Page 313 - Process Equipment and Plant Design Principles and Practices by Subhabrata Ray Gargi Das

11.4 Fractionator 315

D) The bubble point and the dew point equations, respectively, are
8 9 8 9
> > > >
> n > > n >
1 X 1 X
< = < =
>P n K i;j L i;j 1 ¼ 0 and >P n V i;j = K i;j 1 ¼ 0 (11.26)
L i;j V i;j
> i¼1 > > i¼1 >
>
>
: ; : ;
i¼1 i¼1
The bubble and the dew point equations are used in some methods to determine the stage
temperatures.
E) Energy balance equations
The overall energy balance equation for a column with one feed and only bottom and top product is
FH F DH D BH B WH W þ Q Reboiler Q Condenser ¼ 0 (11.27)
Energy balance for the simple stage j (Fig. 11.13B)is

V jþ1 H V;jþ1 þ L j 1 H L;j 1 V j H V;j L j H L;j ¼ 0 (11.28)
The enthalpies are function of the stage temperature, pressure and composition

H V;j ¼ H V;j T j ; P j ; y i;j and H L;j ¼ H L;j T j ; P j ; x i;j (11.29)
Energy balance for the feed stage and the product draw off stage are to include the stream flow rates
shown in Fig. 11.13C and D, respectively, multiplied by the corresponding enthalpy terms.
The energy balances for the reboiler is
L N 1 H L;N 1 V N H V;N BH L;N þ Q Reboiler ¼ 0 (11.30)

and the same for a partial condenser with a vapour and a liquid product is
V 2 H L;2 L 1 H L;1 L top H L;1 V top H V;1 Q Condenser ¼ 0 (11.31)
The efficiency equations can also be written in terms of the tray flow and composition parameters
but are not presented here.
The foregoing set of equations would constitute the rigorous model of the distillation unit. Solution
of the system of equations with specified number of variables would provide the steady-state operating
conditions. The number of specified variables must be equal to the degrees of freedom of the system.
Intuitively one can say that the model would simulate the column operation when its configuration
(number of stages, feed stage and product draw stages) is specified along with the operating conditions
(say temperature and pressure on top and bottom stages, pressure on all other stages, temperature and
pressure of feed, feed composition, product flow rates) is known.
In order to use the model for design, the model is run with different input parameters selected as
design variables to generate solutions that respect the constraints on product compositions, limits on
top and bottom temperature, etc.
Several solution procedures for the rigorous model are available in literature and are implemented
in process simulators. However, these are not discussed here.