Question 3.15: In the batch preparation of an aqueous solution the water is...

The rate of heat transfer from the jacket to the water will be given by the following expression (see Volume 1, Chapter 9):

\ \frac{dQ}{dt} =UA(t_{s} -t)       (a)

where dQ is the increment of heat transferred in the time interval dt, and

U = the overall-heat transfer coefficient,

\ t_{s} = the steam-saturation temperature,

t = the water temperature.

The incremental increase in the water temperature dt is related to the heat transferred dQ by the energy-balance equation:

\ dQ=WC_{p} dt         (b)

where \ WC_{p} is the heat capacity of the system.

Equating equations (a) and ( b)

\ WC_{p}\frac{dt}{dt} =UA(t_{s} -t)

Integrating            \ \int\limits_{0}^{t_{B} }{dt} =\frac{WC_{p}}{UA}\int\limits_{t_{1} }^{t_{2}}{\frac{dt}{(t_{s} -t)} }

Batch heating time

\ t_{B} =-\frac{WC_{p}}{UA}\ln \frac{t_{s}-t_{2}}{t_{s}-t_{1}}

For this example

\ WC_{p}=4.18\times 4545\times 10^{3} JK^{-1}

\ UA=285×27 WK^{-1}

\ t_{1}=15ºC,  t_{2}=80ºC,  t_{s}=130ºC  

\ t_{B} =-\frac{4.18\times 4545\times 10^{3}}{285\times 27.9} \ln \frac{130-80}{130-15} =1990s=\underline{\underline{33.2min} }

In this example the heat capacity of the vessel and the heat losses have been neglected for simplicity. They would increase the heating time by 10 to 20 per cent.