Nomenclature (Use consistent units.) A = heat-transfer surface; C, c = specific heats of hot and cold fluids respectively; L0 = flow rate of liquid added to tank; M = mass of fluid in tank; T, t = temperature of hot and cold fluids respectively; Ti, ti = temperatures at beginning of heating or cooling period or at inlet; T2, t2 = temperature at end of period or at outlet; T0, t0 = temperature of liquid added to tank; U = coefficient of heat transfer; and W, w = flow rate through external exchanger of hot and cold fluids respectively.
Applications One typical application in heat transfer with batch operations is the heating of a reactor mix, maintaining temperature during a reaction period, and then cooling the products after the reaction is complete. This subsection is concerned with the heating and cooling of such systems in either unknown or specified periods.
The technique for deriving expressions relating time for heating or cooling agitated batches to coil or jacket area, heat-transfer coefficients, and the heat capacity of the vessel contents was developed by Bowman, Mueller, and Nagle [Trans. Am. Soc. Mech. Eng., 62, 283294 (1940)] and extended by Fisher [Ind. Eng. Chem., 36, 939-942 (1944)] and Chaddock and Sanders [Trans. Am. Inst. Chem. Eng., 40, 203-210 (1944)] to external heat exchangers. Kern (Process Heat Transfer, McGraw-Hill, New York, 1950, Chap. 18) collected and published the results of these investigators.
The assumptions made were that (1) U is constant for the process and over the entire surface, (2) liquid flow rates are constant, (3) specific heats are constant for the process, (4) the heating or cooling medium has a constant inlet temperature, (5) agitation produces a uniform batch fluid temperature, (6) no partial phase changes occur, and (7) heat losses are negligible. The developed equations are as follows. If any of the assumptions do not apply to a system being designed, new equations should be developed or appropriate corrections made. Heat exchangers are counterflow except for the 1-2 exchangers, which are one-shell-pass, two-tube-pass, parallel-flow counterflow.
Coil-in-Tank or Jacketed Vessel: Isothermal Heating Medium ln (T1 - t1)/(T1 -t2) = UA9 /Mc (11-35)
Cooling-in-Tank or Jacketed Vessel: Medium
Isothermal Cooling ln (T1 - ti)/(T2 - ti) = UAQ/MC (11-35«)
Coil-in-Tank or Jacketed Vessel Medium ln
W -11 = WC / W - 1 ^ W -t2 Mc \ K1 where ] = eUA/WC
Coil-in-Tank: Nonisothermal Cooling Medium ln WW = WW \WW/ e (11-35c)
where K2 = eUA/wc
External Heat Exchanger: Isothermal Heating Medium
T1 -12 Mc \ K2 External Exchanger: Isothermal Cooling Medium
External Exchanger: Nonisothermal Heating Medium W -11 _( K3-1\ I wWC
External Exchanger: Nonisothermal Cooling Medium
Ti-11 = / KW 1 \ / Wwc T2 -11 = \ M / \ Wwc t WC l where K4 = eUA(1WC - Vwc
External Exchanger with Liquid Continuously Added to Tank: Isothermal Heating Medium
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