Computational and Experimental Studies on a Moving Ion Exchange Bed Shinichiro Gondo' Departmeni of Chemical Engineering, Kyzishit Cniversity, Fitkuokn, Japan
Masato ltai Jtitsiti Sliipbuilding and h'tigineeriiig Co., Lid., T o k ~ oJapan , Koichiro Kusunoki Department oj" Chciiiical Engineeri?ig, k ' y u s k u Cnirersiiy, Fitkuoku, J a p a n
Transient and steady-state conditions of a moving ion exchange bed were studied using an ion exchange system of 0.1 N NaCl and Amberlite IR-l20(H+). The experimental results showed that at steady state the concentration of the outlet liquid i s only a function of the ratio of the rate of flow of liquid to that of the resin particles. For the transient state the time required for obtaining steady state i s 130 to 400 times as much as the residence time of the liquid in the column, increasing with a decrease in the ratio of flow rates. To simulate a moving ion exchange b e d b y digital computations, a numerical method was derived for solving the partial differential equations representing a material balance. Simulations b y this method were able to predict fairly well the outlet liquid concentrations observed in the experiments.
IVhcti tlic iioritivc tlirertioti of the axis of the tiioviiig lied is t:ikcii aloitg that of the liquid flow throiigh the ( ~ o l ~ i i i i tthe i, iii:iterinl 1):il:iiiccs aroutid t8he iiifiiiite~~iinnl bed height,, Z . ore represeiited as follo~vs: eoliitiiit
Theory
Numerical Treatment of Partial Differential Equations of Material Balance. 'I'lie i o i i r s r 1 i : i i i g ~ 1~ 1 1 ' o c (1i:is ~ lwei1 (,oii,.i(lri~ul I ( , (a(~ii>i,-t
of
~'(iiiis , trl)G,
(I
140 Ind.
Eng. Chem. Fundam., Vol. 10, NO. 1, 1971
-
e ) Q du = k1aCr(z at
2.J
(4)
Accordiiig to the aesuriilition t h a t the first' step is ratedeterniiniiig, there should lie aii equilibrium relation lietweeii zs aiid y. Froin Equatioii 2 we olit'siii
(5)
=
In tipplyiiig these equatioiis to the nioviiig lied operatioii, Equatioiis 3, 4, niicl 5 niust lie yolved siiiiultaiieously for the liquid flow period :iiiti theii the coiiceiitratioii distriljutioiis :it the elid of the liquid flow period iiiust be moved aloiig the tlirectioii of I d nioviiig t'o the bed height, Z,, which is occul)ietl b\. the fre,sh rc3iii pnrticles fed after every portion of liquid flo\v. I11 other word.*;, the coiiceiitratioii distributions o1)t:iiiied iii this \yay liecome the iiiit'ial coiiditioiis for the liest period. T h e 1)heiioiiieiin of the nioviiig lied operatioil r a n be relmwnted theoretically by repeating these procedures. A s the equilibriuiii relation giveii hy Equ:itioii 5 is iioiiliiiear and the iiiitid coiiditioiis for each period riiny be giveii oiily by nuiiieric:il values, the siiiiultaiieous 3olutioii of Equations 3, 4, : i i i d 5 iiiuqt be 1)erforiiied iiuiiieric:illy. Helice, these eqiiatioiis m i s t lie imxitteii iii differeiice form. I11 the following, At :tiid A 2 nie:iii the time niid the nxixl iiicremeiit, respectively, aiitl i :riid ,j iiienii the i i u i i i l ~ e rof i~icremeiitsof A 2 aiicl At, respectively. Figure 1 is :I tlingr:ini of the l)rocess, using iiotatioiis th:it a l ) p ( w iii the followiiig equ:itioiis, R q h c i i i g the sccoiitl term of the left side of Equatioii 3 by the right side of Equntioii 4,we ol)taiii the following difference ec1ri:itioii.
[-Ir
1 I
Z
,u
\
1.2
,
7 ,[at 3 Diagram of ion exchange column
Figure 1 .
The coiitlitioiis for y t , j + l given liy Equ:itioii 10 to lie positive ;iiid iiot grexter thnii iuiity will I x found by coiiibiniiig Equatioii 10 Tvith Equatioii 8 :is folloiv;.:
I:
- be Ivhy tlic coluiiiii height h:id i i o effect oil espci~iiiiciit:ilv:ilucs of zn,:is sIio\~iiiii Figure 5 . *\s for the effect of T F on the pheiioineiia, iiuir~ericnlcnlcul:it,ioii gives clear differelices of the co~iceiitrntioiiprofile. betiveeli 7 p = 41 aiid 378 secoiids, a.: s h o ~ i iin Figure 9, a i i t l coiii~):ii~i~o~i of Figure S,C, with Figure 8 , D , shon-s that ~ n i n l l e rT~ value> give larger 0, values, :is o h e r v e d in t'he esl~eriiiieiits. A s A?,. :ipl)roaches 30, the coiiceiitratioti profile< teiid to sl)reatl all over the coliiniii. -111esninple is >howl i i i Figure 10. Conclusions
Esl~ci~iiiieiit~ : i l i d coiiil,iit:itioii:il studies were lierforiiietl for the iiioviirg ioii esc1i:iiige lied uiiiig the ioii eschaiige ays144 Ind. Eng. Chem. Fundam., Vol. 10, No. 1, 1971
-
X
!
VF = 0.7 00 VP 0.0 1 2 2 , TF,=3,7 8, 100 2 00
,
100 TIME, M I N
0
2 00
T I M E , M I N.
0 5cc
V~z1.24 VP 0.0 2 6 3
VF = 1.0 7 Vo 0.0 246
200
100 TIME, M I N
0
0
100
200
TIME, M I N
Figure 8. Comparisons of numerical calculations and experiments
c2: = 0.1,XI =
0 5 -E
O II J '
d t = l 89
Atz082
dZ=O 101
A Z = O 090
1
'
'
'
1
1
200
100
0
tP OF'
V~z288 1 VP 0.0 5 6 7 ' a,= 3,76,
'
'
TIME,MIN.
1
0
1 , X J = 0,Yz = 0 Calculations b y digital computer Experimental d a t a
V~z1.53 V ~ = 0 . 0 2 3 2i ??F,=4,1 ,
1
'
100 TIME, M I N.
0
-
2 00
f---
05-H At 1.89
t i
VF = 1.7 6 VP = 0.02 5 2 , T F , = 3,7 8,
, TIMET
100 MI
I
i
200
N.
X or Y
.
0
L
100,
I
9 5~
9 or
.9
1L
5r For 2 et c a n d I
8o c
At e t c
,
60
refer t o the notes shown in Figure 8-C
75
distribution
Patterns just before b e d starts moving downward
85 *-
8
Figure 9. Concentration curves at steady state
For Z e t c and At e t c , r e f e r t o the notes shown i n Figure 8 - D
I
75t
L
Ind. Eng. Chem. Fundam., Vol. 10, No. 1, 1971
145
F1,F2, i
=
I I, j
= = = =
kF
2 =10 VF =0.300 V p = O . O l 03 T~=41 Rvz29.1 dt=1.64
AZ = 0.0 80
K
=
Pe
2! Re
= = =
t
=
ts
=
At 5
= = = =
Y
=
2
=
AZ
UF
V Figure 10. Spreading of concentration distribution curves a t RV close to Q/Cz Patterns just before bed starts moving downward
tern of the sodiuiii ion, the hydrogeii ion, and the stroiig catioii exchange resin, Amberlitr IR-120. The results show that the outlet liquid cuiiceiitratioii depeiids 011 R\. but not 011 2, r F jaiid the absolute value of V p or J"p. They also show that 8,) the noiidiiiieii~ionaltime required for the steady state, ranges froin 130 to 400, decreasing with an iricrease of RVjaiid that sixdler rI7tends to give larger 8,.
T o simulate the phenoiiiciin observed iii the experiments, numerical calculatioiis of the ninterial balance equntioiis were derived and carried out with k F giveii by Carberryla correlxtion, assuiiiiiig that the resistance of liquid phase is ratecoiitrolliiig. The results of simulation were satisfactory for large I , values, but for small I , values there were corisiderable differences betweeii theory and experiments. However, the method of simulatioii derived here might be useful for apliropriate predictions of the pheriomena. Nomenclature a
=
C
= =
CZ
d,
DF DP
=
= =
surface of particles per unit volume of packed bed, equal to 6(1 - e ) / d , for spheres d, in diameter, cm2/cc liquid coiicentratioii of Ka+, meq/cc total cation coiicent'ration of liquid phase, meq/cc average dianiet,er of resin particles, cm ionic diffusivit,y in liquid phase, cm2/sec ionic diffusivity in resin particles, cm2/sec
146 Ind. Eng. Chem. Fundam., Vol. 10, No. 1 , 1971
= = =
R,
sc
b
= = = =
0,
=
TF
=
TP
= = =
a, B e
I.(
P
coiistaiits defined by Equation 11, dimeiisioiiless
= iiumber of AZ, diinensionless
Z/AZ Z,/AZ iiuniber of At, diineiisioiiless inass t'raiisfer coefficient' of liquid phase, ciii/sec selectivity coefficient, dimensioiiless Peclet nuinl)er, d , u ~ / 6 ( 1 - DE., dime~isio~iless coiiceiitratioii of Ea+ in resin liarticle, iiieq/cc ion exchange capacity, mcq/cc, of 6n.elled resiii iiart'icles R&iiolds iiumber, d,zi=p/p, dimeiisioiiless
VF/T', Schmidt, iiumber, p , ' p D p , dimeiisioiiless time, sec time required for hteady state, sec time iiicreineiit, aec superficial velocit'y of liquid, cm/sec volunietric flow rate, cc/sec equivaleiit fraction of S a + iii liquid phase, dinieii~ioiiless equivalent fractioii of Nn+ in resiii phase, dimeiisioiiless columii height, or coordinate along axis of coIulnll, CIII. incremelit of columii height, cin factors in Equatioii 15, dinieiisioiiless void fraction of packed bed, dimeiisioiiless parameter defined hy Equatioii 22 diineiisionless t'irne required for steady state, diineusioiiles< time of liquid flon- period, sec time of lied moveriicift, 1)eriod. sec vihco,
