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Introduction
In June 2008 I retired from UC Davis and now Professor Emeritus of Chemical Engineering at UC Davis. I also hold Visiting Professorships at Hanoi University of Mining and Geology, and Tokyo University of Science. My area of research is the fluid mechanics and stability of thin film coating processes. I was the President of the International Society of Coating Science and Technology (http://www.iscst.com), 2008-2010, and 2017-2018. I am a active user of Mathematica, and was a content developer for WolframAlpha.
Current institution
Additional affiliations
January 1986 - present
Independent Researcher
Position
- Professor Emeritus
Education
September 1975 - June 1980
January 1973 - June 1975
January 1969 - December 1972
Publications
Publications (145)
This text has been written for use in the first course in a typical chemical engineering program. That first course is generally taken after students have completed their studies of calculus and vector analysis, and these subjects are employed throughout this text. Since courses on ordinary differential equations and linear algebra are often taken...
In this study, we consider the drying of a thin film that contains a stable dispersion of colloidal particles so that a coating of these particles is formed after the liquid is driven off by evaporation. For sufficiently thin films, we show that evaporative cooling can drive a Marangoni flow that results in surface deformation of the drying film. A...
The coupled, unsteady Navier-Stokes, convective diffusion, and thermal energy equations that describe spin coating of colloidal suspensions are solved numerically. The theoretical model, absent of any adjustable parameters, is used to explore the effects of angular velocity, initial solvent weight fraction, solvent properties and spin coating proto...
In this report I present an introductory review of the properties and solutions of 1st order ordinary differential equations. In section 2 I introduce the concept of 1-parameter family of level curves which we show in later sections forms the building block for understanding the solution properties of 1st order ODEs. I then review some basic definit...
A collection of fluid mechanics problems (44) with detailed solutions. These problems ( or variants of them) are addressed in a typical upper division or introductory graduate class in fluid mechanics.
This report reviews the seminal work of Fred Trouton when a fluid is subject to a pure extensional flow. We re-examine Trouton's derivation of a profile equation for filament thinning. In particular, we frame our discussion by considering the general formulation of the integrated form of the axial momentum balance for a thinning liquid filament. We...
A successful numerical solution of stiff nonlinear ODEs or PDEs often requires that the user specify a suitable computational grid. In this report we illustrate how one can useMathematica's kernel function to automatically generate the appropriate grid for the computation, especially if the numerical solution exhibits sharp boundary layer structure...
In the analysis of reacting systems, one often has to deal with mass, momentum and energy transfer in a system involving multiple species. In this report we first derive the thermal energy balance for multicomponent systems. We show how to derive the enthalpy version of the energy equation, and illustrate how the energy equation can be applied to C...
In thermodynamics and reaction engineering, the extent of reaction represents the amount of progress made towards equilibrium in a chemical reaction. In this report we will show that the Pivot Theorem given by Higgins & Whitaker (2012) provides all the necessary information on a reacting system to construct the extents of reaction variables. We wil...
In this report we discuss the solution of the following matrix problem:
A · x = λ x, where A is a nxn matrix, which may have complex eigenvalues. Matrix problems of this form occur frequently in the solution of differential equations in the physical sciences and engineering. The scalar λ is called an eigenvalue of A if there exists a nonzero column...
In this study we will examine a process to produce methanol from synthesis gas (mixture of hydrogen and carbon monoxide). The reactor used in the process was shown to obey Langmuir-Hinshelwood kinetics. We illustrate how to preform a nonlinear regression analysis on a set of data that can be modeled using Langmuir-Hinshelwood kinetics
A common task in engineering is to determine a formula that describes the functional dependence of given physical quantity on various parameters. In this report we will examine the functional dependence of heat capacity on temperature. We will illustrate how to use Mathematica to compute a linear regression analysis on a heat capacity/temperature d...
In polynomial regression the relationship between the independent variable x and the dependent variable y is modeled as n^th degree polynomial in x. We illustrate how such an analysis can be undertaken using Mathematica.
In this study, the scalar convection-diffusion equation, (also referred to as advection-diffusion equation) is solved using a finite difference method (based on the flux formulation). The numerical scheme is developed in Mathematica and the numerical results are compared with the analytical solution.
The reaction mechanism (or reaction path) can be thought of as a chemical road map that describes the precise, step-by-step process by which a reaction proceeds. The reaction steps that describe the reaction mechanism are at the molecular level; hence they are referred to as elementary reactions. In this report we will consider a reaction network t...
n this report we will analyze the properties of reactions in series. A simple example of a irreversible series reaction involving 3 species is A->B->P. Here B is viewed as the intermediate product, while species P is the final product of the reaction. We suppose that these reactions take place in an isothermal well mixed batch reactor with constant...
In this report we analyze the batch reactor from different view points. For convenience we will assume our batch reactor has a fixed volume V. We showed how an overall rate expression can be derived from a set of elementary reaction steps. Each reaction step defines a kinetic schema with appropriate constraints (defined by elementary stoichiometry...
In this report we will present the key steps in formulating how a reaction proceeds in a porous slab with heating effects.We consider a chemical reaction A -> B in a porous catalyst pellet. The reactant A is a fluid surrounding the catalyst particle at a specified concentration C_f, and temperature T_f. Species A diffuses into the catalyst and reac...
In these notes we extend the analysis of axisymmetric stagnation flow originally analyzed by Homann (1936) on the semi-infinite domain to an equivalent flow that is confined between two parallel disks. A full account of the early work on stagnation flow can be found in the monograph on laminar boundary layers edited by Rosenhead (1963), starting wi...
Plane stagnation flow ( also known as Hiemenz flow) is a classical 2-D flow in fluid mechanics that can be solved exactly using a similarity solution. The development of the theory can be found in many classical fluid mechanics text books. In this report we show how Hiemenz's flow, a nonlinear boundary value problem, can be solved using Mathematica...
In the analysis of free surface flows, the liquid-gas interface is usually not known a priori.Furthermore, the interface generally does not conform to a co-ordinate surface. Thus to apply the appropriate boundary conditions at the prescribed interface, it is often necessary to draw on concepts from differential geometry.
In these notes we review t...
Wilson (1993) suggested a simple model for analyzing the leveling behavior of paint films. His model consists of a non-volatile resin and a solvent. In this report we revisit the basic assumptions used to derive the Wilson's equation. We show how one can derive an exact integral representation for the species balance for the resin in the evaporatin...
In this study we will be concerned with the heat that is generated locally in the fluid due to viscous heating. The extent of the heat generated is in proportion to the square of the local velocity gradient. Ultimately, any heat generating within the flow must be eventually conducted to the boundary walls. If the boundary walls are held at a fixed...
Our goal of this report is to study the solution properties of Bessel's equation and to illustrate how Bessel's equation arises in the solution of heat conduction problems. The asymptotic behavior of Bessel functions is discussed. We then discuss the steady state temperature distribution u(r, z) in a rod of radius unity and length L. The mathematic...
The mathematical structure and solution for laminar flow over a heated/cooled flat plate is analyzed. The thermal boundary layer is computed and the heat flux at the surface of the plate is determined from the thermal boundary. Correlation for the Nusselt Number for arbitrary Prandtl Numbers is determined.
Transient heat conduction in a finite cylinder requires solving a 2-D eigenvalue problem. Consequently the orthogonality conditions for satisfying the BCs and ICs are more complicated. In this notebook we extend the Mathematica analysis given for the transient heat conduction with one spatial variable to multiple spatial variables.
In this report we show how a macroscopic energy balance can be formulated to solve transient heat conduction problems in solids involving multi-mode heat transfer effects, viz., convection and radiation. The example is a two-step manufacturing process that accounts for curing and drying of a coating.
We consider unsteady heat conduction in a rod of length L with a chemical reaction heat source. We derive the differential equation that governs the temperature profile in the rod. At steady state we show that the thermal conduction problem has multiple solutions. This is also known as the Frank-Kamenetskii problem.
In this report we illustrates how one can solve 1-D conduction problems involving composite layers with different thermal properties. We show how Mathematica can be used to handle the algebraic manipulations and plot the temperature profiles in the composite rod.
In this report we consider unsteady heat conduction in a rod of length L with a chemical reaction heat source. We derive the differential equation that governs the temperature profile in the rod. At steady state we show that the thermal conduction problem has multiple solutions. This is also known as the Frank-Kamenetskii problem.
In this report we show how to solve a 1-D heat conduction problem where the thermal conductivity is a function of temperature.
In this report we consider multicomponent diffusion in a Stefan tube apparatus. At the bottom of the tube we specify a binary liquid mixture of n-hexane and ethanol with a given composition. Air is the carrier gas. The mathematical framework needed to predict the composition profiles in the tube as well as the diffusion path L(t) is discussed. The...
Solutions curves that exhibit turning points occur in many branches of science and engineering. As a rule solution curves with turning points are not known a priori, as they are normally the solution to complex nonlinear differential equations. In these notes we will illustrate numerical techniques to compute solution curves with turning points. Th...
In this report we consider flow in a duct with a cross-section defined by an equilateral triangle that has a side of length a. We describe how the flow in the duct can can be computed and visualized using Mathematica's Finite Element Method. The accuracy of the Finite Element method is assessed by comparing the solution with a known analytical solu...
On this report we consider the following surface mediated reaction on a catalytic surface:
A+B -> C. The reaction is represented with an adsorption step and a desorption step. We illustrate how the rate limiting step can be computed using Mathematica's symbolic algebra features.
In this study we consider a spherical drop that initial contains fluid A. Species C absorbs and then diffuses into the interior of the drop and undergoes a decomposition reaction to form product P. We assume that the kinetics for this reaction are nC ->P. We solve the nonlinear reaction diffusion problem numerically using the Method of Lines. The c...
In this report we explore through numerical computations the periodic behavior of autonomous equations. We examine numerically using Mathematica an example problem that exhibits steady solutions, bifurcations from those steady solutions to time-periodic solutions and then bifurcations from the time-periodic solutions to quasi-periodic solutions.
In this report we consider unsteady heat conduction in a rod of length L with chemical reaction. We assume that the cross-section varies along the axis of the rod. Thus at any axial distance the rod radius is a specified function R(z), where z is the axial coordinate. We show how to derive rigorously the Frank-Kamenetskii thermal energy equation st...
The central theme for most numerical methods is to approximate a function u(x) by a set of polynomials called trial functions. The type of numerical method depends on the choice of the trial functions. For example, various finite difference methods are based on choosing local polynomials of low degree. Spectral methods, on the other hand, such a Le...
In this report notes we give a brief introduction to the spectral method for solving differential equations, as well as approximating functions over a specified interval. The particular spectral method we will focus on in these notes is called the pseudo-spectral method, which is a collocation method using Chebyshev polynomials as the approximating...
In the analysis of chemical reacting systems, a kinetic scheme is often postulated for a reacting system based on mass action kinetics. An alternative approach is to assume that the kinetic steps can be approximated to be quasi-steady. Mathematically, this approximation is implemented by assuming that the rates of production of the radical species...
In this report we examine carefully all the steps that lead to the solution of a binary diffusion problem involving heterogeneous catalysis. We consider a species A that diffuses through a gas layer of thickness δ and reacts then at a catalytic surface according to the following reaction schema A -> nB. We compute the concentration profile for vari...
Eigenvalues are often introduced in the context of linear algebra, but as noted above the theory finds application in many branches of science (e.g., quantum mechanics, and optics, mathematics, statistics) as well as in various engineering disciplines. In these notes we lay out the theory for the algebraic eigenvalue problem, and focus on how one c...
In this report we illustrate how Legendre Polynomials can be used to (i) approximate a continuous function and (ii) to obtain a least squares fit to a given data set. First, we give a brief overview of Legendre polynomials, and show one can use them in Mathematica. Then we illustrate how to fit an exponential function with a set of Legendre polynom...
In this report we study a constant volume, non-isothermal, non-adiabatic CSTR ( constant stirred tank reactor). Heat is removed from the reactor by use of a cooling coil with surface area A and overall heat transfer coefficient U. The cooling medium maintains a constant temperature of T_a. The inlet temperature and composition is given by T_0 and C...
In this report we illustrate how to compute periodic solutions to autonomous equations that exhibit period doubling behavior. The example used in the study was used by Lorenz in his early studies of chaotic behavior. The goal of the study is to determine numerically the periodic solutions and then use Floquet theory to assess stability, as well as...
In this notebook we study the solution space for a nonlinear reaction diffusion equation. Reaction diffusion equations in often arise in the study of population dynamics of a species subject to a predator. In this study we analyze (both theoretical and numerically) the solution space of a well known phenomenological model that describes logistic gr...
The Stefan-Maxwell equations are used to describe multicomponent diffusion in an ideal gas mixture. In this report we derive limiting forms for the Stefan- Maxwell equations for the following cases:
(i)Binary system
(ii) Mixture of almost identical species
(iii) Ternary Mixture
(iv) Ternary mixture with dilute species
In this report we apply the film model to heterogeneous reaction at a catalytic surface. he key assumption in the film model is that all the fluxes are taken to be normal to the interface. As result the diffusion process depends solely on the coordinate in the direction normal to the interface. At the catalytic surface we have a generic decompositi...
In this report we study multicomponent diffusion in a two-bulb apparatus.The analysis is based on the Stefan-Maxwell equations. We analyze the binary diffusion problem as well as the ternary diffusion problem. We use Mathematica to solve the governing ODEs and illustrate how the composition in the opposing bulbs evolve over time.
We consider 2-D transient diffusion of a species A is a slab with inclusions. At some inclusions, species A in absorbed, according to either a Neumann or Robin boundary condition. At the external boundary of the slab, the concentration of species A is held fixed. A time animation illustrates how the diffusion process proceeds.The problem is solved...
In this report we solve the steady 2-D convective diffusion equation by the Finite Element Method using Mathematica. Dirichlet or Neumann boundary conditions are specified on the boundary. The details of the method are outlined and we show how such problems can be solved in Mathematica.
In this report we examine the steady heat conduction in a cylindrical rod with a specified heat source. We show that the solution can be found using the method of separation of variables, once the problem is reformulated as a Laplace problem. The Fourier series solution is calculated and plotted for a specified heat source. We also show how the sam...
In this report we study the steady rectilinear flow of a Newtonian fluid in a pipe that contains a series of parallel tubes. The flow direction is taken to be parallel to the axis of the tubes. Of interest here is to evaluate details of the interstitial flow between the tube arrays. In our calculations we will assume the flow within the tube arrays...
In this report we study the steady 2-D flow of a Newtonian fluid in a channel with a contraction or expansion. We use Mathematica's Finite Element implementation to assess the features (size and strength) of the recirculation zone downstream of the contraction/expansion.
The Gelfand-Bratu problem is a well known nonlinear BVP that admits at least two solutions when the parameter lambda is greater than zero . More precisely it is a nonlinear eigenvalue problem. We illustrate how this problem can be solved using a collocation method that can be implemented with Mathematica .
The dynamics of a reaction-diffusion equation is used to illustrate a chemical switch.
In this paper we solve the nonlinear evolution equation that describes a Rayleigh-Taylor Instability with periodic boundary conditions over the interval -L < x < L. The model equation used in this study is related to the one used by Hammond(1983), Yiantsios & Higgins(1989) and Lister et al. (2006). The main goal of this notebook is to examining the...
First order partial differential equations in chemical engineering arise in models of processes that are dominated by convection rather than by dissipative processes. In this paper we consider a process that can be described by the 1-D convective diffusion equation with chemical reaction as model for chromatography. We show how solutions to the con...
The purpose of this report is to provide an introduction to flows that arise in a rotating frame of reference. In particular, we will examine geostrophic flows, Coriolis effects, and Ekman boundary layer flows.The report includes a mathematical description of rotating frames of reference, and the derivation of how the equations of motion for a Newt...
In this notebook, we illustrate how to compute the stream function from velocity data. The flow problem we will use is the lid driven cavity problem, for which there is an analytical solution based on Fourier series (separation of variables for the Biharmonic equation) when the Reynolds number is small. The computational tool we will use in this st...
In these notes we will use Mathematica to explore the properties of 1-D maps.The mathematical properties and features of nonlinear systems such as bifurcation, stability, and chaos can be readily appreciated by studying the dynamics of difference equations, the simplest of nonlinear systems.
It is well known that physico-chemical systems far from thermodynamic equilibrium can exhibit complex spatio-temporal organizations. A reaction diffusion system is an example of such systems that displays spatio-temporal patterns in which fluid convection plays no role. In these notes we will be concerned with activator-inhibitor reaction-diffusion...
The purpose of these notes is to give a brief introduction to the spectral method for solving differential equations, as well as approximating functions over a specified interval. The particular spectral method we will focus on in these notes is called the pseudo-spectral method, which is a collocation method using Chebyshev polynomials as the appro...
In the notebook we lay out the mathematical principles for analyzing reaction sequences using metabolic flux analysis. The challenge is given a specified reacting system that has an exchange flux with its surroundings, in which certain fluxes are measured. The challenge then can the remaining fluxes can be calculated from the known measured fluxes....
In this notebook we study the eddy structure in a lid driven container with a concave bottom. The container structure has sidewalls that are straight and of height h, but the bottom is a concave surface with radius R centered at the mid-plane of the container which makes an angle \[Alpha] with the sidewalls. We solve this problem using the Finite E...
In this study we consider a liquid phase elementary reaction A->B, where the rate constant has an exponential dependence on temperature. We will illustrate how the conversion of species A depends on temperature. The system has multiple steady states. We also use Mathematica's Manipulate function to study interactively the behavior of the reactor as...
The simplest model of population growth is due to Malthus who assumed that a population N(t) experiences a constant birth rate b per capita and a constant death rate d per capita. In this notebook we examine the solution properties of a partial differential equation that combines diffusion with logistic population growth, known as the Fisher equati...
In this notebook we analysis the dynamics of system of ODEs. The issues we will be concerned with are the number of steady states, the stability of the steady states, bifurcation points, eigenvalues of the linear stability, and continuation methods for tracking steady solutions.
This preprint shows how to compute standard Heats of Reaction using Mathematica
In these notes we consider an irreversible conversion of a precursor P into a final product C through two intermediate species A and B. The reaction occurs in a batch reactor, under non-isothermal conditions.
We will analyze the dynamics of a batch reaction in which the irreversible conversion of a precursor P into a final product C occurs through two intermediate species A and B. The reaction occurs under isothermal conditions.
In these notes we will explore the possibility of multiple steady states in an isothermal CSTR reactor. The kinetics selected for the study is based on a model autocatalytic reaction: A+2B->3B
This text has been written for use in the first course in a typical chemical engineering program. That first course is generally taken after students have completed their studies of calculus and vector analysis, and these subjects are employed throughout this text. Since courses on ordinary differential equations and linear algebra are often taken...
In coating related industries there is a growing need to utilize complex team activity in the development of new products. Although simulation driven product development using computational fl uid dynamics (CFD) and other related computational tools has taken root in the industry, knowledge integration of coating fundamentals among all team members...
We describe a parameter estimation algorithm for determining the rheological parameters for isothermal polymer melt flow in a tube. The viscosity shear rate data is fitted to a Carreau model. The method requires solving a nonlinear boundary value problem at each step of the parameter estimation algorithm. The Mathematica code requirements to implem...
Linear stability analysis of a rectilinear flow U0(y) in a 2-D channel requires solving the Orr-Sommerfeld equation for the eigenvalues (i.e., the growth rates) of the disturbance. This is a well-known eigenvalue problem in fluid mechanics. When heat effects are important, then the momen-tum equations are coupled to the thermal energy equation ofte...
The numerical solution of a fourth-order eigenvalue problem using a pseudo-spectral collocation method is described. The algorithm is im-plemented in Mathematica and the computed eigenvalues are compared with an analytical solution. The Mathematica code for implementing the collocation method is given in the Appendix.
In coating related industries there is a growing need to utilize complex team activity in the development of new
products. Although simulation driven product development using computational fl uid dynamics (CFD) and
other related computational tools has taken root in the industry, knowledge integration of coating fundamentals
among all team members...
The extent of reaction initially introduced by De Donder in 1920 is a dimensioned property that describes the progress of a chemical reaction. In this paper we illustrate how the extent of reaction can be generalized to describe the progress of any arbitrary reacting system involving N reacting species. The formulation is independent of the reactio...
in Wiley Online Library (wileyonlinelibrary.com). Stoichiometry refers to conservation of atomic species. In this article, local refers to a point at the continuum level, global refers to the macroscopic balance level, and elementary refers to conservation of atomic species associated with distinct kinetic steps. The role of stoichiometry in the de...
Coating flows are known to develop regions of high curvature when the interface is subject to a combination of viscous/pressure and capillary forces that produce large surface deformations locally. A well-known example is the cusp-like interface that develops when a liquid film on a rotating cylinder is dragged into a reservoir of similar liquid. A...
In premetered slot or extrusion coating and related sheet coating, a “bead” of liquid is held between the coating die and the moving sheet by capillary forces, which depend on gap clearances, surface tension, contact line attachment, and dynamic contact angle, by viscous forces, which depend on clearances, viscosity, meniscus location, and coating...
We describe a simple method for tracking solutions of nonlinear equations f (u, \mu) through turning points
(also known as limit or saddle-node bifurcation points). Our implementation makes use of symbolic software
such as Mathematica to derive an exact system of nonlinear ODE equations to follow the solution path, using a parameterization closely...
When a porous medium, such as a piece of paper, is placed into contact with a liquid reservoir, capillary action drives the liquid through the porous medium. The penetration distance L(t) of the liquid/air interface is typically described by the Lucas-Washburn equation, with any deviations normally occurring on a length scale set by the average por...
In film coating and other applications involving thin liquid films, surfactants are typically employed to suppress the usually undesirable instabilities driven by surface phenomena. Yet, in the present study a mechanism of Marangoni instability in evaporating thin films is presented and analyzed, which has its origin on the effects of a soluble sur...
