Bernoulli's Principle and Fluid Lift

Bernoulli's Principle

• Discovered by Daniel Bernoulli in the 18th century.
• For inviscid flow, an increase in fluid speed occurs simultaneously with a decrease in pressure or potential energy.

How Wings Generate Lift

• Airplane wings are shaped to make air move faster over the top than underneath.
• Faster-moving air exerts less pressure.
• Higher pressure below the wing and lower pressure above create net upward force, also known as lift.

Factors Affecting Lift

• Lift is proportional to the square of the air speed.
• Larger wings generate more lift.
• Lift increases with the angle of attack up to a point.
• Denser air generates more lift.

Mathematical Representation

• Bernoulli's Equation: $P + \frac{1}{2} \rho V^2 + \rho g h = constant$
• P is the static pressure of the fluid
• $\rho$ is the density of the fluid
• $V$ is the speed of the fluid
• $g$ is the acceleration due to gravity
• $h$ is the height of the fluid above a reference point
• If fluid speed (V) increases, pressure (P) decreases, where height (h) is constant.

Applications of Bernoulli's Principle

• Airplanes: Wings are designed for faster airflow above, creating lift from the pressure difference.
• Race Cars: Spoilers create downward force by manipulating airflow.
• Spray Bottles: Airflow over a tube creates low pressure, drawing liquid upwards.
• Chimneys: Wind across the top reduces pressure, helping smoke rise.
• Pitot Tubes: Measure aircraft airspeed comparing static with dynamic pressure.
• Venturi Meters: Used for measuring flow rate in pipes.

Regular Expressions (Reguläre Ausdrücke)

• Character strings that describe a pattern
• Used for searching, replacing text, and validating inputs

Components

• Literal characters: Characters that match directly (e.g., a, b, 1, 2)
• Metacharacters: Special characters with specific meanings (e.g., ., *, +, ?, [], (), ^, $, |)

Metacharacter and meaning

Character Classes

Quantifiers

Application Examples

• E-Mail Validation: ^[a-zA-Z0-9._%+-]+@[a-zA-Z0-9.-]+\.[a-zA-Z]{2,}$
• Password Validation: ^(?=.*[a-z])(?=.*[A-Z])(?=.*\d)(?=.*[@$!%*?&])[A-Za-z\d@$!%*?&]{8,}$ (At least 8 characters, 1 uppercase, 1 lowercase, 1 number, 1 special character)
• Telephone Number Validation (German): ^(\+49|0)[2-9][0-9]{1,2}[0-9]{3,8}$
• Useful Link: https://regexr.com/ (For testing and visualizing regular expressions)

Chapter 2: Sorting Algorithms

Introduction

• Sorting algoriths sort input Lists
• Output is an ordered permutation list

Utility

• List ordered
• Used as sub-routine in other algorithsm including searcing etc

Complexity

• Cost/Complexity of sorting algorithm measured in # of operations to sort a list of size n

Types of Sorts

Internal Sorts

• Data to sort located in RAM

External Sorts

• Data to sort too big, stored externally on hard drive, etc

Comparative Sorts

• Sorting depends on comparing lists to sort

Non-Comparative Sorts

• Based on comparing elements

Simple Sorts

• Quadradic Complexity

Sort by Selection

Principle

• Find smallest elemtn from list
• Exchange with the first element
• Find smallest element rest of list
• repeat....

Pseudo-Code

procedure tri_selection(tableau T)
  pour i allant de 0 à taille(T)-2 faire
    min ← i
    pour j allant de i+1 à taille(T)-1 faire
      si T[j] < T[min] alors
        min ← j
      fin si
    fin pour
    si min ≠ i alors
      echanger T[i] et T[min]
    fin si
  fin pour
end procedure

Complexity

• $n-1 + n-2 +... + 1 = \sum_{i=1}^{n-1} i = \frac{n(n-1)}{2} = O(n^2)$

Chemical Engineering: Phase Equilibrium

Vapor Pressure

• Pressure when gas phase is in equilibrium with the liquid phase

Vapor Pressure Empirical Equations

Antoine Equation

• $\log_{10}P^* = A - \frac{B}{T + C}$
• $\text{P}^*$ is Vapor Pressure
• T is Temperature
• A, B, and C are Antoine Constants
• Units:
  • P* in mmHg, T in °C
  • P* in bar, T in Kelvin

Cox Chart

• $\log P^*$ versus $\frac{1}{T}$ yields a straight line.

Clausius-Clapeyron Equation

• $\frac{dP^*}{dT} = \frac{\Delta H_v}{T\Delta V}$
• Assumptions:
  • $\Delta V = V_{gas}$
  • $V_{gas}$ is in accordance with the ideal gas law
• $\Delta H_v$ = heat of vaporization
• $\frac{dP^}{P^} = \frac{\Delta H_v}{R} \frac{dT}{T^2}$
• $\ln P^* = -\frac{\Delta H_v}{RT} + B$
• If $\Delta H_v$ is constant, then $\ln \frac{P_2^}{P_1^} = -\frac{\Delta H_v}{R} (\frac{1}{T_2} - \frac{1}{T_1})$

Gibbs Phase Rule

• $DOF = 2 + N - \pi - r$
• DOF = Degrees of freedom
• N = Number of species
• $\pi$ = Number of phases
• r = Number of reactions