Page 1 Page 2 明治大学工学部研究報告 N。・ 55 (ー988) rati。n at the

（明治大学工学部研究報告第55号・1988年12月）
IV−44 A Simple Design Method of a Fixed Bed
Adsorber by Use of a Personal
Computer
Eiji FURUYA
Yasushi TAKEUCHI
Abstract
Analytical solution is presented to obtain a lcngth of adsorption zone， based on L． D． F． approxi−
mation for a systcm whose equilibrium obeys either Langmuir or Freundl量ch equation． Also， a simple
design method was obtained by applying the solution to thc simplest MTZ method， but the method
provided morc accurate results than the original MTZ method． The solution can be applied to the
Extended．MTZ．Method for multicomponent system and hence break times in multicomponent fixed．bed
adsorption can be easily cstimated by a personal computer using the simple procedure with numerical
or analytical adsorption cquilibria． In order to prove usefulness of the Extended．MTZ−Method， it is
demonstratcd that the break times obtaincd by the method agrce wcll with those obtained from rigorous
ら
Dumerical breakthrbugh curvcs for ternary component system． Furthermore， operational conditions which
satisfy assumptions used to derive the Extended−MTZ−Method are discusscd．
Introduction
Many design methods werc already prescntcd fbr single componcnt fixed bed adsorberlD．
Thcse methods can be classified into two groups， i． e．，（1）amethod which can be uscd to
estimate only brcak time based on the calculation of adsorption zone of a constant lcngth，
and（2）amethod which provides a whole calculation of breakthrough curves and then the
break timc can be estimated．
Furthermore， the mcthod（1）is divided into（1−1）analytical solution using an approxi−
mation which considers an ovcrall mass transfcr resistancclo）（MTZ mcthod）and（1−2）graphi−
cal solution based on L．D．F． approXimation（r一ζmcthod for Langmuir−type isotherm‘）and
l／n一ζmethod f〈）r Freundlich−typc isotherm system3＞）． Thc method（2）is also divided into
（2−1）analytical solution using L．D．F． approximation5）and（2−2）the most rigorous numerical
solution of partial differential equations2・7）（hereafter， the results will be called the numerical
breakthrough curv6）． Thc mcthod（1−1）is the simplest among those mcthods， while the dif−
ficulty as well as the accuracy in the estimation of the brcak time incrcases toward the
method（2−2）through（1−2）and（2−1）．
On the othcr hand， the method（2）provides whole brcakthrough curves but the curvcs
are not alwayS necessary to know。 Only a knowledge of the break time under certain opcra−
tional conditions is cnough to design a fixed−bed adsorbcr． As shown in Equation（1）ll）， the
break timc， tB， is a function of thc bed length，ζ， thc lcngth of the so−called adsorption zonc，
Ca， fractional residual capacity of the adsorption zone， f， lincar flow ratc of fluid， u， slope of
the operational line，β， and bcd density，γ．
tB＝βr（9−Lノつζα）／u ・・一・・・・・・・・・… 一・・一一・・・・・・・・・・・・・・・・… 9・一・・… 一一・・・・・・・・・・・・・・・・・・・・・・・… （1）
Because the value ofβcan be obtaincd from thc adsorption equilibrium and fluid
（35）
concent一
’．明治大学工学部研究報告 No．55（1988）
ration at the inlet of the bed， one can estimate the exact break time from Equation（1），
provided that values l）f f and 4αare obtained accurately・
Regarding．calculation by a personal computer， genefally， graphical solution is not adapt−
able． It is thought t6 be the bcst to derive an analytical solution of 4αby use of L D． F．
approximation instead．of the overall mass transfcr approximation． Thc solution should be
applied to thc Extend6d−MTZ−Mcthod， and hcnce fixed bed multicomponent adsorbcr can
be designed by a personal． computer．
The purpose of this paper is to prcscnt an analytical solution of length of adsorption zone
under certain operational conditions f（）r Langmuir−or Freundlich−Type isothcrm system and
to apply the solution to the original MTZ mcthod fbr single component system and to thc
Extcnded−MTZ−Method9，1°）． fbr multicomponent system， rcspectivcly． The algorism to design
the fixed bed adsorber for single−and multi−componcnt system is also mentioned． The valucs
of fractional attainment of equilibrium， s＝1−f， furthcrmorc， were obtaincd under various
constant−pattern conditions fbr single−and multi−componcnt systcms．
1．Theoretical Approach
1．l Single component adsorption
As already well−known， mass transfer equation at fluid−to−solid interface can be given as：
γ（dg／dt）＝kFav（c−Cs）一・一・… 一・… ◆・… 一・・一・一・一一・・・・・・・・・… 一… 一・・・・・・・・・・・・・・・・… （2）
Because the lcft hand side of Equation（2），γ（dq／dt）， can be transf（）rmed intoγ（go／co）（dc／dt）
under the constant pattern conditions（9／c＝qo／co）， variables， c and t， in the equation can bc
scparated． Thcref（）re， the difference， At， between the exhaustion time， tE， and the break time，
tB， can be obtained as：
・t−、ll。∫：1、空，……・・…一……・…・・一……・………・……一………・・（・）
On the other hand， thc time difference， At， is rclated to the length of adsorption zone
by a well known cquation taking mass balancc f（）r the adsorbate species；
9α＝uAt／βγ ・・・… 一・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・・… （4）
From Equations（3）and（4）， the f（）llowing equation can be obtained．
呼〃・一∫：：、隻。…一・……………一……・…・……・…・…・…一一一・（・）
The relationship betwccn 6 and 68 can bc also obtaincd from intraparticlc mass transfer
equation， e． g． Equation（6）， based on the Linear Driving Force（L． D． F，）approximation with
an apPropriatc adsorption equilibria．
（dg／dの＝ksav（9s−9）・・・・・・・・・・・・・・・・・・・・・・・・… 一・・・・・・・・・・・・・・・・・… 一・・・・・・・・・・・… 一・・… （6）
whereん8 denotes the soild phase mass transfer cocfficient andα7 is total surface area of solid
particles pcr unit bed volumc． gs is amount adsorbed at geometric surface of the particles
and g is amount adsorbcd．
Thercfbrc， the analytical solution of the lcngth of adsorption zone should be derived
（36）
Asimple design method of a fixed bed adsorbed by use of a personal computer
from integration of rcsulting equation， provided that cquilibrium relation can be expressed
by a simple equation．
a）Langmuir isotherm system Regarding gas phase adsorption cquilibrium， one of the
simplcst two parameters isotherm is thc Langmuir−equation． Using the equat呈on， the term of
the definite integral in Equation（5）is analytically obtainedn〕， and hence one can obtain the
負）llOWing CqUatiOn：
ζα㌻α7−’響）Ll・爵≡11−（ζ讐71・1：≡i’−1・激・一…（・）
where
κ1＝（CB／Co）十へ／（CB／Co）2−2a3（CB／Co）十a呈
κ2＝（CE／Co）十K／（CE／Co）2−2α3（CE／Co）十ごz呈
al＝（1＋rζ／η）／（1−r）（1＋ζ／η），
7＝1／（1十1動0）
a2＝｛1＋（2ζ／η）−rζ／η｝／（1−r）（1＋ζ／η），
ζ＝c。kFav／90ksav
a3＝（1−rζ／η）／（1−r）（1＋ζ／η），
η＝1。0−0．192（1−r）3
Then， the length of the adsorption zonc under the constant pattern conditions can be est三一
mated．
Figure l shows the relative errors of valucs of z。kFav／u to those obtained from rigorous
numcrical breakthrough curves by the method（2−2）for various separation factors， r・The
range of integration was taken to be cB／6。＝0． l and cE／co＝0．9， rcsperctively， The values of
「
0．25∼
【3．66）
’ コ x
ぐ、“ノV
。．！、51＼、、
， s ノ
／〆
^一＼、
．83）・
嘩・
／’
［承］﹂O﹂﹂山Φ♪＝
理O匡
O O
2
4．0
1＼一／＼
、
0．069
＼
〔2．52｝
一2．0
0．2
0．40．6 1．0
2．0
4．0
く［一］
Figure l Plots of relative error versusζfor Langmuir isotherm
system systemS
Numbers in parenthesis present the values of Nos
mass transfer unit shown in thc figure， Nos， wcre calculatcd from the analytical solution，
ハ「os ＝（2十、Kc。／Kc。）ln（qE／go）／（gB／e。）． The values of gE／g。 and gB／go were take to bc O． l and O．9，
respectively。 Finc longitudinal vibration of the curves in the figure is duc to thc cumulative
error during the calculation・ Figure 2 shows thc rclationships betweenζand fractional at−
tainment of equilibrium， s＝1−．んwhich werc obtained from the numerical breakthrough
curves fc）r various opcrational conditions． From those figures， it can be concluded that thc
above describcd simple method with Equation（1）is useful to dcsign a fixed bcd adsorbcr，
bccausc undcr ordinary operational conditions of solvent rccovery or gas adsorption， 4α is
much smallcr than g and hence the cstimation crror of彦B is small．
The method must be used when thc constant pattern conditions is satisfied． The mini−
mum bed length， Zmin， necessary to reveal a constant pattern breakthrough curve was alrcady
（37）
明治大学工学部研究報告 No．55（1988）
0．7
〃
−／μ
［ー］
0．6
ω
0．5
・ク
”／
0．4
0．2 0．4
1．0 2。0 4．0 8．0
く［一］
Figure 2 Relation between fractional attainment of equilibrium
at the adsorption zone andζfor Langmuir isotherm
systems
（calculated for single component adsorption）
prcsentcd elsewhere by thc authers， namely， Cmin can be determined丘om the fbllowing equa−
tions by adopting the largcr valuc of Cmin2）．
ζmin＝0．06841×1＞bso・3435×1＞8K。−3・o×N°・ 1’452…………………………………（8a）
9min＝0．444×NOs2・‘…・………・・………・………・…一…・………・……・…・…・＜8b）
飾一2芸壽6°1・雛・・／・・一・・19・／・・一・・9
b）Frcundlich isotherm systcm Regarding liquid phase adsorption， Freundlich isotherm is
widely used to present cquilibrium relationship because of simplicity． By usc of the samc
manncr as Miura’s method5）with the equation， the analytical solution of the right hand side
of Equation（5）， i． c． the integral term， can be obtained not as a function of bulk con−
centration，6， but as a funcuion of fluid concentration at the interface， cs． Thcrefbre， the
exhaustion and the break concentrations， respectively， can be estimated from cs．
準一’警）e−1・壽li｝≡1；1；≡｝−Si・美1−一・一………・……・…・（・）
where κ1＝（cs／6。）at break tim［c
κ2＝（cs／6。）at cxhaustion time
（CB／60）＝κ1（1／n）一トζκi／〈ζ＋η）
（CE／Co）＝κ2（1／n）＋ζκ2／（ζ＋η）
ζ＝c。kFa7／qoksa7， η＝0．808−0．192／n
Figure 3 shows the rclative errors of 4αkFav／u to those obtained from the numerical
breakthrough curves fbr various Frcundlich exponents where cB／oo＝0． l and cE／co＝0．9 werc
uscd・Figure 4 shows the relationship betwecn fractional attainment of equilibrium at the
adsorption zonc obtained from the numerical curves andζ． From those figures， it can be
also concluded that thc proposed simple mcthod is ubeful to dcsign a fixed bed adsorber
under ordinary operational condit量ons of waste water trcatmcnt． Thc minimum bcd length
to show constant pattcrn bchavior can be also estimatcd from Equation（8）with the usc of
（38）
A・impl・d・・ign m・th・d・f・fi・・d b・d・d…ber by・・e・f・p・…n・1・・mp・ter
コロ
＼
0 0 0
［承］﹂O﹂﹂四Φ︾一劇幅馴Φ㏄
4 2
・一
黶_
＼（鐸，
、 ●
一
3．02
（3．66）
2 0
0．2 0．4 1。0 2．0 4．0
（［一］
Figure 3 Plots of rclative crror versusζ for Freundlich
isotherm system
Numbers in the parenthesis present the valucs of Nos
0．7
，∫
／づ！
！ ’
／！
／
0．6
！
ノ
’
ノ
！
［1］
P
0．5
一｝
@［
〔3．66，
！
@ ’
@ ’
T・65フ
ノ
3．02
／1 9．93
n
ω
〃
・ノ
！ F
@ ／ ’
／
ｼ2．83）
〔2．52，
0．4
0．2 0。4 1．0 2．0 4．0 8．0
（［一］
Figure 4
Relation between fractional attainmcnt of equilibrium at the
adsorption zone and ζ for Freundlich isotherm systems
（calculated for single component adsorption）
the fbllowing Nos7）；
N・・−1丑。1・｛≡翻il≡：i；llll・・／炉・」e・／・・一・・9
1．2Multicomponent adsorption
a）Estimation of break times Concentration history at the solid phase of a fixed−bcd is
schematically shown in Figure 5・By usc of the constant pattern conccpt， the break timcs
can be given as fbllowsg）：
tBl＝［Z−（1／2）4w］／σ・… 一・・・・・・・… 一・・・・・・・・・・・… 。・… 。・・・・・・・・・・… 一・・・・・・・・・・・… 一・・。（10）
彦B2＝［4−（1／2）ζ四一］／［σ2！十｛1−（902／9e2）｝こノ』1］一・・・… 一・… 一・・・・・・・・・… 一・… 一・・・… （ll）
彦B3＝［ζ一（1／2）ζ［］／乙1』一・・・・・・・・・・・… 一・一・・・・・・・・・・・・・・・・・・・・・… 一・・・・・・・・・・・・・・・・・・・… （12）
σ一σ・！＋｛1−（9’・1／ee1）｝U・’＋［｛1−（e。、／ge1）｝一｛1−（9’el／eel）｝（9。2／9，，）］U，
び1ノーUC・1／reei，σ2ノ＝UC。，／re・，，σ、＝UC。、／γ9。3
Valucs of gol， go2， and go3 can be estimatcd from adsorption equilibria f（）r givcn concen−
（39）
明治大学工学部研究報告 No．55（1988）
IVII
一8＼σ
1ムゼ「
←△t2埠
Time
Figure 5 Schematic breakthrough curves for a ternary
component systcm
trations， col， c。2 and 6。3． The amount adsorbed， gl1， gel and ge2 can be also estimated as
described elsewhcrc8）． The lcngths of the adsorption zone， i． e．，ζII，ζy and鞠， reach to certain
values with the progress of time and the constant pattcrn concept can be applied under these
conditions． Theref（）re， the length ofζw can be given by use of the simplified mcthod mentioned
above with the values of g，、 and ce1． By use of apparent adsorption isotherm as describcd．
clsewhere8）， the values of gy and zl can be estimated in thc．same ma皿er as加． Therefbrc，
all of the variables in Equations（10）to（12）can be knowl1， using multicomponent adsorption
cquilibria and the cstimation平nethod of adsorption zone length mentioned above。
b）Smallest bcd length to show constant pattcrn breakthrough curves The brcak time of
cach componcnt is estimatcd by use of lcngths of constant pattern adsorption zoncs fbr
multicomponcnt adsorption． Theref（）re， thc estimation of the minimum bed length necessary
to satisfy the conccpt of the ExtendOd−M．TZ−Method is vcry important in the application of
thc method．
Afしer N adsorption zones appcar in a fixcd−bcd adsorber for N components adsorption，
the zones show constant pattern behavior， i． e． the timc differcnces， Ati（＝tEi−tBi whcre i＝1，
2，…ノVrespectively）， are independcnt of thc bed length． This mcans， f（）r example， f（）r ternary
systcms， that values of（tB2−tE1）and（tB3−tE2）should be equal to zero or at least positive（scc
Figurc 5）． Taking（tB2−tEl）and／or伽3−tE2）as O， theref（）re， the minimum bed lengths， gmin，
neccssary to fbrm each adsorption zone in a fixed bed of length z， can bc derived as fbllows
f（）rternary SyStemS：
To apPcar Zones IV separatcd from VI，
・一・・一去〔｛U，ノ／乙1』＋1−9。2／9，，｝（之y＋鞠）乙11！／σ一e。1／ee 1−（e3i／9e、）｛σ2！／乙1，−9。，／9e，｝〕・…・…・………一（13）
To apPear Zones II separated from IV，
・…一去〔u2ノ／濃焉／，e2＋・1〔〕…・………・…・・……………一・……………（1・）
Regarding single component fixcd bed adsorption， as already mentioned， thc smallest bed
length necessary to show a constant pattern breakthrough curvc， Zmin， can bc obtaincd from
either empirical Equation（8a）or（8b）7）， by adopting thc larger value to obtain Nos based
on an apParent adsorption isotherm．
（40）
Asimple design method of a fixed bed adsorber by use of a personal computcr
In multicomponent adsorption， the value ofハlos f｛〕r Zone VI is the samc as that f（）r
t
single component adsorption of Con埴onent l and the valucs f（）r Zoncs II and IV， can be
estimated丘om apparcnt isotherms fd’r Componcnt 2 and 3， respectively． Finally， the largcst
value obtaincd from Equations（8a），1（8b），（13）a孕d（14）has to be uscd f（）r the smallest bed
length nccessary to show thc constant pattern adsorption zones．
c）Fractional attainment of equilibrium at the adsorption zonc To use the Extcndcd−
MTZ−Method， the values of飴ctional attainment of equilibrium in all adsorption zones werc
assumed to be 1／2． To test the validity of this assumption， the values were obtained from
the numerical．breakthrough curvcs estimated by thc most rigorous difference equations as
shown in Figure 6・From the figure， it is clear that the fractional attainment of cquilibrium
at the zones f（）r more adsorbable componcnt is about O．6 in any adsorption zones， while
those valucs f（）r less adsorbable component desorbcd by thc more adsorbable component is
about O．4． Therefbre， the values of break time fbr each component obtaincd from Equations
（10）to（12）will be accuratc， provided the sum of the lengths of adsorption zoncs is relatively
small to the total bed length．
0．7
・瀞△Q
0．6
匿’冒
1
一
〇．5
ω
繊
▲▲塾
0．4 8〔ρ
壽
0．3
0．10．20．4 1．02．04．0 10 50
（［一コ
Fignre 6 Plots of fractional attainment of cquilibrium at
respcctive adsorption zolles versusζfor multi−
component systcms
2． ApPlication of the Extended・MTZ・］Method to Several Practical Cases
Experimental and numerical breakthroy gh curves wcre obtained with rcspect to various
bed lengths， for propanol（03）−butanol（64）−pentanol（c5）−activatcd carbon systems． In com−
parison to cxpcrimental values， numerical量．breakthrough curves are shown in Figure 7． All
numerical curves agreed well with expcrimental values and hence numerical rcsults could be
useful to give a critcrion of the validity of the Extendcd−MTZ−Method． Under thc conditions
of coi＝600 g／m3 and u＝0．212cm／sec， the minimum bcd lengths estimated from Equations（13）
and（14）wcre l5． O and 8．3cm， respectivcly． From Equation（8a）or（8b）， thc minilnum
lellgths wcre cstimated to bc 50．6，69． l and 254 cm fbr Component l，2and 3， rcspcctively．
Thcref（）rc， the Extcnded−MTZ−Method seems to be applicable to thc case when the bed length
is larger than about 2．5m・On the other hand， from Figure 8， all adsorption zoncs for thc
longer bcd lengths than l。5m， secm to show the constant pattcrn behavior． It bccame clear
倉om Table l that the brcak timcs obtained・from the Extendcd−M．TZ−Method almost agree
（41）
明治大学工学部研究報告・No．55（1988）
＼
1．5
［1］8＼o
O
t
0．5
0 500 1000 t500 2000
． Time ［min］
Figure 7 Comparison of numerical breakthrough curves with
expcrimental values
冒
／
150
C
！I
Zmin．b ter Zone ll
’萱1．oo
C4、
T
ム
／．rZm・．・…Z・・e lV
J50
イ！r c
面
一
0 100 200
Bed Length ［cm］
Figure 8 Plots of difference between exhaustion and break time
versus bed length
Table 1
Comparison of break times from the Extended−MTZ．Method
with those from numerical breakthrough curves
z
tBl tB2 tB3
tBl 功2 tB3
〔cm〕
from the Extended
−MTZ．Method
from the numerical
breakthrough curves
100
285．
595
144
423
935
200
191
561
250
237
694
300
282
831
1930
150
96．4
252
653
ll7
393
1000
1270
165
530
1330
1593
203
659
1631
250
799
1967
70．l
with those frbm thc numerical breakthrough curves． Thc difference betwccn both scts of
values seems to be caused mainly．by crrors in the assumption of thc fractional attainment of
equilibria at the adsorption zoncs and in prediction of lqngths of adsorption zones obtained
by the simplificd卑ethod mcntioncd above．・
Conclusion
Based on L． D． F． approximation， analytical solution of length of adsorption zone was
（42）
Asimple design method of a fixed bed adsorber by use of a pcrsQnal computer
presentcd f（）r a system obeyed Langmuir or Freundlich．cquilibrium equation． By adopting
the solution to be the simplest MTZ method， a simplified dcsign mcthod was obtained and
thc mcthod gave more accurate results than the original MTZ method．
Thc abovc solution can be also applicd to multicomponent systems in thc name of the
Extended−MTZ−Method． For ternary component’adsorp専ion， the estimation method of the
smallest bed length necessary to satisfy the Extended−MTZ−Method， was proposed by use of
the mcthod． For a longcr bed lcngths than the value obtained from the proposed method，
the break times obtained by the Extended−MTZ−Method with the method agrced well with
those from the rigorous numerical breakthrough curves．
Thc assumption used to dcrivc the Extended−MTZ−Method， that fractional attainment of
equilibria was l／2， was evidenccd to bc satisfied．
NOMENCLATURE
6
の
＝COnCentratlOn
［kg MH3］
6，乞，c2i
＝conccntratlon ln respectlve zones
Lkg m−3］
＝Langmuir constant
［m3kg−1］
k
＝Freundlich pre−exponent
ハJOs
＝mass tansfer unit based on amount adsorbed
［一］
l／n
＝Freundlich cxponcnt
卜］
9
＝amount adsorbcd
［kg kg−1］
9・i，9≦，
＝amount adsorbed in equilibrium with c，i and cli， rcspectivcly
［kg kg−1］
［kg kg−1］
9。。
＝Langmuir constant
［kg kg−i］
r
＝scparation factor
［一］
t
＝tlme
＝break time
［sコ
＝cxhaustion time
［s］
＝linCar flOW ratC
［cms−1］
彦Bi
tEi
［s］
あ
ヒζ
ら βγ﹂
ζ
＝b6d length
［cm］
＝length of respective zones
［cm］
＝slope of operational．line
［m3 kg−1］
＝bed bulk density
［kgm−3］
＝time difFerence bctwccn彦Ei and彦Bi
［s］
＜subscripts＞
z ＝component or zone・
0 ＝initial or influent
HTERATURE CITli】D
1） Furuya， E． and Y・Takeuchi：J・chcm・Eng・Japan，19，62（1986）
2）Furuya， E． and Y． Takeuchi：Research Report of the Faculty of Engineering， Mciji Univ．， No．35，95
（1978）．
3） Hashimoto， K． et al．：J． Chem・Eng・Japan，10，26（1977）
4） Kawazoe， K． and Y． Fukuda：Kagaku Kogaku，29，374（1965）
5）Miura， K． and K． Hashimoto：J・chem． Eng． Japan，10，490（1977）
6）Miura， K．， et al．：」． Chem． Eng． Japan，12，281（1979）
（43）
明治大学工学部研究報告 No．55（1988）
7︶ 8︶ 9︶
Takeuchi， Y． and E． Furuya：J． Chem・Eng・Japan，13，500（1980）
Takeuchi， Y．， E． Furuya and Y． Suzuki：Kogyo YosuiジNo．233，30（1978）
Takeuchi， Y．， Y． Suzuki and E． Furuya：J． Chem． Eng・Japan，12，486（1979）
10）
Takeuchi， Y．， T． wasai and S・suginaka：J・chem、・Eng・Japan，11，458（1978）
11）
The Soc。 Chem、． Engrs．， Japan（ed．）：“Chem． Engrs． Handbook”，5th cd．， Maruzen（1988）
（44）

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