Multiplicity features of porous catalytic pellets—I. Single zeroth-order reaction case

The steady state multiplicity features of porous catalytic pellets, the temperature of which is uniform but different from that of the surrounding, are determined for a zeroth-order reaction. Simple algebraic expressions are derived which divide the global parameter space into regions with different number of solutions as well as with different types of bifurcation diagrams of surface temperature vs ambient Thiele modulus. The model predicts that up to five steady state solutions exist. The region of parameters with five solutions shrinks with increasing external mass transfer resistance and as the geometry of the pellet changes from a slab to a sphere.     

Steady-state multiplicity in a solid oxide fuel cell: Practical considerations

Steady-state multiplicity in a solid oxide fuel cell (SOFC) in three modes of operation, constant ohmic external load, potentiostatic and galvanostatic, is studied using a detailed first-principles lumped model. The SOFC model is derived by accounting for heat and mass transfer as well as electrochemical processes taking place inside the fuel cell. Conditions under which the fuel cell exhibits steady state multiplicity are determined. The effects of operating conditions such as convection heat transfer coefficient and inlet fuel and air temperatures and velocities on the steady state multiplicity regions are studied. Depending on the operating conditions, the cell exhibits one or three steady states. For example, it has three steady states: (a) at low external load resistance values in constant ohmic external load operation and (b) at low cell voltage in potentiostatic operation.     

On shape normalization for non-uniformly active catalyst pellets—II

A shape normalization is developed for non-uniformly active catalyst pellets with an isothermal first-order reaction. A variety of activity profiles, including step-distribution and those which increase or decrease towards the pellet surface, are considered. Mass transfer in the outer inert zone is found to be characterized by a single dimensionless parameter, called the internal Biot number. The effect of finite external mass transfer is also taken into account. Analytic solutions and numerical comparisons in terms of the normalized significant parameters are reported for various activity profiles and pellet shapes. The shape normalized curves, obtained for different geometries, activity profiles and external mass transfer, match perfectly for small or large values of the Thiele modulus. However, as in part I of this paper (and as for positive-order reactions in a uniformly active slab catalyst where a normalization for the reaction order is sought), the normalized curves can differ significantly for intermediate values of the Thiele modulus. The particular case of activity distributions which drop continuously to zero within the pellet is excluded from the shape normalization, when mass transfer resistance—both in the bulk phase external to the pellet and in the outer inert region of the pellet—is negligible.     

Steady state multiplicity behavior of an isothermal axial dispersion fixed-bed reactor with nonuniformly active catalyst

An isothermal, heterogeneous fixed-bed reactor packed with nonuniformly active catalyst pellets where a biomolecular Langmuir-Hinshelwood reaction occurs, is studied using an axial dispersion model. A catalyst activity distribution given by a Dirac delta function, where the active catalyst is deposited at a specific location within the pellet, is considered. This includes the common case of externally coated pellets with external mass transfer resistance. The steady state multiplicity behavior of this reactor, and its limiting cases: CSTR, PFR and pseudohomogeneous axial dispersion, are examined in detail. The nonlinearity of the reaction kinetics provides two sources of multiplicity, through the heterogeneous nature of the reactor and the presence of axial dispersion in the fluid phase. Their roles in determining reactor multiplicity behavior are fully explored. It is shown that this system can admit at most nine steady state solutions. The limiting behavior of the heterogeneous axial dispersion model as Pe → 0 or ∞ is not represented fully by the CSTR or PFR models because of ignition phenomenon. Finally, the effects of mixing on reactor conversion are discussed.     

Approximations for unsteady state diffusion and adsorption with mass transfer resistance in both phases

A new approximate kinetic model, valid for both long and short duration of transient mass (or heat) transfer processes, was proposed. The presented model may be used in the cases of transient diffusion in bodies in the shape of infinite slab, infinite cylinder or sphere. The case where mass transfer resistance is both inside and outside a body (adsorbent pellet) has been considered. In the presented model Laplace transform for diffusion equation is used, the solution is transformed to the form of continued fractions and the resulting expression is truncated to obtain such a number of terms that ensures the required accuracy. This concept, which was presented by J. Lee and D.H. Kim (Chemical Engineering Journal, 2012) for systems without external mass transfer resistance, in this work has been extended on systems where such resistance exists. On the basis of calculations done in this work it was shown that values obtained with the approximate model are almost identical to values derived from the exact model. It is worth mentioning that such consistency refers to all considered shapes regardless of the magnitude of external mass or heat transfer resistance. The model can be used for modelling and design of cyclic adsorption–desorption processes.     

单个液滴蒸发模型中不同质量传递公式的有效性分析

液滴蒸发是液体喷雾的重要物理过程,文献中使用了不同的模型计算液滴蒸发中的质量流率。依据质量传递驱动力HM的处理,可将这些模型归类于三种不同形式,即经典模型、质量类比模型和漂移流动模型。本文对这些模型的来源和假设作了分析和总结,发现漂移流动模型考虑了Stefan效应和质量迁移,应为正确的表述,而经典模型忽略了瞬态效应,质量类比模型忽略了Stefan流效应。最后以两种不同蒸发率条件下的液滴蒸发为例进行计算,研究了不同质量传递驱动力对液滴蒸发的影响, 发现经典模型和质量类比模型都对液滴的蒸发估计过低,且高蒸发率下三种模型对液滴蒸发特性的影响更显著。     

An experimental study of steady-state multiplicity features of two parallel catalytic reactions

A study of the multiplicity features of a single catalytic pellet on which either ethane, carbon monoxide or both reactants were oxidized revealed several new features such as: four different stable states during the co-oxidation of both reactants and three states during the oxidation of carbon monoxide. Very intricate bifurcation diagrams, which describe the dependence of a state variable on an operating condition, were found.It was shown that the experimental finding of a cross-section of the bifurcation set is the rational approach of studying the multiplicity features of systems characterized by a large number of parameters and for which no exact mathematical model is available. Recent theoretical results provide useful guidance in selecting the operating conditions so that all the possible multiplicity features are found. These theoretical results may be used to place constraints on the functional form of any proposed reaction rate expression.     

Uniqueness criteria for first order catalytic reactions with external transport limitations

Uniqueness and multiplicity criteria are developed for a non-isothermal first order reaction occurring in infinite slab or spherically shaped catalyst pellets, with external heat and mass transport limitations. The analysis is done for the analytically resolvable model wherein the catalyst pellet is assumed to be internally isothermal, but with internal gradient of concentration. The results are also compared with those obtained from numerical integration of the complete heat and mass balance equations, including internal temperature gradient as well. The general features of the results will carry over to pellets of any shape.     

Uniqueness and multiplicity criteria for an nth order chemical reaction

New and very strong criteria are presented for a priori prediction of the conditions for which the steady-state lumped parameter model of an nth order chemical reaction (n ≥ 0) in an adiabatic CSTR has either a unique or multiple solutions. The criteria show that the higher the order of the reaction the smaller is the region in the parameters space for which multiplicity can occur.New uniqueness and multiplicity criteria are developed also for an nth order reaction in a porous catalyst using a model, which accounts for intraparticle concentration gradients, while assuming a uniform intraparticle temperature different from the ambient one. The region in the parameters space for which steady state multiplicity can occur for this model is smaller than that for a corresponding lumped model, which ignores the intraparticle concentration gradients.     

Model-based prediction of substrate conversion and protein synthesis and excretion in recombinant Aspergillus niger biopellets

Aspergillus niger grown as biopellets in a submerged culture has been used as an expression system for glucose oxidase (GOD). With regard to mass transfer, substrate utilization and the formation and excretion of the product GOD, the behavior of aggregated cells such as biopellets differed significantly from that of systems with suspended cells. A one-dimensional pellet model has been developed which is capable to describe the relevant processes in the pellet. Experimental results showed a phase with a highly dynamic pellet size distribution during the first 20 h of the cultivation. A stable number of pellets were usually found after this first period. In the model presented this initial process was described by assuming both suspended hyphae and a fixed number of pellets. The suspended hyphae were attached to the pellets and the substrate conversion and product formation was than described with the pellet model. The model predicted the production of GOD by A. niger in a bioreactor very well. The pellet density was identified as a key parameter which has to be provided as input parameter for the simulation runs. A sensitivity analysis with three different classes of pellet sizes showed that the influence of the final pellet size distribution on the simulation result was neglectable. On the other hand, it can be shown that the total pellet number, and thus the average pellet size was an important parameter. The pellet number influenced mainly the pellet surface area and, consequently, the turnover of substrate and formation and excretion of the product.     

An analysis of a nonisothermal one-component sorption in a single adsorbent particle—a simplified model

In this paper on approximation is described making it possible to calculate temporal temperature and adsorbed amount profiles during an adsorption process in a single pellet. The suggested model includes heat transfer in an external film and mass transfer inside a porous structure. The profiles calculated from the approximation and rigorous model are compared and it is shown that for a majority of operating systems the simplified model can be used and approximates very well the results obtained from an exact model.     

Competitive adsorption of two dissolved organics onto activated carbon—III: Adsorption kinetics in fixed beds

Adsorption breakthrough curves for bisolute systems of dissolved organics on activated carbon are measured in fixed beds.Results for strongly adsorbable species indicate that at low liquid concentrations (X<0.1 mmol/l.) only external mass transfer resistance is rate determining.However, at higher liquid concentrations internal mass transfer becomes increasingly significant. Breakthrough behaviour is predicted using alternatively three different models with different assumptions about diffusion in the liquid filled pores and diffusion on the surface in series with external film diffusion.Multi-solute adsorption equilibria are predicted from single-solute data using the ideal adsorbed solution theory developed by Myers and Prausnitz, while the single-solute equilibria are represented by Freundlich isotherms. The external mass transfer coefficient for each component is calculated by a general correlation for heat and mass transfer in fixed beds. The internal diffusion coefficient for each component is determined in batch reactor tests with the single-solute system.Systematic deviations between measured breakthrough curves and those calculated from different models using only single-solute data are observed in all experiments with mixed solutes if there is significant internal diffusional resistance and marked displacement of one component inside the carbon particles. The deviations may be due to mutual interference of diffusing molecules. A better agreement between calculated and observed breakthrough curves can be obtained using an extended model in which mixture data are required.     

EFFECT OF GAS-SIDE RESISTANCE ON STEADY STATE MULTIPLICITY IN GAS-LIQUID CSTRs

The effect of gas-side resistance to mass transfer on the existence and uniqueness of steady states in gas-liquid CSTRs is analyzed. It is shown that there exists an upper bound on gas-side resistance beyond which multiplicity cannot arise.     

余热制冷吸附床的一个新传热传质模型

在既考虑吸附剂颗粒内部的传质阻力,又考虑吸附剂颗粒之间的外部传质阻力的前提下,建立了余热吸附制冷系统中吸附床的三维非平衡传热传质数学模型,它包括加热/冷却流体传热模型、换热管传热模型、肋片传热模型、吸附剂颗粒床传热传质模型4部分;通过对模型的数值求解,获得了吸附床内部的温度演变曲线和性能指标,与实验结果基本吻合。     

A criterion of isothermality and determination of reaction boundary for pore-mouth poisoned catalyst pellet

A criterion of isothermality is derived for the inner core of a catalyst pellet deactivated by pore-mouth poisoning using the pseudo steady-state shell-progressive model. If the criterion is met, the core of the deactivated catalyst is at least as isothermal as the fresh catalyst. Knowledge of activity factor mass Biot number and the reaction boundary is required for the use of the criterion. Activity factor can be determined from the measurements of external temperature difference. For the determination of the reaction boundary, an alternate expression is derived for the activity factor for the case of reaction dominated by diffusional resistance. Determination of the reaction boundary requires knowledge of an observable, the Biot numbers and measurements of external temperature difference. Results are applied to some literature data.     

环柱形颗粒填充床传热参数

在一可同时测定轴向与径向温度分布的壁冷式固定床实验装置上 ,通过实验测定了球形、圆柱形和环柱形填充颗粒的径向有效导热系数和壁给热系数 .采用计及内、外环外表面积的颗粒等比表面积当量直径 ,将这些传热参数与Reynolds数相关联 .     

THE ROLE OF LATTICE OXYGEN IN THE DYNAMIC BEHAVIOR OF OXIDE CATALYSTS

The lattice of an oxide catalyst used for oxidation reactions can act as a reservoir for oxygen, storing and releasing it for reactions at the catalyst surface under appropriate conditions. The implication of this oxygen storage property of an oxide catalyst on its dynamic response characteristics has been investigated through an experimental study of 2-butene oxidation over vanadium oxide as a model reaction. Isothermal reaction rate measurements in a differential reactor and nonisothermal studies in a single pellet reactor have been carried out. Following a step increase in the feed butene concentration, isothermal reaction rate overshoot and pellet temperature overshoot were observed. These observations could be modelled in a qualitatively correct way by a very simple model accounting for the participation of lattice oxygen in the catalytic reactions under dynamic conditions. It is demonstrated through model simulations that the ignition characteristics of a catalyst pellet are significantly affected by the participation of the lattice oxygen, when steady state multiplicity is present.     

DRYING KINETICS OF APRICOTS

The kinetics of apricot dehydration was examined using a factorial experimental design in order to evalute the effect of air temperature, air velocity and pretreatment of the sample on the drying time and the transport coefficients of apricots. A finite element method was used to solve the differential equations describing the transfer of moisture in the irregular shape of the apricot. The two transport coefficients of the system were estimated using non-linear regression analysis. It was found that the moisture transfer is entirely controlled by the external resistance to mass transfer for the air velocities examined. A new method is proposed for the evaluation of the relative significance of the external and the internal mass transfer during, which should repiace the well known Biot number criterion.     

Mathematical Model for Batch Drying in an Inert Medium Fluidized Bed

A modified three-phase model is proposed for batch drying of fine powders in an inert medium fluidized bed. The overall heat and mass transfer coefficients between the interstitial gas and solid phases have been determined by the proposed surface-stripping model in which the Biot number is a governing parameter. The effects of gas velocity, inlet gas temperature and mass ratio of starch to inert particles on the drying characteristics of starch in a 0.083 m ID × 0.80 m high medium fluidized bed have been determined. Based on the proposed model, the internal resistance of mass transfer at the powder is equal to the external resistance. The model predicts well the bed temperature, humidity of outlet gas, moisture content of solid particles, heat and mass transfer in an inert medium fluidized bed.