DYNAMICS OF PACKED TOWER DISTILLATION GlULlO
TOMMASI'
A N D
P H I L I P
RICE
Syracuse Uniuersity, Syracuse, N . Y . 13210 The transient response of the enriching section of a packed distillation column to a step change in the reflux ratio was investigated. The solution in the case of a straight-line equilibrium curve was adapted to the real case ( w i t h a nonlinear equilibrium curve). This solution gives the liquid concentration, x, at any height of column, z, after the step change in the reflux ratio, in the form of a series of exponential functions of the time, f :
where the S,'S are negative constants which depend only on the parameters characteristic of the final steady state, but not on the magnitude of the step disturbance, and.the coefficients C,'s depend on z only. A series of exponential terms is a convenient form of the solution for computing transfer functions, which are used in the control theory. Since the experimental data available in the literature were insufficient, a laboratory scale column was built and new data were taken. The basic results of the linear solution seem to be valid also for nonlinear systems. The first time constant, st, is either independent, or a slowly varying function of column height. It does not depend on the magnitude of the step disturbance or the value of the initial concentration, but only on the parameters characteristic of the final steady state.
accurate prediction of transfer functions which describe the dynamic response of distillation columns to step disturbances in the reflux ratio and input concentration is important for the design of column control systems. Accurate transfer functions can also be used to predict the composition us. time behavior during the startup of distillation columns, during their approach to steady state after an unwanted disturbance, or after a resetting of the column variables. The theoretical prediction of the unsteady-state behavior of packed distillation columns requires the solution of the nonlinear differential equations which describe the dynamic response of tray or packed columns. The numerical solution of these equations is difficult and may require a large amount of computation time for each set of process conditions. An analytical solution can be obtained, but only for the simplifying assumptions necessary to linearize the equations. The use and applicability of the linear solution are discussed below. THE
literature Survey
Although considerable effort has been devoted to describing the dynamic behavior of a tray distillation column, very little similar work has been done for packed distillation columns. Mathematical models for the transient response of a packed distillation column in the reflux Present address, Montecatini Edison SPA, Milan, Italy
234
Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 2, 1970
ratio have been given by Cohen (1940), Marshall and Pigford (1948), Jawson and Smith (1954), and Heinke and Wagner (1966). Cohen (1940) and Marshall and Pigford (1948) considered the approach to steady state a t total reflux for the case of a straight-line equilibrium curve. Jawson and Smith (1954) reported a solution in the case of a straight-line equilibrium curve describing the transient behavior of a continuous packed distillation column but for an unusual set of boundary conditions. The most general analytical solution was presented by Heinke and Wagner (1966), who developed a solution by separation of variables and series expansions for a curved equilibrium line. Their solution is valid if the changes in column parameters throughout the column are small enough so that the section of the equilibrium line over which the concentration a t a given column height can be approximated by a straight line. In this paper a less general-but simpler-solution is used, obtained with the Laplace transform. This solution has the form of a series of exponential functions, and consequently is convenient for computing directly the transfer functions of a column response. The only experimental work on the transient response of packed columns was done by Heinke et al. (1965). They studied the response of the composition to a step change in the reflux ratio using benzene-trichloroethylene mixtures. These mixtures have a constant relative volatility over the entire composition range.
Object of Work
The object of this work was to accumulate further transient response data for step changes in reflux ratio in packed enriching columns, and to compare these data with the predictions of a linear theoretical model. The solutions of the partial differential equation for a linear equilibrium line were used to calculate the response of the column to step changes in reflux ratio for some ethanolwater mixtures. I n addition, some representative empirical second-order transfer functions were calculated from the experimental data and compared with those predicted by the linear theory. Mathematical Model
Linear Solution. If the molar heats of vaporization of the substances being distilled are approximately equal, and hence the liquid and vapor flow rates within the enriching section of a packed column are constant, if the holdup in the condenser can be neglected. and if the transfer of material between phases can be described as a product of an over-all gas mass transfer coefficient and the difference between the equilibrium vapor mole fraction, y*, and the actual vapor mole fraction, y, then the material balances on a differential section of column yield two simultaneous differential equations:
ax = L dx H- kGa (y* az
at
h
~~
= -
at
- y)
If the vapor-liquid equilibrium curve is linear, of the form
+
y* = xx 3 (10) the two partial differential equations are linear, and a solution can be found with the use of the Laplace transform for the initial and boundary conditions
0) Y ( l , 0)
=
?(;)
=
YC)
Y(0, 4)
=
760
x i l , 4)
=
y(1, 4)
X(L,
The two initial conditions given the initial concentration distribution in the column before the change in reflux ratio: the super bar indicates the initial steady state. The third condition describes a column which is fed by vapor of constant composition from the bottom. The last condition says that the liquid refluxed to the column is of the same composition as the vapor condensed a t the column head. The solution for the liquid mole fraction with the above boundary conditions is:
x ( ~ 0,)
=
x ( ~ m, )
-
C,
eslo
where the s,'s are solutions of the equation
tan ( b N1) = 2 x
vaY + kca (y* - y)
(11)
2b
- (1.
+ 1) - 5'
with
dz
If the behavior after a step change in the reflux ratio is studied, it is advantageous to change the time variable, t , to
t'
= t - (2,
-
2)
H L
(3)
since (2, - z ) H L is the time required for the disturbance in the liquid stream t o reach an arbitrary column height, 2.
Introducing the dimensionless variables
(4) (5) Equations 1 and 2 can be written in the dimensionless form
The major steps of the derivation of the above solution, the constants C,(c), and the other symbols are given in the appendix. This solution is a series of exponential decay terms in which the coefficient of each term is a function of the column position only, while the time constants are independent of column height. Moreover, the time constants are predicted to depend only on parameters characteristic of the final steady state of the column. I n particular, they are functions of R , Nl, and (Y only, and they are independent of R , a5 well as 3 and 3 . Use of Linear Solution. T o use effectively the solution obtained for the linear partial differential equations, a choice of the parameters (Y and 3, which describe the equilibrium line, must be made. I n general, a prudent choice of all the parameters which occur in the equations and their boundary conditions could be made. These are a , 13, N1, R , R , and z0. However, if the initial and final operating lines for the column and the bottom concentration are to remain unchanged, the parameters left are 0 ,p, N l , and N,. For these calculations, N 1 and N,were chosen equal to the experimental values, and a and i3 selected so that the bottom and overhead concentrations were equal t o the observed experimental values in both the initial and final steady states. This choice assured that the calculated concentration profile would pass through the experimental points a t the top and a t the bottom of the column. at
ml,
where R is the reflux ratio. here defined as
R = -L
V
and N, is the total number of transfer units:
(9)
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 2, 1970 235
both t = 0 and t = E , but could be different from the experimentally observed profile at the intermediate points of the column. The number of transfer units for the case of a linear equilibrium curve is
N1 =
1
w
CONDENSERS
Y*O - Yo In yl* - y1
I_____
where y* = ZX
+5
Solving Equation 16 for 3 a t both the initial and final steady state, the two required conditions are obtained:
3
= fl(Z)
S I LVEREO VACUUM JACKET
=
REDISTRIBUTOR -SAMPLE 'TUBES
xl(l
-
R)
x1 - Yo +~ ; 1if x ~ R, =
TIMER SWITCH HEATING MANTEL
'"B
vA R iAc%
Figure 1. Packed distillation column
First a can be found by solving the equation
fl(4 - f d Z )
=
0
(18)
and then p calculated, by substituting the value of 0 in either of Equations 17. With N and P determined this way, the time constants, sL', and the coefficients, C,, in Equation 12 can be determined from Equation 13. Then, the course of the concentration change in response to a change in reflux ratio can be predicted. Usually, two time constants are sufficient to give an adequate description of the column behavior and consequently only the first two terms of Equation 12 were used. Experimental Work
Description of Equipment. A sketch of the packed column is shown in Figure 1. This column is made of four glass sections of 60-mm internal diameter, connected to each other, to the reboiler, and to the column head, by five 70160 ground, standard-taper joints. The column pipe is 172 cm long, but the effective packing length is only 120 cm. Each section has a jacket evacuated t o a pressure of 3 x mm of Hg and silvered except for a narrow observation window front and back extending the full effective length. The packing, consisting of 4-mm ceramic Berl saddles, is supported in each section by an inverted cone having 3.5-mm holes punched in it about 5 mm from the outside edge. The function of these cones is also to redistribute the liquid from the wall to the center of the bed, and to prevent channeling. This redistribution occurs every five column diameters, and the ratio of column diameter to Berl saddle size is 15. With this 236
Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 2, 1970
ratio and redistribution, the fraction of liquid on the wall should be small enough to ensure an efficient performance of the column (Jameson, 196'7). Under each redistributor there is an 8-mm sample tube which collects some of the liquid and conveys it out to a three-way stopcock. Here the liquid can be sent back into the column, or part of it withdrawn as a sample. Along the length of the distillation column there are three withdrawal and return positions. The connections between the four sections of column were well insulated with asbestos. t o minimize any heat losses from the column. Under normal operating conditions, about 3 liters per hour were vaporized in the reboiler and one sample of 3.5 cc was taken every 15 minutes a t each of the three sample ports within the column. This removed 42 cc per hour, or less than 1.5'; of the total flow rate. Samples of much less than 3.5 cc were taken when the index of refraction could be used as a measure of the concentration, but generally the density was determined by a 3-cc pycnometer. At the top of the distillation column a distilling head with automatic reflux control was used. With this, the condensate could be divided between the reflux and the product by a swinging funnel with a soft iron core. This was moved by an electromagnet connected to an electric timer. Fastened to the bottom of the column is the reboiler, a 50-liter flask with a side inlet tube and a thermometer well heated by an electric heating mantle. The total vapor flow rate in the column was regulated by regulating the power to the heating mantle with a Variac. The overhead product that was withdrawn was always sent immediately to the boiler through a 10-mm. recycle
pipe. At a constant reflux ratio, the boiler concentration remained constant with time. I t did change slightly if the reflux ratio was changed, because then the average concentration of the liquid within the column was changed. The volume of the boiler was chosen large enough to keep all these changes in the boiler concentration very small-no more than 1 mole SC in the present experiments. This corresponds t o an even smaller change of the input concentration, yo. Two more sample ports were set along the recycle lineone for taking samples of the condensate from the top of the column, and one, just a t the flask inlet, for taking samples of the liquid in the boiler. These samples were sucked from the boiler with a large syringe. Experimental Procedure. A binary mixture of ethanol and water was chosen. I t s atmospheric-pressure liquidvapclr equilibrium curve (Carey and Lewis, 1932) is shown in Figure 2. The molar heats of vaporization of ethanol and water are within 3% of each other; ensuring that the liquid and vapor flow rates are constants throughout the column. The steady-state characteristics of the packed column were investigated first. The boiler was filled with 20 kg (859 gram-moles) of a mixture of 18.9541 of ethanol and 81.05‘~ of water, by mole. The column was brought to an equilibrium with a total vapor flow rate of 78.1 grammoles per hour and a reflux ratio, R = 0.503. After 10 hours, the concentrations were measured, and from these, by a graphical method (Treybal, 1968), came the number of transfer units and the height of a transfer unit for each section: Ethanol, Mole r( Boiler Section Section Section Section
I I1 I11 IV Concentration at top of
18.44 48.8 58.0 66.1 78.3
NTC‘
H.T V , Cm
...
...
5.1 5.2 5.2 4.5
5.9 5.8 5.8 6.7
each section
The standard error in the concentration determination is estimated below. If the error from the concentration determination is taken into account, the separation efficiency of each section can be seen to be equal. Therefore, a total number of transfer units equal to 20 was taken, in all the calculations, and the height of a transfer unit was considered constant, equal to 6.0 cm, throughout the column. These values were confirmed by all the successive experiments. A parameter which does not appear in the solution of the steady-state equations, but is important in the unsteady-state problem, is the liquid holdup. An independent measurement of the liquid holdup was not made. However, measuring the concentration in the boiler during the steady-state operation and writing a mass balance give the liquid holdup with sufficient accuracy, if it is assumed to be constant throughout the column:
9
Figure 2. Ethanol-water liquid-vapor equilibrium curve
where n6 are the gram-moles of liquid in the boiler, nb* the gram-moles of vapor in the boiler, n, the gram-moles of liquid and vapor in the ccmdenser and in the recycle pipe, and nob the total number of gram-moles of liquid originally put into the boiler. The vapor holdup, h , was calculated from the total void volume of the column-the void fraction of the dry Berl saddle packing being equal to O.j’i+Since the liquid holdup is a strong function of the liquid flow rate, this calculation must be repeated for every experiment. As a first approximation, if we neglect n,, nb’, and hz, compared t o n06, Equation 19 becomes
H
=
const. (xob-
xb)
Therefore, for H to be accurate to two significant figures, the difference, X o b - IO,must be accurate to two significant figures. This requires a fairly high accuracy in the determination of the values of x,,t and xb, which are very close. This was obtained by taking always a large number of samples from the boiler-at least 9-and calculating the average of all the concentrations. A fairly high accuracy in the experimental value of H is important because the time constants are proportional to l i o ~ and , hence to L H because, for R ( h H)
