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Numerical simulation of a semi-continuous counter-current adsorption unit for fructose-glucose separation
The Chemzcal Engmeermg
Journal, 33 (1986)
B55
B55 - B61
Numerical Simulation of a Semi-continuous Counter-current Adsorption Unit for Fructose-Glucose Separation K HIDAJAT
Department
and C B CHING
of Chemical Engmeerzng,
Natzonal
Unzverslty
of Smgapore,
Kent Ridge 0511, (Smgapore)
D M RUTHVEN
Department (Canada) (Received
of Chemzcal Engmeermg, July 18, 1985,
Unwerszty
m final form January
of New Brunswick,
Frederzck,
New Brunswzck
E3B 5A3
21, 1986)
ABSTRACT
The transaent and steady state behavwur of a semg-contmuous counter-current adsorption umt, szmdar to the UOP ‘Sarex”process, for the separataon of fructose-glucose maxtures has been stmulated numeracally and studled expertmentally Both transzent and steady state concentra tlon pro files are m good agreement with the predlctlons of a simple model m which the system 1s considered as a cascade of ideal theoretical stages, with the number of stages determined a prlorl from pulse chroma tographic measurements
useful representation, it 1s based on a rather severe ldeahzatlon and cannot therefore be expected to account fully for the detaded behavlour of the real system. An alternatlve approach, which 1s followed here, 1s to simulate the system numencally, treating each column as a cascade of ideal mlxmg stages This approach requires far more bulky numerical calculations but has the advantage that it provides a more detailed descnptlon of the system behavlour and yields mformatlon on the transient behavlour as well as on the final cychc steady state
2 EXPERIMENTAL
DETAILS
1 INTRODUCTION
Simulated counter-current adsorption processes, m which the fluid inlet and outlet pomts are switched at mtervals through a fixed adsorbent bed m order to simulate the effect of continuous counter-current contact, have been developed for a number of mdustnally important separations mcludmg the separation of glucose and fructose (the Sarex process). The subJect has been revlewed by de Rossett et al [ 11 and Ruthven [2]. In previous papers (Chmg et al [3,4]) we reported the results of an experimental and theoretical study of a small scale “Sarex” unit and showed that the expenmentally observed behavlour could be understood by modellmg the system m terms of the equivalent true counter-current system. Although such a model provides a practically 0300-9467/86/$3
50
The experimental system, shown schematically m Fig 1, consisted of 12 identical columns (of internal diameter 5 1 cm and length 100 cm) packed with Duohte ion exchange resin m the Ca*+ form The columns were connected m series through pneumatically operated switch valves and there were liquid sampling points between each pour of columns Additional pneumatic valves allowed the feed and eluent to be introduced or the raffmate and extract product streams to be withdrawn between any pax of columns To control the flow rates of extract, raffmate and eluent streams, addltlonal metermg pumps were added, m addltlon to the feed The system was operated as sketched, with four columns between eluent inlet and extract outlet, three columns between extract and feed, three columns @ Elsevler
Sequola/Prmted
m The Netherlands
between feed point and raffmate and two columns between raffmate draw-off and eluent inlet (sections I, II, III and IV respectively) Simulated counter-current operation was achieved by advancing the eluent, extract, feed and raffmate points at fured time mtervals by one column m the dlrectlon of the hquld flow. At approximately the mldpomt of the swltchmg interval, liquid samples were withdrawn from the sample pomts and analyzed for glucose and fructose by high pressure liquid chromatography (HPLC) using a “Sugarpak” column
Extract, \
QE Sectjon
II 7
Direction oi fluld flowand port swlchlng
4
TABLE
1
Summary values a
Eluent
of experimental
condltlons
42 2 22 8 153 90 16 5 0 88 0 50 30 5
QR~ (ml mn-l) &a (ml mn-*) QE (mlmn-‘) QF (ml mu-l) QR (ml rnln-‘) KF KG
Switch
time (mm)
and parameter
7
aTen theoretlcal stages per column, twelve columns, feed contams 5 wt % glucose and 5 wt % fructose Note that the experimental condltlons (and therefore the steady state proflle) are slmllar to those of Run 2a described In our previous paper [ 41
Details of the experimental condltlons are given m Table 1 Poor to the run the system was fully purged with dlstllled water and at time zero the feed solution, containing 5% fructose and 5% glucose, was introduced between the seventh and eighth columns, as indicated m Fig 1 Details of how the experimental condltlons were chosen have been described elsewhere
[41
MODEL
Pulse chromatographlc measurements reported previously (Chmg and Ruthven [ 3 ] ) suggest that, m the experimental system, mass transfer resistance 1s relatively small and
QRe
Rafflnate,
QR
Fig 1 Schematic diagram of simulated counter-current adsorptlon system showing notation for flow rates, concentration and theoretlcal stages
dispersion of the response arises mamly from axial mlxmg The chromatographlc height equivalent to a theoretical plate (HETP) was found to be about 10 cm for both fructose and glucose and essentially independent of fluid velocity Adopting the plate model, we can therefore say that each column may be consldered as equivalent to ten theoretical stages To model the simulated countercurrent umt, each theoretical stage 1s consldered as an ideal murmg cell of total volume equal to one tenth of the column volume (204 ml) dlvlded between solid (lmmoblle) and liquid (mobile) phases m accordance with the bed voldage (z e V, = 204 (1 - e) = 122 4, V, = 2046 = 816 ml ) The transient differential mass balance for an mdlvldual stage 1s
Qc,_ , =
dc,
Qc,+ V, __
dt
dqz
+ V, -dt
where Q IS the volumetric flow rate and c, _ 1, and outlet concentrations respectively The adsorption equlilbrmm constants for fructose and glucose (defined by q = Kc) are 0 88 and 0 5 respectively [3, 41 and since the adsorption equlllbnum relatlonships for these species are linear and nonmteractmg, the mass balance relations for each component may be treated mdec, are the inlet
3 MATHEMATICAL
QE,
pendently Assuming equilibrium m each stage, eqn (1) may be written for each component m the form
used a standard numerical method (Gear’s algorithm) described m the IMSL manual [6], an analytical Jacobian matrix of partial derivatives was supplied and stiff options were required to solve these equations. At each switch, the positions of the eluent, extract, feed and raffmate streams were advanced by one column m the direction of fluid flow, Just as m the real system
The flow rate Q is constant for all stages within a given section but varies from section to section as a result of mtroduction of feed or removal of product streams The full set of equations (see Appendix A) may be written in matrix form. dx
-=Ax+b dt
4 RESULTS AND DISCUSSIONS
Figures 2 - 4 show the theoretical profiles, calculated at the beginning, midpomt and end of the switchmg interval (30.5 mm), for cycle 1 (366 mm), cycle 3 (1098 mm) and at quasi-steady state, together with the experimental profiles, measured close to the mid-time of the switchmg mtervals. It is evident that there is good agreement between the experimental and predicted profiles, the measured profiles lying close to the theoretical profiles for the mid-point of the
(3)
This represents a set of first-order linear ordmary differential equations which may be solved analytically as described by Amundson [ 51, however, considerable mathematical mampulations are needed and computer solutions will still be required Instead we
J”
Glucose
Fructose
0
2
3
4
5 Column
6 7 Number
8
9
10
11
12
Fig 2 Comparison of theoretlcal concentration profiles after one complete cycle (12 switches) for glucose and for fructose at various stages of the swltchmg interval (numbers on curves denote time m mm from previous switch) with the experlmental profile (curve denoted by 0) measured near the middle of the switching interval
B58
Glucose
.xv
Fructose
-0
1
2
3
4
5
6
7
8
9
10
11
12
Column Number
Fig 3 Comparison of theoretlcal concentratzon proflles after three complete cycles (36 switches) for glucose and for fructose at various stages of the swltchmg interval (numbers on curves denote time tn mm from previous switch) with the experlmental profile (curve denoted by 0) measured near the middle of the swltchmg interval
50, Glucose 40-
2
30-
g :
zo-
lo-
o-
0 0
1
2
3
4
5
6
7
8
9
10
11
12
Column Number
Fig 4 Comparison of theoretical concentration profiles for glucose and for fructose at steady state at various stages of the swltchmg interval (numbers on curves denote time m mm from previous switch) with the experlmental profile (curve denoted by 0) measured near the middle of the switching Interval
B59
Fructose
0
1
2
3
4
5 Column
6
7
8
9
10
11
4l 12
Number
Fig 5 Theoretical profdes showing development of fructose concentration profile from a pre-loaded lmtlal condotlon (X) The curves denote profiles at the middle of the switch intervals after 1, 2 and 5 complete cycles The steady state IS almost comcldent with the curve for 5 cycles
switchmg Interval Since several minutes are required to take the necessary samples it is not possible to measure the experimental profile precisely at the mid-pomt of the switchmg interval and the differences between the theoretical and experimental profiles are no greater than is to be expected as a result of small differences m samphng time The numerical simulation showed that to reach the quasi-steady state, startmg from a clean bed, requires about ten complete cycles for fructose and six cycles for glucose (the less strongly adsorbed species) It 1s of obvious practical interest to reduce the transient time and this may be achieved, m principle, by pre-loading the system m accordance with the final steady state profile To pre-load the system to the precise profile is not a simple task but much the same effect can be achieved very simply, merely by equihbratmg columns 3 - 7 with a solution containing the feed concentration of fructose (5% m this case). This is illustrated m Fig 5 which shows the simulated profiles, startmg from the pre-loaded system. (The number of cycles required to establish the steady state glucose profile may be similarly reduced by pre-loadmg columns 7 - 10.)
steady state behavlour of the simulated counter-current adsorption unit Under the expelnmental operating conditions it requires about 60 h of operation to approach steady state conditions for fructose and about 37 h for glucose These times may be substantially reduced by partially pre-loading the system.
REFERENCES A J de Rosset, R W Neuzd and D B Broughton, m A E Rodrlgues and D Tondeur (eds ), Percolatzon Processes Theory and Applzcatzons, NATO AS1 No 3, SiJthoff and Noordhoff, Alphen van den Rijn, 1981, p 249 D M Ruthven, Prznczples of Adsorptzon and Adsorptzon Processes, Wdey, New York, 1984, p 380 C B Chmg and D M Ruthven, Chem Eng Scz ) 40 (1985) 877 C B Chmg, D M Ruthven and K HidaJat, Chem Eng Scz, 40 (1985) 1411 N R Amundson, Mathematzcal Methods zn Chemzcal Engzneerzng, Matrzces and Thezr Applzcatzon, Prentice-Hall, Englewood Chffs, NJ, 1966 IMSL Manual, International Mathematics and Statlstlcs Libraries Inc , Houston, TX
5 CONCLUSIONS APPENDIX A
The slmphfied numerical simulation based on representing the system as a cascade of equihbrmm stages provides an excellent representation of both the transient and the
The relevant notation is indicated m Fig. 1, and columns are numbered from the eluent mlet
B60
dc NII
Sectton I (N, theoretical stages) The concentration of the fluid entering section I 1s given by cNQRe/(QRe + Qm) where cN is the concentration leaving the last column m section IV, and the flow rate through sectlon I is (QRe + QEl) At z = 1 dc, -=
-a2cl
+ alcN
dc,
z-
= a2(cz
- 1-
b = QPA~& For NII+l <
dc, dt
(4)
ziere a2 = (Q ~1 + QR~)/(& (V,+KV,) Forl
+ KU,
dc, -=
(5)
dt
-
= a3(c,
dt where
- * -
z < Nm -l-c,)
a1(c, pl--
Equations form as dx =Ax+b iii where
Sectaon II (N,, theoretical stages) The flow rate m Section II IS (QEI + QRe QE) For N1 < l< N,,
dcz
=u4(c,
c,)
c,)
(4 - 8) may be written
X=
QE)
in matrix
1
Cl (a~1 + QR~ -
+ KU,
+ KV,)
Sectaon IV (NIV theoretical stages) The flow rate through section IV 1s QRe ForNm
al = QRJ
c,)
+I
+ b = U3CNII - a4cN&l dt where a4 = (QEl + QRe - QE + QF)/(&
2
b=
c2
u3 = -(v>vF
Nn+ 1 Sectton III (NIII theoretacal stages) The flow rate through section III IS (QEl + R Re - QE + QF) For 1 = N,, + 1 (feed plate)
N
.CN.
and 1
2
3 -
1
-a2
-
2
(12
-=2-
3
-
Ni
A= NI
a2
N
-
-
-
--
--
-
-
-
-
-
-
-
-
-
-
-
---
-
-
-
---
-
-
-
-
---
-
-
-
-
---
-
-
-
-
-
-
-
--
-
-
-
-
---
-
-
-
-
-
-
---
-
-
--_
-
---
-
-----
-
-
-
-
-
---
--
-
-
-
-
-
---
-
-
---
-
-
-
-
---
-
--__
-
-
-
----
-
---
-
-
-
-
-
--
-
---
--
--
-
--
-
-
-
N1:
--
--a2
41
NI ---
--
a2 -a2 a3
-a3 a3
-
-
-
-3
-
-
-
--
-
a3
-
--
--
-
--
--
-
--
--
-
---
-a3 a3
a4
-
a4 -a4 -a4 - a4
--
--
a4
N
NIlI -.
__ -
-a4 al
-
-
-
---
-
---
-
---
-
---
-
---
-
---
-
---
-
---
-
a1
-
---
-
---
-
---
-
---
-
-
---
-a1
-
---
-a1
-a,
---
--
a,
-
a
B61 APPENDIXB QI -a4
A b
b
CF
K KG,
N
KF
NOMENCLATURE
constants defined m Append= A square matrix defined m Appendur A constant defined m Append= A vector matrix defined m Appendur A concentration of solute leavmg stage i (g 100 ml-‘) feed concentration (g 100 ml-‘) adsorptlon equlllbnum constant adsorptlon equlllbnum constants for glucose and fructose respectively total number of theoretical stages m system
N, &II>
NI, NIV
9
L,
QE,
QF, QR, QR~ t
6 K
Greek symbol E
numbers of theoretical stages m Sections I - IV respectively adsorbed phase concentration (g 100 ml-l) flow rate (ml mm-‘) flow rates of eluent, extract, feed, raffmate and recycle streams respectively (ml mm-‘) time (mm) liquid volume m theoretical stage (ml) solid phase volume m theoretical stage (ml)
bed voldage
Journal, 33 (1986)
B55
B55 - B61
Numerical Simulation of a Semi-continuous Counter-current Adsorption Unit for Fructose-Glucose Separation K HIDAJAT
Department
and C B CHING
of Chemical Engmeerzng,
Natzonal
Unzverslty
of Smgapore,
Kent Ridge 0511, (Smgapore)
D M RUTHVEN
Department (Canada) (Received
of Chemzcal Engmeermg, July 18, 1985,
Unwerszty
m final form January
of New Brunswick,
Frederzck,
New Brunswzck
E3B 5A3
21, 1986)
ABSTRACT
The transaent and steady state behavwur of a semg-contmuous counter-current adsorption umt, szmdar to the UOP ‘Sarex”process, for the separataon of fructose-glucose maxtures has been stmulated numeracally and studled expertmentally Both transzent and steady state concentra tlon pro files are m good agreement with the predlctlons of a simple model m which the system 1s considered as a cascade of ideal theoretical stages, with the number of stages determined a prlorl from pulse chroma tographic measurements
useful representation, it 1s based on a rather severe ldeahzatlon and cannot therefore be expected to account fully for the detaded behavlour of the real system. An alternatlve approach, which 1s followed here, 1s to simulate the system numencally, treating each column as a cascade of ideal mlxmg stages This approach requires far more bulky numerical calculations but has the advantage that it provides a more detailed descnptlon of the system behavlour and yields mformatlon on the transient behavlour as well as on the final cychc steady state
2 EXPERIMENTAL
DETAILS
1 INTRODUCTION
Simulated counter-current adsorption processes, m which the fluid inlet and outlet pomts are switched at mtervals through a fixed adsorbent bed m order to simulate the effect of continuous counter-current contact, have been developed for a number of mdustnally important separations mcludmg the separation of glucose and fructose (the Sarex process). The subJect has been revlewed by de Rossett et al [ 11 and Ruthven [2]. In previous papers (Chmg et al [3,4]) we reported the results of an experimental and theoretical study of a small scale “Sarex” unit and showed that the expenmentally observed behavlour could be understood by modellmg the system m terms of the equivalent true counter-current system. Although such a model provides a practically 0300-9467/86/$3
50
The experimental system, shown schematically m Fig 1, consisted of 12 identical columns (of internal diameter 5 1 cm and length 100 cm) packed with Duohte ion exchange resin m the Ca*+ form The columns were connected m series through pneumatically operated switch valves and there were liquid sampling points between each pour of columns Additional pneumatic valves allowed the feed and eluent to be introduced or the raffmate and extract product streams to be withdrawn between any pax of columns To control the flow rates of extract, raffmate and eluent streams, addltlonal metermg pumps were added, m addltlon to the feed The system was operated as sketched, with four columns between eluent inlet and extract outlet, three columns between extract and feed, three columns @ Elsevler
Sequola/Prmted
m The Netherlands
between feed point and raffmate and two columns between raffmate draw-off and eluent inlet (sections I, II, III and IV respectively) Simulated counter-current operation was achieved by advancing the eluent, extract, feed and raffmate points at fured time mtervals by one column m the dlrectlon of the hquld flow. At approximately the mldpomt of the swltchmg interval, liquid samples were withdrawn from the sample pomts and analyzed for glucose and fructose by high pressure liquid chromatography (HPLC) using a “Sugarpak” column
Extract, \
QE Sectjon
II 7
Direction oi fluld flowand port swlchlng
4
TABLE
1
Summary values a
Eluent
of experimental
condltlons
42 2 22 8 153 90 16 5 0 88 0 50 30 5
QR~ (ml mn-l) &a (ml mn-*) QE (mlmn-‘) QF (ml mu-l) QR (ml rnln-‘) KF KG
Switch
time (mm)
and parameter
7
aTen theoretlcal stages per column, twelve columns, feed contams 5 wt % glucose and 5 wt % fructose Note that the experimental condltlons (and therefore the steady state proflle) are slmllar to those of Run 2a described In our previous paper [ 41
Details of the experimental condltlons are given m Table 1 Poor to the run the system was fully purged with dlstllled water and at time zero the feed solution, containing 5% fructose and 5% glucose, was introduced between the seventh and eighth columns, as indicated m Fig 1 Details of how the experimental condltlons were chosen have been described elsewhere
[41
MODEL
Pulse chromatographlc measurements reported previously (Chmg and Ruthven [ 3 ] ) suggest that, m the experimental system, mass transfer resistance 1s relatively small and
QRe
Rafflnate,
QR
Fig 1 Schematic diagram of simulated counter-current adsorptlon system showing notation for flow rates, concentration and theoretlcal stages
dispersion of the response arises mamly from axial mlxmg The chromatographlc height equivalent to a theoretical plate (HETP) was found to be about 10 cm for both fructose and glucose and essentially independent of fluid velocity Adopting the plate model, we can therefore say that each column may be consldered as equivalent to ten theoretical stages To model the simulated countercurrent umt, each theoretical stage 1s consldered as an ideal murmg cell of total volume equal to one tenth of the column volume (204 ml) dlvlded between solid (lmmoblle) and liquid (mobile) phases m accordance with the bed voldage (z e V, = 204 (1 - e) = 122 4, V, = 2046 = 816 ml ) The transient differential mass balance for an mdlvldual stage 1s
Qc,_ , =
dc,
Qc,+ V, __
dt
dqz
+ V, -dt
where Q IS the volumetric flow rate and c, _ 1, and outlet concentrations respectively The adsorption equlilbrmm constants for fructose and glucose (defined by q = Kc) are 0 88 and 0 5 respectively [3, 41 and since the adsorption equlllbnum relatlonships for these species are linear and nonmteractmg, the mass balance relations for each component may be treated mdec, are the inlet
3 MATHEMATICAL
QE,
pendently Assuming equilibrium m each stage, eqn (1) may be written for each component m the form
used a standard numerical method (Gear’s algorithm) described m the IMSL manual [6], an analytical Jacobian matrix of partial derivatives was supplied and stiff options were required to solve these equations. At each switch, the positions of the eluent, extract, feed and raffmate streams were advanced by one column m the direction of fluid flow, Just as m the real system
The flow rate Q is constant for all stages within a given section but varies from section to section as a result of mtroduction of feed or removal of product streams The full set of equations (see Appendix A) may be written in matrix form. dx
-=Ax+b dt
4 RESULTS AND DISCUSSIONS
Figures 2 - 4 show the theoretical profiles, calculated at the beginning, midpomt and end of the switchmg interval (30.5 mm), for cycle 1 (366 mm), cycle 3 (1098 mm) and at quasi-steady state, together with the experimental profiles, measured close to the mid-time of the switchmg mtervals. It is evident that there is good agreement between the experimental and predicted profiles, the measured profiles lying close to the theoretical profiles for the mid-point of the
(3)
This represents a set of first-order linear ordmary differential equations which may be solved analytically as described by Amundson [ 51, however, considerable mathematical mampulations are needed and computer solutions will still be required Instead we
J”
Glucose
Fructose
0
2
3
4
5 Column
6 7 Number
8
9
10
11
12
Fig 2 Comparison of theoretlcal concentration profiles after one complete cycle (12 switches) for glucose and for fructose at various stages of the swltchmg interval (numbers on curves denote time m mm from previous switch) with the experlmental profile (curve denoted by 0) measured near the middle of the switching interval
B58
Glucose
.xv
Fructose
-0
1
2
3
4
5
6
7
8
9
10
11
12
Column Number
Fig 3 Comparison of theoretlcal concentratzon proflles after three complete cycles (36 switches) for glucose and for fructose at various stages of the swltchmg interval (numbers on curves denote time tn mm from previous switch) with the experlmental profile (curve denoted by 0) measured near the middle of the swltchmg interval
50, Glucose 40-
2
30-
g :
zo-
lo-
o-
0 0
1
2
3
4
5
6
7
8
9
10
11
12
Column Number
Fig 4 Comparison of theoretical concentration profiles for glucose and for fructose at steady state at various stages of the swltchmg interval (numbers on curves denote time m mm from previous switch) with the experlmental profile (curve denoted by 0) measured near the middle of the switching Interval
B59
Fructose
0
1
2
3
4
5 Column
6
7
8
9
10
11
4l 12
Number
Fig 5 Theoretical profdes showing development of fructose concentration profile from a pre-loaded lmtlal condotlon (X) The curves denote profiles at the middle of the switch intervals after 1, 2 and 5 complete cycles The steady state IS almost comcldent with the curve for 5 cycles
switchmg Interval Since several minutes are required to take the necessary samples it is not possible to measure the experimental profile precisely at the mid-pomt of the switchmg interval and the differences between the theoretical and experimental profiles are no greater than is to be expected as a result of small differences m samphng time The numerical simulation showed that to reach the quasi-steady state, startmg from a clean bed, requires about ten complete cycles for fructose and six cycles for glucose (the less strongly adsorbed species) It 1s of obvious practical interest to reduce the transient time and this may be achieved, m principle, by pre-loading the system m accordance with the final steady state profile To pre-load the system to the precise profile is not a simple task but much the same effect can be achieved very simply, merely by equihbratmg columns 3 - 7 with a solution containing the feed concentration of fructose (5% m this case). This is illustrated m Fig 5 which shows the simulated profiles, startmg from the pre-loaded system. (The number of cycles required to establish the steady state glucose profile may be similarly reduced by pre-loadmg columns 7 - 10.)
steady state behavlour of the simulated counter-current adsorption unit Under the expelnmental operating conditions it requires about 60 h of operation to approach steady state conditions for fructose and about 37 h for glucose These times may be substantially reduced by partially pre-loading the system.
REFERENCES A J de Rosset, R W Neuzd and D B Broughton, m A E Rodrlgues and D Tondeur (eds ), Percolatzon Processes Theory and Applzcatzons, NATO AS1 No 3, SiJthoff and Noordhoff, Alphen van den Rijn, 1981, p 249 D M Ruthven, Prznczples of Adsorptzon and Adsorptzon Processes, Wdey, New York, 1984, p 380 C B Chmg and D M Ruthven, Chem Eng Scz ) 40 (1985) 877 C B Chmg, D M Ruthven and K HidaJat, Chem Eng Scz, 40 (1985) 1411 N R Amundson, Mathematzcal Methods zn Chemzcal Engzneerzng, Matrzces and Thezr Applzcatzon, Prentice-Hall, Englewood Chffs, NJ, 1966 IMSL Manual, International Mathematics and Statlstlcs Libraries Inc , Houston, TX
5 CONCLUSIONS APPENDIX A
The slmphfied numerical simulation based on representing the system as a cascade of equihbrmm stages provides an excellent representation of both the transient and the
The relevant notation is indicated m Fig. 1, and columns are numbered from the eluent mlet
B60
dc NII
Sectton I (N, theoretical stages) The concentration of the fluid entering section I 1s given by cNQRe/(QRe + Qm) where cN is the concentration leaving the last column m section IV, and the flow rate through sectlon I is (QRe + QEl) At z = 1 dc, -=
-a2cl
+ alcN
dc,
z-
= a2(cz
- 1-
b = QPA~& For NII+l <
dc, dt
(4)
ziere a2 = (Q ~1 + QR~)/(& (V,+KV,) Forl
+ KU,
dc, -=
(5)
dt
-
= a3(c,
dt where
- * -
z < Nm -l-c,)
a1(c, pl--
Equations form as dx =Ax+b iii where
Sectaon II (N,, theoretical stages) The flow rate m Section II IS (QEI + QRe QE) For N1 < l< N,,
dcz
=u4(c,
c,)
c,)
(4 - 8) may be written
X=
QE)
in matrix
1
Cl (a~1 + QR~ -
+ KU,
+ KV,)
Sectaon IV (NIV theoretical stages) The flow rate through section IV 1s QRe ForNm
al = QRJ
c,)
+I
+ b = U3CNII - a4cN&l dt where a4 = (QEl + QRe - QE + QF)/(&
2
b=
c2
u3 = -(v>vF
Nn+ 1 Sectton III (NIII theoretacal stages) The flow rate through section III IS (QEl + R Re - QE + QF) For 1 = N,, + 1 (feed plate)
N
.CN.
and 1
2
3 -
1
-a2
-
2
(12
-=2-
3
-
Ni
A= NI
a2
N
-
-
-
--
--
-
-
-
-
-
-
-
-
-
-
-
---
-
-
-
---
-
-
-
-
---
-
-
-
-
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41
NI ---
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B61 APPENDIXB QI -a4
A b
b
CF
K KG,
N
KF
NOMENCLATURE
constants defined m Append= A square matrix defined m Appendur A constant defined m Append= A vector matrix defined m Appendur A concentration of solute leavmg stage i (g 100 ml-‘) feed concentration (g 100 ml-‘) adsorptlon equlllbnum constant adsorptlon equlllbnum constants for glucose and fructose respectively total number of theoretical stages m system
N, &II>
NI, NIV
9
L,
QE,
QF, QR, QR~ t
6 K
Greek symbol E
numbers of theoretical stages m Sections I - IV respectively adsorbed phase concentration (g 100 ml-l) flow rate (ml mm-‘) flow rates of eluent, extract, feed, raffmate and recycle streams respectively (ml mm-‘) time (mm) liquid volume m theoretical stage (ml) solid phase volume m theoretical stage (ml)
bed voldage