🧊Thermodynamics II Unit 4 – Gas Power Cycles: Otto, Diesel, and Brayton

Key Concepts and Definitions

• Thermodynamic cycle: A series of thermodynamic processes that a system undergoes, returning to its initial state
• Heat engine: A device that converts thermal energy into mechanical work by operating in a cyclic process
• Thermal efficiency: The ratio of the net work output to the heat input during a thermodynamic cycle
• Compression ratio: The ratio of the maximum volume to the minimum volume in a reciprocating engine
• Isentropic process: A thermodynamic process in which the entropy of the system remains constant
• Adiabatic process: A thermodynamic process in which no heat is exchanged between the system and its surroundings
• Isochoric process: A thermodynamic process that occurs at constant volume
• Isobaric process: A thermodynamic process that occurs at constant pressure

Thermodynamic Principles Review

• First Law of Thermodynamics: Energy cannot be created or destroyed, only converted from one form to another
  • Mathematically expressed as $\Delta U = Q - W$, where $\Delta U$ is the change in internal energy, $Q$ is the heat added to the system, and $W$ is the work done by the system
• Second Law of Thermodynamics: The entropy of an isolated system always increases over time
  • Implies that heat engines cannot achieve 100% efficiency and that some energy is always lost as waste heat
• Ideal gas law: Relates pressure, volume, and temperature of an ideal gas ($PV = nRT$)
  • Assumes gas molecules have negligible volume and no intermolecular forces
• Specific heat capacity: The amount of heat required to raise the temperature of a unit mass of a substance by one degree
  • Varies depending on whether the process is at constant volume ($c_v$) or constant pressure ($c_p$)
• Adiabatic process equation: Relates pressure and volume during an adiabatic process ($PV^\gamma = \text{constant}$)
  • The value of $\gamma$ is the ratio of specific heats ($c_p/c_v$) and depends on the gas composition

Otto Cycle Breakdown

• Four-stroke spark-ignition internal combustion engine cycle
• Consists of four processes: isentropic compression, isochoric heat addition, isentropic expansion, and isochoric heat rejection
• Process 1-2: Isentropic compression of the air-fuel mixture
  • Piston moves from bottom dead center (BDC) to top dead center (TDC), compressing the mixture
• Process 2-3: Isochoric heat addition (constant volume)
  • Spark plug ignites the compressed mixture, causing a rapid increase in pressure and temperature
• Process 3-4: Isentropic expansion
  • High-pressure gases push the piston down, performing work on the piston
• Process 4-1: Isochoric heat rejection (constant volume)
  • Exhaust valve opens, and the burnt gases are expelled from the cylinder
• Thermal efficiency depends on the compression ratio and the specific heat ratio of the working fluid
  • Higher compression ratios lead to higher efficiencies

Diesel Cycle Explained

• Four-stroke compression-ignition internal combustion engine cycle
• Similar to the Otto cycle but with key differences in the heat addition process
• Process 1-2: Isentropic compression of air only (no fuel)
  • Compression ratios are higher than in the Otto cycle, leading to higher temperatures
• Process 2-3: Isobaric heat addition (constant pressure)
  • Fuel is injected into the compressed hot air, causing it to ignite and burn at nearly constant pressure
• Process 3-4: Isentropic expansion
  • High-pressure gases push the piston down, performing work on the piston
• Process 4-1: Isochoric heat rejection (constant volume)
  • Exhaust valve opens, and the burnt gases are expelled from the cylinder
• Thermal efficiency depends on the compression ratio, specific heat ratio, and the cutoff ratio (ratio of volumes at the end and start of the heat addition process)
  • Higher compression ratios and lower cutoff ratios lead to higher efficiencies

Brayton Cycle Analysis

• Open gas turbine cycle used in jet engines and power generation
• Consists of four processes: isentropic compression, isobaric heat addition, isentropic expansion, and isobaric heat rejection
• Process 1-2: Isentropic compression in the compressor
  • Ambient air is drawn into the compressor and compressed to a higher pressure
• Process 2-3: Isobaric heat addition in the combustion chamber
  • Fuel is injected and burned at constant pressure, increasing the temperature of the gas
• Process 3-4: Isentropic expansion in the turbine
  • Hot, high-pressure gases expand through the turbine, generating power to drive the compressor and provide useful work
• Process 4-1: Isobaric heat rejection (exhaust)
  • Exhaust gases are released to the atmosphere at constant pressure
• Thermal efficiency depends on the pressure ratio (ratio of compressor outlet to inlet pressures) and the specific heat ratio of the working fluid
  • Higher pressure ratios lead to higher efficiencies

Efficiency Comparisons

• Otto, Diesel, and Brayton cycles have different ideal thermal efficiencies due to their unique operating principles
• Otto cycle efficiency: $\eta = 1 - \frac{1}{r^{\gamma-1}}$, where $r$ is the compression ratio and $\gamma$ is the specific heat ratio
  • Efficiency increases with higher compression ratios and specific heat ratios
• Diesel cycle efficiency: $\eta = 1 - \frac{1}{r^{\gamma-1}} \left(\frac{r_c^\gamma - 1}{\gamma(r_c - 1)}\right)$, where $r_c$ is the cutoff ratio
  • Efficiency increases with higher compression ratios and lower cutoff ratios
• Brayton cycle efficiency: $\eta = 1 - \frac{1}{r_p^{(\gamma-1)/\gamma}}$, where $r_p$ is the pressure ratio
  • Efficiency increases with higher pressure ratios and specific heat ratios
• In general, Diesel engines have the highest efficiency, followed by Otto engines and then Brayton engines
  • However, actual efficiencies are lower than ideal due to various irreversibilities and losses

Real-World Applications

• Otto cycle: Used in gasoline-powered vehicles, small engines (lawnmowers, generators), and some aircraft piston engines
• Diesel cycle: Used in diesel-powered vehicles (trucks, buses, trains), heavy machinery, and marine propulsion systems
  • Preferred for high-torque, low-speed applications due to their high compression ratios and fuel efficiency
• Brayton cycle: Used in jet engines for aircraft propulsion and in gas turbines for power generation
  • Combined cycle power plants use a gas turbine (Brayton cycle) in conjunction with a steam turbine (Rankine cycle) to achieve higher overall efficiencies
• Modifications to ideal cycles: Real engines incorporate additional features to improve efficiency and performance
  • Turbochargers (Otto and Diesel) use exhaust gases to compress intake air, increasing power output
  • Regenerative Brayton cycles use heat exchangers to preheat the compressed air before combustion, improving efficiency

Problem-Solving Strategies

• Identify the type of cycle (Otto, Diesel, or Brayton) and the given parameters
• Determine the process paths and their corresponding thermodynamic relations
  • Use isentropic, isochoric, and isobaric process equations as appropriate
• Apply the First Law of Thermodynamics to each process, considering heat transfer and work interactions
• Calculate the heat input, heat rejection, and net work output for the cycle
  • Use specific heat capacities, temperature changes, and pressure ratios as needed
• Determine the thermal efficiency using the appropriate formula for the cycle
• Consider the effects of varying parameters (compression ratio, cutoff ratio, pressure ratio) on efficiency and performance
• Analyze the results and compare them to the ideal cycle efficiencies and real-world expectations
  • Discuss potential sources of irreversibilities and losses that reduce actual efficiency