The prediction of the transient time required to heat and/or cool the contents of a jacketed vessel is dependent upon many variables: jacket configuration—plain jacket or dimpled jacket (with or without directional vanes) or a limpet coil; heat transfer medium in the jacket—isothermal, nonisothermal or pump-around fluid and an external heat exchanger; type of vessel, its geometrical dimensions and the type, speed and dimensions of the stirrer; and last but not least, the fluid in the vessel and its physical properties.
Heggs and Hills (1995) have developed a unifying set of equations for these predictions based on a number of assumptions: fluid in the vessel is perfectly mixed; an overall heat transfer coefficient applies and remains constant; flow rates are steady; physical properties remain fixed; heat losses or gains are negligible; the heat capacities of the jacket, vessel, stirrer are small relative to that of the liquid contents of the vessel; and the thermal response of the jacket and any external heat exchanger and pipework are instantaneous.
The Overall Heat Transfer Coefficient is given by:
where the film and fouling heat transfer coefficients, α and αf, are predicted from correlations and tables of empirical data: Kern (1965), Fletcher (1987) and Foumeny and Ma (1991). The symbol Rw is the conductive resistance of the vessel wall, and subscripts m and p apply to the outside and inside of the vessel respectively. (See also Agitated Vessel Heat Transfer, Tank Coils and Fouling for methods of calculating α and αf.)
A proportionality factor, χ, is used in the unifying set of equations to account for the configuration and the means of affecting the transfer of heat, so that for an isothermal heating (condensing) or cooling (boiling) medium in the jacket (Figure 1) with Tm,1 = Tm,2,
For a nonisothermal transfer medium, Figure 1 with Tm,1 ≠ Tm,2,
For a pump-around fluid and an external heat exchanger, (see Figure 2)
where ε is the thermal effectiveness of the external heat exchanger and Cmin = min [(cp)m,( cp)int], and χ2 is evaluated using the flow rate and specific heat capacity, (cp)int, of the pump-around fluid in Equation 3.
The transient time of heating or cooling is given by:
the contents temperature after a time t is given by:
the total heat requirements up to a time t is given by:
total heat requirements to achieve a temperature T p
the rate of heat transfer at any instant of time t is given by:
Heggs, P. J. and Hills, P. D. (1995) The design of heat exchangers for batch reactors. Ch. 18 in Heal Exchange Engineering. 4: 297-313.
Fletcher, P. (1987) Heat transfer coefficients for stirred batch reactor design. The Chemical Engineer. 435. April: 33-37.
Foumeny, E. A. and Ma, J. (1991) Design correlations for heat transfer systems. Ch. 11 in Heat exchange engineering: I, Design of Heal Exchangers. Ellis Horwood Limited. London. 159-178.
Kern, D. Q. (1965) Process Heat Transfer. McGraw-Hill Book Co. New York.