Linear Algebra: Vectors and Products

Algèbre Linéaire

Vecteurs

• A vector has a direction, a sense, and a magnitude (length).
• Addition: $\vec{u} + \vec{v} = (u_1 + v_1, u_2 + v_2)$
• Subtraction: $\vec{u} - \vec{v} = (u_1 - v_1, u_2 - v_2)$
• Scalar Multiplication: $k\vec{u} = (ku_1, ku_2)$, where $\vec{u} = (u_1, u_2)$, $\vec{v} = (v_1, v_2)$, and $k$ is a scalar.

Produit Scalaire

• The dot product of two vectors is $\vec{u} \cdot \vec{v} = ||\vec{u}|| \cdot ||\vec{v}|| \cdot \cos(\theta)$
• $||\vec{u}||$ and $||\vec{v}||$ are the magnitudes of the vectors
• $\theta$ is the angle between them.
• In Cartesian coordinates: $\vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2$

Produit Vectoriel

• The cross product of two vectors $\vec{u}$ and $\vec{v}$ in $\mathbb{R}^3$ results in a vector that is perpendicular to the plane containing $\vec{u}$ and $\vec{v}$.
• Magnitude: $||\vec{u} \times \vec{v}|| = ||\vec{u}|| \cdot ||\vec{v}|| \cdot \sin(\theta)$
• The direction is given by the right-hand rule.
• In Cartesian coordinates: $\vec{u} \times \vec{v} = (u_2v_3 - u_3v_2, u_3v_1 - u_1v_3, u_1v_2 - u_2v_1)$

Matrices

• A matrix is a rectangular array of numbers
• An $m \times n$ matrix has $m$ rows and $n$ columns.
• Addition: $A + B = C$, where $c_{ij} = a_{ij} + b_{ij}$
• Scalar Multiplication: $kA = B$, where $b_{ij} = ka_{ij}$
• Matrix Multiplication: $(AB){ij} = \sum{k=1}^{n} a_{ik}b_{kj}$
• A and B are matrices, k is a scalar.

Transposition

• The transpose of a matrix $A$, denoted $A^T$, is obtained by swapping $A$'s rows and columns.
• If $A$ is $m \times n$, then $A^T$ is $n \times m$.

Inverse

• Inverse of a square matrix $A$, denoted $A^{-1}$, satisfies $AA^{-1} = A^{-1}A = I$, where $I$ is the identity matrix.

Déterminant

• The determinant of a square matrix $A$, denoted $\det(A)$ or $|A|$, is a scalar.
• For a $2 \times 2$ matrix, $A = \begin{bmatrix} a & b \ c & d \end{bmatrix}$, $\det(A) = ad - bc$

Systèmes d'équations linéaires

• Can be represented as $Ax = b$, where $A$ is the coefficient matrix, $x$ is the vector of unknowns, and $b$ is the vector of constants.
• Solution Methods:
  • Gaussian elimination
  • Matrix inversion: $x = A^{-1}b$
  • Cramer's rule

Valeurs propres et vecteurs propres

• For a square matrix $A$, an eigenvector $v$ and eigenvalue $\lambda$ satisfy $Av = \lambda v$.
• Eigenvalues are roots of the characteristic polynomial $\det(A - \lambda I) = 0$.

The Pythagorean Theorem

• Useful to workout missing length / side of a right triangle
• Equation: $a^2 + b^2 = c^2$
• a/b are the legs
• c is the hypotenuse
• Some triangles sizes that follow this rule are
• 3, 4, 5
• 5, 12, 13
• 8, 15, 17
• 7, 24, 25
• Applications of this theorum includes, geometry, navigation, engineering and physics

Partial Differential Equations

• Relates an unknown function $u$ of at least two variables and its partial derivatives.
• General form for linear, second-order PDEs in two independent variables: $Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y + Fu = G$
  • $A, B, C, D, E, F,$ and $G$ are functions of $(x, y)$.
• The equation is linear due to the linear appearance of $u$ and its derivatives.
• Second-order because the highest derivative is of second order.

The Discriminant

• Key for PDE identification and simplification
• $\Delta = AC - B^2$ can identify the form of PDE, and its form
• If $\Delta > 0$, the PDE is elliptic.
• If $\Delta = 0$, the PDE is parabolic.
• If $\Delta < 0$, the PDE is hyperbolic.

Canonical Form

• A simplified form of a PDE attained via a change of independent variables.
• The type of canonical form depends on the discriminant $\Delta$.

Elliptic PDEs $(\Delta > 0)$

• Example: Laplace's equation: $u_{xx} + u_{yy} = 0$
• Canonical form: $\xi_{\eta\eta} + \xi_{\xi\xi} = \phi(\xi, \eta, \xi_{\xi}, \xi_{\eta})$

Parabolic PDEs $(\Delta = 0)$

• Example: Heat equation: $u_t = ku_{xx}$
• Canonical form: $\xi_{\eta\eta} = \phi(\xi, \eta, \xi_{\xi}, \xi_{\eta})$

Hyperbolic PDEs $(\Delta < 0)$

• Example: Wave equation: $u_{tt} = c^2u_{xx}$.
• Canonical form: $\xi_{\xi\eta} = \phi(\xi, \eta, \xi_{\xi}, \xi_{\eta})$

Lab 3: Working with Data

Objectives

• You will achieve
• the ability to import data into a Pandas DataFrame
• clean and prepare data for analysis
• perform exploratory data analysis (EDA) using Pandas
• visualize data using Matplotlib and Seaborn

Introduction

• Task focuses on a dataset of customer transactions from an online retail store
• It contains details on each transaction, such as customer ID, product ID, transaction date, and amount.
• Pandas is used to import, clean, and prepare the data.

Task 1: Data Import and Cleaning

• Steps to create setup your dataframe

1. Import transactions.csv into a Pandas DataFrame using read_csv().
2. Inspect data structure and content using head(), info(), and describe().
3. Clean data by handling:
   • Missing values using isnull().sum() to identify and dropna() to remove.
   • Duplicate rows using duplicated().sum() to identify and drop_duplicates() to remove.
   • Inconsistent data types by converting transaction_date to datetime using pd.to_datetime().

Task 2: Exploratory Data Analysis (EDA)

• Calculate descriptive statistics for numerical columns using describe().
• Group data by columns (e.g., customer_id) and calculate summary statistics (e.g., total transaction amount) using groupby() and sum().
• Identify outliers using box plots created with Matplotlib or Seaborn.

Task 3: Data Visualization

• Visualization methods includes:
• Create histograms of numerical columns using plt.hist().
• Create scatter plots of numerical columns using plt.scatter().
• Create bar charts of categorical data using plt.bar() with value counts.

Submission

• Submit code and report summarizing findings and insights from the EDA.
• Explore advanced visualizations, in-depth EDA, and predictive modeling as bonus tasks.

Chemical Engineering Thermodynamics

Chapter 10 Vapor-Liquid Equilibrium (VLE)

• Equilibrium of a liquid and its vapor

10.1 Basic Thermodynamics of VLE

• When two or more phases are brought into contact, spontaneous transport of species from one phase to another occurs.
• Vapor-Liquid Equilibrium occurs when the rate of evaporation equals the rate of condensation.
• The chemical potential of each component is the same in the liquid and vapor phases.

$$\hat{f}{i}^{V} = \hat{f}{i}^{L}$$

• $\hat{f}_{i}^{V}$ is the fugacity of component i in the vapor phase
• $\hat{f}_{i}^{L}$ is the fugacity of component i in the liquid phase

Vapour & Solutions equations

• For an ideal gas: $\hat{f}{i}^{V} = y{i}P$

  • $y_{i}$ is the mole fraction of component i in the vapor phase, P is the total pressure

• For an ideal solution: $\hat{f}{i}^{L} = x{i}P_{i}^{sat}$

  • $x_{i}$ is the mole fraction of component i in the liquid phase
  • $P_{i}^{sat}$ is the vapor pressure of pure component i at the system temperature

From these two points:

• $y_{i}P = x_{i}P_{i}^{sat}$
  • This is known as Raoult's Law.
• $y_{i}P = x_{i}\gamma_{i}P_{i}^{sat}$
  • $\gamma_{i}$ is the activity coefficient of component i in the liquid phase.
  • Corrects for non-ideality of the liquid phase, in Modified Raoult's Law.

10.2 VLE Calculations

• Types of calculations, solving the system

• Bubble Point Calculation -- At a given T & $x_{i}$ find P & $y_{i}$

• Dew Point Calculation -- At a given T & $y_{i}$ find P & $x_{i}$

• Bubble Temperature Calculation -- At a given P & $x_{i}$ find T & $y_{i}$

• Dew Temperature Calculation -- At a given P & $y_{i}$ find T & $x_{i}$

10.3 Flash Calculations

• Flash Calculation -- At a given P, T & $z_{i}$ find $x_{i}$, $y_{i}$, V, L
• $z_{i}$ is the overall mole fraction of component i in the feed
• V is the molar flow rate of the vapor phase
• L is the molar flow rate of the liquid phase
• Example: Flash calculation for a binary mixture obeying Raoult's Law -- Given values and using this law, we can workout the state of the material
• Antoine Equation: -- $\ln(P^{sat}) = A - \frac{B}{T + C}$
• A, B, and C are Antoine coefficients for a specific component.
• This equation estimates the saturation pressure at different temperatures.

10.4 VLE in Non-Ideal Systems

• For non-ideal systems, activity coefficients are used
• Correct for deviations from ideality in the liquid phase and estimate activity $$\gamma_{i} = \frac{\hat{f}{i}^{L}}{x{i}P_{i}^{sat}}$$
• Coefficient for different conditions can be estimated using Van Laar, Wilson, or NRTL models

10.5 Azeotropes

• No change in properties by distillation
• Occurs when the vapor composition is equal to the liquid composition at a specific temperature and pressure. $$x_{i} = y_{i}$$
• Two liquids, minimum-boiling or maximum-boiling

10.6 Immiscible Liquids

• Liquids does not mix
• Each liquid exerts its own vapor pressure
• The total pressure is the sum of the vapor pressures of the individual liquids. $$P = P_{1}^{sat} + P_{2}^{sat}$$
• High-boiling substance can be distilled at a lower temperature by adding water (steam distillation).

Economía

Producción

• Producción breakdown includes
• Trabajo (L): human efforts and time
• Capital (K): durable goods used in production
• Tierra (T): natural resources
• Tecnología (A): knowledge and techniques

Función de producción

• Maximum output with available factors

$Q = f(L, K, T, A)$

• Q = cantidad de producción
• L = trabajo
• K = capital
• T = tierra
• A = tecnología

Producto total, producto marginal y producto medio

• Understanding outputs
• Producto Total (PT): total production
• Producto Marginal (PMg): change in total product per unit increase in a factor
• Producto Medio (PMe): average product per unit of factor used

$PMg = \frac{\Delta PT}{\Delta factor}$

$PMe = \frac{PT}{cantidad ; de ; factor}$

Representación gráfica

• Shows relationships between PT, PMe, and PMg using a graph with a horizontal axis (L) and vertical axis (Q)
• Etapas de la producción:*

1. Stage 1 is from pt 0 to pt where production is optimal on the PMe curve
2. Stage 2 occurs from a pt where PT is at its optimal point, to zero on production on the PMg curve
3. Stage 3: All data after optimal values from the curve are reached

Costos

• Costs to create products
• Costos a corto plazo (short term) -- Costos fijos (CF): fixed costs -- Costos variables (CV): variable costs -- Costo total (CT): total cost -- Costo marginal (CMg): marginal cost related to production of additional units.

$CT = CF + CV$

$CMg = \frac{\Delta CT}{\Delta Q}$

Costos medios

• Costo fijo medio (CFMe): average fixed cost
• Costo variable medio (CVMe): average variable cost
• Costo total medio (CTMe): average total cost

$CFMe = \frac{CF}{Q}$

$CVMe = \frac{CV}{Q}$

$CTMe = \frac{CT}{Q}$

Representación gráfica

• Curves showing the relationship between total costs and production
• The marginal cost curve intersects with the average total and variable cost curves at their lowest points
• CMg is U-shaped.
• CVMe is U-shaped and reaches its minimum before CTMe.
• CTMe is U-shaped and is above the CVMe curve.
• CFMe is decreasing.

Statics

Chapter 1 General Principles

• Mechanics: study of the state of rest or motion of bodies under forces.

1.2 Fundamental Concepts

• Basic Quantities:

  • Length: position in space
  • Time: succession of events
  • Mass: quantity of matter
  • Force: a "push" or "pull"

• Idealizations:

• Particle: A body of negligible size

• Rigid Body: A combination of a large number of particles in which all the particles remain at a fixed distance from one another

• Concentrated Force: The effect of a loading which is assumed to act at a point on a body

• Newton's Three Laws of Motion:

  1. First Law: inertia (bodies at rest remain at rest, bodies in motion remain in motion)
  2. Second Law: $F = ma$ (Force = mass * acceleration)
  3. Third Law: mutual forces between two particles are equal, opposite, and collinear.

• Newton's Law of Gravitational Attraction: $$ F = G \frac{m_1 m_2}{r^2} $$

• $F$ is force of gravitation

• $G = 66.73(10^{-12}) m^3/(kg \cdot s^2)$

• $m_1, m_2$ are mass

• $r$ is the distance

• Weight is $W = mg$ where $g = 9.81 m/s^2$.

1.3 Units of Measurement

• SI Units:
  • Length (meter)
  • Time (second)
  • Mass (kilogram)

Force is $F = ma = kg \cdot m/s^2 = Newton (N)$.

• Prefixes are use to scale and read metrics e.g. kilo, mega, micro

1.4 The International System of Units

• Rules for use of SI units Algebraic operations follow the same rules as for numeric values, and any physical quantity can be expressed by appropriate SI units

1.5 Numerical Calculations

• Dimensional Homogeneity: each term in an equation must have the same units.
• Significant Figures: Answer precision should match the least precise data entry.
• Rounding Numbers: increase the last digit to be retained by one if the first digit dropped is 5 or greater.

1.6 General Procedure for Analysis

• Understand the problem, draw diagrams, apply principles, solve equations, report the answer, and analyze solution.