Engineering Thermodynamics and Energy Systems

Introduction and Context

Thermodynamics is the study of energy, its transformations, and the limits imposed on those transformations. For mechatronics engineers, thermodynamics connects mechanical motion, electrical power, sensors, actuators, control systems, manufacturing processes, and thermal management. Motors generate heat, batteries store and release energy, compressors require shaft work, and electronic controllers must operate within safe temperature limits.

This chapter develops the essential thermodynamic tools used to analyze closed systems, open systems, power cycles, refrigeration systems, and electromechanical thermal loads. The emphasis is on ideal-gas models, the first and second laws, entropy, energy balances, and engineering interpretation.

Learning Objectives

By the end of this chapter, the reader should be able to:

1. Define thermodynamic systems, properties, states, processes, and cycles.
2. Apply the first law to closed and open systems.
3. Use ideal-gas relationships and property data consistently.
4. Calculate heat, work, internal-energy change, and enthalpy change.
5. Explain entropy and apply the second law to real processes.
6. Evaluate the performance of turbines, compressors, pumps, engines, refrigerators, and heat pumps.
7. Interpret the operation of common power and refrigeration cycles.
8. Estimate thermal loads in electromechanical systems.
9. Identify practical, safety, environmental, and control implications of energy conversion.

1. Thermodynamic Foundations

1.1 Systems, surroundings, and boundaries

A system is the quantity of matter or region selected for analysis. Everything outside it is the surroundings, and the separating surface is the boundary.

A closed system, or control mass, contains a fixed amount of matter. Energy may cross its boundary as heat or work, but mass does not. A gas inside a piston-cylinder is a common example.

An open system, or control volume, permits mass and energy to cross its boundary. Turbines, compressors, pumps, heat exchangers, and nozzles are open systems.

A system is isolated when neither mass nor energy crosses its boundary. An idealized thermos is an approximation to an isolated system.

1.2 Properties and state

A property is a measurable macroscopic characteristic. Pressure P, temperature T, volume V, density ρ, internal energy U, enthalpy H, and entropy S are thermodynamic properties.

Intensive properties do not depend on system size, such as temperature and pressure. Extensive properties depend on the amount of matter, such as volume and total internal energy. Specific properties are extensive properties per unit mass, for example specific volume v = V/m and specific internal energy u = U/m.

A system is in a state when its properties have definite values. The state postulate states that the state of a simple compressible substance is fixed by two independent intensive properties. For example, pressure and temperature generally specify the state of an ideal gas.

A process is a change from one state to another. A cycle is a sequence of processes returning the system to its initial state. Because properties are state functions, their changes depend only on the initial and final states. Heat and work are path functions and depend on how the process occurs.

1.3 Equilibrium and quasi-equilibrium

Thermodynamic equilibrium requires thermal, mechanical, and chemical equilibrium. A quasi-equilibrium process proceeds sufficiently slowly that the system remains approximately uniform and close to equilibrium. This assumption permits the use of a well-defined system pressure at each stage and is useful in deriving boundary-work expressions.

2. Properties and Ideal-Gas Models

2.1 Pressure, temperature, and density

Absolute pressure must be used in thermodynamic equations involving temperature or gas properties. Gauge pressure is measured relative to local atmospheric pressure:

where P_abs is absolute pressure, P_gauge is gauge pressure, and P_atm is atmospheric pressure. Pressure is measured in pascals, where 1 Pa = 1 N/m².

Temperature differences may be expressed in kelvins or degrees Celsius, because 1 K = 1 °C as an interval. Absolute temperature in gas equations must be in kelvins:

2.2 Ideal-gas equation of state

For a sufficiently dilute gas at moderate pressure, the ideal-gas model is often appropriate:

Here P is absolute pressure in pascals, V is volume in cubic metres, m is mass in kilograms, R is the specific gas constant in joules per kilogram-kelvin, and T is absolute temperature in kelvins. The equation is applicable when intermolecular effects are small. For air, R ≈ 287 J/(kg·K) and k ≈ 1.4 near room temperature.

For a fixed mass of ideal gas:

Special cases include:

• Constant temperature: P₁V₁ = P₂V₂ (isothermal process).
• Constant pressure: V₁/T₁ = V₂/T₂.
• Constant volume: P₁/T₁ = P₂/T₂.

2.3 Specific heats and enthalpy

For an ideal gas, internal energy and enthalpy depend primarily on temperature. With constant specific heats:

where c_v and c_p are specific heats at constant volume and constant pressure, respectively, in J/(kg·K). For an ideal gas:

and

where k is the specific-heat ratio. These constant-property approximations are useful over moderate temperature ranges; property tables or temperature-dependent specific heats are preferable over large ranges.

Enthalpy is defined as:

or, per unit mass,

Enthalpy is particularly useful in open systems because the PV term accounts naturally for flow work.

3. The First Law of Thermodynamics

The first law is conservation of energy. Energy cannot be created or destroyed, although it can be transferred and transformed.

3.1 Closed-system energy balance

For a closed system:

where ∆E is the change in total system energy, Q is heat transferred to the system, and W is work done by the system. The total energy is:

Thus:

The convention used here is heat into the system positive and work by the system positive. Other conventions are acceptable if used consistently.

For a stationary system with negligible changes in kinetic and potential energy:

Boundary work for a quasi-equilibrium process is:

For constant pressure, this becomes W_b = P(V₂ − V₁). Electrical work, shaft work, and stirring work may also cross a closed-system boundary.

Worked Example 1: Heating air in a rigid tank

A rigid tank contains 2 kg of air. Its temperature increases from 300 K to 500 K. Assume c_v = 0.718 kJ/(kg·K), negligible kinetic and potential-energy changes, and no boundary work. Determine the heat transfer.

Because the tank is rigid, W_b = 0. Therefore:

The heat transferred to the air is approximately 287 kJ.

3.2 Steady-flow energy equation

For a control volume operating at steady state, the general energy balance is:

Here Q̇ is heat-transfer rate, Ẇ_s is shaft-work rate, ṁ is mass-flow rate, V is flow speed, g is gravitational acceleration, and z is elevation. Specific energies are in joules per kilogram, and rates are in watts.

For many devices, the equation simplifies:

4. The Second Law and Entropy

The first law does not determine whether a process can occur naturally or how efficiently energy can be converted. The second law supplies this direction and establishes performance limits.

Heat flows spontaneously from a higher-temperature body to a lower-temperature body. No heat engine can convert all heat supplied into work while rejecting no heat. Real processes generate entropy because of friction, mixing, electrical resistance, finite temperature differences, and unrestrained expansion.

4.1 Entropy balance

For a closed system:

where S_gen ≥ 0 is entropy generation. For an isolated system, ∆S = S_gen ≥ 0. The equality S_gen = 0 applies only to an internally reversible process.

For a control volume, the entropy-rate balance is:

where T_j is the boundary temperature associated with heat-transfer rate Q̇_j, s is specific entropy, and Ṡ_gen ≥ 0.

4.2 Ideal-gas entropy changes

For an ideal gas with constant specific heats:

An equivalent form is:

The logarithm arguments are dimensionless ratios. Entropy has units of kJ/(kg·K) or J/(kg·K).

A reversible adiabatic process is isentropic:

For an ideal gas with constant k, the isentropic relations are:

These relations apply only to ideal-gas, internally reversible, adiabatic processes.

5. Heat Engines, Refrigerators, and Performance Measures

A heat engine receives heat Q_H from a high-temperature reservoir, produces work W_net, and rejects heat Q_L to a low-temperature reservoir. Over a cycle:

Thermal efficiency is:

The Carnot efficiency is the maximum possible efficiency for an engine operating between absolute temperatures T_H and T_L:

It is a limit, not a prediction of actual engine performance.

A refrigerator removes heat from a low-temperature region. Its coefficient of performance is:

A heat pump delivers heat to a high-temperature region:

For the same reversible cycle:

Unlike efficiency, a coefficient of performance may exceed one because the device transfers heat rather than converting all input work into useful output.

6. Common Energy-Conversion Cycles

6.1 Rankine cycle

The ideal Rankine cycle models steam power plants and consists of:

1. Isentropic pumping of saturated liquid to boiler pressure.
2. Constant-pressure heat addition in the boiler.
3. Isentropic expansion through a turbine.
4. Constant-pressure heat rejection in the condenser.

Its approximate net work is:

and its thermal efficiency is:

Superheating, reheating, and regenerative feedwater heating can improve performance, but they increase equipment complexity and cost.

6.2 Otto and Diesel cycles

The ideal Otto cycle represents spark-ignition engines. Heat addition occurs at constant volume. The ideal efficiency depends on compression ratio r:

The formula assumes an air-standard cycle, constant specific heats, reversible processes, and complete combustion represented by external heat addition.

The Diesel cycle represents compression-ignition engines. Heat addition is modeled at constant pressure, and efficiency depends on compression ratio and cutoff ratio. Actual engine performance is lower than ideal predictions because of friction, heat loss, incomplete combustion, pumping work, and finite-rate processes.

6.3 Vapor-compression refrigeration cycle

The basic cycle contains a compressor, condenser, expansion valve, and evaporator. Refrigerant vapor is compressed, rejects heat in the condenser, expands through the valve, and absorbs heat in the evaporator. The expansion valve is approximately isenthalpic, while the compressor requires significant electrical input.

The cycle is central to industrial cooling, air conditioning, battery thermal management, and electronics cooling.

7. Electromechanical Thermal Loads

Thermal analysis is essential in mechatronics because electrical and mechanical losses eventually become heat. For a motor:

Copper loss in a winding is:

where I is current in amperes and R is resistance in ohms. For a three-phase balanced motor, the stator copper loss is commonly 3I²R_phase when I is phase current and R_phase is phase resistance.

Mechanical shaft power is:

where τ is torque in newton-metres and ω is angular speed in radians per second. Motor efficiency is:

A first-order lumped thermal model is:

where C_th is thermal capacitance in joules per kelvin, θ is temperature rise above ambient in kelvins, R_th is thermal resistance in kelvins per watt, and P_loss is heat-generation rate. At steady state:

This model assumes approximately uniform component temperature and constant thermal parameters. It is useful for preliminary sizing but may be inaccurate when heat paths are distributed or temperature-dependent.

Worked Example 2: Motor thermal load

A motor receives 2.0 kW electrical input and delivers 1.7 kW mechanical output. Its thermal resistance to ambient is 0.20 K/W. Estimate the steady-state temperature rise.

The losses are:

Therefore:

The estimated winding or motor-body temperature rise is 60 K, subject to the validity of the lumped thermal model. If ambient temperature is 25 °C, the predicted steady temperature is approximately 85 °C. Actual winding temperature may be higher than housing temperature, so sensor placement and insulation class must be considered.

8. Applied Case: Thermal Management of a Servo-Driven Robotic Joint

Background

This illustrative case considers a robotic joint driven by a servo motor. The joint performs repeated pick-and-place motion. During acceleration, current is high; during steady motion, current is moderate; during braking, regenerative electrical energy may return to the drive.

Problem

The motor has a peak electrical input of 1.8 kW during acceleration and an average input of 0.75 kW over a duty cycle. Average mechanical output is 0.62 kW. The motor and gearbox are installed in a compact enclosure with an estimated thermal resistance of 0.28 K/W. Ambient temperature may reach 35 °C. Determine the approximate continuous heat load and assess whether passive cooling is adequate for a maximum allowable motor temperature of 100 °C.

Analysis and solution

Average heat generation is estimated from the difference between average electrical input and useful mechanical output:

The approximate steady temperature rise is:

The predicted motor temperature is therefore:

Under average operation, passive cooling appears adequate. However, peak acceleration causes transient heating, and the gearbox, bearings, drive electronics, and motor windings may have different thermal resistances. A transient model should be used for the actual duty cycle:

for a constant heat load starting from ambient. During rest or low-load periods, the temperature decays rather than remaining at the average-load value.

Outcome and learning points

The analysis shows that average power alone is insufficient when thermal time constants and peak loads are important. A robust design should:

• Measure winding or internal temperature rather than relying only on housing temperature.
• Include overload duration and duty-cycle data.
• Limit current through the drive controller.
• Provide fault shutdown for overtemperature.
• Account for blocked ventilation, dust, altitude, and ambient temperature.
• Consider regenerative braking and its effect on the DC bus.
• Validate calculations with thermal testing and finite-element or network simulation when necessary.

9. Engineering Practice, Limitations, and Contemporary Developments

Thermodynamic calculations are models, not measurements. The ideal-gas model may fail at high pressure, low temperature, or near phase-change conditions. Constant specific heats introduce errors over broad temperature ranges. Steady-state assumptions can conceal dangerous transient temperatures. The use of average efficiency can also hide short overloads that damage insulation or bearings.

Good practice requires absolute pressure and temperature where appropriate, consistent units, clearly stated sign conventions, and an energy-balance check. Engineers should compare calculated heat loads with measured temperatures, power meters, flow meters, and manufacturer data.

Modern mechatronic systems increasingly use embedded temperature sensors, digital twins, model-predictive control, variable-speed drives, heat pumps, liquid cooling, and wide-bandgap semiconductor power electronics. Artificial intelligence can assist in predicting thermal faults from current, vibration, and temperature data, but its predictions must be validated against physical energy balances. Safety-critical thermal protection should not depend solely on an opaque model.

Energy systems also involve environmental and regulatory responsibilities. Engineers must consider refrigerant global-warming potential, pressure-vessel safety, electrical isolation, fire risk, battery thermal runaway, noise, waste heat, and lifecycle energy use. Efficient operation is not only a cost issue; it reduces resource consumption and emissions.

Chapter Summary

Thermodynamics provides the framework for analyzing energy storage, transfer, and conversion in mechatronic systems. A system may be closed, with fixed mass, or open, with mass crossing a control volume. State properties such as pressure, temperature, internal energy, enthalpy, and entropy describe its condition.

The ideal-gas equation is useful for gases under appropriate conditions. The first law provides energy conservation: heat and work alter internal, kinetic, and potential energy. For steady-flow devices, enthalpy provides a convenient account of flow energy. The second law introduces entropy generation and establishes the direction and limits of processes. Isentropic models are useful ideal references for turbines, compressors, and nozzles.

Heat engines are evaluated by thermal efficiency, while refrigerators and heat pumps are evaluated by coefficients of performance. Rankine, Otto, Diesel, and vapor-compression cycles provide idealized models of important energy systems. In electromechanical equipment, electrical, magnetic, mechanical, and fluid losses become heat. Thermal resistance-capacitance models support preliminary temperature and cooling calculations, but transient behavior, sensor location, environmental conditions, and safety limits must also be considered.

Key Terms

System: Selected matter or region analyzed in thermodynamics.

Closed system: System with fixed mass across whose boundary energy may transfer.

Open system: Control volume through which mass and energy may pass.

Property: Measurable characteristic defining a system state.

State: Condition of a system specified by its properties.

Process: Change from one thermodynamic state to another.

Cycle: Sequence of processes returning to the initial state.

Internal energy: Microscopic molecular energy stored within a substance.

Enthalpy: Property defined by H = U + PV, especially useful for flowing fluids.

Specific heat: Energy required per unit mass to raise temperature by one kelvin under specified conditions.

Heat: Energy transfer caused by a temperature difference.

Work: Energy transfer associated with force, displacement, shaft rotation, or other generalized effects.

Entropy: Property used to quantify energy dispersal and irreversibility.

Entropy generation: Entropy produced by irreversibilities; it cannot be negative.

Isentropic process: Constant-entropy process, ideally reversible and adiabatic.

Thermal efficiency: Net useful work divided by heat supplied.

Coefficient of performance: Useful refrigeration or heating effect divided by work input.

Thermal resistance: Temperature difference per unit heat-transfer rate.

Thermal capacitance: Energy required to change a body's temperature by one kelvin.

Steady state: Condition in which control-volume properties do not change with time.

Review Questions

5 Short Answer Questions

1. Distinguish between a closed system and an open system.
2. Why must absolute temperature be used in the ideal-gas equation?
3. State the first-law energy balance for a stationary closed system.
4. What does entropy generation indicate about a process?
5. Why is enthalpy especially useful in open-system analysis?

5 Long Answer Questions

1. Explain the ideal-gas model, including its equation of state, assumptions, and limitations.
2. Derive the simplified steady-flow energy equations for a turbine and a compressor.
3. Compare thermal efficiency with coefficient of performance and explain why a coefficient of performance can exceed one.
4. Describe the four principal components and processes of a vapor-compression refrigeration cycle.
5. Explain how electrical and mechanical losses produce thermal loads in a servo motor and how a thermal-resistance model can be used for preliminary design.

3 Analytical / Application Questions

1. A compressor receives air at 300 K and discharges it at 600 K at a mass-flow rate of 0.08 kg/s. Using c_p = 1.005 kJ/(kg·K) and neglecting kinetic and potential-energy changes, estimate the adiabatic power input.
2. A refrigerator removes 4 kW from a cabinet while consuming 1.25 kW of electrical power. Calculate its coefficient of performance and the heat rejected to the room.
3. A motor operates with 18 A through a winding resistance of 0.35 Ω. Calculate its copper loss and discuss two design changes that could reduce the resulting thermal load.

Multiple Choice Questions with Answers

Question 1

Which quantity is a property of a thermodynamic system?

A. Heat transfer

B. Boundary work

C. Temperature

D. Path followed

Correct Answer: C. Temperature

Question 2

A rigid tank contains a gas. Which boundary-work condition applies?

A. Work is always positive

B. Boundary work is zero

C. Boundary work equals heat transfer

D. Boundary work is infinite

Correct Answer: B. Boundary work is zero

Question 3

For an ideal gas, which equation is correct?

A. PV = mRT

B. P + V = mRT

C. PV = m/T

D. P/T = mRV

Correct Answer: A. PV = mRT

Question 4

What is the minimum possible value of entropy generation?

A. −1

B. Zero

C. The ambient temperature

D. The system pressure

Correct Answer: B. Zero

Question 5

A turbine is commonly approximated as steady, adiabatic, and with negligible velocity changes. Its specific work output is approximately:

A. h₂ − h₁

B. h₁ − h₂

C. u₂ − u₁

D. Pv₂ − Pv₁ only

Correct Answer: B. h₁ − h₂

Question 6

A heat engine receives 1,000 kJ and rejects 600 kJ per cycle. Its thermal efficiency is:

A. 20%

B. 40%

C. 60%

D. 167%

Correct Answer: B. 40%

Question 7

A refrigerator removes 5 kW of heat and consumes 1 kW of work. Its coefficient of performance is:

A. 0.2

B. 4

C. 5

D. 6

Correct Answer: C. 5

Question 8

A motor has electrical input power of 3.0 kW and shaft output of 2.4 kW. Its loss is:

A. 0.6 kW

B. 1.25 kW

C. 2.4 kW

D. 5.4 kW

Correct Answer: A. 0.6 kW

Question 9

Copper loss in a winding is proportional to:

A. I/R

B. IR²

C. I²R

D. I + R

Correct Answer: C. I²R

Question 10

If a device has a thermal resistance of 0.15 K/W and dissipates 200 W at steady state, its temperature rise is:

A. 13.3 K

B. 30 K

C. 133 K

D. 1,333 K

Correct Answer: B. 30 K