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# 熱傳導

## 傅立葉定律

### 微分形式

${\displaystyle {\overrightarrow {q))=-k{\nabla }T}$

${\displaystyle {\overrightarrow {q))}$ 是熱通量密度，單位W·m−2
${\displaystyle {\big .}k{\big .))$ 是這種材料的熱導率，單位W·m−1·K−1
${\displaystyle {\big .}\nabla T{\big .))$ 是溫度梯度，單位K·m−1

${\displaystyle q_{x}=-k{\frac {dT}{dx))}$

### 積分形式

${\displaystyle P={\frac {\partial Q}{\partial t))=-k\oint _{S}((\overrightarrow {\nabla ))T\cdot \,{\overrightarrow {dA))))$

• ${\displaystyle {\big .}P={\frac {\partial Q}{\partial t)){\big .))$ 是熱傳導功率，即單位時間通過面積S的熱量，單位W，而
• ${\displaystyle {\overrightarrow {dA))}$ 是面元向量，單位m2

${\displaystyle {\big .}P={\frac {\Delta Q}{\Delta t))=-kA{\frac {\Delta T}{\Delta x))}$

A 是介質的截面積，
${\displaystyle \Delta T}$ 是兩端溫差，
${\displaystyle \Delta x}$ 是兩端距離。

## 熱導

${\displaystyle {\big .}U={\frac {kA}{\Delta x)),\quad }$

${\displaystyle {\big .}P={\frac {\Delta Q}{\Delta t))=U\,(-\Delta T).}$

${\displaystyle {\big .}R={\frac {1}{U))={\frac {\Delta x}{kA))={\frac {-\Delta T}{P)).}$

${\displaystyle {\big .}{\frac {1}{U))={\frac {1}{U_{1))}+{\frac {1}{U_{2))}+{\frac {1}{U_{3))}+\cdots }$

${\displaystyle {\big .}P={\frac {\Delta Q}{\Delta t))={\frac {A\,(-\Delta T)}((\frac {\Delta x_{1)){k_{1))}+{\frac {\Delta x_{2)){k_{2))}+{\frac {\Delta x_{3)){k_{3))}+\cdots )).}$