Numerical Methods, Transient & 2D Conduction

2. Numerical Methods: steady state

A thin metallic foil of thickness 0.25 mm with a pattern of extremely small holes serves as an

acceleration grid to control the electrical potential of an ion beam. Such a grid is used in a

chemical vapor deposition (CVD) process for the fabrication of semiconductors. The top surface

of the grid is exposed to a uniform heat flux caused by absorption of the ion beam, qs = 600

W/m². The edges of the foil are thermally coupled to watercooled sinks maintained at 300 K.

The upper and lower surfaces of the foil experience radiation exchange with the vacuum

enclosure walls maintained at 300 K. The effective thermal conductivity of the foil material is 50

W/mK, and its emissivity is 0.40.

-Vacuum enclosure, Tsur

lon beam, q

↓↓↓ ↓

Grid-

-X

L = 115 mm

$²00

× ×‚×

00

Grid hole

pattern

Water-cooled electrode

sink, Tsin

sink

a) Comment on the temperature gradients in the x and y directions. Is one larger? Why?

b) Assuming one-dimensional conduction and using a finite-difference method representing the

grid by 10 nodes in the x-direction, estimate the temperature distribution for the grid.

Hint: For each node requiring an energy balance, use the linearized form of the radiation rate

equation that was used for resistor networks (and valid for low AT, to find an effective radiation

thermal resistance, Rrad". Equivalently, you may use this same approach to define a radiation

coefficient hrad.