IEEE Circuits and Systems Magazine - Q1 2023 - 46

In the vented liquid case, the contained mass is represented
by a level ℓ, so that denoting by A the base area
of the volume (assumed cylindrical for simplicity) we get
M = ρAℓ
whence
np
ρAwi
i
 =
∑
=1
while pressure depends on level by the Stevinus law
p = ρgℓ,
(17)
(16)
pdv = pd(1/ρ)-where v = 1/ρ is specific volume-which
is often termed " pressure work " , or maintain the fluid
flow across the volume, thus be viewed as dp/ρ which is
named " impelling work. " Summing the two differentials
we obtain d(pv) = d(p/ρ), which explains why enthalpy
is the natural thermodynamic variable to associate with
energy transfers that occur by mass transfer.
Denoting by w the mass flow rate through a port and by
h the enthalpy on the side of the port from which the fluid
is coming, the energy rate at the port is therefore wh (with
the sign dictated by the fluid direction as just noted).
(18)
g being the gravity acceleration. Note that p in (18) is relative
(or gage) pressure: if absolute pressure is needed
one has to add the atmospheric one.
C. Mass Transfer
The transfer of fluid mass can happen spontaneously
owing to pressure differences like in pipes and valves, or
be caused by some impelling mechanism like in pumps
and fans [26]. In both cases the pressure drop Δp across
the element is related to the mass flow rate w through it
by an equation grounded in the conservation of momentum.
In any situation relevant to us it is acceptable to
neglect the effect of the fluid inertia, which allows writing
algebraic equations in the form
Δp = Δp0(θ, u) + f (w, θ, u)
(19)
where Δp0 is a flow-rate-independent term due to external
forces like gravity and/or the zero-flow-rate pressure
difference in active elements like pumps, while function
f accounts for friction losses and can take various forms
(owing e.g. to laminar or turbulent flow) with f (0,θ, ·) = 0;
vector θ contains the physical parameters of interest,
relative to the fluid, to the element in which the motion
occurs (e.g., the flow coefficient of a valve or the static
head of a pump) and sometimes to the installation conditions
(e.g., the inlet-outlet height difference for a pipe).
Input u may be present to represent a command when
one exists (e.g., valve opening or pump speed set point).
D. Energy Transfer With Mass
A means to transport energy, of paramount importance
in cooling, is by transporting mass [26]. When a mass
stream enters or exits a volume, there are two effects:
one is the energy contribution owing to the internal energy
e of the entering or exiting fluid, the other is the
(signed) mechanical work exerted by the stream on the
fluid in the volume. This work can alter the pressure
in the volume, thus be viewed in differential form as
46
IEEE CIRCUITS AND SYSTEMS MAGAZINE
E. Energy Storage
Dynamic energy balances can refer to the same cases
we classified mass balances with, and take the form
d
dt
()=+
==
Me ∑∑Q ,
i 11
nppm
whi
ii
j
(20)
where M is the mass in the volume, e its specific energy,
wi and hi are the exchanged flow rate and specific enthalpy
at the i-th port (out of np) where energy is given
or taken by transport of mass, see Section VI-D, and Qi
the exchanged power at the j-th port (out of mp) where
energy is given or taken without mass transport, see
Sections VI-F and VI-G.
For solid elements M is ρV-hence it is constant-and e
is cT, which makes temperature the natural state variable.
For the filled liquid volume case M is constant, but for
the reasons in Section VI-D the natural state variable is
h, hence e is expressed as h − p/ρ (although the p/ρ term
is largely dominated by e, that is cT, in any case relevant
for us).
The situation is analogous for the filled (ideal) gas
volume case, expressing M as ρV where ρ, p, e, h and T
are related by (12). The energy balance can still be written
in the form (20), however, thanks to the symbolic
manipulation capabilities of Modelica tools.
For the open liquid volume case, finally, M is not constant,
see (16), but here too (and for the same reason
above) the balance can be written in the form (20), with
e = h − p/ρ.
F. Spontaneous Heat Exchange
Heat flows spontaneously by conduction, convection
and radiation (which we do not treat in this work owing
to its low relevance in MPSoC cooling).
In our context, conduction is relevant only within solids
because fluids move around, and energy transfers
with mass make conduction within them negligible [27].
Also, in MPSoC cooling circuits the only solid elements
are fluid containment ones, and for them conduction is
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IEEE Circuits and Systems Magazine - Q1 2023

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