2.1 One-Dimensional Steady-State Conduction

Heat Flux and Temperature Distribution

Fourier's Law and One-Dimensional Heat Flux

Fourier's Law is the starting point for almost every conduction problem you'll solve. It says that heat flows from hot to cold, and the rate of that flow depends on how steep the temperature gradient is and how well the material conducts heat.

The one-dimensional form is:

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

where $q''$ is the heat flux (W/m²), $k$ is the thermal conductivity (W/m·K), and $\frac{dT}{dx}$ is the temperature gradient in the direction of heat flow.

The negative sign matters: heat flows in the direction of decreasing temperature, so the negative sign makes $q''$ positive when heat moves from hot to cold.

To get the total heat transfer rate through a surface, multiply the flux by the cross-sectional area perpendicular to the heat flow:

$Q = -kA \frac{dT}{dx}$

You'll apply Fourier's Law across plane walls, cylindrical shells, and spherical shells. The geometry changes how you set up the math, but the underlying principle is always the same.

Steady-State Conduction and Boundary Conditions

Steady-state means nothing changes with time. The temperature at every point in the material stays constant, so all the heat entering a region also leaves it (no energy storage).

For a plane wall with constant $k$ and no internal heat generation, the temperature profile is a straight line between the two surface temperatures. In cylindrical and spherical coordinates, the profile is logarithmic or hyperbolic, respectively, because the area through which heat flows changes with radius.

To actually solve for the temperature distribution, you need boundary conditions at each surface. The most common types are:

• Constant temperature (Dirichlet): The surface is held at a known temperature, e.g., $T(0) = T_s$
• Constant heat flux (Neumann): A known heat flux is applied at the surface, e.g., $-k \frac{dT}{dx}\bigg|_{x=0} = q''_s$
• Convection (Robin): The surface exchanges heat with a fluid, e.g., $-k \frac{dT}{dx}\bigg|_{x=0} = h[T_\infty - T(0)]$
• Insulated surface: A special case of constant heat flux where $q'' = 0$, meaning $\frac{dT}{dx} = 0$ at that boundary

You pick two boundary conditions (one for each side of a wall, for instance), plug them into the general solution of the heat diffusion equation, and solve for the unknown constants.

Thermal Conductivity of Materials

Definition and Dependence on Material Properties

Thermal conductivity $k$ quantifies how easily a material conducts heat. A high $k$ means heat passes through readily; a low $k$ means the material resists heat flow.

Some representative values to keep in mind:

• Metals conduct heat very well. Copper has $k \approx 401$ W/m·K, and aluminum is around $k \approx 237$ W/m·K.
• Insulators have very low thermal conductivity. Fiberglass insulation is roughly $k \approx 0.04$ W/m·K, and polyurethane foam is similar.
• Gases are poor conductors. Still air has $k \approx 0.026$ W/m·K, which is why trapped air pockets make good insulation.

The value of $k$ depends on the material's composition, internal structure (crystalline vs. amorphous), and temperature. Choosing the right material for a given application often comes down to whether you want to promote heat transfer (heat sinks, exchangers) or resist it (insulation, thermal barriers).

Temperature Dependence

For many problems you'll treat $k$ as constant, but in reality it varies with temperature. The trend depends on the type of material:

• Metals: $k$ generally decreases slightly with increasing temperature (due to increased lattice vibrations scattering electrons).
• Nonmetallic solids and liquids: behavior varies; some increase, some decrease.
• Gases: $k$ increases with temperature (faster-moving molecules transport energy more effectively).

When $k$ varies significantly over the temperature range in your problem, you can use an average value or integrate Fourier's Law directly with $k(T)$. Thermal conductivity data for specific materials is found in property tables in your textbook or engineering handbooks.

Conduction with Heat Generation

Heat Generation and Its Effects

Some systems generate heat internally. Common sources include:

• Electrical resistance heating (Joule heating): Current flowing through a wire or resistive element produces heat at a rate $\dot{q} = I^2 R / V$, where $V$ is volume.
• Nuclear reactions: Fuel rods in a reactor generate heat volumetrically.
• Exothermic chemical reactions: Heat released within a reacting material.

When heat generation is present, the governing equation for 1-D steady-state conduction (constant $k$) in Cartesian coordinates becomes:

$\frac{d^2T}{dx^2} + \frac{\dot{q}}{k} = 0$

where $\dot{q}$ is the volumetric heat generation rate (W/m³). The $\dot{q}$ term is what makes the temperature profile curved (parabolic for a plane wall with uniform generation) rather than the straight line you get without generation.

Solving Conduction Problems with Heat Generation

For a plane wall of thickness $2L$ with uniform heat generation $\dot{q}$ and both surfaces held at temperature $T_s$, the solution is:

$T(x) = \frac{\dot{q}}{2k}(L^2 - x^2) + T_s$

The maximum temperature occurs at the center ($x = 0$) because heat must flow outward to both surfaces.

Steps for solving these problems:

1. Write the appropriate form of the heat diffusion equation for your geometry (Cartesian, cylindrical, or spherical).
2. Integrate twice to get the general solution with two unknown constants.
3. Apply your two boundary conditions to solve for those constants.
4. Plug the constants back in to get the full temperature distribution $T(x)$.
5. Use Fourier's Law to find the heat flux at any location if needed.

If $k$ varies with temperature, the equation becomes nonlinear and you may need numerical methods (finite difference, finite element) to solve it. For simple cases, though, analytical solutions are available and are what you'll use most in this course.

Multi-Layer Conduction with Resistance

Multilayer Systems and Thermal Resistance

Real walls and pipes are rarely a single material. A building wall might have drywall, insulation, plywood, and siding. Analyzing these composite systems is where the thermal resistance concept becomes extremely useful.

The idea is analogous to electrical resistance. Just as $V = IR$ in circuits, heat transfer follows:

$Q = \frac{\Delta T}{R_{total}}$

where $\Delta T$ is the overall temperature difference driving heat flow and $R_{total}$ is the sum of all resistances in the path.

The thermal resistance for each type of element:

For layers in series (heat passes through one after another), you add the resistances:

$R_{total} = R_1 + R_2 + R_3 + \cdots$

This lets you find the total heat transfer rate quickly without solving the temperature profile in every layer. Once you have $Q$, you can work backward through each resistance to find the temperature at any interface.

Thermal Contact Resistance

When two solid surfaces are pressed together, they don't make perfect contact. Microscopic air gaps and surface roughness create an additional resistance at the interface called thermal contact resistance, $R_{t,c}$.

This resistance is modeled as another term in your series resistance network:

$R_{total} = R_{layer\,1} + R_{t,c} + R_{layer\,2} + \cdots$

Contact resistance depends on:

• Surface roughness: Rougher surfaces trap more air and have higher $R_{t,c}$
• Contact pressure: Higher pressure deforms surface peaks and improves contact
• Interstitial materials: Filling gaps with a conductive substance lowers $R_{t,c}$

A practical example: the junction between a microprocessor and its heat sink. Without thermal interface material (thermal paste or a thermal pad), the contact resistance can be large enough to cause the chip to overheat. Applying thermal paste fills the microscopic air gaps and can reduce $R_{t,c}$ dramatically.

In many textbook problems, contact resistance is given as $R''_{t,c}$ in units of m²·K/W. To get the actual resistance for your geometry, divide by the contact area: $R_{t,c} = R''_{t,c} / A$.