Jethro Akroyd

Stochastic solution of population balance equations for reactor networks

This work presents a sequential modular approach to solve a generic network of reactors with a population balance model using a stochastic numerical method. Full-coupling to the gas-phase is achieved through operator-splitting. The convergence of the stochastic particle algorithm in test networks is evaluated as a function of network size, recycle fraction and numerical parameters. These test cases are used to identify methods through which systematic and statistical error may be reduced, including by use of stochastic weighted algorithms. The optimal algorithm was subsequently used to solve a one-dimensional example of silicon nanoparticle synthesis using a multivariate particle model. This example demonstrated the power of stochastic methods in resolving particle structure by investigating the transient and spatial evolution of primary polydispersity, degree of sintering and TEM-style images. An algorithm is presented to solve reactor networks with a population balance model.A stochastic method is used to solve the population balance equations.The convergence and efficiency of the reported algorithms are evaluated.The algorithm is applied to simulate silicon nanoparticle synthesis in a 1D reactor.Particle structure is reported as a function of reactor length and time.

First-Principles Thermochemistry for the Thermal Decomposition of Titanium Tetraisopropoxide.

The thermal decomposition of titanium tetraisopropoxide (TTIP) is investigated using quantum chemistry, statistical thermodynamics, and equilibrium composition analysis. A set of 981 Ti-containing candidate species are proposed systematically on the basis of the thermal breakage of bonds within a TTIP molecule. The ground state geometry, vibrational frequencies and hindrance potentials are calculated for each species at the B97-1/6-311+G(d,p) level of theory. Thermochemical data are computed by applying statistical thermodynamics and, if unknown, the standard enthalpy of formation is estimated using balanced reactions. Equilibrium composition calculations are performed under typical combustion conditions for premixed flames. The thermodynamically stable decomposition products for different fuel mixtures are identified. A strong positive correlation is found between the mole fractions of Ti species containing carbon and the TTIP precursor concentration.

A moment projection method for population balance dynamics with a shrinkage term

A new method of moments for solving the population balance equation is developed and presented. The moment projection method (MPM) is numerically simple and easy to implement and attempts to address the challenge of particle shrinkage due to processes such as oxidation, evaporation or dissolution. It directly solves the moment transport equation for the moments and tracks the number of the smallest particles using the algorithm by Blumstein and Wheeler (1973) 41. The performance of the new method is measured against the method of moments (MOM) and the hybrid method of moments (HMOM). The results suggest that MPM performs much better than MOM and HMOM where shrinkage is dominant. The new method predicts mean quantities which are almost as accurate as a high-precision stochastic method calculated using the established direct simulation algorithm (DSA).

Ab initio Variational Transition State Theory and Master Equation Study of the Reaction (OH)3SiOCH2 + CH3 ⇌ (OH)3SiOC2H5

Abstract In this paper we use variable reaction coordinate variational transition state theory (VRC-TST) to calculate the reaction rate constants for the two reactions, R1: (OH)3SiOCH2 + CH3 ⇌ (OH)3SiOC2H5, and R2: CH2OH + CH3 ⇌ C2H5OH. The first reaction is an important channel during the thermal decomposition of tetraethoxysilane (TEOS), and its rate coefficient is the main focus of this work. The second reaction is analogous to the first and is used as a basis for comparison. The interaction energies are obtained on-the-fly at the CASPT2(2e,2o)/cc-pVDZ level of theory. A one-dimensional correction to the sampled energies was introduced to account for the energetic effects of geometry relaxation along the reaction path. The computed, high-pressure rate coefficients were calculated to be, R1: k1 = 2.406 × 10−10T−0.301 exp (− 271.4/T) cm3 molecule–1 s–1 and R2: k2 = 1.316 × 10−10T−0.189 exp (− 256.5/T) cm3 molecule–1 s–1. These rates differ from each other by only 10% – 30% over the temperature range 300–2000 K. A comparison of the computed rates with experimental data shows good agreement and an improvement over previous results. The pressure dependency of the reaction R1 is explored by solving a master equation using helium as a bath gas. The results obtained show that the reaction is only weakly pressure dependent over the temperature range 300–1700 K, with the predicted rate constant being within 50% of its high-pressure limit at atmospheric pressure.

The Polarization of Polycyclic Aromatic Hydrocarbons Curved by Pentagon Incorporation: The Role of the Flexoelectric Dipole

Curvature in polyaromatic hydrocarbons (PAHs), due to pentagon inclusion, produces a dipole moment that contributes significantly to self-assembly processes and adsorption at the surface of carbon materials containing curved structures. This work presents electronic structure calculations of the dipole moment for 18 different curved PAH molecules for various numbers of pentagons and the total number of aromatic rings. A significant dipole moment was found that depends strongly on the number of aromatic rings (4–6.5 debye for ring count 10–20). The main cause for the dipole is shown to be the π-electron flexoelectric effect. An atom-centered partial charge representation of the charge distribution in these molecules is insufficient to correctly describe their electrostatic potential; distributed multipoles were required instead.

Extension of moment projection method to the fragmentation process

The method of moments is a simple but efficient method of solving the population balance equation which describes particle dynamics. Recently, the moment projection method (MPM) was proposed and validated for particle inception, coagulation, growth and, more importantly, shrinkage; here the method is extended to include the fragmentation process. The performance of MPM is tested for 13 different test cases for different fragmentation kernels, fragment distribution functions and initial conditions. Comparisons are made with the quadrature method of moments (QMOM), hybrid method of moments (HMOM) and a high-precision stochastic solution calculated using the established direct simulation algorithm (DSA) and advantages of MPM are drawn.

Theoretical Study of the Ti–Cl Bond Cleavage Reaction in TiCl4

Abstract In this work the kinetics of the TiCl4 ⇌ TiCl3 + Cl reaction is studied theoretically. A variable-reaction coordinate transition-state theory (VRC-TST) is used to calculate the high-pressure limit rate coefficients. The interaction energy surface for the VRC-TST step is sampled directly at the CASPT2(6e,4o)/cc-pVDZ level of theory including an approximate treatment of the spin-orbit coupling. The pressure-dependence of the reaction in an argon bath gas is explored using the master equation in conjunction with the optimised VRC-TST transition-state number of states. The collisional energy transfer parameters for the TiCl4–Ar system are estimated via a “one-dimensional minimisation” method and classical trajectories. The Ti–Cl bond dissociation energy is computed using a complete basis set extrapolation technique with cc-pVQZ and cc-pV5Z basis sets. Good quantitative agreement between the estimated rate constants and available literature data is observed. However, the fall-off behaviour of the model results is not seen in the current experimental data. Sensitivity analysis shows that the fall-off effect is insensitive to the choice of model parameters and methods. More experimental work and development of higher-level theoretical methods are needed to further investigate this discrepancy.

Research Data Supporting "Theoretical study of the TiCl4=TiCl3+Cl reaction"

The data come from the simulations performed in the paper. They are divided into four folders: 1) Folder: Quantum-Data A) Subfolder: 2Dplot - Contains energies obtained in Molpro2012 for different positions of the TiCl3 and Cl fragments. This was used to create the 2D contour plot. B) Subfolder: Geoms-Freqs-Energies - Contains input/output files from the Gaussian09 software giving all the information about the species geometries, frequencies and energies. - Contains input/output files from the Molpro2012 software giving all the information about the species improved energies. C) Subfolder: SOC - Contains input/output files from the Molpro2012 software giving spin-orbit-coupling energies for the selected points along the minimum energy pathway. 2) Folder: Gorin-Model - Contains input/output files needed to reproduce all the Gorin Model simulations 3) Folder: VRC-TST - Contains input/output files needed to reproduce VRC-TST calculations 4) Folder: ME - Contains input/output files needed to reproduce Master Equation calculations including the sensitivity analysis as stated in the paper. Readme.txt - Contains detailed description of all the data.