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 thin-walled tubing and surface area As. The flow rate, mh, mean inlet temperature, Th,i, and specific heat. cp,K of the hot fluid are known, as are the initial temperature, Tc,i

(a) Starting from basic principles, derive expressions that can be used to determine the variation of T, and Th.o with time during the heating process.

(b) Consider a pharmaceutical of volume Vc = 1m3, density Pc = 1100 kg/m3, specific heat cv.c = 2000 J/kg ∙ K, and an initial temperature of Tc,i = 25°C. A coiled 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.; = 200°C and mh = 2.4 kg/so 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, μh = 0.002 N ∙ s/m2, kh = 0.260 W/m ∙ K, and Prh = 20. For the foregoing conditions compute and plot the pharmaceutical temperature T( and the outlet temperature Tm,o as a function of time over the range 0

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Hot fluid Tha () - Coiled tubing Pharmaceutical -Containment vessel Insulation