Computational Fluid Dynamics (CFD)

Introduction to CFD

• Computational Fluid Dynamics (CFD) predicts fluid flow, heat and mass transfer, chemical reactions, and related phenomena.
• CFD uses numerical solutions to solve governing mathematical equations.

What Does CFD Do?

• CFD simulations offer engineering data for conceptual studies, product development, troubleshooting, and redesign.
• CFD analysis gives flow parameter distributions like velocity, pressure, temperature, concentration, species, and forces.

How Does CFD Work?

• Problem setup includes geometry, mesh generation, physics definition, fluid properties, and boundary conditions.
• Equations are solved iteratively to achieve a converged solution.
• Results analysis and report generation occur during post-processing phase.

CFD Applications

• Aerospace: Aircraft aerodynamics relating to drag and lift.
• Automotive: Aerodynamics for drag reduction, and engine cooling processes.
• Biomedical: Studies airflow in lungs, blood flow in arteries.
• Chemical Processing: Application to mixing processes, and separation techniques.
• HVAC: Used for heating, ventilation, and air conditioning design and analysis.
• Hydrology: Simulates open channel flow, and sediment transport.
• Marine: Assesses loads on offshore structures.
• Nuclear: Examines thermal-hydraulics of reactors.
• Power Generation: Used for combustion analysis, and heat transfer optimization.

Conservation Laws

• CFD solves conservation equations for mass, momentum, and energy.

Mass Conservation

• Equation: $\frac{{\partial \rho }}{{\partial t}} + \nabla \cdot \left( {\rho V} \right) = 0$, where $\rho$ is density and $V$ is the velocity vector.

Momentum Conservation

• Equation: $\frac{{\partial \left( {\rho V} \right)}}{{\partial t}} + \nabla \cdot \left( {\rho VV} \right) = - \nabla p + \nabla \cdot \tau + \rho g + F$, where $p$ is pressure, $\tau$ is stress tensor, $g$ is gravitational acceleration, and $F$ is external force.

Energy Conservation

• Equation: $\frac{{\partial \left( {\rho h} \right)}}{{\partial t}} + \nabla \cdot \left( {\rho Vh} \right) = \nabla \cdot \left( {k\nabla T} \right) + S_h$, where $h$ is enthalpy, $k$ is thermal conductivity, $T$ is temperature, and $S_h$ is the heat source.

Turbulence Modeling

• Turbulence involves random, chaotic changes in fluid flow parameters over time and space.
• Turbulent flows feature 3D vorticity, fluctuations, dissipation, and diffusivity.
• Most engineering flows exhibit turbulence.
• Direct Numerical Simulation (DNS): Solves Navier-Stokes equations without any turbulence models.
• Reynolds Averaged Navier-Stokes (RANS): Solves Navier-Stokes equations with turbulence models.
• Large Eddy Simulation (LES): Solves filtered Navier-Stokes equations, resolving only the large eddies.
• Detached Eddy Simulation (DES): Hybrid RANS/LES model.

Numerical Methods

• Discretization methods in CFD include Finite Difference Method (FDM), Finite Volume Method (FVM), and Finite Element Method (FEM).
• Solution algorithms include iterative methods, explicit/implicit schemes, and coupled/segregated algorithms.

CFD Advantages

• CFD reduces lead times and costs.
• It enables study of systems under hazardous conditions.
• Offers a practically unlimited level of detail in results.
• Provides comprehensive information on parameters.

CFD Disadvantages

• Approximations to real physics are inevitable.
• Accuracy depends on user skill.
• Results are always subject to a degree of uncertainty.
• CFD is a tool and not a replacement for physical experiments.

Chemical Engineering Design - Third Edition

• "Principles, Practice and Economics of Plant and Process Design"
• Written by Towler & Sinnott, published by Butterworth-Heinemann, an imprint of Elsevier.

Análisis de Fourier (Fourier Analysis)

• Fourier analysis is a mathematical tool to decompose a function into simpler sinusoidal functions.
• It aids in analyzing signals and solving differential equations.

Series de Fourier (Fourier Series)

• A Fourier series represents a periodic function as a weighted sum of sines and cosines.
• For function $f(x)$ with period $2L$, the series is: $f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \left( a_n \cos \left( \frac{n \pi x}{L} \right) + b_n \sin \left( \frac{n \pi x}{L} \right) \right)$.
• Coefficients $a_n$ and $b_n$ are given by Euler's formulas.
• $a_0 = \frac{1}{L} \int_{-L}^{L} f(x) , dx$
• $a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos \left( \frac{n \pi x}{L} \right) , dx$
• $b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin \left( \frac{n \pi x}{L} \right) , dx$

Convergencia de la Serie de Fourier (Convergence of Fourier Series)

• If $f(x)$ and $f'(x)$ are piecewise continuous on $[-L, L]$, the Fourier series converges to $f(x)$ if $f$ is continuous at $x$.
• If $f$ is discontinuous at $x$, it converges to the average $\frac{f(x^+) + f(x^-)}{2}$.

Funciones Pares e Impares (Even and Odd Functions)

• If $f$ is even, $b_n = 0$ for all $n$.
• If $f$ is odd, $a_n = 0$ for all $n$.

Ejemplos de Series de Fourier (Examples of Fourier Series)

• Función Escalonada (Step Function):

  $f(x) = \begin{cases} 0, & -L < x < 0 \ E, & 0 < x < L \end{cases}$.

  $f(x) = \frac{E}{2} + \frac{2E}{\pi} \sum_{n=1}^{\infty} \frac{\sin((2n-1)\pi x/L)}{2n-1}$.

• Función Periódica Lineal (Linear Periodic Function):

  $f(x) = x, \quad -L < x < L$.

  $f(x) = \frac{2L}{\pi} \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \sin\left(\frac{n \pi x}{L}\right)$.

Transformada de Fourier (Fourier Transform)

• An extension of Fourier series to non-periodic functions, transforming time-domain to frequency-domain.

Definición (Definition)

• $F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-j \omega t} , dt$, where $F(\omega)$ is frequency representation of $f(t)$, $\omega$ is angular frequency, and $j$ is the imaginary unit.

Transformada Inversa de Fourier (Inverse Fourier Transform)

• Retrieves original time-domain function $f(t)$ from its frequency representation $F(\omega)$.
• $f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega) e^{j \omega t} , d\omega$.

Propiedades de la Transformada de Fourier (Properties of Fourier Transform)

• Linealidad (Linearity): $\mathcal{F}{af(t) + bg(t)} = aF(\omega) + bG(\omega)$.
• Escalamiento (Scaling): $\mathcal{F}{f(at)} = \frac{1}{|a|}F\left(\frac{\omega}{a}\right)$.
• Desplazamiento en el Tiempo (Time Shift): $\mathcal{F}{f(t - t_0)} = e^{-j \omega t_0} F(\omega)$.
• Desplazamiento en Frecuencia (Frequency Shift): $\mathcal{F}{e^{j \omega_0 t} f(t)} = F(\omega - \omega_0)$.
• Convolución en el Tiempo (Time Convolution): $\mathcal{F}{(f * g)(t)} = F(\omega)G(\omega)$.
• Convolución en Frecuencia (Frequency Convolution): $\mathcal{F}{f(t)g(t)} = \frac{1}{2\pi}(F * G)(\omega)$.
• Derivación en el Tiempo (Time Differentiation): $\mathcal{F}\left{\frac{df}{dt}\right} = j\omega F(\omega)$.
• Integración en el Tiempo (Time Integration): $\mathcal{F}\left{\int_{-\infty}^{t} f(\tau) , d\tau\right} = \frac{F(\omega)}{j\omega} + \pi F(0) \delta(\omega)$.

Ejemplos de Transformadas de Fourier (Examples of Fourier Transforms)

• Función Impulso (Delta de Dirac) (Impulse Function):

  $f(t) = \delta(t)$,

  $F(\omega) = 1$.

• Función Constante (Constant Function):

  $f(t) = 1$,

  $F(\omega) = 2\pi \delta(\omega)$.

• Función Escalón Unitario (Unit Step Function):

  $f(t) = u(t) = \begin{cases} 0, & t < 0 \ 1, & t > 0 \end{cases}$,

  $F(\omega) = \pi \delta(\omega) + \frac{1}{j\omega}$.

• Función Exponencial (Exponential Function):

  $f(t) = e^{-at}u(t), \quad a > 0$,

  $F(\omega) = \frac{1}{a + j\omega}$.

Aplicaciones (Applications)

• Signal processing, physics, engineering, and mathematics.
• Includes spectrum analysis, filter design, data compression, solving differential equations, and medical imaging.
• It facilitates analysis and manipulation of signals and systems.

Chemical Kinetics

• Studies reaction rates and mechanisms.

Reaction Rate

• Quantifies how fast reactant concentration decreases or product concentration increases over time.
• Rate law expresses the rate as a function of concentrations and temperature.

General Form

• For $aA + bB \rightarrow cC + dD$,
• $rate = -\frac{1}{a}\frac{\Delta[A]}{\Delta t} = -\frac{1}{b}\frac{\Delta[B]}{\Delta t} = \frac{1}{c}\frac{\Delta[C]}{\Delta t} = \frac{1}{d}\frac{\Delta[D]}{\Delta t}$

Rate Law

• Rate law must be determined experimentally.
• Rate constant ($k$) is temperature-dependent.
• Exponents ($x, y$) are the order of the reaction.
• $rate = k[A]^x[B]^y$

Integrated Rate Laws

• Relate reactant concentration to time.

Collision Theory

• Reaction occurs when reactant molecules collide with sufficient energy and proper orientation.

Arrhenius Equation

• Relates rate constant ($k$) to activation energy ($E_a$) and temperature ($T$).
• $k = Ae^{-E_a/RT}$.
• A is frequency factor related to collisions with proper orientation.
• $ln(k) = ln(A) - \frac{E_a}{RT}$.
• $ln(\frac{k_1}{k_2}) = \frac{E_a}{R}(\frac{1}{T_2} - \frac{1}{T_1})$.

Reaction Mechanisms

• Series of elementary steps.
• The sum of steps must give overall balanced equation.
• Mechanism must agree with experimentally determined rate law.
• Intermediate is formed then consumed in a subsequent step.
• Rate-determining step is the slowest step, therefore dictates overall reaction rate law.
• Catalyst speeds up rate without being consumed; it is consumed in one step, regenerated in another.

Tema: Geometría Analítica (Topic: Analytic Geometry)

1. Vectores en $\mathbb{R}^2$ y $\mathbb{R}^3$ (Vectors in $\mathbb{R}^2$ and $\mathbb{R}^3$)

1.1. Definiciones (Definitions)

• Vector: Directed line segment with magnitude and direction.
• Componentes: Representation in terms of projections on coordinate axes.
• Magnitud: Length of the vector, denoted by $\lVert \overrightarrow{v} \rVert$.
• Dirección: Angle the vector makes with the positive x-axis.

1.2. Operaciones (Operations)

• Given vectors $\overrightarrow{u} = (u_1, u_2)$ and $\overrightarrow{v} = (v_1, v_2)$:
• Suma (Addition): $\overrightarrow{u} + \overrightarrow{v} = (u_1 + v_1, u_2 + v_2)$
• Resta (Subtraction): $\overrightarrow{u} - \overrightarrow{v} = (u_1 - v_1, u_2 - v_2)$
• Producto escalar (Scalar Multiplication): $k\overrightarrow{u} = (ku_1, ku_2)$, where $k \in \mathbb{R}$

1.3. Producto escalar (punto) y vectorial (cruz) (Scalar and Vector Products)

• Producto escalar (Scalar Product): $\overrightarrow{u} \cdot \overrightarrow{v} = \lVert \overrightarrow{u} \rVert \lVert \overrightarrow{v} \rVert \cos(\theta) = u_1v_1 + u_2v_2 + u_3v_3$
• Propiedades (Properties):
• Conmutativo (Commutative): $\overrightarrow{u} \cdot \overrightarrow{v} = \overrightarrow{v} \cdot \overrightarrow{u}$
• Distributivo (Distributive): $\overrightarrow{u} \cdot (\overrightarrow{v} + \overrightarrow{w}) = \overrightarrow{u} \cdot \overrightarrow{v} + \overrightarrow{u} \cdot \overrightarrow{w}$
• Ortogonalidad (Orthogonality): If $\overrightarrow{u} \cdot \overrightarrow{v} = 0$, then $\overrightarrow{u}$ and $\overrightarrow{v}$ are orthogonal.
• Producto vectorial (Vector Product): $\overrightarrow{u} \times \overrightarrow{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ u_1 & u_2 & u_3 \ v_1 & v_2 & v_3 \end{vmatrix}$
• Propiedades (Properties):
• Anticonmutativo (Anticommutative): $\overrightarrow{u} \times \overrightarrow{v} = -\overrightarrow{v} \times \overrightarrow{u}$
• Distributivo (Distributive): $\overrightarrow{u} \times (\overrightarrow{v} + \overrightarrow{w}) = \overrightarrow{u} \times \overrightarrow{v} + \overrightarrow{u} \times \overrightarrow{w}$
• Area del paralelogramo $\lVert \overrightarrow{u} \times \overrightarrow{v} \rVert$ (Parallelogram Area): is the area of the parallelogram formed by $\overrightarrow{u}$ y $\overrightarrow{v}$.
• Ortogonalidad (Orthogonality): $\overrightarrow{u} \times \overrightarrow{v}$ es ortogonal a ambos $\overrightarrow{u}$ y $\overrightarrow{v}$ (is orthogonal to both $\overrightarrow{u}$ y $\overrightarrow{v}$).

2. La Recta (The Line)

2.1. Ecuaciones (Equations)

• Vectorial (Vector Form): $\overrightarrow{r} = \overrightarrow{r_0} + t\overrightarrow{v}$, where $\overrightarrow{r_0}$ is a point on the line, $\overrightarrow{v}$ is the direction vector, and $t \in \mathbb{R}$.
• Paramétrica (Parametric Form):
• $x = x_0 + at$
• $y = y_0 + bt$
• $z = z_0 + ct$
• Simétrica (Symmetric Form): $\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c}$
• General (en el plano) (General Form): $Ax + By + C = 0$
• Pendiente-ordenada al origen (Slope-Intercept Form): $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
• Punto-pendiente (Point-Slope Form): $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a known point and $m$ is the slope.

2.2. Angulo entre do rectas (Angle between two lines)

• $\tan(\theta) = \left| \frac{m_2 - m_1}{1 + m_1m_2} \right|$, donde $m_1$ y $m_2$ (where $m_1$ and $m_2$ are the slopes of the lines.)

2.3. Distancia de un punto a una recta (Distance from a point to a line)

• $d = \frac{\lvert Ax_1 + By_1 + C \rvert}{\sqrt{A^2 + B^2}}$, where $(x_1, y_1)$ is the point and $Ax + By + C = 0$ is the equation of the line.

3. El Plano (The Plane)

3.1. Ecuaciones (Equations)

• Vectorial (Vector Form): $\overrightarrow{n} \cdot (\overrightarrow{r} - \overrightarrow{r_0}) = 0$, where $\overrightarrow{n}$ is the normal vector to the plane, $\overrightarrow{r_0}$ is a known point on the plane, and $\overrightarrow{r}$ is any point on the plane.
• General (General Form): $Ax + By + Cz + D = 0$, where $\overrightarrow{n} = (A, B, C)$ is the normal vector.

3.2. Ángulo entre dos planos (Angle between two planes)

• $\cos(\theta) = \frac{\overrightarrow{n_1} \cdot \overrightarrow{n_2}}{\lVert \overrightarrow{n_1} \rVert \lVert \overrightarrow{n_2} \rVert}$, where $\overrightarrow{n_1}$ and $\overrightarrow{n_2}$ are the normal vectors of the planes.

3.3. Distancia de un punto a un plano (Distance from a point to a plane)

• $d = \frac{\lvert Ax_1 + By_1 + Cz_1 + D \rvert}{\sqrt{A^2 + B^2 + C^2}}$, where $(x_1, y_1, z_1)$ is the point and $Ax + By + Cz + D = 0$ is the equation of the plane.

4. Secciones Cónicas (Conic Sections)

4.1. Circunferencia (Circle)

• Ecuación estándar (Standard Equation): $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.

4.2. Parábola (Parabola)

• Ecuación estándar (Standard Equation):
• $(y - k)^2 = 4p(x - h)$ opens right
• $(y - k)^2 = -4p(x - h)$ opens left
• $(x - h)^2 = 4p(y - k)$ opens up
• $(x - h)^2 = -4p(y - k)$ opens down
• Where $(h, k)$ is the vertex and $p$ is the distance from the vertex to the focus and from the vertex to the directrix.

4.3. Elipse (Ellipse)

• Ecuación estándar (Standard Equation):
• $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$ (major axis horizontal)
• $\frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1$ (major axis vertical)
• $(h, k)$ is the center, $a$ is the length of the semi-major axis, $b$ is the length of the semi-minor axis, and $c$ is the distance from the center to each focus, with $c^2 = a^2 - b^2$.

4.4. Hipérbola (Hyperbola)

• Ecuación estándar (Standard Equation):
• $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$ (transverse axis horizontal)
• $\frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1$ (transverse axis vertical)
• $(h, k)$ is the center, $a$ is the distance from the center to each vertex, $b$ is related to the distance from the center to the foci by $c^2 = a^2 + b^2$, and the asymptotes have slope $\pm \frac{b}{a}$ or $\pm \frac{a}{b}$.

Guía completa de mapas mentales (Complete guide of Mind Maps)

¿Qué es un mapa mental? (What is a mind map?)

• A mind map is a diagram used to represent words, ideas, tasks, or other elements linked and organized around a central key word or idea.
• It helps organize information, generate ideas, solve problems, and make decisions.

Características principales de un mapa mental: (Main features of a mind map:)

• Idea central: The main topic is placed in the center of the map.

• Ramas: Subtopics or secondary ideas connect to the main idea through branches.

• Palabras clave: Keywords are used instead of long phrases to represent each idea.

• Imágenes y símbolos: Images and symbols can be used to make the mind map more visual and memorable.

• Color: The use of colors helps to organize and highlight the information.

¿Para qué sirve un mapa mental? ( What is a mind map for?)

• Toma de notas (Note-taking): Captures and organizes information quickly and efficiently.

• Lluvia de ideas (Brainstorming): Facilitates generating new ideas and exploring different perspectives.

• Planificación (Planning): Helps organize tasks, projects, and events.

• Resolución de problemas (Problem solving): Visualizes the problem from different angles and finds creative solutions.

• Memorización (Memorization): Facilitates memorizing information by associating it with images and colors.

• Presentaciones (Presentations): Can be used as a visual script for presentations.

¿Cómo crear un mapa mental? ( How to create a mind map?)

• Start with the central idea in the middle of a page.

• Add main branches with primary related ideas.

• Develop secondary branches from each main branch with more specific ideas.

• Use keywords, not long phrases.

• Incorporate images and symbols.

• Use different colors to organize information.

• Be creative and find your own style.

Herramientas para crear mapas mentales (Mind Map Tools)

• Analog: Use paper and pencils as standard and simple method.
• Software specific to Mind Maps: MindManager, XMind, FreeMind, offer digital map creations.
• Online Platforms: Miro, Lucidchart or MindMeister permit collaborative maps in the cloud.

Consejos para crear mapas mentales efectivos (Tips for creating effective mind maps)

• Be clear and concise using phrases and keywords.
• Use visuals and memorable symbols.
• Organize and highlight information using color.
• Be creative.
• Review and update.

General guide on Mind Mapping

• Provides a method for visualizing information and relationships in a non-linear format.

Tema central: Planificación de un viaje (Central theme: Travel Planning)

• Ramas principales: - include Destino( Destinations), Transporte(Transport) , Alojamiento(Accommodation) , Actividades(Activities), Presupuesto(Budget)
• Ramas secundarias: - further refine, for example Destino( Destinations) > Ciudad(city), Pais( country), Clima( weather)..Presupuesto(Budget) Comida (food

Chapter 14 - Theories of Personality

What is Personality?

• Personality is an individual's characteristic patterns of thought, feeling, and behavior.
• It is derived from "persona," referring to the mask.
• Personality is stable across time and situations.

Four Approaches to Personality

1. Psychoanalytic Approach: Focuses on unconscious mind and childhood experiences.
   • Key figures: Freud, Jung, Adler, Horney.
   • Importance of unconscious processes and defense mechanisms.
2. Trait Approach: Focuses on identifying and measuring individual differences in personality traits.
   • Key figures: Allport, Cattell, Eysenck, Costa, McCrae.
   • Personality is composed of stable, enduring traits that can be measured.
3. Humanistic Approach: Focuses on personal growth, self-actualization, and inherent goodness of people.
   • Key figures: Maslow, Rogers.
   • Importance of subjective experience.
4. Social-Cognitive Approach: Focuses on how cognitive processes, social interactions, and situational factors influence personality.
   • Key figures: Bandura, Mischel.
   • Personality is shaped by reciprocal interactions.

The Psychoanalytic Approach - Freud

• Focus on Psychic determinism, internal structure (Id, Ego, Superego), psychic conflict, and psychic energy (libido).

Freud's Stages of Psychosexual Development

• Oral Stage (0-18 months): Focus on mouth; theme is dependency.
• Anal Stage (18 months-3 years): Focus on anus; theme is self-control.
• Phallic Stage (3-7 years): Focus on sexual organs; theme is gender identity.
• Latency Stage (7 years-puberty): A break from development.
• Genital Stage (puberty on): Focus on genitals and maturity.

The Psychoanalytic Approach - Defense Mechanisms

• Denial: Refusing to acknowledge stimuli.
• Repression: Threats banished from memory.
• Reaction Formation: Impulse expressed as its opposite as a reaction.
• Projection: Attributing thoughts, behavior and impulses to others.
• Rationalization: False justifications for behavior.
• Displacement: Redirecting impulses to other channels to provide satisfaction.
• Sublimation: Channeling unacceptable impulses into acceptable activities.
• Regression: Faced with stress, reverting to behviors from earlier stages of developemnt.

The Trait Approach

• Traits are stable patterns of behavior used to describe, explain, and predict.

The Big Five Personality Traits (OCEAN)

• Openness to Experience
• Conscientiousness
• Extraversion
• Agreeableness
• Neuroticism

The Humanistic Approach

• Carl Rogers and Actualizing tendency: The drive to become one's actual self.
• Key components are the organismic valuing process, conditions of worth, unconditional and conditional positive regard.
• Abraham Maslow and Hierearchy of needs - focus on needs based motivation like physiological and self actualization needs Self-Actualization and peak experiences are the process and fulfillment in the context of needs.

The Social-Cognitive Approach

• Albert Bandura and Reciprocal Determinism with observation, learning and self efficacy, as demonstrated with the Bobo doll experiment.
• Julian Rotter, with expectancy theory and locus of control.