Presentation of representative examples of chemical processes. Definition of raw material, process and product. Presentation of chemical processes as interdependent sequences of operations and main pieces of equipment of the process industry. Usual parameters for the description of process flows: composition, temperature, pressure, density, viscosity. Dimensions, unities and unit systems. 

Chemical equilibrium and phase equilibrium. Chemical potential. Basic concepts and definitions: kinetic, potential and internal energy, heat and work. State functions. Reversible and irreversible processes. Heat capacities under constant pressure and volume. Formulation of the First Law of Thermodynamics for flow systems. 

Quantity of motion balance. Laminar and turbulent flow in ducts. Fundamentals of Fluid Rheology. Charge loss in pipes. Centrifuge pump. Boundary layer. Classic equations of fluid mechanics, heat and mass transfer. Means of transport of thermal energy and mass: equations of rates for conduction, convection and radiation for thermal transport. Definition of mass and molar flow. Mass diffusivity. Heat and mass transfer coefficients.  

Mass and energy balances. Formulation and application of the Law of Conservation of Mass and Energy to physical and chemical processes. Mass balances: global and by components. Conceptualization of systems and processes: batch, semi-batch and continuous; permanent and transient regimes; concentrated and distributed systems. Processes with reaction systems. Homogeneous and heterogeneous kinetics. Consecutive and parallel reactions, limiting reagent, conversion, degree of advancement of the reactions, percent excess, selectivity and performance. 

Introduction. Basic laws of thermodynamics. Microscopic view of energy and entropy. Thermodynamic potentials and auxiliary variables (Legendre transformations), basic relations and equilibrium criteria. Canonical, microcanonical and grand canonical partition functions. Volumetric and calorimetric properties. Phase equilibrium: partial molar properties. Chemical potential. Fugacity and fugacity coefficient in mixtures through equations of state. Fugacity of liquid and solid mixtures through excess free energy. Phase equilibrium and stability method. Algorithms for calculation of equilibrium of multicomponent systems (emphasis on VLE). Chemical equilibrium criteria. Calculation of equilibrium constants. Chemical equilibrium in homogeneous systems. Chemical and phase equilibrium (heterogeneous systems).  Calculation of system equilibrium with several reactions through direct minimization of free energy. 


Smith, J.M.; Van Ness, H.C. e Abbott, M.M. Introdução à Termodinâmica da Engenharia Química, 7ª Edição [WINDOWS-1252?]- 2007, Editora: LTC, Rio de Janeiro. 

John M. Prausnitz, Rudiger N. Lichtenthaler, Edmundo Gomes de Azevedo, Molecular Thermodynamics of Fluid-Phase Equilibria (3rd Edition), Prentice Hall, 1999. 

Jefferson W. Tester, Michael Modell, Thermodynamics and Its Applications (3rd Edition) Prentice Hall, 1996. 

Herbert B. Callen, Thermodynamics and an Introduction to Thermostatistics, 2nd Edition, (2nd edition) John Wiley 1985. 

Bruce E. Poling, John M. Prausnitz, John P. O'Connell. The Properties of Gases and Liquids. Fifth Edition, John Wiley, New York (2001). 

Hill, T., Introduction to Statistical Thermodynamics, Dover (1960). 

McQuarrie, D.A., Statistical Mechamics, Harper and Row Publishers, New York (2000).

*Mandatory Subject for the Master’s Course

Credit Hours: 45h

Phase stability: local and global criteria. Equilibrium calculation: chemical and/or phase equilibrium - modern flash algorithms. Saturation points. Phase diagrams: tracing and characteristics of different types of diagrams. Critical point: characterization and calculation algorithms. Equations of State: virial and its extensions. Van der Waals and its extensions. Solutions theories: Van Laar, Regular Solution, Flory-Huggins. Chemical theories of solution. Local composition models - Wilson, NRTL, UNIQUAC, UNIFAC. Application of local composition models for the obtention of mixtures rules for equations of state. Solubility of solids and gases in liquids. Thermodynamic Formalism for semi-continuous mixtures. Vapor-liquid Equilibrium in the presence of electrolytes. 

Reaction systems of polymerization, homogeneous and heterogeneous. Polymerization mechanisms and kinetics. Distribution of molecular weighs and particle sizes in condensation and chain polymerization processes (through free-radical, ionic, coordination). Diffusive effects. Polymerization in suspension and emulsion. Polymerization reactors. Experimental characterization methods.

Basic concepts: nature and characterization of macromolecules and polymeric systems. Polymer industry and the Brazilian market. Classification and properties of polymers according to molecular structure, topology of mers and supramolecular morphology: copolymers, segmental mobility, spatial isomerism, crystallinity, etc. Thermal and mechanical transitions. Polymer solutions and permeability. Polymer characterization techniques. Polymerization as a process. Step and chain polymerization. Classic kinetics of free radicals. Copolymerization. Kinetics of heterogeneous, Ziegler Natta and metallocene polymerization. Homogeneous and heterogeneous reaction systems (suspension, dispersion, emulsion, mud). Tehcnological problems in polymerization systems. Polymer processing (extrusion, calendering, molding). 

The reduction of environmental impact related to industrial and urban growth is obtained through the use of advanced separation processes or the combination of different processes. Membrane separation processes such as reverse osmosis and microfiltration, the combination of these with biological processes, such as bioreactors with membranes, or with traditional unit operations have been implemented in large scale, in general, aiming at effluent reuse. This course presents the most utilized membrane separation processes, as well as their combination with biological and traditional processes. Membrane processes still in implementation phase, such as pervaporation and gas separation are also presented. 


BAKER, R. W., Membrane Technology Applications, 2 ª Edição, Jonh Wiley & Sons, Ltda., 2004.

BYRNE, W., Reverse Osmosis - A practical guide for Industrial Users, Tall Oaks Publishing Inc., 1995.

HO, W.S.W., SIRKAR, K. K., Membrane Handbook, Van Nostrand Reinhold. NY, 1992. 

NOBREGA, R., BORGES, C.P., HABERT, A.C., “Processos de Separação com Membrana”, Escola Piloto PEQ/COPPE/UFRJ, 1997.

Nunes, S. P., Peinemann, K.-V. , “Membrane Technology in the Chemical Industry”, Wiley, 2001

Classification of membrane processes and their applications. Preparation techniques of different kinds of polymer membranes. Transport mechanisms and models. Kinds of modules and their main characteristics. Reverse osmosis and ultrafiltration: theoretical fundamentals, membrane synthesis by phase inversion; influence of synthesis variables on the transport characteristics of membranes. Concentration Polarization. Influence of operational variables; applications. Project for a specific application. Pervaporation and gas separation: theoretical fundamentals; synthesis of dense and composite membranes. Influence of operational variables. Project for a specific application.

Vector and tensor algebra. Tensor calculation. Vector fields theory: divergence, Stokes and circulation theorems. Kinematics of fluid flow: streamlines, streaklines, pathlines, Reynolds transport theorem, continuity equation, deformation and rotation tensors, vorticity. Introduction to Continuum Mechanics: types of forces, hydrostatic equilibrium, Kelvin’s theorem, potential flow of ideal fluid. Flow function. Euler’s and Cauchy’s Laws: surface forces, conservation of linear and angular momentum, stress tensor. Shear stress in fluids: Newtonian fluids, some non-Newtonian fluid models, Navier-Stokes equations. Slow flow, boundary layer and turbulence. Energy equation for pure substances: conservation of internal and kinetic energy, methods of heat transfer, functional form of the heat flow vector, entropy inequality. Conservation of mass of chemical species in a multicomponent mixture: theorem of transport of a chemical species, mass balance for a component, basic definitions of concentrations, mass velocity and flow, review of the postulates for the multicomponent mixture. Heat and mass transfers in binary mixtures: Fourier’s and Fick’s Laws, thermal mass analogies. Applications using diffusive and connective problems. Mass transfer in multicomponent mixtures: interactions forces, Maxwell-Stefan equation, estimation of the mass transfer coefficients. Applications. 

* Mandatory Subject for the Master’s Course

Credit Hours: 45h

Description of surface and interface, thermodynamics of surfaces. Surfaces and forces. Ionic and covalent solids. Physical and chemical adsorption forces. Gas-solid interface: physical adsorption of gases and vapors, isotherms, adsorption heat, capillary condensation. Chemisorption: mechanism, distinction between chemical and physical adsorption. Liquid-solid interface: isotherms, dilute solutions, preferential adsorptions, surface chemistry aspects in chromatography. Electrical aspects of surfaces. The Electrical Double Layer Theory. Electrokinetic phenomena. 

* Mandatory Subject for the Master’s Course

Credit Hours: 45h

Superficial segregation. Definition and characteristics of interfaces. Interfacial orientation and excess. Interfacial segregation theoretical prediction. Characterization of surfaces by chemisorption. Examples of segregation: Alloys. Polymeric mixtures. Experimental quantification of the superficial composition (Chemisorption and XPS). Solid/liquid adsorption. Forces involved in adsorption. Adsorption isotherms. Examples of applications. Experimental development for determination of kinetics and construction of adsorption isotherms. 

Introduction to the numerical methods of discretization: Finite Differences, Finite Volumes, Finite Elements, Spectral Methods, Generalized Integral Transform Method. Properties of the discretized equations. Finite volumes: simulation of diffusive processes. Interactive methods for the solution of systems of algebraic equations. Temporal integration: implicit and explicit methods. Method of lines. Equations of fluid motion. Simulation of convective processes. Interpolation functions. Simulation of flow: segregated solution and pressure-velocity coupling. Mismatched and collocated pipeline systems. Simulation of flow with heat and/or mass transfer. Introduction to the solution of hyperbolic systems: TVD and ENO methods. Solution of the advection equation.  

Ordinary differential equations. Finite difference equations. Series of orthogonal functions. Integral functions. Green’s Function. The Sturm-Liouville problem. Residue theory. Integral transforms: Fourier and Laplace. Partial difference equations. Systems of ordinary differential equations. Characteristic values and vectors. State space representation. Disturbance methods. State transition matrix. 

OBJECTIVES: Deepening of the theoretical concepts and mathematical methods for the solution and analysis of differential equations. 


1. Ordinary differential equations

1.1 First order differential equations

1.2 Differential operator

1.3 Second order differential equations

1.4 Existence and uniqueness of solutions

2. Difference equations

2.1 Methods for the solution of linear difference equations 

2.2 Discrete Ricatti Equation

2.3 Z-Transform

3. Orthogonal Series and Functions

3.1 Integral functions

3.2 Power series method

3.3 Legendre’s equations

3.4 Method of Frobenius 

3.5 Bessel’s equations

3.6 Sturm-Liouville problems

3.7 Euler’s operator

4. Residue theory

4.1 Complex numbers and functions

4.2 Complex integration

4.3 Complex series

4.4 Residue Theorem

4.5 Real Integrals 

5. Fourier and Laplace transforms

5.1 Laplace transform and its inverse

5.2 Laplace transform of derivatives and integrals

5.3 Differentiation and integration of transforms 

5.4 Convolution theorem

5.5 Fourier series and integrals

5.6 Fourier transform and its inverse

5.7 Fourier transform of derivatives

5.8 Convolution transform

6. System of ordinary differential equations

6.1 Characteristic values and vectors

6.2 Nonhomogeneous linear systems

6.3 Nonlinear systems

6.4 Disturbance methods applied to the solution of ordinary differential equations

7. Partial differential equations

7.1 Basic concepts and classifications

7.2 Separation of Variables method

7.3 Asymptotic Solution

7.4 Solution via Laplace transform

7.5 Solution via Fourier transform

7.6 Solution through similarity

7.7 Classic equations

WORK METHOD: expositive classes with performance of in-class exercises and extra-classroom exercise lists.  

PROCEDURES AND/OR ASSESSMENT METHODS: assessment based on the grades of two exams with unlimited consultation and on the weekly exercise lists.   


• Abramowitz, M. & Stegun, I.A. Handbook of Mathematical Functions, Dover, 1964. []

• Amundson, N. R. Mathematical Methods in Chemical Engineering, Prentice Hall, Inc., 1966.

• Kreider, D., Ostberg, D.R., Kuller, R.C. & Perkins, F.W. Introdução à Analise Linea, vols. 1, 2 e 3. Ao Livro Técnico  S. A., 1972.

• Kreyszig, E. Advanced Engineering Mathematics, 8ª ed. John Wiley & Sons, Inc., 2001.

• Özisik, M. N. Heat Conduction, John Wiley & Sons, 1980.

• Rice, R. R. & Do, D.D.- Applied Mathematics and Modeling for Chemical Engineers, John Wiley & Sons, 1995.

• Richards, D. - Advanced Mathematical Methods with MAPLE, 1a ed. Cambridge, USA, 2002.

• Spiegel, W.M.R. & Liu, J. Manual de Fórmulas e Tabelas Matemáticas, 2ª ed. Coleção Schaum-Bookman, 2004.

• Wiley, C. R. & Barrett, L.C. - Advanced Engineering Mathematics, 5ª ed. McGraw Hill, 1985.

* Mandatory Subject for the Master’s Course

Credit Hours: 45h

Complex and simple reaction rates in a homogeneous system. Determining steps. Enzymatic process kinetics; notions of enzyme kinetics, Michaelis-Menten model and its variants, product inhibition models. Polymerization process kinetics; basic growth mechanisms in polymerization reactions, polycondensation reactions kinetics, addition reaction kinetics (chain reactions). Kinetics of reactions in heterogeneous systems; rate equation models considering adsorption, reaction and desorption in isolated particles, determining step, global reaction rates in a gas-solid system: inter- and intraparticle heat and mass transfer, determining step, reaction rates in two- and three-phase systems. 

* Mandatory Subject for the Master’s Course

Credit Hours: 45h

The isolated particle: heat and mass transfer. Adsorption and chemical reaction. Modeling for non-porous solids. Controlling steps. Non-isothermal systems. Modeling for porous solids. Controlling steps. Langmuir-Hinshelwood rate expressions. Experimental techniques.

Basic concepts and definitions. Adsorption. Adsorption isotherms. Preparation of catalysts: precipitation, impregnation, drying, calcination, reduction. Catalyst forms. Physicochemical characterization: nature of the structure, texture, active surface, electronic properties. Sensitive and insensitive reactions. Evaluations. Selectivity and activity. Applications in selective hydrogenation processes, C1 chemistry and reformation. 

Forms of pollution. Quality standards. Characterization of domestic and industrial effluents. Forms of primary treatment (grating, decantation, neutralization/equalization). Biological treatment of effluents. Modeling of bacterial growth, substrate consumption and oxygen consumption. Aerobic processes: activated sludge and its variants. Aerated lagoons, biological filters and discs. Treatment of effluents: fundamentals; kinds of digesters (UASBR, AF, FFAR). Biological nitrogen removal. Phosphorus removal.

Fundamentals of biochemistry and microbiology: general aspects and metabolic ways for making products of industrial interest. Kinetics of microbial growth and metabolite production. Bioreactors: types and forms of operation. Monitoring of biotechnological processes: measurement and control of variables of interest. 

Introduction to process mathematical modeling. Creating linear time-invariant models with concentrated parameters. The input/output representation related to time (continuous and discrete) and Laplace and Z-transforms for linear systems. State space representation. Introduction to probability and statistics. Introduction to optimization. Sensibility analyses. Batch reactions and reactions in tubular reactors. Nonlinear nature of the chemical systems. The complex behavior of dynamic systems.  


J. C. Friedly, “Dynamic Behavior of Processes”, Prentice Hall, 1972.

B. W. Bequette, “Process Dynamics. Modeling, Analysis, and Simulation”, Prentice Hall, l998.

T. F. Edgar, D. M. Himmelblau, “Optimization of Chemical Processes”, MacGraw-Hill, 1989.

* Mandatory Subject for the Master’s Course

Credit Hours: 45h

Mathematical models of processes and concentration and distribution parameters. Numerical simulation of static and dynamic behaviors of processes. Flow, reaction and equilibrium systems. Population balance modeling. Weighted residue method.

Review of classic control. Digital control. Multivariable systems control. Introduction to data reconciliation. Introduction to flaw detection and isolation. Introduction to statistical process control. 


Aström, K, Wittenmark, B., “Computer-Controlled Systems”

Deshpande, P. (editor), “Multivariable Process Control”

Koppel, L., “Introduction to Control Theory”

Ray, W., “Advanced Control”

Narasimhan, S., Jordache, C., “Data Reconciliaton & Gross Erro Detection”

Chiang, L., Russell, E., Braatz, R. “Faul Detection and Diagnosis in Industrial Systems”

Carga Horária: 45hra

The fundamental objective of this subject is the presentation of advanced polymerization processes, which combine the production process of polymeric materials with other simultaneous stages of processing and/or modification of the structure of such materials. In this context, we are going to discuss techniques for in-situ incorporation of polymeric charges during polymerization, for the production of composites and blends. Techniques which allow the incorporation of bioactive charges will also be discussed, with the objective of developing bioactive materials, for biomedical and biotechnological applications. This class will also discuss opportunities for the production of special materials through heterogeneous and homogeneous polymerization techniques, by mechanisms such as free radicals and polycondensation. 

Mathematical modeling of a simulated moving bed separation unit. Application of optimization techniques in problems of separation with simulated moving bed units.  

Interactions between small molecules and polymers in dilute and concentrated solutions. Theories of polymeric solutions. Extension for partially swollen and reticulated polymers. Sorption of gases, vapors and liquids in elastomers and vitreous polymers. Sorption kinetics. Diffusion in elastomeric polymers: molecular and free volume models. Diffusion in vitreous polymers. Experimental methods for determination of solubility, diffusion and permeability coefficients. Industrial and analytical instrumentation applications. 

Permeation of gases and liquids in polymer membranes. Basic transport models: activated solution-diffusion, flow in pores, approaches of thermodynamics of irreversible processes. Effects of geometry and flow dynamics on the fluid/membrane interface: concentration polarization, gel layer. Experimental methods for determination of transport coefficients. Non-isothermal systems. Project of permeators. Applications.

Theoretical part: 

Introduction to multiphase flows. Industrial applications. Flow regimes. Stratified and dispersed multiphase flows. 

Properties of dispersed multiphase flows. Coupling between phases. Particle size reduction.  

Interaction between fluid and particles. Particle conservation equations. Amount of movement, mass and energy transfer. 

Mathematical description of multiphase flows: temporal, volumetric and sample averages.

Practical part:

Experimental analysis of bubble columns: measurement of gas retention and bubble sizes.

Measurements of terminal velocity of isolated particles.

Granulometric analysis of particulate suspensions.


Crowe, C.; Sommerfeld, M.; Tsuji, Y. (1998) Multiphase Flows with droplets and Particles. CRC Press.

Brennen, C. (2005) Fundamentals of Multiphase Flow. Cambridge University Press.

Drew, D. A.; Passman, S. L. (1999) Theory of Multicomponent Fluids. Springer.

Sirignano, W. A.. (1999) Fluid Dynamics and Transport of Droplets and Sprays. Cambridge University Press.

Gidaspow, D. (1994) Multiphase Flow and Fluidization. Academic Press.

Clift, R.; Grace, J. R.; Weber, M. E. (1978) Bubbles, drops and particles. Dover.

Ishii, M. (1975) Thermo-Fluid Dynamic Theory of Two-Phase Flow. Eyrolles.

Massarani, G. (2002) Fluidodinâmica em Sistemas Particulados. E-papers.

Numerical solution of systems of algebraic equations: direct and interactive methods, continuation method. Multiplicity of solutions. Numerical solution of ordinary differential equations: initial value problems.  Explicit and implicit methods. Single and multiple steps methods. Disturbance method. System rigidity. Parametric stability and sensitivity analysis. Solution of differential-algebraic equation systems. Problem of index and consistency of initial conditions. 

OBJECTIVES: Deepening of the theoretical concepts and numerical methods for the solution and analysis of algebraic and differential equations.  


1. Solution of algebraic equations

   1.1 Matrices, factorization, characteristic and singular values and vectors

   1.2 Direct methods for dense and sparse linear systems

   1.3 Interactive methods for dense and sparse linear systems

   1.4 Conditioning and error analysis

   1.5 Nonlinear systems and multiplicity of solutions

   1.6 Continuation method

2. Solution of ordinary differential equations 

   2.1 Explicit vs. implicit methods  

   2.2 Single-step methods vs. multiple steps methods 

   2.3 Disturbance method

   2.4 Polynomial Interpolation

   2.5 System rigidity

   2.6 Stability analysis

   2.7 Parametric sensitivity

3. Solution of differential-algebraic equation systems

   3.1 Resolvability

   3.2 Index problem

   3.3 Consistency of initial conditions

   3.4 Index reduction

   3.5 Automatic and symbolic differentiation

   3.6 Single-step methods vs. multiple steps methods

WORK METHOD: theoretical-practical classes in computer lab with interactive student participation, with one computer per student available. 

PROCEDURES AND/OR ASSESSMENT METHODS: assessment based on individual extra-classroom tasks and an oral presentation of a final paper. 


•    Fröberg, C.E. Introduction to Numerical Analysis, Addison-Wesley, 1965.

•    Carnahan, B., Luther, H.A. & Wilkes, J.O. Applied Numerical Methods, Wiley, 1969.

•    Finlayson, B.A. Nonlinear Analysis in Chemical Engineering, McGraw Hill, 1980.

•    Rice, J. R. Numerical Methods, Software and Analysis, McGraw-Hill, 1983.

•    Davis, M.E. Numerical Methods and Modeling for Chemical Engineers, John Wiley & Sons, 1984.

•    Dekker, K. & Vermer, J.G. Stability of Runge-Kutta Methods for Stiff Non Linear Differential Equations, North-Holland, 1985.

•    Butcher, J.C. The Numerical Analysis of Ordinary Differential Equations, Wiley Interscience Publications, 1987.

•    Brenan, K.E., Campbell, S.L. & Petzold, L.R. Numerical Solution of Initial-Value Problems in Differential Algebraic Equations, North-Holland, 1989.

•    Golub, G.H. & Ortega, J.M. Scientific Computing and Differential Equations, Academic Press, 1992.

•    Kreyszig, E. Advanced Engineering Mathematics, Wiley, 1993.

•    Rice, R.R. & Do, D.D. Applied Mathematics and Modeling for Chemical Engineers, John Wiley & Sons, 1995

•    Müllges, G.E. & Uhlig, F. Numerical Algorihtms with C, Springer, 1996.

•    Golub, G.H. & Van Loan, C.F. Matrix Computations, Johns Hopkins, 1996.

•    Stewart, G.W. Matrix Algorithms, Vol. 1 a 4, SIAM, 1998.

Numerical Methods in Process Modeling and Simulation - Numerical solution of ordinary differential equations. Initial value problems. Runge-Kutta and Predictor-Corrector methods. Stiff problems. Boundary condition problems. Numerical solution of partial derivative equations. Order reduction techniques applied to differential-difference equations. Applicative examples in process models. 

Introduction: polymerization mechanisms, important properties of polymeric solutions and materials. Homogeneous systems: solution polymerization in CSTR, gel effect, thermal effects, complex dynamics in homogeneous systems, reactions in tubular reactors, extruder reactors, azeotropic distillation reactor. Heterogeneous systems: emulsion polymerization; suspension polymerization; catalytic polymerization (Ziegler-Natta): mud bed reactors, fluidized bed reactors. Break-up and coalescence kinetics. Complex dynamics in heterogeneous systems. Monte Carlo simulations.

Direct methods of parameter estimation, nonlinear regressions, interpretation and analysis of results: covariance matrix, confidence intervals and statistic tests. Sequential project of experiments: discrimination criteria between models. Criteria for parameter estimation of a mixed criteria model. Examples applied to kinetic model discrimination. 

Adsorption in solid surfaces. Chemisorption in metal and oxide surfaces: theory, characterization methods and quantitative aspects. Acid surface. Basic surface. Thermal methods of analysis: desorption/reduction/oxidation under programmed temperature. Catalytic activity test. Model reactions.

Fundamentals: rates and mechanisms of superficial reactions. Catalysis by metals, oxides and sulfates. Geometric factors; chemical adsorption/desorption in metals, oxides and sulfates. Activity patterns in model reactions. Microscopic theory of heterogeneous catalysis. Dissociative adsorption - the role of electrons in transition states of superficial reactions. Kinetic models in non-uniform surfaces.  

Enzymes, conceptualization and classification. Enzyme production. Utilization of enzymes as catalysts. Introduction to enzyme kinetics. Enzyme immobilization techniques. Kinetics and diffusion in immobilized enzymes. Enzyme reactors, characteristics and project equations. 

Effluent control parameters. Ozonation. Treatment by ultraviolet radiation. Treatment by peroxides. Principles and applications of photocatalysis. Principles and applications of Fenton and photo-Fenton reactive. Combination of processes and comparative process evaluations. 

Biological aspects of animal cells. Animal cell culture technology. Product separation and purification. Approval and licensing of processes and products. 

Biological processes with biofilms, population dynamics in biological systems, microscopy, impacts of recalcitrant chemical substances in population dynamics, case study on chemical industry effluents. 

Formulation of mathematical models for reaction and separation systems. Homogeneous and heterogeneous systems. Microbial growth, substrate consumption and product formation kinetics. Enzyme kinetics. Systems with immobilized cells and/or enzymes. 

Introduction. Elements of probability and statistics. Different types of models. Methods of nonparametric linear identification in time and frequency. Estimation of parameters. Recursive methods for parameter estimation. Nonlinear identification methods. Project of experiments, selection of structures. Validation of models.

Analysis of multivariable processes: Relative gain array (RGA). Analysis via singular value decomposition: use of methods of differential geometry (relative order). Decentralized control: methods of controller adjustment. Uncoupling: different treatments. Multivariable control: Linear quadratic optimal control. Predictive control based on a linear model. Nonlinear control: Predictive control based on a nonlinear model. Methods based on concepts of differential geometry.

Ordinary differential equations and mappings: existence and uniqueness of solutions. Flows and characterization of trajectories. Linear and nonlinear systems. Permanent solution. Dissipative and conservative systems. Study of stability: Lyapunov functions. Static bifurcations. Fixed-point theorem. Hopf theorem. Higher order degenerations. Stability of periodic solutions. Secondary dynamic bifurcations. Chaos: characterization and universality. Crises and bifurcations. Routes to chaos. Global bifurcations: homo and heteroclinic orbits. Silnikov cycles. 

Professors Príamo Melo and Heloísa Sanches

Chapter 0 - Basic Definitions

Theory of sets, relations and functions.

Abstract algebra. Operations. Abstract systems: groupoid, semi-group, monoid, group, ring, field.

Chapter 1 - Linear Algebra

Vector space, subspace. Linear dependence and independence. Bases. Dimension of a space. Linear transformations: isomorphisms, matrices, linear equations. Functionals. Algebraic duals. Algebraic dual space, dual base. The transpose of a linear operation. Scalar product. Representation theorem in finite dimension spaces. Base and co-base. Adjoint of a transformation. Covariant and contravariant components of tensors. 

Chapter 2 - Lebesgue Measure and Integration Theory

Lebesgue measure theory. Construction and characterization of such measure. Lebesgue integration theory. Measurable functions. Riemann integrals.  Lebesgue integrals. Lp spaces. 

Chapter 3 - Topological Spaces

Topology. Topological space. Open and closed sets. Neighborhood. Interior and exterior point. Open ball, closed ball, sphere.  Accumulation point. Closure. Continuous mappings. Dense subset, separable space. Boundary. Compact sets. Sequences. Convergence of a sequence, limited set. Limited sequence. Cauchy’s convergence criterion. Cauchy sequences. 

Chapter 4 - Metric Spaces

Definition of metric space. Characteristics of a metric. General triangular inequality. Induced metric. Examples of metric spaces and metrics. Inequalities (Holder/Cauchy-Schwarz, Minkowski). Some topological properties of metric spaces. Complete and incomplete metric spaces. Completion of metric spaces. Isometric spaces. Compact metric spaces. 

Chapter 5 - Normed Spaces and Banach Spaces 

Normed space. Banach space. Norm and its properties. Metric induced by norm. Norm as continuous mapping. Incomplete normed spaces. Subspace of a normed space. Closed subspace. Complete subspace of a Banach space. Convergence of sequences in a normed space. Separability. Completion of Banach spaces. Closure of a normed space. Equivalent norms. Compact spaces and subspaces. Linear operators and their properties. Continuous and limited linear operators. Operator norm. Examples of continuous and limited operators. Null space. Equality of operators. Extension of an operator. Limited or unlimited linear functionals. Continuity and limitation of linear functionals. Hahn-Banach theorem. Linear operators and functionals in finite dimension spaces. Representation of a linear operator by a matrix. Banach fixed point theorem. 

Chapter 6 - Inner Product Spaces. Hilbert Spaces. 

Properties of an inner product. Definition of Hilbert space. Orthogonality and orthogonal projections. Orthonormal bases. Fourier series. Orthonormal sets and sequences. Orthogonal polynomials: Legendre, Hermite and Laguerre.

Chapter 7 - Approximation Theory

Approximation in normed spaces. Uniqueness and convexity. Uniform approximation. Chebyshev polynomials. Approximations in Hilbert spaces.  Splines. 

Chapter 8 - Operator Theory

Compact linear operators in normed spaces. Unlimited operators. 

Chapter 9 - Spectral Theory

Characteristic values. Resolvent set and spectrum. Spectrum of continuous operators, compact operators and self-adjoint operators. 


Kreyszig, E. Introductory Functional Analysis with Applications. John Wiley and Sons, 1978.

Ramkrishna, D., Amundson, N. Linear Operator Methods in Chemical Engineering with Applications to Transport and Chemical Reaction Systems, Prentice Hall, 1985.

Oden, J.T., Demkowicz, L.F. Applied Functional Analysis. CRC Press, 1996.

Griffel, D.H. Applied Functional Analysis. Prentice Hall, 1981.

Pinto, J.C. Notas de Aulas sobre Álgebra Linear.

The diversity of optimization problems in Chemical Engineering. Theoretical fundamentals. Univariate and multivariate unrestricted search. Single- and multiple-objective linear programing. Nonlinear programming. Dynamic and heuristic programming. Linear and nonlinear programming with integers. Computational aspects. Applications to equipment and processes. Analysis of recent literature.

The problem of data reconciliation. Definition, mathematical nature and numerical nature of data reconciliation problems. The importance of measurement errors in the formulation of objectives. Examples. Steps of the data reconciliation problem. Classification of variables: redundant, observable and unobservable variables. Removal of gross errors. Preliminary data reconciliation: linear techniques. Nonlinear data reconciliation. Real time applications. Data reconciliation in control environments. Sequential parameter estimation. Adaptive control and reconciliation techniques. On-line parameter estimation.