Answer: Batch processes are often used in chemical and

Chapter , Problem 8.32

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Batch processes are often used in chemical and pharmaceutical operations to achieve a desired chemical composition for the final product. Related heat transfer processes are typically transient, involving a liquid of fixed volume that may be heated from room temperature to a desired process temperature, or cooled from the process temperature to room temperature. Consider a batch process for which a pharmaceutical (the cold fluid, c) is poured into an insulated, highly agitated vessel (a stirred reactor) and heated by passing a hot fluid (h) through a submerged heat exchanger coil of thinwalled tubing and surface area As. The flow rate, , mean inlet temperature, Th,i, and specific heat, cp,h, of the hot fluid are known, as are the initial temperature, Tc,i Th,i, the volume, Vc, mass density, c, and specific heat, cv,c, of the pharmaceutical. Heat transfer from the hot fluid to the pharmaceutical is governed by an overall heat transfer coefficient U. (a) Starting from basic principles, derive expressions that can be used to determine the variation of Tc and Th,o with time during the heating process. Hint: Two equations may be written for the rate of heat transfer, q(t), to the pharmaceutical, one based on the logmean temperature difference and the other on an energy balance for flow of the hot fluid through the tube. Equate these expressions to determine Th,o(t) as a function of Tc(t) and prescribed parameters. Use the expression for Th,o(t) and the energy balance for flow through the tube with an energy balance for a control volume containing the pharmaceutical to obtain an expression for Tc(t). (b) Consider a pharmaceutical of volume Vc 1 m3 , density c 1100 kg/m3 , specific heat cv,c 2000 J/kgK, and an initial temperature of Tc,i 25 C. A coiled Co tube of length L 40 m, diameter D 50 mm, and coil diameter C 500 mm is submerged in the vessel, and hot fluid enters the tubing at Th,i 200 C and m . h 2.4 kg/s. The convection coefficient at the outer surface of the tubing may be approximated as ho 1000 W/m2 K, and the fluid properties are cp,h 2500 J/kg K, h 0.002 N s/m2 , kh 0.260 W/m K, and Prh 20. For the foregoing conditions, compute and plot the pharmaceutical temperature Tc and the outlet temperature Th,o as a function of time over the range 0 t 3600 s. How long does it take to reach a batch temperature of Tc 160 C? The process operator may control the heating time by varying m . h. For 1 m . h 5 kg/s, explore the effect of the flow rate on the time tc required to reach a value of Tc 160 C. 8.33 The ev