Thermal Circuits and Conduction

Thermal Conductivity

Thermal conductivity k has various meaning depending on the phase of a material
For solids:
$k=k_L+k_e$
For liquids:

• Thermal conductivity is based on diffusion and collision

For gasses:
$k\alpha n\overline{c}\lambda$
Where n is the number density, c is the mean thermal speed, and lambda is the mean free path

*If we take a look at the transport equations they simplify to the conduction (diffusion) term only!…

$k{\nabla }^{2}T$

Conservation

Applying conservation of mass, momentum, energy over a control volume of differential size can yield useful equations

*Assumptions we make

• One direction only
• No energy generated or consumed
• Steady state

Some Applied examples

We make the assumption from the 1D Laplacian that
$\frac{d^{2}T}{d{x}^2}=0$
Which means
$\frac{dT}{dx}=const$
and…
$T\left(x\right)=c_{1}x+c_2$
Where you can apply boundary conditions to solve for T.

This analysis yields local values, can use different coundaries conditions other than T = fixed like change in T = fixed

Thermal Circuit Boundary Conditions

There’s a |KCL equivalent in thermodynamics where…
$Q_{cond}^{\prime }=Q_{conv}^{\prime }+Q_{rad}^{\prime }$
Also, you can have parallel thermal resistances where can be combined in parallel like in circuits.

Iteration

If $T_s$ is unknown, the problem needs to be solved by iteration, this needs to be done on the assignment

Note that:

• Any plane wall normal to the x-axis is isothermal (varies in the x-direction only)
• Any plane parallel to the x-axis is adiabatic (assume heat transfer only occurs in the x-direction)

Thermal Contact Resistance

• In the ideal case, theres no temperature drop across different layers but in reality there is which is the thermal contact resistance $R_c=\frac{1}{h_c}={\frac{\Delta Tinterface}{\frac{Q^{\prime }}{A}}}_{}$
  This depends on various factors:
• Roughness
• Material properties
• Fluids at the interface, and other extensive properties at the interface