[aspect-devel] questions about heat flux

[Wolfgang Bangerth]

Max,

> In geophysics, it seems to be common to refer to q=-k grad(T) as the
> heat flux. See for instance chapter 4 of Turcotte and Schubert "... the
> heat flux q, or the flow of heat per unit area and per unit time, at a
> point in a medium is directly proportional to the temperature gradient
> at the point." Similarly, the Darcy flux in groundwater flow is flow per
> unit area (m^3/s)/m^2 = m/s. Certainly this is at odds with the
> nomenclature used in electromagnetism, but nevertheless I think that
> this terminology is in widespread use in our field.

Fair enough. How about this to clarify what we do:
   https://github.com/geodynamics/aspect/pull/1238


> You cannot measure the heat flow across the boundary of a 2D box in W.
> The heat flux across the boundary is q=-k dT/dz. This has units
> (W/m/K)*(K/m)=W/m^2. You calculate the boundary heat flow by integrating
> this across the (1D) boundary, so the result has dimensions of
> Power/Length or W/m, where the 'm' in the denominator indicates 'per
> unit thickness perpendicular to the 2D domain'. Perhaps ASPECT assumes
> out-of-plane thickness of 1 m when reporting 'heat flux'?

Not quite correct. The units of q are "flow of Joules per unit cross 
section per second per unit thermal gradient". In 2d, unit cross section 
has units m, so the whole thing is J/m/s/(K/m)=J/(s K). In 3d, unit 
cross section has units m^2, so q has units J/(s m K). Integration over 
the boundary of -q dT/dz then again yields a heat flux in J/s=W.

Of course, you can argue that a material that is only 2-dimensional can 
not actually hold any heat because it has mass zero. On the other hand, 
I would argue that the density then also needs to be given in kg/m^2.

Best
  Wolfgang


-- 
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Wolfgang Bangerth          email:                 bangerth at colostate.edu
                            www: http://www.math.colostate.edu/~bangerth/