IEEE Circuits and Systems Magazine - Q1 2023 - 43

variety of solutions trespassing the said coverage, the
D2I translation must be re-done each time. This can
be effort-heavy, and apparently requires knowledge of
both the modeling principles to apply, and how to solve
the obtained equations numerically.
Figure 3. D2I example 2-the modeled system.
where T1,2 are the resistor temperatures, α1,2 their temperature
coefficients, and R01,2 the resistance values at
the reference temperatures T01,2. We neglect heat storage
in the air, assume the duct walls to be adiabatic,
and indicate with wa(t) and Tai(t) the air mass flow rate
and inlet temperature, respectively. This said, we can
express the time derivatives of T1 and T2 and then obtain
the imperative model by approximating these with
incremental ratios as we did above.
For this example, however, we do not really need to
perform the D2I translation. It is enough to look at the
expressions of T˙1 (t) and T˙2 (t), that respectively read
2
.
Tt =
1()
1
C ()01 11
vt
t1 RT tT(( ))
()



+
cw tR 1 α (( ))Tt T+− 01
and
.
Tt =
2()
1
C ()02 22
vt
t2 RT tT(( ))
()



+
cw tR +−01 11
2
+
cw tR
aa
aa() ()
()−Tt T202
02
1
1
+ 2
α
α ((( ))
(( ))Tt T
01
() ()
()
()
vt


,


(11)
to make two immediate considerations. First, the obtained
expressions are nonlinear, hence the path to the
imperative code will be inherently more complicated
and error-prone than it was in Example 1 above. Second,
and again most relevant, changing the number of
cascaded " resistor in duct " elements changes the form
of the equations, not only the size of some matrix: one
just needs to observe (10) and (11) to conclude that the
derivative of each temperature depends on that temperature
and on all the upstream ones in the air duct.
Also, the derivative expressions would change significantly
should one for example insert a resistance in
the electric supply line between R1 and R2 instead of
assuming them perfectly in parallel or place them side
by side (instead of downstream one another) in the air
duct. Summing up, then, one could still get to an algorithm
that can be used by just compiling configuration
files, but this would entail accepting a coverage that is
narrow indeed. If the analyst has to experiment with a
FIRST QUARTER 2023
1+− 02
2
α
vt
2
()
+−
GT tT t
tai() ()

22


1
1
aa() ()01 11
()
1+− 01
2
α
vt
()
+−
GT tT t
tai() ()

11







1
1
(10)
C. Examples 1 and 2 the declarative way
1) Example 1: As shown in Listing 2, making the rod
a Modelica component and using it to assemble
an example model is quite straightforward once
the HP (heat port) connector is defined; notice the
similarity to the electric pin. Notice also the natural
separation between the models of the rod and
of the boundary conditions, incidentally.
2) Example 2: Besides nonlinearity, the presence of a
moving fluid gives rise in example 2 to the transport
of thermal properties with the fluid itself.
This requires some equations to be conditional
with respect to the direction (i.e., the sign) of flow
Listing 2. D2I example 1, the declarative way.
IEEE CIRCUITS AND SYSTEMS MAGAZINE
43
IEEE Circuits and Systems Magazine - Q1 2023

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