[aspect-devel] questions about heat flux

[Max Rudolph]

Wolfgang,

On Fri, Sep 30, 2016 at 1:56 PM, Wolfgang Bangerth <bangerth at tamu.edu>
wrote:

>
> 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


This seems better, but it is inconsistent still to report it in units of W
in a 2D calculation if indeed you calculate this quantity by integrating
\int -k dT/dz dx (along z=0, for instance). The units must be W/m, unless
you redefine the units of k to be W/K instead of W/m/K. In the ASPECT
manual on page 23, it says that thermal conductivity is in units of W/m/K
with no caveat about 2D vs 3D. On page 322, the formula for integrating
heat flux density is given, and it is apparent that this integral would
yield a quantity with dimensions W/m in 2D unless the dimensions of the
unit normal to the boundary are taken to be area instead of distance. There
are many places in the ASPECT manual where these terms continue to be used
interchangeably or inconsistently. On page 13, a quantity with units W/m^2
is referred to as a heat flux. The heat flux is also defined mathematically
on page 161 using Fourier's law. Elsewhere in the manual, e.g. page 322,
the same formula is used and this quantity is referred to as 'flux
density'. It is easy to see how one might become confused!

I became somewhat interested in this 'flux' definition issue and I did take
a quick look at Fourier (1822). To me it appears even there to be a rate of
heat transport per unit area, though my French is not super good. I
completely agree that it's bad terminology in the sense that it conflicts
with the definition of flux used in other branches of physics, but it does
seem that casually at least, flux is often used to mean rate of transport
per unit area. See for instance 'solute flux', 'heat flux', and 'Darcy
flux'. Some authors when using Fick's law for chemical diffusion do write
'flux per unit area' when defining q=-k grad(C).


> 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".


OK, fair enough, but this still do not the definition given anywhere in the
documentation for ASPECT and it's not consistent with the formulae for q in
the manual or the literature, provided that k is defined the same way in 2D
as it is in 3D. The units of q have to be derived from [units of k] *
[units of thermal gradient], right?


> 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.
>

Is it wrong to think about 2D calculations as 3D calculations with imposed
translational symmetry in the direction perpendicular to the plane? If this
is not the case, then as you say, material properties and transport
properties need to be redefined.


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