learning Outcomes

After reading this chapter, the reader will be able to

• Understand the meaning of three processes of heat flow: conduction, convection, and radiation

• Know about thermal conductivity, diffusivity, and steady-state condition of a thermal conductor

• Derive Fourier's one-dimensional heat flow equation and solve it in the steady state

• Derive the mathematical expression for the temperature distribution in a lagged bar

• Derive the amount of heat flow in a cylindrical and a spherical thermal conductor

• Solve numerical problems and multiple choice questions on the process of conduction of heat

6.1 Introduction

Heat is the thermal energy transferred between different substances that are maintained at different temperatures. This energy is always transferred from the hotter object (which is maintained at a higher temperature) to the colder one (which is maintained at a lower temperature). Heat is the energy arising due to the movement of atoms and molecules that are continuously moving around, hitting each other and other objects. This motion is faster for the molecules with a largeramount of energy than the molecules with a smaller amount of energy that causes the former to have more heat. Transfer of heat continues until both objects attain the same temperature or the same speed. This transfer of heat depends upon the nature of the material property determined by a parameter known as thermal conductivity or coefficient of thermal conduction. This parameter helps us to understand the concept of transfer of thermal energy from a hotter to a colder body, to differentiate various objects in terms of the thermal property, and to determine the amount of heat conducted from the hotter to the colder region of an object. The transfer of thermal energy occurs in several situations:

• When there exists a difference in temperature between an object and its surroundings,

• When there exists a difference in temperature between two objects in contact with each other, and

• When there exists a temperature gradient within the same object.

Introduction

Statistical mechanics bridges the gaps between the laws of thermodynamics and the internal structure of the matter. Some examples are as follows:

• 1. Assembly of atoms in gaseous or liquid helium.

• 2. Assembly of water molecules in solid, liquid, or vapor state.

• 3. Assembly of free electrons in metal.

The behavior of all these abovementioned assemblies is totally different in different phases. Therefore, it is most significant to relate the macroscopic behavior of the system to its microscopic structure.

In this mechanics, most probable behavior of assembly are studied instead of individual particle interactions or behavior.

The behavior of assembly that is repeated a maximum time is known as most probable behavior.

hase Space

Six coordinates can fully characterize the state of any system:

• 1. Three for describing the position x, y, z and three for momentum Px, Py, Pz.

• 2. The combined position and momentum space (x, y, z, Px, Py, Pz) is called phase space.

• 3. The momentum space represents the energy of state,

For a system of N particles, there exists 3N position coordinates and 3N momentum coordinates. A single particle in phase space is known as a phase point, and the space occupied by it is known as µ-space.

olume Element ofµ-Space

4. Consider a particle having the position and momentum coordinates in the range.

13.1 Implementation of k-means Clustering and Hierarchical Clustering

In the previous chapter, we discussed various clustering algorithms. We learned that clustering algorithms are broadly classified into partitioning methods, hierarchical methods, and density-based methods. The k-means clustering algorithm follows partitioning method; agglomerative and divisive algorithms follow the hierarchical method, while DBSCAN is based on density-based clustering methods.

In this chapter, we will implement each of these algorithms by considering various case studies by following a step-by-step approach. You are advised to perform all these steps on your own on the mentioned databases stated in this chapter.

The k-means algorithm is considered a partitioning method and an unsupervised machine learning (ML) algorithm used to identify clusters of data items in a dataset. It is one of the most prominent ML algorithms, and its implementation in Python is quite straightforward. This chapter will consider three case studies, i.e., customers shopping in the mall dataset, the U.S. arrests dataset, and a popular Iris dataset. We will understand the significance of k-means clustering techniques to implement it in Python through these case studies. Along with the clustering of data items, we will also discuss the ways to find out the optimal number of clusters. To compare the results of the k-means algorithm, we will also implement hierarchical clustering for these problems.

We will kick-start the implementation of the k-means algorithm in Spyder IDE using the following steps.

• Step 1: Importing the libraries and the dataset—The dataset for the respective case study would be downloaded, and then the required libraries would be imported.

• Step 2: Finding the optimal number of clusters—We will find the optimal number of clusters by the elbow method for the given dataset.

• Step 3: Fitting k-means to the dataset—A k-means model will be prepared by training the model over the acquired dataset.

• Step 4: Visualizing the clusters—The clusters formed by the k-means model would then be visualized in the form of scatter plots.

Introduction

Humans have had a lengthy history of understanding electricity and magnetism. The tangible characteristics of light have also been studied. But in contrast to optics, electricity and magnetism—now known as electromagnetics—have been believed to be governed by different physical laws. This makes sense because optical physics as it was previously understood by humans differs significantly from the physics of electricity and magnetism. For instance, the ancient Greeks and Asians were aware of lode stone between 600 and 400 BC. Since 200 BC, China has been using the compass. The Greeks described static energy as early as 400 BC. But these oddities had no real effect until the invention of telegraphy. The voltaic cell or galvanic cell was created by Luigi Galvani and Alesandro Volta in the late 1700s, which led to the development of telegraphy. It quickly became clear that information could be transmitted using just two wires attached to a voltaic cell. The development of telegraphy was therefore prompted by this potential by the early 1800s. To learn more about the characteristics of electricity and magnetism, Andre-Marie Ampere (1823) and Michael Faraday (1838) conducted tests. Ampere's law and Faraday's law are consequently called after them. In order to comprehend telegraphy better, Kirchhoff voltage and current rules were also established in 1845. The data transmission mechanism was not well comprehended despite these laws. The cause of the data transmission signal's distortion was unknown. The ideal signal would alternate between ones and zeros, but the digital signal quickly lost its shape along a data transmission line.

Learning Outcomes

After reading this chapter, the reader will be able to

• Express the meaning of sphere of influence and collision frequency

• Derive the distribution function for the free paths among the molecules and demonstrate the concept of mean free path

• Calculate the expression for mean free path following Clausius and Maxwell

• Derive the expression for pressure exerted by a gas using the survival equation

• Calculate the expressions for viscosity, thermal conductivity, and diffusion coefficient of a gaseous system

• Demonstrate Brownian motion with its characteristics and calculate the mean square displacement of a particle executing Brownian motion

• State the idea of a random walk problem

• Solve numerical problems and multiple choice questions on the mean free path, viscosity, thermal conduction, diffusion, Brownian motion, and random walk

4.1 Introduction

The molecules of an ideal gas are considered as randomly moving point particles. From the concept of kinetic theory of gases (KTG), it is well established that even at room temperature, such point molecules of the ideal gas move at very large speeds. The average value of this speed can be determined assuming that the molecules obey Maxwell's speed distribution law and is given by the following expression

Color is a psycho-physiological property of humanvisual experiences when the eyes look at objects andlight. Color is not a physical property of thoseobjects or light, rather, it is the result of aninteraction between physical light in theenvironment and human visual system (Palmer, 1999).For processing color images, it is required todevelop an understanding on how colors arerepresented following human perception.

3.1 Light sources

A broad range of electromagnetic spectrum, shown inFig. 3.1, consists of electromagnetic waves rangingfrom very long wavelengths at radio waves to veryhigh frequency at gamma waves. A very narrowinterval in this spectrum, toward the higher end ofspectral frequencies, accounts for the visible raysand it is called the visiblespectrum. The light and colors that ahuman eye perceives relate to the frequencies ofwaves that fall under the visible spectrum. Apictorial representation of the correspondence ofwavelengths in the visible range of the spectrum todifferent perceived colors has been shown in Fig.3.1. There are seven distinguishable colors in thefigure, violet, indigo, blue, green, yellow, orange,and red, usually known in order of their increasingwavelengths by the acronym of VIBGYOR. The luminancesensitivity function that is shown as a curve inFig. 3.1 is a function of the wavelength. It isempirically observed that the sensitivity of thehuman visual system is maximum in the green zone ofthe visible spectrum. The luminance sensitivityfunction gradually decays toward violet (higherfrequencies) and red (lower frequencies) from thegreen zone, as shown in the figure by the whitecurve.

Operational Amplifiers: Introduction

An operational amplifier (op-amp) is a very prominent active device used in analog integrated circuit (IC) design. Prominence is due to the widespread and diverse areas of applications of the op-amps as its parameters are very close to ideal in a certain range of operating frequencies. Apart from basic arithmetic operations such as addition, multiplication, and integration, op-amps are also widely employed as amplifiers, wave shaping circuits, active filters, log/anti-logarithmic amplifiers, nonlinear function generators, and in analog-to digital and digital-to-analog conversion, and so on.

Figure 8.1(a) shows a pin connection diagram of the most commonly used type-741 op-amp; it needs a dual power supply, has two terminals for inverting and non-inverting inputs, one terminal for the output, and three terminals without any connections for simple applications. Dual op-amps and quad op-amp ICs with matching characteristics are also available.

Op-amp is essentially a high-gain differential amplifier (DA) that can be shown in its simplest form as represented in Figure 8.1(b). The output voltage of the op-amp is the difference between the two input voltages multiplied by the high-gain factor A, so the output voltage is expressed as:

The differential gain A is frequency dependent in a practical op-amp. Therefore, as a first approximation, it is represented by a single-pole roll-off model given below.

Key Concepts

• • The growing share of electricity in the energy sector

• • The connection of electricity and global warming

• • Important terms related to electricity

• • Conventional sources of electricity generation

• • Green and renewable sources of electricity generation

• • Smart grid

Introduction

Electricity is the fundamental driver for growth of the modern society. The availability of reliable electric supply is a priority for any residential, industrial, or commercial setup. With the rapid proliferation of digital appliances and the critical role they are playing in our daily life, the dependence on high-quality electric power supply has further increased manifold.

Electricity started as a source of energy for lighting, replacing oil and gas-based lamps. But at that time very few people would have realized that slowly this new source of energy will ‘capture’ the whole residential, industrial, and workplace setup. It is difficult to imagine our lives without electricity now – starting from heating our meals, washing and drying of our clothes, heating the water, keeping the house or office cool or hot to running all kinds of entertainment and communication appliances. This source of energy has turned into an omnipresent phenomenon in our lives. Electricity is the main driver behind technologies related to the Internet and communication also. A major part of the railways is already running on electricity, and the transition of road transport is also imminent in the near future.

Key Concepts

• • Three-D path for green energy transition

• • Operation and maintenance of solar PV power plants using digital technologies

• • Role of energy storage in the decarbonization of power grid

• • Microgrid, nanogrid, and vehicle to grid

• • Peer-to-peer energy trading

• • Latest trends in solar cell and module technology

• • Role of smart grid in enabling integration of solar and wind energy sources with the power grid

• • Wind-solar hybrid, floating solar, agro photovoltaics

• • Role of education, training, research and development in successful transition to green energy

Introduction

Rapid transition of the energy system with growing utilization of green and renewable sources has come up with a number of challenges and opportunities. This transition will continue and completely alter the whole energy network. These developments have come up at a time when a number of new and established technologies are available which need to be used and integrated in this changed network. Artificial intelligence (AI), ML, Big Data, cloud computing, blockchain, and so on are some of these important technologies.

Operation and maintenance of solar and wind plants and the role of AI, ML, Big Data and so on; peer-to-peer energy transactions and the role of blockchain in them; grid integration challenges and their solutions; off-grid applications with and without battery storage; handling of PV waste; and solar energy derivatives such as green hydrogen are the areas which are set to play very important roles in the successful transition to the green and distributed energy network.

Apart from these technologies, other important developments are underway, such as solar PV modules of higher efficiency with new technology and material, a new shape, a lesser effect on ambient temperature, requiring less water for cleaning, and so on.

Semiconductor Diodes

A semiconductor diode is a two-terminal device. Ideally, the diode behaves as a short circuit in one direction for current flow; it is called the forward direction, and the same diode behaves as an open circuit for the current flow in the opposite direction, which is called the reverse direction.

Ideal Silicon p–n Junction Diode

One of the most important characteristics of an ideal diode is that it behaves as an ideal switch, and the switching action is controlled by the direction of the current that flows in one direction only. Figure 1.1(a) shows a symbol for an ideal diode as a switch, where the arrow is in the forward direction. It also shows the positive direction of the current and voltage drop across the diode. Figure 1.1(b) shows the v, i characteristics of the ideal diode, which conducts only in one direction. The names assigned to the diode terminals are anode and cathode. The direction of the forward current in the diode is from the anode to the cathode inside the diode.

Fabricating an ideal diode in practice is not possible. Still, its idealized model in Figure 1.1(b) serves as a good approximation of a practical diode for basic analysis purposes.

At an early stage, diodes were realized using vacuum tubes with filament inside. However, now solid-state diodes are fabricated using semiconductors.

Introduction

Figure 10.1 shows a simple block diagram of a typical analog signal processing system. The first step in the development of an analog signal processing system is to divide the given specifications into analog and digital parts. Based on the advantages in VLSI technology, a variety of signal processing systems have been developed and will continue to be developed with the format of Figure 10.1. One such example is the advent of sampled data technique and MOS technology, which enabled the fabrication of a general signal processor. However, analog-sampled techniques moved onward from MOS technology toward CMOS technology, as it was highly suitable for combining analog and digital systems.

In most of the practical cases, input signals are of analog types like speech signals, sensor output, radar signals, and so on. The first block in Figure 10.1 is a pre-processing block, which usually consists of analog filters, sample and hold process, and an analog-to-digital converter (ADC). Depending on the nature of the input signal, the input block may require signal processing with strict speed and accurate specifications. After the conversion of analog signals to digital signals, the next block is mostly a microprocessor. The advantage of using a microprocessor is that its function can easily be controlled and modified. Post-processing is done in the final stage, wherein the signal is converted back to the analog form in most cases, and this stage uses digital-to-analog converter (DAC) and some filtering. Proper interfacing is required between the three building blocks of Figure 10.1; placement of the the inter facing is indicated symbolically by the arrow heads.

The climate change challenge, mainly reflected as global warming, has emerged as an existential crisis not only for humanity but for the planet itself. This challenge and the need for sustainable development are, therefore, the most talked about issues of recent times. Ensuring development without causing harm to nature is the basic idea behind sustainable development. In line with this principle, there is a need to review and reset the energy sector and make it more environment friendly.

The need for electrical energy is a basic requirement of modern society. But meeting this requirement has contributed a large part to the climate change also. Conventional methods of generating electricity have been major contributors to CO2 emissions. In order to rectify this, generating the required energy in an environment friendly way has to be implemented. The decarbonization of the electric energy system, therefore, is an integral part of reworking the electrical energy sector in the spirit of sustainable development.

India, over the years, has shown unwavering commitment to contributing towards attaining the sustainable development goals. The country has taken excellent steps in this area under the leadership of Hon’ble Prime Minister Shri Narendra Modi. A major boost to these efforts was announced in terms of the Panchamrit promises declared at the 26th Conference of Parties at Glasgow, UK. Making 50% of the total installed capacity based on non-fossil fuel sources, reducing the carbon emission intensity in its GDP by 45%, and the installation of 500 GW of green energy plants by 2030 are indicative targets of the national resolve.