.
. -.... . .: ~
Cation-Exchange Water Softening Rates John du Domaine, R . L. Swain, 0.A . Hougen
Experiments on the softening of water by catian exchangers and the regeneration of the exchanger by a salt solution haw been conducted o n suflciently thin beds to permit the differential rate equations t o be established in terms of operating variables. The usual method of conducting tests on thick beds does ~t a l h the differential equations to be obtained. The differential rate equations ham been determined and solved for the variable boundary wnditiom encountered in the changing activity of the stationary bed. Mathematical equations and charts are & e n for the performance of commercial beds of zedite under conditions of variable thickness, rate of water flow, initial hardness of w t e r , and initial base-exchange capacity, particle sise, and activity of exchanger.
UNIVERSITY OF WISCONSIN. MADISON, WIS.
HE objective of this investigation were to obtain experimentally the di8ecentisl rate equations for the snftening of water by a cation exchanger and by mathe m a t i d treatment to apply the reaulta to the construction of performance charts for the operation of commercial beds of this exchanger. A d u m aluminum &cate sjnthetic cation exchanger was selected, of the type extenaively used for the rapid removal of dcium and msgneaium ions fmm hard waters. This excapacity of 15,000 changer has a guaranteed b-change grains of calcium csrbonate per cubic foot, a density of 28 pounds per cubic foot, and a cornpsition compnding to
T
1
2
Time.
C.in&O
Ebpy. m. 0
25 60
75 1MI ias 150 176
aw
546
aCsRa
~s.viw Bed."
moved from
0.74 1.40
3.40 a.74 2.14
a m
2.68
3.06 3.4a 3.64 3.78 a.&p
H,O,YI
1.66
1.08 0.7~ 0.66 0.36 0.80
4k Av.Cs in H.O. y.
a.u 2.77 3.07 a 36 3.60 3.78 a.89 3.96 3.99
6s
C a b
rnmed/Timc 1nts.7.l.
u.
0
3.08
a.u 1.84 i.3a
I.tedC.
%moved.
Z"*
The exparimentsl apparahu, in OUtIined in Figure 1: The softener mnsinted of ,a g l a p cylinder 5.5 inches (14 om.) in internal diameter. In thls c hder a h d of e x c h 8 of ~ 100 ~ gram (dry basis) about 0.5 incg thick was sup rted on a brass meen. A perforated plate was set above the be% the to of +he chamber to distniute the water w ly and to avoid &e dis turbance which a single straam mente. Tbe hard water
-I#
7.
C.in ~=obn.m,
PXlWo
movedfro~ E=O+
L X 0 1.m
1'884 0.814 0.562
0.678 1.038
8.88
1.630 1.W
1.630 1.800 1.912 1 wa a.or*,
0.4a
io.ga
i.gga 2.064
R, x 1 0
2.380 2.180
0.678 1.088 1.384 1.912
-
0
0
10.69
101
SI
Na Rb
0
3.08 5.6a 7.36 8.57 10.17
o.aa
WATER SOFTWING EXPSBWENTS
6d
Aooumu-
0.88
0.60
the formula NarO.A401.6SiG Teata were conducted on four grain sises--on 14 mesh, 14 to 20 meah, 20 to 28 mesh, and 25 to 35 mesh (Tyler 6crem). Inasmuch as the wmp i t i o n of both water and the exchanger vsry from p i n t to pint in the bed as well as with time, it in desirable to conduct tests on thin beds to estsbliah instantaneous and p i n t valuea of reaction rata. U E ~tests Y ace conducted on thick beds, hut E U C ~tests give integrated resulta only and do not permit eatsblishment of the ditierential rate equation or the mechanism of softening e x q t through the highly u n e pmcedure of asa rate equation which, upon integration, gives the eqmimentsl d t a . A bed thickness of 0.5 inch (1.27 cm.) wra ussd which in su5cientJy thin compared to w m m d beds (averaging 3 feet or 91.4 cm.in thicknea~)to pmnit the UBB of the arithmetic maan valuea of entnrnce and exit stream conditions in eutablinhing mean p i n t dues and thw to arrive directly at the desired differential rate quations. A mthetic hard water wra made fmm distilled water and oalcium chloride, pmducing a hardness equivalmt to about 200 parte of calcium carbonate per million psrts of water. Freeh zeolite fmm the aamelot wan used in all tests.
0.880 0.m 0.128
INDUSTRIAL AND ENGINEERING CHEMISTRY
%x
Q*
lli
10
0,976
0.780 O.M¶ 0.387 0.m
dfi 3.120 2.mo a.810 1.887 1.504 1.208
0.146 0.088
0,990
O.oB9 0.067
0.7116
0.851
Val. 35, No. 9
WM
essential to prevent
The time for dispIacinsthsinitinldiatined aster content by the bard water supplyat the abmmding.
0fesohrun~estlms from the void content and rate of flow. The rab of
water flow nas m d and contmlled by a rotameter accurate to within 0.5 per aeot. Hardwaterwwlpasaed until the exehanga matsrisl wasnear1 spent. Thebardnaa of a i -des ~f wa* WM meamred by htratmn with standard potasdurn palmitate solution IIOM) to ~ u t s r ' smodification% of the method of Blscher. Granberg, and I ( i S M ( 1 ) . Interpdation of Experimental Data The meohsnim of s o f t m h g water oonsista in the m o v a l of metallic ions from the water in exchange for Fisun 1. Flow Chart for fiprimenta on Water softenin# sodium ions from the exohangar. The reverse reaction predominates in the regeneration of the spent exchanger with salt solutio ater; the flow rate waa 752 grams per minute, and the tamThe most plausible theory of the rate of cation exchange perature 85' F. (29.4' C.). The times reoorded in Table I aonsistent with the d e n t a l data is that the btmtanewere coirected for the times requid to displace the W e d oum rate of softening it constant tempersture and constant water initidly present in the bed. This correctedtime waa esflowrata is proprtiond to the pmduct of the concentration timated from the void content of the interchanger and the of calcium ions in the water and the square of the concentrarate of flow of water. tion of d b l e sodium in the exchsnger. All the sodium Interpretation of Sotlening Data. The method of inin the exohan@r might not be repheable. tiiae the bed terpreting the experimental data on softening in terms of of exchanger wag stationmy and the water wm paeaed thmugh Equation 1 is illustrated in columns 3 to 11 of Table I. in a steady contiiuoum atream, it is simpler mathematidy From Equation 1, to basa the reaction upon the rate of accumulation of calcium ionsby the bxehmger ratherthanu p n ita rate of removal from the watar. Because the inkchange of sodium and (4) calcium is ionic, it in more convenient to expm exchange rates on the brs& of gram eqniwlents rather than u p n gram Values of are plotted against u,, the sodium r e m o d maes. a# P In this discussion concentrations of sodium and calcium from the exchanger, in Figure 5. In general, for sll runa a ions in the bard water are expressed aa gram equivalents per straight line most consistently fitsthe results, even though in million grsma of water. The sodium and dcium contenta Figue 5 a slight curvature better fits the data for this parof the exchanger are e x p d in gram equivalents per ticular run. The interssotion of this straight line witb the gram of ex-r. The rate equations consistent with zero ordinate line corresponds to a condition of sero softening rate and indicates that sll the available sodium hae been m theory and testa are aa follows: moved; hence vo = u, = 0.00262 (Figure 5). The slope of Fonwmn F~EACI-ION Rnm, thii line gim the d u e of dK,and the intarcept with zero
4%
-
dE
-
(3)
absciasa line gim the initial value of Thus for ba, P' run 21, Figure 5: 4, 1.43, k, = 2.04, and g 0.00262. In this manner the d u e s of k~ were obtainad for all softaning runs covering merent mesh sizes and Werent rates of water flow. The &e& of particle size and rata of water flow are listed in Tables I1 and 111.
Typical experimentaldata for run 21 am given in column 2 ofTable I and in Figura2. About twenty-five softsning teste were conducted, but the deteiled data are shown only for run 21, selected at random. The initial hardness of the water, tb,wag 4.14 gram equidents of calcium per million grams of
Reaction Velocity Constant of Revase Reaction. To values of the reaction velooity constant k, of the ob* revera reaction, the qent exhsnger wag regenersted after each softening run by w i n g d t solution t h r o d the bed. The brine eamples were snalysd for calcium content and the
b w w ~ A I X I O N RATE,
* - R. -
a=-??- rc, (?d*P NB7T !bACXION RATE.
R.
May. 1943
= RI
61 uv'
- k4w)'p
(2)
INDUSTRIAL A N D ENGINEERING CHEMISTRY
547
18
11-80
a
ai
o.m1w
0 o . m 1 m,
~a m
0.omml
o.mm54
2.04
1.88
Av. 1.99
When water issofteared by flowingsteadily tluvugh abed of cation adauger, the mpwitiona of both water and exhsn(Fa vary pmgsssively with both time and location in the bed. "hue, the variables u, u, p , and q are functim of z and y. and may be denoted more precieely 88 u(z,y), etc., with initialvalues 4 = u(0, O), etc. W d e r an element of the bed, at depth z,dz om. thick and 1sq.cm.in crcm d o n ( n o d to the direction of water Row). At time y, the concentration of dcium ions in the water ent4ring in u; #at in water leaving is u (du/b)&. In time dg. then, the'nmber of gram equidenta of calcium adsorbed by the exchanger is
+
and fromthin equation of material balance Equation 5 is ob-
tained:
a w 46
1104 la10 1809
O.Wp14
0.00164
results intmpreted in a manner Bimilar to the softening te&. The individual test runa were erratic, but from about t m t y tests an average value of !24l,O00 'IPS obtained for the equilibrium constant: K = k t h Wl,?
-
Summary of Results. The net rate equation combining both softening and regeneration can be e x p r d by Equation 3:
R. = kl w'
- kr p(vd'
I
I
FIG.3. W l U M REUDVED FROU WATER
1
The values of the forwd reaction velocity constant E for didterent particle am of mlite are aa follows: M a 6 She
h
am3
a m (at.)
%as
icao
On 14
.
a.1o
1.91 1.81
Au might be Bxpeded, the reactiop velooity constent8 based umn unit &t of exchamr - inclsssa with incrseaed he. n e s s of particli. Eauilibrium constant K of the reaction wan found to be a b u t ~1,000. The mreaction oonstsnt kr can thw be obtained from the equilibrium constent. The deed of water &w rate waa inappmiable for either of repmdudble forward or reverse reaction within the results. The vuriable results obtsined are perhaps due to chaluleling which mssLed any de& of flowrate.
e
EATB EQUAllOn Bop 8 o E T B " G PROCESS
In thin development the rate Equation 1 for the forward d o n isinte&mted. A aimilsr development mube mde for the m v e nesction, ~ ~ Equation 2. Tbe net rate Eqsustion 3 WM not integrated. The following mathematid d e velopmeat is limited to d t i m where the mlsso tionisrFhtiWlya. 548
INDUSTRIAL AND ENGINBERING CHEMISTRY
VoL SS. No. 3
or
whem *(z)
-
arbitrary h t i o n of z
By Equatiom 9 and 10, f ( z ,0) of the letter in Equation la gives
-
a w
+ C; sukditution
0(2)-(Ukz+C-- 1
..&&f w hc
-
-
4
'
oarz
a
+ (c - k) - f
(la)
arbitrary colllltsnt
The ordinary difTerenW equation coriw~~~nding to Equation 13 in:
TheveriableacanbeaeparatedbylettingG= F -ao# C (l/@),and the solution is:
Esoh gram equivalent of celom ion added to the interreplaced by onegsm equivulent of sodium in the reolite, Ap
-
-aO,sothat
- +
F%
- ~n(F - m.2
- c + $)
-
e mnstgnt
It followsthat the general solutionof Equation la in:
- (1- ); (1- &) 1
u.
The finsl dution of following form:
7 may now be w&en in the
- 1)
(*;-l)--ke L(l-;)(l-e)
(
1
w
It has barn veri6ed h t wution 18 sa* 8ptam 7 8nd boundary d t i o m 9. Fwhmore, it follons fmm the method that thesolutionis unique. =Y,
1943
INDUSTRIAL A N D ENQINEERINQ CHEMISTRY
58
I
I IN D U S T R I A L R E A C TI0 N R AT E S
Hence from Equation 23,
For &pli6mtion, the following new d i m d o n l a variables are intmduced:
It should slso be noted that far Gred 2,u andv ara incresing and decregaing f u n ~ t i o of~s,, respeotively; for ked #, u and o are decregaing and incrssSing functions of 2, leepeetively. Also u (2, -) = a; o(z, -)- 0 ; u(-, y) = 0; 4-,y) = u, Abo, u and D are single-valued functions of
Wtive) Iand 11. These properties of the aolutiou are BP
Fmm Equations 19 and ZO,
expected.
-
-
Comparison with experiment is gratifying. Thus, the data of softening nm 21 give q = 4.14, R 0.00162, @A 752, PAZ = 100, and k~ 2.04, where A is the surface area of the seolitebed. Thesegiver = 1.88ands 3 0.022ly. Compruiaon of o b d results (Column 2, Table I) and calculated resulk follows (aee also Figure 2).
-
In any sped60 problem r and 8 may be oalculated from the given data. &nation 21 may be quickly solved for z by B table of natural logarithms or by a graph of the fundiou %
on#+#). Of Bpecisl interest is the behavior of u/uo
-
BP
-
# approaches
zero (s, + O), z being 6xed (for the value of u at s, 0 cannot be determined by direct substitution in &uations 19 or 23). Then r h constSnt and 8 0 , m that by Equation 21, I+0. From Equation 17,
lim (1n Y-0
3
=
lim (a
v-n
- E - r)
-
--I
Other mftening' rnns yield no worse eompsrimns.
and some give better
In mftening luud wster i d W y free of d u m , the re-
vem reaction can for practid purposes be neglected. The dative magnitude of the revem d o n now be wtimated. The forward and reverse mtm are RI = k, w' and
INDUSTRIAL A N D ENGINEERING CHEMISTRY
ss1
-
4- u - u with Equations ZO, 21, and 23, it=can be
lh = kdw)*,respectively. Using the relation p
and q & shown that
U = 4 X 8.33 uo
w
-
33.32 lb./eq. ft./min.
- (%,ooo)(&)(A) -
-
0.00255 Ib. equivdent NaAb. mohsnger
6.W Ib. equivalent,a ion/million Ib. water y-16XBo-gBonun.
Fmm Equation 20, ~n the typical conmerciaio888ihetrated in the foUowing pages, &./It1 1.12 per mt. This mE&lum d u e of the revane rata is attaioed only at the exit faca of the-bedwhen thebedisneariypabausted. Thismesnsthat,whenhsrd water whioh is dum-* upon pmtrancs flows through B bed of ~~, the exchanger becorn wmpletely q.mnt, the mvem rate is nearly nediejble and under such aircumstenae of steady flow and initial rem sodium content, chemical equilibrium ia not attained.
s
APPUCATION OF EQUATIONS
The prformance of a cation exchanger in softening water can now be calculated for conditiom where the rate of the revem reaction rate is relatively dight. For example, it is deaired to know the reaidual hardness of water after steady operation of a softener for 16 h o q with.water flowing through the bed at a rate of 4 gsuons per square foot per minute, with a bed 3 feet thick. The initial llardnm of the water corresponds to 300 parts calcium carbonste per million partm water. This mpreanta 6.00 pound equivslomts of
10'
p
kt u:
ZI
(10)' (28) (1.91) (0.00266)' (3)
r-o33.32
--
In I + e
h c ee
h a
0.521
+ 8 -r
-
Ln28.1
+ 28.1 - 31.3
-
31,8
-0.136
-
Hen= u 0.366 (6.00) = 2.12 pound equivalentm of calcium ion in water leavingsoftener at end of 16 hours. In this m e r , values of &dual MWbe ~ C U Esloium ion per million pounds of water. The initial repla04 able d u m content of the exchanger ia equivalent to 2 6,W lated for any time, tbioLneasof bed, rate of water flow, cationexchange oapaciw of exchanger, and reaction velocity cong d n a calcium carbonate per cubic foot. The density of the stent. m i n d is 28 p u n & per cubic foot and grain Bise, 14 to 20 mesh. Under theee conditione: p
, T
kt
--
-
PBBPOPM*NCECB*BTS
28 Ib./au. ft.
3.0 ft. 1.91 per min.
To expedite calculations, perfornunee charta have bsu expming the efieot of all variablesgroupe~in the dimmionless groups r, a, ?/a, and u/%. In Figure 6, u/m is plotted against 8/r for equal value8 of r. In FEguFe 7, value8 of r am plotted against a for such conditions that the reaidual hardnm of the water leaving the bed is 7 pert3 calcium oarbonate per million (0.41 grain per gallan). Cnrvea are constructedfor a e r ent values of initial hardness The problem pnevioudy dved algebraically aan now be readily ~ l d b ~ e - 6 . Fmaample, at r 31.3, a = 28.1, a/? 0.898, and u/uo = 036 h ment with calculatiom. For the particular conditiom of tJ& problem, r is conateat and a Vsrig onlysstime. Figure6showsthst 8/r from 0.68 to 0.w) h at r 31.3, the hardnea of water leaving rapidly haease froma u/uo value of 0.001 to 0.37 or from 0.3 to 111 parta a s l b carbonateper million. The vdufa of a am dire& proportional to time, for the given amdition 8 (28.1 v/e60) o.ozga U. Thus the hardners of the water leavingthe ;eoliterapidly changes
-
-
-
-
INDUSTRIAL A N D BNQINBBBING CHEMISTRY
-
Vd. 35, No. 5
from 0.3 to 111 p. p. m. calcium oarbonate BLI the time elspaed ohangw fmm 12.2 to 16.1 hum. The mineral ahould be regewaated when the water leaving reaahea a hardness of 7 p. p. m. cslcium csrbonate. From li5gure 7 an rvalue of 318 corresponds to an svalue of 24.6, or the time elapse3 is 24.6/0.0282 = 844 minutea (14.1 hours). The water mftened by 8 cubic feet of m i n d duringthi3pricdis844x4=3376g4Uons; thecalcium carbonate removed is (3376 X 8.34 X 3W7)/10s = 8.24 pounda caloium carhoda (S7,ssO praina), or 19,227 grsinS per cubic foot. The total water softened by one oubic foot of mineral under the above conditions is (4 X 844)/3 1130
-
gallons. For any paaiculas zeolite values of kl, p and tb are constant. For any particular zeolite it is more convenient to plot z/Q e&& w, inutead of r e&& 8 as in -, 7, where Zp3 can be expressed in thiclmass of bed in feet divided by rate of water Bow in gauonS per quare foot per minuta, and can be expressed as initial hardness of water mpply in grains calcium carbonate per gallon multiplied by a o f h h g time in hours. Such a plot is neither general nor e x p d in dimenniodess mi%but for a particular zeolite it has the advantage of reading directly in commmial tam. In theae calrmktions the mvem reaction ww neglected. However, even in the OBBB where the bed WBB uaed until only So per cent of thehardrieeawasremoved,the revemreaction reached a maximumratio of only 1.1 per cent of the forward reaation at the exit side of the bed at the end of 16 h o w For dl ehorter dietences in the bed and for dl shorter inof time, the revem rate would ha correepondingly less than 1.1 per cent of the forward rate. REACTION VEUIUTY h FROM Pl3BEVJUMANCE DATA
.
g e n d procedure, the mathematical treatment, and the
perfornunee CM p ~ n t x here l ean he used to obtain the complete performance cbsracteriatics of a bad of ration eaohanger with the minimum of experimental data; in general the method should be npplicable to any watera for the removal of magnesium as well 88 calaium. The effeds of pH, temperature, and mined composition of the water,and the &e& of grain ah, chemical cornpition, and structure of the mineral ean be evaluated by this procedure and the m l t a extrapolate3 with the minimum of I?XPE~~menM d a h The mathematical treatment ean slso be a p plied to aimilsr heterogPmeaus d o n a in a batch process whexa one atom or ion in the h i d p h e reads with two atom or active spots in the solid phase. ACKNOWLEDGMENT
credit is gratefully &owledged to E. C. Carhn nud D. X. Starr for technical earvices in eaperimentsl work; to N.A. W e for oonstruetionof pertormanee charts; to R. E. Langer for the mathod of deriving Equation 11 from system 7; to Milton Shoemaker for information on industrial performance; and to the Wisaonsin Alumni Reaearch Foundation for finsncisl support.
c
b
G
k,
K p
II 1
---
=
NOMWCL4TUIIE
-(lO'#kJ/U -kt
conagnt maes velocity of water Bow, Ib./sq. ft./min. reeotion velmity oonstant of softening 88 dehed by Fnimtinu 1
eloeity constant of mpneration 88 delined by n2 eauicbrium constant k J k cliloium content of , para equivalents/pm zeolite,andlb. equi=. d t e d i m m n h t of h e . mam .eouivalents/miUion a n w&. ~ ~red lb. edvelenttl&llion Ib. water
-
--
-
Z(33.32 X 3.0 X 8ae) (10' X 28 X 0.00266 X 8)
From Fylre 6 the inteweotion of r/r 0.0283 gives an I vdue of 32. Thus,
revme d o m but not for the net reaction rate. Where the iuitlal water supply has a high sodium content, the rew m readion becornea appreoiable and the given equations do not strictly apply. It is believed, however, that thii
a
With the kineti0 quatiom estgblisbed for water s o h ing and ita mat he ma^ development, it henomea paesible to calrmkta reaction velocity oonstant 4 from the performance of a thioL bed as in commermsl opepation. Itisnearasaryto start either with a fully wgentmted or freuh cation exelmgar and to h o w ita density, maximum catiorwxchangecapdty, and thialmpss of bed. The initial hsrdness, of the water supply, rate of water hw,G, and exit hdnea~, u, at the end of a given time, y, must bamesaured. For example, a bed of mchmgar is used whichis3 f d thicl, has a d€maity of 28 pounds per cubic toot,and hae a Ihed. mum cathachange c a p d t y of 25,000 gmim &um oarbonate per cubic foot. Water haskg an initial bdneea of aoopsrtscalcimoarbonstepermillionispsaaed~~ exctuuqpr at a rata o f 4 @ns per squan,foot prminuta for 14.1 hours when the water leaving hae a hsrdntQ of 7 parts calcium oarbonste per million. Thpee dado am similar to praviousiuustration,but his now tobcslaulatad. undes theseconditions, uo = 6.0, cb = 0.00265, Q 33.83 pounds pa quare foot per minute, u/* 0.0283; y 848 minutea: 8
@nerdvalue and has been dev#opd for both the forward and
-
o.79
0.78 with u/u.
-
INDUSTRIAL A N D ENGINEERING ClfEMlSTRY
993
