Modeling Material Flow
Background: What is “Material
C&R Thermal Desktop® and
SINDA are very capable of modeling steady and unsteady
heat transfer problems including conduction, convection,
thermal radiation, etc. for moving and stationary parts.
When a batch process is to be simulated, or when
discrete parts move (such as ingots through a furnace,
or ground-tracking antennae on satellites), the part
itself can be translated or rotated within a transient
solution. But when the motion is continuous, such
that a steady-state solution is possible, different
modeling methods are available and should be employed.
Examples of such continuous
motion include a sheet of glass solidifying as
it is lowered through a temperature-controlled
zone, a gypsum board moving through a drier, and
a conveyor belt carrying baked goods through a continuous
oven. In those circumstances, a fixed model of the
both stationary parts (heat lamps, ovens, driers,
etc.) and the moving parts (rollers, sheets, belts,
etc.) is built. Then, an advection or “material
flow” term is superimposed on the rotating
or translating parts.
For example, below is an open mesh conveyor belt with rollers moving under a heat lamp (more like a laser: collimated). Ray plots have been superimposed to show the lamp rays passing through the mesh belt.
Example applications for this capability include:
conveyor furnaces, conveyor ovens
and aluminum sheet metal manufacturing
making (especially plate glass)
making, fiber products, particle board and flakeboard
drying and curing, drywall (wallboard) manufacturing
fiber manufacturing (drawing fiber optic cable
through a furnace)
- rotary furnaces
foam and metal foam heat exchangers, geothermal
disk heat exchangers and dehumidifiers
belt heat exchangers, moving belt radiators
Hot Wire: Material Flow Example
A large rectangular
copper “wire” passes
through a continuous-flow tubular furnace used to
harden a thermoset polymer coating. A pair of cooled
rollers at the exit of the furnace help to both position
the wire and smooth the coating.
A 8.2cm by 3.8cm copper bar
or “wire” at
275°C enters a tubular furnace at 1000°C.
The wire has been coated with a 0.25mm thick polymer
coating (1 W/m-K estimated thermal conductivity,
and 0.8 estimated infrared emissivity). The coating
must be heated to at least 315°C at all locations,
both not more than 350°C at any one location.
The furnace, which operates in a vacuum environment,
is nominally 100cm long and 10cm in diameter, but
its exact sizing is one of the purposes of the analysis.
The nominal speed of the wire through the furnace
is 10 cm/s.
Located 15cm past the exit
of the furnace are two 10cm wide copper rollers,
10cm in diameter and 0.5cm thick (9 cm inner diameter).
They are exposed to 20°C room temperature environment,
and so operate considerably colder than the wire
due to radiative cooling. The exact nature of the
thermal contact with the wire is uncertain, though
clamping pressures are high and the polymer coating
is still relatively soft at that point. The rollers
are assumed to be lightly coated with polymer and
hence to exhibit the same emissivity as the bar.
It is desired to know how hot the coating will
be by the time it reaches the rollers, how hot
the rollers will be, and whether the roller causes
any significant cooling of the coating as it passes
The CAD drawing below depicts the problem as a cut-away
along one of the two planes of symmetry, with the
wire (green) traveling to the right and into the
page, the furnace (in red), and the rollers at the
Thermal Desktop Model
The Thermal Desktop model is depicted below, exploiting
the aforemented plane of symmetry to both reduce
the model and to create a cut-away view. This model
includes a semi-cylinder for the furnace, a finite
difference solid “brick” to represent
half the wire, and a pair of solid finite difference
cylinders representing half (5cm width) of the rollers.
Though there is no
“beginning” nor “end” to
the wire, a 10cm entrance section is modeled to included
radiation from the end of the open furnace to the
coated bar as it enters the furnace, and 30cm of
wire is modeled after the furnace to include the
effects of the roller.
Symbols are used to make the model
parametric, such that variations in geometry can
be explored, the effect of uncertainties in properties
can be gauged, etc. The size of the furnace (and
therefore the wire), for example, can be changed,
with the positions of the rollers moving accordingly
to stay 15cm away from the exit of the furnace (1).
Key symbols are presented below. Note that the contact
conductance of between the roller and the coating
has been estimated as 30,000 W/m2-K (3
W/cm2-K), applied over a width of 1 cm.
(The model will reveal that these values do not have
a strong influence on the results.)
Results and Discussion
In the postprocessed graphic below, the color scale
has been set such that any temperature above the
350°C limit would appear off-the-scale (as purple),
which is the color that the furnace tube becomes.
However, no temperature on the wire appears above
this threshold: that portion of the design requirements
has been met.
The rollers are much cooler than the wire, and despite
the high contact conductance between them, the amount
of energy transferred is very small: no significant
cooling of the coating occurs other than radiative
cooling as the wire passes next to the cool rollers.
The rollers themselves are not appreciably heated
by the wire, being dominated more by their exposure
to the ambient temperature. The spinning of the rollers
helps unify their temperature, as if they were “perfectly
mixed,” as verified by the small temperature
differences exhibited in that component (shown below).
The fluid analogy can be continued with the wire,
where the cut-away section reveals temperature profiles
roughly mimic velocity profiles in a fluid.
To verify that the coating
has been heated adequately to the requirement of
315°C, the drawing is zoomed
and the color scale changed again, as depicted below.
Since any portion of the coating that had been heated
above 315°C now appears purple (off scale), it
can be seen that the furnace was somewhat undersized:
a stripe about 2cm wide along the center of the wire
is only reaching about 311°C (depicted as red).
The lower temperature scale has also been raised
such that the cool rollers are now below the threshold
of 280°C, revealing the temperature contours
in the coating as it exits in greater detail.
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1 - A Thermal Desktop “assembly” has
been used for this purpose, with the translation
of the rollers in the X axis being controlled as
a function of the furnace length (symbol FurnLen)