A dynamic model for plug flow reactor state profiles

A dynamic model for state profiles of a plug flow reactor is developed, including multiple fluid and solid phases. The model is based on conservation of reactor state profile moments along the spatial dimension of the reactor. These moments are transformed analytically into a polynomial approximation at each time step. The method is flexible, and low as well as high order numerical schemes are resulted in by appropriate choice of parameters. A significant advantage of the present method is that boundary conditions of the partial differential equation reactor model are implicitly satisfied via the moment transformation, whereas the polynomial profile in the numerical solution does not have to be forced to satisfy the boundary conditions. The method is tested numerically against analytical solutions in three numerically challenging benchmark cases: prediction of breakthrough curve in packed bed adsorbers; simulation of chromatographic separation; and feeding a step impulse in a plug flow dimerization reactor. It is shown that the high resolution methods result in considerably smaller numerical errors than a simple low-order assumption of piecewise continuous solution.     

Catalyst mixing in bubble column slurry reactors

The reported experimental data of Pandit and Joshi (1984) on axial and radial steady-state catalyst concentration in a semibatch bubble column slurry reactor is interpreted by the dispersion model. The elliptic partial differential equation with its associated boundary conditions is solved analytically for catalyst concentration by the method of separation of variables. The proposed model adequately fits the experimental data.     

BOUNDARY CONDITIONS FOR A TUBULAR NONISOTHERMAL NONADIABATIC PACKED BED REACTOR WITH WALL HEAT TRANSFER IN THE FORE AND AFTER SECTIONS

An analysis of the boundary conditions for a nonadiabatic steady-state flow reactor with axial dispersion is presented. Conclusions regarding reactor behavior are reached as a result of solution of six differential equations for the reaction section, fore and after sections. Axial dispersion and thermal conductivity as well as wall heat transfer occur in the fore and after sections. It is shown that the new boundary conditions exert only a weak effect on the shape of temperature and concentration profiles and that the classical Danckwerts boundary conditions represent a good approximation.     

BOUNDARY CONDITIONS FOR A TUBULAR NONISOTHERMAL NONADIABATIC PACKED BED REACTOR WITH WALL HEAT TRANSFER IN THE FORE AND AFTER SECTIONS

An analysis of the boundary conditions for a nonadiabatic steady-state flow reactor with axial dispersion is presented. Conclusions regarding reactor behavior are reached as a result of solution of six differential equations for the reaction section, fore and after sections. Axial dispersion and thermal conductivity as well as wall heat transfer occur in the fore and after sections. It is shown that the new boundary conditions exert only a weak effect on the shape of temperature and concentration profiles and that the classical Danckwerts boundary conditions represent a good approximation.     

A technique for analysing integral rate data in non-ideal flow reactors to estimate unknown parameters

A technique is described for analysing integral rate data taken from a non-ideal flow chemical reactor characterised by the axial dispersion model, using a weighted residual approach. This approach avoids the formidable computational difficulties involved with the solution of the boundary value problem associated with the model differential equations. The measured concentration profile is approximated by a series of orthogonal polynomials. Model parameters are estimated by specifying that the fitting polynomial satisfy certain moment relations derived from the mass balance equation and boundary conditions. If the form of the rate equation is known, a priori, the estimation of the rate constant and the dispersion coefficient becomes explicit. In the more general case, in which the reaction order is also unknown, the estimation problem reduces to a uni-dimensional search for the reaction order parameter (or the adsorption parameter in the case of complex kinetics). The method is applied to both linear and non-linear systems and conclusions are drawn regarding the sensitivity of the chemical kinetic and transport parameters to experimental error.     

The application of flow dispersion models to the FM01-LC laboratory filter-press reactor

The flow distribution in the rectangular channel of a laboratory filter-press electrochemical reactor was evaluated using three flow models namely: (a) axial dispersion, (b) sum of two phases and (c) fast and stagnant zones. In the case of the axial-dispersion model, several methods have been used to calculate the Peclet number; the moment method, the non-linear least-squares and the Laplace transform technique. Several boundary conditions, involving different physical and experimental assumptions of the flow were used to solve the partial differential equation that describes the flow behaviour. A total of nine expressions to examine flow dispersion has been used. The comparison of experimental and predicted response signals was made by evaluating the root mean squared error. A data fit in real time has been found to be a better choice as solutions based on the evaluation of moments are prone to error due the overweight of the signal at long times. Data fitting in the Laplace plane is very accurate but it does not guarantee a good fit in real time. Models based on the sum of a fast and a slow or stagnant phase resulted in solutions having very low values of the extension of the slow and stagnant phases, the assumption of a single phase with some degree of dispersion was considered more appropriate.     

Comparison of two methods of optimal control synthesis: Partial differential equation approach and integral equation approach

This study shows how the optimal control theory for distributed parameter systems can be implemented for a problem of tubular reactor with axial dispersion described by partial differential equations. Two methods are implemented. One is based on differential equation approach and the other is based on integral equation approach. It was found that the approach with partial differential equations is preferable to the one with integral equations for the type of problems treated in this study. Computation algorithms and programs for both cases are developed.     

Incorporating Danckwerts’ boundary conditions into the solution of the stochastic differential equation

The one-dimensional dispersion model has been solved analytically as well as numerically to describe flow in continuous “closed” boundary systems using the celebrated Danckwerts boundary conditions. Nevertheless, a continuous state stochastic approach can sometimes be more appropriate especially in cases when input fluctuations are of the same order as the time scale of the system and in such cases an accurate treatment of the boundary conditions is indispensable for the successful application of the method. A deterministic approach was carried out in which the differential equation was solved using Fourier's method and the Laplace transform. These solutions were used as a yardstick to assess the precision of the stochastic solution with its proposed boundary conditions conforming to Danckwerts’ boundary conditions. Our problem is somehow simplified if we assume that the convection term and the dispersion term are constants independent of space and time. A stochastic differential equation was thus employed, governed by the Wiener process and solved using the Euler-Maruyama method.     

Simulation of discontinuous population balance equation with integral constraint of growth rate expression

An effective method based on the concept of continuous characteristics is developed to solve the continuous population equation with integral constraint of growth rate expression. This method can also be extended to solve a general form of a first order partial differential equation. A typical example of a class II MSMPR crystallization process at transient state is modelled and analyzed. The system which possesses originally a discontinuous population density function is transformed into a continuous one by appropriate treatment of the initial condition. The partial differential equation of the continuous population density function is solved by the shifted Legendre polynomials approximation and moments method simultaneously. The original discontinuous population density function is then transformed back from the calculated continuous one by the system characteristics. Very satisfactory computational results are obtained.     

A solution to the distributed parameter model of a continuous grinding mill at steady state

A distributed parameter model of an open-circuit size reduction device at steady state is considered. The model is applicable to tumbling or vibratory mills. Material transport is described in terms of a size-dependent axial convective velocity and a size-dependent axial dispersion coefficient. For n size fractions, the model consists of a set of n coupled ordinary differential equations of second order with two-point boundary conditions. An explicit solution to the model equations is derived for the case in which these equations are linear with constant coefficients. This corresponds to the circumstance in which the axial convective velocities, axial dispersion coefficients and the quantities used to describe the size reduction process are independent of the axial position or of the dependence of the material holdup or size distribution on axial position in the device. These assumptions are not too restrictive and the solution appears to be a useful approximation for a broad range of material-mill combinations. Applications to other process unit operations are suggested.     

Simulation of packed-bed separation processes using orthogonal collocation

The two-point boundary value problem resulting from the heat and material balance equations of a packed separation column are solved using polynomial approximation techniques. The model equations are based on the two-film theory of mass transfer. The resulting partial differential equations are first reduced to ordinary differential equations and then integrated using semi-implicit Runge-Kutta method of integration. Application of orthogonal collocation simplifies the solution of the two-point boundary value problem. For the examples studied, the algorithm is found to converge rapidly with respect to the number of collocation points used in the polynomial approximation.     

Modeling and simulation of a small-scale trickle bed reactor for sugar hydrogenation

Laboratory-scale trickle bed reactor was modeled and simulated, taking into account axial dispersion, gas–liquid, liquid–solid and internal mass transfer as well as catalyst deactivation under isothermal conditions. For catalyst particles dynamic and steady state models were developed, including both mass and heat balances. Catalyst deactivation was included in the model by using the final activity concept for the catalyst particles. A well-working numerical algorithm (method of lines) was applied for solving the reactor model with Matlab 7.1 and the results followed experimental trends very well. The steady-state reactor model was based on simultaneous solution of mass balances. The aim was to illustrate how these parabolic partial differential equations could be solved with a step-by-step calculation for a selected geometry. The final model verification was done against experimental data from the hydrogenation of arabinose to arabitol on a ruthenium catalyst.     

Heat and mass transfer in packed beds—IV: Use of weighted moments to determine axial dispersion coefficients

The method of weighted moments is analysed on the basis of measured axial dispersion in a methanehydrogen mixture flowing through a bed of glass beads at various velocities. The optimum value of the Laplace parameter S is chosen as the one giving the minimum deviation between experiment and model in the integrand of the second weighted moment. The corresponding axial dispersion coefficients are lower than those determined with ordinary moments, and agree very closely with the best-fit values in the time domain. An equation proposed by Anderssen and White based on other considerations is shown to give a relatively good approximation of the optimum value of S. With the aid of this near-optimum S, calculating time for weighted moments can be shorter than for curve fitting in the time domain.     

用单纯形法同时确定轴向混合系数与传质系数

在内径5cm的梯形波空气脉冲柱内,测定了30%TBP(煤油)-Th(NO_3)_4-HNO_3(H_2O)体系在两种板段结构条件下钍的浓度剖面.用扩散模型描述萃取过程,用单纯形法直接由浓度剖面同时确定轴向混合系数、传质系数与真实传质单元高度.由此获得的计算的浓度剖面与实验测定的浓度剖面比较符合.结果表明:梯形波脉冲柱的轴向混合系数较小,用于补偿轴向混合的柱高约占表观传质单元高度的32—44%;用单纯形法寻优,对原始微分方程采用差分近似得两组线性代数方程组,对每组线性代数方程组用追赶法直接解,两组方程之间用迭代法,程序简单,收敛较快.在PDP11/23小型计算机上,约1分钟即算得一组结果.     

Model predictive control of a second-order hyperbolic transport-reaction process

The model predictive controller (MPC) design is developed for a tubular chemical reactor, considering a second-order hyperbolic partial differential equation as the model of the transport-reaction process with boundary actuation. Without loss of generality, closed–closed boundary conditions and relaxed total flux are assumed. At the same time, the model is discretized in time by the Cayley–Tustin method, and, under the assumption that only the reactor's output is measurable, the observer design for the state reconstruction is addressed and integrated with the MPC design. The Luenberger observer gain is obtained by solving the operator Ricatti equation in the discrete-time setting, while the MPC accounts for constrained and optimal control. The simulations show that the output-based MPC design stabilizes the system under the input and output constraints satisfaction. In addition, to address the models' disparities, the results for both parabolic and hyperbolic equations are presented and discussed.     

Function space methods for analysis of nonadiabatic tubular reactors with axial mixing-I. Uniqueness criteria of the steady state and computation of the steady state profiles

Use is made of abstract function spaces to investigate the uniqueness of the steady state of nonadiabatic tubular reactors with axial mixing. For the study, the system differential equations are first transformed into integral equations. The properties of the integral operators are then investigated in the space of square integrable functions in order to obtain sufficient conditions for the uniqueness of solution of the system equations. A uniqueness criterion is obtained by use of the Contraction Mapping theorem, which permits one to compute the steady state profiles of reactor temperature and concentration by means of successive iteration with the transformed integral system equations. The computation involved here is quite simple compared to the numerical solution of the system differential equations with split boundary conditions. Numerical examples are given to investigate the effects of various system parameters on the steady state profiles of reactor temperature and concentration.     

The operation and modelling of a periodic,countercurrent, solid—liquid reactor

Several models are proposed for the Cloete—Streat stage-wise solid—liquid reactor in which the solids are periodically transferred from stage to stage countercurrent to the net flow of liquid. The behaviour predicted by the models is compared with experimental data on an ion exchange reaction system. The results illustrate a range of operating conditions for which unsteady-state operation (with periodic solid transfer) is superior to continuous steady-state operation.The various models differ in the treatment of the composition distribution of the ion exchange resin. They include (a) a discretisation of the distribution; (b) use of the continuous distribution function and solution of the resulting hyperbolic partial differential equations; and (c) approximation of the state of the resin by the leading moments of the distribution.     

Liquid distribution and redistribution in packed columns-I Theoretical

When liquid is uniformly distributed at the top of a packed column it is found that there is a preferential flow to the wall, and early theoretical work has suggested that the observed radial velocity is proportional to the radial gradient of axial velocity. A set of consistent boundary conditions has not been found. In this paper the experimental observation of preferential liquid flow is interpreted as a difference in permeability between the wall and bulk region of packing, and the existence of a potential for liquid redistribution is inferred from an examination of experimental work on two-phase flow in porous media, and of the internal consistency of the early relation between radial velocity and the radial gradient of axial velocity. The existence of a potential for liquid redistribution, and a difference in permeability between the wall and bulk regions are shown to lead to a differential equation describing liquid distribution in the packing, defined by consistent and well-posed boundary conditions that are determined from the physical analysis.The solution to the partial differential equation describing redistribution from an initial axisymmetric distribution is given.     

BEHAVIOUR OF AN ADIABATIC PACKED BED REACTOR PART 2: MODELLING

The steady state and dynamic behaviour of an adiabatic packed bed reactor for the selective hydrogenation of mixtures of ethyne and ethene is studied. A heterogeneous model with axial dispersion of heat is solved numerically by means of a fully implicit discretisation scheme. From an analysis of the inlet and outlet boundary conditions, it follows that in order to avoid an influence of the type of boundary condition on the reactor behaviour, sufficiently long inert zones before and after the active bed should be present. Using preliminary kinetic expressions adapted from Mcn'shchikov ct al. (1975), the model calculations show a reasonable agreement with the experiments performed in a laboratory scale reactor and demonstrate the importance influence of small amounts of carbon monoxide both on the steady state as well as on the transient behaviour of the reactor. However, using the rate expressions derived from our own kinetic experiments in a Berty reactor, the behaviour under runaway conditions and the transition from non-runaway to runaway conditions can not be described well. Several factors that contribute to this discrepancy are discussed. This work shows that, in particular for a complex reaction system such as the selective hydrogenation of ethyne in ethene, the limited accuracy of the kinetic model and kinetic parameters dominates the precision attainable with packed bed reactor models.