I am looking for a computational model of for multilayered thermal
In cryogenics, this is a simple radiative calculation as there will be
no thermal conduction (vacuum) nor convection, only radiative exchange.
Things seem to get complicated when you add a bit of conduction as well.
Has anybody modelized such a system?
I would be glad for any pointers...
Newton's Law of cooling: [Only registered users see links. ]
Allow that K will include terms for each layer, and that K acts like
conductivity (the reciprocal of resistivity).
For a series of barriers that resist heat transfer, use
1/K(tot) = 1/K(1) + 1/K(2) + 1/K(3) + ...
Each K needs to be determined empirically.
Good superinsulation alternates a set of diaphanous poor conductors
(tissue paper like layers) with thin (low Cp) reflective layers to
resist radiation, all in a convention-proof vacuum chamber. Even
radiative enchange is poor. We used to keep a 100L helium Dewar for a
week before it went empty from filling flasks several times a day.
Untapped our helium Dewar would keep for a month.
Thank you for your answer. Actually, I am slightly further in my
modeling. The reference you give is relevant to conduction only. I need
to model with simultaneous radiation exchange.
Yes, I know. I worked at Hel temperatures quite some years. I had
occasion to redo the superinsulation on several cryostats. I found that
there is the kind you describe (with alternating insulating and
reflecting spacers) but also, much easyer to handle, "wafered" aluminum
foil. The structuring reduces thermal contact to very small values.
I should have written "almost no thermal conduction".
In any case, even if I can model a single layer, the case of combined
radiative and conductive heat transfer in a multilayered system seems to
be a big programming effort.
I just wanted to know if there is already such a program or if I have to
reinvent the wheeel?
I think you missed my point. If K is modeled empirically for each type
of layer, then it does not matter how the heat is transferred. There
is no sensitivity in Newtons equation to the method of heat flow.
K(tot) can simply be determined up front for your actual configuration,
and then used as if there were only one very efficient layer to your
I haven't seen such off-the-shelf software, but that is a little
outside my current work (I left the cryo lab in '97).
"tadchem" <[Only registered users see links. ]> wrote in message
I think that a bulk model would be fine as long as the
time periods of interest are sufficient for essentially
steady state to obtain. For short time periods, radiative
transfer will precede conduction and convective effects.
It's much like the difference between transient response
and steady state response in electrical circuit analysis.
I am not sure I can follow you on this.
K is a value derived for conductive (or convective) heat exchange and
Newton's law (or Fourier's for purely conductive exchange) does not at
all consider radiative exchange.
See : [Only registered users see links. ]
K describes the energy transfer proportinal to Delta T. If you try to
include radiative transfer in K, then K would be a function of Delta T^3
(I think, have to verify this).
But your idea could be extended in a finite difference calculation by
including conduction and radiation as input/output of a cell. A complex
system could then be modelled by stepwise solution.
I shall try to go like this.
Notwithstanding the technical engineering detail in the article,
Newton's Law of cooling [Only registered users see links. ]
is a *empirical* equation.
dT/dt = -K*(T - T_s)
and since K is an empirical constant, it is *operationally* defined as
K = -dT/dt * 1/(T - T_s)
Converting the algebra into descriptive english, K is the "fractional
decrease in the temperature difference per unit time."
As such there is no distinction between a temperature change due to
conduction and a temperature change due to convection or radiation.
All that *counts* is how much the temperature changes, within a unit
Now you may feel there is some theoretical justification for believing
that when radiative heat transfer is concerned K_rad varies inversely
with (T - T_s) cubed
in which for small temperature differences (T/T_s << 1) is
approximately the constant -1/T_s^2
Given the empirical nature of the equation, even if there is a
justification for a cubic relationship, the linear relationship
provides an adequate description of the behaviour.
K_rad ~= dT/dt * (1/T_s)^2 * 1/(T - T_s)
with a slightly different value for K.
Given that your large temperatrure difference will be sustained by
*multiple* layers of insulation, each of which in practice will
maintain only a small temperature difference on its own, I think this
would be an adequate description of the effects of a single layer.
The combined effects can be modelled as I described before, as thermal
resistors acting in series to maintain a higher difference with a lower
thermal current, analogously to DC electricity.