- Email: [email protected]
An output multiplicity in binary distillation: Experimental verification
Computers chem. Engng Vol.20, Suppl..pp. $835-$840, 1996
Pergamon
S0098-1354(96)00147-0
Copyright© 1996 ElsevierScienceLtd Printed in Great Britain.All rightsreserved 0098-1354/96 $15.00+0.00
An Output Multiplicity in Binary Distillation: Experimental Verification A. Koggersb¢l, T.R. Andersen, J. Bagterp and S.B. Jcrgensen* Department of Chemical Engineering, Technical University of Denmark, DK-2800 Lyngby, Denmark. A b s t r a c t : Earlier it was believed that distillation multiplicity only could occur as a result of separating a multicomponent mixture. It has, however, been proven theoretically using very simple distillation models that measuring process flows in units of mass or volume (as is always the case in practical applications) instead of moles may give rise to a bifurcation in the mapping between volumetric reflux flowrate and distillate purity from the case of a single steady state to that of two stable and one unstable steady states being possible for a specific set of actuator settings. This paper verifies the existence of multiple steady states on a semi-industrial scale distillation column separating a mixture of methanol and isopropanol. Simulation studies show that the column will exhibit multiplicity in central parts of the operating window. An experiment is designed by which the unstable branch is mapped and the singular points identified under stable conditions. The results show that distillation column multiplicity is in fact a phenomenon of practical relevance which may lead to operational problems even for binary distillation columns,
1
Introduction
As a side product of comparing gains of models estimated from experimental data with gains of simulation models derived from first engineering principles Nielsen (1990) discovered for the specific distillation process she was analysing, that for certain operating conditions a plot of molar reflux rate (L) versus mass or volumetric reflux flowrate (Lw) could go through a singular point (of infinite gradient) at which the gradient would change sign. For values of L lower than at the singular point the slope O L / O L w would be negative and for higher values it would be positive. The implication of this was that for specific values of L~, there would be two feasible values for L. General conditions for the existence of multiple steady states for binary columns were later given by Jacobsen and Skogestad (1991). They analysed simple models and found that multiplicity could arise from the conversion between molar flowrates and volumetric or mass flowrates, from the conversion of reboiler heat duty into molar boil-up, and from the influence of an energy balance (per tray) upon the internal flowrates L and V. They found multiplicity to be related to the existence of singular points in these conversions. The case studied here is a semi-industrial scale distillation column separating a mixture of methanol and isopropanol. Simulations show that the column will exhibit multiple steady states in central parts of the operating window. An experiment is designed by which the L ~ - x o curve can be mapped and the singular points identified under stable conditions. The paper is organized as follows. First section briefly lists the results of Nielsen (1990) and Jacobsen and Skogestad (1991). Then follows an analysis of the operating region for occurrences of multiplicity, and an experiment is designed and performed for practical verification of the simulation results.
2
Earlier results
During her study of different distillation-column model structures suitable for parameter estimation ajtd control design Nielsen (1990) observed that for certain operating conditions the signs of those model gains related to reflux flowrate were dependent on whether the flowrate was measured in units of moles or volume. Figure 1 shows the relation between the two flowrate measurements L (mol/sec) and L~ (litre/min) and between L~o and top purity =o for a specific vicinity of the operation region. This shows that for a given volumetric reflux flowrate there may exist more than one steady state value for the distillate purity. This has never been observed when using flowrates on molar basis. In the context of model parameter estimation this led to the conclusion that handling flow variables in terms of moles should be preferred for identification and control purposes to avoid problems with the sudden change of sign of certain gains. $835
S836
European Symposiumon Computer Aided Process Engineering--& Part B Reflux 6.11
rote
(mole/sec)
Top
purity
0.96
6.10
(
6.09
(molefraction)
J
0.95
6.08 6.07
0.94 6.05 6.05
0.95
6,04 6.03 15.20
I 5.25 Reflux rote
15.50 (I/min)
I 5.55
0.92 . . . . 15.20
, . . . . , . . . . 15.25 15.50 R e f l u x rate ( I / m i n )
15.2,5
Figure 1: Molar reflux flowrate and top purity versus volumetric reflux flowrate. Reproduced from Nielsen (1990). Obviously the implications of this observation reaches a lot further considering that molar flowrates of liquids are never accessible for direct manipulation. A column operated in the (L~, V)-configuration under closed loop SISO product purity control would for instance become unstable if the distillate purity setpoint was changed from one side of the singular point to the other such that the process gain changed sign. Jacobsen and Skogestad (1991), however, showed that multiplicity is related to a much more fundamental and until then denied property (Doherty and Perkins, 1982) of binary distillation columns: Multiplicity occurs between singular points at which a pole of the system crosses the imaginary azis. This means that not only do gains change sign, but the entire column actually becomes open loop unstable along the part of the L~ - XD curve which has a negative gradient. Using the negative gradient as an indicator for instability they derived necessary and sufficient conditions for instability for the (Lw,V) and (Dw,V)-configurations. They found that for binary mixtures for which the molecular weight of the light component is smaller than the molecular weight of the heavy component (i.e. M1 < M2), a column operated in the (Lw,V)-configuration will be unstable if the following condition is satisfied. ( OXD ~ Ms XD + L \ on ] v > M2 - M---~
(1)
z o is the molefraction of the light component in the distillate stream. Jacobsen and Skogestad (1991) further found that if the same column is operated in the (Dw,V)-configuration it will remain stable under all operating conditions. To apply equation 1 to a specific distillation column one needs to know the value of the gain (OXD/OL) at constant molar boil-up. This will usually be difficult to obtain from stored process data since operation at constant molar boil-up is hardly ever a goal in process industry. If, however, one manages to find data for two steady state operating points with equal molar boil-up the formula is easily used to check for potential instability of the (Lw,V)-configuration, i.e. the condition is satisfied or the left hand side is only slightly smaller than the right hand side. The right hand side of equation 1 equals 2.14 for the methanol-isopropanol system separated in the column to be investigated here. Process data at constant molar boil-up is not readily available but simulation data at medium pressure, medium to high boil-up rate, and distillate purity varying from 0.95 to 0.98 is easily generated and used to evaluate the left hand side of equation 1. See table 1. These results show that instability of the (Lto,V)-configuration should quite easily be established by operating the column under standard conditions with top purity lower than 97 - 98 mole-% methanol. Apart from being difficult to apply in practise the use of equation 1 is also limited by the fact that it only predicts instability and not multiplicity. Jacobsen and Skogestad (1991) showed that instability need not be accompanied by multiple steady states, since a column may have exclusively unstable operating points in certain areas of the operating region. The opposite, however, is always the case: Lw-multiplicity implies one unstable solution for each set of inputs.
3
E x p e r i m e n t a l verification
The preceding subsection leads to the following expectations. (1) Operated in the (L~,V)-configuration the column is expected to be unstable at numerous operating points and with any luck this instability will be a result of multiple steady states. (2) In the (Dw,V)-configuration it is stable at all operating points
European Symposium on Computer Aided Process Engineering--6. Part B
$837
B:
A:
Figure 2: The span (W) of the unstable branch in terms of volumetric reflux flowrate [m3/h] plotted as a function of boil-up (V) [kmole/h], column pressure (P) [kPa], feed flowrate (F) [m3/h], and feed composition (z) [mole/mole]. Subplot A: F = 5.95 kmole/h = 0.342 m3/h and z -- 0.5. Subplot B: V = 24.75 kmole/h and P = 60 kPa. and the (Lw,V) multiplicity can therefore be mapped under stable conditions by operating in the (D~,V)configuration. Based on this an experiment is designed. 3.1
Experiment
design
The goal of validating experimentally the existence of steady state multiplicity is approached through a simulation study which reveals the dissemination of multiplicity in the entire operation region of the column. From this study suitable operating conditions for the verification experiment are decided upon, taking into account operational restrictions and resemblance to realistic ordinary operating conditions. A measure of how distinct multiplicity is at a certain point in the operation region is introduced: W = Distance between the singular points of the L-Lw transformation, measured in terms of volumetric flowrate. Or if only one singular point exists in practice, W = Width of the unstable branch of the curve of molar versus volumetric reflux flowrate for cases with multiplicity, measured in terms of volumetric flowrate. With a comprehensive simulation effort the measure W is computed for the entire operating region of the distillation plant (:Iergensen, 1992). Figure 2 shows W as a function of boil-up, column top pressure, V [kmole/h] 34.20
24.75
L [kmole/h] 31.7552 31.8006 31.8208 22.4428 22.4885 22.5086 22.5421 22.6048 22.6724
xD 0.95060 0.96562 0.97075 0.94983 0.96429 0.96910 0.97470 0.98014 0.98286
OXD/BL [h/kmol~ 0.3308 0.2540
LHS ofeq. 1 (unstable i f > 2 . 1 4 ) 11.4564 9.0417
0.3164 0.2393 0.1672 0.0868 0.0402
8.0510 6.3459 4.7317 2.9305 1.8897
Table 1: Evaluation of the (L~,V) instability criterion using simulation data from central parts of the operating region. Column top tray pressure is 92.2 kPa, and the feed is 0.240 ma/h of a 50:50 [mole/mole] mixture of methanol and isopropanol.
$838
European Symposium on Computer Aided Process Engineering---& Part B Top p u r i t y 1.oo
Limit
Molar 2~.o0
D = 0
reflux Limit
D =
0
......................
/
0.90
24.00
\
_
~
~
0.80
23.00
0.70
\
0.60
0.50
22.00 . . . . . . . . . . . . . . . . . . . . . . . . .
Limit
B =
Limit
0
B = 0
21 .O0
0.40 0.30 , ....,,,,......,,,,......,,,j.........,,,,,.....i o.80
o.9o
1 .oo
1.1 o
Volumetric
"f.2o
20.00 0.86 . . . . . . 6:~6 . . . . . .
1.3o
reflux
1:66 . . . . . .
Volumetric
i:;6 ......
~:~6 . . . . . .
;:~
reflux
Figure 3: Simulation of the (Lw,xD) trajectory proposed for experimental verification feed flowrate, and feed composition. It is seen that only at very low boil-up rates does W become zero, indicating that only unique stable solutions occur. The weeping limit at the lowest pressure attainable, which is approximately 60 kPa, is V --- 12 kmole/h (100% methanol). At higher pressure such as 100 kPa the limit has increased to some 17 kmole/h. The design value for the feed flowrate is F = 4 litre/min 0.240 m3/h with a 50:50 feed mixture. Subplot A is calculated for a higher feed flowrate (F = 0.342 m3/h) and subplot B shows that for the design feed flowrate and lower flowrates the value of W will be even higher than depicted in subplot A. So, with the feasible operating region roughly limited by 12 < V < 40 kmole/h and 60 < P < 110 kPa it is clear that multiplicity in the (Lw,V)-configuration can occur in the entire operating range of the column. Since experiment design apparently is not as critical as first anticipated with respect to locating unstable regions, the two primary objectives of the design becomes (a) to operate in a region where W has a substantial value such that the unstable branch will be easy to find, and (b) to operate in a well known region where operation limits as well as more complex column behaviour will not interfere with the experiment. The following operating conditions are found to be suitable. P V
= =
87.5 24.75
kPa ]F kmole/h z
= =
0.180 0.50
m~/h I mole/mole
For this case only the unstable and the upper stable branches in the L~-xo plot exist. This means that as top product flowrate is decreased from Dw = F (B = 0 and xD = z) the process follows the unstable branch towards lower Lw and higher XD. At the (single) singular point the curve turns onto the stable branch along which XD is close to but still approaching 1.0 as Lw is increased. Figure 3 shows the simulated relation between volumetric and molar reflux flowrate and top purity. In the experiment it will be the intension to follow this trajectory from D~ ~, 0 to ZD ~, 0.9 by operation in the (D~,V)-configuration, preferably by controlling XD with Dw in closed loop to speed up transients. 3.2
Experimental
equipment
Figure 4 shows the outline of the plant. The column has 19 sieve trays, a thermosyphon reboiler, a total condenser and a reflux drum. It separates a mixture of methanol and isopropanol. On trays 1, 5, 10, 15 , and 19 PT100 temperature sensors are located in the liquid hold-up. In combination with pressure measurements these are used for concentration estimates. All flows in and out of the system and the reflux flowrate are measured on volumetric basis. Feed, bottom product, and distillate can be sampled manually for gas chromatograph analysis. The heat pump has an expansion valve (Exp.valve) which throttles high pressure liquid freon to a lower pressure (PL) suitable for evaporation in the condenser. After the condenser there is a control valve ( c v g ) by which the freon vapour flowrate can be manipulated. After superheating the vapour the compressor elevates the pressure to a high pressure (PH) suitable for condensation in the reboiler. A small part of the condensation takes place in a secondary condenser which by a cooling water circuit is connected to a set of air-fan coolers. The cooling rate can be manipulated by the control valve CV8, thus controlling PH. Through a storage tank (Rec) and the super heater heat exchanger the freon circuit is closed at the expansion valve. The basic control configuration for the plant is as follows: The high pressure PH is controlled by CV8, the low pressure PL is controlled by CV9, accumulator level is controlled by L, and reboiler level is controlled by B. QB and Qc are then manipulated through the setpoints to PH and PL since these indirectly affect
European Symposium on Computer Aided Process Engineering---6. Part B
~E
CV9Pi
$839
xp~ 1 v a l v e
Super heating before compression
~
AIR COOLERS
Sec ndary condenser
B
Figure 4: Schematic flowsheet of the plant. the temperature gradients of the reboiler and the condenser. The concentration profile is manipulated in the high gain direction by D, and in the low gain direction by a combination of PH and PL by which column pressure is also controlled. Thus, this configuration resembles the standard (D,V)-configuration which is openloop stable as shown previously. 3.3
Experiment
and Results
It was attempted to operate the column at constant molar vapour flowrate. The volumetric vapour flowrate out of the column top was estimated from measurements of reflux and distillate flowrates and the rate of change of accumulator level. From temperature and pressure on the top tray the liquid composition here was estimated using Henry's law and Antoine equations. Based on the setpoint to the molar vapour flowrate and the top tray composition estimate, the desired volumetric vapour flowrate was calculated and passed on as setpoint to a 6 by 2 MIMO controller which controlled volumetric vapour fiowrate and column pressure by manipulation of the heat pump high and low pressures. The column was taken through 10 steady state operating points, at which distillate, bottom product, and feed flow were sampled for later analysis on a gas chromatograph. The process was considered at steady state when all concentration estimates had been stationary for one hour.
4
Discussion
Figure 5 shows the data points obtained in the experiment, surrounded by a 95% rectangular confidence band estimated from measurement standard variations. The data point at (0.95 , 0.994) which apparently has a lower distillate composition than neighboring points was obtained during a shift between feed tanks where too low an inlet concentration was fed to the column. The existence of multiple steady states is clearly verified. With 95% confidence bands there clearly exist two separate branches in the plot of distillate purity versus volumetric reflux, such that for all values of the volumetric reflux there are two possible values for the distillate purity. This multiplicity result implies that the column would be unstable when operating at points on the a negative slope in the xD vs. L,~ plot in figure 5 if the other operating conditions indeed are constant. These aspects are investigated in figure 5-B and C. The plot of molar versus volumetric reflux flowrate at constant molar boil-up rate should indicate the stability characteristics for the (L,V)-configuration. A positive slope indicates a stable process while a negative slope indicates an unstable process. On the A-branch boil-up is constant and the reflux rates indicate a stable process. On the B-branch, however, boil-up rate could only be held within 20£ deviation from the initial value. This seemingly small deviation is unfortunately of the same magnitude as the measured changes in the molar reflux flowrate on the B-branch. Subtraction of the change in boil-up rate from the change in molar reflux rate results in zero slope of the B-branch in the molar reflux plot. With any reasonable uncertainty level it is therefore not possible to conclude whether operating points on the B-branch are stable or unstable.
5
Conclusions
The existence of steady state multiplicity has been verified experimentally, and that for our example column which is a well designed process operated at or near the design specifications and separating a rather simple
$840
European Symposiumon Computer Aided Process Engin~ringm-6. Part B
Dislillote purity (molefroction)
Molor reflux (kmole/h) 250
i ol
Molor boil-up 2EoA
k
1.00
rote
(kmole/h)
, i 25.5 -
24.5
•
i
B
0.99 ,
i
24.0
25.0 -
23.5
24.5
0.g8 -
A
0.97
23.0
24.0 -
o.g6
,l,,,,,,,l,
0.93
,,,,,,,,I,,,,,,,,II,,I,,,,,,I,,,,,,,,,I,,,,,,H,
0.95 0.96 0.~)7 0.98 0.99 Volumetrix reflux (m i/h)
0,94
0.92 0.9,3 0.94 0.95 0.96 30.97 0,98 0.99
volumetrixreflux (m/h)
0.92 0.93 0.g4 0.95 0.96 ~).97 0.98 0.99
Volumetrixreftux (m/h)
Figure 5: Experimental results. Distillate purity measured by off-line GC analysis; molar reflux flowrate calculated from measured volumetric reflux and distillate purity; and molar boil-up rate calculated from measured volumetric boil-up rate. All plotted versus volumetric reflux flowrate. Points represents experimentally obtained datapoints. Curves represents 95% rectangular confidence bands estimated from process variations and GC-analysis variations. binary mixture it has not been difficult to obtain conditions for multiplicity. It was not possible to verify the expected instability for operation in (L,V)-configuration on the lower (or intermediate) branch of the L,o - ZD plot, partly due to difficulties with maintaining a constant molar boil-up rate throughout the experiment. This result could indicate that for practical operation of distillation columns it may be possible to experience steady state multiplicity without instability if other conditions such as boil-up rate are not carefully controlled. The most significant condition for multiplicity is that the molecular weight of the heavy component (Ms) must be significantly larger than the molecular weight of the light component (M1). In the investigated case the ratio M2/M1 is approximately 2. In many industrial distillation processes, though, molecular weights differ only slightly and the ratio thus approaches 1. For these processes steady state multiplicity will not occur. There remains, however, a large number of cases where the ratio M~/M1 is significantly larger than 1, and which therefore may exhibit steady state multiplicity and possibly instability. For these processes the results found here definitely suggest that the discoveries of Nielsen (1990) and the analyses by Jacobsen and Skogestad (1991) are of practical significance.
6
References
Doherty, M.F.; Perkins, J.D. (1982); 'On the Dynamics of Distillation Processes: IV. Uniqueness and Stability of the SteadyState in Homogeneous Continuous Distillation'; Chem. Eng. Sci., vol.37(4}, p.381. Jacobsen, E.W.; Skogestad, S. (1991); 'Multible Steady States in Ideal Two-Product Distillation'; AIChE Journal, vol. 37(4), pp.499-511. J~rgensen, L. (1992); 'Dynamic Simulation - and Experimental Verification of Multiplicity in Distillation'; M.Sc. Thesis, Institut for Kemiteknik, DTH, Building 229, DK-2800 Lyngby. Nielsen, C.S. (1990); 'Multivariable Identification and Control of an Experimental Distillation Column with Heat pump'; Ph.D. Thesis, Institut for Kemiteknik, DTH, Building 229, DK-2800 Lyngby.
Pergamon
S0098-1354(96)00147-0
Copyright© 1996 ElsevierScienceLtd Printed in Great Britain.All rightsreserved 0098-1354/96 $15.00+0.00
An Output Multiplicity in Binary Distillation: Experimental Verification A. Koggersb¢l, T.R. Andersen, J. Bagterp and S.B. Jcrgensen* Department of Chemical Engineering, Technical University of Denmark, DK-2800 Lyngby, Denmark. A b s t r a c t : Earlier it was believed that distillation multiplicity only could occur as a result of separating a multicomponent mixture. It has, however, been proven theoretically using very simple distillation models that measuring process flows in units of mass or volume (as is always the case in practical applications) instead of moles may give rise to a bifurcation in the mapping between volumetric reflux flowrate and distillate purity from the case of a single steady state to that of two stable and one unstable steady states being possible for a specific set of actuator settings. This paper verifies the existence of multiple steady states on a semi-industrial scale distillation column separating a mixture of methanol and isopropanol. Simulation studies show that the column will exhibit multiplicity in central parts of the operating window. An experiment is designed by which the unstable branch is mapped and the singular points identified under stable conditions. The results show that distillation column multiplicity is in fact a phenomenon of practical relevance which may lead to operational problems even for binary distillation columns,
1
Introduction
As a side product of comparing gains of models estimated from experimental data with gains of simulation models derived from first engineering principles Nielsen (1990) discovered for the specific distillation process she was analysing, that for certain operating conditions a plot of molar reflux rate (L) versus mass or volumetric reflux flowrate (Lw) could go through a singular point (of infinite gradient) at which the gradient would change sign. For values of L lower than at the singular point the slope O L / O L w would be negative and for higher values it would be positive. The implication of this was that for specific values of L~, there would be two feasible values for L. General conditions for the existence of multiple steady states for binary columns were later given by Jacobsen and Skogestad (1991). They analysed simple models and found that multiplicity could arise from the conversion between molar flowrates and volumetric or mass flowrates, from the conversion of reboiler heat duty into molar boil-up, and from the influence of an energy balance (per tray) upon the internal flowrates L and V. They found multiplicity to be related to the existence of singular points in these conversions. The case studied here is a semi-industrial scale distillation column separating a mixture of methanol and isopropanol. Simulations show that the column will exhibit multiple steady states in central parts of the operating window. An experiment is designed by which the L ~ - x o curve can be mapped and the singular points identified under stable conditions. The paper is organized as follows. First section briefly lists the results of Nielsen (1990) and Jacobsen and Skogestad (1991). Then follows an analysis of the operating region for occurrences of multiplicity, and an experiment is designed and performed for practical verification of the simulation results.
2
Earlier results
During her study of different distillation-column model structures suitable for parameter estimation ajtd control design Nielsen (1990) observed that for certain operating conditions the signs of those model gains related to reflux flowrate were dependent on whether the flowrate was measured in units of moles or volume. Figure 1 shows the relation between the two flowrate measurements L (mol/sec) and L~ (litre/min) and between L~o and top purity =o for a specific vicinity of the operation region. This shows that for a given volumetric reflux flowrate there may exist more than one steady state value for the distillate purity. This has never been observed when using flowrates on molar basis. In the context of model parameter estimation this led to the conclusion that handling flow variables in terms of moles should be preferred for identification and control purposes to avoid problems with the sudden change of sign of certain gains. $835
S836
European Symposiumon Computer Aided Process Engineering--& Part B Reflux 6.11
rote
(mole/sec)
Top
purity
0.96
6.10
(
6.09
(molefraction)
J
0.95
6.08 6.07
0.94 6.05 6.05
0.95
6,04 6.03 15.20
I 5.25 Reflux rote
15.50 (I/min)
I 5.55
0.92 . . . . 15.20
, . . . . , . . . . 15.25 15.50 R e f l u x rate ( I / m i n )
15.2,5
Figure 1: Molar reflux flowrate and top purity versus volumetric reflux flowrate. Reproduced from Nielsen (1990). Obviously the implications of this observation reaches a lot further considering that molar flowrates of liquids are never accessible for direct manipulation. A column operated in the (L~, V)-configuration under closed loop SISO product purity control would for instance become unstable if the distillate purity setpoint was changed from one side of the singular point to the other such that the process gain changed sign. Jacobsen and Skogestad (1991), however, showed that multiplicity is related to a much more fundamental and until then denied property (Doherty and Perkins, 1982) of binary distillation columns: Multiplicity occurs between singular points at which a pole of the system crosses the imaginary azis. This means that not only do gains change sign, but the entire column actually becomes open loop unstable along the part of the L~ - XD curve which has a negative gradient. Using the negative gradient as an indicator for instability they derived necessary and sufficient conditions for instability for the (Lw,V) and (Dw,V)-configurations. They found that for binary mixtures for which the molecular weight of the light component is smaller than the molecular weight of the heavy component (i.e. M1 < M2), a column operated in the (Lw,V)-configuration will be unstable if the following condition is satisfied. ( OXD ~ Ms XD + L \ on ] v > M2 - M---~
(1)
z o is the molefraction of the light component in the distillate stream. Jacobsen and Skogestad (1991) further found that if the same column is operated in the (Dw,V)-configuration it will remain stable under all operating conditions. To apply equation 1 to a specific distillation column one needs to know the value of the gain (OXD/OL) at constant molar boil-up. This will usually be difficult to obtain from stored process data since operation at constant molar boil-up is hardly ever a goal in process industry. If, however, one manages to find data for two steady state operating points with equal molar boil-up the formula is easily used to check for potential instability of the (Lw,V)-configuration, i.e. the condition is satisfied or the left hand side is only slightly smaller than the right hand side. The right hand side of equation 1 equals 2.14 for the methanol-isopropanol system separated in the column to be investigated here. Process data at constant molar boil-up is not readily available but simulation data at medium pressure, medium to high boil-up rate, and distillate purity varying from 0.95 to 0.98 is easily generated and used to evaluate the left hand side of equation 1. See table 1. These results show that instability of the (Lto,V)-configuration should quite easily be established by operating the column under standard conditions with top purity lower than 97 - 98 mole-% methanol. Apart from being difficult to apply in practise the use of equation 1 is also limited by the fact that it only predicts instability and not multiplicity. Jacobsen and Skogestad (1991) showed that instability need not be accompanied by multiple steady states, since a column may have exclusively unstable operating points in certain areas of the operating region. The opposite, however, is always the case: Lw-multiplicity implies one unstable solution for each set of inputs.
3
E x p e r i m e n t a l verification
The preceding subsection leads to the following expectations. (1) Operated in the (L~,V)-configuration the column is expected to be unstable at numerous operating points and with any luck this instability will be a result of multiple steady states. (2) In the (Dw,V)-configuration it is stable at all operating points
European Symposium on Computer Aided Process Engineering--6. Part B
$837
B:
A:
Figure 2: The span (W) of the unstable branch in terms of volumetric reflux flowrate [m3/h] plotted as a function of boil-up (V) [kmole/h], column pressure (P) [kPa], feed flowrate (F) [m3/h], and feed composition (z) [mole/mole]. Subplot A: F = 5.95 kmole/h = 0.342 m3/h and z -- 0.5. Subplot B: V = 24.75 kmole/h and P = 60 kPa. and the (Lw,V) multiplicity can therefore be mapped under stable conditions by operating in the (D~,V)configuration. Based on this an experiment is designed. 3.1
Experiment
design
The goal of validating experimentally the existence of steady state multiplicity is approached through a simulation study which reveals the dissemination of multiplicity in the entire operation region of the column. From this study suitable operating conditions for the verification experiment are decided upon, taking into account operational restrictions and resemblance to realistic ordinary operating conditions. A measure of how distinct multiplicity is at a certain point in the operation region is introduced: W = Distance between the singular points of the L-Lw transformation, measured in terms of volumetric flowrate. Or if only one singular point exists in practice, W = Width of the unstable branch of the curve of molar versus volumetric reflux flowrate for cases with multiplicity, measured in terms of volumetric flowrate. With a comprehensive simulation effort the measure W is computed for the entire operating region of the distillation plant (:Iergensen, 1992). Figure 2 shows W as a function of boil-up, column top pressure, V [kmole/h] 34.20
24.75
L [kmole/h] 31.7552 31.8006 31.8208 22.4428 22.4885 22.5086 22.5421 22.6048 22.6724
xD 0.95060 0.96562 0.97075 0.94983 0.96429 0.96910 0.97470 0.98014 0.98286
OXD/BL [h/kmol~ 0.3308 0.2540
LHS ofeq. 1 (unstable i f > 2 . 1 4 ) 11.4564 9.0417
0.3164 0.2393 0.1672 0.0868 0.0402
8.0510 6.3459 4.7317 2.9305 1.8897
Table 1: Evaluation of the (L~,V) instability criterion using simulation data from central parts of the operating region. Column top tray pressure is 92.2 kPa, and the feed is 0.240 ma/h of a 50:50 [mole/mole] mixture of methanol and isopropanol.
$838
European Symposium on Computer Aided Process Engineering---& Part B Top p u r i t y 1.oo
Limit
Molar 2~.o0
D = 0
reflux Limit
D =
0
......................
/
0.90
24.00
\
_
~
~
0.80
23.00
0.70
\
0.60
0.50
22.00 . . . . . . . . . . . . . . . . . . . . . . . . .
Limit
B =
Limit
0
B = 0
21 .O0
0.40 0.30 , ....,,,,......,,,,......,,,j.........,,,,,.....i o.80
o.9o
1 .oo
1.1 o
Volumetric
"f.2o
20.00 0.86 . . . . . . 6:~6 . . . . . .
1.3o
reflux
1:66 . . . . . .
Volumetric
i:;6 ......
~:~6 . . . . . .
;:~
reflux
Figure 3: Simulation of the (Lw,xD) trajectory proposed for experimental verification feed flowrate, and feed composition. It is seen that only at very low boil-up rates does W become zero, indicating that only unique stable solutions occur. The weeping limit at the lowest pressure attainable, which is approximately 60 kPa, is V --- 12 kmole/h (100% methanol). At higher pressure such as 100 kPa the limit has increased to some 17 kmole/h. The design value for the feed flowrate is F = 4 litre/min 0.240 m3/h with a 50:50 feed mixture. Subplot A is calculated for a higher feed flowrate (F = 0.342 m3/h) and subplot B shows that for the design feed flowrate and lower flowrates the value of W will be even higher than depicted in subplot A. So, with the feasible operating region roughly limited by 12 < V < 40 kmole/h and 60 < P < 110 kPa it is clear that multiplicity in the (Lw,V)-configuration can occur in the entire operating range of the column. Since experiment design apparently is not as critical as first anticipated with respect to locating unstable regions, the two primary objectives of the design becomes (a) to operate in a region where W has a substantial value such that the unstable branch will be easy to find, and (b) to operate in a well known region where operation limits as well as more complex column behaviour will not interfere with the experiment. The following operating conditions are found to be suitable. P V
= =
87.5 24.75
kPa ]F kmole/h z
= =
0.180 0.50
m~/h I mole/mole
For this case only the unstable and the upper stable branches in the L~-xo plot exist. This means that as top product flowrate is decreased from Dw = F (B = 0 and xD = z) the process follows the unstable branch towards lower Lw and higher XD. At the (single) singular point the curve turns onto the stable branch along which XD is close to but still approaching 1.0 as Lw is increased. Figure 3 shows the simulated relation between volumetric and molar reflux flowrate and top purity. In the experiment it will be the intension to follow this trajectory from D~ ~, 0 to ZD ~, 0.9 by operation in the (D~,V)-configuration, preferably by controlling XD with Dw in closed loop to speed up transients. 3.2
Experimental
equipment
Figure 4 shows the outline of the plant. The column has 19 sieve trays, a thermosyphon reboiler, a total condenser and a reflux drum. It separates a mixture of methanol and isopropanol. On trays 1, 5, 10, 15 , and 19 PT100 temperature sensors are located in the liquid hold-up. In combination with pressure measurements these are used for concentration estimates. All flows in and out of the system and the reflux flowrate are measured on volumetric basis. Feed, bottom product, and distillate can be sampled manually for gas chromatograph analysis. The heat pump has an expansion valve (Exp.valve) which throttles high pressure liquid freon to a lower pressure (PL) suitable for evaporation in the condenser. After the condenser there is a control valve ( c v g ) by which the freon vapour flowrate can be manipulated. After superheating the vapour the compressor elevates the pressure to a high pressure (PH) suitable for condensation in the reboiler. A small part of the condensation takes place in a secondary condenser which by a cooling water circuit is connected to a set of air-fan coolers. The cooling rate can be manipulated by the control valve CV8, thus controlling PH. Through a storage tank (Rec) and the super heater heat exchanger the freon circuit is closed at the expansion valve. The basic control configuration for the plant is as follows: The high pressure PH is controlled by CV8, the low pressure PL is controlled by CV9, accumulator level is controlled by L, and reboiler level is controlled by B. QB and Qc are then manipulated through the setpoints to PH and PL since these indirectly affect
European Symposium on Computer Aided Process Engineering---6. Part B
~E
CV9Pi
$839
xp~ 1 v a l v e
Super heating before compression
~
AIR COOLERS
Sec ndary condenser
B
Figure 4: Schematic flowsheet of the plant. the temperature gradients of the reboiler and the condenser. The concentration profile is manipulated in the high gain direction by D, and in the low gain direction by a combination of PH and PL by which column pressure is also controlled. Thus, this configuration resembles the standard (D,V)-configuration which is openloop stable as shown previously. 3.3
Experiment
and Results
It was attempted to operate the column at constant molar vapour flowrate. The volumetric vapour flowrate out of the column top was estimated from measurements of reflux and distillate flowrates and the rate of change of accumulator level. From temperature and pressure on the top tray the liquid composition here was estimated using Henry's law and Antoine equations. Based on the setpoint to the molar vapour flowrate and the top tray composition estimate, the desired volumetric vapour flowrate was calculated and passed on as setpoint to a 6 by 2 MIMO controller which controlled volumetric vapour fiowrate and column pressure by manipulation of the heat pump high and low pressures. The column was taken through 10 steady state operating points, at which distillate, bottom product, and feed flow were sampled for later analysis on a gas chromatograph. The process was considered at steady state when all concentration estimates had been stationary for one hour.
4
Discussion
Figure 5 shows the data points obtained in the experiment, surrounded by a 95% rectangular confidence band estimated from measurement standard variations. The data point at (0.95 , 0.994) which apparently has a lower distillate composition than neighboring points was obtained during a shift between feed tanks where too low an inlet concentration was fed to the column. The existence of multiple steady states is clearly verified. With 95% confidence bands there clearly exist two separate branches in the plot of distillate purity versus volumetric reflux, such that for all values of the volumetric reflux there are two possible values for the distillate purity. This multiplicity result implies that the column would be unstable when operating at points on the a negative slope in the xD vs. L,~ plot in figure 5 if the other operating conditions indeed are constant. These aspects are investigated in figure 5-B and C. The plot of molar versus volumetric reflux flowrate at constant molar boil-up rate should indicate the stability characteristics for the (L,V)-configuration. A positive slope indicates a stable process while a negative slope indicates an unstable process. On the A-branch boil-up is constant and the reflux rates indicate a stable process. On the B-branch, however, boil-up rate could only be held within 20£ deviation from the initial value. This seemingly small deviation is unfortunately of the same magnitude as the measured changes in the molar reflux flowrate on the B-branch. Subtraction of the change in boil-up rate from the change in molar reflux rate results in zero slope of the B-branch in the molar reflux plot. With any reasonable uncertainty level it is therefore not possible to conclude whether operating points on the B-branch are stable or unstable.
5
Conclusions
The existence of steady state multiplicity has been verified experimentally, and that for our example column which is a well designed process operated at or near the design specifications and separating a rather simple
$840
European Symposiumon Computer Aided Process Engin~ringm-6. Part B
Dislillote purity (molefroction)
Molor reflux (kmole/h) 250
i ol
Molor boil-up 2EoA
k
1.00
rote
(kmole/h)
, i 25.5 -
24.5
•
i
B
0.99 ,
i
24.0
25.0 -
23.5
24.5
0.g8 -
A
0.97
23.0
24.0 -
o.g6
,l,,,,,,,l,
0.93
,,,,,,,,I,,,,,,,,II,,I,,,,,,I,,,,,,,,,I,,,,,,H,
0.95 0.96 0.~)7 0.98 0.99 Volumetrix reflux (m i/h)
0,94
0.92 0.9,3 0.94 0.95 0.96 30.97 0,98 0.99
volumetrixreflux (m/h)
0.92 0.93 0.g4 0.95 0.96 ~).97 0.98 0.99
Volumetrixreftux (m/h)
Figure 5: Experimental results. Distillate purity measured by off-line GC analysis; molar reflux flowrate calculated from measured volumetric reflux and distillate purity; and molar boil-up rate calculated from measured volumetric boil-up rate. All plotted versus volumetric reflux flowrate. Points represents experimentally obtained datapoints. Curves represents 95% rectangular confidence bands estimated from process variations and GC-analysis variations. binary mixture it has not been difficult to obtain conditions for multiplicity. It was not possible to verify the expected instability for operation in (L,V)-configuration on the lower (or intermediate) branch of the L,o - ZD plot, partly due to difficulties with maintaining a constant molar boil-up rate throughout the experiment. This result could indicate that for practical operation of distillation columns it may be possible to experience steady state multiplicity without instability if other conditions such as boil-up rate are not carefully controlled. The most significant condition for multiplicity is that the molecular weight of the heavy component (Ms) must be significantly larger than the molecular weight of the light component (M1). In the investigated case the ratio M2/M1 is approximately 2. In many industrial distillation processes, though, molecular weights differ only slightly and the ratio thus approaches 1. For these processes steady state multiplicity will not occur. There remains, however, a large number of cases where the ratio M~/M1 is significantly larger than 1, and which therefore may exhibit steady state multiplicity and possibly instability. For these processes the results found here definitely suggest that the discoveries of Nielsen (1990) and the analyses by Jacobsen and Skogestad (1991) are of practical significance.
6
References
Doherty, M.F.; Perkins, J.D. (1982); 'On the Dynamics of Distillation Processes: IV. Uniqueness and Stability of the SteadyState in Homogeneous Continuous Distillation'; Chem. Eng. Sci., vol.37(4}, p.381. Jacobsen, E.W.; Skogestad, S. (1991); 'Multible Steady States in Ideal Two-Product Distillation'; AIChE Journal, vol. 37(4), pp.499-511. J~rgensen, L. (1992); 'Dynamic Simulation - and Experimental Verification of Multiplicity in Distillation'; M.Sc. Thesis, Institut for Kemiteknik, DTH, Building 229, DK-2800 Lyngby. Nielsen, C.S. (1990); 'Multivariable Identification and Control of an Experimental Distillation Column with Heat pump'; Ph.D. Thesis, Institut for Kemiteknik, DTH, Building 229, DK-2800 Lyngby.