Foundations — Thermal analysis — conduction in structures, thermal stress

This page builds every symbol, word, and picture the parent note Thermal Analysis leans on — starting from a smart 12-year-old's level, with nothing assumed. Read it top to bottom: each idea is used only after it is earned.

1. Temperature $T$ — "how hot, as a number"

Before anything else we need a way to say how hot a spot is.

Why the topic needs it: a spacecraft in orbit sees +120°C in Sun and −150°C in shadow (see Thermal Environment in Orbit). Every later quantity is a reaction to $T$ not being the same everywhere.

2. Position $x$ and length $L$ — "where along the bar"

To say how the temperature changes across the bar, we need an address for each point.

Look at the figure: the bar is a horizontal strip. The blue arrow is the $x$-axis running along it. $x=0$ is the hot (orange) end, $x=L$ is the cold (blue) end. Later, "the temperature at position $x$" is written $T(x)$ — the colour you'd read off if you stood at address $x$.

Why the topic needs it: heat flow is a left-to-right story. Without $x$ we could not say "hotter here, cooler there."

3. The temperature gradient $\frac{dT}{dx}$ and $\nabla T$ — "steepness of the colour change"

We now have $T$ at every $x$. The next question is: how fast does $T$ change as you walk along the bar? That rate is the star of conduction.

In the figure the temperature curve is drawn. The orange line is the slope $dT/dx$ at one point — literally how tilted the curve is there. Steeper tilt = larger $|dT/dx|$.

Why the topic needs it: Fourier's Law (next) says heat flow is driven by this steepness. No gradient, no heat flow.

4. Heat flux $\vec{q}$ and Fourier's Law — "how much heat crosses, and which way"

Let us earn every piece:

• $\nabla T$ points toward hotter. But heat flows toward colder. The minus sign flips the arrow so $\vec{q}$ points down the hill, hot → cold. That minus is physics, not decoration.
• $k$ (next section) sets how much heat flows for a given steepness.

Red arrow = $\vec{q}$, always pointing from the orange (hot) end toward the blue (cold) end, opposite to the temperature-increasing direction.

5. Thermal conductivity $k$ — "how easily this material passes heat"

Picture: for the same temperature steepness, a high-$k$ bar carries a fat stream of heat, a low-$k$ bar a trickle. See Material Selection for Spacecraft and Thermal Control Subsystems for how engineers pick $k$ deliberately.

6. Constant $A$ — "cross-sectional area"

Why the topic needs it: energy conservation in the derivation compares total heat in one face against total heat out the next — that comparison needs area.

7. Steady state and $\frac{d^{2}T}{d{x}^2}$ — "settled down, no more change"

8. Strain $\epsilon$ — "fractional stretch"

Now we cross from heat into mechanics.

The green bar is the original length $L$; the outline shows it grown by $\Delta L$. Strain is the fraction of that growth.

• $\Delta L$ = change in length (metres). $\Delta$ ("delta") always means "the change in."

9. Coefficient of thermal expansion $\alpha$ — "growth per degree"

Why the topic needs it: $\alpha$ is the bridge from the temperature field to the wants-to-stretch amount. Low-$\alpha$ materials like the carbon composites of Composite Materials in Spacecraft barely move — a design lever.

10. Stress $\sigma$ and Young's modulus $E$ — "internal push, and stiffness"

11. Constraint — "can it move or not?"

Prerequisite map

The left branch (heat) produces the temperature field; the right branch (mechanics) turns temperature into stress. The topic is the meeting of these two rivers.

Equipment checklist

Self-test: cover the right side and answer before revealing.