Radiative Heat Transfer: Radiation Properties

Radiative Heat Transfer

Radiation Properties

Definitions

• Radiosity ($J$) signifies the total energy leaving a surface per unit area and time, measured in $[W/m^2]$.
• Radiosity can be expressed as $J = \epsilon E_b + \rho G + \tau G$, where $\epsilon$ is emissivity, $E_b$ is blackbody emissive power, $\rho$ is reflectivity, $G$ is irradiation, and $\tau$ is transmissivity.
• For opaque surfaces, transmissivity ($\tau$) is zero, simplifying the equation to $J = \epsilon E_b + \rho G$.
• For diffuse and gray surfaces, emissivity ($\epsilon$) equals absorptivity ($\alpha$).
• Applying energy balance to a surface yields $\alpha G = J - G$, which can be rearranged to express irradiation as $G = \frac{J - \epsilon E_b}{\rho}$.

Blackbody Radiation

• Planck's law defines the spectral radiance of blackbody radiation: $E_b(\lambda, T) = \frac{C_1}{\lambda^5 [exp(C_2/\lambda T) - 1]}$.
• Within Planck's law:
  • $C_1 = 2\pi h c_o^2 = 3.742 \times 10^8 [W \mu m^4/m^2]$.
  • $C_2 = h c_o / k = 1.439 \times 10^4 [\mu m \cdot K]$.
  • $h = 6.6256 \times 10^{-34} J \cdot s$ represents Planck's constant.
  • $k = 1.3805 \times 10^{-23} J/K$ represents Boltzmann constant.
  • $c_o = 2.998 \times 10^8 m/s$ is the speed of light.
• Wien's displacement law relates the peak wavelength of emitted radiation to temperature: $\lambda_{max} T = 2898 \mu m \cdot K$.
• Stefan-Boltzmann law states that the total energy radiated per unit surface area of a blackbody is proportional to the fourth power of its absolute temperature: $E_b = \sigma T^4$, where $\sigma = 5.670 \times 10^{-8} W/m^2 \cdot K^4$.

Radiation Exchange

• For two surfaces $A_i$ and $A_j$, the reciprocity relation is $A_i F_{i \rightarrow j} = A_j F_{j \rightarrow i}$.
• $F_{i \rightarrow j}$ is the view factor, representing the fraction of radiation leaving surface $i$ intercepted by surface $j$.
• The summation rule dictates that for any surface $i$ in an enclosure, the sum of view factors to all surfaces $j$ (including itself) is 1: $\sum_{j=1}^N F_{i \rightarrow j} = 1$.
• For an enclosure of N surfaces, the net radiation heat transfer from surface i is $q_i = A_i (J_i - G_i) = \frac{E_{bi} - J_i}{(1 - \epsilon_i)/(\epsilon_i A_i)}$.
• The net radiation heat transfer between surfaces $i$ and $j$ is $q_{i \rightarrow j} = A_i F_{i \rightarrow j} (J_i - J_j) = \frac{J_i - J_j}{1/(A_i F_{i \rightarrow j})}$.
• For gray-diffuse surfaces, radiosity is $J_i = \epsilon_i E_{bi} + \rho_i G_i = \epsilon_i E_{bi} + (1 - \epsilon_i) G_i$.
• The net rate of radiation transfer from surface $i$ to all other surfaces is $q_i = A_i (J_i - G_i) = A_i J_i - \sum_{j=1}^N F_{i \rightarrow j} J_j A_i$.

Simplified Relations

• For a small object in a large enclosure, $q = \epsilon_1 A_1 \sigma (T_1^4 - T_2^4)$.
• For two large parallel plates, $q = \frac{\sigma (T_1^4 - T_2^4)}{\frac{1}{\epsilon_1} + \frac{1}{\epsilon_2} - 1} A$.
• For long concentric cylinders or concentric spheres, $q = \frac{\sigma (T_1^4 - T_2^4)}{\frac{1}{\epsilon_1} + \frac{A_1}{A_2} (\frac{1}{\epsilon_2} - 1)} A_1$.

Python Data Types, Operators, and Control Flow

Data Types

• Python has several data types, including integers (int), floating point numbers (float), strings (str), and booleans (bool).
• Integers are whole numbers without a fractional part and can be positive, negative, or zero.
• Floating point numbers have a decimal point.
• Strings are sequences of characters enclosed in single or double quotes.
• Booleans are either True or False.

Type Conversion

• Data types can be converted using functions like int(), float(), str(), and bool().
• int() converts a value to an integer.
• float() converts a value to a floating point number.
• str() converts a value to a string.
• bool() converts a value to a boolean.

Operators

Arithmetic Operators

• + performs addition.
• - performs subtraction.
• * performs multiplication.
• / performs division.
• // performs floor division, which returns the integer part of the division.
• % performs modulo, which returns the remainder of the division.
• ** performs exponentiation.

Comparison Operators

• == checks for equality.
• != checks for inequality.
• > checks if one value is greater than another.
• < checks if one value is less than another.
• >= checks if one value is greater than or equal to another.
• <= checks if one value is less than or equal to another.

Logical Operators

• and returns True if both operands are True.
• or returns True if at least one operand is True.
• not returns the opposite boolean value.

Control Flow

If Statements

• if statements execute a block of code if a condition is true.

Else Statements

• else statements execute a block of code if the condition in the if statement is false.

Elif Statements

• elif statements allow checking multiple conditions in sequence.

For Loops

• for loops iterate over a sequence of values.

While Loops

• while loops execute code as long as a condition is true.

Functions

Defining Functions

• Functions are defined using the def keyword, followed by the function name, parentheses, and a colon.

Calling Functions

• Functions are called by using their name followed by parentheses.

Return Values

• Functions can return values using the return keyword.

Lecture 24: Hypothesis Testing I

Statistical Hypothesis

• A statement about the parameters of a population distribution.
• Example: $H_0: \mu = 0$, $H_1: \mu \neq 0$.

Hypothesis Testing

• A procedure for deciding whether to accept or reject a hypothesis based on sample data.

Steps in Hypothesis Testing

• State the null and alternative hypotheses.
• Choose a significance level $\alpha$.
• Compute the test statistic.
• Make a decision based on the p-value compared to $\alpha$.

Null Hypothesis ($H_0$)

• A statement about the population parameter assumed to be true unless there is convincing evidence to the contrary.
• Example: $H_0: \mu = 0$.

Alternative Hypothesis ($H_1$ or $H_a$)

• A statement that contradicts the null hypothesis.
• Example: $H_1: \mu \neq 0$.

Types of Errors

• Type I Error: Rejecting the null hypothesis when it is actually true.
  • $P(\text{Type I error}) = \alpha$.
• Type II Error: Failing to reject the null hypothesis when it is actually false.
  • $P(\text{Type II error}) = \beta$.

Power of a Test

• The probability of correctly rejecting the null hypothesis when it is false.
• $P(\text{Reject } H_0 \mid H_0 \text{ is false}) = 1 - \beta$.

Significance Level ($\alpha$)

• The probability of rejecting the null hypothesis when it is true.
• Common values: 0.01, 0.05, or 0.10.

Test Statistic

• A value computed from the sample data used to determine whether to reject the null hypothesis.
• Examples:
  • $z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}$.
  • $t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}$.

P-value

• The probability of observing a test statistic as extreme as, or more extreme than, the one computed from the sample data, assuming that the null hypothesis is true.
• If p-value $\leq \alpha$, reject $H_0$.
• If p-value $>\alpha$, fail to reject $H_0$.

Decision Rule

• Reject $H_0$ if the p-value $\leq \alpha$; otherwise, fail to reject $H_0$.

Errors in Hypothesis Testing Summary

• $H_0$ True:
  • Reject $H_0$: Type I Error.
  • Fail to Reject $H_0$: Correct Decision.
• $H_0$ False:
  • Reject $H_0$: Correct Decision.
  • Fail to Reject $H_0$: Type II Error.

Lecture 25: Multi-Layer Perceptron

Multi-Layer Perceptron (MLP)

• MLP is also known as a feedforward neural network.

Perceptron

• $\hat{y} = \sigma(\mathbf{w}^{\top}\mathbf{x})$, where $\sigma(z) = \begin{cases} 1 & \text{if } z \geq 0 \ 0 & \text{otherwise} \end{cases}$
• Perceptron Limitation: It can only learn linearly separable functions.

Activation Function

• Sigmoid Function: $\sigma(z) = \frac{1}{1 + e^{-z}}$
• Tanh Function: $\tanh(z) = \frac{e^z - e^{-z}}{e^z + e^{-z}}$
• ReLU Function: $\text{ReLU}(z) = \max(0, z)$

Forward Propagation

• Process of computing the output of the network given an input $\mathbf{x}$.
• Given $\mathbf{x}$, we can compute $\hat{y}$ by applying the following equations:
  • $\mathbf{h} = \sigma(\mathbf{W}\mathbf{x} + \mathbf{b})$
  • $\hat{y} = \sigma(\mathbf{w}^{\top}\mathbf{h} + b)$

Loss Function

• For example, Mean Squared Error (MSE) loss function: $\mathcal{L} = \frac{1}{2} (\hat{y} - y)^2$.

Training

• Gradient descent can be used to find the optimal weights and biases that minimize the loss function.
• Update the weights and biases by taking a step in the opposite direction of the gradient of the loss function with respect to the weights and biases:
  • $\mathbf{w} \leftarrow \mathbf{w} - \eta \frac{\partial \mathcal{L}}{\partial \mathbf{w}}$
  • $b \leftarrow b - \eta \frac{\partial \mathcal{L}}{\partial b}$
• $\eta$ is the learning rate.

Backpropagation

• Backpropagation computes the gradient of the loss function with respect to the weights and biases.
• Chain Rule: if $y = f(u)$ and $u = g(x)$, then $\frac{dy}{dx} = \frac{dy}{du} \frac{du}{dx}$.

Backpropagation Example

• $\mathcal{L} = \frac{1}{2} (\hat{y} - y)^2$
• $\hat{y} = \sigma(\mathbf{w}^{\top}\mathbf{h} + b)$
• $\mathbf{h} = \sigma(\mathbf{W}\mathbf{x} + \mathbf{b})$
• Goal: Compute $\frac{\partial \mathcal{L}}{\partial \mathbf{w}}$, $\frac{\partial \mathcal{L}}{\partial b}$, $\frac{\partial \mathcal{L}}{\partial \mathbf{W}}$, and $\frac{\partial \mathcal{L}}{\partial \mathbf{b}}$.
• Calculation: $\frac{\partial \mathcal{L}}{\partial \mathbf{w}} = \frac{\partial \mathcal{L}}{\partial \hat{y}} \frac{\partial \hat{y}}{\partial \mathbf{w}}$. $\frac{\partial \mathcal{L}}{\partial b} = \frac{\partial \mathcal{L}}{\partial \hat{y}} \frac{\partial \hat{y}}{\partial b}$. $\frac{\partial \mathcal{L}}{\partial \mathbf{W}} = \frac{\partial \mathcal{L}}{\partial \hat{y}} \frac{\partial \hat{y}}{\partial \mathbf{h}} \frac{\partial \mathbf{h}}{\partial \mathbf{W}}$. $\frac{\partial \mathcal{L}}{\partial \mathbf{b}} = \frac{\partial \mathcal{L}}{\partial \hat{y}} \frac{\partial \hat{y}}{\partial \mathbf{h}} \frac{\partial \mathbf{h}}{\partial \mathbf{b}}$.

Computational Graph

• A directed graph where each node corresponds to an operation to visualize forward and backward propagation.

Algorithmic Game Theory

Game Theory

• Classical Game Theory: Mathematical study of strategic interactions between rational agents.
• Algorithmic Game Theory: Focuses on algorithmic and computational issues in game theory and economics, incorporating computer science aspects like algorithm design and computational complexity.

Routing

• Model includes a network of nodes and edges.
• Each edge $e$ has a non-negative, non-decreasing cost function $l_e(x)$ representing the cost incurred by each unit of traffic traversing it with a traffic volume of $x$.
• $k$ source-destination pairs $(s_i, t_i)$ exist, each with traffic $r_i$ to be routed.
  • Routing $f$ defines traffic flow on paths between source-destination pairs.
    • $f_p$: flow on path $P$.
    • $\sum_{P: s_i \rightarrow t_i} f_p = r_i$
  • $f_e$: flow on edge $e$.
    • $f_e = \sum_{P: e \in P} f_p$
• The latency of a path $P$ is the sum of individual edge latencies: $C_P(f) = \sum_{e \in P} l_e(f_e)$.
• The total cost is the total latency: $C(f) = \sum_{P} f_p \cdot C_P(f) = \sum_{e} f_e \cdot l_e(f_e)$.

Wardrop Equilibrium

• A Wardrop equilibrium is a state where no agent can unilaterally improve their cost.
• Wardrop equilibrium exists and is unique if latency functions are continuous.

Braess's Paradox

• Adding a link to a network can lead to higher total latency at equilibrium.

Price of Anarchy (PoA)

• Measures the degradation of social welfare due to selfish behavior.
• $\text{PoA} = \frac{\text{Cost of the worst-case Nash equilibrium}}{\text{Optimal social cost}}$.
• A lower PoA signifies a more efficient system.

Parallel Links

• n users connect to the Internet via parallel links with capacities $c_1$ and $c_2$.
• User $i$ minimizes their cost $\frac{x_j}{c_j}$ by choosing link $j$.
• Pure Nash Equilibrium conditions: $\frac{x_1}{c_1} \leq \frac{x_2 + 1}{c_2}$ and $\frac{x_2}{c_2} \leq \frac{x_1 + 1}{c_1}$.Social cost: $\sum_{j=1}^{2} \frac{x_j^2}{c_j}$.

Heat Treatment Furnaces

Types of Heat Treatment Furnaces

• Batch Furnaces:
  • Box Furnaces: Used for small to medium-sized parts.
  • Pit Furnaces: Vertical furnaces for long parts.
  • Car-Bottom Furnaces: For large, heavy components.
• Continuous Furnaces:
  • Roller Hearth Furnaces: For continuous processing of long products.
  • Pusher Furnaces: Parts are pushed through the furnace.
  • Walking Beam Furnaces: Parts move through the furnace on walking beams.

Furnace Atmospheres

• Inert Atmospheres: Argon, nitrogen, helium to prevent oxidation.
• Reducing Atmospheres: Hydrogen, carbon monoxide to remove oxides.
• Vacuum Atmospheres: For high-purity heat treatment.

Applications

• Automotive: Hardening gears, axles, and other components.
• Aerospace: Annealing and hardening of aircraft parts.
• Tool and Die: Heat treatment of tool steels.

Temperature Control

• Thermocouples are used to measure temperature inside the furnace.
• PID Controllers control the heating process to maintain desired temperature.

Heating Methods

• Electric Resistance Heating: Using heating elements.
• Gas-Fired Heating: Using burners with natural gas or propane.
• Induction Heating: Using electromagnetic induction.

Furnace Components

Heating Elements

• Resistance Heating Elements are made of alloys like Ni-Cr or Fe-Cr-Al.
• Radiant Tubes are used in gas-fired furnaces.

Refractory Materials

• Firebricks: Insulating bricks to retain heat.
• Ceramic Fibers: Lightweight insulation.

Quenching Systems

• Oil Quenching is for hardening steel parts.
• Water Quenching is for rapid cooling.
• Air Quenching is for slower cooling rates.

Safety Features

• Temperature Interlocks: To prevent overheating.
• Flame Monitoring: For gas-fired furnaces.
• Emergency Shutdown Systems: To quickly shut down the furnace in case of a problem.

Advanced Technologies

• Vacuum Carburizing: Carburizing in a vacuum environment.
• Plasma Nitriding: Surface hardening using plasma.

Typical Heat Treatment Furnace Components

• Outer Shell
• Insulation
• Heating Elements
• Workpiece
• Thermocouple
• Control Panel
• Door
• Exhaust
• Support Structure
• Heat Flow

Lecture 24: Max Flow / Min Cut

Max-Flow Problem

• Input: A directed graph $G = (V, E)$, edge capacities $c: E \rightarrow R^+$, a source node $s \in V$ and a sink node $t \in V$.
• Output: Find the maximum amount of flow from $s$ to $t$ satisfying:
  • $0 \le f(e) \le c(e)$ for each edge $e \in E$.
  • $\sum_{u:(u,v) \in E} f(u, v) = \sum_{w:(v,w) \in E} f(v, w)$ for each vertex $v \in V \setminus {s, t}$.

Ford-Fulkerson Algorithm

• Initialize flow $f(e) = 0$ for all edges $e \in E$.
• While an $s \rightarrow t$ path $P$ exists in residual graph $G_f$:
  • Compute bottleneck capacity $c_f(P) = \min_{e \in P} c_f(e)$, where $c_f(e) = c(e) - f(e)$ if $e \in E$ or $c_f(e) = f(e)$ for backward edges.
  • Augment flow along $P$ by $c_f(P)$.
• Return the flow $f$.

Residual Graph

• The residual graph contains edges with remaining capacity ($c(e) - f(e)$) and edges with flow ($f(e)$) in the reverse direction.

Max-Flow Min-Cut Theorem

• Cut: A cut $(S, T)$ is a partition of $V$ such that $s \in S$ and $t \in T$.
• Capacity of a Cut: $c(S, T) = \sum_{u \in S, v \in T} c(u, v)$.
• Theorem Statement: The maximum flow from $s$ to $t$ is equal to the minimum capacity over all $s$-$t$ cuts: $\max_{f} |f| = \min_{S,T} c(S, T)$.

Applications of Max-Flow

Bipartite Matching

• Given a bipartite graph $G = (L \cup R, E)$, find a maximum-size set of edges.
• Reduction to Max-Flow: Add a source $s$ connected to $L$ and a sink $t$ connected to $R$. Assign all edge capacities to 1.

Image Segmentation

• Problem: Separate foreground from background.
• Reduction to Min-Cut:
  • Create a graph with pixels (nodes), a source $s$ (foreground), and a sink $t$ (background).
  • Edges:
    • $s$ to pixel $i$: capacity $\alpha_i$ (cost of assigning pixel $i$ to background).
    • Pixel $i$ to $t$: capacity $\beta_i$ (cost of assigning pixel $i$ to foreground).
    • Between neighboring pixels $i$ and $j$: capacity $p_{ij}$ (penalty for assigning differently).

Circulation with Demands

• Input: Directed graph $G=(V, E)$, edge capacities $c: E \rightarrow R^+$, and node demands $d: V \rightarrow R$ (positive for demand, negative for supply), such that $\sum_{v \in V} d(v) = 0$.
• Goal: Find a flow $f$ such that $0 \le f(e) \le c(e)$ for all $e \in E$, and for each node $v$, the net flow into $v$ equals $d(v)$.

Reduction to Max-Flow

• Add a source node $s$ and a sink node $t$.
• For nodes $v$ with $d(v) > 0$, add an edge from $s$ to $v$ with capacity $d(v)$.
• For nodes $v$ with $d(v) < 0$, add an edge from $v$ to $t$ with capacity $-d(v)$.

Survey Design

• Problem: Design a survey asking customers about subsets of products such that each customer is asked a bounded number of products and information is gathered from a bounded number of customers per product.

Reduction to Circulation:

• Graph Nodes: Customers and Products.
• Graph Edges: Customer i buys product j.
• Set lower and upper bounds on edges:Edge from $s$ to customer $i$: $[c_i, c_i']$.
• Edge from product $j$ to $t$: $[p_j, p_j']$.
• Edges $(i, j)$: $[0, 1]$.

Algèbre Linéaire et Géométrie Analytique I

Vecteurs de $\mathbb{R}^{n}$ :

L'espace vectoriel $\mathbb{R}^{n}$:

$\mathbb{R}^{n}$ est l'ensemble de tous les n-uplets ordonnés de nombres réels $(x_{1}, x_{2},..., x_{n})$, où chaque $x_{i}$ est un nombre réel.

Opérations sur $\mathbb{R}^{n}$:

• Addition Vectorielle : $u + v = (u_{1} + v_{1}, u_{2} + v_{2},..., u_{n} + v_{n})$.
• Multiplication Scalaire : $cu = (cu_{1}, cu_{2},..., cu_{n})$.

Propriétés des Opérations Vectorielles:

Pour tous $u, v, w \in \mathbb{R}^{n}$ et $c, d \in \mathbb{R}$:

• Commutativité: $u + v = v + u$.
• Associativité: $(u + v) + w = u + (v + w)$.
• Vecteur Nul: Il existe un vecteur $0 \in \mathbb{R}^{n}$ tel que $u + 0 = u$.
• Inverse Additive: Pour chaque $u \in \mathbb{R}^{n}$, il existe $-u \in \mathbb{R}^{n}$ tel que $u + (-u) = 0$.
• Distributivité (scalaire sur somme de vecteurs): $c(u + v) = cu + cv$.
• Distributivité (somme de scalaires sur vecteur): $(c + d)u = cu + du$.
• Associativité (scalaire): $c(du) = (cd)u$.
• Identité Multiplicative: $1u = u$.

Combinaisons Linéaires:

Un vecteur $v \in \mathbb{R}^{n}$ est une combinaison linéaire des vecteurs $v_{1}, v_{2},..., v_{k} \in \mathbb{R}^{n}$ s'il existe des scalaires $c_{1}, c_{2},..., c_{k} \in \mathbb{R}$ tels que : $v = c_{1}v_{1} + c_{2}v_{2} +... + c_{k}v_{k}$

Indépendance Linéaire:

Les vecteurs $v_{1}, v_{2},..., v_{k} \in \mathbb{R}^{n}$ sont linéairement indépendants si l'équation $c_{1}v_{1} + c_{2}v_{2} +... + c_{k}v_{k} = 0$ a seulement la solution triviale $c_{1} = c_{2} =... = c_{k} = 0$.

Base et Dimension:

• Une base de $\mathbb{R}^{n}$ est un ensemble de vecteurs linéairement indépendants qui engendrent $\mathbb{R}^{n}$.
• La dimension de $\mathbb{R}^{n}$ est le nombre de vecteurs dans une base de $\mathbb{R}^{n}$, et la dimension de $\mathbb{R}^{n}$ est $n$.