Online proceedings - EDA Publishing Association

7-9 October 2009, Leuven, Belgium
profile), if the inlet profile was reversed w.r.t. to the
developed zone profile.
Q w = m c (T out – T in )
Q w = H (T w – T b )
(5.8)
(5.9)
There can be different tracks to incorporate profile
dependence in H in order to let it be BCI (see section 3). One
of them is the flexible profile approach [7]. But in all cases, this
self-contained H that would be capable of adjusting itself to
meet each usage case cannot be composed of a single
number, but rather a whole set of numbers, which can be
arranged in a matrix that encompasses matrix H ij mentioned
Where Q w is the heat power added at the wall, m is the mass
flow rate while T out , T in , T w and T b are respectively outlet,
inlet, wall and bulk temperatures. How many temperatures,
and hence how many nodes do we need to consider The
minimum is 3 because bulk temperature can be considered as
a weighted average of inlet and outlet temperatures:
earlier.
T b = T out + (1 – ) T in (5.10)
B. HTC is symmetric, while convection is not
Convection is by nature a non symmetric phenomenon
because upstream and downstream conditions have different
effects on heat transfer. This can be seen from the governing
energy equation, which reads for steady flow with uniform
physical properties:
2
c v T
T
(5.6)
where v is a given velocity vector field (we only consider
forced convection), c and are respectively fluid density,
heat capacity and thermal conductivity. Finally q v is the rate of
volumetric heat generation. In case we transform this
equation into a dimensionless one, we get, keeping the same
symbols for dimensionless v T and q v as their dimensional
counterparts:
2
q v
q v
Pe v T
T
(5.7)
where Pe is the Peclet number Pe = v D/ (where v is a
characteristic fluid velocity, D a characteristic length and
thermal diffusivity). The operator acting on the unknown T
contains an odd order derivative (the LHS) which means it is
NOT a self similar operator. Hence, the temperature field
cannot manifest the same dependence on upstream and
downstream conditions (w.r.t. to the direction of v). It is wellknown
that the higher is Pe the less would downstream
conditions be able to affect heat transfer. For vanishingly
small Pe, we recover the conduction case where both
upstream and downstream conditions have the same effect on
heat transfer.
Modeling this non symmetric phenomenon with H is
inadequate because H is the inverse of a resistance, which is a
perfectly symmetric element. Let us look at the implications
by considering the simplest ever problem of a straight circular
duct heated at its walls according to any known profile. The
complete model should involve a conservation equation (the
first law) and a constitutive equation (the one involving HTC)
like for instance:
where, for simplicity, is considered as a known coefficient
(0, 1). If we wish to confine ourselves to resistive networks
(as suggests the usage of H), then we can have up to 3
resistors relating the three nodes: R w_in , R w_out and R in_out .
Values of these resistors are:
R in_out = 1 / (m c)
R w_out = 1 / ( H)
R w_in = 1 / ((1 – H)
(5.11.a)
(5.11.b)
(5.11.c)
Now, if heat is added at the walls, then T out > T in , hence the
model above predicts that heat should flow from outlet to
inlet with an amount that is proportional to the mass flow
rate! This unphysical behavior is due to the choice of a
resistive based model, which does not comply with the
problem nature. In case we use the matrix representation,
then this matrix should be non-symmetric. More will be given
on the form of this proposed matrix in section 3. Please note
that this matrix should contain, among other parameters, the
parameter Pe, in a form that transforms the matrix into
symmetric if Pe was vanishingly small (i.e. a pure conductive,
hence a pure resistive, i.e. a perfectly symmetric case).
C. HTC cannot model transient effects
The HTC largely depends on the boundary layer thickness,
which for transient problems is time dependent. A sudden
heating of the walls, of an otherwise isothermal flow, creates
a time dependent thermal boundary layer of initial thickness 0
all over the duct. The HTC is thus virtually infinite at start of
heating. It will gradually damp to the steady state values with
a time constant that depends mainly on fluid (and wall!)
thermal capacity.
Some researchers have proposed to use the quasi-static
approximation, i.e. an instantaneous H(t) based on available
steady state correlations in which we would insert
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ISBN: 978-2-35500-010-2