Mechanical Engineering

\documentclass[12pt]{report} \setcounter{tocdepth}{5} \setcounter{secnumdepth}{5} \usepackage{geometry} \geometry{ a4paper, total={170mm,257mm}, left=20mm, top=20mm, } \setlength{\footskip}{45pt} \renewcommand{\familydefault}{\sfdefault} \usepackage{euler} \usepackage{parskip} \usepackage{graphicx} \usepackage{amsmath} \everymath{\displaystyle} \begin{document} \tableofcontents \chapter{Thermodynamics} \section{Thermodynamic’} \subsection{Continuum} [pressure], [volume], [specific density],~ [temperature], [internal energy], [enthalpy], [entropy] \subsection{Systems:} [a body confined in space by defined permeabilities, which separate it from its surroundings]\par \includegraphics[width=\textwidth]{sys.PNG} \subsection{State Variables:} [set of variables that describes the mathematical state of a dynamical system enough about the system to determine its future behaviour in the absence of any external forces affecting the system] \begin{enumerate} \item Intensive: [a bulk property that does not depend on the amount of material in the system] \item Extensive: [additive for subsystems because they increase and decrease as they grow] \end{enumerate} \subsection{Equilibrium} [intensive variables independent of time][no net macroscopic flows of matter or of energy, either within a system or between system] \par {\bfseries MINUS FIRST LAW OF THERMODYNAMICS:}\par \emph{“if a collection of matter entirely isolated from its surrounding has been left undisturbed for an indefinitely long time, classical thermodynamics postulates that it is in a state in which no changes occur within it, and there are no flows within it”}\par~\ {\bfseries ZEROTH LAW OF THERMODYNAMICS:}\par \emph{“if two thermodynamic systems are each in thermal equilibrium with a third one, then they are in thermal equilibrium with each other”} \subsection{Quasi-Static Processes:} [the passage of a thermodynamic system from an initial to a final state of thermodynamic equilibrium]\par [a process may take place slowly or smoothly enough to allow its description to be usefully approximated by a continuous path of equilibrium thermodynamic states then it may be approximately described by a process function that does depend on the path idealized as a quasi-static process, which is infinitely slow, enough for the system to remain in internal equilibrium]\par [baric,~choric,~thermal,~energetic,~entropic,~enthalpic]\par [special polytropic process:]
$P{V}^n=C\to W_{1-2}=\frac{P_1{V}_1-P_2{V}_2}{n-1}$ \begin{center} \begin{tabular}{|c|c|}\hline Index & Effects\\hline n=$\infty$& isochoric process\ n=0& isobaric process\ n=1&isothermal for ideal gases (from equation of state)\ n=$\gamma$& isentropic (or adiabatic) for ideal gases (from first law, energy or entropy [constant cs])\\hline \end{tabular}\end{center} \section{Pure Substances} [“the state of a simple system of pure substances at equilibrium is completely specified by two independent, intensive properties”]\par [pressure], [specific volume], [temperature]\par \begin{center} \includegraphics[width=0.49\textwidth]{pvt.png} \includegraphics[width=0.49\textwidth]{water.png}\end{center} \begin{center} \emph{-pvT diagram for pure substances, anomalous behaviour of water} \end{center} \begin{center} \includegraphics[width=0.5\textwidth]{pt.png}\par \emph{-PT diagram for pure substances}\par \includegraphics[width=0.49\textwidth]{pv.png} \includegraphics[width=0.49\textwidth]{tv.PNG} \emph{-Pv and Tv diagram for liquid-vapour region of pure substances} \end{center} ~\par [in saturation regions, pressure and temperature are not independent so to fix the state] $\text{quality}(x)=\frac{m_{\text{g}}}{m_{\text{l}}+m_{\text{g}}}$ then, $[\text{SE}]=[\text{SE of liquid}]+x[\text{change in SE on vaporization}]$ [the dependent variables are related by]\par [property tables] [equation of states]\par \begin{enumerate} \item Ideal Gas:\par [roughly accurate for weakly polar gases at low pressures and moderate temperatures] $Pv=[\text{specific gas constant}]T$ \item In-compressible Liquid:\par $P=[\text{constant}]$ \end{enumerate} \section{First Law} {\bfseries FIRST LAW OF THERMODYNAMICS}\par \emph{“the total energy of an isolated system is constant”} \subsection{Control Mass} \begin{itemize} \item Mass Conservation: ${\dot{m}}_{\text{CM}}=0$ \item Energy Conservation: ${\dot{Q}}_{\text{CM}}={\dot{E}}_{\text{CM}}+{\dot{W}}_{\text{CM}}$ ${\dot{E}}_{\text{CM}}[\text{contained in the system}]=\dot{U}+\dot{KE}+\dot{PE}$ Q_{CM}[\text{through heat}]$$$$\dot{W}_\text{CM}[\text{through work}] \end{itemize} {\bfseries Applications:} \subsubsection{Isochoric} $Q_{12}=\Delta U_{12}\left[\text{specific heat capacity},c_v={\left(\frac{\partial u(T,v)}{\partial T}\right)}_v\right]$ \subsubsection{Isobaric} $Q_{12}=\Delta H_{12}\left[\text{specific heat capacity},c_p={\left(\frac{\partial h(T,p)}{\partial T}\right)}_p\right]$ {\bfseries NOTE:}\par if [incompressible ][solid-liquid] $c_p=\frac{\partial }{\partial T}[u(T)+pv]=c_v=c$ $u_2-u_1={\int }_{T_1}^{T_2}c(T)dT,h_2-h_1=v(p_2-p_1)+{\int }_{T_1}^{T_2}c(T)dT$ if [ideal gas][vapor], $c_p=\frac{\partial }{\partial T}[u(T)+rT]=r+c_v$ $u_2-u_1={\int }_{T_1}^{T_2}{c}_v(T)dT,h_2-h_1={\int }_{T_1}^{T_2}{c}_p(T)dT$ \subsection{Control Volume} \begin{itemize} \item Mass Conservation: ${\dot{m}}_{\text{CV}}={\dot{m}}_{\text{in}}-{\dot{m}}_{\text{out}}$ $\left[\text{for 1D flow},\dot{m}=\frac{[\text{area of exhaust}][\text{velocity of exhaust}]}{[\text{specific volume}]}\right]$ \item Energy Conservation: ${\dot{Q}}_{\text{CV}}={\dot{E}}_{\text{CV}}+\dot{W}-{\dot{E}}_{\text{net}}$ ${\dot{E}}_{\text{CV}}[\text{contained in the system}]=\dot{U}+\dot{KE}+\dot{PE}$ $Q_{CV}[\text{through heat}]$ ${\dot{E}}_{\text{net}}[\text{accompanying mass flow}]={\dot{m}}_{\text{in}}[u_{\text{in}}+k{e}_{\text{in}}+p{e}_{\text{in}}]-{\dot{m}}_{\text{out}}[u_{\text{out}}+k{e}_{\text{out}}+p{e}_{\text{out}}]$ $\dot{W}[\text{through work}]=[{\dot{m}}_{\text{in}}(p_{\text{in}}{v}_{\text{in}})-{\dot{m}}_{\text{out}}(p_{\text{out}}{v}_{\text{out}})][\text{mass flow}]+{\dot{W}}_{\text{CV}}[\text{across boundary}]$ finally, ${\dot{Q}}_{\text{CV}}={\dot{E}}_{\text{CV}}+{\dot{W}}_{\text{CV}}+{\dot{m}}_{\text{out}}[h_{\text{out}}+k{e}_{\text{out}}+p{e}_{\text{out}}]-{\dot{m}}_{\text{in}}[h_{\text{in}}+k{e}_{\text{in}}+p{e}_{\text{in}}]$ \end{itemize} {\bfseries Steady Analysis} $[{\dot{m}}_{\text{CV}}={\dot{E}}_{\text{CV}}=0]$ then, ${\dot{Q}}_{\text{CV}}={\dot{W}}_{\text{CV}}+{\dot{m}}_{\text{i/o}}[(h_{\text{out}}-h_{\text{in}})+(k{e}_{\text{out}}-k{e}_{\text{in}})+(p{e}_{\text{out}}-p{e}_{\text{in}})]$ \section{Second Law} {\bfseries SECOND LAW OF THERMODYNAMICS:}\par \emph{“total entropy of an isolated system can never decrease over time, and is constant if and only if all processes are reversible”} \subsection{Other Statements} \subsubsection{Clausis’s} \emph{“heat can never pass from a cold reservoir to a warm reservoir without some other change, connected therewith, occurring at the same time”} \subsubsection{Kelvin’s} \emph{“it is impossible to devise a cyclically operating heat engine, the effect of which is to absorb energy in the form of heat from a single thermal reservoir and to deliver an equivalent amount of work”}\par \subsubsection{Planck’s} \emph{“every process occurring in nature proceeds in the sense in which the sum of the entropies of all bodies taking part in the process is increased, in the limit, i.e. for reversible processes, the sum of the entropies remains unchanged”} \par {\bfseries Corollaries:} \begin{enumerate} \item The thermal efficiency of an irreversible cycle is always less than that of reversible when both operates between the same thermal reservoirs. \item All reversible cycles operating between two thermal reservoirs have same thermal efficiency. \end{enumerate} \subsubsection{The Equality} \emph{“the algebraic sum of all the transformations occurring in a cyclical process can only be less than zero, or, as an extreme case, equal to nothing”} $dS=\frac{dQ}{T}[\text{for a reversible process}]$ $[TdS=dU+pdV]\sim [TdS=dH-Vdp]$

\subsection{Entropy Generation}

$[\text{entropy generated}]=[\text{entropy change}]-[\text{entropy change if reversible}]$ {\bfseries NOTE: (for internally reversible)}\par if $v$ = [constant][solid-liquid], $s_2(T_2)-s_1(T_1)={\int }_{T_1}^{T_2}\frac{c(T)}{T}dT$ if [ideal gas][vapor], $s(T_2,v_2)-s(T_1,v_1)={\int }_{T_1}^{T_2}\frac{c_v(T)}{T}dT+r\ln \left(\frac{v_2}{v_1}\right)$ $s(T_2,p_2)-s(T_1,p_1)={\int }_{T_1}^{T_2}\frac{c_p(T)}{T}dT-r\ln \left(\frac{p_2}{p_1}\right)$ \section{Cyclic Processes} [a linked sequence of thermodynamic processes that involve transfer of heat and work into and out of the system, while varying state variables within the system, and that eventually returns the system to its initial state in which the working system may convert heat from a warm source into useful work, and dispose of the remaining heat to a cold sink, thereby acting as a heat engine or conversely, the cycle may be reversed and use work to move heat from a cold source and transfer it to a warm sink thereby acting as a heat pump]\par \begin{center} \begin{tabular}{|c|c|c|c|c|}\hline Cycle&Compression (1$\to$2)&Heat (+) (2$\to$3)&Expansion (3$\to$4)&Heat (-) (4$\to$1)\\hline Carnot (Ideal)&isentropic&isothermal&isentropic&isothermal\ Rankine (V)&isentropic&isobaric&isentropic&isobaric\ Brayton (G) &isentropic&isobaric&isentropic&isobaric\ Otto (G)&isentropic&isochoric&isentropic&isochoric\ Diesel (G)&isentropic&isobaric&isentropic&isochoric\\hline \end{tabular}
\end{center} \subsection{Carnot Cycle} \begin{enumerate} \item Isothermal Heat Addition:\par [the pressure drops while the temperature of the gas does not change during the process because it is in thermal contact with the hot reservoir, resulting in an increase in the entropy of the gas] \item Isentropic Expansion: \par [the gas in the engine is thermally insulated from both the hot and cold reservoirs yet continues to expand by reduction of pressure, doing work on the surroundings, and losing an amount of internal energy equal to the work done causing it to cold reservoir so entropy remains unchanged] \item Isothermal Heat Rejection: \par [the gas is in thermal contact with the cold reservoir so the surroundings do work on the gas causing increasing pressure, heat to leave the system to the low temperature reservoir and the entropy of the system to decrease by same amount of entropy absorbed in first step] \item Isentropic Compression: \par [gas is thermally insulated from the reservoirs so, the surroundings do work on the gas compressing it, increasing its internal energy, and causing its temperature to rise back to initial due solely to the work added to the system, but the entropy remains unchanged] \end{enumerate} since, $\frac{T_{\text{hot}}}{T_{\text{cold}}}=\frac{Q_{\text{hot}}}{Q_{\text{cold}}}$ $\to \eta =\frac{\text{output work}}{\text{heat supplied}}=\frac{T_H-T_L}{T_H},\text{COP}=\frac{\text{desired output}}{\text{supplied work}}=\frac{T_{?}}{T_H-T_L}$ \subsection{Rankine Cycle} \begin{enumerate} \item Isentropic Compression (1$\to$ 2): [working fluid is pumped in a boiler] \item Isobaric Heat Addition (2$\to$ 3): [the compressed liquid enters a boiler, where it is heated by an external heat source to become a dry saturated (or super heated) vapour] \item Isentropic Expansion (3$\to$ 4): [the vapour expands through a turbine, generating power] \item Isobaric Heat Rejection (4$\to$ 1): [wet vapour then enters a condenser, where it is condensed to become a saturated liquid] \end{enumerate} $\eta =\frac{\text{output work}}{\text{input work}}=\frac{w_{34}-w_{12}}{q_{23}}$ \subsection{Brayton Cycle} \begin{enumerate} \item Isentropic Compression (1$\to$ 2): [ambient air is drawn into the compressor, where it is compressed] \item Isobaric Heat Addition (2$\to$ 3): [the compressed air then runs through a combustion chamber, where it is heated] \item Isentropic Expansion (3$\to$ 4): [air gives up its energy expanding through a turbine and some of which is used to drive the compressor] \item Isobaric Heat Rejection (4$\to$ 1): [reject heat through exchanger] \end{enumerate} $\eta =\frac{\text{output work}}{\text{heat supplied}}=\frac{w_{34}-w_{12}}{q_{23}}=1-\frac{T_4-T_1}{T_3-T_2}=1-{\left(\frac{1}{r_p}\right)}^{\frac{\gamma -1}{\gamma }}$ \subsection{Otto Cycle} \begin{enumerate} \item Isentropic Compression (1$\to$ 2): [compression of air mass as the piston moves from BDC to TDC] \item Isochoric Heat Addition (2$\to$ 3): [heat transfer to the working gas from an external source (through ignition) while the piston is at TDC] \item Isentropic Expansion (3$\to$ 4): [air gives up its energy expanding through BDC to TDC] \item Isochoric Heat Rejection (4$\to$ 1): [completes the cycle as heat is rejected from the air while the piston is at BDC] \end{enumerate} $\eta =\frac{\text{output work}}{\text{heat supplied}}=\frac{w_{34}-w_{12}}{q_{23}}=1-\frac{T_4-T_1}{T_3-T_2}=1-{\left(\frac{1}{r_v}\right)}^{\gamma -1}$ \subsection{Diesel Cycle} \begin{enumerate} \item Isentropic Compression (1$\to$ 2): [piston moves from BDC to TDC compressing the gas] \item Isobaric Heat Addition (2$\to$ 3): [heating the working gas from an external source (self burnt) while the piston moves from TDC to BDC] \item Isentropic Expansion (3$\to$ 4): [on impact of burning gas exanpands further to BDC] \item Isochoric Heat Rejection (4$\to$ 1): [heat is rejected to surroundings from the air while the piston is at BDC] \end{enumerate} $\eta =\frac{\text{output work}}{\text{heat supplied}}=\frac{w_{34}-w_{12}}{q_{23}}=1-\frac{1}{\gamma }\left(\frac{T_4-T_1}{T_3-T_2}\right)=1-\frac{1}{\gamma }{\left(\frac{1}{r_v}\right)}^{\gamma -1}\left[\frac{{\alpha }_{co}^{\gamma }-1}{{\alpha }_{co}-1}\right]$ \begin{center} \includegraphics[width=0.6\textwidth]{fran.PNG}\par \begin{center} Rankine Cycle \end{center} \includegraphics[width=0.5\textwidth]{cyc.PNG}
\end{center} \section{Heat Transfer} \subsection{Conduction} [an object is at a different temperature from another body or its surroundings, heat flows so that the body and the surroundings reach the same temperature, at which point they are in thermal equilibrium]\par {\bfseries Fourier’s Law (in 1D)} $[\text{rate of heat flow}]=-[\text{thermal conductivity}][\text{cross sectional area}][\text{temperature gradient}]$ \begin{enumerate} \item Plane Wall: $\dot{Q}=\frac{kA}{L}(T_2-T_1)$ \item Hollow Cylinder: $\dot{Q}=\frac{2\pi kL}{\ln \left(\frac{r_2}{r_1}\right)}(T_2-T_1)$ \end{enumerate} \subsection{Convection} [bulk flow of a fluid (gas or liquid) carries heat along with the flow of matter in the fluid]\par {\bfseries Newton’s Law of Cooling} $[\text{rate of heat flow}]=-[\text{convection coefficient}][\text{cross sectional area}][\text{temperature difference}]$ \subsection{Radiation} [transfer of energy by means of electromagnetic waves governed by the same laws]\par {\bfseries Stefan’s Law:} $[\text{rate of heat flow}]=-[\text{emissivity}][\text{stefan’s constant}][\text{cross sec area}][\text{4th power temperature diff}]$ \subsection{Electric Analogy} [in series adds, in parallel reciprocals] \end{document}