This thesis concerns the development and application of wavelet methods for the numerical solution of two key classes of problems in process engineering. The primary aim is to develop an advanced numerical framework to solve the models for population balance systems and transient chemical processes with moving steep gradients. A standard wavelet-based collocation method and an adaptive wavelet-based collocation method are developed and applied to these problems.
The first area of application relates to transient chemical processes with moving steep gradients. Multiphase systems and chemical reacting flows are the basis of several important applications in chemical engineering. They often lead to field quantities that develop sharp gradients and highly localized phenomena. Several features of the dynamics of these processes make the modelling solution particularly demanding. These include strong nonlinearities, coupling effects, mass transfer resistance and fluid-dynamic dispersion phenomena.
The second area relates to population balance systems. There are many systems both in nature and in industrial practice whose natural modelling framework leads to a population balance equation (PBE), which is described by a nonlinear partial integro-differential equation including integrals over the entire particle-size spectrum. The nonlinear behaviour due to complex growth, nucleation, agglomeration and breakage mechanisms, and the distributed nature of population balances make the equations difficult to solve.
In addition, numerical solutions for both these problem classes constitute an important first step in model reduction and controller design. Tools for accurate and fast numerical solution play an important role in bridging realistic model development with model-based control techniques.
Wavelet analysis is a relatively new numerical concept that allows one to represent a function in term of a set of base functions, called wavelets. Wavelets have many good properties, such as localization both in space (time) and frequency. They also have compact support, possess hierarchical structures and treat integral terms easily. Compactly supported wavelets are localized in space, which means that the solution can be refined in regions of high gradient without having to regenerate the mesh for the entire problem. Due to the good properties of wavelets, the induced operational matrices are sparse. This significantly reduces the number of function evaluations required to generate the Jacobian and accelerates integration. Nonlinear operators in process models, which are already in the physical space, can be handled easily. Therefore no extra computation is required for the passage between wavelet coefficients and the physical space.
The two numerical methods based on wavelet collocation have been developed for the solution of models for the problems previously described. The wavelet-collocation approaches convert these problems to the solution of a system of algebraic and differential equations for the unknown wavelet series coefficients. In practice, we usually group the terms in the engineering models and develop operational matrices for some typical operators such as derivative terms, integral terms and other nonlinear terms. The operational matrices in the wavelet domain can be calculated in advance.
A standard wavelet-based collocation method was developed as a general and fundamental approach to the solution of these problems. The level of resolution is an important numerical parameter. It relates to the accuracy and oscillation of the solution. It makes the algorithms particularly useful for developing hierarchical solutions for chemical processes. At an early modelling stage, we can use low order to obtain preliminary information about the solution with modest computing time. We can use the high-order approximations to get an accurate solution when necessary.
A more powerful adaptive wavelet-based collocation method was developed. The interpolating wavelet transform (IWT) and sparse point representation (SPR) were used to reduce the number of collocation points without significant loss of accuracy. The grid is adjusted according to the singularity of the solution based on IWT and SPR. The solution is approximated in an optimal way, which is capable of taking advantage of the local smoothness of the solution using a coarse grid in the largest part of the domain, but capable on the other hand of coping with any singularity. Therefore, although we choose a higher level of resolution, the number of collocation points does not increase dramatically. The solution times for the adaptive wavelet-based method are 1/3 to 1/6 of those of the uniform grid method.
A new strategy is developed to select the optimal level of resolution based on process characteristics. In practice, we usually know some information about the singularity of the process, for example, the most irregular distribution in a granulation process. This saves significant computation time and memory.
The proposed techniques have been demonstrated for some typical case studies involving an axial dispersion model, a diffusion-reaction model and population balances with nucleation, growth, agglomeration and breakage. In all cases the numerical results agree very well with known solutions. The presence of moving sharp fronts can be addressed without the prior investigation of the characteristics of the processes. For all case studies, the algorithms are stable and fast. The numerical framework is a general and direct tool for the solution of these models in chemical engineering. The most important advantage of this numerical framework is that it solves the models of population systems and chemical processes with steep fronts more effectively and conveniently than conventional numerical methods.