Thermodynamics I: Master Cheat Sheet
Mastering thermodynamics hinges on accurately identifying system types, understanding state properties, and applying conservation laws and process path equations to analyze energy transfer and system changes.
Core Principles
- Closed System: Mass is conserved (dm=0), but energy (Q, W) can cross the boundary.
- Open System (SSSF): Mass flow rate in equals mass flow rate out (m_dot_in = m_dot_out).
- First Law (Closed): Energy balance is Q - W = Delta_U.
- First Law (Open): Energy balance includes flow work: Q_dot - W_dot = m_dot * (h_2 - h_1 + KE + PE).
- Second Law (Carnot Efficiency): Maximum theoretical efficiency is limited by temperature: eta_th_max = 1 - T_L/T_H.
Action Steps
- 1. Identify System Type: Determine if the system is closed or open to apply correct mass and energy conservation principles.
- 2. Identify Substance & State: Use property tables for pure substances; check saturation temperature for phase; use ideal gas law (P*v << 1) or real gas equation (Z factor) otherwise.
- 3. Determine Process Path: Select the correct boundary work formula based on the process (isochoric, isobaric, isothermal, polytropic).
- 4. Apply First Law: Write the energy balance equation (Q - W = Delta_U for closed systems; Q_dot - W_dot = m_dot * Delta_H + KE + PE for open systems).
- 5. Apply Second Law (for cycles): Calculate theoretical efficiency using Carnot limits (eta_th_max = 1 - T_L/T_H) or COP for refrigerators.
Formulas
- Mass Flow Rate: $m_{\dot{}} = \rho \cdot A \cdot V$
- Ideal Gas Law: $P \cdot V = m \cdot R \cdot T$
- Real Gas Deviation: $Z = v_{\text{actual}} / v_{\text{ideal}}$
- Isochoric Process Work: $W_b = 0$
- Isobaric Process Work: $W_b = P \cdot (V_2 - V_1)$
- Isothermal Process Work: $W_b = P_1 \cdot V_1 \cdot \ln(V_2/V_1)$
- Polytropic Process Work: $W_b = \frac{P_2 V_2 - P_1 V_1}{1-n}$
- First Law (Closed System): $Q - W = \Delta U$
- First Law (Open System SSSF): $Q_{\dot{}} - W_{\dot{}} = \dot{m} \cdot (h_2 - h_1 + KE + PE)$
- Nozzle/Diffuser Energy Balance: $h_1 + \frac{V_1^2}{2} = h_2 + \frac{V_2^2}{2}$
- Heat Engine Efficiency: $\\eta_{\text{th}} = 1 - Q_L/Q_H$
- Refrigerator COP: $COP_R = Q_L / (Q_H - Q_L)$
- Carnot Efficiency: $\\eta_{\text{th, max}} = 1 - T_L/T_H$
- Ideal Gas Entropy Change: $s_2 - s_1 = c_v \ln(T_2/T_1) + R \ln(v_2/v_1)$
- Turbine Efficiency: $\\eta_t = W_{\text{actual}} / W_{\text{isentropic}}$
Key Terms
- Closed System: A system where mass cannot cross the boundary, but energy (heat and work) can.
- Open System (SSSF): A system where both mass and energy can cross the boundary, operating at steady state (mass flow in equals mass flow out).
- Pure Substance: A substance that is uniform in chemical composition throughout and exists in one phase or a combination of phases (e.g., water, ammonia).
- Isentropic Process: A reversible adiabatic process where entropy remains constant.
- Carnot Cycle: The most efficient possible thermodynamic cycle operating between two heat reservoirs, consisting of two isothermal and two isentropic processes.
Pro Tips
- Use property tables for pure substances instead of the ideal gas law (P*v = R*T) unless the pressure is significantly lower than saturation pressure (P_r << 1).
- For n=1 in a polytropic process, use the isothermal work formula as it simplifies to that case.
- Recognize that a piston on stops implies constant volume (V=const), meaning boundary work (W_12) is zero.
Pitfalls to Avoid
- Assuming ideal gas behavior for real gases at high pressures or low temperatures, leading to inaccurate calculations of volume and work.
- Forgetting to account for kinetic and potential energy changes in open system energy balances, especially in nozzles and diffusers.
- Confusing heat engine efficiency with refrigerator COP, resulting in incorrect performance assessments.
Myth vs Reality
- The ideal gas law (PV=mRT) is always applicable.: The ideal gas law is an approximation valid only when the gas is at low pressure and high temperature (P_r << 1). Real gases deviate, requiring the compressibility factor Z (Z = v_actual / v_ideal).
- All processes involving no heat transfer (Q=0) are isentropic.: Only reversible adiabatic processes are isentropic. Irreversible adiabatic processes (Q=0 but with friction or other irreversibilities) result in an increase in entropy (Clausius Inequality: Integral(dQ/T) <= 0).
Real World Examples
- A piston-cylinder device containing steam expands at constant pressure.: Use the isobaric process work formula: W_b = P * (V_2 - V_1).
- Steam flowing through a turbine.: Apply the open system first law: Q_dot - W_dot = m_dot * (h_2 - h_1 + KE + PE), often assuming negligible KE/PE changes and Q=0 for adiabatic turbines.
- A gas undergoing compression where P*V^n = constant.: Use the polytropic work formula: W_b = (P_2*V_2 - P_1*V_1)/(1-n).
Statistics
- Ideal Gas Condition: P_r << 1 (Reduced pressure significantly less than 1)
More like this