MTE 309: Introduction to Thermodynamics and Heat Transfer

Sources and References

• Yunus A. Cengel and Michael A. Boles, Thermodynamics: An Engineering Approach (McGraw-Hill), property tables, cycle analyses, and worked examples.
• Michael J. Moran, Howard N. Shapiro, Daisie D. Boettner, and Margaret B. Bailey, Fundamentals of Engineering Thermodynamics (Wiley), for control-volume formulation and second-law accounting.
• Frank P. Incropera, David P. DeWitt, Theodore L. Bergman, and Adrienne S. Lavine, Fundamentals of Heat and Mass Transfer (Wiley), for conduction, convection correlations, and transient charts.
• Yunus A. Cengel and Afshin J. Ghajar, Heat and Mass Transfer: Fundamentals and Applications (McGraw-Hill), for practical correlations and electronics-cooling framing.
• Allan D. Kraus, Avram Bar-Cohen, and Abhay A. Wativar, Thermal Analysis and Control of Electronic Equipment, for heat sinks, package resistances, and TIM selection.
• Gordon N. Ellison, Thermal Computations for Electronics: Conductive, Radiative, and Convective Air Cooling, for compact thermal models and chassis-level analysis.
• MIT OpenCourseWare 2.005 Thermal-Fluids Engineering I and 2.006 Thermal-Fluids Engineering II, for alternative pedagogical perspectives on the First and Second Laws and on convective transport.
• NIST Webbook (webbook.nist.gov/chemistry), for authoritative pure-substance property data beyond the textbook tables.

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Course Overview

MTE 309 is a single-term survey that introduces mechatronics students to two related but distinct subjects: classical engineering thermodynamics and the three modes of heat transfer. Thermodynamics supplies the bookkeeping for energy as it moves through systems and changes form, while heat transfer supplies the rate equations that determine how fast that energy crosses temperature boundaries. The combination is central to virtually every mechatronic device that dissipates power, harvests energy, or interacts with a thermal environment — motor controllers, battery packs, sensor enclosures, additive manufacturing printheads, thermoelectric stages, and industrial process actuators.

The course adopts the macroscopic (continuum) viewpoint. We do not model individual molecules; instead, we treat matter as a continuum characterized by a small number of properties — pressure, temperature, specific volume, internal energy, enthalpy, entropy — and we write balance equations for conserved quantities across control surfaces. This produces a compact mathematical toolkit: the First Law, the Second Law, the steady and unsteady conduction equations, convective correlations, and the radiation laws. The remainder of the notes works through these tools in a narrative sequence, beginning with vocabulary and culminating in electronics cooling as a synthetic application.

1. Thermodynamic Concepts and Vocabulary

1.1 System, Boundary, Surroundings

A thermodynamic system is a quantity of matter or a region in space selected for analysis. Everything outside the system is the surroundings. The boundary is the real or imaginary surface that separates them. Boundaries may be rigid or deformable, stationary or moving, permeable or impermeable.

Two system idealizations dominate engineering practice. A closed system, also called a control mass, contains a fixed quantity of matter; mass does not cross its boundary, but energy may in the form of heat and work. A sealed piston-cylinder assembly is the canonical example. An open system, or control volume, is a region of space through which mass may flow; a pump, a nozzle, a turbine, and a heat exchanger are all analyzed as control volumes. An isolated system exchanges neither mass nor energy with its surroundings.

1.2 Properties, States, Processes, Cycles

A property is a macroscopic characteristic of a system that is independent of path — pressure \( P \), temperature \( T \), volume \( V \), internal energy \( U \), enthalpy \( H \), entropy \( S \). Properties are classified as extensive (proportional to mass, such as \( V \) or \( U \)) or intensive (independent of mass, such as \( T \) or \( P \)). Dividing an extensive property by mass yields a specific property, e.g. specific volume \( v = V/m \) or specific internal energy \( u = U/m \), which is intensive.

A state is the complete set of property values at a specified instant. A process is any change from one equilibrium state to another, and the sequence of states traversed is the path. A quasi-equilibrium or quasi-static process proceeds slowly enough that the system remains infinitesimally close to equilibrium at every instant; such processes are idealized but serve as the foundation for reversible-work integrals. A cycle is a process whose end state coincides with its initial state.

1.3 The Zeroth Law and Temperature

The Zeroth Law of Thermodynamics states that if two systems are each in thermal equilibrium with a third, they are in thermal equilibrium with each other. This transitivity is what makes temperature measurement meaningful: a thermometer measures its own temperature, but the Zeroth Law guarantees that this also equals the temperature of anything in thermal equilibrium with it. The absolute Kelvin scale \( T \left[ \text{K} \right] = T \left[ ^\circ\text{C} \right] + 273.15 \) is used in all thermodynamic equations; Rankine \( T \left[ ^\circ\text{R} \right] = T \left[ ^\circ\text{F} \right] + 459.67 \) is the absolute scale associated with Fahrenheit.

1.4 Pressure

Pressure is the normal force per unit area exerted by a fluid. In thermodynamic equations \( P \) is always absolute pressure, measured relative to a perfect vacuum. Gauge pressure is measured relative to local atmospheric pressure: \( P_{\text{gauge}} = P_{\text{abs}} - P_{\text{atm}} \). A vacuum pressure is the deficit below atmospheric. For a column of fluid at rest the hydrostatic variation is

\[ \frac{dP}{dz} = -\rho g, \]

from which manometry relations follow.

2. Properties of Pure Substances

2.1 Phase and the Two-Property Rule

A pure substance has a fixed chemical composition throughout. Water, nitrogen, and refrigerant R-134a are pure substances; a nonreacting mixture of gases, such as dry air, also behaves as one over normal engineering ranges. For a simple compressible pure substance in equilibrium, the state postulate asserts that the thermodynamic state is fixed by any two independent intensive properties. “Independent” is the key word: within the two-phase region of water, \( P \) and \( T \) are not independent.

A substance may exist as solid, liquid, or vapor, and at the transition points as a two-phase mixture. The saturation temperature \( T_{\text{sat}} \) at a given pressure is the temperature at which phase change occurs; equivalently, \( P_{\text{sat}} \) at a given temperature is the saturation pressure. A subcooled (or compressed) liquid is below \( T_{\text{sat}} \) at the prevailing pressure; a superheated vapor is above it; a saturated mixture contains both phases in equilibrium.

2.2 P-v-T Surface and Projections

The equilibrium states of a simple compressible substance form a surface in \( P \)-\( v \)-\( T \) space. Its projections onto the \( P \)-\( T \), \( T \)-\( v \), and \( P \)-\( v \) planes are the phase diagrams used for engineering work. Critical features:

• The critical point is the state above which liquid and vapor are indistinguishable. For water, \( T_{\text{cr}} \approx 374\,^\circ\text{C} \), \( P_{\text{cr}} \approx 22.06\,\text{MPa} \).
• The triple line is the locus of states at which all three phases coexist; its projection on \( P \)-\( T \) is a point.
• The saturation dome on a \( T \)-\( v \) or \( P \)-\( v \) diagram is bounded by the saturated-liquid line and the saturated-vapor line; inside the dome lies the two-phase region.

2.3 Quality and Property Tables

Inside the two-phase region, a new property — quality — is introduced to locate the state between the saturation lines:

\[ x = \frac{m_{\text{vapor}}}{m_{\text{total}}}. \]

Any specific property \( y \) (where \( y \in \left\{ v, u, h, s \right\} \)) in the mixture is

\[ y = y_f + x \left( y_g - y_f \right) = y_f + x \, y_{fg}, \]

where the subscript \( f \) denotes saturated liquid, \( g \) saturated vapor, and \( fg \) the difference. Thermodynamic tables (the “steam tables” for water and analogous tables for refrigerants) list \( v_f, v_g, u_f, u_{fg}, u_g, h_f, h_{fg}, h_g, s_f, s_{fg}, s_g \) as functions of \( T \) or \( P \) along the saturation curve, and also list \( v, u, h, s \) for superheated vapor and compressed liquid as functions of \( \left( T, P \right) \).

2.4 Enthalpy

Enthalpy is defined as

\[ H \equiv U + P V, \quad h \equiv u + P v. \]

It is a derived property, introduced because the combination \( u + P v \) arises naturally in control-volume analyses and in constant-pressure processes.

3. Ideal Gas Behaviour

At sufficiently low densities the intermolecular interactions of a gas become negligible and its behaviour approaches that of an ideal gas, described by

\[ P v = R T, \quad P V = m R T = N R_u T, \]

where \( R = R_u / M \) is the specific gas constant, \( R_u = 8.314\,\text{kJ} / \left( \text{kmol} \cdot \text{K} \right) \) is the universal gas constant, and \( M \) is the molar mass. For air, \( R \approx 0.287\,\text{kJ} / \left( \text{kg} \cdot \text{K} \right) \).

The internal energy of an ideal gas depends on temperature only: \( u = u \left( T \right) \). Therefore enthalpy also depends on temperature only: \( h = u + R T = h \left( T \right) \). Define the specific heats at constant volume and at constant pressure:

\[ c_v \equiv \left. \frac{\partial u}{\partial T} \right|_v, \quad c_p \equiv \left. \frac{\partial h}{\partial T} \right|_P. \]

For an ideal gas these are functions of \( T \) alone, and

\[ c_p - c_v = R, \quad k \equiv \frac{c_p}{c_v}. \]

Integrating gives \( \Delta u = \int c_v \, dT \) and \( \Delta h = \int c_p \, dT \). When \( c_p \) and \( c_v \) can be treated as constant (the “cold-air-standard” approximation), these reduce to \( \Delta u = c_v \Delta T \) and \( \Delta h = c_p \Delta T \).

The ideal-gas assumption fails near the critical point and at high pressures. Compressibility factor charts \( Z = P v / \left( R T \right) \) and equations of state such as van der Waals, Redlich-Kwong, and Beattie-Bridgeman extend the treatment to real gases, but for this course the ideal-gas model suffices for air, nitrogen, oxygen, and combustion products at typical pressures.

4. Energy, Work, and Heat

4.1 Forms of Energy

The total energy of a system is the sum of internal, kinetic, and potential contributions:

\[ E = U + \text{KE} + \text{PE} = U + \frac{1}{2} m V^2 + m g z. \]

Internal energy aggregates the microscopic kinetic and potential energies of molecules — translational, rotational, vibrational, and intermolecular. For most stationary closed systems \( \Delta \text{KE} = \Delta \text{PE} = 0 \), and only \( \Delta U \) matters.

4.2 Work

Work is energy transfer associated with a force acting through a distance, or more generally with any generalized force acting through its conjugate displacement. The sign convention used throughout these notes is work done by the system is positive. The principal mode of work in closed systems is moving-boundary (or \( PdV \)) work:

\[ W_b = \int_1^2 P \, dV. \]

This is the area under the process path on a \( P \)-\( V \) diagram, which makes clear that \( W_b \) is path-dependent. For common quasi-equilibrium processes:

Process	Constraint	Moving-boundary work
Isobaric	\( P = \text{const} \)	\( W_b = P \left( V_2 - V_1 \right) \)
Isochoric	\( V = \text{const} \)	\( W_b = 0 \)
Isothermal, ideal gas	\( T = \text{const} \)	\( W_b = m R T \ln \left( V_2 / V_1 \right) \)
Polytropic \( P V^n = C \), \( n \ne 1 \)	—	\( W_b = \left( P_2 V_2 - P_1 V_1 \right) / \left( 1 - n \right) \)
Polytropic, \( n = 1 \)	\( P V = C \)	\( W_b = P_1 V_1 \ln \left( V_2 / V_1 \right) \)

Other work modes — shaft, spring, electrical, surface-tension, magnetic — may enter specific problems and are handled analogously: each is the integral of a generalized force over its displacement.

4.3 Heat

Heat is energy transfer driven by a temperature difference. The sign convention is heat into the system is positive. Like work, heat is a path function; it is not a property, and a system does not “contain” heat. The rate of heat transfer is \( \dot Q \), and the total heat for a process is \( Q = \int \dot Q \, dt \). The three transport mechanisms — conduction, convection, and radiation — are studied quantitatively in Sections 10–14.

An adiabatic process has \( Q = 0 \), either because the boundary is insulated or because the process is so fast that no appreciable heat has time to transfer.

5. First Law for Closed Systems

The First Law of Thermodynamics is a statement of energy conservation. For a closed system between states 1 and 2,

\[ Q_{\text{net, in}} - W_{\text{net, out}} = \Delta E_{\text{system}} = \Delta U + \Delta \text{KE} + \Delta \text{PE}. \]

On a rate basis,

\[ \dot Q_{\text{net, in}} - \dot W_{\text{net, out}} = \frac{dE_{\text{system}}}{dt}. \]

For a stationary closed system \( \Delta \text{KE} = \Delta \text{PE} = 0 \) and the law collapses to \( Q - W = \Delta U \). In differential form, \( \delta q - \delta w = du \) per unit mass, where the inexact differentials \( \delta q, \delta w \) emphasize path dependence.

5.1 Specific Heats in Context

For a stationary closed system undergoing an isochoric process with only \( PdV \) work, \( W = 0 \) and the First Law gives \( Q = \Delta U \). In differential form \( \delta q = du = c_v \, dT \), which is the operational origin of \( c_v \). For an isobaric process, \( W = P \Delta V \), so \( Q = \Delta U + P \Delta V = \Delta H \), and \( \delta q = dh = c_p \, dT \). This reveals the utility of enthalpy: constant-pressure heat transfer equals the change in enthalpy.

5.2 Worked Illustration: Rigid Tank Cooling

Consider a rigid insulated tank of air at \( \left( P_1, T_1 \right) \) stirred by a paddle wheel that adds shaft work \( W_{\text{sh}} \). Because the volume is fixed and the tank is adiabatic,

\[ -W_{\text{sh, on system}} = 0 - W_{\text{sh, out}} = \Delta U, \quad \text{so} \quad W_{\text{sh, in}} = m c_v \left( T_2 - T_1 \right). \]

The paddle-wheel work raises the air temperature; this is mechanical-energy dissipation through viscous shear and illustrates the directionality that the Second Law later formalizes.

6. First Law for Control Volumes

6.1 Conservation of Mass

For a control volume, conservation of mass states

\[ \frac{dm_{CV}}{dt} = \sum_{\text{in}} \dot m_i - \sum_{\text{out}} \dot m_e. \]

The mass flow rate through a surface of area \( A \) carrying fluid of density \( \rho \) at average normal velocity \( V \) is \( \dot m = \rho A V \). The volumetric flow rate is \( \dot{\mathcal{V}} = A V \).

6.2 Energy Equation

Each stream crossing a control surface carries with it internal, kinetic, and potential energy, and also performs flow work \( P v \) to push itself into or out of the control volume. Combining \( u + P v = h \) yields the control-volume energy balance:

\[ \frac{dE_{CV}}{dt} = \dot Q_{CV} - \dot W_{CV} + \sum_{\text{in}} \dot m_i \left( h_i + \frac{V_i^2}{2} + g z_i \right) - \sum_{\text{out}} \dot m_e \left( h_e + \frac{V_e^2}{2} + g z_e \right). \]

Here \( \dot W_{CV} \) excludes flow work (already absorbed into \( h \)) and normally represents shaft work. At steady state all time derivatives vanish and mass in equals mass out.

6.3 Common Steady-Flow Devices

Under steady-state conditions, each of the following devices reduces the energy equation to a compact form:

• Nozzle / diffuser. \( \dot W = 0, \dot Q \approx 0, \Delta \text{PE} \approx 0 \). Result: \( h_1 + V_1^2 / 2 = h_2 + V_2^2 / 2 \). A nozzle trades enthalpy for kinetic energy; a diffuser does the reverse.
• Turbine. \( \dot Q \approx 0 \), kinetic and potential changes usually negligible. Result: \( \dot W_{\text{out}} = \dot m \left( h_1 - h_2 \right) \).
• Compressor / pump. Same as a turbine with work in: \( \dot W_{\text{in}} = \dot m \left( h_2 - h_1 \right) \).
• Throttling valve. \( \dot Q \approx 0, \dot W = 0 \), velocity changes small. Result: \( h_1 = h_2 \). An isenthalpic process; it drops pressure at essentially constant enthalpy and is the workhorse of refrigeration cycles.
• Heat exchanger. Each stream obeys \( \dot Q = \dot m \Delta h \) if work and KE/PE changes are neglected; the two streams are coupled by \( \dot Q_{\text{hot, out}} = \dot Q_{\text{cold, in}} \) if the exchanger shell is adiabatic.
• Mixing chamber. Multiple streams combine adiabatically; mass and energy balances give \( \sum \dot m_i h_i = \dot m_e h_e \).

6.4 Unsteady Analysis

Charging a rigid tank from a supply line and discharging a pressurized tank to atmosphere are classic unsteady problems. The time-integrated form of the control-volume balance — often called the “uniform-flow, uniform-state” model — assumes properties at each port are constant over the charging interval and properties within the tank are uniform (though time-varying). This yields an algebraic relation between initial and final tank states and the integrated inflow enthalpy.

7. Second Law of Thermodynamics

The First Law says energy is conserved, but it does not say which processes actually occur. Hot objects cool; cold ones do not spontaneously heat. A paddle wheel heats water, but water never spontaneously stirs a paddle wheel. The Second Law encodes this directionality.

7.1 Thermal Reservoirs and Heat Engines

A thermal reservoir is an idealized body with such large thermal capacity that heat transfer to or from it does not change its temperature — the atmosphere, a large lake, a high-temperature furnace. A heat engine is a device operating in a cycle that accepts heat \( Q_H \) from a high-temperature reservoir, rejects heat \( Q_L \) to a low-temperature reservoir, and produces net work \( W_{\text{net}} = Q_H - Q_L \). Its thermal efficiency is

\[ \eta_{\text{th}} = \frac{W_{\text{net}}}{Q_H} = 1 - \frac{Q_L}{Q_H}. \]

7.2 Kelvin-Planck and Clausius Statements

Two classical statements are logically equivalent:

• Kelvin-Planck. It is impossible for any device operating in a cycle to receive heat from a single reservoir and produce a net amount of work. Equivalently, no heat engine can achieve \( \eta = 100\% \): some heat must be rejected to a low-temperature reservoir.
• Clausius. It is impossible to construct a device operating in a cycle whose sole effect is the transfer of heat from a cooler body to a hotter body. Heat pumps and refrigerators can move heat uphill, but only by consuming work input.

7.3 Reversibility

A reversible process is one whose direction can be reversed without leaving any net change in either system or surroundings. All real processes are irreversible: friction, unrestrained expansion, mixing, heat transfer across a finite \( \Delta T \), inelastic deformation, and electrical resistance are all sources of irreversibility. Reversible processes are the ideal limit against which real processes are measured.

7.4 Refrigerators and Heat Pumps

A refrigerator operates in a cycle to remove heat \( Q_L \) from a cold space by consuming work \( W_{\text{in}} \) and rejecting \( Q_H = Q_L + W_{\text{in}} \) to a warmer environment. Its coefficient of performance is

\[ \text{COP}_R = \frac{Q_L}{W_{\text{in}}} = \frac{1}{Q_H / Q_L - 1}. \]

A heat pump is the same device with the desired effect relabeled: its objective is to deliver \( Q_H \) to a warm space. Its COP is

\[ \text{COP}_{HP} = \frac{Q_H}{W_{\text{in}}} = \text{COP}_R + 1. \]

8. The Carnot Cycle

The Carnot cycle is a totally reversible cycle composed of four reversible processes: an isothermal heat addition at \( T_H \), a reversible adiabatic (isentropic) expansion from \( T_H \) to \( T_L \), an isothermal heat rejection at \( T_L \), and a reversible adiabatic compression from \( T_L \) to \( T_H \). Its efficiency is

\[ \eta_{\text{Carnot}} = 1 - \frac{T_L}{T_H}, \]

where \( T_H \) and \( T_L \) are absolute temperatures. Two Carnot principles follow from the Second Law:

1. The efficiency of an irreversible heat engine operating between two reservoirs is always less than that of a reversible engine operating between the same reservoirs.
2. All reversible engines operating between the same two reservoirs have the same efficiency, independent of working fluid.

These principles justify using the Carnot efficiency as the thermodynamic upper bound on any real engine. The corresponding COP bounds are

\[ \text{COP}_{R, \text{Carnot}} = \frac{T_L}{T_H - T_L}, \quad \text{COP}_{HP, \text{Carnot}} = \frac{T_H}{T_H - T_L}. \]

9. Entropy

9.1 Clausius Inequality and the Definition of \( S \)

For any closed system executing a cycle,

\[ \oint \frac{\delta Q}{T} \le 0, \]

with equality for totally reversible cycles. This motivates the definition of entropy as the property whose differential is

\[ dS = \left( \frac{\delta Q}{T} \right)_{\text{rev}}. \]

Entropy is a property because its cyclic integral along a reversible path is zero. For an irreversible process from state 1 to state 2,

\[ \Delta S \ge \int_1^2 \frac{\delta Q}{T}, \]

and the difference, entropy generation \( S_{\text{gen}} \ge 0 \), quantifies irreversibility.

9.2 The Increase-of-Entropy Principle

For an isolated system, \( \delta Q = 0 \) and therefore \( \Delta S_{\text{isolated}} \ge 0 \). Considering any system together with its surroundings as isolated,

\[ S_{\text{gen}} = \Delta S_{\text{system}} + \Delta S_{\text{surroundings}} \ge 0. \]

Entropy is not conserved; it can only be generated, never destroyed. This is perhaps the sharpest expression of the Second Law.

9.3 Gibbs (\( Tds \)) Relations

Combining the First and Second Laws for a reversible process yields

\[ T \, ds = du + P \, dv, \quad T \, ds = dh - v \, dP. \]

These equations relate properties only, so once derived under reversible assumptions they are valid for any process between equilibrium states. They underlie all subsequent entropy calculations.

9.4 Entropy Change of Ideal Gases

Integrating the \( Tds \) relations for an ideal gas with constant specific heats,

\[ s_2 - s_1 = c_v \ln \frac{T_2}{T_1} + R \ln \frac{v_2}{v_1} = c_p \ln \frac{T_2}{T_1} - R \ln \frac{P_2}{P_1}. \]

An isentropic process has \( \Delta s = 0 \). For an ideal gas with constant specific heats this implies

\[ \frac{T_2}{T_1} = \left( \frac{P_2}{P_1} \right)^{\left( k - 1 \right) / k} = \left( \frac{v_1}{v_2} \right)^{k - 1}. \]

For variable specific heats, tabulated functions \( s^\circ \left( T \right) \) and the reduced-pressure / reduced-volume shortcut \( P_r, v_r \) are used.

9.5 Isentropic Efficiencies

Real adiabatic devices are not isentropic because of internal irreversibilities. Isentropic efficiency compares actual performance to the reversible ideal:

\[ \eta_{T} = \frac{h_1 - h_{2a}}{h_1 - h_{2s}}, \quad \eta_{C} = \frac{h_{2s} - h_1}{h_{2a} - h_1}, \quad \eta_N = \frac{V_{2a}^2}{V_{2s}^2}. \]

Typical values fall in the range 0.75–0.95 for well-designed turbomachinery.

10. Power and Refrigeration Cycles

10.1 Rankine Cycle

The Rankine cycle is the workhorse of steam power plants. It has four components: boiler, turbine, condenser, pump. In the ideal cycle, each component operates reversibly: the pump and turbine isentropically, the boiler and condenser isobarically. Steam is generated at high pressure, expanded through the turbine to produce work, condensed to liquid, and pumped back to the boiler. Reheat and regeneration are practical modifications that raise efficiency by reducing moisture at the turbine exit and by preheating feedwater with bled steam.

10.2 Brayton Cycle

The Brayton cycle models gas-turbine engines. Air is compressed, combusted (modeled as constant-pressure heat addition), expanded through a turbine, and either exhausted (open cycle) or cooled back to inlet conditions (closed cycle). For cold-air-standard analysis,

\[ \eta_{\text{Brayton}} = 1 - \frac{1}{r_p^{\left( k - 1 \right) / k}}, \]

where \( r_p = P_2 / P_1 \) is the pressure ratio. Regeneration, intercooling, and reheating improve performance.

10.3 Otto and Diesel Cycles

These are the air-standard models of spark-ignition and compression-ignition reciprocating engines respectively. The Otto cycle comprises two isentropic and two isochoric processes; its efficiency is

\[ \eta_{\text{Otto}} = 1 - \frac{1}{r^{k - 1}}, \]

where \( r = V_{\text{max}} / V_{\text{min}} \) is the compression ratio. The Diesel cycle replaces the isochoric heat addition with an isobaric one, giving

\[ \eta_{\text{Diesel}} = 1 - \frac{1}{r^{k - 1}} \left[ \frac{r_c^k - 1}{k \left( r_c - 1 \right)} \right], \]

where \( r_c = V_3 / V_2 \) is the cutoff ratio.

10.4 Vapor-Compression Refrigeration

The standard refrigeration cycle reverses the Rankine layout: a compressor raises the refrigerant vapor pressure, a condenser rejects heat at high temperature, a throttling valve drops the pressure isenthalpically, and an evaporator absorbs heat at low temperature. The working fluid (R-134a, R-410A, ammonia, propane, or CO\(_2\), depending on application) is selected for its saturation curve and environmental profile. The COP is

\[ \text{COP}_R = \frac{h_1 - h_4}{h_2 - h_1}, \]

with state 1 at compressor inlet and state 2 at compressor outlet.

11. Conduction Fundamentals

With thermodynamics providing energy balances, heat transfer supplies the rate equations that quantify how fast heat actually moves. The three mechanisms — conduction, convection, radiation — often act simultaneously.

11.1 Fourier’s Law

Conduction is energy transport by molecular interaction, dominant in solids and stagnant fluids. Fourier’s law states that the local heat flux is proportional to the negative temperature gradient:

\[ \vec q'' = -k \, \nabla T, \]

where \( k \left[ \text{W} / \left( \text{m} \cdot \text{K} \right) \right] \) is the thermal conductivity. Typical values: pure copper \( \approx 400 \), aluminum \( \approx 237 \), steel \( \approx 15{-}50 \), ceramics \( 1{-}30 \), polymers \( 0.1{-}0.5 \), gases \( 0.02{-}0.2 \), insulators \( \approx 0.03 \).

11.2 Heat Equation

Applying conservation of energy to a differential control volume in a solid with volumetric generation \( \dot e_{\text{gen}} \left[ \text{W} / \text{m}^3 \right] \) yields the heat diffusion equation:

\[ \rho c \frac{\partial T}{\partial t} = \nabla \cdot \left( k \, \nabla T \right) + \dot e_{\text{gen}}. \]

For constant \( k \),

\[ \frac{\partial T}{\partial t} = \alpha \nabla^2 T + \frac{\dot e_{\text{gen}}}{\rho c}, \quad \alpha \equiv \frac{k}{\rho c}. \]

The thermal diffusivity \( \alpha \left[ \text{m}^2 / \text{s} \right] \) measures how quickly temperature disturbances propagate through a material.

11.3 One-Dimensional Steady Conduction

For a plane wall with no generation and fixed surface temperatures \( T_{s,1} \) and \( T_{s,2} \) over thickness \( L \),

\[ T \left( x \right) = T_{s,1} + \left( T_{s,2} - T_{s,1} \right) \frac{x}{L}, \quad q'' = \frac{k \left( T_{s,1} - T_{s,2} \right)}{L}. \]

In cylindrical and spherical geometries the profiles become logarithmic and reciprocal, respectively.

11.4 Thermal Resistance

By analogy with Ohm’s law, heat flow through a medium can be written \( q = \Delta T / R_{\text{th}} \). The resistance of a plane wall is \( R_{\text{cond}} = L / \left( k A \right) \); of a cylindrical shell, \( R_{\text{cond}} = \ln \left( r_o / r_i \right) / \left( 2 \pi L k \right) \); of a convective film, \( R_{\text{conv}} = 1 / \left( h A \right) \); of a radiative film linearized about a mean temperature, \( R_{\text{rad}} = 1 / \left( h_r A \right) \). Series and parallel combinations follow circuit rules, giving the thermal resistance network method used extensively in electronics cooling.

A contact resistance \( R''_{tc} \left[ \text{m}^2 \cdot \text{K} / \text{W} \right] \) accounts for the imperfect microscopic mating of two solid surfaces; it is reduced by thermal interface materials.

11.5 The Critical Radius of Insulation

For a cylindrical pipe, adding insulation increases conduction resistance but also increases outer surface area, reducing convective resistance. The total resistance reaches a minimum at the critical radius of insulation

\[ r_{\text{cr, cyl}} = \frac{k_{\text{ins}}}{h}. \]

Below \( r_{\text{cr}} \), adding insulation increases heat loss. This is relevant for thin wires where the insulation actually helps dissipate heat.

12. Extended Surfaces (Fins)

Fins are extended surfaces that augment heat transfer by increasing the area in contact with a fluid. The general fin equation, for a fin of uniform cross section \( A_c \), perimeter \( p \), base temperature \( T_b \), and ambient \( T_\infty \), is

\[ \frac{d^2 \theta}{dx^2} - m^2 \theta = 0, \quad m^2 \equiv \frac{h p}{k A_c}, \quad \theta \equiv T - T_\infty. \]

For an infinite fin, \( \theta / \theta_b = e^{-m x} \) and

\[ q_{\text{fin}} = \sqrt{h p k A_c} \, \theta_b. \]

For a fin of length \( L \) with an insulated tip,

\[ q_{\text{fin}} = \sqrt{h p k A_c} \, \theta_b \, \tanh \left( m L \right). \]

The fin efficiency \( \eta_{\text{fin}} = q_{\text{fin}} / \left( h A_{\text{fin}} \theta_b \right) \) and fin effectiveness \( \varepsilon_{\text{fin}} = q_{\text{fin}} / \left( h A_c \theta_b \right) \) quantify the benefit; a fin is worthwhile only when \( \varepsilon_{\text{fin}} > 2 \) or so. Design guidance: use high-\( k \) materials, thin long fins, and low-\( h \) situations (fins help most when convection is weak).

13. Transient Conduction

13.1 The Biot Number and the Lumped Capacitance Model

The Biot number

\[ \text{Bi} = \frac{h L_c}{k}, \quad L_c \equiv \frac{V}{A_s}, \]

compares internal conduction resistance to external convection resistance. When \( \text{Bi} < 0.1 \), internal gradients are negligible, and the body can be modeled as lumped: a single uniform temperature. Energy conservation then gives

\[ \rho c V \frac{dT}{dt} = -h A_s \left( T - T_\infty \right), \quad \frac{T \left( t \right) - T_\infty}{T_i - T_\infty} = \exp \left( -\frac{t}{\tau} \right), \quad \tau = \frac{\rho c V}{h A_s}. \]

The time constant \( \tau \) is the central design quantity for thermally responsive systems.

13.2 One-Term and Heisler-Chart Solutions

When \( \text{Bi} \ge 0.1 \), internal gradients matter and the heat equation must be solved as a boundary-value problem. For a plane wall, long cylinder, and sphere of characteristic length \( L \), subjected to convection with \( h \) on the surfaces, the dimensionless temperature \( \theta^* = \left( T - T_\infty \right) / \left( T_i - T_\infty \right) \) admits a Fourier-series solution. For Fourier number \( \text{Fo} = \alpha t / L^2 > 0.2 \), a one-term approximation suffices:

\[ \theta^* \left( x, t \right) \approx A_1 \, e^{-\lambda_1^2 \text{Fo}} \, f \left( \lambda_1 x / L \right), \]

where \( \lambda_1 \) and \( A_1 \) are tabulated as functions of Biot number, and \( f \) is cosine, Bessel \( J_0 \), or sinc, depending on geometry. Heisler charts graph this solution for engineering use.

For a semi-infinite solid with step surface temperature or step surface heat flux, the similarity variable \( \eta = x / \left( 2 \sqrt{\alpha t} \right) \) reduces the PDE to an ODE whose solution involves the complementary error function \( \text{erfc} \left( \eta \right) \).

14. Convection Fundamentals

14.1 Newton’s Law of Cooling

Convection is heat transfer between a surface and a moving fluid. The surface heat flux is

\[ q'' = h \left( T_s - T_\infty \right), \]

where \( h \left[ \text{W} / \left( \text{m}^2 \cdot \text{K} \right) \right] \) is the convection coefficient, an empirical quantity that depends on flow geometry, fluid properties, velocity, and surface condition. Typical magnitudes: free-convection air \( 2{-}25 \), forced-convection air \( 25{-}250 \), forced-convection water \( 50{-}2{,}000 \), boiling water \( 2{,}500{-}100{,}000 \), condensing steam \( 5{,}000{-}100{,}000 \).

14.2 Boundary Layers

When a fluid flows over a surface, momentum and thermal boundary layers develop adjacent to the wall. Transition from laminar to turbulent flow occurs at critical Reynolds numbers (for a flat plate, \( \text{Re}_{x, \text{cr}} \approx 5 \times 10^5 \); for internal pipe flow, \( \text{Re}_D \approx 2300 \)). The flow regime profoundly affects heat transfer: turbulent flow enhances \( h \) substantially because of eddy transport.

14.3 Dimensionless Groups

Convection correlations are written in terms of dimensionless groups derived from the governing equations:

Group	Definition	Physical meaning
Reynolds	\( \text{Re} = \rho V L / \mu \)	Inertia / viscous forces
Prandtl	\( \text{Pr} = \mu c_p / k = \nu / \alpha \)	Momentum diffusivity / thermal diffusivity
Nusselt	\( \text{Nu} = h L / k \)	Dimensionless convection coefficient
Grashof	\( \text{Gr} = g \beta \Delta T L^3 / \nu^2 \)	Buoyancy / viscous forces
Rayleigh	\( \text{Ra} = \text{Gr} \cdot \text{Pr} \)	Strength of free convection
Peclet	\( \text{Pe} = \text{Re} \cdot \text{Pr} \)	Advection / diffusion (heat)

For forced convection the Nusselt number is a function of \( \text{Re} \) and \( \text{Pr} \); for natural convection, of \( \text{Ra} \) and \( \text{Pr} \).

14.4 Forced-Convection Correlations

Selected benchmark correlations:

• Flat plate, laminar, local: \( \text{Nu}_x = 0.332 \, \text{Re}_x^{1/2} \, \text{Pr}^{1/3} \) for \( \text{Pr} \gtrsim 0.6 \).
• Flat plate, turbulent, local: \( \text{Nu}_x = 0.0296 \, \text{Re}_x^{4/5} \, \text{Pr}^{1/3} \).
• Pipe flow, turbulent, fully developed (Dittus-Boelter):

\[ \text{Nu}_D = 0.023 \, \text{Re}_D^{4/5} \, \text{Pr}^{n}, \]

with \( n = 0.4 \) for heating and \( n = 0.3 \) for cooling. Valid for \( 0.7 \le \text{Pr} \le 160 \), \( \text{Re}_D > 10{,}000 \), \( L / D > 10 \).

• Pipe flow, laminar, fully developed: \( \text{Nu}_D = 3.66 \) for uniform wall temperature; \( \text{Nu}_D = 4.36 \) for uniform wall heat flux.
• Flow across a cylinder (Churchill-Bernstein): a composite correlation spanning all \( \text{Re}_D \) and \( \text{Pr} \ge 0.2 \).

14.5 Natural-Convection Correlations

• Vertical plate (Churchill-Chu): valid for all \( \text{Ra}_L \).

\[ \text{Nu}_L = \left\{ 0.825 + \frac{0.387 \, \text{Ra}_L^{1/6}}{\left[ 1 + \left( 0.492 / \text{Pr} \right)^{9/16} \right]^{8/27}} \right\}^2. \]

• Horizontal plate, hot surface up: \( \text{Nu}_L = 0.54 \, \text{Ra}_L^{1/4} \) for \( 10^4 \le \text{Ra}_L \le 10^7 \); \( \text{Nu}_L = 0.15 \, \text{Ra}_L^{1/3} \) for \( 10^7 \le \text{Ra}_L \le 10^{11} \).
• Horizontal cylinder (Churchill-Chu): analogous composite form.
• Enclosures and parallel vertical plates: correlations due to MacGregor-Emery, Elenbaas, and Bar-Cohen-Rohsenow, relevant to electronics enclosures.

15. Radiation

15.1 Blackbody Emission

A blackbody is an idealized surface that absorbs all incident radiation and emits the maximum possible radiation at every wavelength and direction. Its total emissive power follows the Stefan-Boltzmann law:

\[ E_b = \sigma T^4, \quad \sigma = 5.670 \times 10^{-8} \, \text{W} / \left( \text{m}^2 \cdot \text{K}^4 \right). \]

The spectral distribution follows Planck’s law, and the peak wavelength shifts with temperature per Wien’s displacement law \( \lambda_{\text{max}} T = 2898\,\mu\text{m} \cdot \text{K} \).

15.2 Real Surfaces

Real surfaces are characterized by an emissivity \( \varepsilon \le 1 \), defined as the ratio of emitted radiation to blackbody emission at the same temperature:

\[ E = \varepsilon \sigma T^4. \]

By Kirchhoff’s law, under thermal equilibrium \( \varepsilon = \alpha \) (absorptivity). For real surfaces \( \varepsilon \) generally depends on wavelength and direction, but engineering calculations often use gray-surface (wavelength-independent) and diffuse (direction-independent) assumptions. Typical values: polished aluminum \( 0.05 \), anodized aluminum \( 0.8 \), painted surfaces \( 0.85{-}0.95 \), human skin \( 0.97 \).

15.3 Radiative Exchange

For two gray, diffuse, opaque surfaces in a vacuum, the net exchange depends on a view factor \( F_{ij} \), the fraction of radiation leaving surface \( i \) that strikes surface \( j \). For a small convex object in a large enclosure,

\[ q_{\text{rad}} = \varepsilon \sigma A \left( T_s^4 - T_{\text{surr}}^4 \right). \]

For small temperature differences, this linearizes to \( q = h_r A \Delta T \) with \( h_r = 4 \varepsilon \sigma \bar T^3 \), allowing radiation to enter a resistance network as a parallel path to convection.

16. Thermal Management of Electronic Devices

Mechatronic systems are typically rich in electronics that dissipate power as heat. Reliability is temperature-dependent; many failure mechanisms accelerate with an Arrhenius \( \exp \left( -E_a / k_B T \right) \) dependence, so a rule of thumb is that every 10 K rise in junction temperature roughly halves device lifetime. This section synthesizes the preceding material into a design vocabulary.

16.1 Thermal Resistance Networks for Packages

A packaged IC has a stack of resistances from silicon junction to ambient:

\[ R_{\theta JA} = R_{\theta JC} + R_{\theta CS} + R_{\theta SA}. \]

Here \( J \) is the semiconductor junction, \( C \) the package case, \( S \) the heat sink, and \( A \) the ambient. \( R_{\theta JC} \) is set by the package — die attach, leadframe, molding compound. \( R_{\theta CS} \) is the interface between case and sink. \( R_{\theta SA} \) is the sink-to-air resistance, dominated by convection and fin geometry. Given maximum allowable junction temperature \( T_{J, \max} \) and dissipation \( P_D \),

\[ T_{J, \max} - T_A \ge P_D \cdot R_{\theta JA}. \]

This algebraic relation is the starting point for nearly every electronics-cooling problem in practice.

16.2 Thermal Interface Materials

The microscopic roughness between a chip case and a heat sink leaves most of the apparent contact area occupied by air gaps; air’s low conductivity makes the joint thermally poor. A thermal interface material (TIM) — thermal grease, phase-change pad, graphite sheet, or sintered metal — fills these gaps. Good TIM design minimizes bond-line thickness and maximizes through-plane conductivity. TIM specifications are usually given in units of thermal resistance per unit area \( R''_{tc} \left[ \text{K} \cdot \text{cm}^2 / \text{W} \right] \); representative values range from \( 0.1 \) for high-end metal-based products to \( 1.5 \) for generic greases.

16.3 Heat Sinks and Forced-Air Cooling

A heat sink is an array of fins intended to maximize \( \eta_{\text{fin}} A_{\text{fin}} h \) within packaging and acoustic constraints. Key design levers:

• Base material. Aluminum (\( k \approx 200 \)) for mass-produced extrusions; copper (\( k \approx 400 \)) where its density is acceptable; aluminum with embedded copper slug or vapor chamber in high-flux regions.
• Fin density and aspect ratio. More fins give more area but also thicker boundary layers and increased pressure drop; optimization balances \( h \) against flow resistance.
• Flow direction and fan selection. Axial fans pair with straight-fin arrays; radial blowers with pin-fin sinks. Fan curves (pressure vs flow) intersect system curves at the operating point; always design the system curve into the fan choice.
• Nusselt correlations for fin arrays. Elenbaas-type correlations govern free convection between vertical parallel plates; forced-convection correlations extend the flat-plate or pipe formulas with a channel-specific characteristic length.

16.4 Liquid and Two-Phase Cooling

When heat flux exceeds the capability of forced air — typically above \( 50{-}100 \, \text{W} / \text{cm}^2 \) at the package — liquid cooling becomes necessary. Options include direct-liquid cold plates (water or glycol mixture pumped through a finned channel), immersion cooling in dielectric fluids, and two-phase systems that exploit latent heat at a phase change. Heat pipes and vapor chambers are passive two-phase devices with extraordinary effective conductivity (thousands of W/(m·K)) that move heat from a localized source to a spread area where conventional convection can handle it.

16.5 Thermoelectric (Peltier) Coolers

A Peltier device is a solid-state heat pump based on the Peltier effect: passing current through a junction of dissimilar semiconductors transports heat from one face to the other. The maximum temperature difference \( \Delta T_{\max} \) of a commercial single-stage module is around 65–70 K; the COP is low (\( \sim 0.3{-}0.6 \)) compared to vapor-compression refrigeration, so Peltiers are used where silence, compactness, precise control, or sub-ambient cooling of a small load outweighs energy efficiency — laser diode stabilization, CCD sensor cooling, analytical-chemistry instruments, consumer beverage coolers. Careful heat-sink design on the hot side is essential; a Peltier alone does not remove heat from a system, it merely relocates it plus the electrical input.

16.6 Transient Thermal Events

Short power transients are handled with transient thermal impedance \( Z_{\theta JA} \left( t \right) \), often provided as a chart for a given package. Under pulsed operation, the effective resistance is lower than the DC value because thermal capacitance stores heat. Lumped-capacitance analysis (Section 13.1) applied to the heat sink gives a first-cut estimate of junction temperature response to a load step; more accurate design uses RC-ladder network models identified from package datasheets.

16.7 System-Level Considerations

Good thermal design at the system level:

• Budgets power dissipation by component and allocates a temperature rise to each resistance in the chain.
• Plans airflow so that inlet air reaches hot components before being heated by upstream dissipators.
• Uses radiation where surfaces run hot and can emit to cold surroundings; inside a closed chassis, radiation exchange among interior walls and boards can be non-negligible.
• Validates with thermocouples, IR imaging, or embedded temperature sensors; numerical models (CFD, compact thermal models) support iterative redesign.

The tools required — First Law for heat-balance budgets, resistance networks for steady analysis, Biot and Fourier numbers for transient response, convection correlations for \( h \), Stefan-Boltzmann for radiation — are exactly the contents of this course. Thermal management of electronics is therefore a natural capstone application of MTE 309.

17. Summary of Core Equations

• Ideal gas: \( P v = R T \), \( c_p - c_v = R \).
• First Law, closed: \( Q - W = \Delta U \).
• First Law, control volume steady: \( \dot Q - \dot W = \sum \dot m_e \left( h + V^2 / 2 + g z \right)_e - \sum \dot m_i \left( h + V^2 / 2 + g z \right)_i \).
• Carnot: \( \eta = 1 - T_L / T_H \).
• Entropy generation: \( S_{\text{gen}} = \Delta S_{\text{system}} + \Delta S_{\text{surr}} \ge 0 \).
• Fourier: \( \vec q'' = -k \nabla T \).
• Heat equation: \( \rho c \, \partial T / \partial t = \nabla \cdot \left( k \nabla T \right) + \dot e_{\text{gen}} \).
• Fin, insulated tip: \( q = \sqrt{h p k A_c} \, \theta_b \tanh \left( m L \right) \).
• Lumped capacitance: \( \theta / \theta_i = \exp \left( -t / \tau \right) \), \( \tau = \rho c V / \left( h A_s \right) \).
• Newton’s law of cooling: \( q'' = h \left( T_s - T_\infty \right) \).
• Stefan-Boltzmann: \( E_b = \sigma T^4 \); with emissivity \( q_{\text{rad}} = \varepsilon \sigma A \left( T_s^4 - T_{\text{surr}}^4 \right) \).
• Thermal resistance: \( R_{\text{cond, plane}} = L / \left( k A \right) \), \( R_{\text{conv}} = 1 / \left( h A \right) \), \( R_{\text{cyl}} = \ln \left( r_o / r_i \right) / \left( 2 \pi L k \right) \).