Modeling Material Flow

Background: What is “Material Flow?”

C&R Thermal Desktop® and SINDA are very capable of modeling steady and unsteady heat transfer problems including conduction, convection, 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:

• Belt conveyor furnaces, conveyor ovens
• Steel and aluminum sheet metal manufacturing
• glass making (especially plate glass)
• paper making, fiber products, particle board and flakeboard drying and curing, drywall (wallboard) manufacturing
• optical fiber manufacturing (drawing fiber optic cable through a furnace)
• pebble bed reactors
• coke furnaces
• rotary furnaces
• carbon foam and metal foam heat exchangers, geothermal storage systems
• rotating disk heat exchangers and dehumidifiers
• moving 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.

Problem Statement

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 underneath.

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 exit (blue).

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.

Customization and Consulting

C&R also provides consulting and custom software solutions to specifically meet your needs.

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)