of the KNO3 injection. This study shows that an assessment of the efficiency of nitrate addition for stimulating bioremediation has to consider possible reactions of nitrate with reduced sulfur components and ferrous iron.

Momentum and heat transfer in the adsorbent of a waste-heat adsorption cooling system

This paper presents a numerical and experimental analysis of the coupled heat and mass transfer mechanisms in the adsorbent of a waste-heat adsorption cooling system. A new three-dimensional non-equilibrium model is developed and used to investigate the simultaneous transport of heat and mass. In the model, a linear driving force equation is used to account for mass-transfer resistance within the pallets (internal resistance), while Darcy's law is introduced to describe the adsorbate flows in the interparticle voids (external resistance). An experiment has been conducted to validate this model. The temperature, pressure, velocity and water uptake fields of the bed and their effects on system performance are discussed.

Effects of coupled heat and mass transfers in adsorbent on the performance of a waste heat adsorption cooling unit

This paper presents a numerical analysis of the performance of a waste heat adsorption cooling system associated with the coupled heat and mass transfer mechanisms in the adsorbent. A new three-dimensional non-equilibrium model is developed and used to investigate and optimize the simultaneous transport of heat and mass in the waste heat adsorber. In the model, a linear driving force equation is used to account for the mass transfer resistance within the pallets (internal resistance), while the Darcy's law is introduced to describe the adsorbate flows in the interparticle voids (external resistance). Using the model, the effects of the bed configurations such as the number of fins, the bed dimensions, the heat transfer coefficients, the permeability and the adsorption rate on the system performance are extensively investigated. it is found that the performance of the system can be seriously deteriorated by poorer mass transfers in the adsorbent if its permeability is smaller than a critical value. On the other hand, for systems with short cycle times, the internal mass transfer resistance can be a significant limiting factor as well.

Modelling of a pressure swing adsorption process for oxygen enrichment with carbon molecular sieve

AbstractAdsorptive separation of oxygen from nitrogen and argon is carried out during the desorption steps of a pressure swing adsorption (PSA) process which uses carbon molecular sieves developed by Bergbau‐Forschung GmbH. The adsorption isotherms of the three main components of air are very similar. On account of the pore size distribution of CMSN2, the diffusion coefficient of oxygen is more than eight times those of nitrogen and argon so that air separation occurs by adsorption kinetics. Experimental results for the individual steps and cyclic operation of the PSA process are presented and compared with the predictions of an isothermal plug‐flow model. Adsorption rate is represented by a linear driving force equation. If the diffusion coefficients are adapted separately to every step, a good agreement is observed between the model calculations and experimental results.

A simplified driving force model for activated carbon adsorption

AbstractThe use of two simplified driving force expressions are investigated for describing the intraparticle rate behaviour during adsorption onto activated carbon. A previously developed numerical solution to the surface diffusion model is used to compare the linear and quadratic driving force approximation when non‐linear isotherms and time variant boundary conditions are present. The results show that while the linear driving force equation is a poor approximation of the exact solution, the quadratic driving force is a good approximation, and its use in a kinetic model of activated carbon adsorption gives as good a description of adsorption behaviour as is given by the more detailed surface diffusion model. The driving force model has the advantage of being easier to solve, and the numerical solution requires significantly less computational effort.