10.3 Laws of Thermodynamics

10.3 Laws of Thermodynamics

Quick Answer: The zeroth law defines temperature via thermal equilibrium. The first law is energy conservation for heat and work: ΔU = Q − W (with a declared sign convention). The second law limits direction and efficiency: heat will not spontaneously flow cold→hot, and no engine can convert heat entirely into work without rejecting some heat. Heat-engine efficiency η = 1 − Q_c/Q_h < 1 always for a cyclic engine between two reservoirs.

Thermodynamics turns the bookkeeping of energy into laws that every heat engine—including aircraft propulsion and environmental control—must obey. At FSc level for the PAF Aeronautical Engineer Initial exam, you need clear statements of the 0th–2nd laws, confident use of the first-law equation, and the standard heat-engine picture with efficiency.

System, Surroundings, and Internal Energy

A system is the part of the universe we study (gas in a cylinder, working fluid in an engine model). Everything else is the surroundings. Internal energy U is the microscopic energy of the system (molecular KE + PE). For an ideal gas, U depends only on temperature: higher T → higher U.

State variables (P, V, T, U) describe the condition; heat Q and work W are path-dependent transfers, not properties stored “inside” the system as named piles of heat.

Zeroth Law of Thermodynamics

Zeroth law: If system A is in thermal equilibrium with system B, and B is in thermal equilibrium with system C, then A is in thermal equilibrium with C.

This transitive rule justifies thermometers. A mercury or electronic sensor in equilibrium with the cabin air shares that air’s temperature; two objects reading the same thermometer temperature are in equilibrium with each other. Without the zeroth law, “temperature” would not be a consistently measurable scalar.

Aero link: Engine oil temperature, EGT/ITT, and OAT gauges all assume that the sensor and the fluid/metal of interest can be treated as reaching a meaningful shared temperature for monitoring and limits.

First Law of Thermodynamics

First law: Energy is conserved. For a closed system, the change in internal energy equals heat added to the system minus work done by the system (common FSc/physics convention):

$\Delta U = Q - W$

Some engineering texts write ΔU = Q + W with W as work done on the system—always check the sign convention in the question. In this guide we use:

• Q > 0 — heat absorbed by the system
• W > 0 — work done by the system (e.g., gas expanding a piston)

Consequences:

1. You cannot create energy from nothing inside the system—only convert forms.
2. If a gas expands and does work without heat input (adiabatic idealization with Q = 0), ΔU = −W → temperature falls for an ideal gas.
3. If heat is added at constant volume (W = 0 for PdV work), ΔU = Q → temperature rises.

Worked example — first law. A gas absorbs Q = 800 J of heat and expands, doing W = 300 J of work. ΔU = 800 − 300 = 500 J. Internal energy rose; not all heat became work—some stayed as increased microscopic energy.

Worked example — adiabatic expansion (Q = 0). Gas does 120 J of work with no heat exchange. ΔU = −120 J. For an ideal gas, temperature decreases.

These four labels appear constantly in heat-engine cycle sketches (Otto, Diesel, Brayton—names may appear in later engineering study; FSc focuses on the energy balance idea).

Second Law of Thermodynamics

The first law allows any energy balance that conserves energy; the second law forbids many of those balances in the real world.

Common FSc statements (equivalent ideas):

1. Clausius: Heat does not flow spontaneously from a colder body to a hotter body.
2. Kelvin–Planck: It is impossible to build a cyclic engine that converts heat completely into work with no other effect (no 100% efficient heat engine).
3. Entropy: The entropy of an isolated system never decreases; irreversible processes increase entropy.

Refrigerators and air-conditioners move heat from cold to hot, but only by consuming work—they do not violate Clausius because the “other effect” (work input) is present. Aircraft environmental control packs similarly need power to refrigerate cabin air.

Heat Engines — The Core Picture

A heat engine operates in a cycle between a hot reservoir at temperature T_h and a cold reservoir at T_c:

1. Absorbs heat Q_h from the hot reservoir
2. Performs net work W
3. Rejects heat Q_c to the cold reservoir

Energy conservation for a cycle (ΔU = 0 over a full cycle):

$W = Q_h - Q_c$

Thermal efficiency:

$\eta = \frac{W}{Q_h} = 1 - \frac{Q_c}{Q_h}$

Because Q_c cannot be zero for a real cyclic engine (second law), η is always less than 1. The theoretical Carnot limit η_Carnot = 1 − T_c/T_h (temperatures absolute) is the maximum possible between those reservoirs; real engines are lower due to friction, incomplete combustion, heat losses, and non-ideal cycles.

Worked example — efficiency. An engine takes in 2000 J per cycle from the hot side and rejects 1200 J to the cold side. W = 800 J; η = 800/2000 = 0.40 = 40%. About 60% of the input heat is dumped to the exhaust/atmosphere—not a design failure alone, but a thermodynamic necessity plus real losses.

Worked example — Carnot ceiling. Between T_h = 600 K and T_c = 300 K, η_Carnot = 1 − 300/600 = 50%. No engine operating only between these two temperatures can exceed 50% even in the ideal reversible case.

Aeronautical Interpretation

• Propulsion: Chemical energy in fuel becomes thermal energy; a fraction becomes useful propulsive work/kinetic energy of the exhaust; the rest leaves as hot exhaust and thermal losses—first law accounting plus second-law efficiency limits.
• Why engines have exhaust heat: Rejecting Q_c is mandatory for cyclic conversion of heat to work.
• APU / ECS: Cooling loops are heat pumps; COP definitions differ from engine η, but the same laws apply—work input required to move heat uphill.
• Brakes: Kinetic energy becomes internal energy (heat) in discs; energy is conserved, but recovering it fully as organized work is not what the brake is designed to do.

Irreversibility — Exam Intuition

Friction, unrestrained expansion, and heat flow across a finite temperature difference generate entropy. They make processes irreversible: you cannot restore both system and surroundings to the exact initial state without leftover effects. That is why perpetual-motion machines of the second kind fail even when they seem to satisfy ΔU bookkeeping.

Mistake Checklist

• Claiming a cyclic engine with η = 100%
• Using Celsius in Carnot’s 1 − T_c/T_h formula
• Mixing sign conventions for W mid-solution
• Saying the first law “forbids” heat flowing cold→hot (that is the second law; the first only tracks energy totals)
• Forgetting ΔU = 0 for a complete engine cycle when computing W = Q_h − Q_c

Hold the triad firmly: zeroth → temperature meaning; first → energy ledger; second → direction and efficiency cap. That triad is the thermodynamics core of this chapter.