Prediction of strongly-heated internal gas flows Page: 5 of 17
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appropriate assumptions [McEligot, 1963], one can
approximate the increase in bulk temperature along the
tube as
(Tb/Tin) = 1 + 4 q+in (xD)
and the wall-to-bulk temperature difference as
((Tw - Tb) / Tin) = q+in Rel-a pr0.6 / C
The non-dimensional heating rate q+ in =
q"w/Gcp,inTin = q"wAcs/ it cp,inTin evolves naturally
from non-dimensionalizing the governing equations and
boundary conditions in pipe flow with an imposed wall
heat flux distribution [Bankston and McEligot, 1970].
Property effects come in via the (non-dimensional)
exponents in the power law representations. With
"large" heat fluxes (large q+), the resulting temperature
range causes significant variation of the gas properties,
invalidating the use of design relations such as the
popular Dittus-Boelter correlation.
If one defines the Reynolds number based on
bulk fluid properties as Re = GD/ b = 4 mit /(f-IDb),
then its value will continuously decrease as the axial
distance increases with heating. The perfect gas
approximation shows the bulk density likewise to
decrease as x increases. From an integral continuity
relationship for steady flow, rn = PbVbAcs, one sees
the velocity increases in the streamwise direction. That
is, the flow accelerates spatially.
In strongly-heated, internal gas flows the
pressure drop can be dominated by the induced
acceleration as suggested by McEligot, Smith and
Bankston [1970]. Assumptions and approximations
involved are steady state, one-dimensional flow,
constant cross section and low Mach number. The
momentum equation then can be arranged in a non-
dimensional form,
-{2pxgcDh(dp/dx)/G2} ~8q+ + 4f + 2(Gr* /Rex2)
where the subscript x indicates evaluation of properties
at the local bulk temperature. The Grashof number
appearing in the body force term is defined as Gr* =
gDh /vx2 and g is taken as directed opposite to the
flow direction (i.e., upflow).
Thus, general effects of strong heating of a gas
are variation of the transport properties, reduction of
density causing acceleration of the flow in the central
core, and - in some cases - significant buoyancy forces.
Growth of the internal thermal boundary layer leads to
readjustment of any previously fully-developed turbulent
momentum profile, i.e., no truly fully-established
conditions are reached because the temperature rises --
leading, in turn, to continuous axial and radial variation
of properties such as the gas viscosity. For
calculations, the property variation couples the
momentum equation to the thermal energy equation so
they must be solved "simultaneously."In an application such as the High Temperature
Engineering Test Reactor (HTTR) in Japan, or reduction
of flow scenarios in other plants, another complication
arises. To obtain high outlet temperatures, design gas
flow rates are kept relatively low. For example, at the
exit of the HTTR cooling channels, the Reynolds
number is about 3500. In this range, the heat transfer
parameters may appear to correspond to turbulent flow
or to laminar flow or to an intermediate behavior,
depending on the heating rate (Bankston, 1970], with
consequent differences in their magnitudes (Figure 4). If
the designer is to have confidence in a CTFD code, its
turbulence model must demonstrate the "proper"
predictions in these conditions. The situation where
laminar values are measured at Reynolds numbers
typifying turbulent flow is called "laminarization" by
some authors [Perkins, 1975]. Several authors have
developed approximate criteria for laminarization by
heating using graphical correlations of the heating rate
as q+in{Rein) [McEligot, 1963; Fujii et al., 1991] as
in Figure 5; these have been shown by McEligot,
Coon and Perkins [1970] to correspond to an
acceleration parameter, Kv = (v2/Vb)/(dVb/dx)
4q+in/Rein, which varies approximately as 1/ it 2.St
o aL.
v v o lb G.UI
I-
* a:MReb
Figure 4. Measurements of Bankston [1965].0.01
qIn0.001
2000_ _ 10a
Re
inFigure 5. Suggested laminarization criteria [Ezato et
al., 1997].3
McEligot [1963]
"Laminarizing"
0-
Ogawa et al. [19821 " 0 --
Akino [19781 1Coon
0" [19681
Turbulent
,,.I A,Gl =
s m
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McEligot, D. M.; Shehata, A. M. & Kunugi, Tomoaki. Prediction of strongly-heated internal gas flows, article, December 31, 1997; Idaho Falls, Idaho. (https://digital.library.unt.edu/ark:/67531/metadc691775/m1/5/: accessed July 16, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.