Chapter 2: Introduction to Conduction

Source files used: late Heat Transfer (03), Heat Transfer (04), Heat Transfer (05); textbook Chapter 2 as support. No worked examples are included.

1. Big Picture

Chapter 1 introduced conduction with Fourier’s law. Chapter 2 answers the next question:

$$\text{How do we find the temperature distribution inside a solid?}$$

This is Biddle’s repeated conduction logic:

$$\boxed{\text{First find }T(x,y,z,t),\text{ then use Fourier’s law to find heat transfer.}}$$

In Chapter 1, simple wall problems already had known surface temperatures, so heat transfer could be found directly. In Chapter 2, we learn how to pose conduction problems using the heat diffusion equation plus boundary and initial conditions.

The chapter is not mainly about solving every PDE by hand. It is about knowing what equation applies and what conditions must be specified for the problem to be well posed.

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2. Core Ideas

2.1 Temperature Distribution Comes First

In conduction, heat flux depends on the temperature gradient:

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

So if we do not know how temperature varies with position, we cannot calculate heat flux. Chapter 2 gives the governing equation that determines $T$.

The general sequence is:

Physical problem → assumptions → heat diffusion equation → boundary/initial conditions → temperature distribution → Fourier’s law → heat rate or heat flux

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2.2 Thermal Conductivity $k$

Thermal conductivity measures how easily a material conducts heat.

• Large $k$: metal-like behavior, temperature drop is small for a given heat flow.
• Small $k$: insulation-like behavior, temperature drop is large for a given heat flow.

Units:

$$[k]=\text{W/(m\cdot K)}$$

Biddle emphasizes using property tables correctly. Metals, nonmetals, building materials, insulation materials, liquids, and gases appear in different appendices/tables. Some tabulated values, especially thermal diffusivity, may be written in scaled form such as $\alpha\times10^6$. Read the table heading carefully.

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2.3 Thermal Diffusivity $\alpha$

Thermal diffusivity is:

$$\alpha=\frac{k}{\rho c_p}$$

where:

• $k$: thermal conductivity, W/(m·K),
• $\rho$: density, kg/m³,
• $c_p$: specific heat, J/(kg·K),
• $\alpha$: thermal diffusivity, m²/s.

Physical meaning:

$$\text{thermal diffusivity} = \frac{\text{ability to conduct heat}}{\text{ability to store heat}}$$

Large $\alpha$ means the material responds quickly to thermal changes. Small $\alpha$ means the material responds slowly because it stores energy strongly or conducts poorly.

Do not confuse thermal diffusivity $\alpha$ with absorptivity $\alpha$ from radiation. Same Greek letter, different context.

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2.4 Heat Generation

Some solids generate heat internally. Examples:

• electrical resistance heating,
• nuclear fuel,
• chemical reaction,
• volumetric energy generation in a material.

The volumetric generation term is usually written:

$$\dot{q}'''$$

with units:

$$\text{W/m}^3$$

Generation changes the temperature distribution. With no generation and constant $k$, a 1D steady plane wall has a linear temperature profile. With generation, the profile becomes curved.

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2.5 Boundary and Initial Conditions

A conduction differential equation is not enough by itself. Biddle emphasizes that a problem is properly posed only when the governing equation and the correct boundary/initial conditions are specified.

For a second-order spatial equation, you need two boundary conditions per spatial direction. If time appears, you also need an initial condition.

Typical boundary conditions:

1. specified surface temperature,
2. specified heat flux,
3. insulated or symmetry boundary,
4. convection boundary,
5. radiation boundary,
6. interface between two materials.

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3. Main Governing Equations and Formulas

3.1 Thermal Diffusivity

$$\alpha=\frac{k}{\rho c_p}$$

Use when transient conduction appears or when comparing how quickly materials respond thermally.

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3.2 General Heat Diffusion Equation: Cartesian Coordinates

For constant properties:

$$\frac{\partial^2 T}{\partial x^2}+\frac{\partial^2 T}{\partial y^2}+\frac{\partial^2 T}{\partial z^2}+\frac{\dot{q}'''}{k}=\frac{1}{\alpha}\frac{\partial T}{\partial t}$$

where:

• $T$: temperature,
• $x,y,z$: spatial coordinates,
• $\dot{q}'''$: volumetric heat generation, W/m³,
• $k$: thermal conductivity,
• $\alpha$: thermal diffusivity,
• $t$: time.

Use this for transient, multidimensional conduction with constant properties.

Special cases:

Steady state:

$$\frac{\partial T}{\partial t}=0$$

No generation:

$$\dot{q}'''=0$$

One-dimensional, steady, no generation:

$$\frac{d^2T}{dx^2}=0$$

One-dimensional, steady, with generation:

$$\frac{d^2T}{dx^2}+\frac{\dot{q}'''}{k}=0$$

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3.3 Cylindrical Heat Diffusion Equation

For constant properties:

$$\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial T}{\partial r}\right) +\frac{1}{r^2}\frac{\partial^2 T}{\partial \phi^2} +\frac{\partial^2 T}{\partial z^2} +\frac{\dot{q}'''}{k} =\frac{1}{\alpha}\frac{\partial T}{\partial t}$$

Common radial-only steady form:

$$\frac{1}{r}\frac{d}{dr}\left(r\frac{dT}{dr}\right)+\frac{\dot{q}'''}{k}=0$$

If no generation:

$$\frac{1}{r}\frac{d}{dr}\left(r\frac{dT}{dr}\right)=0$$

Use cylindrical coordinates for pipes, tubes, rods, wires, and cylinders.

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3.4 Spherical Radial Heat Diffusion Equation

For radial conduction in a sphere with constant properties:

$$\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dT}{dr}\right)+\frac{\dot{q}'''}{k}=\frac{1}{\alpha}\frac{\partial T}{\partial t}$$

Steady, no generation:

$$\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dT}{dr}\right)=0$$

Use for spheres and spherical shells.

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3.5 Fourier’s Law in Vector Form

$$\vec q''=-k\nabla T$$

In Cartesian components:

$$q_x''=-k\frac{\partial T}{\partial x},\qquad q_y''=-k\frac{\partial T}{\partial y},\qquad q_z''=-k\frac{\partial T}{\partial z}$$

Use after $T$ is known.

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3.6 Boundary Condition: Specified Temperature

$$T_s=T_0$$

Use when the surface is maintained at a known temperature.

Example language:

• “surface is kept at 100°C,”
• “wall is maintained at $T_1$,”
• “constant surface temperature.”

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3.7 Boundary Condition: Specified Heat Flux

$$-k\frac{\partial T}{\partial n}=q_s''$$

where $n$ is the outward normal direction.

Use when a known heat flux is imposed at the surface, such as a heater attached to a wall.

For adiabatic or insulated surfaces:

$$q_s''=0$$

so

$$\frac{\partial T}{\partial n}=0$$

This same mathematical condition also appears at a symmetry plane.

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3.8 Boundary Condition: Convection

$$-k\frac{\partial T}{\partial n}=h(T_s-T_\infty)$$

Use when conduction inside the solid reaches a surface and then heat leaves by convection to a fluid.

This is one of the most important boundary conditions in the course. It connects conduction in the solid to convection at the surface.

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3.9 Boundary Condition: Radiation

$$-k\frac{\partial T}{\partial n}=\varepsilon\sigma\left(T_s^4-T_{\text{sur}}^4\right)$$

Use when a surface loses or gains heat by radiation to surroundings.

Temperatures must be in K.

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3.10 Interface Conditions

For perfect contact between materials A and B:

Temperature continuity:

$$T_A=T_B$$

Heat flux continuity:

$$k_A\frac{\partial T_A}{\partial n}=k_B\frac{\partial T_B}{\partial n}$$

Use at material interfaces.

If thermal contact resistance is specified, use the Chapter 3 contact resistance model instead of perfect temperature continuity.

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4. Problem-Solving Workflow

1. Identify the geometry. Plane wall, cylinder, sphere, or multidimensional shape.
2. Choose coordinates. Cartesian for plane walls/rectangles; cylindrical for pipes/tubes; spherical for spheres.
3. List assumptions. Steady/transient, 1D/2D/3D, generation/no generation, constant properties.
4. Simplify the heat diffusion equation. Remove terms that do not apply.
5. Write boundary conditions. Use physical information from the problem statement.
6. Write initial condition if transient. Usually $T(x,y,z,0)=T_i$.
7. Solve or recognize the required method. Analytical solution, resistance method, transient chart, or numerical method.
8. Use Fourier’s law after temperature is known. Do not skip directly to heat rate unless a resistance/rate formula applies.

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5. Decision Rules / Decision Trees

5.1 Which Governing Equation?

Plane wall / rectangular coordinates? → Cartesian heat equation

Pipe, tube, rod, cylinder? → Cylindrical heat equation

Sphere or spherical shell? → Spherical heat equation

5.2 Which Terms Stay?

Steady state? → remove time derivative

No generation? → remove $\dot{q}'''/k$ term

1D in x only? → keep only $d^2T/dx^2$

Radial cylinder only? → keep only $(1/r)\,d/dr(r\,dT/dr)$

5.3 Boundary Condition Selection

Surface temperature given? → $T=\text{specified value}$

Surface heat flux given? → $-k\,dT/dn=q_s''$

Insulated surface? → $dT/dn=0$

Symmetry line/plane? → $dT/dn=0$

Surface exposed to fluid with $h$ and $T_\infty$? → $-k\,dT/dn=h(T_s-T_\infty)$

Surface exchanges radiation? → $-k\,dT/dn=\varepsilon\sigma(T_s^4-T_{\text{sur}}^4)$

5.4 When Is a Problem Properly Posed?

Spatial second-order equation? → need two boundary conditions per spatial coordinate

Transient problem? → need initial condition

Boundary conditions missing? → cannot solve uniquely

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6. Important Tables / Correlations Needed

6.1 Property Table Use

Biddle’s table-use logic:

Material/Property Need	Where to Look Conceptually
Metals	metallic property tables
Nonmetallic solids	nonmetallic property tables
Building materials and insulation	structural/insulation tables
Liquids	liquid property tables
Gases	gas property tables

Key warning: when a table heading says $\alpha \times 10^6$, the listed number is not $\alpha$. It means:

$$\alpha=(\text{listed number})\times10^{-6}\ \text{m}^2/\text{s}$$

6.2 Common Material Behavior

Material Type	Typical Behavior
Metals	high $k$, conduct heat well
Insulation	low $k$, large temperature drop for small heat rate
Liquids	moderate thermal properties; convection often important
Gases	low $k$; convection coefficients often small

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7. Key Takeaways

• In conduction, find $T$ first, then use Fourier’s law.
• Thermal conductivity $k$ measures conduction ability.
• Thermal diffusivity $\alpha=k/(\rho c_p)$ measures how fast temperature responds.
• The heat diffusion equation is simplified based on assumptions.
• Boundary conditions are physical statements translated into math.
• A convection boundary condition connects conduction inside the solid to convection outside.
• An insulated boundary and a symmetry boundary both give zero normal temperature gradient.
• For radiation boundary conditions, use K.
• Read property tables carefully, especially scaled columns.