Page 120 - Chemical engineering design

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The application of a differential energy balance is illustrated in Example 3.13.
Example 3.15 CHEMICAL ENGINEERING
Differential energy balance
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In the batch preparation of an aqueous solution the water is first heated to 80 C in a jacketed,
Ž
agitated vessel; 1000 Imp. gal. (4545 kg) is heated from 15 C. If the jacket area is 300 ft 2
2
(27.9 m ) and the overall heat-transfer coefficient can be taken as 50 Btu ft 2 h 1 Ž 1
F
1
(285 W m 2 K ), estimate the heating time. Steam is supplied at 25 psig (2.7 bar).
Solution
The rate of heat transfer from the jacket to the water will be given by the following
expression (see Volume 1, Chapter 9):
dQ
D UA t s t a
dt
where dQ is the increment of heat transferred in the time interval dt,and
U D the overall-heat transfer coefficient,
t s D the steam-saturation temperature,
t D the water temperature.

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

where WC p is the heat capacity of the system.
Equating equations (a)and (b)
dt
WC p D UA t s t
dt
t B t 2

WC p dt
Integrating dt D
0 UA t 1 t s t
Batch heating time
WC p t s t 2
t B D ln
UA t s t 1
3
For this example WC p D 4.18 ð 4545 ð 10 JK 1
UA D 285 ð 27 WK 1
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Ž
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t 1 D 15 C,t 2 D 80 C,t s D 130 C
4.18 ð 4545 ð 10 3 130 80
t B D ln
285 ð 27.9 130 15
D 1990s D 33.2min
In this example the heat capacity of the vessel and the heat losses have been neglected
forsimplicity. Theywouldincrease the heatingtimeby10to20per cent.