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1. A lens is used to focus the illumination energy (i.e., radiation) that is required to develop the resist in a lithographic manufacturing process, as shown in the Figure. The lens can be modeled as a plane wall with thickness L = 1.0 cm and thermal conductivity k = 1.5 W/mK. The lens is not perfectly transparent but rather absorbs some of the illumination energy that is passed through it; the absorption coefficient of the lens is a = 0.1 mm-1. The flux of radiant energy that is incident at the lens surface (x = 0) is d'ad = 0.1 W/cm2. The top and bottom surfaces of the lens are exposed to air at T = 20 C and the average heat transfer coefficient on these surface is ñ = 20 W/m²K. = incident radiant energy, "ad = 0.1 W/cm? T = 20°C h = 20 W/m²-K 9.com.x=0 condx.0 19 L = 1.0 cm dx + 9x+de 9 condotel k= 1.5 W/m-K a=0.1 mm 9 cow.x2 T. = 20°C h = 20 W/m²-K transmitted radiant energy Figure 1: The volumetric rate at which absorbed radiation is converted to thermal energy in the lens (q" ) is pro- portional to the local intensity of the radiant energy flux, which is reduced in the x-direction by absorption. The result is an exponentially distributed volumetric generation that can be expressed as:
ar q" = Pradae-2 = (1) (a) Determine and plot the temperature distribution within the lens. (b) Determine the location of the maximum temperature (Imax) and the value of the maximum temperature (Tmax) in the lens. (c) Determine the het flux at x = 0 and at x = L.