Introduction

Daniel FitzGreen

Thermal Conductivity of Gases

Introduction

The thermal conductivity of a material is a measurement of how well that material conducts heat. Thermal conductivity is measured in units of Watts per meter-Kelvin (W/m·K), and typically depends on the temperature at which it’s measured. The rate of heat transfer $(\dot Q)$ is given by

(1) $\begin{equation*} \dot Q =-\lambda A \cfrac{dT}{dx}, \end{equation*}$

where $\lambda$ is the thermal conductivity of the material, $A$ is the cross-sectional area where heat transfer can take place, and $dT/dx$ is the rate of temperature change in the $x$ direction.

Heat flow in cylinders is a particularly common situation since cooling and heating fluids flow through pipes. One can consider the one-dimensional problem of two concentric circles. From the equation above, the heat transfer rate per unit length is

(2) $\begin{equation*} \cfrac{\dot Q}{2\pi r}dr = -\lambda dT.\end{equation*}$

Note that the equation has been rearranged with spatial terms on the left and temperature terms on the right. One could easily derive the equation relating input power to the temperature difference between the cylinders:

(3) $\begin{equation*} P = \cfrac{2\pi \lambda L\Delta T}{\ln(b/a)},\end{equation*}$

where $P$ is the total input power, $\lambda$ is the thermal conductivity, $L$ is the length of the cylinders, and $a$ and $b$ are the inner and outer cylinder radii.

Introduction

License

Share This Book