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Moreover, the a leg of the assumed loop (see figure on side)
will be at constant temperature T (outlet and inlet refers to the
core flow) and b has to be at almost (within 1%) constant
temperature T inlet
to optimise the loop efficiency.
Thus, the ∆P evaluation can be particularly easy under some
additional hypotheses on the heat exchanger height and the
temperature distribution. In literature, the same height (and
therefore h a
= h b
) and linear temperature behaviour is often assumed
in the two components. Generally, we will not need such an
approximation for temperatures, nevertheless it can be worthwhile
to recall that it would lead to:
∆P
= g = gh
g
b
( −ρ
) = −ρgh
β T −T
)
ρ (11)
b
a
b
(
out in
where β is the volumetric dilatation coefficient of the coolant.
Actually, if we know the coolant temperature (i.e. density)
distributions along the components, the steady state equation (11)
can be substituted by more precise formulations, for instance the
following one:
∆ P t = 0)
= g
g
( h ρ + h ρ −h
ρ −h
ρ )
( (12)
exc
exc
b
b
core
core
a
a
2.3 The transient
Under time-dependent conditions, besides pressure drops and gravitational pull variations, also
kinetics energy variations must be considered. The general method is the classical one relevant to the
mechanics: infinitesimal work relevant to external forces equals the infinitesimal kinetics-energy
variation. However, practically, a detailed knowledge of the loop is needed. In fact, it is easy to verify
that, in any loop volume comprised between two sections and having length l and flow area σ, an
infinitesimal kinetics energy variation can be written as:
2
⎛ 1 2 ⎞ ⎛ 1 Q ⎞
m l
(13)
d⎜
mv ⎟ = d
⎜ l = QmdQm
ρσ
⎟
⎝ 2 ⎠ ⎝ 2 ⎠ ρσ
Owing to the fact that we are dealing with liquid coolant and we will apply our model to slow
temperature transients, the time derivative of equation (1) can be neglected to evaluate current flow
rates. Equation (13) shows that, if Q m
does not depend on the considered circuit section, the
infinitesimal variation of kinetics energy will depend on the considered section area σ. Practically, this
leads to the fact that in a loop characterized by several portions l i
having different flow area (σ i
), the
kinetics-energy variation has to be expressed as a summation of terms depending on the loop
geometry:
Qm
ρσ
( ∆P g
−∆P ) dV ≡σ
( ∆P
g
−∆P ) vdt=
∑li
dQm
= v dQm
∑li
= vLdQm
i
ρ σ
i
i
i
ρ σ
i
i
(14)
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