DESIGN OF BATCH DISTILLATION COLUMNS FOR BINARY MIXTURES

In this note analytical equations are presented that describe the separation obtained for binary batch distillation. Formulas are obtained for both the constant distillate composition and constant reflux methods of operation. The results are strictly for distillation with an infinite number of stages ( corresponding to minimum reflux) however, it is shown that the equations derived closely describe the operation with a small finite number of stages.     

On the global and efficient solution of stochastic batch plant design problems

The presence of uncertainty in product demands of batch plant design formulations with fixed structure and continuous equipment sizes transforms them into large-scale nonconvex nonlinear programs. This paper describes recent developments towards the efficient solution of such mathematical models. Two global optimization algorthms, a specialized GOP algorithm and a reduced space branch and bound algorithm, are presented and applied to this class of batch plant design models. It is shown that, by taking advantage of the special structure of the resulting mathematical formulations, encouraging computational results can be obtained from both algorithms for problem sizes that would otherwise be practically unsolvable with conventional global optimization techniques. An efficient, specialized Gaussian quadrature technique is also described for the case of product demands following normal probability distribution functions with which reduced model size and improved estimation of the expected profit integral are achieved. These developments are tested on example problems from the literature covering single batch plant configuration with various scheduling policies and flexible configurations with alternative production sequences.     

A CONCEPT OF BASIC FUNCTIONS IN THE SERIES REPRESENTATIONS OF TRANSPORT PROBLEMS

This paper deals with a concept of closed form basic functions which is necessary in the representation of series solutions of transport problems. These closed form basic solutions allow us to rearrange the series solutions obtained by either separation of variable technique or finite integral transform in such a way that the resulting series solutions converge much faster and also permit derivative evaluations at the boundary. The procedure is presented to a general class of differential equations and is illustrated in various chemical engineering problems.     

Optimal operation of multicomponent batch distillation—multiperiod formulation and solution

A method is proposed to determine optimal multiperiod operation policies for binary and general multicomponent batch distillation of a given feed mixture, with several main products and intermediate off-specification cuts. A two-level optimal control formulation is presented so as to maximize a general profit function for the multiperiod operation, subject to general constraints. The solution of this problem determines the optimal amount of each main and off cut, the optimal duration of each distillation step and the optimal reflux ratio profiles during each production period. The outer level optimization maximizes the profit function by manipulating carefully selected decision variables. These are chosen in such a manner that the need of specifying the mole fractions of all the components in the products, as required by previous methods is avoided. For values of the decision variables fixed by the outer loop, the multiperiod operation is decomposed into a sequence of independent optimal control problems, one for each production step. In the inner loop, a minimum time problem is then solved for each step to generate the optimal reflux ratio values, reflux switching times and duration of the step. The procedure permits the use of very general distillation models described by differential and algebraic equations, including rigorous thermodynamics if desired. The model equations are integrated by using an efficient Gear's type method, the inner loop optimal control problems are solved using a variational method, and all optimisations are solved using a robust and efficient successive quadratic programming code (Chen, Ph.D. Thesis, Imperial College, 1988).

Several example problems (involving binary and multicomponent mixtures) are used to demonstrate the idea and to show the effect of the cost functions used (in particular the value of the main products) on the optimal solutions.     

Transport processes and conversion in isothermal slurry reactors

Transient mass transfer and chemical reaction in isothermal continuous-flow and batch agitated reactors is studied mathematically. Analytical solutions are derived in the form of infinite integrals. The solutions include the effects of fluid-to-particle diffusion, intraparticle diffusion and reversible first-order adsorption for a first-order reaction on the particle surface. The conversion at steady-state is explored in some detail. An efficient numerical scheme for evaluating the infinite integral in the transient solution is given. A large number of calculations are presented, for both the continuous feed and the batch reactor, for the case with negligible adsorption rate resistance. The influences of the different mass transfer resistances on the conversion are clearly demonstrated.     

General mathematical models of transport processes with and without chemical reactions

Based on the surface renewal and penetration concept, a number of fairly general mathematical models for transport systems without chemical reaction or heat generation and transport systems with zero and first order reactions have been developed. Solutions of the governing transport equations are presented for cases in which the number of capacitors of the multiple capacitance age distribution is finite or infinite. Some of their limiting solutions are reduced to those proposed by the previous investigators.     

Boundary value problems in transport with mixed and oblique derivative boundary conditions-II: Reduction to first order systems

Mixed second derivative or oblique derivative boundary conditions for the steady state heat conduction equation with heat generation in the two-dimensional plane are of importance in applications[1]. In general, they lead to non-selfadjoint boundary value problems or to singular integral equations.In this paper, the second order partial differential equations have been decomposed into first order systems, which under suitable circumstances can be conveniently adapted to satisfy the mixed or oblique derivative boundary conditions. Furthermore, the differential operator with respect to one of the variables is shown to be symmetric in its domain, possessing a denumerable set of eigenvalues and a complete set of eigenvectors. The solution to the boundary value problems is obtained by expansion in terms of these eigenvectors.The method works in infinite regions with separable coordinates, where it gives the same solution as that by infinite Fourier transform. It also works on finite regions, when periodic boundary conditions are used such as in right circular cylinders. Solutions are presented for an infinite slab and concentric annulus for both mixed and oblique derivative boundary conditions.     

An analysis of chemical reactor stability and control—XVI: The stochastic stirred pot

A continuous stirred tank reactor subject to random fluctuations in flow rate and inlet temperature was simulated on a hybrid computer. The stochastic response of the reactor about unique stable deterministic steady states was studied as a function of the damping coefficient and response time of the deterministic system and the power spectrum of the stochastic input. It was found that the stochastic response could be classified into categories similar to those used for forced periodic systems according to the relationship between the deterministic system response time and the 90% cut-off frequency of the stachastic input. The nature of the stochastic response is predictable for relatively low frequency inputs but unexpected results may occur at intermediate frequencies. The magnitude of reactor state fluctuations was seen to be dependent on the deterministic damping coefficient. The distribution of reactor states was studied as a function of input process variance and it was found that the distribution can become bimodal even when the associated deterministic steady state is unique.The concept of stochastic stability is discussed and several practical stochastic stability definitions are proposed. The stochastic stability of the random systems was seen to be well described by the stochastic regions of operation predicted by the input process power spectrum and the deterministic system response time. The input variance levels necessary to produce stochastic instability can be estimated in the Quasi Steady region of operation. It was found that exposure of an autonomous limit cycle about a unique unstable deterministic steady state to high frequency random inputs may lead to effective stabilization.     

Fault detection and diagnosis with parametric uncertainty using generalized polynomial chaos

This paper presents a new methodology to identify and diagnose intermittent stochastic faults occurring in a process. A generalized polynomial chaos (gPC) expansion representing the stochastic inputs is employed in combination with the nonlinear mechanistic model of the process to calculate the resulting statistical distribution of measured variables that are used for fault detection and classification. A Galerkin projection based stochastic finite difference analysis is utilized to transform the stochastic mechanistic equation into a coupled deterministic system of equations which is solved numerically to obtain the gPC expansion coefficients. To detect and recognize faults, the probability density functions (PDFs) and joint confidence regions (JCRs) of the measured variables to be used for fault detection are obtained by substituting samples from a random space into the gPC expansions. The method is applied to a two dimensional heat transfer problem with faults consisting of stochastic changes combined with step change variations in the thermal diffusivity and in a boundary condition. The proposed methodology is compared with a Monte Carlo (MC) simulations based approach to illustrate its advantages in terms of computational efficiency as well as accuracy.     

分批精馏最小过渡馏分量及过渡馏分脉冲控制法的研究

研究了分批精馏最小过渡馏分量 ,在此基础上针对分批精馏过渡馏分段的特点采用了全回流 全馏出交替进行的脉冲控制法。该法以动态模拟确定了全回流时间 ,统计法确定了全馏出时间。采用正己烷 环己烷物系进行对比实验结果表明 :与传统的恒回流比操作法相比 ,脉冲操作法可明显减少过渡馏分量、缩短操作时间和提高分离效率     

ANALYTICAL TEMPERATURE DISTRIBUTION FOR MULTIDIMENSIONAL BODIES EXPOSED TO A CONSTANT SURFACE HEAT FLUX

For many decades, solutions for transient temperature distributions in multidimensional objects were determined by combining as a product the solutions of one-dimensional objects necessary to delimit the contour of these multidimensional objects. These product solutions are usually restricted to two types of boundary conditions: a constant wall temperature and a constant heat transfer coefficient

This paper considers the case of an object exposed to a constant surface heat flux. It is shown that when the surface of multidimensional object is submitted to a constant heat flux density, its temperature distribution can be obtained by the simple addition of one-dimensional temperature distributions. Only three one-dimensional solutions are necessary to solve all possible multidimensional problems. These are the solutions for a semi-infinite slab, an infinite plate and an infinite cylinder. The equations describing the temperature profile within each of these one-dimensional objects are presented as well as their graphical representations in a generalized form for rapid determination of temperatures.     

Design and optimisation of batch and semi-batch reactors

This paper addresses a systematic methodology for batch and semi-batch reactor design and optimisation for both ideal and non-ideal mixing. It can be applied to non-isothermal and multiphase systems. The method starts from a general representation in the form of a temporal superstructure based on the similarity of between plug flow reactors and ideal batch reactors. The temporal superstructure of a batch reactor exists in both the space and time dimensions. For non-ideal mixing, this paper addresses a mixing compartment network model to represent mixing inside reactors. The mixing compartment network is then included into the temporal superstructure to model non-ideally mixed batch reactors and the mixing pattern optimised with the other variables. Besides the operation variables for batch reactors, this method can also suggest the optimum mixing pattern and promising reactor configurations for mechanical design. A profile-based approach is proposed to make a search of the profiles for temperature, pressure and feed addition. This approach starts from a set of initial profiles of temperature, pressure and feed addition. Then the performance of the batch reactor is evaluated against the objective function under different profiles. An optimal set of profiles is then found by this profile searching process. A stochastic optimisation technique based on simulated annealing is employed to obtain optimal solutions. This method is also extended to multiphase reaction systems based on the concept of shadow reactor compartments. A number of case studies are presented to illustrate the use of the proposed methodology.     

Method of calculating molecular weight distribution function from gel permeation chromatograms

An integral equation taking account of the limited resolution of the chromatographic columns is given to relate the gel permeation chromatogram and the true molecular weight distribution function. Three approaches to solve the integral equation are described. The first approach provides a special solution for the log-normal molecular weight distribution function; the other two approaches give two numerical solutions for general distribution functions. The use of these solutions in the treatment of gel permeation chromatography data is discussed.     

Equations of an Unsteady-State Laminar Boundary Layer: General Transformations and Exact Solutions

New classes of exact solutions and transformations of the unsteady-state laminar boundary layer equations are described. Particular attention is given to general solutions that depend on arbitrary functions. Newtonian and non-Newtonian liquids characterized by arbitrary relationships between the shear stress and the shear rate are considered. A transformation is presented that preserves the form of the three-dimensional unsteady-state boundary layer equations in an arbitrary orthogonal curvilinear coordinate system. Three-dimensional nonisothermal and diffusion boundary layers are analyzed.     

Finite element solution for gas–solid reactions: Application to the moving boundary problems

Gas–solid reactions are very important in the chemical and metallurgical process industries. Several models described these reactions such as volume reaction model, grain model, and nucleation model. These models give two coupled partial differential equations (CPDEs). In this work an integral transformation and subsequent finite element method is used for solving the coupled partial differential equations of these reactions. In each mesh the Rayleigh–Ritz method is applied. Finally the results of this work are compared with the existing numerical solutions and experimental data successfully.     

Effect of Surfactant Concentration on the Dispersion Coefficient for the Comparison of Model and Experimental Results in Emulsion Polymerization of Butadiene

In this work, a mechanistic model was developed on the basis of population balance equations for butadiene emulsion polymerization and validated with experimental data in a batch reactor. A combination of finite difference and weighted essentially nonconciliatory (WENO) schemes was used as a precise technique for the discretization of the population balance equations. The obtained values during the evolution of conversion and the average particle diameter values were in good agreement with the measured results for different surfactant concentrations. However, the simulated particle-size distributions (PSDs) were much narrower than the experimental PSDs. Therefore, to match the experimental PSDs, a stochastic term was included in the zero–one model. The effect of surfactant concentration, which is an important factor with a direct impact on the dispersion coefficient, was also investigated and confirmed with experimental results.     

Model parameter estimation and feedback control of surface roughness in a sputtering process

This work focuses on model parameter estimation and model-based output feedback control of surface roughness in a sputtering process which involves two surface micro-processes: atom erosion and surface diffusion. This sputtering process is simulated using a kinetic Monte Carlo (kMC) simulation method and its surface height evolution can be adequately described by the stochastic Kuramoto-Sivashinsky equation (KSE), a fourth-order nonlinear stochastic partial differential equation (PDE). First, we estimate the four parameters of the stochastic KSE so that the expected surface roughness profile predicted by the stochastic KSE is close (in a least-square sense) to the profile of the kMC simulation of the same process. To perform this model parameter estimation task, we initially formulate the nonlinear stochastic KSE into a system of infinite nonlinear stochastic ordinary differential equations (ODEs). A finite-dimensional approximation of the stochastic KSE is then constructed that captures the dominant mode contribution to the state and the evolution of the state covariance of the stochastic ODE system is derived. Then, a kMC simulator is used to generate representative surface snapshots during process evolution to obtain values of the state vector of the stochastic ODE system. Subsequently, the state covariance of the stochastic ODE system that corresponds to the sputtering process is computed based on the kMC simulation results. Finally, the model parameters of the nonlinear stochastic KSE are obtained by using least-squares fitting so that the state covariance computed from the stochastic KSE process model matches that computed from kMC simulations. Subsequently, we use appropriate finite-dimensional approximations of the identified stochastic KSE model to design state and output feedback controllers, which are applied to the kMC model of the sputtering process. Extensive closed-loop system simulations demonstrate that the controllers reduce the expected surface roughness by 55% compared to the corresponding values under open-loop operation.     

任意形状电熔玻璃窑流动和传热的二维数值模拟

任意形状电熔玻璃窑流动和传热的二维数值模拟祁建伟（清华大学工程力学系１０００８４）陈越南（浙江大学力学系３１００２７）Ｔｗｏ－ＤｉｍｅｎｓｉｏｎａｌＮｕｍｅｒｉｃａｌＭｏｄｅｌｌｉｎｇｏｆａｎＡｒｂｉｔｒａｒｙＳｈａｐｅｓＥｌｅｃｔｒｉｃＧｌａｓｓＦ...     

多产品间歇化工过程最优设计的研究——启发法

修正了modi等人提出的多产品间歇化工过程最优设计的混合整数非线性规划(MINLP)模型,使之不仅允许中间贮罐的存在,还可调整以同步或异步方式操作的平行单元数.还提出了对以上模型求解的启发算法,其特点是对决策变量的初值要求低,计算速度快,所需内存小,与严格的数学规划法相比较,其相对误差小于1%.以实例说明了方法的有效性.