The goal of this course is to achieve conceptual understanding and to retain the best traditions of traditional calculus. The syllabus is designed to provide the basic tools of calculus mainly for the purpose of modeling the engineering problems mathematically and obtaining solutions. This is a foundation course that mainly deals with topics such as single variable and multivariable calculus and plays an important role in the understanding of science, engineering, economics, and computer science, among other disciplines.
UNIT I: DIFFERENTIAL CALCULUS
Representation of functions - Limit of a function - Continuity - Derivatives - Differentiation rules - Maxima and Minima of functions of one variable.
UNIT II: FUNCTIONS OF SEVERAL VARIABLES
Partial differentiation – Homogeneous functions and Euler’s theorem – Total derivative – Change of variables – Jacobians – Partial differentiation of implicit functions – Taylor’s series for functions of two variables – Maxima and minima of functions of two variables – Lagrange’s method of undetermined multipliers.
UNIT III: INTEGRAL CALCULUS
Definite and Indefinite integrals - Substitution rule - Techniques of Integration - Integration by parts, Trigonometric integrals, Trigonometric substitutions, Integration of rational functions by partial fraction, Integration of irrational functions - Improper integrals.
UNIT IV: MULTIPLE INTEGRALS
Double integrals – Change of order of integration – Double integrals in polar coordinates – Area enclosed by plane curves – Triple integrals – Volume of solids – Change of variables in double and triple integrals.
UNIT V: DIFFERENTIAL EQUATIONS
Higher-order linear differential equations with constant coefficients - Method of variation of parameters – Homogenous equation of Euler’s and Legendre’s type – System of simultaneous linear differential equations with constant coefficients - Method of undetermined coefficients
This course is designed to cover topics such as Matrix Algebra, Vector Calculus, Complex Analysis, and Laplace Transform. Matrix Algebra is one of the powerful tools to handle practical problems arising in the field of engineering. Vector calculus can be widely used for modeling the various laws of physics. The various methods of complex analysis and Laplace transform can be used for efficiently solving the problems that occur in various branches of engineering disciplines.
UNIT I : MATRICES
Eigen values and Eigenvectors of a real matrix – Characteristic equation – Properties of Eigen values and Eigenvectors – Cayley-Hamilton theorem – Diagonalization of matrices – Reduction of a quadratic form to canonical form by orthogonal transformation – Nature of quadratic forms.
UNIT II : VECTOR CALCULUS
Gradient and directional derivative – Divergence and curl - Vector identities – Irrotational and Solenoidal vector fields – Line integral over a plane curve – Surface integral - Area of a curved surface - Volume integral - Green’s, Gauss divergence and Stoke’s theorems – Verification and application in evaluating line, surface and volume integrals.
UNIT III : ANALYTIC FUNCTIONS
Analytic functions – Necessary and sufficient conditions for analyticity in Cartesian and polar coordinates - Properties – Harmonic conjugates – Construction of analytic function - Conformal mapping – Mapping by functions 1 2 z z w z c, cz, , - Bilinear transformation.
UNIT IV: COMPLEX INTEGRATION
integral - Cauchy’s integral theorem – Cauchy’s integral formula – Taylor’s and Laurent’s series – Singularities – Residues – Residue theorem – Application of residue theorem for evaluation of real integrals – Use of circular contour and semicircular contour.
UNIT V : LAPLACE TRANSFORMS
Existence conditions – Transforms of elementary functions – Transform of a unit step function and unit impulse function – Basic properties – Shifting theorems -Transforms of derivatives and integrals – Initial and final value theorems – Inverse transforms – Convolution theorem – Transform of periodic functions – Application to solution of linear second-order ordinary differential equations with constant coefficients.
To develop the capacity to predict the effect of force and motion in the course of carrying out the design functions of engineering.
UNIT I : STATICS OF PARTICLES
Introduction – Units and Dimensions – Laws of Mechanics – Lami’s theorem, Parallelogram and triangular Law of forces – Vectorial representation of forces – Vector operations of forces -additions, subtraction, dot product, cross product – Coplanar Forces – rectangular components – Equilibrium of a particle – Forces in space – Equilibrium of a particle in space – Equivalent systems of forces – Principle of transmissibility.
UNIT II : EQUILIBRIUM OF RIGID BODIES
Free body diagram – Types of supports –Action and reaction forces – stable equilibrium – Moments and Couples – Moment of a force about a point and about an axis – Vectorial representation of moments and couples – Scalar components of a moment – Varignon’s theorem – Single equivalent force -Equilibrium of Rigid bodies in two dimensions – Equilibrium of Rigid bodies in three dimensions
UNIT III : PROPERTIES OF SURFACES AND SOLIDS
Centroids and center of mass – Centroids of lines and areas - Rectangular, circular, triangular areas by integration – T section, I section, - Angle section, Hollow section by using standard formula – Theorems of Pappus - Area moments of inertia of plane areas – Rectangular, circular, triangular areas by integration – T section, I section, Angle section, Hollow section by using standard formula – Parallel axis theorem and perpendicular axis theorem – Principal moments of inertia of plane areas – Principal axes of the inertia-Mass moment of inertia –mass moment of inertia for prismatic, cylindrical and spherical solids from the first principle – Relation to area moments of inertia.
UNIT IV : DYNAMICS OF PARTICLES
Displacements, Velocity, and acceleration, their relationship – Relative motion – Curvilinear motion - Newton’s laws of motion – Work Energy Equation– Impulse and Momentum – Impact of elastic bodies.
UNIT V : FRICTION AND RIGID BODY DYNAMICS
Friction force – Laws of sliding friction – equilibrium analysis of simple systems with sliding friction – wedge friction-. Rolling resistance -Translation and Rotation of Rigid Bodies – Velocity and acceleration – General Plane motion of simple rigid bodies such as cylinder, disc/wheel, and sphere
To familiarize the students to understand the fundamentals of thermodynamics and to perform thermal analysis on their behavior and performance. (Use of Standard and approved Steam Table, Mollier Chart, Compressibility Chart, and Psychrometric Chart permitted)
UNIT I: BASIC CONCEPTS AND FIRST LAW
Basic concepts - the concept of continuum, comparison of microscopic and macroscopic approach. Path and point functions. Intensive and extensive, total and specific quantities. System and their types. Thermodynamic Equilibrium State, path, and process. Quasi-static, reversible, and irreversible processes. Heat and work transfer, definition and comparison, sign convention. Displacement work and other modes of work. P-V diagram. Zeroth law of thermodynamics – the concept of temperature and thermal equilibrium – relationship between temperature scales –new temperature scales. The first law of thermodynamics –application to closed and open systems – steady and unsteady flow processes.
UNIT II : SECOND LAW AND AVAILABILITY ANALYSIS
Heat Reservoir, source, and sink. Heat Engine, Refrigerator, Heat pump. Statements of the second law and its corollaries. Carnot cycle Reversed Carnot cycle, Performance. Clausius inequality. Concept of entropy, T-s diagram, Tds Equations, entropy change for - pure substance, ideal gases - different processes, the principle of increase in entropy. Applications of II Law. High and low-grade energy. Available and non-available energy of a source and finite body. Energy and irreversibility. Expressions for the energy of a closed system and open systems. Energy balance and entropy generation. Irreversibility. I and II law Efficiency.
UNIT III : PROPERTIES OF PURE SUBSTANCE AND STEAM POWER CYCLE
Formation of steam and its thermodynamic properties, p-v, p-T, T-v, T-s, h-s diagrams. p-v-T surface. Use of Steam Table and Mollier Chart. Determination of dryness fraction. Application of I and II law for pure substances. Ideal and actual Rankine cycles, Cycle Improvement Methods - Reheat and Regenerative cycles, Economiser, preheater, Binary, and Combined cycles
UNIT IV : IDEAL AND REAL GASES, THERMODYNAMIC RELATIONS
Properties of Ideal gas- Ideal and real gas comparison- Equations of state for ideal and real gases- Reduced properties. Compressibility factor-.Principle of Corresponding States. -Generalised Compressibility Chart and its use-. Maxwell relations, Tds Equations, Difference and ratio of heat capacities, Energy equation, Joule-Thomson Coefficient, Clausius Clapeyron equation, Phase Change Processes. Simple Calculations.
UNIT V : GAS MIXTURES AND PSYCHROMETRY
Mole and Mass fraction, Dalton’s and Amagat’s Law. Properties of the gas mixture – Molar mass, gas constant, density, change in internal energy, enthalpy, entropy, and Gibbs function. Psychrometric properties, Psychrometric charts. Property calculations of air vapour mixtures by using charts and expressions. Psychrometric process – adiabatic saturation, sensible heating and cooling, humidification, dehumidification, evaporative cooling, and adiabatic mixing. Simple Applications.
To understand the concepts of stress, strain, principal stresses, and principal planes. To study the concept of shearing force and bending moment due to external loads indeterminate beams and their effect on stresses. To determine stresses and deformation in circular shafts and helical spring due to torsion. To compute slopes and deflections indeterminate beams by various methods. To study the stresses and deformations induced in thin and thick shells.
UNIT I : STRESS, STRAIN AND DEFORMATION OF SOLIDS
Rigid bodies and deformable solids – Tension, Compression, and Shear Stresses – Deformation of simple and compound bars – Thermal stresses – Elastic constants – Volumetric strains –Stresses on inclined planes – principal stresses and principal planes – Mohr’s circle of stress.
UNIT II : TRANSVERSE LOADING ON BEAMS AND STRESSES IN BEAM
Beams – types transverse loading on beams – Shear force and bending moment in beams – Cantilevers – Simply supported beams and over – hanging beams. Theory of simple bending– bending stress distribution – Load carrying capacity – Proportioning of sections – Flitched beams – Shear stress distribution.
UNIT III : TORSION
Torsion formulation stresses and deformation in circular and hollows shafts – Stepped shafts– Deflection in shafts fixed at both ends – Stresses in helical springs – Deflection of helical springs, carriage springs.
UNIT IV : DEFLECTION OF BEAMS
Double Integration method – Macaulay’s method – Area moment method for computation of slopes and deflections in beams - Conjugate beam and strain energy – Maxwell’s reciprocal theorems.
UNIT V : THIN CYLINDERS, SPHERES, AND THICK CYLINDERS
Stresses in thin cylindrical shells due to internal pressure circumferential and longitudinal stresses and deformation in thin and thick cylinders – spherical shells subjected to internal pressure – Deformation in spherical shells – Lame’s theorem.
To understand the mechanisms of heat transfer under steady and transient conditions. To understand the concepts of heat transfer through extended surfaces. To learn the thermal analysis and sizing of heat exchangers and to understand the basic concepts of mass transfer. (Use of standard HMT data book permitted)
UNIT I : CONDUCTION
General Differential equation of Heat Conduction– Cartesian and Polar Coordinates – One Dimensional Steady State Heat Conduction –– plane and Composite Systems – Conduction with Internal Heat Generation – Extended Surfaces – Unsteady Heat Conduction – Lumped Analysis – Semi-Infinite and Infinite Solids –Use of Heisler’s charts.
UNIT II : CONVECTION
Free and Forced Convection - Hydrodynamic and Thermal Boundary Layer. Free and Forced Convection during external flow over Plates and Cylinders and Internal flow through tubes.
UNIT III : PHASE CHANGE HEAT TRANSFER AND HEAT EXCHANGERS
Nusselt’s theory of condensation - Regimes of Pool boiling and Flow boiling. Correlations in boiling and condensation. Heat Exchanger Types - Overall Heat Transfer Coefficient – Fouling Factors - Analysis – LMTD method - NTU method.
UNIT IV : RADIATION
Black Body Radiation – Grey body radiation - Shape Factor – Electrical Analogy – Radiation Shields. Radiation through gases
UNIT V : MASS TRANSFER
Basic Concepts – Diffusion Mass Transfer – Fick’s Law of Diffusion – Steady-state Molecular Diffusion – Convective Mass Transfer – Momentum, Heat, and Mass Transfer Analogy – Convective Mass Transfer Correlations.
To introduce the concepts of Mathematical Modeling of Engineering Problems. To appreciate the use of FEM in a range of Engineering Problems.
UNIT I: INTRODUCTION
Historical Background – Mathematical Modeling of field problems in Engineering – Governing Equations – Discrete and continuous models – Boundary, Initial and Eigen Value problems– Weighted Residual Methods – Variational Formulation of Boundary Value Problems – Ritz Technique – Basic concepts of the Finite Element Method.
UNIT II: ONE-DIMENSIONAL PROBLEMS
One Dimensional Second-Order Equations – Discretization – Element types- Linear and Higher-order Elements – Derivation of Shape functions and Stiffness matrices and force vectors- Assembly of Matrices - Solution of problems from solid mechanics and heat transfer. Longitudinal vibration frequencies and mode shapes. Fourth Order Beam Equation –Transverse deflections and Natural frequencies of beams.
UNIT III: TWO DIMENSIONAL SCALAR VARIABLE PROBLEMS
Second-Order 2D Equations involving Scalar Variable Functions – Variational formulation –Finite Element formulation – Triangular elements – Shape functions and element matrices and vectors. Application to Field Problems - Thermal problems – Torsion of Non-circular shafts –Quadrilateral elements – Higher Order Elements.
UNIT IV: TWO DIMENSIONAL VECTOR VARIABLE PROBLEMS
Equations of elasticity – Plane stress, plane strain, and axisymmetric problems – Body forces and temperature effects – Stress calculations - Plate and shell elements
UNIT V: ISOPARAMETRIC FORMULATION
Natural co-ordinate systems – Isoparametric elements – Shape functions for isoparametric elements – One and two dimensions – Serendipity elements – Numerical integration and application to plane stress problems - Matrix solution techniques – Solutions Techniques to Dynamic problems – Introduction to Analysis Software.
To introduce Governing Equations of viscous fluid flows To introduce numerical modeling and its role in the field of fluid flow and heat transfer To enable the students to understand the various discretization methods, solution procedures, and turbulence modeling. To create confidence to solve complex problems in the field of fluid flow and heat transfer by using high-speed computers.
UNIT I: GOVERNING EQUATIONS AND BOUNDARY CONDITIONS
Basics of computational fluid dynamics – Governing equations of fluid dynamics – Continuity, Momentum and Energy equations – Chemical species transport – Physical boundary conditions – Time-averaged equations for Turbulent Flow – Turbulent–Kinetic Energy Equations – Mathematical behavior of PDEs on CFD - Elliptic, Parabolic, and Hyperbolic equations
UNIT II: FINITE DIFFERENCE AND FINITE VOLUME METHODS FOR DIFFUSION
Derivation of finite difference equations – Simple Methods – General Methods for first and second-order accuracy – Finite volume formulation for steady state One, Two and Three - dimensional diffusion problems –Parabolic equations – Explicit and Implicit schemes – Example problems on elliptic and parabolic equations – Use of Finite Difference and Finite Volume methods.
UNIT III: FINITE VOLUME METHOD FOR CONVECTION DIFFUSION
Finite volume methods -Representation of the pressure gradient term and continuity equation – Staggered grid – Momentum equations – Pressure and Velocity corrections – Pressure Correction equation, a SIMPLE algorithm, and its variants – PISO Algorithms.
UNIT V: TURBULENCE MODELS AND MESH GENERATION
Turbulence models, mixing length model, Two equation (k-Є) models – High and low Reynolds number models – Structured Grid generation – Unstructured Grid generation – Mesh refinement – Adaptive mesh – Software tools.