FE Exam Thermo

Cheatsheet Content

### Thermodynamic Properties - **Specific Volume:** $v = V/m = 1/\rho$ (m$^3$/kg) - $V$: total volume (m$^3$) - $m$: mass (kg) - $\rho$: density (kg/m$^3$) - **Specific Internal Energy:** $u = U/m$ (kJ/kg) - $U$: total internal energy (kJ) - **Specific Enthalpy:** $h = u + Pv$ (kJ/kg) - $P$: pressure (kPa) - **Specific Entropy:** $s = S/m$ (kJ/(kg·K)) - $S$: total entropy (kJ/K) - **Ideal Gas Law:** $Pv = RT$ or $PV = mRT$ or $PV = N\bar{R}T$ - $T$: absolute temperature (K) - $R$: specific gas constant (kJ/(kg·K)) - $\bar{R}$: universal gas constant (8.314 kJ/(kmol·K)) - $N$: number of moles (kmol) - $M$: molar mass (kg/kmol) - **Compressibility Factor:** $Z = Pv/(RT)$ (for real gases) - $Z = 1$ for ideal gases - **Specific Heats:** - $c_v = (\partial u / \partial T)_v$: specific heat at constant volume (kJ/(kg·K)) - $c_p = (\partial h / \partial T)_p$: specific heat at constant pressure (kJ/(kg·K)) - For ideal gases: $c_p - c_v = R$, $k = c_p / c_v$ - $k = c_p / c_v$: specific heat ratio (dimensionless) - $\Delta u = c_v \Delta T$, $\Delta h = c_p \Delta T$ (for ideal gases) ### First Law of Thermodynamics: Closed Systems - **Energy Balance:** $Q - W = \Delta E = \Delta U + \Delta KE + \Delta PE$ - $Q$: heat transfer (kJ) - $W$: work done (kJ) - $\Delta E$: change in total energy (kJ) - $\Delta KE$: change in kinetic energy (kJ) - $\Delta PE$: change in potential energy (kJ) - For stationary systems: $Q - W = \Delta U$ - **Work:** - **Boundary Work:** $W_b = \int P dV$ - For constant pressure: $W_b = P(V_2 - V_1)$ - **Heat Transfer Modes:** - **Conduction:** $\dot{Q} = -kA \frac{dT}{dx}$ - $\dot{Q}$: heat transfer rate (W) - $k$: thermal conductivity (W/(m·K)) - $A$: area (m$^2$) - $dT/dx$: temperature gradient (K/m) - **Convection:** $\dot{Q} = hA(T_s - T_\infty)$ - $h$: convection heat transfer coefficient (W/(m$^2$·K)) - $T_s$: surface temperature (K) - $T_\infty$: fluid temperature (K) - **Radiation:** $\dot{Q} = \epsilon \sigma A (T_s^4 - T_{surr}^4)$ - $\epsilon$: emissivity (dimensionless) - $\sigma$: Stefan-Boltzmann constant (5.67 x 10$^{-8}$ W/(m$^2$·K$^4$)) - $T_{surr}$: surrounding temperature (K) ### First Law of Thermodynamics: Open Systems (Control Volumes) - **Mass Balance:** $\sum \dot{m}_{in} - \sum \dot{m}_{out} = \frac{dm_{CV}}{dt}$ - $\dot{m}$: mass flow rate (kg/s) - $dm_{CV}/dt$: rate of change of mass in control volume (kg/s) - For steady flow: $\sum \dot{m}_{in} = \sum \dot{m}_{out}$ - $\dot{m} = \rho A V_{avg} = A V_{avg} / v$ - $V_{avg}$: average velocity (m/s) - **Energy Balance (Steady Flow):** $\dot{Q} - \dot{W} = \sum \dot{m}_{out}(h + \frac{V^2}{2} + gz) - \sum \dot{m}_{in}(h + \frac{V^2}{2} + gz)$ - $\dot{Q}$: rate of heat transfer (kW) - $\dot{W}$: rate of work done (power) (kW) - $g$: acceleration due to gravity (m/s$^2$) - $z$: elevation (m) - For single inlet/outlet: $\dot{Q} - \dot{W} = \dot{m}[(h_2 - h_1) + \frac{V_2^2 - V_1^2}{2} + g(z_2 - z_1)]$ - **Common Devices:** - **Nozzle/Diffuser:** $\dot{W}=0, \dot{Q}=0, \Delta PE=0$. $\frac{V_2^2 - V_1^2}{2} = -(h_2 - h_1)$ - **Turbine:** $\dot{Q}=0, \Delta KE=0, \Delta PE=0$. $\dot{W}_{out} = \dot{m}(h_1 - h_2)$ - **Compressor/Pump:** $\dot{Q}=0, \Delta KE=0, \Delta PE=0$. $\dot{W}_{in} = \dot{m}(h_2 - h_1)$ - **Throttling Valve:** $\dot{W}=0, \dot{Q}=0, \Delta KE=0, \Delta PE=0$. $h_1 = h_2$ - **Heat Exchanger:** $\dot{W}=0, \Delta KE=0, \Delta PE=0$. $\dot{Q}_{in} = \dot{m}_{hot}(h_{1,hot} - h_{2,hot}) = \dot{m}_{cold}(h_{2,cold} - h_{1,cold})$ ### Second Law of Thermodynamics & Entropy - **Clausius Statement:** No process is possible whose sole result is the transfer of heat from a colder body to a hotter body. - **Kelvin-Planck Statement:** No system can operate in a cycle and produce a net amount of work while exchanging heat with a single thermal reservoir. - **Entropy Change:** $\Delta S = \int \frac{\delta Q}{T} + S_{gen}$ - For internally reversible process: $\Delta S = \int \frac{\delta Q}{T}$ - For adiabatic reversible (isentropic) process: $\Delta S = 0$ - $S_{gen}$: entropy generation (kJ/K) - **Isentropic Efficiency:** - **Turbine:** $\eta_T = \frac{W_a}{W_s} = \frac{h_1 - h_{2a}}{h_1 - h_{2s}}$ - $\eta_T$: turbine isentropic efficiency (dimensionless) - **Compressor/Pump:** $\eta_C = \frac{W_s}{W_a} = \frac{h_{2s} - h_1}{h_{2a} - h_1}$ - $\eta_C$: compressor/pump isentropic efficiency (dimensionless) - **Nozzle:** $\eta_N = \frac{V_{2a}^2}{V_{2s}^2}$ - $\eta_N$: nozzle isentropic efficiency (dimensionless) - Subscript 'a': actual - Subscript 's': isentropic - **Carnot Cycle:** - **Thermal Efficiency:** $\eta_{th,Carnot} = 1 - T_L/T_H$ - $\eta_{th,Carnot}$: Carnot thermal efficiency (dimensionless) - $T_L$: low temperature reservoir (K) - $T_H$: high temperature reservoir (K) - **COP (Refrigerator):** $COP_{R,Carnot} = T_L / (T_H - T_L)$ - $COP_{R,Carnot}$: Carnot coefficient of performance for refrigerator (dimensionless) - **COP (Heat Pump):** $COP_{HP,Carnot} = T_H / (T_H - T_L)$ - $COP_{HP,Carnot}$: Carnot coefficient of performance for heat pump (dimensionless) ### Power & Refrigeration Cycles - **Rankine Cycle (Vapor Power):** 1. Pump (Isentropic compression) 2. Boiler (Constant pressure heat addition) 3. Turbine (Isentropic expansion) 4. Condenser (Constant pressure heat rejection) - **Brayton Cycle (Gas Power):** 1. Compressor (Isentropic compression) 2. Combustion Chamber (Constant pressure heat addition) 3. Turbine (Isentropic expansion) 4. Heat Rejection (Constant pressure) - **Otto Cycle (Spark-Ignition Internal Combustion):** 1. Isentropic compression 2. Constant volume heat addition 3. Isentropic expansion 4. Constant volume heat rejection - **Diesel Cycle (Compression-Ignition Internal Combustion):** 1. Isentropic compression 2. Constant pressure heat addition 3. Isentropic expansion 4. Constant volume heat rejection - **Vapor-Compression Refrigeration Cycle:** 1. Evaporator (Constant pressure heat absorption) 2. Compressor (Isentropic compression) 3. Condenser (Constant pressure heat rejection) 4. Expansion Valve (Isenthalpic expansion) ### Psychrometrics - **Relative Humidity:** $\phi = P_v / P_g$ (dimensionless) - $P_v$: partial pressure of water vapor (kPa) - $P_g$: saturation pressure of water at dry-bulb temperature (kPa) - **Specific Humidity (Humidity Ratio):** $\omega = 0.622 P_v / (P_{atm} - P_v)$ (kg water/kg dry air) - $P_{atm}$: atmospheric pressure (kPa) - **Dew-Point Temperature:** Temperature at which condensation begins ($T_{dp}$) (K or $^\circ$C) - **Wet-Bulb Temperature:** Temperature measured by a thermometer with a wet wick ($T_{wb}$) (K or $^\circ$C) - **Enthalpy of moist air:** $h = h_a + \omega h_g$ (kJ/kg dry air) - $h_a$: enthalpy of dry air (kJ/kg dry air) - $h_g$: enthalpy of saturated water vapor at dry-bulb temperature (kJ/kg water) - **Adiabatic Saturation Process:** Basis for psychrometric charts