Problem Statement Consider an array of heated tubes submerged in a vessel with fluid flowing past them. Neglecting end effects, the flowfield can be assumed 2-D in planes with normals parallel to the tube axes. Further, for modest fluid onset velocity, a steady state solution can be sought. Finally, because of resultant planes of symmetry, the flowfield solution domain W can be the small region between the hatched lines shown in Figure 1.
Figure 1. Schematic of the DE simplified analysis problem. This problem statement requires that DM, DP and DE all be addressed. In FEMLAB nomenclature, these conservation laws are expressed as
DM:
DP:
DE:
where r
is density,
Your
classical undergraduate coursework characterizes heat transfer modes via
non-dimensional groups. For the typical case of onset flow at small Mach
number (Ma < 0.3), the fluid density can be assumed constant (ro)
everywhere except in the body force term F in (2). Then one invokes
the Boussinesq model for thermal buoyancy effects, whereby
Assuming all other thermal data are constant, and that the flowfield
is laminar, the non-dimensional forms of DM, DP, DE
equivalent to (1)-(3) are
DM:
DP:
DE:
The definitions for the non-dimensional groups Re (Reynolds number),
Gr (Grashoff number), and Pr (Prandtl number) in terms of FEMLAB variables
are
wherin the reference scales are: U Þ
velocity, D Þ
tube diameter, b
Þ
thermal coefficient of expansion (Tabs)-1,
and g Þ
gravity constant, and the non-dimensional (potential) temperature definition
is
Finally, the convection heat transfer coefficient h, in the BC for DE, becomes replaced by Nusselt number Nu, and natural convection sometimes employs the Rayleigh number Ra. These definitions are
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The heat transfer modes as defined in your textbooks are
natural convection for
mixed convection for
The resultant convective heat transfer is correlated by Nusselt number, Nu.
Available literature correlations for natural and forced convection, NuN
and NuF, are:
The
conservation law system (5)-(13) describes the analysis problem at hand.
Note that any consistent sets of units are admissible, and the input to
FEMLAB is dimensional. Hence Re, Pr, Gr, and Nu will be computed in the code
from input data. Setting up the problem in FEMLABThe base case data specification is:
In the subdomain settings define the buoyancy body force Fy as Fy = rogb (T-To) The steady incompressible Navier-Stokes state variable solution {Q} is established via GWSh + qTS implementation using the FE k = 1 natural coordinate basis on a mesh of triangles. |