Chapter 1: Introduction

Source files used: Heat Transfer (01), Heat Transfer (02), early Heat Transfer (03); textbook Chapter 1 as support. No worked examples are included.

1. Big Picture

Chapter 1 answers the question that thermodynamics usually avoided: how do we calculate heat transfer?

In thermodynamics, heat transfer often appeared as a known value in the first law. In heat transfer, that value is no longer simply handed to us. The purpose of the course is to calculate heat transfer rates from physical mechanisms.

The first rule is simple:

$$\text{Heat transfer requires a temperature difference.}$$

No temperature difference means no heat transfer. Thermal energy moves from hotter regions toward colder regions. Biddle divides first-course heat transfer into three modes:

1. Conduction — thermal energy transfer through matter by interaction between particles.
2. Convection — thermal energy transfer between a surface and a moving fluid.
3. Radiation — thermal energy transfer by electromagnetic waves.

The structure of the course follows this split. Chapter 1 gives the introductory picture. Chapters 2, 3, 4, and 5 develop conduction; Chapters 6, 7, 8, and 9 develop convection; radiation is developed more deeply later, but only the Chapter 1 radiation basics are needed in this study guide.

The Chapter 1 problem-solving mindset is:

$$\boxed{\text{Understand the heat path first, then write the rate equations and energy balance.}}$$

Do not start by hunting formulas. First ask what physical process is occurring.

---

2. Core Ideas

2.1 Heat Rate vs Heat Flux

Biddle emphasizes the notation because many mistakes come from mixing total heat rate with heat flux.

Symbol	Meaning	Units
$\dot{q}$, $\dot{Q}$	heat transfer rate	W
$q''$	heat flux, heat transfer rate per unit area	W/m²

$$q''=\frac{\dot{q}}{A}$$

or

$$\dot{q}=q''A$$

The area must be the area through which the heat actually passes. For a plane wall, this is the wall face area normal to the direction of heat flow.

---

2.2 Conduction

Conduction is heat transfer through a material because energetic particles interact with less energetic particles. If one side of a steel plate is heated, atoms and electrons on the hot side have more energy. They transfer energy to neighboring particles, and energy moves through the solid.

Chapter 1 uses only the simplest conduction picture: one-dimensional conduction through a plane wall.

Important intuition:

• Larger temperature difference gives larger heat transfer.
• Larger area gives larger heat transfer.
• Larger thermal conductivity $k$ gives larger heat transfer.
• Larger wall thickness $L$ gives smaller heat transfer.

---

2.3 Convection

Convection occurs between a surface and a fluid. The surface is at $T_s$, and the fluid away from the surface is at $T_\infty$.

The key quantity is the convection coefficient $h$. Biddle’s warning is that $h$ is not a simple material property. You usually do not just look it up in a table. It depends on:

• fluid type,
• fluid temperature,
• gas pressure if relevant,
• geometry,
• velocity,
• laminar/turbulent/free/forced flow regime.

In Chapter 1, $h$ is often given. Chapters 6–9 are mostly about finding $h$.

---

2.4 Radiation

Radiation is thermal energy transfer by electromagnetic waves. It does not require matter, which is why radiation still works in vacuum.

Chapter 1 only needs the basic radiation ideas:

• A surface emits radiation because of its temperature.
• A blackbody is the ideal maximum emitter.
• A real surface emits less than a blackbody, represented by emissivity $\varepsilon$.
• Radiation formulas must use absolute temperature in kelvin.

Conduction and convection often use temperature differences, so °C and K differences are equivalent. Radiation contains $T^4$, so the actual absolute temperature matters.

---

2.5 Energy Balance

The main tool in Chapter 1 problems is the energy balance.

For a control volume:

$$\dot{E}_{\text{in}}-\dot{E}_{\text{out}}+\dot{E}_g=\dot{E}_{st}$$

In words:

$$\text{in} - \text{out} + \text{generated} = \text{stored}$$

For steady state, storage is zero. If there is no generation, generation is zero. Then the balance often reduces to:

$$\dot{E}_{\text{in}}=\dot{E}_{\text{out}}$$

For a surface energy balance, the control volume has no meaningful volume. Usually there is no storage term and no volumetric generation term; you balance heat arriving at and leaving the surface.

---

3. Main Governing Equations and Formulas

3.1 Fourier’s Law of Conduction

Differential one-dimensional form:

$$q_x''=-k\frac{dT}{dx}$$

where:

• $q_x''$: heat flux in the $x$direction, W/m²,
• $k$: thermal conductivity, W/(m·K),
• $dT/dx$: temperature gradient, K/m or °C/m.

The negative sign means heat flows in the direction of decreasing temperature.

For a plane wall with steady, one-dimensional conduction, constant $k$, no generation, and surface temperatures $T_1$ and $T_2$:

$$q''=k\frac{T_1-T_2}{L}$$ $$\dot{q}=kA\frac{T_1-T_2}{L}$$

Use when:

• wall is plane,
• heat flow is 1D,
• steady state,
• constant $k$,
• no internal generation,
• surface temperatures are known.

---

3.2 Newton’s Law of Cooling

$$q''=h(T_s-T_\infty)$$ $$\dot{q}=hA(T_s-T_\infty)$$

where:

• $h$: convection coefficient, W/(m²·K),
• $A$: surface area exposed to fluid, m²,
• $T_s$: surface temperature,
• $T_\infty$: fluid temperature away from the surface.

Use when heat is exchanged between a surface and a fluid.

If $T_s>T_\infty$, heat leaves the surface. If $T_s<T_\infty$, heat enters the surface.

---

3.3 Blackbody Emission

$$E_b=\sigma T_s^4$$

where:

• $E_b$: blackbody emissive power, W/m²,
• $\sigma=5.67\times10^{-8}\ \text{W/(m}^2\cdot\text{K}^4)$,
• $T_s$: absolute surface temperature, K.

Use this for ideal blackbody emission.

---

3.4 Real-Surface Emission

$$E=\varepsilon\sigma T_s^4$$

where $\varepsilon$ is emissivity, with $0\le\varepsilon\le1$.

If $\varepsilon=1$, the surface behaves as a blackbody. If $\varepsilon<1$, it emits less than a blackbody.

---

3.5 Small Object in Large Surroundings

$$\dot{q}_{\text{rad}}=\varepsilon\sigma A\left(T_s^4-T_{\text{sur}}^4\right)$$

where:

• $A$: area of the small object,
• $T_s$: object surface temperature, K,
• $T_{\text{sur}}$: surrounding wall temperature, K.

Use this when a small object is inside a much larger enclosure.

---

3.6 Absorbed Irradiation

$$G_{\text{abs}}=\alpha G$$

or as a heat rate:

$$\dot{q}_{\text{abs}}=\alpha GA$$

where:

• $G$: irradiation, incoming radiation, W/m²,
• $\alpha$: absorptivity,
• $A$: absorbing area.

Use for solar/incident-radiation type problems.

---

3.7 Control-Volume Energy Balance

$$\dot{E}_{\text{in}}-\dot{E}_{\text{out}}+\dot{E}_g=\dot{E}_{st}$$

Special cases:

Steady state:

$$\dot{E}_{st}=0$$

No generation:

$$\dot{E}_g=0$$

Steady, no generation:

$$\dot{E}_{\text{in}}=\dot{E}_{\text{out}}$$

---

3.8 Surface Energy Balance

Common form:

$$q''_{\text{in}}=q''_{\text{out}}$$

Examples:

$$q''_{\text{cond}}=q''_{\text{conv}}$$ $$q''_{\text{absorbed}}=q''_{\text{conv}}+q''_{\text{rad}}$$

Use surface balances when the important physics happens at a boundary or interface.

---

4. Problem-Solving Workflow

1. Draw the heat path. Decide whether conduction, convection, radiation, generation, or storage appears.
2. Choose the control volume or control surface. Pick something that makes the energy balance simple.
3. Write the energy balance. Use in − out + generation = storage.
4. Replace each energy term with the correct rate equation.
5. Use correct areas. Plane-wall area, cylinder surface area, etc.
6. Use kelvin for radiation. Do not use °C inside $T^4$.
7. Check the result physically. Heat should flow hot to cold; thicker walls reduce conduction; larger areas increase heat transfer.

---

5. Decision Rules / Decision Trees

5.1 Which Mode?

Heat through a solid or stationary material? → conduction

Heat between surface and fluid? → convection

Heat by emission/absorption across space? → radiation

Multiple modes? → include each term in an energy balance.

5.2 Heat Rate or Heat Flux?

Problem asks W? → $\dot{q}$

Problem asks W/m²? → $q''$

Given $q''$ and need $\dot{q}$? → $\dot{q}=q''A$

Given $\dot{q}$ and need $q''$? → $q''=\dot{q}/A$

5.3 Radiation Temperature Rule

Conduction/convection temperature difference? → °C difference or K difference both work.

Radiation T^4? → absolute temperature in K only.

5.4 Surface Balance Rule

At a wall surface or interface? → balance what reaches the surface with what leaves the surface.

Example: conduction into surface = convection out of surface

---

6. Important Tables / Correlations Needed

6.1 Typical Convection Coefficient Ranges

Situation	Typical $h$, W/(m²·K)
Free convection, gases	2–25
Free convection, liquids	50–1000
Forced convection, gases	25–250
Forced convection, liquids	100–20,000
Boiling/condensation	very large; pool boiling excluded from this guide

These are rough ranges only. Detailed $h$ calculations come later.

6.2 Radiation Constants and Properties

Symbol	Meaning
$\sigma$	Stefan–Boltzmann constant
$\varepsilon$	emissivity; emission ability
$\alpha$	absorptivity; absorption ability
$G$	irradiation; incoming radiation
$E$	emissive power; emitted radiation

---

7. Key Takeaways

• Heat transfer requires a temperature difference.
• Chapter 1 introduces conduction, convection, and radiation.
• $\dot{q}$ is W; $q''$ is W/m².
• Fourier’s law handles conduction.
• Newton’s law handles convection.
• Stefan–Boltzmann law handles radiation.
• Radiation requires temperatures in K.
• Most Chapter 1 problems are energy-balance problems.
• Draw the heat path before writing equations.