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A 1 m length squаre plаte mаde оf steel (ASTM A203 B) that has a thickness оf 1 cm is part оf the equipment for a fluid flow process. The plate has a thermal conductivity of 52 W/m⋅K, a specific heat of 470 J/kg⋅K, and a density of 7.9 g/cm3. Both large square surfaces of the plate are exposed to a cryogenic fluid at -50° C with a convection heat transfer coefficient of 50 W/ m2 ⋅K. The minimum suitable temperature for the ASTM A203 B steel plate is -30 °C, and the plate has an initial temperature of 20 °C. (a) Can the lumped capacitance method be applied to this problem? Justify your answer by calculating the Biot number. (b) Conduct an energy balance and derive an equation to calculate the temperature of the steel plate as a function of time. (c) Estimate the time required for the plate to reach its minimum suitable temperature of -30°C when exposed to the cryogenic fluid. (d) Suppose the whole amount of material of the plate is reshaped into a solid sphere and exposed to the same cryogenic fluid at -50°C. Will the time required to reach the minimum suitable temperature be the same as in part (b)?
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