Comparison of experimental and predicted breakthrough curves for adiabatic adsorption in fixed bed

COMPARISON OF EXPERIMENTAL AND PREDICTED BREAKTHROUGH CURVES FOR ADIABATIC ADSORPTION IN FIXED BED LIS MARCUSSEN Departmentof ChemicalEngineering, The Technical University of Denmark, Building 229, DK-2800Lyngby, Denmark (Received 1I June1980;accepted 6 Juuly1981) Abstracr-Mass and heat transfer in adiabatic fixed bed adsorbers is described by a model which considers resistance to mass and heat transfer simultaneouslywithin the porous adsorbent particles and in the fluid flowing past the pellets. The adsorption equilibriumis described by a temperature dependent Freundlich isotherm. The mathematical model which is given in dimensionless form is solved numericallyfor different values of the dimensionless parameters. An example of predicted concentration and temperature breakthrough curves for an adiabatic integral bed is given. This prediction is performed solely on the basis of data from an isothermal differential bed. The predicted breakthroughtime is within 8% of the experimental value when the dimensionlessparameters are assumed constant and within 2% when their concentration and temperature dependence is taken into account. INTRODUCTION

Breakthrough curves for the experiments described in[l]

are predicted by means of different models for adsorption in integral fixed bed. First, the breakthrough time is predicted by some models assuming isothermal adsorption. The method described by Treybal[2] is based on the assumptions of a constant-pattern mass transfer zone and an overall driving force for the transfer of adsorbate from the flowing fluid to the adsorbent. The model of Hougen and Marshall[3] considers mass transfer controlled by gas film resistance and describes the adsorption equilibrium by a linear relationship. Kyte[4] also assumes film controlled mass transfer, but allows for a nonlinear adsorption equilibrium by using a Freundlich isotherm. RosenDl maintains Hougen and Marshall’s assumption of a linear adsorption equilibrium, but includes a diffusion resistance within the adsorbent in addition to the fluid film resistance. Next, nonisothermal adsorption is considered. Work on nonisothermal adsorption in integral beds has been performed, e.g. by Meyer and Weber[7] and Carter[g]; their solutions are given for parameter values different from those in the experiments mentioned above. The method of Leavitt[9] for nonisothermal adsorption which is further developed by Pan and Basmadjian [lo] assumes the existence of two constant-pattern transfer zones separated by an equilibrium zone; mass transfer from fluid to adsorbent is described by an overall transfer coefficient. The model for nonisothermal adsorption which is used in the present work is described below.

considered to be adiabatic. Axial dispersion and radial concentration and temperature gradients in the fluid are assumed to be negligible. The nonlinear Freundlich isotherm q = k( T)*c”

The gas phase accumulation of adsorbate may be neglected according to [12]. Initial and boundary conditions: C=Ci.it~attSO,

OGlCR,

$=Oatr=O,

O
OCi?CL

(3)

t>O

(4)

at r=R, O=ZzGL, cff’ ;ac 1R =he(c,-cCn)

D

[

t>O. (5)

Mass transfer from the flowing fluid to the surface of the adsorbent particles:

MODEL DEVELOPMENT

The model which is used for the prediction of breakthrough curves considers resistance to both heat and mass transfer in the fluid film surrounding the particles as

(1)

is used to describe the adsorption equilibrium. The constant k is a function of temperature whereas n is taken to be constant which is justified experimentally in the temperature range 9544°C for the system water vapouractivated alumina [l I]. The mass transfer within the spherical adsorbent particles is described by the differential equation:

V.$+h[c,-cIJ=-t.

??5 at.

(6)

with initial and boundary conditions

well as within the particles. Resistance to adsorption on the pore surface is neglected. The adsorption process is

c, = ceit at t c 0, 0 G 2 d L 299

(7)

LISMARCUSSEN

300 c, = cfa

t > 0.

at z = 0,

(8)

Heat transfer in the adsorbent particles: kc*.

g+~.~]=pp.cp.~-wH.~ E

w,

Y=

4o

(9)

CfO”

’ t.

PP .

CP

with initial and boundary conditions T= Timitat r&O, OsrGR, g=Oat

r=O,

OG.z
OczaL,

I>0

(10) (11) Sh, = $$

n=ffH *(c-T,)

at r=R,

OGzSL,

t> 0.

err

= Modified Sherwood number

Ri=$_% = Biot number err

(12)

w, . h4 A = R, . T,,

Heat transfer from the flowing thud to the outer surface of the adsorbent: “.aT,

&I’& +--JT~-TR)=-~$ az p

0 = viv,

(13)

n

with the initial and boundary conditions yinit= Km T, = rni, at t s 0, 0 c z C L

(14)

Tr = T,, at 2 = 0, t Z 0

(15)

The differentia1 equation (2) may now be written in It is assumed that it is not necessary to consider dimensionless form: momentum transport. Equations (D-o-(5) are solved simultaneously. The problem has: 4 dependent variables c, cr, i” and T,; 3 independent variables z, r, t; 18 parameters k(T), cro, cinrr,n, Dee h, a,,, K R 6, L, Go, Tnit, WH, kR, aH~ c,

1

A Pll ’ G

The number of parameters is reduced by introduction of the following dimensionless parameters and variables: Independent variables:

From eqn (9):

x=rjR

(16) is multiplied by y and added to (17): r=-

vo

t

r*L

(18)

Dependent variables: y = c/c,,

(16) leads to

ay’_

z-S.K.n.y”-l

1

.V2y-~.~.

(19)

where (&Y/&)= (X/a@ * (J@/JT) is found from K(B), which is estimated by means of the ClausiusClapeyron equation: Parameters:

W,.M=R,T

Comparisonof experimentaland predictedbreakthrough curved

301

and boundary conditions $oat

*=o,

$=Sh,.[Y-y,_,l

OSZGl.

at

x=1,

r>o

(31)

04261,

7>0 (32)

Y=latZ=O, g=Oatx=O, W, is assumed to be independent of temperature in the actual temperature interval.

7>0 OGZGl,

~=Bi*(QI-B,,,)atx=1,

OGZGl,

*= 1 at Z=O, K=t=exp

7>0.

(33) 7>0

(34)

r> 0

(35) (36)

w;;.;;“.(;-l)+alne] =exp{[~,(f-l)+lnA].nJ

(21)

where ( W, *M)/(R, - T,,) = A is a dimensionless parameter. (21) fits the experimental values of k(T) in [ll] well.

which is inserted into (19). The set of differential equations to be solved simultaneously is then: &av= Jr K.,:,._I.[~+~.n]-y.(-~+~),6,~

(23)

o.~=r.[3+~.~]+~.[~+~.~]

(24)

“*g+a.[Y-y,_,l=-$f

OCXSl,

0~2~1,

7QO (27)

d = Binit= Tinid50, O
O
7GO (28)

Yinir=yi”;t, OCZ91, @inil= &a,

SOLUTtON

Y(X,7. z)

Y(7,?I

(25)

with the initial conditions y=yi,it=Ci,i,/CfO,

NUMHUCAL

The eqns (23)-(36) are solved numerically using an IBM 370 computer. The partial differential eqns (23) and (24) are written at N selected radial positions x1, x2.. . xN according to the orthogonal collocation method, which was introduced by Villadsen and Stewart[l31. In this way the partial differential equations are split into 2 x N coupled ordinary differential equations. These collocation equations and the fluid phase equations are solved simultaneously using the method of characteristics and a second order predictor-corrector method for integration in the (Z, I)-plane. An exception from this is the first integration step where a first order predictorcorrector method must be used. The results printed out are:

7~0

0 Q Z < 1, 7 4 0

(29) (30)

The computer program is checked by computation of: (I) Isothermal adsorption with linear adsorption isotherm and different values of S/t, and a. The results are in close agreement with the results of Rosen[Sl. (2) Isothermal adsorption in integral bed with nonlinear adsorption isotherm. The solution y(x, 7) at Z = 0 is identical with the solution for adsorption in differential bed[l4]. (3) Adiabatic adsorption with nonlinear isotherm for a case studied by Carter[8, Fii. 31. The results agree with Carters work. Some of the numerical results are shown in Figs. l-7, where the predicted breakthrough times are given for different values of each dimensionless parameter, keeping the other parameters fixed at the following

302

La

MAR~USSEN

values:

a . Z = 23.8 g=o.914 a

(40)

= 0.731

Y

-=_ a@

A

‘,

(37)

YiGr =

=

1

yi& = CJCf*z 0

Bi., = #bi, = 1

(42) where

I

I

in many cases if the fluid is a gas.

The ordinate in Figs. l-7 is (T-Z)/(S), since the introduction of U=(T-Z)/(S), w=(Y*Z and v=l in the eqns (23x26) leads to

8 _ ko. c,,,-’ . kc, K PP. CP .Dee The breakthrough time (7 -Z)/(S) at the bed outlet is 7-l

(38) where K is given by (21).

(41)

cuZB-y6LnBiSh)=0 t&' ,)(It , t *

=22

n = 0.4737 V

(Y

showing that the solution for constant fluid velocity (V= 1) may be given as:

Sk, = 18.0 Bi = 0.947

E W- !%=,I

a0

; = 3%.6

-z-z

s

7

a7.t

S R2. k, cton-”

since 7 9 1 for practical applications.

06-

20 axZ=a,xLxjDxS;2~3xZ

Fig. 1. Calculated“breakthroughtime” vs dimensionlessposition in adsorbcr.

304

hS bfARCUSEN

Y 06 0

05 04 03 02 7-Z

01

T 0.

C

Fig. 4.

Sh, Fig. 5.

305

Comparisonof experimental and predicted breakthroughcurves

Y

0.6

I

I

I

2

4

6

Bi Fig. 6.

A Y

04-

/-

0.6

r_Z 8 oz-

0

1 IO

1 05 n Fii.7.

D

-

306

LISMARCLISSEN

Anyhow, the breakthrough time predicted for adiabatic adsorption (curve number 2) is still larger than the experimental value (curve number 5). Consequently, a breakthrough curve (number 3, not shown in Fig. 8) is predicted, allowing the effective diffusivity to vary with temperature as given in[ll]. This improves the agreement between experimental and predicted breakthrough curves. A still closer agreement is obtained when the breakthrough curve (number 4 in Fig. 8) is predicted by means of a modified computer program which allows the phy. sical parameters to vary with concentration and temperature. Curve No. 4 shows that this variation of the parameters has an effect which is similar to the effect of axial dispersion. Figure 9 shows the experimental and predicted temperature vs time curve at the bed outlet z = L. The curve numbers have the same meaning as in Fig. 8. Breakthrough curves are also predicted by means of other published models, among them the models of Hougen and MarshaU31, Kyte[41 and Rosen[Sl, which are limiting cases of the model used in this work. The predicted breakthrough times 7prsdic,sdare compared to the experimental values 7.,, An example is given in Table 1 for experiment number 1 at Y = 0.05. Inspection of Table 1 shows that only the methods of Rosen, Pan and Basmadjian predict a “safe” value of the breakthrough time c for this special case, while the other methods overestimate the capacity of an existing bed (or underestimate the necessary bed height if the column is to be designed).

DLWUWON

Figure 8 shows breakthrough curves which are predic-

ted by means of the computer program using data from adsorption in isothermal differential beds[ll] and from literature. No information from adsorption experiments in integral beds is used for the prediction_ The calculations shown in Fig. 8 are performed for the conditions of experiment number 1 in[l], using a number of internal collocation points N = 5 and various simplifying assumptions:

1

Curve Number 2 3t No No Yes Yes

4 No Yes

Isothermal adsorption Adiabatic adsorption with temperature dependent adsorption isotherm

Yes No

Temperaturedependent effective diffusivity E&n(T) Variable parameters: &(ZJ KM C&I), h(T), QUIP,(T)

No

No

Yes

Yes

No

NO

No

Yes

tNot shown in Fig. 8 for the sake of clearness. Curvenumber3 is located between number 2 and 4. Figure 8 shows that the assumption of isothermal adsorption (curve number 1) as expected leads to a larger

breakthrough time than predicted when adiabatic adsorption is considered (curve number 2).

Comparison of experiment01 breakthrough Predicted number

and

predicted

curves. curve

Assumption Isothermal adsorption, Adiabatic adsorption. with constant parameters. Adiabatic absorption with variable parameters.

h 4

5 texperlment)

IOCQ

2000

3000 t, SBC

4000

WOO

Fig. 8. Comparisonof experimentaland predicted breakthroughcurves.

6000

Comparison of experimental

and predicted

breakthrough

307

curves

Comparison of experimental and predicted gas ot bed outlet. Predicted for adiabatic adsorption with Curve no. 2: constant paramatsrs Curve no. 4: Predicted for adiabatic adsorption with variable parameters.

4

0

1000 Time

I500 t,

m

temperature

0

2000

WC

Fig. 9. Comparison of experimental and predicted gas temperature at bed outlet. 0, experiment. Curve No. 2: predicted for adiabatic adsorption with constant parameters. Curve No. 4: predicted for adiabatic adsorption with variable parameters.

Table 1. Isothermal adsorption, nonisothermal adsorption Reference

'predicted- 'exe.

Resistance to mass -transfer

'exp. %

Treybal[2]

overall [6,151 (Linear driving force in a8 and solid7

Nonlinear

51

44

Hougen and Marshall[ 33

Gas film

Linear

KYte

Gas film

Nonlinear

102

Linear

-30

Rosen

[41 [5]

Model used Ln this work (Curve No. 1 in fig. 8 ).

Simultaneous gas film and solid diffusion resistance

Simultaneous gas film and pore diffusion resistance

Overall 16,151

Non1 inear

Nonlinear

37

-73

(Linear driving force in as and solid f

Model used in this work. (Curve No. 2 in fig. 8). constant parameters

Simultaneous gas film and pore diffusion resistance

Nonlinear

Model used in this work. (Curve No. 4 in fig. 6). Variable parameters.

Simultaneous gas film and

Nonlinear

pore diffusion resistance

8

a.05

LISMARCLWEN

3@3

Table 2. Comparison of experiments and approximate prediction of dimensionless breaktbrougb time (7 - I)/@). The predicted breaktbrougb times are found by means of the numerical solutions using the simplifyingassumption (43) 'predicted

- 'exp

7 number Experiment

exP

+4.4 +8.7 +6.5 -2.4

1

2 3. 4. 5* 6 7

-5.3 +13.0

-2.8 -1.2

10,

+16.3 +12.2 -0.4

Il.

t11.1

+l.a +4.0

9.

+15.5 +11.7

LlO.0

14

+3.1

-11.4

solution

extrapolated.

ma1 adsorption is within 8% of the experimental value if the parameters are assumed constant and within 2% if they are allowed to vary with concentration and temperature. A rough prediction of the breakthrough time for

experiments number 1-14 is performed by means of the numerical solutions shown in Figs. l-7, assuming that there is no coupling between the dimensionless parameters: -7-l L 6

-17.1

12

Numerical

1

ref

. n fi f

s

-16.2

13

Consequently, it might be an advantage to use, e.g. Rosen’s method if this was true for all cases. Comparison of the predictions by Rosen’s method with those of the model used in this work (Figs. l-7, at Y = 0.1 to avoid extrapolation of Rosen’s results at lower values of Y) shows, however, deviations in the range -43% to + 120% for the cases considered here. This is caused by the fact that Rosen’s assumptions of isothermal adsorption and linear isotherm have opposite influence on the breakthrough time: Isothermal adsorption gives larger r than adiabatic adsorption, and linear isotherm leads to smaller r than nonlinear adsorption equilibrium. The deviations found between predicted and experimental values in Table 1 reflect the fact that the assumptions of the models are not fulhlled for this experiment. This refers especially to the assumptions of isothermal adsorption, linear isotherm, constant-pattern transfer zone, linear driving force in the particles and only one controlling resistance to mass transfer. In Table 1 is also shown the breakthrough time for experiment number 1 predicted by the model used in this work. The predicted breakthrough time for nonisother-

7-l -= s

100

-5.6 -2.5 -16.5 -19.7 -10.7 -9.4

+11.8

a

*

.1

(43)

where [fr - l)/(S)lM is the breakthrough time for the reference set of parameter values given in (37).

(r- l)/(S) is the dimensionless breakthrough time read from Fig. l-7 for the actual value of parameter i keeping the other parameters at the reference values. The results are shown in Table 2 at two values of the adsorbate depletion ratio Y. The results indicate that a first estimate of the breakthrough time may be obtained by means of Figs. l-7 in this way. NOTATION

outer surfaceof adsorbent particles, rn’lrn’bed Biotnumber = R (IJt adsorbate concentration in the fluid phase of the adsorbent particleat distance r from the center, kg adsorbate/m3 gas adsorbate concentration in the flowing medium, kg adsorbatclm3gas very small initial concentration in the bed, c.
Comparisonof experimentaland predicted breakthrough curves

wka

!i

zk Y

li

P Pr 4

r : SC Sh,

I T q V ? Y y z z

effective thermal conductivity of the adsorbent particles, kW/ImK) thermal conductivity of tbe fluid, kW/(mK) bed length, m Lewis number = Sc/Pr molecular weight of adsorbate, k&no1 exoontnt of the Freundlich isotherm number of collocation points in the xdirection porosity of the adsorbent, (m3pore vohtme)/(m’particle) Prandtl number = cp. v/k, adsorbate concentration on the solid at distance r from the center, (kg adsorbate)/(m’particle) distance from the sphere center, m radius of the spherical adsorbent particles, m gas k.l/(kmqlK) _ -. constant ._ Schmidt number = q/(p .Db) modiied Sherwood number = R . h/D, time,s temperature of the adsorbent at distance r from the center, K temperature of the fluid, K superficialvelocity of fluid, m/s tiy of adsorption, kJ/kg clcfo c&0 axial distance from bed entrance, m JL

Greek symbols a &. . L . hY(v,) an heat transfer coefficient in tbe Auid.tXm surroundinn the

particles, kW/(m*K) B (a.D. L ’ n,)/(V ’ p ’ cp) wlf-

%*

Cfo”~I~Tfcl’PP

.CP)

; (b era”- . V,,. R’)/(r *L *Dd)

porosity of the bed 1) viscosity of the fluid, k&m s) c

e

flTf0

309

A (WI *W(% * Tfo)

(pp . C, V, * R2)I(6 *L * k,#) K p density of the fluid, kg/m3

pp r u @

density of the adsorbent kgjrn’particle ( V,+t)/(r *L) (number of residence times) viv, TAT,,

SubsctiDts

value at bed entrance If

value at outer surface of the adsorbent particle init initial value

REFERENCliS

Ul Marcussen L. and Vinding C., Chem. Engng Sci. (MS No. 147). [21 Treybal R. E., Muss Transfer Operations. McGraw-Hill. New York 1%8. 0. A. and Marshall W. R., Chem. Engng Pmg. 1947 I31 y;;p Kyte W. S., Chem.Engng Sci. 197328 1853. ;; Rosen J. B., lnd. Engng Chem. 195446 1590, J. Chem. Phys. 195220 387. 161Lightfoot E. N., Sanchez-Palma R. J. and Edwards D. O., New Chemical Engineering Separation Techniques. Interscience. New York 1962. Meyer 6. A. and Weber T. W., A.1Ch.E.J. i%7 13 457. t:l Carter J. W., Trans. Inst. Chem. Engrs 196644 253. 191 L.1 Leavitt F. W., Chem. Engng Frog. 196258 54. _ _ Sci 1%7 22 HOIPan C-Y. and Basmadjian D., Chem. Engw 285. Marcussen L., Chem. Engng Sci. 1974 29 2061. Marcussen L., Chem. Engng Sci. 1970 25 1487. VilladsenJ. V. and Stewart W. E., Chem. Engng Sci. 1%7 22

1483. 1141Marcussen L., Acta Po/ytech. Stand., Chemistry including MetallurgySeries No. 94 1970. WI Glueckauf E., Trans. Faraday Sot. 1955 51 1540.