Study Notes on Second Law of Thermodynamics

Kwame Nkrumah University of Science & Technology

Introduction to Engineering Thermodynamics

• Course: Engineering Thermodynamics (MME 365)

• Instructor: Kofi Owura Amoabeng, PhD

• Department of Mechanical Engineering

• University: Kwame Nkrumah University of Science and Technology, Kumasi, Ghana

Second Law of Thermodynamics

Fundamental Concepts

• Directional Processes:

  • Processes occur in a specific direction, which cannot be reversed.

  • Example: Water flows down a waterfall (high to low potential energy).

  • Example: Heat naturally flows from a high temperature to a low temperature.

• Gas Expansion:

  • Gases expand from areas of high pressure to areas of low pressure.

  • Example: Heat generation when electrical current flows.

Thermal Energy Reservoir (Heat Reservoir)

• Definition:

  • A hypothetical body with a large heat capacity.

  • Can supply or absorb finite amounts of heat without changing temperature.

• Properties:

  • The temperature of a heat reservoir remains constant, ensuring reversible processes occur within it.

  • A reservoir supplying heat is called a source; one absorbing heat is a sink.

• Examples of Sinks:

  • Atmosphere, rivers, oceans.

• Examples of Sources:

  • Boiler, furnace, nuclear reactor, the sun.

Heat Engine

• Definition:

  • A closed system that operates in a cycle, producing work from heat.

• Important Notes:

  • It is impossible for a heat engine to convert heat entirely into work (i.e. 100% thermal efficiency is not achievable).

  • Example of Heat Conversion:

  • Types of Heat Flow:

    • High Temperature (hot reservoir) to Low Temperature (cold reservoir).

  • Thermal Efficiency is defined as:
    $\eta_{th} = \frac{W_{net}}{Q_{in}}$ where:

  • $W_{net}$ = Work output,

  • $Q_{in}$ = Heat input.

• Conditions for Heat Engine Operation:

  1. Receives heat from a high-temperature source.

  2. Converts part of this heat to work.

  3. Rejects remaining heat to low-temperature sink.

  4. Completes a thermodynamic cycle.

• Example:

  • A heat engine with a heat input of 400 MW and a heat output of 160 MW produces 240 MW of work.

• Operation of Common Heat Engines:

  • Types include steam power plants, gas power plants, and automobile engines.

Thermal Efficiency of a Heat Engine

• Thermal Efficiency Calculation: For a heat engine: $\eta_{th} = \frac{W_{net}}{Q_{in}}$

  • Given inputs:

  • $Q_{in} = 400 \, MW$

  • $Q_{out} = 165 \, MW$

  • $W_{out} = 240 \, MW$

  • Thus,
    $\eta_{th} = \frac{160}{400} = 0.40 = 40\%$

Kelvin-Planck Statement of the Second Law

• Statement:

  • It is impossible for any device operating in a cycle to receive heat from a reservoir and do an equivalent amount of work without some waste heat being rejected.

Heat Engine Examples

1. Example 1:

   • Rate of heat transfer to heat engine = 80 MW

   • Rate of waste heat rejection = 50 MW

   • $W_{net} = Q_{in} - Q_{out} = 30 \, MW$

   • $\eta_{th}= \frac{30}{80} = 37.5\%$

2. Example 2:

   • Steam power plant produces 75 kW, rejects 190 kW.

   • Find: a) Heat supplied = 265 kW; b) Thermal efficiency = 28.3%.

Reversed Heat Engine

• Definition:

  • A closed system that operates in a cycle, extracting heat from a low-temperature reservoir and rejecting it to a high-temperature reservoir, while doing work on the system.

• Types:

  • Refrigerators (extract heat) and heat pumps (reject heat).

Performance of Refrigerators and Heat Pumps

• Coefficient of Performance (COP):

  • For refrigerators:
    $COP_{R} = \frac{Q_{C}}{W_{in}}$

  • For heat pumps:
    $COP_{HP} = \frac{Q_{H}}{W_{in}}$

Clausius Statement of the Second Law

• Statement:

  • Impossible to operate a system in a cycle, transferring heat from cooler to hotter body without work being done.

Examples of Reversed Heat Engine Performance

1. Example 1:

   • Refrigerator removing heat = 360 kJ/min, power input = 2 kW.

   • Determine COP and heat rejection rate. Suggested COP = 3, heat rejection = 8 kW.

Entropy

• Definition:

  • Quantitative measure of a system’s disorder; indicates energy unavailable for work.

• Mathematical Expressions:

  • For reversible process:
    $dS = \frac{\delta Q_{rev}}{T}$

  • For irreversible process:
    dS > \frac{\delta Q}{T}

Key Theorems and Principles

• Carnot Principle:

  • A reversible heat engine cycle maximizes net work output between two temperature limits.

• Carnot Efficiency:

  • Maximum efficiency of a reversible heat engine:
    $\eta_{th,rev} = 1 - \frac{T_C}{T_H}$

Conclusion

• Integration of the Laws:

  • Every cyclic process must comply with both the first law (energy conservation) and the second law (entropy considerations).