IEEE Circuits and Systems Magazine - Q1 2023 - 45

electro-mechanical impelling, or multiple interacting
fluid circuits. Here, as the example should have proven,
there is simply no " writing a general code and then specializing
it with configuration files " ; the D2I translation
can be even more costly than in Example 1, owing to
nonlinearities, and is needed for every case.
The so outlined scenario explains why EBM-that conversely
can manage heterogeneous model complexity by
suitably abstracting modular declarative components-
needs bringing into play for modeling modern IC cooling.
Of course, doing so requires a bit more consciousness
of modeling principles than just compiling configuration
files, but first this is inevitable given the variety of cases
to address, and then as the burden of manual D2I translation
is avoided-it can be made accessible to many, as we
are going to discuss in the following.
VI. MPSoC/IC Cooling-Modeling Principles
In this section we define and discuss the principles on
which a modeling solution for the cooling of modern
ICs-like MPSoCs-must be grounded, and how to apply
those principles to the addressed domain. The reader
interested in more details about the used modeling
methodologies can refer, e.g., to [17], [18], [19] or many
analogous works. In the following we just report a few
references relative to the considered phenomena and
their dynamic description.
A. Substances
Any first-principle equation has to do with properties of
materials. In MPSoC cooling these include the following.
■ Solids, whose thermodynamic state in the conditions
of interest for us is fully described by their
temperature T, and that are physically characterized
by a density ρ, a specific heat c and a thermal
conductivity λ that one can assume constant in
the said conditions [20].
■ Single-phase (subcooled) liquids, whose state is
described by their pressure p and specific enthalpy
h, and whose physical properties (density, specific
heat, thermal conductivity) can be assumed
constant in any case of interest [21].
■ Gases that behave (almost) ideally, i.e., whose
state is described by their pressure p and temperature
T (or equivalently, specific enthalpy h) and
whose behavior is ruled by
p
== =+
* ,, ,
RT ecvTh e
p
ρρ
where ρ is density, e is internal energy, and R∗, cv are respectively
the considered gas constant and its constantvolume
specific heat [21].
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■ Phase-transitioning species such as refrigerating
fluids, not treated in this tutorial owing to
complexity and space limits. For these the same
modeling considerations given above apply,
however, provided relationships are available
to compute their properties based on pressure
p and specific enthalpy h. Notable examples are
the water-steam tables, included in the MSL, or
the ExternalMedia package [22] to interface
Modelica with external fluid property calculation
software like, e.g., REFPROP [23] and CoolProp
[24].
■ Fluid mixtures, most notably moist air, not treated
herein either. These are handled in the same way
above once analogous relationships are available
(such as Mollier-like ones, to compute moist air
properties based on pressure p, temperature T
and absolute humidity X).
We now proceed to write the most important equations
for the phenomena of interest: the hypotheses
just made apply throughout, hence are not repeated
for brevity.
B. Mass Storage
Dynamic mass balances can refer to closed volumes
filled with fluid (e.g., a pipe), to closed ones not completely
filled (e.g., a gas pressuriser for a liquid circuit),
or to open ones (e.g., a vented tank). Here we only treat
the completely filled case with liquid and gas, and the
vented case with liquid [25].
In the filled liquid case the equation is trivial, as the contained
mass is constant. Denoting by wi the generic mass
flow rate through the i-th out of np volume port (e.g., the
flanges of a pipe or of a header) the equation is therefore
∑ =
1
np
=
wi
i
while pressure is determined by some of the other components
of which the complete model is made.
In the filled gas case, denoting by V the volume, the
contained mass is ρV, hence (recall that the dot means
derivative with time) the equation reads
np
Vwi
i
that recalling (12) becomes
V
R
d
* dt



p
T


np
 =
∑
i=1
In (15) T can be considered constant or come from
some thermal equation, see Section VI-E.
IEEE CIRCUITS AND SYSTEMS MAGAZINE
45
wi .
(15)
ρ =∑ ,
=1
(14)
0.
(13)
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