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Unformatted text preview: PROBLEM 1.41 KNOWN: Hot platetype wafer thermal processing tool based upon heat transfer modes by conduction through gas within the gap and by radiation exchange across gap. FIND: (a) Radiative and conduction heat fluxes across gap for specified hot plate and wafer temperatures and gap separation; initial time rate of change in wafer temperature for each mode, and (b) heat fluxes and initial temperaturetime change for gap separations of 0.2, 0.5 and 1.0 mm for hot plate temperatures 300 < T h < 1300 C. Comment on the relative importance of the modes and the influence of the gap distance. Under what conditions could a wafer be heated to 900 C in less than 10 seconds? SCHEMATIC: ASSUMPTIONS: (1) Steadystate conditions for flux calculations, (2) Diameter of hot plate and wafer much larger than gap spacing, approximating plane, infinite planes, (3) Onedimensional conduction through gas, (4) Hot plate and wafer are blackbodies, (5) Negligible heat losses from wafer backside, and (6) Wafer temperature is uniform at the onset of heating. PROPERTIES: Wafer: = 2700 kg/m 3 , c = 875 J/kg K; Gas in gap: k = 0.0436 W/m K. ANALYSIS: (a) The radiative heat flux between the hot plate and wafer for T h = 600 C and T w = 20 C follows from the rate equation, ( 29 ( 29 ( 29 ( 29 4 4 4 4 8 2 4 4 2 rad h w q T T 5.67 10 W / m K 600 273 20 273 K 32.5kW / m  = + + = = < The conduction heat flux through the gas in the gap with L = 0.2 mm follows from Fouriers law, ( 29 2 h w cond 600 20 K T T q k 0.0436W / m K 126 kW / m L 0.0002 m = = = < The initial time rate of change of the wafer can be determined from an energy balance on the wafer at the instant of time the heating process begins, w in out st st i dT E E E E c d dt  = = where out E = and in rad cond E q o r q . = Substituting numerical values, find 3 2 w r a d 3 i,rad dT q 32.5 10 W / m 17.6 K /s dt cd 2700kg / m 875 J / kg K 0.00078 m = = = < w cond i,cond dT q 68.4 K /s dt cd = = < Continued .. PROBLEM 1.41 (Cont.) (b) Using the foregoing equations, the heat fluxes and initial rate of temperature change for each mode can be calculated for selected gap separations L and range of hot plate temperatures T h with T w = 20 C. In the lefthand graph, the conduction heat flux increases linearly with T h and inversely with L as expected. The radiative heat flux is independent of L and highly nonlinear with T h , but does not approach that for the highest conduction heat rate until T h approaches 1200 C. The general trends for the initial temperaturetime change, (dT w /dt) i , follow those for the heat fluxes....
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 Fall '07
 SAVASYAVUZKURT

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