Units for Thermal PDE thermalProperties Function

I am using the thermalProperties function to specify the density, thermal conductivity, and specific heat capacity of a handful of materials. an example is:

thermalProperties(ObjectName,'ThermalConductivity',0.385,'SpecificHeat',380,MassDensity=0.0000089);

From searching online for the units for these values, the best I can find is this:

https://www.mathworks.com/matlabcentral/answers/858085-what-units-is-thermalproperties-requesting-for-thermalmodel-of-the-pde-toolbox

which states that I can use any units I want to, as long as I stay consistent. This does not line up with what my simulation is showing however. Running the parameters using mm instead of m, for example, gives very different results, which makes sense as I never specify which I am using, I only change the values inputted into the thermalProperties function.

The answer in the linked thread is also just speculation. If anyone knows for sure, an answer would be much appreciated.

Thank you,

Simon

Answer 1

[David Goodmanson]

Hi Simon,

The heat conduction equation is

dT/dt = alpha (d^2/dx^2+d^2/dy^2+d^2/dz^2) T

where alpha, the thermal diffusivity, has to have units of length^2/time, and alpha = k/(c*rho).

Suppose at first that SI units are used. Then

k thermal conductivity ~~ W / (m K)
c heat capacity ~~ J / (kg K)
rho density ~~ kg / m^3.

and alpha ~~ m^2/sec.

If your constants involve mixed units, you still end up with length^2/time.

The numerical value of alpha is the only constant that the PDE really knows about. If alpha is in m and sec, then without any other considerations** the time and space grids have that spacing. Suppose alpha has a certain value a0 in mks units. Then

alpha = a0 m^2/sec = (10^6*a0) mm^2 / sec       (grids in mm, sec)
                   = (10^3*a0) mm^2 / msec      (grids in mm, msec)
                   =       a0  mm^2 / usec      (grids in mm, usec)

and so forth. (If your boundary conditions or source terms involve heat flux for example, those units would have to match the choice of units used in alpha).

** In the last example, scaling the space and time grid units by appropriate factors still has the same numerical value that it had with SI units.

[Simon]

I think I understand. Are you saying that the only acceptable unit conversion (for the last example) from m^2/sec is mm^2/usec, as this is the only one that does not scale the alpha value, as these are scaled by "appropriate values"?

[David Goodmanson]

Hi Simon,

No, you can use any of the three examples and similar ones. In the first two, instead of a0 you would plug in a different numerical value of alpha into the pde expession. (Although highly large or highly small values for alpha might make things difficult on the solver).