4.1 Thermal diffusivity and further simulations
Thermal diffusivity of any material is a measure of how fast the
material temperature adapts to the surrounding temperature. In other
words, it is the material’s ability to transfer heat energy relative to
the ability to store it. This explains how fast heat is propagated (in
the direction of heat flow) into the heat shield as a function of
temperature and time. Thermal diffusivity is the inverse of thermal
inertia. A heat shield with high thermal diffusivity causes heat to move
rapidly through it because the heat shield conducts heat quickly in
relation to its thermal bulk (volumetric heat capacity). Applying the
differential form of Fourier law [17, 18], the local heat flux
density is proportional to the negative temperature gradient \((\nabla T)\).
Where T is the scalar temperature in the x, y, and z directions T(x,y,z)
\(\nabla\) is is the 3D “del” operator; and k is the thermal conductivity of the material. Applying the law of conservation of energy and eliminating convective and radiative effects, then the electrical heat gained through conduction is equal to \(_{cp}\rho\nabla T\). Where cp is the specific heat capacity at constant pressure, ρ is the material’s mass density, and \(\Delta T\) is temperature rise. Hence,
\(\Delta Q=_{cp}\rho\Delta T\) Eq. 4.2
Equating Eq.4.1 to 4.2 describes the correlation between local
temperature change and spacial variation of temperature at a certain
location within the heat shield given as:
\(\frac{_{cp}\rho\nabla T}{dt}=\frac{k\nabla T}{dt}\rightarrow\ _{cp}\rho\frac{dT}{dt}=k\nabla^2T\)
\(\therefore\frac{dT}{dt}=\left(\frac{k}{_{cp}\rho}\right)\nabla^2T\)
Where \(\left(\frac{k}{_{cp}\rho}\right)\) is the thermal diffusivity
δ of the material. Transient thermal simulation has been chosen because
it helps determine the variation of temperature as a function of time
and position T(x,t) in the heat shield and electrical connections. This
makes it possible (at any point in time) to predict the evolution of
temperature field thus, obtaining a reliable temperature field T(x,t)
for the computation of heat flows by mathematical derivations. The
result of increasing the current in Fig.3.2 to 400A and repeating the
Ansys simulation (‘Steady-State Electric Conduction’ coupled with
‘Transient Thermal’) in duration of 30 seconds for tungsten diameter
2mm, 4mm, 6mm, 8mm, and 10mm is shown in Fig.4.1.