2.1 Thermodynamic principles and laws

Thermodynamics is the backbone of aerospace propulsion. It's all about energy and how it moves between things. Understanding these principles helps us figure out how engines work and how to make them better.

The laws of thermodynamics are like the rules of the game. They tell us what's possible and what's not when it comes to energy. Knowing these laws helps us design more efficient engines and understand their limits.

Thermodynamics Fundamentals

Thermodynamic Systems and Properties

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• Thermodynamics studies energy and its interactions with matter, focusing on the relationships between heat, work, temperature, and energy
• A thermodynamic system is a specific portion of the universe being studied, with everything outside the system considered the surroundings
  • The system and surroundings are separated by a system boundary, which can be fixed or movable and can allow the transfer of matter, energy, or both
• Thermodynamic properties are characteristics that describe the state of a system, such as temperature, pressure, volume, internal energy, enthalpy, and entropy
  • These properties are determined by the state of the system and are independent of the path taken to reach that state
  • Intensive properties are independent of the system size (temperature, pressure), while extensive properties depend on the size of the system (volume, mass, internal energy)

Thermodynamic Equilibrium and Quasi-Static Processes

• Thermodynamic equilibrium is a state in which a system's properties remain constant over time, with no net exchange of energy or matter between the system and its surroundings
• There are three types of equilibrium:
  • Thermal equilibrium: no net heat transfer between the system and surroundings
  • Mechanical equilibrium: no net work done by or on the system
  • Chemical equilibrium: no net change in the chemical composition of the system
• Quasi-static processes are idealized processes that occur slowly enough for the system to remain in thermodynamic equilibrium at each step
  • This allows the process to be analyzed using equilibrium thermodynamics

Energy Conservation in Propulsion

First Law of Thermodynamics and Energy Balance

• The first law of thermodynamics states that energy cannot be created or destroyed, only converted from one form to another, representing the conservation of energy principle
• For a closed system, the change in internal energy ($ΔU$) is equal to the heat added to the system ($Q$) minus the work done by the system ($W$): $ΔU = Q - W$
• In an open system, such as a propulsion system, the first law must also account for the energy entering and leaving the system with the flow of matter
  • The steady-flow energy equation (SFEE) is used to analyze open systems: $Q̇ + Ẇ + ṁ(h₁ + v₁²/2 + gz₁) = ṁ(h₂ + v₂²/2 + gz₂)$, where $h$ is specific enthalpy, $v$ is velocity, $g$ is gravitational acceleration, and $z$ is elevation

Work and Heat Transfer in Propulsion Systems

• Work in propulsion systems can be in the form of shaft work (turbines, compressors) or flow work (thrust)
• Heat transfer in propulsion systems can occur through conduction, convection, and radiation, significantly impacting the performance and efficiency of the system
• The first law can be applied to individual components (combustion chambers, nozzles) or the entire propulsion system to analyze energy balances, heat transfer, and work output

Efficiency and Feasibility of Propulsion Cycles

Second Law of Thermodynamics and Entropy

• The second law of thermodynamics states that the total entropy of an isolated system always increases over time, and heat cannot spontaneously flow from a colder body to a hotter body
• Entropy is a measure of the disorder or randomness of a system, quantifying the amount of energy unavailable for useful work
  • The change in entropy ($ΔS$) is equal to the heat transfer ($Q$) divided by the absolute temperature ($T$): $ΔS = Q/T$
• The second law imposes limits on the efficiency of propulsion cycles, as it is impossible to create a heat engine (or propulsion system) that converts all input heat into useful work, with some energy always lost as waste heat

Carnot Cycle and Real Propulsion Cycles

• The Carnot cycle represents the most efficient theoretical heat engine, operating between two thermal reservoirs at different temperatures
  • The efficiency of a Carnot cycle ($η$) is given by: $η = 1 - (T_cold/T_hot)$, where $T_cold$ and $T_hot$ are the absolute temperatures of the cold and hot reservoirs, respectively
• Real propulsion cycles, such as the Brayton cycle (gas turbine engines) and the Otto cycle (reciprocating engines), have lower efficiencies than the Carnot cycle due to irreversibilities (friction, heat loss, incomplete combustion)
• The second law can be used to identify sources of inefficiency in propulsion systems and guide the design of more efficient cycles and components
• Exergy analysis, combining the first and second laws, can quantify the available energy in a system and identify the locations and magnitudes of irreversibilities

Pressure, Volume, Temperature, and Entropy Relationships

Ideal Gas Law and Thermodynamic Processes

• Thermodynamic processes are characterized by changes in the state variables of a system (pressure ($P$), volume ($V$), temperature ($T$), and entropy ($S$))
• The ideal gas law relates pressure, volume, and temperature for an ideal gas: $PV = nRT$, where $n$ is the number of moles and $R$ is the universal gas constant
• Isobaric processes occur at constant pressure, with work done given by: $W = P(V₂ - V₁)$
• Isochoric (or isometric) processes occur at constant volume, with no work done ($W = 0$)
• Isothermal processes occur at constant temperature, with work done given by: $W = nRT ln(V₂/V₁)$
• Adiabatic processes occur with no heat transfer between the system and surroundings ($Q = 0$), and the pressure and volume are related by: $PV^γ = constant$, where $γ$ is the ratio of specific heats ($c_p/c_v$)

Polytropic Processes and Entropy Changes

• Polytropic processes follow the relationship $PV^n = constant$, where $n$ is the polytropic exponent
  • Isobaric, isochoric, isothermal, and adiabatic processes are all special cases of polytropic processes
• The change in entropy for a reversible process is given by: $ΔS = ∫(dQ/T)$, where $dQ$ is the infinitesimal heat transfer and $T$ is the absolute temperature
• For an ideal gas undergoing a reversible process, the change in entropy is given by: $ΔS = c_v ln(T₂/T₁) + R ln(V₂/V₁)$, where $c_v$ is the specific heat at constant volume
• The T-s (temperature-entropy) diagram is a useful tool for visualizing thermodynamic processes and cycles, with the area under a process curve representing the heat transfer