Laws of Thermodynamics

The work done by a gas in a thermodynamic process is:

$W = \int_{V_1}^{V_2} p\,dV$

Key principles:

• Work equals the area under the curve in a p-V representation
• For constant pressure processes: $W = p(V_2 - V_1)$
• Positive work occurs when a system expands (gas does work on surroundings)
• Negative work occurs when a system is compressed (surroundings do work on gas)

Example: For a gas expanding at constant pressure of 120 kPa from 200 cc to 450 cc, the work done is: $W = (120 \times 10^3 \text{ Pa})(450 - 200) \times 10^{-6} \text{ m}^3 = 30 \text{ J}$

The First Law of Thermodynamics is expressed as: $\Delta U = \Delta Q - \Delta W$

Where:

• $\Delta U$ = change in internal energy of the system
• $\Delta Q$ = heat added to the system
• $\Delta W$ = work done by the system

Physical meaning:

• Represents conservation of energy
• Internal energy changes only through heat transfer or work
• A system's internal energy increases when heat is added ($+\Delta Q$) or work is done on it ($-\Delta W$)
• When a system returns to its initial state after a cyclic process, $\Delta U = 0$

Note: When work is done by the system, $\Delta W$ is positive; when work is done on the system, $\Delta W$ is negative.

Heat transfer occurs through molecular collisions:

1. Gas molecules strike the higher-temperature wall
2. During collision, energy transfers from higher kinetic energy wall molecules to lower kinetic energy gas molecules
3. These energized gas molecules share kinetic energy with other gas molecules through subsequent collisions
4. This increases the total internal energy of the gas system
5. The process continues until thermal equilibrium is reached

Example: Air in a metal container placed on a hot surface receives energy through this collision-based transfer mechanism, gradually increasing the overall gas temperature.

A steam engine converts thermal energy to mechanical motion through these key steps:

1. Steam enters the steam chest at high pressure
2. The D-valve alternately directs steam to the upper and lower parts of the main cylinder
3. The piston is pushed up and down by the alternating steam pressure
4. The piston-rod transfers this linear motion to the crank mechanism
5. The crank converts linear motion to rotational motion of the flywheel

The D-valve's critical function:

• Controls the flow direction of steam within the engine
• Slides up and down to alternately cover/uncover steam inlet ports
• When in upper position: directs steam to bottom of cylinder, pushing piston up
• When in lower position: directs steam to top of cylinder, pushing piston down
• Allows spent steam to exit to the condenser
• Creates the reciprocating cycle necessary for continuous engine operation

Note: The D-valve's movement is synchronized with the piston through the eccentric mechanism, ensuring precise timing of steam delivery.

The eccentric mechanism in a steam engine:

1. Primary purpose: Converts the rotary motion of the crankshaft into the linear motion needed to operate the D-valve

2. Components:

   • Eccentric: An off-center disc fixed to the main shaft
   • Eccentric-rod: Connects the eccentric to the valve-rod
   • Crosshead (C₂): Provides a hinge point for the valve-rod

3. Synchronization function:

   • Creates a 90° phase difference between piston and valve movements
   • When piston is at top/bottom positions, valve is at mid-stroke
   • When piston is at mid-stroke, valve is at extreme positions

4. Operation sequence:

   • As crankshaft rotates, eccentric rotates with it
   • Eccentric's offset position creates reciprocating motion in the eccentric-rod
   • This motion transfers to the valve-rod, moving the D-valve up and down
   • The D-valve position is automatically synchronized with piston position, ensuring steam enters the cylinder at precisely the right moment

This timing mechanism creates the self-regulating cycle essential for continuous engine operation without requiring external valve adjustment.

A heat engine operates by:

• Taking heat Q₁ from a high-temperature reservoir
• Converting a portion of this heat into mechanical work W
• Rejecting the remaining heat Q₂ to a low-temperature reservoir

The efficiency is defined as: η = W/Q₁ = (Q₁-Q₂)/Q₁ = 1-Q₂/Q₁

By the First Law of Thermodynamics: W = Q₁ - Q₂

Example: In a steam engine, high-temperature steam provides heat Q₁, some energy converts to mechanical motion of a piston (W), and the remaining heat Q₂ is released to the environment.

Note: No heat engine can convert all thermal energy into work (100% efficiency) as required by the Second Law of Thermodynamics.

The Otto cycle consists of four strokes:

1. Charging Stroke:

   • Inlet valve opens while outlet valve remains closed
   • Piston moves downward, creating low pressure
   • Fuel-air mixture is drawn into the cylinder
   • Crankshaft rotates 180° during this process

2. Compression Stroke:

   • Both inlet and outlet valves are closed
   • Piston moves upward, compressing the fuel-air mixture
   • Temperature rises to approximately 500°C
   • Pressure increases significantly
   • Crankshaft completes a full 360° rotation from start

3. Working Stroke:

   • Both valves remain closed
   • Spark plug ignites the compressed mixture
   • Combustion raises temperature to ~2000°C and pressure to ~15 atm
   • Expanding gases force the piston downward
   • This is the only power-producing stroke
   • Crankshaft reaches 540° of rotation

4. Exhaust Stroke:

   • Outlet valve opens while inlet valve remains closed
   • Piston moves upward, expelling burnt gases
   • Cylinder is cleared for the next cycle
   • Crankshaft completes full 720° rotation (two revolutions)

The working stroke is distinct because:

• It's the only power-producing phase among the four strokes
• Energy from fuel combustion is converted to mechanical work
• High temperature (~2000°C) and pressure (~15 atm) push the piston downward
• The crankshaft receives rotational force during this stroke
• All other strokes consume energy rather than produce it

Engine efficiency is directly related to the working stroke because:

• The ratio of useful work output to energy input occurs here
• Incomplete combustion during this stroke wastes fuel energy
• Heat losses during combustion reduce the pressure acting on the piston
• The design of the combustion chamber affects flame propagation
• Maximum pressure achieved determines the force applied to the piston

Note: In practical engines, only about 25-30% of the fuel's chemical energy is converted to useful mechanical work during this stroke, with the remainder lost as heat and friction.

In a p-V diagram of a Carnot cycle:

Work:

• The area enclosed by the entire cycle represents the net work output (W) of the engine
• Clockwise cycles produce positive work (heat engines)
• During expansion processes (a→b and b→c), work is done by the system (positive)
• During compression processes (c→d and d→a), work is done on the system (negative)

Heat:

• During isothermal expansion (a→b): Heat Q₁ enters the system at temperature T₁
• During isothermal compression (c→d): Heat Q₂ leaves the system at temperature T₂
• During adiabatic processes (b→c and d→a): No heat exchange (Q = 0)
• Net heat input = Q₁ - Q₂

Entropy changes:

• Isothermal expansion (a→b): ΔS = Q₁/T₁ (positive)
• Isothermal compression (c→d): ΔS = -Q₂/T₂ (negative)
• Adiabatic processes (b→c and d→a): ΔS = 0
• For the complete cycle: ΔS = 0 (entropy is a state function)

Mathematical relationships:

• Efficiency: η = W/Q₁ = 1 - Q₂/Q₁ = 1 - T₂/T₁
• The ratio Q₁/T₁ = Q₂/T₂ (a fundamental property of the Carnot cycle)

The enclosed area's significance extends beyond work done - it represents the maximum possible conversion of heat to work for any engine operating between temperatures T₁ and T₂, as established by the Second Law of Thermodynamics.

Kelvin-Planck Statement

"It is impossible to design a heat engine which works in a cyclic process and whose only result is to take heat from a body at a single temperature and convert it completely into mechanical work."

Fundamental Implications

1. Efficiency Limitations:

   • No heat engine can achieve 100% efficiency
   • Maximum theoretical efficiency is limited to Carnot efficiency: $\eta_{max} = 1 - \frac{T_C}{T_H}$
   • Some heat must always be rejected to a cold reservoir

2. Temperature Requirements:

   • Heat engines must operate between two different temperatures
   • A temperature differential is mandatory for converting heat to work
   • Cannot extract useful work from objects at uniform/ambient temperature

3. Directionality of Energy Conversion:

   • Work can be completely converted to heat (e.g., friction)
   • Heat cannot be completely converted to work in a cyclic process
   • Spontaneous processes that convert heat entirely to work are impossible

4. Practical Consequences:

   • Perpetual motion machines of the second kind are impossible
   • Cannot extract unlimited energy from environment without temperature differences
   • Explains why we need fuel in engines despite abundant thermal energy in surroundings

Example: A hypothetical device that extracts heat from ocean water and converts it entirely to mechanical energy without creating any other changes would violate this principle.