Online proceedings - EDA Publishing Association

7-9 October 2009, Leuven, Belgium
uniform heat flux q i (r) = q, the heat flux density can be taken
M
1 u u
T ri
Tref
u0
Ti
i ri
out of the integral in (3.4). The integral of the known modified
(4.1)
Green’s function G will give a known space function that will
M
be expressed as 1/H(r):
1 u u
q ri
u0
q i i ri
(4.2)
q = H (r) T (r) (3.5) where M is the number of modes retained (this number need
This looks like the Newton’s law of cooling (1.1), but now we not be the same on all nodes, but we will assume it constant for
know the assumptions. Only 2 nodes and uniform heat flux. In
u
simplicity), i is the element of order u of the known function
the other extreme of an imposed uniform temperature
u
difference T, with still only 2 nodes, equation (3.4) becomes a series (Fourier, Legendre …) for the node i and finally T i and
Helmholtz integral equation of the first kind in q i (r). It does not u
qi
are expansion coefficients. Hence, temperature and heat
matter at all how we can solve it at this level (we can always do
flux density fields are replaced by two vectors of size NxM
that numerically at least), all what matters that is solvable and
each (M series elements on each of the N nodes). Let us denote
will give an expression of the form:
these vectors by T and q. Substituting (4.1) and (4.2) in (3.4),
q (r) = H' (r) T (3.6)
Evidently, H and H' are not the same. Both were obtained via
an assumption that depends on the wall thermal conductivity
(respectively much lower or much higher than that of the
fluid). This assumption depends on the nature of surrounding
objects (the wall), which makes the HTC model boundary
conditions dependent, an undesirable property as explained
above. It would also work only for 2-node objects which is too
restrictive.
What have we gained by passing through the most general
form (3.4) before getting the ‘over’ simplified expression (3.5)
or (3.6) Mainly, that we can now make a less crude
assumption transforming (3.4) into a slightly more complicated
form than (1.1) but significantly closer to reality. This will be
done in the next section, but before doing so, let us comment
on the following property of (3.4). Temperatures T i at any point
in each node depend on q at all points in all nodes. In other
physical words, the local value of the thermal boundary layer,
hence the local thermal ressistance depends on all the past
history of heat transfer in the upstream section. Likewise, from
the Helmholtz integral equation, q at any point depends on T at
all points. In (3.4) profile information of both T i and q i are
captured, but not in (3.5, 6). They DO influence each other as
shows (3.4).
IV. PRACTICAL SIMPLE FORMS
In practice, people will not use sophisticated Green's notion to
obtain "engineering" models. However, a direct analog of the
Green's function is the Matrix, to which the Green's function
can be reduced to in any numerical approximation. But before
we sketch a method to construct this matrix, we have to find a
simple way to express profile information on each node. There
are usually two approaches. The first is the so called nodal
approach, in which both temperature and heat power density
are approximated by an interpolation function using
temperature values at many points inside the node. The second
approach is the ‘modal’ approach, in which the temperature, as
well as heat power density, along each node are approximated
by a series expansion over a given well-known functional
series, preferably orthogonal (Fourier, Legendre, …). The
second approach will be selected, giving:
multiplying by
T
R
u
i and integrating over each node gives:
q
where matrix R contains the elements
R
uv
u v
ij
G r
i j
j , ri
i j dr
j dri
(4.3)
uv
R ij , which are given by:
(4.4)
Elements of R are of dimension 1/HTC. In case R was
diagonal, we get the Newton’s law of cooling. It can only be
true if heat transferred at a point depends on the local
temperature difference, regardless of upstream or downstream
conditions. This is in flagrant contradiction with boundary
layer theory as will be developed in next section. This
approach represents a generalization of the "adiabatic heat
transfer coefficient" suggested earlier [3] as well as the flexible
profile approach also suggested earlier [7] for conduction.
Available HTC data can be reused to get at least few of the
elements in the sought for R matrix or its inverse the H matrix.
In fact, the order zero element in any functional series is
0
0 0
usually the uniform one: i const.
. Hence, T i and q i are
respectively the uniform parts of temperature and heat power
densities. This means that in case we had a uniform
temperature profile Ti u 0 for u 0.
(or a uniform heat flux
profile q u i 0 for u 0.
), then it is expected to find the well
known uniform T (or uniform Q) HTC in R (or its inverse H).
V. WHY IS HTC INADEQUATE
a. HTC cannot model multiple heat sources
Let us first introduce the concept of a “node”. It is defined as a
region in space (a surface or a volume) which participates in
heat transfer, by receiving a certain heat power Q [in W] that
depends on its own temperature T as well as that of other
neighboring nodes. Of course the region may have a
distribution of temperatures (also called a profile); but this
issues will be postponed to the next subsection. In case we
only have two nodes, labeled 1 and 2, then we may write the
so called Newton’s law of cooling:
Q 1 = h 1 S 1 (T 2 – T 1 ) (5.1)
©EDA Publishing/THERMINIC 2009 4
ISBN: 978-2-35500-010-2