2.3 Applications of the First Law

Energy analysis using the First Law is crucial for understanding how systems conserve and transfer energy. This concept applies to closed systems like piston-cylinders and open systems like turbines, helping us calculate changes in internal energy, heat, and work.

Heat engines, refrigerators, and heat pumps are practical applications of energy analysis. By using the First Law, we can determine their efficiency or coefficient of performance, which is essential for optimizing these devices in real-world scenarios.

Energy Analysis using the First Law

Energy balances in systems

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• First Law of Thermodynamics states energy is conserved in a system
  • Change in internal energy ($\Delta U$) equals sum of heat added to system ($Q$) and work done on system ($W$)
  • Mathematical representation: $\Delta U = Q + W$
• For closed system (no mass flow across boundaries):
  • $\Delta U = Q - W$, where $W$ is work done by system (piston-cylinder device)
• For open system (mass flow across boundaries):
  • First Law modified to include energy associated with mass flow
  • $\Delta U = Q - W + \sum m_i h_i - \sum m_e h_e$, where $m_i$ and $m_e$ are masses entering and exiting system, and $h_i$ and $h_e$ are respective specific enthalpies (turbine, compressor)

Heat engines and refrigerators

• Heat engines convert thermal energy into mechanical work
  • Efficiency ($\eta$) of heat engine: $\eta = \frac{W_{net}}{Q_H} = 1 - \frac{Q_C}{Q_H}$, where $W_{net}$ is net work output, $Q_H$ is heat input from hot reservoir, and $Q_C$ is heat rejected to cold reservoir (internal combustion engine, steam turbine)
• Refrigerators and heat pumps transfer heat from cold reservoir to hot reservoir
  • Coefficient of Performance (COP) for refrigerator: $COP_R = \frac{Q_C}{W_{net}}$ (household refrigerator, air conditioner)
  • Coefficient of Performance (COP) for heat pump: $COP_{HP} = \frac{Q_H}{W_{net}}$ (geothermal heat pump, reverse cycle air conditioner)
• Apply First Law to determine heat transfer, work, and efficiency or COP for these devices

Thermodynamic Cycles and Efficiency

Efficiency of thermodynamic cycles

• Carnot cycle: idealized, reversible cycle operating between two thermal reservoirs
  • Consists of four processes: isothermal expansion, adiabatic expansion, isothermal compression, and adiabatic compression
  • Carnot efficiency: $\eta_{Carnot} = 1 - \frac{T_C}{T_H}$, where $T_C$ and $T_H$ are absolute temperatures of cold and hot reservoirs
• Rankine cycle: practical vapor power cycle used in steam power plants
  • Consists of four processes:
    1. Isentropic compression (pump)
    2. Constant-pressure heat addition (boiler)
    3. Isentropic expansion (turbine)
    4. Constant-pressure heat rejection (condenser)
  • Rankine cycle efficiency: $\eta_{Rankine} = \frac{W_{net}}{Q_{in}} = \frac{W_t - W_p}{Q_{in}}$, where $W_t$ is turbine work, $W_p$ is pump work, and $Q_{in}$ is heat input in boiler

Performance of real-world systems

• Real-world systems involve irreversibilities and energy losses, such as:
  • Friction (bearings, seals)
  • Heat transfer across finite temperature differences (heat exchangers)
  • Unrestrained expansion (throttling valves)
• Irreversibilities reduce efficiency of real-world systems compared to ideal, reversible systems
• To analyze real-world systems:
  1. Identify sources of irreversibility and energy losses
  2. Apply First Law, accounting for these losses
  3. Calculate actual efficiency or COP of system
• Strategies to improve efficiency:
  • Minimize irreversibilities (reduce friction, optimize heat transfer)
  • Recover waste heat for useful purposes (cogeneration, regeneration)
  • Use high-performance materials and components (advanced alloys, ceramics)