IEEE Circuits and Systems Magazine - Q1 2023 - 47

only relevant in the direction orthogonal to the surface
in contact with the fluid. The reason is again that fluids
move around: considering for example a pipe wall as a
sequence of annular elements, the heat transported from
an element to the downstream one by the moving fluid by
convection largely dominates the heat transferred by the
former to the latter through solid-to-solid contact.
As a result, in cooling circuits we only have to do with
one-dimensional conduction between two surfaces, in
planar or cylindrical geometry. For the planar case we
can neglect border effects and compute the thermal
conductance Gt between two identical surfaces of area
A, separated by a layer of thickness s of a material with
thermal conductivity λ as
Gt = λ
A
s
(21)
while for the cylindrical case, considering an annulus of
length L, inner radius ri and outer radius re, we get
Gt = λ
2πL
.
log( /)
rr
ei
In both cases, denoting by T1 and T2 the two surface
temperatures and by Q12 the conductive heat rate from
surface 1 to surface 2, we have
Q12 = Gt (T1 − T2).
(23)
Convection is a more complex phenomenon, as it involves
the motion of a fluid in contact with a solid. In
particular, we need to distinguish laminar and turbulent
flow. In a nutshell and thinking of a pipe, laminar flow is
when fluid particles proceed aligned in the direction of
the pipe axis, pushed by pressure difference and subject
to mutual friction with the adjacent ones (or the wall).
This happens at low speeds and results in a parabolic
speed profile, with the fastest particles at the pipe center.
At higher speeds, the order of laminar flow breaks. Particles
still move on average along the tube axis, but instead
of proceeding pretty much like in parallel lanes they also
continuously mix up with the neighboring ones, resulting
in an almost flat speed profile. As turbulence apparently
helps giving or taking heat, owing to fast fluid particle
turnover in the vicinity of the wall, turbulent convection
is significantly more efficient than laminar one.
We thus compute the fluid to solid convective heat
rate Qfs referring to a bulk (average) fluid temperature Tf
and a surface one Ts for the contacting solid, in the form
Qfs = γA(Tf − Ts)
(24)
where A is the contact surface and γ the convective heat
transfer coefficient, computed from the fluid properties
FIRST QUARTER 2023
(22)
and motion conditions with correlations like the well-established
Dittus-Bölter one. Notice that contrary to the
conduction case γ is not constant; it depends on other
variables in the system, most typically fluid speed and
thus flow rate, making the phenomenon nonlinear.
G. Work-Driven Heat Exchange
Heat pumps, which include Peltier elements, are machines
that employ mechanical or electrical power to
seemingly violate the second principle of thermodynamics
by transferring heat from the colder to the hotter.
For simulating and evaluating a cooling system there
is no need to represent the internal physics of such devices:
it is more practical to describe them based on the
nominal external characteristics provided by manufacturers.
These are an efficiency and a Coefficient of Performance
(COP), either constant or-more frequently and
realistically-depending on the operating conditions.
Denoting by Qc the heat rate taken from the cold side
of the pump, by Qh the heat rate released to the hot side
and by Wp the net power to the pump, we define the COP
as the ratio of the desired effect to the spent effort, i.e.,
COP ,, (25)
heat==
Q
W
c
p
COPcool
Q
W
h
p
depending on whether the pump is devoted to heating
or cooling; notice that COPheat = COPcool + 1, since obviously
Qh = Qc + Wp. In our context we are always cooling,
hence we drop the subscript and define COP = Qc/Wp.
The above established, we can describe a heat pump as
QQ W
QCOP W
WW
hc p
cp
p
=
=
=
+
⋅
η
COPf TT
= COPh c(, )
(26)
where Tc and Th are the cold and hot side temperatures,
and fCOP a function-most frequently the interpolation
of some measured points-to obtain the COP; η, in general
reasonably constant, accounts for losses that cause
the absorbed power W to be greater than Wp. The COP
is limited by the Carnot efficiency, that expressing temperatures
in Kelvin degrees for the cooling case reads
COPCarnot =
TT
c
T
.
hc
−
Real heat pumps rarely achieve a COP above 40%-
50% of the Carnot one (a useful information for first-cut
sizing). Moreover, notice that the COP-not only the
Carnot one, clearly diminishes as the pump temperature
difference ΔT = Th − Tc
increases. This brings into
play the heat exchangers connecting the pump to the
cold and hot environments, as summarized in Fig. 4.
IEEE CIRCUITS AND SYSTEMS MAGAZINE
47
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IEEE Circuits and Systems Magazine - Q1 2023

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