Batch processesare often used in chemical and

Chapter 8, Problem 8.32

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Batch processesare often used in chemical and phar-maceutical operations to achieve a desired chemicalcomposition for the final product. Related heat transferprocesses are typically transient, involving a liquid offixed volume that may be heated from room tempera-ture to a desired process temperature, or cooled fromthe process temperature to room temperature. Considera batch process for which a pharmaceutical (the coldfluid, c) is poured into an insulated, highly agitated ves-sel (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, ,mean inlet temperature, Th,i, and specific heat, cp,h, ofthe hot fluid are known, as are the initial temperature,Tc,iTh,i, the volume, Vc, mass density, ?c, and specificheat, cv,c, of the pharmaceutical. Heat transfer from thehot fluid to the pharmaceutical is governed by an over-all heat transfer coefficient U.(a) Starting from basic principles, derive expressions thatcan be used to determine the variation of Tcand Th,owith time during the heating process. Hint: Twoequations may be written for the rate of heat transfer,q(t), to the pharmaceutical, one based on the log-mean temperature difference and the other on anenergy balance for flow of the hot fluid through thetube. Equate these expressions to determine Th,o(t) asa function of Tc(t) and prescribed parameters. Use theexpression for Th,o(t) and the energy balance for flowthrough the tube with an energy balance for a controlvolume containing the pharmaceutical to obtain anexpression for Tc(t).(b) Consider a pharmaceutical of volume Vc?1m3, den-sity ?c?1100 kg/m3, specific heat cv,c?2000 J/kg?K,and an initial temperature of Tc,i?25C. A coiled ube of length L?40 m, diameter D?50 mm, andcoil diameter C?500 mm is submerged in the vessel,and hot fluid enters the tubing at Th,i?200C andm.h?2.4 kg/s. The convection coefficient at theouter surface of the tubing may be approximatedasho?1000 W/m2?K, and the fluid propertiesarecp,h?2500 J/kg?K, ?h?0.002 N?s/m2, kh?0.260 W/m?K, and Prh?20. For the foregoingconditions, compute and plot the pharmaceuticaltemperature Tcand the outlet temperature Th,oas afunction of time over the range 0?t?3600 s.How long does it take to reach a batch temperatureof Tc?160C? The process operator may controlthe heating time by varying m.h. For 1?m.h?5kg/s, explore the effect of the flow rate on the timetcrequired to reach a value of Tc?160C