Thermodynamics Quick Reference

Cheatsheet Content

1. Basic Concepts (Thermodynamics) Thermodynamic System: A region of space or a quantity of matter chosen for study. Properties of a System: Characteristics that describe the system. Intensive Properties: Independent of mass (e.g., $T, P, \rho$ (density)). Extensive Properties: Dependent on mass (e.g., $V, U, H, S$). State: Condition of a system described by its properties. Process: Change in state of a system. Cycle: A series of processes where the system returns to its initial state. Equilibrium: No tendency for further change. Thermal Equilibrium: Uniform temperature. Mechanical Equilibrium: Uniform pressure. Chemical Equilibrium: No change in chemical composition. 2. Types of Systems (Thermodynamics) Closed System: No mass transfer, but energy transfer is possible. Open System: Both mass and energy transfer are possible. Isolated System: Neither mass nor energy transfer. Adiabatic System: No heat transfer, but work transfer is possible. Homogeneous System: Uniform chemical composition and physical state. Heterogeneous System: Non-uniform chemical composition or physical state. 3. Temperature & Pressure (Thermodynamics) Temperature Units: Celsius ($^\circ C$): Freezing point $0^\circ C$, Boiling point $100^\circ C$. Fahrenheit ($^\circ F$): $T_{^\circ F} = 1.8 \times T_{^\circ C} + 32$. Kelvin ($K$): Absolute temperature, $T_K = T_{^\circ C} + 273.15$. Rankine ($^\circ R$): $T_{^\circ R} = T_{^\circ F} + 459.67$. Pressure: Force per unit area. $P = F/A$. Absolute Pressure ($P_{abs}$): Measured relative to perfect vacuum. Gauge Pressure ($P_g$): Measured relative to atmospheric pressure. $P_{abs} = P_{atm} + P_g$. Vacuum Pressure ($P_{vac}$): Pressure below atmospheric. $P_{abs} = P_{atm} - P_{vac}$. Units: Pascal ($Pa = N/m^2$), Bar ($1 \text{ bar} = 10^5 \text{ Pa}$), $mmHg$, $H_2O$. $P = \rho g h$. 4. Energy, Work, and Heat (Thermodynamics) Energy: Capacity to do work. Internal Energy ($U$): Energy stored in a system due to molecular activity. $U \propto T$. Potential Energy ($PE$): Energy due to position. $PE = mgh$. Kinetic Energy ($KE$): Energy due to motion. $KE = \frac{1}{2}mv^2$. Flow Energy ($FE$): Energy associated with mass flow. $FE = PV$. Work ($W$): Energy transfer due to a force acting through a distance. P-dV Work: $W = \int P dV$. Positive work: System does work on surroundings (expansion). Negative work: Surroundings do work on system (compression). Heat ($Q$): Energy transfer due to temperature difference. Positive heat: System absorbs heat. Negative heat: System rejects heat. 5. Laws of Thermodynamics 5.1. Zeroth Law If two systems are each in thermal equilibrium with a third system, then they are in thermal equilibrium with each other. This defines temperature. 5.2. First Law (Conservation of Energy) Energy cannot be created or destroyed, only transformed. For a cycle: $\oint \delta Q = \oint \delta W$. For a process: $\delta Q = dU + \delta W$. Internal Energy Change: $dU = m C_v dT$. Enthalpy ($H$): Total heat content of a system. $H = U + PV$. Enthalpy Change: $dH = m C_p dT$. 5.3. Second Law Kelvin-Planck Statement: It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work. Clausius Statement: It is impossible to construct a device that operates in a cycle and produces no effect other than the transfer of heat from a lower-temperature body to a higher-temperature body. Entropy ($S$): Measure of disorder or randomness. For a reversible process: $dS = \frac{\delta Q}{T}$. For an irreversible process: $dS > \frac{\delta Q}{T}$. The entropy of an isolated system never decreases; it either increases or remains constant ($\Delta S_{isolated} \ge 0$). Carnot Cycle (Reversible Cycle): Most efficient cycle operating between two temperature reservoirs. Consists of two isothermal and two adiabatic processes. Carnot Efficiency: $\eta_{Carnot} = 1 - \frac{T_C}{T_H}$. ($T_C$ and $T_H$ are absolute temperatures). Availability (Exergy): Maximum useful work that can be obtained from a system as it comes into equilibrium with its surroundings. 6. Ideal Gases (Thermodynamics) Ideal Gas Law: $PV = nRT$ or $PV = mRT$. $P$: Absolute pressure, $V$: Volume, $n$: Moles, $m$: Mass. $R$: Specific gas constant ($R = R_u/M$, where $R_u$ is universal gas constant and $M$ is molar mass). $T$: Absolute temperature. Boyle's Law: $PV = \text{constant}$ (at constant $T$). Charles's Law: $V/T = \text{constant}$ (at constant $P$). Specific Heats: $C_p$: Specific heat at constant pressure. $C_v$: Specific heat at constant volume. Relation: $C_p - C_v = R$. Ratio of Specific Heats: $\gamma = C_p / C_v$. 7. Non-Flow Reversible Processes (Thermodynamics) Constant Volume (Isochoric): $\delta W = 0$ $Q = \Delta U = m C_v \Delta T$ Constant Pressure (Isobaric): $W = P \Delta V$ $Q = \Delta H = m C_p \Delta T$ Constant Temperature (Isothermal): $\Delta U = 0$ (for ideal gas) $Q = W = P_1 V_1 \ln\left(\frac{V_2}{V_1}\right) = P_1 V_1 \ln\left(\frac{P_1}{P_2}\right)$ Adiabatic (Isentropic for Reversible): No heat transfer ($\delta Q = 0$). $PV^\gamma = \text{constant}$ $T_1 V_1^{\gamma-1} = T_2 V_2^{\gamma-1}$ $T_1 P_1^{(1-\gamma)/\gamma} = T_2 P_2^{(1-\gamma)/\gamma}$ $W = \frac{P_1 V_1 - P_2 V_2}{\gamma - 1} = \frac{mR(T_1 - T_2)}{\gamma - 1}$ $\Delta U = -W$ Polytropic: $PV^n = \text{constant}$. ($n$ is polytropic index). $W = \frac{P_1 V_1 - P_2 V_2}{n - 1} = \frac{mR(T_1 - T_2)}{n - 1}$ $Q = \left(\frac{\gamma - n}{\gamma - 1}\right) W$ 8. Steam Properties (Thermodynamics) Specific Enthalpy of Water ($h_f$): Enthalpy of saturated liquid. Latent Heat of Vaporization ($h_{fg}$): Enthalpy change during phase transition from liquid to vapor. Specific Enthalpy of Dry Saturated Steam ($h_g$): $h_g = h_f + h_{fg}$. Dryness Fraction ($x$): Mass of dry steam / Total mass of wet steam. Specific Enthalpy of Wet Steam ($h_{ws}$): $h_{ws} = h_f + x h_{fg}$. Specific Enthalpy of Superheated Steam ($h_{sup}$): $h_{sup} = h_g + C_p (T_{sup} - T_{sat})$. 9. Internal Combustion (IC) Engines Definition: Engines where combustion of fuel and an oxidizer occurs in a confined combustion chamber. Components: Cylinder, Piston, Connecting Rod, Crankshaft, Spark Plug/Injector, Valves, Flywheel. Key Terms: TDC (Top Dead Center): Piston's uppermost position. BDC (Bottom Dead Center): Piston's lowermost position. Stroke: Distance between TDC and BDC. Bore: Diameter of the cylinder. Swept Volume ($V_s$): Volume displaced by piston ($V_s = \frac{\pi}{4} B^2 L$). Clearance Volume ($V_c$): Volume above piston at TDC. Total Volume ($V_t$): $V_t = V_s + V_c$. Compression Ratio ($r$): $r = V_t / V_c = (V_s + V_c) / V_c$. 10. Four-Stroke Engine Completes one power cycle in four strokes of the piston (two crankshaft revolutions). Intake Stroke: Piston moves from TDC to BDC. Intake valve open, exhaust valve closed. Air-fuel mixture (petrol) or air (diesel) drawn into cylinder. Compression Stroke: Piston moves from BDC to TDC. Both valves closed. Mixture/air compressed, increasing pressure and temperature. Ignition (spark or injection) occurs near TDC. Power (Expansion) Stroke: Piston moves from TDC to BDC. Both valves closed. Combustion of fuel-air mixture generates high pressure, pushing piston down. This is the only power-producing stroke. Exhaust Stroke: Piston moves from BDC to TDC. Exhaust valve open, intake valve closed. Combustion products expelled from cylinder. Applications: Cars, trucks, most modern vehicles. Advantages: Higher fuel efficiency, lower emissions, smoother operation, more torque at lower RPM. Disadvantages: More complex design, heavier, less power per displacement compared to 2-stroke. 11. Two-Stroke Engine Completes one power cycle in two strokes of the piston (one crankshaft revolution). Upward Stroke (Compression & Intake/Transfer): Piston moves from BDC to TDC. Piston compresses the fresh charge above it. Simultaneously, the rising piston creates a vacuum in the crankcase, drawing in fresh air-fuel mixture through the intake port. Downward Stroke (Power & Exhaust/Scavenging): Piston moves from TDC to BDC. Ignition occurs near TDC, pushing piston down (power stroke). As piston descends, it first uncovers the exhaust port, allowing exhaust gases to escape. Then it uncovers the transfer port, allowing the fresh mixture from the crankcase to enter the cylinder, pushing out remaining exhaust gases (scavenging). Applications: Motorcycles, lawnmowers, chain saws, marine outboards. Advantages: Simpler design (no valves), lighter, more power per displacement, able to operate in any orientation. Disadvantages: Higher fuel consumption (fuel-oil mix), higher emissions (unburnt fuel exits with exhaust), less durable, rougher idle. 12. Engine Cycles Otto Cycle (Spark Ignition Engines): 1-2: Isentropic Compression 2-3: Constant Volume Heat Addition 3-4: Isentropic Expansion 4-1: Constant Volume Heat Rejection Efficiency: $\eta_{Otto} = 1 - \frac{1}{r^{\gamma-1}}$, where $r$ is compression ratio. Diesel Cycle (Compression Ignition Engines): 1-2: Isentropic Compression 2-3: Constant Pressure Heat Addition 3-4: Isentropic Expansion 4-1: Constant Volume Heat Rejection Efficiency: $\eta_{Diesel} = 1 - \frac{1}{r^{\gamma-1}} \left[ \frac{r_c^\gamma - 1}{\gamma(r_c - 1)} \right]$, where $r_c$ is cutoff ratio.