A furnace is located next to a dense array of cryogenic fluid piping. The ice-covered piping approximates a plane surface with an average temperature of T_p = 0?C and an emissivity of ?_p = 0.6. The furnace wall has a temperature of T_? = 200?C and an emissivity of ?_? = 0.9. To protect the refrigeration equipment and piping from excessive heat loading, reflective aluminum radiation shielding with an emissivity of ?_s = 0.1 is placed between the piping and the furnace wall, as shown in the schematic. Assume all surfaces are diffuse-gray. If the temperature of the shield closest to the piping T_{s, N} must be less than 30?C, how many radiation shields, N, must be installed between the piping and the furnace wall?

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Dense piping Furnace wall T = 200°C, e= 0.9 array. T.- O°C, 0.6 Installation criterion Radiation shields, N Ts 30"C E, = 0.1