Online proceedings - EDA Publishing Association

7-9 October 2009, Leuven, Belgium
the inclusion of nonlinear effects would be inappropriate at that
level. In a domain , bounded by the closed surface , the
following equation holds at steady state:
with node i[1, N] is denoted i . Its extent, which can be an
area or a volume, will be denoted S i .
The following modified Green’s function G has been proposed
2
c v T
T q v
(2.1)
earlier which satisfies:
where v is a given velocity vector field, 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
Pe v T
T q v
(2.2)
where Pe is the Peclet number Pe = v D/ (where v is a
characteristic fluid velocity, D a characteristic length and
thermal diffusivity).
Let us analyze implications of (2.2) in its own on the sought for
compact model. Boundary conditions will be postponed to a
subsequent step in order to concentrate on what is general,
embodied by (2.2) as compared to what is problem dependent,
which is specified by the particular set of boundary conditions.
First of all, linearity of (2.2) implies linearity of the sought for
compact model. The HTC model (1.1) is linear, but it is only a
very particular form of a linear relation between the flux (Q)
and the potential difference (T). There are many other linear
forms like the matrix form or the integro-differential form:
qi
j
hij
T
j Tref
(2.3)
T
a Q d
bQ dx
cQdx
(2.4)
In (2.3), q i and T j refer respectively to the heat flux density and
the temperature at different points, while T ref is any suitable
reference temperature. In (2.4), x is a relevant space coordinate,
while a, b, and c can be either constants or space dependent
functions. Any of these forms are linear. We still have to
choose a form that respects problem physics while keeping the
level of complexity relatively low, to let the compact model be
practical. Note that 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.
III. DERIVING THE MOST GENERAL FORM
The shape of the general linear relation between flux and
potential will be derived here without specifying boundary
conditions in order to obtain a general BCI model. Boundary
conditions will be introduced later in the analysis. Let us define
a node as being a surface or a volume across which heat flows
into or out from the system. The sub-domain of , associated
Pe v G
2
r,
r'
Gr,
r'
r
r'
in (3.1)
together with the following set of boundary conditions on the
auxiliary function G (not T) on :
1
S1
r 1
n G
Pen
vG
0 r 1
(3.2)
where 1 is the sub-domain associated with node 1, which will
be called henceforth the reference node, while S 1 is its extent.
The function G can be analytically obtained only in very
simple cases. But this is not an obstacle, as it may also be
obtained numerically in all cases. In fact, what matters is that it
exists! Multiplying (2.2) by G and (3.1) by T, subtracting and
integrating over the whole problem domain with respect to r
we get after some algebra:
T
r'
Tref
Gr,
r'
q
v rdr
Gr,
r'
q
s r
in which q s
T ref Tdr A
1 is the reference temperature.
1
dr
(3.3)
n T
is the dimensionless heat flux density and
Equation (3.3) can be simplified further by recognizing that
surface heat flux density q s or volume internal heat generation
q v are only nonzero over domain nodes, hence will both be
denoted by the unified symbol q i for any node i. Let us denote
by r i values of r i . Equation (3.3) becomes:
N
ri
Tref
j
Gr
ri
T T
1
, q dr
i
j
j
i[1, N] (3.4)
This equation relates node temperatures T i = T(r i ) – T ref with
node heat power densities q i (per unit area q s or per unit volume
q v , whichever is relevant) on all problem nodes. It incorporates
the intrinsic object behavior based on governing partial
differential equation (2.2) but NOT externally applied
boundary conditions on T i or q i , which were not specified to
obtain it. It holds for any set of boundary conditions Dirichlet,
Neumann or Robin. It contains thus all problem physics. There
can be other techniques to express the general solution of (2.2)
(‘general’ in the sense of ‘regardless of the particular set of
boundary conditions’) without passing explicitly or implicitly
by the Green’s function. However, since the solution is unique,
other approaches should yield an expression perfectly
equivalent to (3.4). As no ‘physical’ simplifying assumption
has yet been introduced (apart from that of forced convection
with uniform physical properties) (3.4) is the most general
form of the steady functional relation between heat power q i
(by unit volume q v or by unit area q s ) and T i . The only
approximation needed is that of obtaining G, which can readily
be done either numerically or experimentally.
In case we had only two nodes (for instance wall and fluid at
infinity) one of them can be the reference, the other will be the
space dependent difference T (r). If in addition we had a
©EDA Publishing/THERMINIC 2009 3
ISBN: 978-2-35500-010-2