ch01_41-73 - PROBLEM 1.41 KNOWN Hot plate-type wafer...

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PROBLEM 1.41 KNOWN: Hot plate-type 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 temperature-time 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) Steady-state conditions for flux calculations, (2) Diameter of hot plate and wafer much larger than gap spacing, approximating plane, infinite planes, (3) One-dimensional 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: ρ = 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 σ - ′′ = - × + - + = = < The conduction heat flux through the gas in the gap with L = 0.2 mm follows from Fourier’s 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 cd dt ρ ′′ ′′ ′′ ′′ - = = ± ± ± ± where out E 0 ′′ = and in rad cond E q or q . ′′ ′′ ′′ = Substituting numerical values, find 3 2 w rad 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 ρ ′′ × = = = × × < w cond i,cond dT q 68.4 K /s dt cd ρ ′′ = = < Continued …..
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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 left-hand 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 non-linear 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 temperature-time change, (dT w /dt) i , follow those for the heat fluxes. To reach 900 ° C in 10 s requires an average temperature-time change rate of 90 K/s. Recognizing that (dT w /dt) will decrease with increasing T w , this rate could be met only with a very high T h and the smallest L.
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