ANALYTICAL AND APPROXIMATE SOLUTIONS OF SELECTED CHEMICAL ENGINEERING MODELS

Abstract

In chemical engineering, mathematical models of various processes/systems often appear in the guise of different types of equations, e.g. algebraic equations [AEs], ordinary differential equations [ODEs] and partial- differential equations [PDEs]. For instance, AEs arise in the steady state modeling of lumped parameter systems, such as CSTR, tray in a separation column and evaporators. Several constitutive models are also represented by AEs, e.g. thermodynamic equations of state and friction factor equations. ODEs are ensued in the unsteady state modeling of lumped parameter systems, e.g. CSTR or series of CSTRs, or in the steady state modeling of distributed parameter systems, e.g. PFR, packed bed columns/reactors. Also, steady state modeling of various processes, such as reaction-diffusion process inside a porous catalyst and heat transfer process from a fin, give rise to the formation of ODEs. PDEs may appear in the steady or unsteady state modeling of various processes/systems, such as unsteady reaction-convection-diffusion process, unsteady reaction-diffusion process, unsteady operation of a PFR, steady or unsteady operation of packed bed columns and steady state boundary layer flows. In general, these model equations are nonlinear and since analytical/approximate solutions of these nonlinear equations are difficult and sometimes even impossible to obtain, numerical methods are used to solve them. However, due to their obvious advantages over numerical solutions, analytical/approximate solutions are preferred and thus are explored first. The present research work is basically an attempt in this direction and is concerned with the analytical/approximate solutions of several selected nonlinear model equations, which arise in the modeling of various chemical engineering processes and systems, e.g. heat transfer processes, fluid flow process, reaction-diffusion processes, rotary kiln and tubular chemical reactor. Besides, two constitutive equations, i.e. thermodynamic equation of state and friction factor equation, have also been considered for obtaining their approximate solutions. These selected model equations are mathematically represented by nonlinear AEs and nonlinear ODEs, and from the critical survey of the literature pertaining to the solutions of these equations, it is revealed that the analytical solutions exist for a few of them only. Moreover, these analytical solutions are merely applicable to a few specific situations. Similarly, the approximate solutions existing for some of the processes/systems are valid in a restricted range only. Besides, use of some recently developed efficient approximate methods, e.g. Adomian Decomposition Method [ADM], Homotopy Analysis Method [HAM] and their variants, for obtaining the approximate solutions of some of the processes/systems has not been fully explored. Due to the nonlinear nature of the model equations, only some of them could be solved analytically, and for obtaining the analytical solutions, several well known methods, e.g. separation of variables in combination with the partial fraction decomposition method and derivative substitution method, have been used. On the other hand, model equations, which could not be solved analytically, have been solved in an approximate manner by using ADM, HAM, and/or their variants. In particular, the selected heat transfer processes include transient convective cooling of a lumped spherical body, transient convective-radiative cooling of a lumped spherical body, steady state heat conduction in a metallic rod, radiative heat transfer from a rectangular fin and convective heat transfer from a rectangular fin. The model equations of these processes are represented by nonlinear first and second order ODEs constituting IVPs and BVPs, respectively, and have been solved by using the well known separation of variables method in conjunction with the partial fraction decomposition method and derivative substitution method. For all of these model equations, the obtained analytical solutions have been verified with the corresponding numerical solutions. Besides, the limitations of existing approximate solutions, which have been found to be valid in a restricted range of parameters' values, have also been highlighted by comparing them with the respective analytical solutions. For one of the processes, i.e. the transient convective-radiative cooling of a lumped spherical body, an available experimental study has also been successfully simulated. Likewise, due to the general nature of the model equation of convective heat transfer from a rectangular fin, the criteria of existence, uniqueness/multiplicity and stability of the solutions have also been studied. The model equation of rotary kiln, considered in this work, describes the bed depth profile of solids flowing in the kiln and is represented by a nonlinear first order ODE constituting an IVP. This equation has been solved by using the separation of variables method in conjunction with the partial fraction decomposition method and the obtained analytical solution has been successfully validated with the numerical solution. Effects of various parameters have been studied in detail and it is shown that the present ii analytical solution is accurate in the entire range of E [local fill angle of the solids] unlike the existing approximate solution, which is accurate only in the range, 20.05 deg < e < 65.89 deg. Usefulness of the derived analytical solution has been shown by successfully simulating some of the existing experimental results pertaining to the solid bed depth profile. The model equation selected from the area of fluid flow describes the Poiseuille and Couette-Foiseuille flow of a third grade fluid between two parallel plates, and is given by a nonlinear second order ODE constituting a BVP. This model equation has been solved by using the derivative substitution method, and the analytical solutions of velocity profiles and flow rate have been obtained. Beside the successful validation of analytical solutions, a discussion regarding the effects of various parameters has also been presented. Limitations of the available approximate solutions have also been highlighted by comparing them with the analytical and numerical solutions. It is shown that the velocity profiles obtained by using the existing approximate solutions depict an opposite trend and starts deviating from the true profiles even for moderately higher values of /i a dimensionless parameter dependent on the material moduli. Thermodynamic equation of state is represented by a nonlinear AE and is concerned with the estimation of gas volume at a given pressure and temperature.. Approximate solutions of this equation have been obtained for finding the gas volume by using ADM and one of its variants, i.e. Restarted Adomian Decomposition Method [RADM]. Advantages and limitations of these two methods have been highlighted and the limitations can be avoided by following the proposed guiding principles. The friction factor equation is also expressed by AE and is used to find the friction factor for the laminar and turbulent flow of fluids in smooth pipes. In this work, the fluid considered is a Bingham fluid. This equation has also been solved by using ADM and RADM, and several explicit approximate solutions of friction factor with reasonable accuracy have been derived. These solutions have been successfully compared with the corresponding numerical solution as well as with the available explicit correlations. For turbulent regime, the derived RADM solution of friction factor exhibit an error less than 0.005%, which is smaller than that exhibited by the available correlations. Similarly for laminar regime, the error in RADM solution of friction factor 111 is found to be within ± 5.2 % error, which can further be reduced by considering more terms in the RADM solution. The model equations of reaction-diffusion processes-inside a porous catalyst slab and sphere describe the concentration profile of reactant, and are used in evaluating the effectiveness factor. These equations are represented by second order ODEs constituting BVPs, and have been solved by using ADM and RADM. It is revealed that the ADM solutions yield erroneous results for reaction order n < 1 [n # 0] and for higher Thiele modulus [c > 2]. These limitations can be avoided by using RADM, which not only yields accurate results but is also applicable to other forms of reaction kinetics. The model equation of reaction-diffusion process inside a porous catalyst sphere has also been successfully solved by using another efficient approximate method based on HAM, i.e. Optimal Homotopy Analysis Method [OHAM]. It is shown that due to the presence of convergence control parameter, the error in OHAM solutions can be minimized, which renders the OHAM solutions better than the other existing approximate solutions. Moreover, the approximate solutions obtained by using ADM or its variants can also be obtained by using OHAM as a special case. The axial dispersion model equation of a tubular chemical reactor describes the concentration profile existing in the reactor and is represented by a second order ODE constituting BVP. This equation has also been solved in an approximate manner by using OHAM and the obtained OHAM solutions have been verified with the corresponding numerical solutions. Besides, utility of OHAM has been depicted by trapping multiple solutions which exist for the non-monotonic reaction kinetics. It is our view that the analytical and approximate solutions of selected process models, obtained in the present thesis, will be useful in many ways, i.e. for simulating experimental studies and for the estimation of parameters. Use of techniques and methods adopted for solving AEs and ODEs can also be made to other process models. We feel that if such an attempt is made it would certainly prove to be advantageous in terms of better understanding of the process and also in ease of obtaining solutions.