U.S. patent application number 10/865236 was filed with the patent office on 2004-12-23 for estimation and control in the wet end using co2.
Invention is credited to Duarte, Daniel, Saucedo, Victor M..
Application Number | 20040256069 10/865236 |
Document ID | / |
Family ID | 33519842 |
Filed Date | 2004-12-23 |
United States Patent
Application |
20040256069 |
Kind Code |
A1 |
Saucedo, Victor M. ; et
al. |
December 23, 2004 |
Estimation and control in the wet end using CO2
Abstract
A method for controlling the CO.sub.2 addition in a wet end
process utilizing CO.sub.2 addition is disclosed. The method
includes combining a papermaking composition and CO.sub.2 to create
a CO.sub.2-enriched papermaking composition, measuring or
estimating at least one electrical property of either the
papermaking composition or of the CO.sub.2-enriched papermaking
composition, and controlling the rate of addition of CO.sub.2 to
maintain the at least one electrical property within a pre-selected
range of values.
Inventors: |
Saucedo, Victor M.;
(Willowbrook, IL) ; Duarte, Daniel; (Clarendon
Hills, IL) |
Correspondence
Address: |
Linda K. Russell, Esq.
Air Liquide
Ste 1800
2700 Post Oak Blvd.
Houston
TX
77056
US
|
Family ID: |
33519842 |
Appl. No.: |
10/865236 |
Filed: |
June 10, 2004 |
Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
|
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60479285 |
Jun 18, 2003 |
|
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60479284 |
Jun 18, 2003 |
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Current U.S.
Class: |
162/158 ;
162/198 |
Current CPC
Class: |
D21H 23/08 20130101;
D21H 17/65 20130101 |
Class at
Publication: |
162/158 ;
162/198 |
International
Class: |
D21H 023/08 |
Claims
We claim:
1. In a wet end process utilizing CO.sub.2 addition, a method for
controlling the CO.sub.2 addition comprising, in combination:
combining a papermaking composition and CO2 to create a
CO2-enriched papermaking composition; measuring or estimating at
least one electrical property of either the papermaking composition
or of the CO2-enriched papermaking composition; and controlling the
rate of addition of CO2 to maintain the at least one electrical
property within a preselected range of values.
2. The method of claim 1, wherein the at least one electrical
property is selected from the group consisting of ZP, CD and ion
concentration.
3. The method of claim 2, wherein the electrical property is
estimated by measuring at least one property of either the
papermaking composition or the CO.sub.2 -enriched papermaking
composition and estimating the ZP.
4. The method of claim 3, the electrical property being ZP.
5. The method of claim 3, the electrical property being CD.
6. The method of claim 3, the electrical property being ion
concentration.
7. The method of claim 3, wherein the at least one property is
selected from the group consisting of flowrate, CD, ZP, ion
concentration, Cy, pH, conductivity and alkalinity.
8. The method of claim 4 wherein the at least one property is
selected from the group consisting of flowrate, CD, ion
concentration, Cy, pH, conductivity and alkalinity.
9. The method of claim 5 wherein the at least one property is
selected from the group consisting of flowrate, ZP, ion
concentration, Cy, pH, conductivity and alkalinity.
10. The method of claim 6 wherein the at least one property is
selected from the group consisting of flowrate, CD, ZP, Cy, pH,
conductivity and alkalinity.
11. The method of claim 3, wherein the at least one electrical
property is estimated in the papermaking composition.
12. The method of claim 3, wherein the at least one electrical
property is measured in the CO.sub.2-enriched papermaking
composition.
13. The method of claim 11, wherein the at least one electrical
property is estimated in the papermaking composition.
14. The method of claim 12, wherein the at least one electrical
property is estimated in the papermaking composition.
15. The method of claim 11, wherein the at least one electrical
property is measured in the CO.sub.2-enriched papermaking
composition.
16. The method of claim 12, wherein the at least one electrical
property is measured in the CO.sub.2-enriched papermaking
composition.
17. In a wet end process utilizing CO.sub.2 addition, a method for
controlling the CO.sub.2 addition comprising, in combination:
combining a papermaking composition and CO.sub.2 to create a
CO.sub.2-enriched papermaking composition; measuring or estimating
at least one property of the papermaking composition and generating
papermaking composition property data, and providing the
papermaking composition property data to an advanced controller
constructed and arranged to receive the papermaking composition
property data and generate a papermaking composition output
component; measuring or estimating at least one property of the
CO.sub.2-enriched papermaking composition, generating
CO.sub.2-enriched papermaking composition property data and
providing the CO.sub.2-enriched papermaking composition property
data to a feedback controller constructed and arranged to receive
the CO.sub.2-enriched papermaking composition property data and
generate an outlet controller output component; compensating the
feedback controller by analyzing the inlet controller output
component and the outlet controller output component; and
controlling the inlet flow of CO.sub.2 to maintain at least one
property of the CO.sub.2-enriched papermaking composition within a
preselected range of values.
18. The method of claim 17, wherein the at least one property of
the papermaking composition is selected from the group consisting
of flowrate, ZP, CD, ion concentration, Cy, pH, conductivity and
alkalinity.
19. The method of claim 17, wherein the at least one property of
the CO.sub.2-enriched papermaking composition is selected from the
group consisting of ZP, CD and ion concentration.
20. The method of claim 18, wherein the at least one property of
the CO.sub.2-enriched papermaking composition is selected from the
group consisting of ZP, CD and ion concentration.
21. The method of claim 17, wherein the advanced controller
comprises a feed forward controller.
22. The method of claim 21, wherein the at least one property of
the papermaking composition is selected from the group consisting
of flowrate, ZP, CD, ion concentration, Cy, pH, conductivity and
alkalinity and the at least one property of the CO.sub.2-enriched
papermaking composition is selected from the group consisting of ZP
and CD.
23. In a wet end process utilizing CO.sub.2 addition, a method for
controlling the CO.sub.2 addition comprising, in combination:
combining a papermaking composition and CO.sub.2 to create a
CO.sub.2-enriched papermaking composition; measuring or estimating
at least one property of the papermaking composition selected from
the group consisting of flowrate, ZP, CD, Cy, pH, conductivity, ion
concentration and alkalinity, generating papermaking composition
property data and providing the papermaking composition property
data to an advanced controller constructed and arranged to receive
the papermaking composition property data and generate an inlet
controller output component; measuring or estimating at least one
property of the CO.sub.2-enriched papermaking composition selected
from the group consisting of ZP, CD and salt ion concentration and
generating CO.sub.2-enriched papermaking composition property data,
providing the CO.sub.2-enriched papermaking composition property
data to a feedback controller constructed and arranged to receive
the CO.sub.2-enriched papermaking composition property data and
generate an outlet controller output component; compensating the
feedback controller by analyzing the inlet controller output
component and the outlet controller output component; and
controlling the inlet flow of CO.sub.2 to maintain at least one
property of the CO.sub.2-enriched papermaking composition within a
preselected range of values.
24. The method of claim 23, wherein the advanced controller
comprises a feed forward controller.
25. The method of claim 24 wherein the feed forward controller uses
predictive control.
26. The method of claim 24 wherein the feed forward controller uses
inferential control.
27. In a wet end process utilizing CO.sub.2 addition, a method for
controlling the CO.sub.2 addition comprising, in combination:
combining a papermaking composition and CO.sub.2 to create a
CO.sub.2-enriched papermaking composition; on-line measuring at
least one property of the papermaking composition selected from the
group consisting of flowrate, ZP, CD, ion concentration, Cy, pH,
conductivity and alkalinity, generating papermaking composition
property data and providing the papermaking composition property
data to an observer; on-line measuring at least one property of the
CO.sub.2-enriched papermaking composition selected from the group
consisting of flowrate, ZP, CD, ion concentration, Cy, pH,
conductivity and alkalinity, generating CO.sub.2-enriched
papermaking composition property data and providing the
CO.sub.2-enriched papermaking composition property data to the
observer; the observer receiving the papermaking composition
property data and the CO.sub.2-enriched papermaking composition
property data and generating at least one estimated electrical
property of the CO.sub.2-enriched papermaking composition selected
from the group consisting of ZP, CD and ion concentration, the
observer generating estimated electrical property data for the at
least one electrical property and transmitting the papermaking
composition property data, the CO.sub.2-enriched papermaking
composition property data and the estimated electrical property
data to a controller; and the controller controlling the inlet flow
of CO.sub.2 to maintain at least one electrical property of the
CO.sub.2-enriched papermaking composition within a preselected
range of values.
28. The method of claim 27, the observer comprising a model.
29. The method of claim 27, the papermaking composition property
data, the CO.sub.2-enriched papermaking composition property data
and the estimated electrical property data being incorporated into
a software sensor.
30. The method of claim 28, the papermaking, composition property
data, the CO.sub.2-enriched papermaking composition property data
and the estimated electrical property data being incorporated into
a software sensor.
31. The method of claim 28, the observer further refining the
estimated electrical property data by analyzing inaccuracies
presented by the model.
32. The method of claim 28, the observer further refining the
estimated electrical property data by analyzing expected errors in
measurement.
33. The method of claim 28, the observer further refining the
estimated electrical property data by analyzing inaccuracies
presented by the model and by analyzing expected errors in
measurement.
34. The method of claim 30, the observer further refining the
estimated electrical property data by analyzing inaccuracies
presented by the model.
35. The method of claim 30, the observer further refining the
estimated electrical property data by analyzing expected errors in
measurement.
36. The method of claim 30, the observer further refining the
estimated electrical property data by analyzing inaccuracies
presented by the model and by analyzing expected errors in
measurement.
37. The method of claim 28 further comprising using the estimated
electrical property data to evaluate a set point and implement a
real time closed loop control.
38. The method of claim 30 further comprising using the estimated
electrical property data to evaluate a set point and implement a
real time closed loop control.
39. The method of claim 36 further comprising using the estimated
electrical property data to evaluate a set point and implement a
real time closed loop control.
40. In a wet end process utilizing CO addition, a method for
controlling the CO.sub.2 addition comprising, in combination:
combining a papermaking composition and CO.sub.2 to create a
CO.sub.2-enriched papermaking composition; on-line measuring at
least one property of the papermaking composition selected from the
group consisting of flowrate, ZP, CD, ion concentration, Cy, pH,
conductivity and alkalinity, generating papermaking composition
property data and providing the papermaking composition property
data to an observer; on-line measuring at least one property of the
CO.sub.2-enriched papermaking composition selected from the group
consisting of flowrate, ZP, CD, ion concentration, Cy, pH,
conductivity and alkalinity, generating CO.sub.2-enriched
papermaking composition property data and providing the
CO.sub.2-enriched papermaking composition property data to the
observer; the observer receiving the papermaking composition
property data and the CO.sub.2-enriched papermaking composition
property data and using a model to generate at least one estimated
electrical property of the CO.sub.2-enriched papermaking
composition selected from the group consisting of ZP, CD and ion
concentration; the observer refining the estimate for the at least
one electrical property by analyzing inaccuracies presented by the
model and by analyzing expected errors in measurement; the observer
transmitting the papermaking composition property data, the
CO.sub.2-enriched papermaking composition property data and the
refined estimated electrical property data to a controller; and the
controller controlling the inlet flow of CO.sub.2 to maintain at
least one electrical property of the CO.sub.2-enriched papermaking
composition within a preselected range of values.
Description
CROSS REFERENCE TO RELATED APPLICATION
[0001] This application claims the benefit of U.S. Provisional
Application No. 60/479,285, filed Jun. 18, 2003 and U.S.
Provisional Application No. 60/479,284, filed Jun. 18, 2003.
BACKGROUND OF THE INVENTION
[0002] 1. Field of the Invention
[0003] The disclosure relates to a method for controlling the
CO.sub.2 addition in a wet end process utilizing CO.sub.2
addition.
[0004] 2. Related Art
[0005] A method for controlling the CO.sub.2 addition in a wet end
process utilizing CO.sub.2 addition is disclosed. The assignee of
this application, Air Liquide, currently has pending a patent
application for CO.sub.2 addition in a wet end process.
[0006] In paper processing, due to industrial competition, changes
in raw materials, environmental concerns and greater customer
demands, higher levels of knowledge and understanding of the
chemistry in the wet end are important to commercial success. It
has been discovered that the zeta potential (ZP) is an important
property in the final paper quality fabrication.
[0007] It has long been understood that zeta potential plays an
important role in paper machine process operability by being
related to flocculation, retention and drainage characteristics of
the pulp. N. Vanderhoek, "Optimizing Paper Machine Performance
Through Electrokinetic Measurement", APPITA, Vol. 47. No. 5, pp
397-405, 1994. Wet-end stability is a key part to getting machine
efficiencies. T. Miyanashi, et. al. studied the effects of zeta
potential on these phenomena, and concluded that the zeta potential
should be controlled to provide better flocculation and drainage
additives. The wet end chemistry of optimized paper machines is not
operated at equilibrium conditions, and the order of chemical
addition is critical. T. Miyanashi, and S. Motegi, "Optimizing
Flocculation and Drainage for Microparticle by Controlling Zeta
Potential," TAPPI Journal, pp. 262-270, January 1997.
[0008] The need for better performance of paper machines has
triggered the development of better retention aids. This has
defined a need for better understanding the effects of these new,
wet-end additives and being able to control in real time the
additions to obtain optimum paper machine and overall wet-end
operation performances. The first step to achieving these
objectives relies on available real-time information. It is
desirable to provide a process that adapts itself to real time
changes. In the wet-end process, fiber characteristics, chemical
additives and water properties can change continuously, and so it
is desired to detect these variations as they occur in real
time.
[0009] As mentioned above, Air Liquide has developed a technology
that is capable of altering the zeta potential of cellulose fibers
using CO.sub.2. M. Muguet and J. M. deRigaurd, "Improvements to
Processes for Manufacturing Paper Products by Improving the
Physico-Chemical Behavior of the Paper Stock", International
Publication No. WO 03/074788 A2. Air Liquide has also a pending
patent application on such technology, Serie 6052 which was filed
as a U.S. provisional application on Sep. 30, 2002, bearing Ser.
No. 60/414,876, and as a U.S. non-provisional on Sep. 6, 2003,
bearing U.S. Ser. No. 10/656,857. Applicant hereby expressly
incorporates by reference the entirety of these disclosures as if
fully set forth herein.
[0010] Generally in process control, a mathematical model is
required correlating input parameters with output variations of the
controlled system. To control wet-end parameters such as retention,
drainage and formation by wet-end additives, a mathematical model
correlating the additive parameters with the zeta potential and
cationic demand is first required. Next, a mathematical model
correlating the parameters affecting flocculation such as zeta
potential, cationic demand and the bridge forming capability of
polymeric additives is needed. Since normally these input-output
relationships are unknown, it is not easy to construct either a
mathematical control model or a simulation model (F. Onabe). F.
Onabe, MEASUREMENT AND CONTROL, Chapter 12. W. Scott explains that
these are non-linear, interacting relationships. W. Scott,
PRINCIPLES OF WET END CHEMISTRY, TAPPI Press, Atlanta, Ga., 1996.
Wang, H. et al., describes a neural network that models the
relationship between the wet end chemicals and the properties of
the resulting paper. H. Wang, B. Oyebande, "On the Application of
Neural Networks Modeling to a Wet End Chemical Process in Paper
Making," IEEE Conference on Control Applications--Proceedings 1995.
IEEE, Piscataway, N.J., USA pp. 657-662, 1995.
[0011] To implement a real-time control optimization scheme in the
wet end, it is useful to have real-time measurements available. The
common lack of immediate feedback measurement about the
effectiveness of the adsorption process is a serious shortcoming,
and this leads to mill practices of minimizing input disturbances
by controlling individual parameters. Most of these parameters are
typically controlled, such as pH, ionic demand, consistency (Cy),
flowrate, etc. However, those properties that are not controlled
due to the lack of on-line instruments present challenges. For
example, ZP measurement is normally measured off-line. Most mills
lack an on-line ZP analyzer, and so it is impossible to attain an
on-line ZP control or optimization.
[0012] Thus, a problem associated with paper processing methods
that precede the present invention is that they do not provide a
method of more closely controlling the performance of wet end
chemistry in paper manufacture by controlled CO.sub.2 addition to
the wet end.
[0013] Still another problem associated with paper processing
methods that precede the present invention is that they do not
provide controlled CO.sub.2 addition into the wet end that permits
the control of unmeasured properties to improve the performance of
wet end chemistry in paper manufacture.
[0014] An even further problem associated with paper processing
methods that precede the present invention is that they do not
provide a control mechanism for CO.sub.2 addition into the wet end
that permits the more reliable control of unmeasured properties, by
providing a predictor of disturbances to the system that
facilitates refinement of data.
[0015] The present invention seeks to overcome these problems while
at the same time providing a cost-effective, simply used mechanism
for controlling CO.sub.2 addition into the wet end of paper
manufacture.
SUMMARY OF THE INVENTION
[0016] A method for controlling the CO.sub.2 addition in a wet end
process utilizing CO.sub.2 addition is disclosed. A papermaking
composition and CO.sub.2 are combined to create a CO.sub.2
-enriched papermaking composition. At least one electrical property
of either the papermaking composition or of the CO.sub.2 -enriched
papermaking composition is either measured or estimated. The rate
of addition of CO.sub.2 to maintain the at least one electrical
property within a pre-selected range of values is then
controlled.
[0017] In one embodiment, the electrical property is selected from
the group consisting of ZP, CD and ion concentration or any
equivalent thereto, and is estimated by measuring at least one
property of either the papermaking composition or the
CO.sub.2-enriched papermaking composition selected from the group
consisting of flowrate, CD, ZP, ion concentration, Cy, pH,
conductivity and alkalinity and using a model. Preferably, the ZP
is estimated. Thus, the measured property can be measured from
either the papermaking composition or the CO.sub.2-enriched
papermaking composition, and is preferably measured from the
papermaking composition. Likewise, the estimated electrical
property can be estimated for either the papermaking composition or
the CO.sub.2-enriched papermaking composition, and is preferably
estimated for the CO.sub.2-enriched papermaking composition.
[0018] Various control schemes are employed to better control the
CO.sub.2 addition by maintaining the at least one electrical
property within a pre-selected range of values.
[0019] Thus, it is an object of the present invention to provide a
method of more closely controlling the performance of wet end
chemistry in paper manufacture by controlling the CO.sub.2 addition
to the wet end.
[0020] It is a further object of the present invention to provide a
control mechanism for CO.sub.2 addition into the wet end that
permits the control of unmeasured properties, thereby improving the
performance of wet end chemistry in paper manufacture.
[0021] It is still another object of the present invention to
provide a control mechanism for CO.sub.2 addition into the wet end
that permits the more reliable control of unmeasured properties, by
providing a predictor of disturbances to the system that
facilitates refinement of data and hence better performance of wet
end chemistry in paper manufacture.
[0022] These and other objects of the present invention will be
apparent from the description of the invention that follows.
BRIEF DESCRIPTION OF THE DRAWINGS
[0023] In the detailed description that follows, reference will be
made to the following figures:
[0024] FIG. 1 is a schematic diagram illustrating an embodiment of
the control method adapted to a specific papermaking process;
[0025] FIG. 2 is a schematic diagram illustrating another
embodiment of the control method;
[0026] FIG. 3. is a schematic diagram illustrating the expected
performance of a feed forward control responding to a normalized
step change and a charge demand disturbance; and
[0027] FIG. 4 is a schematic diagram illustrating another
embodiment of the control method adapted to a specific papermaking
process.
DESCRIPTION OF PREFERRED EMBODIMENTS
[0028] A system and method for controlling the parameters in a wet
end process by means of CO.sub.2 injection in a real-time manner
using an advanced controller is presented. The design of the
advanced controller maintains the desired set point and rejects the
influence of undesirable wet end disturbances. The effects of the
disturbances on the wet end are modeled, and their on-line
measurements are used to compensate the addition of CO.sub.2. This
disclosure provides an effective way of controlling parameter in
the wet end using multi-variable advanced control and CO.sub.2
gas.
[0029] As mentioned above, Air Liquide has developed a technology
that is capable of altering the zeta potential of cellulose fibers
using CO.sub.2. M. Muguet and J. M. deRigaurd, "Improvements to
Processes for Manufacturing Paper Products by Improving the
Physico-Chemical Behavior of the Paper Stock", International
Publication No. WO 03/074788 A2. Air Liquide has also a pending
patent application on such technology, Serie 6052 which was filed
as a U.S. provisional application on Sep. 30, 2002, bearing Ser.
No. 60/414,876, and as a U.S. non-provisional on Sep. 6, 2003,
bearing U.S. Ser. No. 10/656,857. Applicant hereby expressly
incorporates by reference the entirety of these disclosures as if
fully set forth herein.
[0030] The instant disclosure relates to a control method adaptable
therefor.
[0031] The alkalinity in water remains unaltered with the addition
of CO.sub.2 due to the balance of carbonic species. However, when
the CO.sub.2 is added in the presence of CaCO.sub.3, the alkalinity
is changed due to the extra production of bicarbonate.
[0032] The utilization of CO.sub.2 has proven to improve the
efficiency of the wet end process. The design of the application
requires a good knowledge of the interactions caused by the
CO.sub.2 and the extent of these interactions. This knowledge is
covered theoretically by equilibrium analysis of water chemistry.
In order to maintain the efficiency of the process it is helpful to
design control systems that keep the process running under the
designed specifications and rejects major disturbances. The most
important disturbances can be attributed to the change of
concentration in the make up water, temperature, pressure,
consistency, or other chemical addition variations. In order to
design the control system capable of rejecting these variations, it
is helpful to know the dynamic effect on the controlled variables
and their dynamic relationship with the CO.sub.2 supply. These
relationships are obtained either from available process knowledge,
or dynamic tests in the mill, or theoretical models.
[0033] Dynamic modeling of the most important species in the
aqueous system with gaseous CO.sub.2 and CaCO.sub.3 and its effect
on Zeta Potential (ZP) and Charge Demand (CD) can be performed. The
dynamic model assumed a constant alkalinity.
[0034] The CO.sub.2--CaCO.sub.3 system is well studied in
equilibrium conditions, but the studies are limited when it comes
to dynamic (kinetic) conditions. When exploring the dynamic
modeling of this system, one needs to consider a variety of
parameters that play important roles, such as mass transfer and
surface reactions. The operating conditions under which the
CO.sub.2--CaCO.sub.3 are chosen to operate, define how fast the
equilibrium conditions are met, and are worthwhile considering when
designing a model-based control system. The most important effects
of various operating conditions such as partial pressure of
CO.sub.2, temperature, along with system-specific variables such as
calcium carbonate surface area and mass transfer coefficient, are
also presented here.
[0035] The validation of the proposed models requires the off-line
measurements of alkalinity and [Ca.sup.2+] which are tedious and
time consuming. The potential for using conductivity measurements
to predict these off-line measurements is therefore considered.
[0036] MODEL DEVELOPMENT
[0037] Equilibrium
[0038] The dynamic behavior of the species in a
CO.sub.2--CaCO.sub.3 system depends on several variables. The
dynamic behavior will be represented in the form of ordinary
differential equations (ODE), which require initial conditions in
order to have their solution. Some of the initial conditions are
known and measured in a typical (batch) experiment, such as pH.
However, other initial conditions are not known and rarely
measured, such as [CO.sub.3.sup.-2] concentration. Equilibrium
studies can provide the estimates of the values of any component
for any given condition assuming the conditions have not changed in
order to reach equilibrium. Then, the equilibrium calculations can
be used as initial conditions for ODE. Similarly, equilibrium
studies can be used to compare the final or steady state results of
dynamic equations. In steady state, the ODE results should be
consistent with the equilibrium conditions at the new operating
conditions.
[0039] The mass transfer coefficient, kx, has an impact on the time
response of the variables but does not affect the final steady
state or equilibrium values. On the other hand, pCO.sub.2 (partial
pressure of CO.sub.2) directly influences the equilibrium values.
So, the value of the pCO.sub.2 that attains the experimental steady
state parameters needs to be found or closely monitored, if
possible. In order to find the operating pCO.sub.2 that achieves
the equilibrium conditions, an equilibrium problem is then
solved.
[0040] Following the example on p. 168 by Stumm & Morgan
(Stumm, W. and Morgan, J. J. "Aquatic Chemistry: Chemical
Equilibria and Rates in Natural Waters", John Wiley & Sons,
Third Edition, 1996), a Matlab program is utilized to solve the set
of nonlinear algebraic equations defined by: 1 [ H + ] [ HCO 3 - ]
[ H 2 CO 3 * ] = K 1 ( 1 ) [ H + ] [ CO 3 2 - ] [ HCO 3 - ] = K 2 (
2 ) [H.sub.2CO.sub.3*]=K.sub.HpCO.sub.2 (3)
[H.sup.+][OH.sup.-]=K.sub.W (4)
[Ca.sup.+2][HCO.sub.3.sup.-]=K.sub.SO (5)
2[Ca]+[H.sup.30 ]=[HCO.sub.3.sup.-]+2[CO.sub.3.sup.2-]+[OH.sup.-]
(6)
[0041] The equilibrium constants are reported in the
literature.
1 Correlations for temperature dependent equilibrium constants
(Stumm & Morgan) Temp 294.45 21.3 A1 A2 A3 A4 A5 LogK1
-6.374687 -356.3094 -0.06092 21834 126.8339 -1684915 LogK2
-10.36285 -107.8871 -0.032528 5151.79 38.92561 -563713.9 LogKH
-1.423209 108.3865 0.019851 -6919.53 -40.45154 669365 LogKw
-14.12347 -283.971 13323 -0.050698 102.2445 -1119669 LogKso
-8.459868 -171.9065 -0.077993 2839.319 71.595 0 Log(Ki) = A1 + A2 *
T + A3/T + A4Log(T) + A5/T{circumflex over ( )}2
[0042]
2 T (deg C.) LogK1 LogK2 LogKH LogKw LogKso 20 -6.383133 -10.37557
-1.406868 -14.16818 -8.453297 25 -6.353105 -10.32885 -1.46794
-13.99953 -8.47983 30 -6.329367 -10.28788 -1.524416 -13.83949
-8.509751 35 -6.311358 -10.25227 -1.576641 -13.68751 -8.543034 40
-6.298577 -10.22169 -1.624926 -13.54308 -8.579652 45 -6.290573
-10.19583 -1.669556 -13.40575 -8.619574 50 -6.286946 -10.1744
-1.710789 -13.2751 -8.662766 55 -6.287334 -10.15714 -1.748859
-13.15074 -8.709195 60 -6.291415 -10.14381 -1.783981 -13.03233
-8.758826 65 -6.298898 -10.13418 -1.816351 -12.91953 -8.81162 70
-6.309522 -10.12806 -1.846147 -12.81207 -8.867541 Dependence of
Equilibrium constants with Temperature: Stumm & Morgan report
the temperature dependent correlations.
[0043] Equations (1)-(6) describe the equilibrium equations of the
CO.sub.2--CaCO.sub.3 in water at a given CO.sub.2 pressure,
pCO.sub.2. These equations can be manipulated in different forms to
fit different conditions. For instance, if the calcium is not in
equilibrium, then equation (5) is not applicable, and the
[Ca.sup.+2] is estimated from a rate equation.
[0044] Kinetics (Rate)
[0045] The equilibrium studies determine the steady state
conditions or the total changes based on operating conditions. On
the other hand, kinetic studies determine also the rate at which
changes occur. Process control design depends on both results, the
extent of the change and the rate of change. The exchange of
CO.sub.2 from the gas phase to the bulk liquid for constant
alkalinity is presented as the diffusion equation: 2 C T t = klx (
p Co2 H + Alk - C T ) = R CO2 ( 7 )
[0046] where,
[0047] C.sub.T=Total carbonic species,
M=[H.sub.2CO.sub.3*]+[HCO.sub.3.sup- .-]+[CO.sub.3.sup.-2]
[0048] klx=Mass transfer coefficient, min.sup.-1
[0049] pCO2=CO.sub.2 partial pressure in the gas, atm
[0050] H=Henry's equilibrium constant
[0051] Alk=Alkalinity, M (mol HCO.sub.3/L)
[0052] Equation (7) is valid when the concentration of [CO.sub.3--]
is negligible and C.sub.T>>Alk. This occurs when the pressure
is significantly higher than the partial pressure of CO.sub.2 in
the atmosphere. For pressures close to atmospheric, Equation (7) is
not valid, but it is assumed that the operating conditions where it
will be used are in CO.sub.2 rich conditions.
[0053] If the alkalinity is not constant, then the rate of transfer
of CO.sub.2 moles converted to [H2CO.sub.3*] is 3 [ H 2 CO 3 * ] t
= t [ C T - [ HCO 3 - ] ] = C T t - [ HCO 3 - ] t ( 8 )
[0054] then, the rate of total carbonic species becomes 4 C T t = R
CO 2 - [ HCO 3 - ] t = R CO 2 - Alk t ( 9 )
[0055] In order to estimate the time dependant variation of pH, one
starts from the relationship: 5 [ H + ] = ( pK 1 Alk ) ( C T - Alk
) ( 10 )
[0056] which can be rewritten as: 6 [ H + ] = pK 1 C T Alk - pK 1 (
11 )
[0057] Taking the derivative with respect to time on Equation (11)
reveals that the equation is more complex when considering a time
dependent alkalinity: 7 [ H + ] t = pK 1 t ( C T Alk ) ( 12 )
[0058] This becomes, 8 [ H + ] t = pK 1 { Alk * R CO2 - C T d Alk
Alk 2 } ( 13 )
[0059] The evaluation of Equation (13) depends on Equation (9) or
R.sub.CO2, but also depends on dAlk/dt. To the present time, an
analytical expression for dAlk/dt has not been found.
[0060] One tentative expression can be derived from the simplified
proton balance expression when CO.sub.2is added in a system in the
presence of CaCO.sub.3:
2[Ca.sup.++].about.=Alk (14)
[0061] Even though Equation (14) is an equilibrium equation for
proton balance, it can be used as an approximate relationship
between calcium ions and alkalinity at all times. Hence, taking the
time derivative in both sides, 9 Alk t = 2 [ Ca ++ ] t ( 15 )
[0062] The implementation of Equations (13) and (15) has posed
implementation problems and the results have not been successful to
this point due to numerical reasons. When solving Equations (13)
and (15) simultaneously, the hydrogen proton concentration becomes
non-real. So, a simplification is presented.
[0063] It is known that the proton-transfer reactions such as the
carbonic reactions are usually very fast with half-lives less than
milliseconds. This suggests that it may not be necessary to express
in ODEs the equations to estimate the hydrogen proton or the
alkalinity. Instead, as the rate limiting equations are solved in
ODEs, the rest of the chemical species can be determined solving
the appropriate equilibrium equations.
[0064] Equations (9), (13) and (15) describe the kinetic equations
of C.sub.T, pH and Alk, correspondingly. The calcium dissolution
kinetics obtained by Plummer et. al. has the form:
R.sub.Ca=k.sub.Ca,1a.sub.H+k.sub.Ca,2a.sub.H2CO3+k.sub.Ca,3a.sub.H2O-k.sub-
.Ca,4a.sub.Cua.sub.HCO3 (16)
[0065] Plummer, L. N., Wigley, T. M., and Parhurst, D. L. "The
kinetics of calcite dissolution in CO.sub.2-- water systems at
5.degree. to 60.degree. C. and 0.0 to 1.0 ATM CO.sub.2", American
Journal of Science, Vol. 278, p. 179-216, February, 1978. The
activity coefficients, a.sub.i, are equal to concentrations. At
25.degree. C., the parameters are:
[0066] k.sub.Ca,1=5.115e-02;
[0067] k.sub.Ca,2=3.4247e-05;
[0068] k.sub.Ca,3=1.1919e-07;
[0069] k.sub.Ca,4=4.55e-02;
[0070] The units of R.sub.Ca are in mmol/(sec.sup.-1cm.sup.2). The
constants k.sub.Ca,i are temperature dependent and k.sub.Ca,4 is
the backward reaction. Besides converting R.sub.Ca to mol and
min.sup.-1, the kinetic expression depends on the total surface
area of the calcite particles. Denoting the particle area as
a.sub.c, defined as the surface area per unit of mass, provides the
calcium dissolution in mol/min as: 10 [ Ca + 2 ] t = a C [ CaCO 3 ]
V R Ca ( 60 / 1000 ) ( 17 )
[0071] The product [CaCO3]V corresponds to the total grams of CaCO3
in the reactor. Plummer reports a range of polished calcite
particles up to 90 cm.sup.2/gr.
[0072] The rate expressions of CO.sub.2 transfer and calcium
dissolutions are the phenomena that dictate the global rates.
Hence, it is suggested that in order to identify the time dependant
values of all the species in a CO.sub.2--CaCO.sub.3 system, small
increments solving the ODEs of CO.sub.2transfer and calcium
dissolution (Equations (9) and (17)) will return the [Ca.sup.+2]
and total carbon species, C.sub.T. Then, the following algebraic
equations are solved to find alkalinity, [HCO.sub.3], and hydrogen
proton, [H.sup.+]: 11 [ H + ] [ HCO 3 - ] [ H 2 CO 3 * ] = K 1 ( 18
) [ H + ] [ CO 3 2 - ] [ HCO 3 - ] = K 2 ( 19 ) [H.sup.+][OH.sup.31
]=K.sub.W (20)
C.sub.T=[H.sub.2CO.sub.3*]+[HCO.sub.3.sup.-]+[CO.sub.3.sup.2-]
(21)
2[Ca]+[H.sup.+]=[HCO.sub.3.sup.-]+2[CO.sub.3.sup.2-]+[OH.sup.-]
(22)
[0073] Once these equations are solved, the new values are used to
solve the next increment of the ODEs.
[0074] The dynamic modeling of the CO.sub.2--CaCO.sub.3 is then
defined by 2 ODEs (Equations (9) and (17)), and the simultaneous
solution of algebraic equations (Equations (18) through (22)). All
these equations have a variety of parameters that have to be
specified prior to the numerical simulations. These parameters are
the temperature, mass transfer coefficient, the Co.sub.2partial
pressure, and calcium carbonate surface area.
[0075] All the kinetic and equilibrium constants are temperature
dependent. The temperature dependence of the kinetic parameters
have been presented in a previous report (CRC200343) and have shown
above for T=25.degree. C. The temperature dependence of the
equilibrium parameters is shown in Appendix A.
[0076] Experiments
[0077] A series of experiments was performed. The experiments
utilized 8.125 gr of CaCO.sub.3 (equivalent to 20% PCC in pulp) in
a 1.3 L of DI water. Two types of CaCO.sub.3 were used: ALBACAR HO
PCC (Specialty Minerals, Inc) and reagent grade CaCO.sub.3 (Fisher
Scientific). The CaCO.sub.3 stock solution was agitated for 24
hours to reach equilibrium with the CO.sub.2in the atmosphere. The
initial sample (time zero) was taken from the stock. 1.3 L of the
stock was placed in the reactor and agitation was set at 1500 rpm.
Timing started when the dosing of CO.sub.2 started. The total
amount of CO.sub.2 was 10.256 Kg/ton assuming a 2.5% Cy slurry, or
2.6e.sup.-04 Kg CO.sub.2/L. Dosing of CO.sub.2 was completed in
approximately 40 seconds. The dosage was performed in order to
obtain a constant pCO.sub.2 pressure in the reactor head. Close to
10-milliliter samples were withdrawn and immediately filtered with
0.02-micrometer filters. Several experiments with the same
conditions were run in order to take more samples. Conductivity,
pH, alkalinity and calcium concentration, were measured from each
sample.
[0078] Experimental Results
[0079] Table 1 shows the experimental results obtained at
.about.24.degree. C. using ALBACAR PCC. Notably, the concentrations
of calcium and alkalinity increase rapidly to reach equilibrium
values in a few minutes.
[0080] From the proton balance indicated in Equation (6), and since
the concentration of carbonate, [CO.sub.3--], is negligible at this
pH, the relation 2*[Ca.sup.+2].about.=Alk must be satisfied (Stumm
& Morgan). From the experimental results shown in Table 1 it
can be seen that this relationship is not satisfied. These results
represent much higher values of alkalinities compared to the
calcium dissolved equilibrium. These results gave the indication
that the ALBACAR PCC might have some chemicals that affect the
CO.sub.2--CaCO.sub.3 equilibrium
[0081] Table 2 shows the experimental results using reagent grade
CaCO.sub.3. As with ALBACAR, the concentrations of calcium and
alkalinity increase very rapidly towards equilibrium. More in depth
explanations on this behavior will be presented later. What is
interesting regarding equilibrium is the fact that the relationship
between the equilibrium calcium and alkalinity is much closer to
meeting the relationship 2*[Ca.sup.+2].about.=Alk. Excluding the
1-minute sample, the experimental values have an average error of
5% from the analytical expression. The higher error at 1 minute is
attributed to the difficulty to take a representative sample at
such an early stage in the experiment.
[0082] It was noticed that the pH variation of the filtered samples
during the experiment was too small between time zero and the last
sample. The pH measurements of the samples before filtering were
closer to the expected according to equilibrium analysis.
[0083] The results shown in Table 2 are utilized in the remaining
of this study to compare with the equilibrium and dynamic
models.
[0084] Equilibrium Analysis of Results
[0085] The equilibrium equations allow determining the target
steady state conditions of CO.sub.2--CaCO.sub.3 systems. With this
information, one can find in advance what steady state values are
expected in experiments and in dynamic models simulations.
[0086] The first step to confirm the validity of the problem set up
and its results, consisted in duplicating the problem on p. 186 by
Stumm & Morgan. The problem determines the equilibrium
concentrations in the CO.sub.2--CaCO.sub.3 system in the presence
of atmospheric pressure (10.sup.-3.5 atm=3.16e-04 atm). The
solution of this problem was implemented in Matlab (Mathworks, Inc)
using a nonlinear algebraic equation solver command. The results
obtained in Matlab solving the set of equations (I) through (6) are
[Ca]=4.636e-04 M; Alk=9.2721 e-04 M and pH=8.3048. Note also that
2[Ca.sup.2+].about.=Alk.
[0087] In order to compare the experimental results obtained with
CaCO.sub.3 with the equilibrium equations, the partial pressure of
CO.sub.2 is needed. This pressure has not been reliable or
available experimentally. Some early measurements indicated that
the partial pressure of CO.sub.2 at the beginning of the experiment
was around 0.43 psig, equivalent to 0.02 atm, but it declined
continuously and its recording was unreliable. Furthermore, the
partial pressure characteristic to the system in equilibrium, as
represented in Equation (3), assumes to be constant. This is not
the case in the experimental apparatus as the system is a batch
reactor and variables change with time. However, in order to make
the equilibrium equations, one has to find the characteristic CO2
partial pressure that attains the same experimental equilibrium
conditions. Table 3 shows a set of results obtained when solving
Equations (1) through (6) for a range of CO.sub.2 partial
pressures.
[0088] Comparing the equilibrium results obtained from equilibrium
equations as shown in Table 3 (with extrapolation), and the
experimental results from Table 2, it is obtained that the
equilibrium conditions that match the experimental results are:
[0089] CASE A: Theoretical Equilibrium at 25.degree. C.
[0090] i. Alk=5e-03 mol HCO.sub.3/L=250 mg CaCO.sub.3/L=250 ppm
CaCO.sub.3
[0091] ii. [Ca]=2.5e-03 mol Ca/L=100 mg Ca/L
[0092] iii. pCO2=10.sup.-1.049 atm=.about.0.09 atm
[0093] iv. pH=6.5 (Matlab), 6.9-7.0 PHREEQ
[0094] The experimental results using ALBACAR HO (Specialty
Minerals, Inc.) suggested that the equilibrium indicator to match
was the pH. But the partial pressure of CO.sub.2 that achieves a
7.3 equilibrium pH achieves a lower theoretical alkalinity than the
experimental values. The theoretical equilibrium conditions for
this case are labeled as CASE B and are shown below. At this point,
it was suspected that the difference in results could be due to the
unknown components in the PCC used (ALBACAR HO, Specialty Minerals,
Inc).
[0095] CASE B: Final Equilibrium Experimental Results (Dec. 12,
2003)
[0096] i. pH=7.3
[0097] ii. [Ca]=106 ppm=2.7e-03 M
[0098] iii. Alk.about.=300 ppm as CaCO3=6e-03 mol [HCO3]/L
[0099] iv. Initial pCO.sub.2.about.0.4 psig (.+-.0.2 psig)=2.72e-02
atm
[0100] Even though the equilibrium pH values in the theoretical and
experimental cases are the same, the calcium and alkalinity
concentrations deviate significantly. The experimental calcium
concentration almost doubles the theoretical. The theoretical
equilibrium calcium concentration depends on the constant K.sub.SO.
Aside from experimental errors, one possibility for the discrepancy
is that the CaCO.sub.3 utilized is either not pure or different.
Based on the experimental results, and combining some of the
previous equations, one can obtain: 12 [ CO 3 - 2 ] = K 1 K 2 K H p
CO 2 [ H + ] 2 = 2.8637 e - 06 M ( 23 )
[0101] Substituting in Equation (5) one gets the equilibrium
constant K.sub.SO, using the experimental calcium concentration
(2.65e-03 M):
K.sub.SO=[Ca.sup.+2][HCO.sub.3.sup.-]=7.588e-09=10.sup.-8.1198
(24)
[0102] This results in a K.sub.SO, almost twice the reported in the
literature. Running the equilibrium software for the new K.sub.SO
and the other constants at 21.degree. C., the following results are
obtained:
[0103] CASE C: Theoretical with new K.sub.SO
[0104] i. pH=7.3061
[0105] ii. pCO.sub.2=10.sup.-1.985 atm=1.0351e-02 atm
[0106] iii. [Ca]=2.2e-03 M=89.75 ppm
[0107] iv. Alk=4.5e-03 M=274.5 ppm
[0108] Although the predicted equilibrium calcium concentration is
now closer to the experimental, the alkalinity estimation is
negatively influenced. The summaries of the results are shown in
the following table.
3 T pCO.sub.2 Case (.degree. C.) (atm) pH [Ca] (M) Alk (M) Comments
Text 25 10.sup.-3.5 8.304 4.636e-04 9.2721e-04 As shown in Morgan
Text 25 10.sup.-3.5 8.304 4.636e-04 9.2721e-04 Duplicated in Matlab
A 25 10.sup.-1.04 6.8 2.5e-03 5e-03 Theoretical B 21.3 0- 7.3
2.7e-03 6e-03 ALBACAR 10.sup.-1.45 PCC Experi- ment C 21.3
10.sup.-1.985 7.306 2.2e-03 4.5e-03 New Kso D 21.3 10.sup.-2.05 7.3
1.4e-03 2.8e-03 Theoretical E 25 N/A 7.1 (?) 2.46e-03 5.08e-03
Grade CaCO3 ex- periment
[0109] The table shows that the experimental results with chemical
grade CaCO3 meet the calcium and alkalinity relationship
consistently (CASE E). Hence, the dynamic experimental data from
this experiment will be used to compare them with the dynamic
models. The complete set of operating conditions, such as the
pCO.sub.2, will be used from the equilibrium calculations (CASE
A).
[0110] Kinetic (Rate) Analysis of Results
[0111] The mass transfer coefficient, kx, in Equation (9) defines
the speed at which gaseous CO.sub.2 is transferred to the liquid.
It depends on physical properties and equipment design. Several
correlations can be found in the literature for specific designs
and operating conditions. Based on published literature, a typical
mass transfer coefficient in an agitated tank where the gas is
bubbled near the stirrer is in the order of 0.2 min.sup.-1. This
number was used as a reference for the next simulation.
[0112] It was found that the experimental results shown in Table 2
are the equilibrium conditions when the partial pressure is 0.09
atm. This pressure will be utilized as the nominal operating
pressure.
[0113] Finally, the surface area is needed to determine the
dissolution rate according to Equation (16). The surface area of
the ALBACAR OH (Specialty Minerals, Inc) was found to be equal to
11.5 m.sup.2/gr (11.5e+04 cm.sup.2/gr) with mean diameter sizes of
.about.1.6 micrometers. On the other hand, the CaCO3 from Fisher
has a mean diameter of 30 to 50 micrometers. This may represent a
surface area in the range of 10.sup.4 cm.sup.2/gr. Note that the
maximum surface area of the calcium carbonate reported on the
calcium dissolution paper by Plummer et. al. is 90 cm.sup.2/gr,
which is several orders of magnitude smaller compared to the
calcium carbonate tested at CRC. The consequence of the surface
area in calcium dissolution and overall kinetics is presented later
on.
[0114] Dynamic Model Test
[0115] Using the nominal operating conditions:
[0116] i. pCO.sub.2=0.09 atm
[0117] ii. kx=0.2 min.sup.-1
[0118] iii. T=25.degree. C.
[0119] iv. a.sub.c=calcium carbonate surface area=10.sup.3
cm.sup.2/gr
[0120] This set of conditions will be labeled as Conditions Set
A
[0121] The initial conditions correspond to the equilibrium with
the atmospheric CO.sub.2 partial pressure (10.sup.-3.5 atm):
pH=8.96; Alk=3.3e-04molHCO.sub.3/L; [Ca.sup.+2]=1.65e-04 mol Ca/L;
C.sub.T=3.45e-04 mol/L. In this report, all the alkalinities are
expressed on mol/L, meaning mol [HCO.sub.3-]/L, and the calcium
will be expressed in mol/L, meaning mol Ca/L.
[0122] Table 4 shows the simulation model using the condition Set A
compared to the experimental values. The steady state predictions
of the model match the experimental values when the nominal partial
pressure of CO.sub.2 was lowered from 0.09 atm to 0.07 atm. This
difference can be attributed to loss of accuracy when discretizing
the dynamic models in order to solve it with equilibrium equations.
The solution shown in Table 4, furthermore, was difficult to
achieve, as it required some "troubleshooting". When solving the
ODE's (Equations (9) and (17)), it was noticed that the solution of
dCa/dt was resulting in [Ca.sup.2+] values higher than the values
that could be obtained in equilibrium. The dynamic solution of
dCa/dt in this system cannot resolve in higher concentrations than
equilibrium. Under steady state conditions, dCa/dt should only
equalize the equilibrium predictions. This problem led to conclude
that kinetic expression in Equation (16) over predicts the calcium
dissolution when the calcium carbonate surface area is
significantly higher than the one used by Plummer et. al. (90
cm.sup.2/gr).
[0123] When the calcium carbonate surface area is high
(.about.10.sup.4 cm.sup.2/gr and higher), the calcium dissolution
is so fast that it approaches practically instantaneously to
equilibrium, which makes this reaction as fast as the
proton-transfer reactions. The fast calcium dissolution then
suggests to calculate the calcium at any specific time using the
equilibrium equation as given by Equation (5), simultaneously with
the other proton-transfer reactions. So, for high surface areas,
only one differential equation is solved in each time interval
(dC.sub.T/dt). Once C.sub.T is known from Equation (9) at one
sample interval, the following equations are solved to find the
equilibrium calcium concentration: 13 K 1 = [ H + ] [ HCO 3 - ] [ H
2 CO 3 * ] ( 25 ) K 1 = [ H + ] [ HCO 3 - ] [ H 2 CO 3 * ] ( 26 )
K.sub.s=[Ca.sup.2+][CO.sub.3.s- up.2-] (27)
C.sub.T=[H.sub.2CO.sub.3*]+[HCO.sub.3.sup.-]+[CO.sub.3.sup.2-]
(28)
[0124] These equations can be combined to find [Ca.sup.2+] as a
function of C.sub.T only: 14 C T = 4 K 2 K 1 K s [ Ca 2 + ] 3 + 2 [
Ca 2 + ] + K s [ Ca 2 + ] ( 29 )
[0125] Equation (29) is faster to solve since does not require
initial values to iterate, as is the case of solving simultaneous
nonlinear algebraic equations. The simulation results in Table 4
show the calcium calculations from equilibrium, as the kinetics
predictions were higher than equilibrium, which is not believed to
be possible.
[0126] The temperature and partial pressures of CO.sub.2 affect the
rate of dissolution, but also the equilibrium conditions. At this
point, one is only interested in changing the rate, while keeping
the equilibrium (steady state) results according to experimental
results. The only available parameter than can achieve this goal is
the mass transfer coefficient. The experiments were carried out at
very high speeds (1500 rpm), but the mass transfer coefficient is
not known. If the mass transfer coefficient is increased from 0.2
min.sup.-1 to 2.0min.sup.-1, the CO.sub.2 is transferred faster to
the bulk and the proton-transfer reactions occur earlier.
[0127] Table 5 shows the simulation results with the higher mass
transfer coefficient. It can be seen now that the experimental
results have much better agreement with the simulations.
[0128] In conclusion, the dynamic modeling of the
CO.sub.2--CaCO.sub.3 system has been developed to represent the
experimental results obtained at CRC. However, it must be
emphasized that in order to have good modeling results, knowledge
of the right operating conditions, such as partial pressure, and
mass transfer coefficient, along with the nature of the calcium
carbonate, have to be considered in order to have the best model
representation. The experimental example has allowed illustrating
some of the important features of the modeling. Next, a more
detailed explanation of the modeling issues will be shown.
[0129] Surface Area
[0130] The calcium dissolution kinetics is controlled by surface
reaction, which leads to the importance of the surface area. It was
already shown that for high surface calcium carbonate, a good
approximation is to disregard the calcium rate predictions and use
the equilibrium equations instead. Nevertheless, a deeper insight
on the calcium dissolution in case the surface area is small is
presented. From now on, several simulations will be shown using the
conditions defined in the following condition Set B:
[0131] Condition Set B:
[0132] i. pCO.sub.2=0.025 atm
[0133] ii. kx=0.2 min.sup.-1
[0134] iii. T=25.degree. C.
[0135] iv. a.sub.c=1e+0.5 cm .sup.2/gr
[0136] Table 4 and Table 5 show simulations that assume that the
calcium surface is significantly high (10.sup.4 cm.sup.2/gr+),
which makes the dissolution evolve so fast that reaches equilibrium
almost instantly and the calcium is calculated from equilibrium
equations. Table 6 shows simulations when the surface area is
smaller and then the calcium dissolution is described by the
kinetic equation reported by Plummer et al.
[0137] In general, it can be seen that the lowest surface area (90
cm.sup.2/gr) takes several orders of magnitude longer to reach
equilibrium. The highest surface area simulated of 9e+04
cm.sup.2/gr was implemented combining the equilibrium equations and
the kinetic equations. At the beginning of the simulation and until
approximately 5 minutes, the rate predictions were higher than the
equilibrium, which is not possible, so the equilibrium predictions
are reported. After 5 minutes, the rate predictions were lower than
the equilibrium, and hence the rate predictions by Plummer et. al.
are plotted. The switching between one model and another causes
some numerical discrepancies shown as spikes in the
simulations.
[0138] Note the effect that the surface area has on pH. A high
surface area enhances the calcium dissolution and the proton
transfer equations equilibrate to lower [H+] concentration (high
pH). On the other hand, when the surface area is small, the calcium
dissolution is very slow and the equilibrium concentration for the
proton-transfer species, such as [H+], is similar to when barely
there is calcium. For this reason, Table 6 shows the simulation
when there is practically no calcium and there is no CaCO.sub.3 to
dissolve. When CaCO.sub.3 is absent, the alkalinity and calcium
remain constant, and the pH drops rapidly to lower steady state
values.
[0139] Table 7 shows a closer view of the pH of Table 6. It is more
clearly seen that when the surface area is small at initial times,
the system behaves as if there was no calcium dissolution and the
pH decreases rapidly. When the calcium concentration starts
building up, then the pH starts increasing towards the
corresponding equilibrium with CaCO.sub.3.
[0140] Mass Transfer Coefficient
[0141] The mass transfer coefficient does not affect the steady
state or equilibrium conditions. It only affects how fast the
equilibrium is reached, just as the calcium carbonate surface area.
From the process control point of view, these variables have an
effect on the time constant of the process and not the gain. Table
8 shows the effect of different mass transfer coefficients in all
the variables.
[0142] Temperature
[0143] One factor that affects the equilibrium conditions is the
temperature of the system. All the equilibrium and kinetic
constants are temperature dependent, and any variation will have an
impact on all the concentrations. Table 9 shows a couple of
simulations at different temperatures. The highest temperature
shown (50.degree. C.) drives the reactions towards lower calcium
dissolution. The values of the equilibrium constants in Appendix
show that the temperature affects the constants K.sub.1, K.sub.2
and K.sub.s in different directions.
[0144] Regarding the sensitivity of the variables with respect to
temperature, Table 9 shows that the temperature has a bigger effect
on the concentrations than on pH. The small change in pH, however,
represents a big change in the [H+] concentration change (because
of the logarithmic relationship). If small temperature changes
occur in the mill, big performance changes are expected when using
CO.sub.2
[0145] CO.sub.2 Partial Pressure
[0146] Table 10 shows the effect of the CO.sub.2 partial pressure
on the CO.sub.2--CaCO.sub.3 system. The increase in the partial
pressure promotes the transfer of CO.sub.2, the pH reduction and
the calcium dissolution. The partial pressure also has a direct
impact on the steady state conditions (equilibrium). Unlike
temperature, the relatively smaller changes in the CO.sub.2 partial
pressure, the effect on all variables, including the pH, is
greater. This remains as the most important manipulating variable
that specifies the final operating conditions.
[0147] Conductivity
[0148] During the experiments using ALBACAR OH (Specialty Minerals,
Inc), the on-line conductivity measurements were recorded. Table 11
shows the on-line conductivity measurements of one of the
CO.sub.2--CaCO.sub.3 experiments using ALBACAR PCC (Specialty
Minerals, Inc). Conductivity is a measure of the ability of an
aqueous solution to carry an electric current. The ability depends
on the presence of ions; on their total concentration, mobility,
and balance; and on the temperature of measurement. A general
theoretical method to calculate the conductivity is presented by
Clesceri et al. Clesceri, L. S., Greenberg, A. E., and Eaton, A. D.
"Standard Methods for the Examination of Water and Wastewater",
20.sup.th Edition, APHA, AWWA, WEF. First, the infinite dilution
conductivity is calculated:
k.sup.0=.SIGMA..vertline.z.sub.i.vertline.(.lambda..sub.-i.sup.0)(mM.sub.i-
)+.SIGMA..vertline.z.sub.i.vertline.(.lambda..sub.-i.sup.0)(mM.sub.i)
(30)
[0149] where:
[0150] .vertline.zi.vertline.=absolute value of the charge of the
i-th ion
[0151] mMi=millimolar concentration of the i-th ion
[0152] .quadrature..sup.0.sub.+i,
.quadrature..sup.0.sub.-i=equivalent conductance of the i-th
ion
[0153] Then, calculate the ionic strength, IS in molar units:
IS=.SIGMA.z.sub.i.sup.2(mM.sub.i)/2000 (31)
[0154] The monovalent ion activity coefficient, y, is calculated
using the Davies equation for IS<=0.5M and for temperatures from
20 to 30.degree. C. 15 y = 10 - 0.5 ( a - 0.3 / S ) ; a = IS 0.5 1
+ IS 0.5 ( 32 )
[0155] Finally, the conductivity is calculated as:
k.sub.calc=k.sup.0y.sup.2 (33)
[0156] Equivalent ionic conductivities in aqueous solutions can be
found in different sources, including Lange's Handbook of
Chemistry. Dean, J. A. "Lange's Handbook of Chemistry",
McGraw-Hill, Thirteen Edition. The equations to calculate the
conductivity as a function of the molar concentration of the ions
in solution are then used to "back-calculate" the ion
concentrations based on the conductivity measurements. Since DI
water was used in the experiments, it is known that only two main
components can be found in the solution: [Ca.sup.2+] and
[HCO3.sup.-]. However, there are an infinite number of ion
concentration combinations that can produce the same conductivity.
For this, an additional relationship is needed. The proton balance
equation in a CO2--CaCO3 for the range of pH of interest, is
simplified as
2[Ca.sup.2+].about.=[HCO3.sup.-]=Alk (34)
[0157] Then, an optimization problem to calculate the concentration
of the [Ca.sup.+2] and [HCO3.sup.-] ions is used and defined as: 16
min [ Ca 2 + ] , [ HCO 3 - ] J = min ( k exp - k ^ calc ) 2 ( 35
)
[0158] such that Equation (32) is true. The back calculation is
solved by minimizing the objective factor, J. This factor consists
of the squared error of the measured or experimental
conductivities, k.sub.exp, and the estimation of the calculated
conductivities from Equation (k.sub.calc). The solution will be the
minimization of the problem that meets the proton balance equation.
This problem is solved numerically in Matlab with the minimization
of a constrained multivariable function.
[0159] Table 12 shows the experimental on-line conductivity
measurements and off-line [Ca.sup.2+] and alkalinity measurements
of an experiment using ALBACAR PCC (Specialty Minerals, Inc). Aside
from the first off-line sample, which may be considered inaccurate
due to sampling difficulties, the calcium predictions from the
conductivity measurements match very close the off-line
experimental measurements. The alkalinities, on the other hand,
have larger estimation errors. The maximum error is up to 40
units.
[0160] Table 13 shows the estimations of [Ca .sup.2+] and
alkalinity of the experiment using chemical grade CaCO3. The
calcium estimates are almost as good as in the previous case. The
alkalinities, on the other hand, have smaller prediction errors.
This suggests that this experiment consists of ions closer to the
expected. The experiment with ALBACAR could have some components or
ions that affect the conductivity, and may drive Equation (32) off.
In summary, it can be said that the conductivity measurement has
the potential to predict the evolution of the most important ions
in the CO.sub.2--CaCO.sub.3 system.
[0161] In the preferred embodiments thus described, phenomena that
help determine the rate of species in a CO.sub.2--CaCO.sub.3 system
are the CO.sub.2 transfer and calcium dissolution. The other
reactions are proton-transfer reactions that occur quickly and can
be considered in equilibrium, including the alkalinity. This
disclosure presents the calculation of the proton-transfer
reactions calculated and updated at every integration step of the
rate equations.
[0162] The calcium dissolution can be calculated with equilibrium
equations if the calcium carbonate surface area is significantly
large (over 10.sup.4 cm.sup.2/gr). The CaCO.sub.3 used in the
experiments of this report have high surface areas and the best
predictions results considered equilibrium calculations of calcium.
The increase of alkalinity with the supply of CO.sub.2 has a big
impact on the pH, and hence on the calcium dissolution.
[0163] Once the alkalinity was calculated in a time dependent
manner, the CO.sub.2--CaCO.sub.3 system was simulated for a variety
of conditions in CO.sub.2 partial pressure, temperature, mass
transfer coefficient and calcium carbonate surface area. If the
calcium carbonate surface area is large, the most important factor
that controls the rate of the system is the mass transfer
coefficient. If the surface area is small (large particle
diameters), the time constant of the system can be increased
dramatically due to slower calcium dissolution rate.
[0164] Unlike the mass transfer coefficient and the particle
surface area, the temperature of the system and the CO2 partial
pressure have an effect on the equilibrium (steady state)
conditions. The partial pressure has a bigger impact on the
equilibrium conditions than temperature, so it is important to have
a well-controlled operation of CO.sub.2.
[0165] Finally, it is shown that the conductivity measurements can
be used to predict the calcium and alkalinity of
CO.sub.2--CaCO.sub.3 systems. A more detailed development would be
needed if other ionic species were present.
[0166] Referring now to FIG. 1, a schematic diagram illustrating an
embodiment of the control method adapted to a specific papermaking
process is shown. A mixing chest 1 receives a regulated inlet flow
of CO.sub.2 and a papermaking composition such as a fiber flow, 7,
and provides a CO.sub.2--enriched papermaking composition, such as
an outlet fiber flow. Properties of the fiber flow such as, for
example, flowrate, CD (charge demand), ZP (Zeta Potential), ion
concentration, Cy (consistency), pH, conductivity and alkalinity
are measured on-line before the fiber flow reaches the mixing
chest, and measurements 2 are generated. At the same time, on-line
properties of the outlet fiber flow are measured by instrument 3,
and can be chosen from the same set of properties as described
above. An advanced controller, in this case a feed forward
controller, uses the measurements of the inlet flow and compensates
the feedback controller, 5, by adding the controller outputs, 6.
The resulting controller output is used to manipulate the inlet
CO.sub.2 that will maintain the desired wet end properties while
minimizing variations in the inlet fiber flow.
[0167] Note that certain properties can be described as electrical
properties, such as ZP, CD and ion concentration. Since an ion is
an atom or molecule which has gained or lost one or more electrons,
it thereby has a net negative or positive electrical charge. For
example, a fusion plasma is so hot that virtually all the electrons
are stripped from the atoms creating ions that have a net positive
charge equal to the number of protons in their nucleus. Ion
concentration is related to the amount of such ions in per unit
volume. Thus, there is a direct link of a concentration of ions
having an electrical charge.
[0168] However, if there are two different ions A and B with
concentration [A] and [B] mixed in a solution, there is still a
total electrical charge, but measuring this electrical charge, for
example with a conductivity meter in the solution, will not enable
distinction and measurement of [A] and [B]. The present disclosure
anticipates separately distinguishing and measuring [A] and [B]
individually. It is to be generally noted that ion concentration,
as used herein, anticipates performing this operation where
desired.
[0169] FIG. 2 shows the block diagram for the feed forward
controller. It requires the mathematical relationship between the
disturbance and the wet end measure, in this case the Zeta
Potential (ZP), GL, and the relationship between the input CO.sub.2
and the same wet end measure, Gp. The advanced controller design
consists on finding the mathematical relationship of the feed
forward controller F, and the feedback controller Gc.
[0170] This embodiment is applicable in the wet end of pulp mills
to control the properties of the paper by the controlled addition
of additives. More specifically, this embodiment considers the
addition of CO.sub.2 in the wet end in a mixing chest. The
measurement of the most important variables in the inlet and outlet
fiber flows such as Zeta Potential, charge demand, alkalinity, pH,
etc. are taken in real-time, collected and made available to a data
acquisition system. The mathematical relationship in the form of
transfer function or any time dependent form are utilized to relate
the CO.sub.2 addition and the desired wet end property, or
controlled variable, such as Zeta Potential or charge demand. In
addition, similar relationships are utilized between possible
variations or disturbances in the inlet fiber flow and the
controlled variable. Any feedback controller that tends to minimize
the variations between the desired controlled value and the
real-time measurements in the outlet fiber flow is used.
[0171] In addition, the available real-time measurements from the
inlet fiber flow are used to design a predictive controller, such
as feed forward controller, model predictive control (MPC), or any
advanced controller. The feed forward controller modifies in a
coordinated manner the output of the feedback controller. The
feedback and feed forward signals are added and sent to a linear
proportional control valve that controls the CO.sub.2 flow into the
mixing tank.
[0172] FIG. 3 shows the performance of a feed forward control
responding to a normalized step change and a charge demand
disturbance. It can be seen that the feedback controller can
respond with no problems to the set point change, and that due to
the feed forward controller, the disturbance is rejected almost
immediately, returning the plant to the desired set point.
[0173] Illustrating another preferred embodiment, FIG. 4 shows a
mixing chest 11, where a regulated inlet flow of CO.sub.2 is
supplied, 7. Some properties of the fiber flow are measured on-line
before going to the mixing chest by measurements 12. At the same
time, some on-line properties of the outlet fiber flow are measured
by 13. All available on-line measurements are used to compute the
unobservable variables or states, 14, using a real-time model, 15,
and an observer or optimal estimator, such as Kalman Filter,
16.
[0174] The Kalman Filter observer is normally known as a
model-based observer as it relies on the real-time model, 15.
Data-driven observers such as neural networks, do not require a
real-time model, but require larger amounts of data to be trained.
The observer then estimates in real-time the unobserved variables
and passes this information to a controller, 18, that manipulates
the input of CO.sub.2 that changes the unobserved variable to the
desired set point.
[0175] This embodiment can be applied to control and optimization
of the wet end of paper mills. It is intended to control the wet
end in real-time when the intended control variable is not measured
or available in real-time. The available on-line measurements, 12
and 13, are related to control variables or states in what is
called an observation equation. The observation equation is part of
the estimator or observer 6. The observation equation can assume
that there is some noise in the instruments and that the rate of
acquisition varies. This way, the information used by the estimator
can also use off-line information that is available in long time
intervals.
[0176] The observation equation is related to the observer, as is
the state equation. The state equation is the transformation of the
model, 15, which is suitable for an estimation algorithm along with
the observation equation. The state equation also includes some
model uncertainty, which corresponds to the inaccuracies of the
model. There are several estimation algorithms, but one of the most
common is the Kalman Filter (Kalman, R. E.) prediction for being an
optimal estimator. The KF estimator is a real-time estimator that
estimates the unobservable variables using the available
measurements and the available model. The KF observer depends on
the process model, 15, and so it is called a model-based
observer.
[0177] If no model was available, a data-based observer such as
neural network can be employed, but this requires a larger amount
of data. The real-time optimal estimates, 14, become the process
variable measurements required by the controller 18, which compares
the estimates with user-defined set points, and calculates a
control output based on the errors. The control output is sent to a
linear proportional valve that changes the addition of CO.sub.2
into the reactor.
[0178] A method for controlling the CO.sub.2 addition in a wet end
process utilizing CO.sub.2 addition is disclosed. A papermaking
composition and CO.sub.2 are combined to create a CO.sub.2-enriched
papermaking composition. At least one electrical property of either
the papermaking composition or of the CO.sub.2-enriched papermaking
composition is either measured or estimated. The rate of addition
of CO.sub.2 to maintain the at least one electrical property within
a pre-selected range of values is then controlled.
[0179] In one preferred embodiment, the electrical property is
selected from the group consisting of ZP, CD and ion concentration
or any equivalent thereto, and is estimated by measuring at least
one property of either the papermaking composition or the
CO.sub.2-enriched papermaking composition selected from the group
consisting of flowrate, CD, ZP, ion concentration, Cy, pH,
conductivity and alkalinity and using a model. Preferably, the ZP
is estimated. Thus, the measured property can be measured from
either the papermaking composition or the CO.sub.2-enriched
papermaking composition, and is preferably measured from the
papermaking composition. Likewise, the estimated electrical
property can be estimated for either the papermaking composition or
the CO.sub.2-enriched papermaking composition, and is preferably
estimated for the CO.sub.2-enriched papermaking composition.
[0180] In another preferred embodiment, a method for controlling
the CO.sub.2 addition in a wet end process utilizing CO.sub.2
addition includes more sophisticated controls. A papermaking
composition and CO.sub.2 are combined to create a CO.sub.2
-enriched papermaking composition. At least one property of the
papermaking composition is measured or estimated and papermaking
composition property data is generated. The papermaking composition
property data is provided to an advanced controller which generates
a papermaking composition output component.
[0181] At least one property of the CO.sub.2-enriched papermaking
composition is measured or estimated and CO.sub.2-enriched
papermaking composition property data is generated. The
CO.sub.2-enriched papermaking composition property data is provided
to a feedback controller which generates an outlet controller
output component. The feedback controller is compensated by
analyzing the inlet controller output component and the outlet
controller output component. The inlet flow of CO.sub.2 is
controlled to maintain at least one property of the
CO.sub.2-enriched papermaking composition within a pre-selected
range of values.
[0182] The at least one property of the papermaking composition is
preferably selected from the group consisting of flowrate, ZP, CD,
ion concentration, Cy, pH, conductivity and alkalinity. The at
least one property of the CO.sub.2-enriched papermaking composition
is preferably selected from the group consisting of ZP, CD and ion
concentration, and is most preferably ZP. Most prerably, the
advanced controller comprises a feed forward controller.
[0183] In alternatives of this embodiment, the advanced controller
comprises a feed forward controller, and the feed forward
controller uses either predictive control or inferential
control.
[0184] In yet another preferred embodiment, a method for
controlling the CO.sub.2 addition in a wet end process utilizing
CO.sub.2 addition includes somewhat different controls. A
papermaking composition and CO.sub.2 are combined to create a
CO.sub.2-enriched papermaking composition. On-line measurements of
at least one property of the papermaking composition selected from
the group consisting of flowrate, ZP, CD, ion concentration, Cy,
pH, conductivity and alkalinity are made, papermaking composition
property data is generated and transmitted to an observer. On-line
measurements of at least one property of the CO.sub.2-enriched
papermaking composition selected from the group consisting of
flowrate, ZP, CD, ion concentration, Cy, pH, conductivity and
alkalinity are made and CO.sub.2-enriched papermaking composition
data is generated. These data are provided to an observer that
generates at least one estimated electrical property of the
CO.sub.2-enriched papermaking composition selected from the group
consisting of ZP, CD and ion concentration. The observer transmits
the papermaking composition property data, the CO.sub.2--enriched
papermaking composition property data and the estimated electrical
property data to a controller, and the controller controlling the
inlet flow of CO.sub.2 to maintain at least one electrical property
of the CO.sub.2-enriched papermaking composition within a
pre-selected range of values.
[0185] The observer can comprise a model, and can further refine
the estimated electrical property data by analyzing inaccuracies
presented by the model and by analyzing expected errors in
measurement. The papermaking composition property data, the
CO.sub.2-enriched papermaking composition property data and the
estimated electrical property data can be incorporated into a
software sensor. The estimated electrical property data can be used
to evaluate a set point and implement a real time closed loop
control.
[0186] Note that, in all of the embodiments described above, in a
wet end process utilizing CO2 to control chemical properties in the
wet end by injecting CO2 at specific locations in the process, the
CO2 injection can be controlled to impart the desired properties
either to the liquid or to the fibers, both of which can be present
in the papermaking composition. These can be achieved using
properties of the fibers or the liquid and rejecting process
disturbances.
[0187] For example, the CO2 can be injected in at least one point
of the wet end either directly to the fiber flow or to a fiber free
liquid that will mix later on with the fibers. Alternatively, the
CO2 can be injected into a tank. The controlled injection can be
either manual if the measurements are off-line or automatic if the
measurements are on-line.
[0188] The manual control takes off-line samples and the properties
are measured from these samples. The CO2 is then manually regulated
based on some recipes. The recipes for manual control are based on
multivariable linear or nonlinear regressions between the CO2
injection, the measurements and the desired set points.
[0189] The automatic control can either be feedback, feedforward, a
combination or an advanced controller. The feedback control takes
at least one measurement of the process after the mixing the fiber
flow and the CO2 rich flow (pure CO2 or water mixed with CO2). The
feedforward takes at least one measurement of the process before
the CO2 (pure CO2 or water mixed with CO2) is injected and adjusts
the CO2 before the process is affected. An advanced controller is
either a combination of the feedforward and the feedback or an
advanced controller such as but not limited to cascade control,
adaptive control, optimum control, robust control, neural network
controller, fuzzy control, model predictive control, etc. The
advanced controller adjusts the CO2 to maintain the desired
chemical property in the wet end, while rejecting any disturbance
or minimizing the use of chemical additives in the wet end.
[0190] The chemical properties to be maintained by the CO2
controller in the wet are but not limited to ZP (Zeta Potential),
CD (Charge demand), pH, ion concentration, etc. The disturbances
that can be rejected by the CO2 controller are but not limited to
broke recirculation, variations in inlet charge demand, variations
in inlet ZP, variations in pH, temperature, consistency, etc. The
CO2 controller uses properties of the process that can be either
intrinsic or extensive. Intrinsic information can be at least one
but not restricted to pH, conductivity, CD, ZP, ion concentration,
etc. Extensive information can be at least but not restricted to
flow, volume, etc.
[0191] The on-line properties used by the controller can either be
measured or unmeasurable. The measured properties can be at least
one but not limited to pH, conductivity, temperature, flow, CD, ZP,
etc. The unmeasurable or unobservable properties can be at least
one but not limited to CD, ZP, ion concentration, etc. The
unmeasurable or unobservable properties can be estimated on-line
and used by the CO2 controller using an observer or estimator. The
observer or estimator can predict the unmeasurable or unobservable
properties based on algorithms provided by, e.g., Kalman filters or
neural networks. Observers use process knowledge either from
fundamental models, empirical models or heuristic models.
[0192] Note further that, in the embodiments described above, ion
concentrations can be from [H+],[OH-],[Ca+2], [Na+], [HCO3-],
[CO3--], etc.
[0193] Moreover, in the embodiments described above, there are at
least three techniques for performing automatic control: using
properties in the papermaking composition, using properties in the
CO2-enriched papermaking composition, and using properties in both.
For the first case, it is desired to use feed forward that can be
typical feedforward or inferential or predictive control. In the
second case, a typical feedback becomes advanced when using
adaptive, model predictive, robust, optimal, neural networks, fuzzy
control, dynamic matrix control, etc. For the third case,
combinations of both techniques can be employed.
[0194] While in the foregoing specification this invention has been
described in relation to certain preferred embodiments thereof, and
many details have been set forth for purpose of illustration, it
will be apparent to those skilled in the art that the invention is
susceptible to additional embodiments and that certain of the
details described herein can be varied considerably without
departing from the basic principles of the invention.
* * * * *