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

This chapter considers how, with animals recognised as a part of nature, legally enshrined ‘rights of nature’ could provide a basis for animals’ legal subjecthood. The chapter centres on the case of Estrellita, an Ecuadorean woolly monkey who was declared to be a subject of rights under Ecuador’s constitutionally enshrined rights of ‘pachamama’ or ‘Mother Earth’. Yet, while Estrellita’s case highlights the potential for rights of nature to serve as a source of animals’ legal subjectivity, the chapter stresses caution. First, several rights-of-nature provisions have arguably co-opted Indigenous ideas, and served to justify continued resource extraction under the guise of living in balance with nature. Second, rights-of-nature provisions maintain the ontological human/all-other-nature divide that exists in current legal systems. Finally, the rights of nature may operate as a kind of ‘eco-coverture’ by encapsulating the interests of individual animals within the sphere of nature’s interests, thereby limiting the potential scope of animals’ legal protection. The chapter concludes that we can do better than grounding animals’ legal subjecthood in the rights of nature.

This chapter presents an alternative to legal personhood and the rights of nature as the means to better include animals within the scope of legal justice. It offers the Principle of Multispecies Legality as not merely an account of animals’ legal subjectivity but of the legal subjectivity of all those beings and entities that have – or that we might, as a democratic society, choose to recognise as having – interests. The PML holds that interests-bearing entitles one to recognition as a subject of the law, with the capacity to take legal action and have one’s interests considered impartially. In rejecting sentience as the grounds of animals’ politico-legal inclusion, the PML’s account of legal subjectivity provides for animals alongside existing sentient and non-sentient legal subjects, like humans and corporations. It also leaves the door open for other valuable entities that currently lack legal subjecthood, such as plants, fungi, bodies of water, and ecosystems. The chapter argues that the inclusivity of the PML is beneficial not only for animals and other non-human entities but also for those humans whose legal subjectivity remains tenuous under existing personhood paradigms.

Learning Outcomes

After reading this chapter, the reader will be able to

• State the assumptions of kinetic theory of gases (KTG)

• Explain the concept of pressure and calculate the expression for it

• Demonstrate mathematically the gas laws using the expression for pressure derived from KTG

• Present the kinetic interpretation of temperature

• Derive the expression for specific heat at constant volume 𝐶𝑉 and constant pressure 𝐶𝑃

• Explain the concept of degree of freedom

• Solve numerical problems and multiple choice questions on KTG

2.1 Introduction

The kinetic theory of gases (KTG) is a theoretical model that describes the physical properties of a gaseous system in terms of a large number of submicroscopic particles, such as atoms, molecules, and small particles. These constituent elements are in random motion and collide constantly with each other and also with the walls of the container. Considering the molecular composition and characteristic features of such random motion of the molecules, various macroscopic properties of the gaseous system, such as pressure, temperature, viscosity, thermal conductivity, and mass diffusivity can be explained with the help of KTG. In this theory, it is postulated that the pressure exerted by a gas is due to the collision of atoms or molecules moving at different velocities on the walls of a container. It basically attempts to explain the macroscopic properties that are related to the microscopic phenomenon. The physical properties of solids and liquids, in general, are described by their shape, size, mass, volume, etc. Gases, however, have no definite shape, and size. Furthermore, their mass and volume are not directly measurable. In such cases, the KTG can be successfully applied to extract the physical properties of the gaseous system.

7.1 INTRODUCTION [LO1]

The problems of stresses and deformations in disks rotating at high speeds are important in the design of both gas and steam turbines, generators and many such rotating machinery in industry. As discussed in earlier chapters, this is another example of axisymmetric problems in polar coordinates. Although the theoretical treatment of a flat disk is simpler, in many industrial applications, disks are tapered. They are usually thicker near the hub, and their theoretical analysis is slightly more involved. We shall first take up the analysis for flat disks.

In the case of rotating disks with centrifugal force as body force, the equation of equilibrium reduces to as in equation (6.1.3).

Combining this with displacement equations, we have, as in equation (6.1.5), a general equation for determining the stress distribution in axisymmetric problems. This is given as

This is a nonhomogeneous differential equation. The associated homogeneous equation (complementary equation) is

The solution of this equation is Lame's equation as discussed in Chapter 6, equation (6.2.3), and taking into consideration the particular solution, the solution to equation (7.1.2) turns out to be

We may also determine the radial displacement from equation (6.2.11), and this is given as

We may therefore write the stresses and displacement for the rotating disk under one bracket as

With these introductory basic equations, we shall now set out to discuss the stress distribution and displacement in rotating disks.

14.1 Introduction to Association Rule Mining

Association rule mining is a rule-based technique to discover the relation between the attributes of a dataset. It is used to find the relation between the sales of item X and item Y. It is often called a “market basket” analysis, as shown in Figure 14.1. Here, the market analyst examines the items that consumers often purchase together to find the relation between the sale of item X and item Y.

In other words, when customers visit a store, they may buy a certain type of items together during a shopping trip. For example, as shown in Figure 14.1, a database of customer’s transactions (e.g., shopping baskets) is shown where each transaction consists of a set of items (e.g., products) purchased during a visit, machine learning (ML) engineers can use association mining for finding out a group of items which are frequently purchased together (customers purchasing behavior). This is also referred to as an analysis of customer purchasing behavior. For example, “IF one buys bread, THEN there is a high probability of buying butter with it”, as it is common that people who buy bread often buy butter with it. The store manager can use this information and arrange the items accordingly to increase sales and the overall efficiency of the store.

Let us consider a situation where the store manager feels that there is a lot of rush and customers always complain about the slow working of his store. He is exploring different ways to improve the efficiency of his store. He performed an association analysis and prepared a list of associated items like bread and butter. He may decide to put all these associated items together on the same shelf or near each other so that customers can find them quickly, reducing their shopping time. It will also improve the overall efficiency of the store and the sale of the products. To further improve the shopping experience of his customers, he can create different combos and put sales over these combos.

22.1 Intuition of Genetic Algorithm

Genetic algorithm (GA) is inspired by nature, and it plays a vital role in the field of machine learning (ML). It selects the best-optimized solution from all available possible solutions or candidates. As nature selects the best possible candidates using the theory of evolution, in the same way, the GA selects the best possible solution from the available solutions.

One of the applications of GAs in ML is to select the global minima from all possible (local) minima by using natural selection. In earlier chapters, we learned that during the training of an artificial neural network, the main goal is to obtain the weights with a minimum cost function value. The gradient descent algorithm is commonly used to find the local minima of the cost function. But, we must find the global minima to reach the optimal weights. A GA can be used to find the global minima out of all available local minima or possible solutions. In this case, the set of possible local minima becomes the population containing possible candidates.

In this chapter, we will discuss inspiration from nature which is the main driving concept in working of GAs and their implementation. To get a good idea about the GA, we will discuss the basics of natural selection by revisiting the theory of evolution in the next section.

22.2 The Inspiration behind Genetic Algorithm

The concepts discussed in this chapter are also available in the form of the free online Udemy Course, Genetic Algorithm for Machine Learning by Parteek Bhatia,

https://www.udemy.com/course/genetic-algorithm-for-machine-learning/

The GA is one of the first and most well-regarded evolutionary algorithms in computer science literature. John Holland, a researcher at the University of Michigan, gave this algorithm in the 1970s, but it became popular in the ‘90s.

3.1 Need for Data Pre-processing

We live in an age where data is considered oil because we need data to train machine learning (ML) algorithms. The most important job for a data analyst is to collect, clean, and analyze the data and build ML models on the cleaned dataset. But often, the raw data that we obtain is noisy. It consists of many discrepancies, inconsistencies, and often missing values. To understand this situation, let us consider an example.

Suppose we have to predict the house price, and for this, we have collected data from a few previous transactions, as shown in Figure 3.1.

In a perfect situation, the captured data should be of this format, as shown in Figure 3.1. Here, we have the size of the house and the number of bedrooms as input features, while the price is the output attribute. We can predict the price of an unknown instance through regression.

But practically, in most situations, the captured data is not of good quality, and usually, we have a dataset, as shown in Figure 3.2.

You can see that this data is messy. There are a lot of unknown or missing values, and if we trained the model on this data, its prediction would be very poor. Also, you can identify the noise and incorrect labels like the second record price is incorrect and will result in poor model training.

We can also consider some more examples like if someone entered –1 in the “salary credited” column in the case of employee dataset. It does not make any sense and will be considered noise. Sometimes, we may have an unrealistic and impossible combination of data; for example, let us consider a record where we have Gender–Male and Pregnant–Yes.

6.1 INTRODUCTION [LO1]

In earlier chapters, we have discussed axisymmetric problems in two-dimensional (2D) polar coordinate systems. Thick cylinders fall into this class of problems. Cylindrical pressure vessels, hydraulic cylinders, gun-barrels, and pipes carrying fluids at high pressure develop radial and tangential stresses (circumferential). Longitudinal stresses can also be developed if the ends are closed. Therefore, ideally, this is a triaxial stress system as shown in Figure 6.1.

• (a) Circumferential or hoop stress (σθ)

• (b) Longitudinal stress (σz)

• (c) Radial stress (σr)

If the wall thickness of a hollow cylinder is less than about 10% of its radius, it may be treated as a thin cylinder. Cylinders with higher wall thickness are considered to be thick cylinders. Before analyzing the stress in a thick cylinder, we should briefly consider the stress state in thin cylinders, where radial stress is small compared to the other stresses, and this can be neglected. Stress variation across the thin wall is also negligible. Analysis of thin-walled pressure vessels may therefore be carried out on the basis of biaxial stress system. Since the presence of shear stress at the cut section would lead to incompatible distortion, the longitudinal and circumferential stresses in this case are both principal stresses. We now take another section of the cut section as shown in Figure 6.2 (a) to consider the equilibrium of the section, and this is shown in Figure 6.2 (b).

The section is acted upon by internal pressure p and the circumferential stress developed at the cut section is σθ. Force on an infinitesimal small area subtended by angle dθ at θ inclination from the horizontal axis is pridθ.

Today, Python is known to be one of the most in-demand programming languages. As per the stats of GitHub (a provider of Internet hosting for software development), Python is the second most popular programming language, following JavaScript, as shown in Figure 2.1, and soon it may be on the top of the chart. Python surpassed Java, PHP, and other prominent languages in 2019.

Python is easy and versatile. So it is acclaimed as the major programming language to work on many new-age technologies like machine learning (ML), artificial intelligence, data science, and natural language processing. The creator of Python, Guido van Rossum, in 1991, stated that Python is a high-level programming language, and its core design philosophy is about code readability and syntax, which allows programmers to express concepts in a few lines of code. Interestingly, the name Python is inspired by Guido's favorite television show Monty Python's Flying Circus.

In this chapter, we will discuss various programming constructs of Python so that you can easily implement ML algorithms by using it. Before writing the actual code in Python, let us focus on the features of Python that make it so popular and unique.

2.1 Features of Python

Features offered by Python can be visualized in Figure 2.2. Talking about them profoundly, the main features of Python are as follows:

• • Beginner's Language: Python is not only just easy to code and learn, but also fast to grasp, and hence it is a suitable choice for any novice user who wants to learn to program. This is why nowadays this language is introduced to students in schools.

• • Interpreted: Unlike other programming languages such as C or C++, Python does not require you to compile programs before executing them. It is an interpreted language, i.e., the code written in Python gets processed in real-time line by line.

• • Interactive: The interactive feature of Python enables real-time feedback, allowing programmers to experiment, debug, and make adjustments on the go.

5.1 Introduction to Simple Linear Regression

As discussed in earlier chapters, regression predicts a continuous value or real-valued output. This chapter will discuss how regression works (from a mathematical aspect) to predict the continuous value for the given dataset. Our first learning algorithm is simple linear regression. In this section, we will discuss the fundamental concepts and mathematical modeling of simple linear regression.

We usually have a dependent variable having a continuous value whose value we wish to predict based on one or more independent variables. If we have only one independent or input variable, this situation is known as simple linear regression (also called univariate regression). If we have multiple independent or input variables, it is known as multiple linear regression or multivariate regression.

Linear regression could be used for studying patterns in different real-life scenarios. Consider a research lab where a researcher wants to understand how the stipend is effected by the years of experience, or, in simple words, we wish to predict the stipend based on the years of experience of the researcher. Machine learning (ML) is about learning from past experiences or data. Thus, to predict the researcher's stipend, we have to collect some data about past researchers, specifically their stipend and experience.

In the supervised learning models, we need a dataset called a training set. We will use the dataset as given in Table 5.1 for training the model, and our job will be to build the ML model that learns from this data and hence predicts the stipend of a researcher based on his experience. Here, the stipend will be considered the dependent or output variable because it depends on the researcher's years of experience. Thus, years of experience will be considered an independent or input variable. So, we will use simple linear regression to build the ML model. For proceeding with this problem, we will use a dataset of researchers’ stipends with their corresponding years of experience, as shown in Table 5.1.

Learning Outcomes After reading this chapter, the reader will be able to

• Know various types of thermodynamic systems such as open, closed, and isolated, and the surroundings

• Classify between intensive and extensive thermodynamic variables

• Understand various types of equilibrium conditions satisfied by a thermodynamic system

• State the zeroth law of thermodynamics and highlight its physical significance

• Comprehend the idea of temperature from the zeroth law of thermodynamics

• Solve numerical problems and multiple choice questions on thermodynamic equilibrium and the zeroth law of thermodynamics

7.1 Introduction

Heat is a form of energy. It can be transformed from one form to another as well as can be transferred between various objects maintained at suitable temperatures. For example, in an electric motor, heat is transformed into mechanical energy by the turbine to power the motor. This mechanical energy is then transformed into electrical energy by the engine to illuminate light bulbs. “Thermodynamics” is a branch of physics that deals with heat and the transformation of heat from one form to another, work, temperature, and their relation to energy, entropy, and other physical properties of matter and radiation. It establishes the relation between heat and various forms of energy and describes the transformations that occur in thermal energy from one energy state to another and how this transformation affects matter. A thermodynamic system is described within a framework based on the four laws of thermodynamics that facilitate a quantitative description of the average macroscopic properties of the system in equilibrium. Macroscopic matter refers to large objects that consist of many atoms and molecules. The average properties of such macroscopic systems are determined by the physical quantities such as volume, pressure, and temperature that do not depend upon the detailed microscopic positions and velocities of the atoms and the molecules comprising the macroscopic system. In the equilibrium state of a thermodynamic system, these average properties also do not change with time. These physical quantities are called thermodynamic coordinates, variables, or parameters. If a subset of these properties are experimentally measured, the rest of them can be calculated using thermodynamic relations. Thermodynamics not only gives the exact description of the state of equilibrium but also provides an approximate description (to a very high degree of precision!) of relatively slow processes. This branch of physics can be successfully applied to a wide variety of topics in science, such as physics, physical chemistry, biochemistry, chemical engineering, and mechanical engineering, but also in other complex fields, such as meteorology.

5.1 INTRODUCTION [LO1]

In any three-dimensional (3D) elasticity problem, there are 15 unknown parameters: 6 stress components, 6 strain components, and 3 displacements. There are 15 related equations: 3 equations of equilibrium, 6 compatibility equations, and 6 constitutive equations. Solutions to a particular elasticity problem require evaluation of these 15 unknown parameters using 15 equations, satisfying all the boundary conditions. As discussed in the earlier chapters, there may be displacement or stress, or mixed boundary conditions. In many cases, solutions to 3D problems are not easy analytically. Even numerical solutions may be difficult.

There are mainly three methods of simplification of solution techniques:

• (a) If the boundary conditions are in terms of stresses, stress function approach may be made as discussed in the earlier chapter. This makes the solution simpler.

• (b) Assumptions of plane stress and plane strain reduce 3D problems to two-dimensional (2D) ones and this also makes the solution simpler.

• (c) Use of St. Venant's principle and superposition principle also makes the solution of elasticity problems simpler.

An introduction to stress function approach has been discussed in Chapter 4. We therefore start our discussion on plane stress and plane strain approaches.

5.2 PLANE STRESS AND PLANE STRAIN PROBLEMS [LO2]

The idealizations of both plane stress and plane strain states are suitable for certain classes of problems that are made to reduce the complexity of solutions. We shall consider the plane stress state first.

Learning Outcomes

After reading this chapter, the reader will be able to

• Learn the basic concept of the theory of probability

• List the assumptions used in the derivation of Maxwell's speed distribution law

• Derive Maxwell's speed distribution law and test its validity experimentally

• Calculate average, root mean square and most probable speed, energy, and momentum in one, two, and three dimensions, respectively

• State and prove the law of equipartition of energy

• Calculate the specific heat of gases

• Solve numerical problems and multiple choice questions on the distribution of molecular speed, energy, and momentum

3.1 Introduction

In Chapter 2, various characteristic features of a gaseous system based on the model of the kinetic theory of gases (KTG) have been discussed elaborately. Macroscopic properties and various relations among the thermodynamic variables have been explained in terms of this kinetic model.

According to the assumptions used in this model, a gaseous system is composed of a large number of particles (atoms or molecules) with practically no volume occupied by them. Most of the times, these molecules move randomly through empty space at temperatures above absolute zero, and such motions remain unaffected by the presence of other particles. This motion of the molecules is extremely chaotic and is characterized by straight-line trajectories interrupted by collisions with other molecules or with a physical boundary. In such a collision, the transfer of kinetic energy with a change in direction takes place depending on the nature of the relative kinetic energies of the particles. Any individual molecule collides with others at a huge rate, typically of the order of a billion times per second. This chapter is focused to present a comprehensive and quantitative discussion on the distributions of velocities, energies, and momenta of these molecules in various dimensions.

Measurement of the velocities of the molecules at a given time leads to a large distribution of values; some molecules may move very slowly and others very quickly. As these molecules move constantly in different directions, the velocity could be momentarily equal to zero