Numerical solution of multicolumn adsorption processes under periodic countercurrent operation

Chemical Enginwing

Science, 1976, Vol. 31, pp. 345-353.

Per&m

Press.

Printed in Great Britain

NUMERICAL SOLUTION OF MULTICOLUMN ADSORPTION PROCESSES UNDER PERIODIC COUNTERCURRENT OPERATION U. GUNNAR SVEDBERG Department of Chemical Engineering, Royal Institute of Technology, Stockhohn, Sweden (Recked 13January 195; accepted 10June 1975) Abstract-Numerical solutions of the dimensionless equations describing multicolumn adsorption processes are presented. The columns are operated in series and move periodically countercurrent to a continuous fluid flow. Results are given for intraparticle mass transfer resistance separately and in combination with lilm resistance assuming a linear adsorption isotherm. Comparisons are made with columns in periodic operation and in continuous countercurrent operation. The utilization of the particles in a multicolumn arrangement, or equivalently in a pulsed bed system,comparedto that in a column in continuous countercurrent operation is commonly about 80% for two columns in series and more than 90% for four columns in series, when the total column lengths and the fluid tlowrates are the same.

INTRODUCTION

There are several ways to contact a granular solid phase with a fluid phase from which one or more components are adsorbed. The mass transfer is most effective in a continuous countercurrent column. Periodic operation in one column is, however, a more common type of operation. In the latter case the adsorbent is less utilized but the apparatus is cheaper and less complex. The width of the mass transfer front is of great importance for the choice of how to operate an adsorption column. A low diffusivity in the adsorbent particle phase and /or a linear or only slightly favorable isotherm give a broad mass transfer front. In that case a periodic column must be very long to give an acceptable utilization of the adsorbent. Semi-continuous countercurrent operation then becomes an attractive alternative way of operation. When one column is operated in a semi-continuous countercurrent manner part of the most exhausted adsorbent is periodically withdrawn from one end of the column, usually from the bottom. At the same time fresh adsorbent is added at the other end. This is called a pulsed-bed system. Another way to get this type of operation is to use a multicolumn system with two or more columns placed in series. Each time an exhausted bed is removed for regeneration a fresh bed is introduced at the downstream end of the multicolumn system. In both of these cases a more or less continuous countercurrent adsorption process is simulated with one or more fixed beds. The process model that will be used in this paper applies to both systems. However, the discussion will primarily refer to the multicolumn system. Purifying liquids by adsorption on activated carbon is one application of growing interest where the semicontinuous type of operation is common. The most wellknown example is perhaps the Pomona plant in Califomia[l] where a multicolumn system consisting of four columns in series are used for purification of wastewater. In Lake Tahoe, Califomia[2], a pulsed-bed system

with one column in semi-continuous countercurrent operation is used for the treatment of municipal wastewater. Several industrial applications of multicolumn systems and pulsed-bed systems have been described131. The technique is also used in ion-exchange[4]. In spite of the great practical importance of the semicontinuous type of operation, no results, based on the same kind of mass transfer model that is used for periodic columns and continuous columns, seems to exist for designing of such processes. An approximate graphical approach has been outlined by Fomwalt and Hutchins [5]. They assumed that all the columns in the multicolumn system have similar shaped breakthrough curves. This assumption gives, as the authors pointed out, an underestimation of the system effectiveness. A “tanks-in-series” model has been used by Chen et a1.161to simulate a semi-continuous countercurrent adsorption process. Plug flow models for single periodic columns have attracted much attention. For a linear isotherm the classical analytical solution including intraparticle diffusion and film diffusion at the surface of the particles is that of Rosen [7]. The same model but including also axial dispersion has been solved analytically by Babcock et al. [8] for spherical particles and by Pellett [9] for cylindrical particles. Numerical solutions for non-linear, favorable isotherms with the assumption of constant-pattern conditions have also been presented[lO]. A mathematical model for continuous countercurrent operation with spherical particles has been solved analytically by Kasten and Amundson[ll]. The mass transfer resistances included in the model are intraparticle diffusion and 6hn diffusion. An analytical solution for a model also including axial dispersion has been presented by Neretnieks[l2]. Both solutions are based on a linear isotherm. As mentioned above, a semi-continuous system is of great interest when the isotherm is nearly linear. The limiting case of a linear isotherm was assumed in this

345 CES Vol. 31, No. S-B

G. BEDBERG

346

paper because of the resulting computational simplifications. Simulations of semi-continuous countercurrent operation have been made using the plug flow type of model.

CYCLE

NO1

MODEL

CYCLE

NO 3

The

mass transfer effects included in the model are intraparticle diffusion, surface Mm diffusion and axial dispersion. The essential assumptions are constant coefficients of dispersion and diffusion, constant porosities and no radial gradients in the bed. The intraparticle diffusion is assumed to take place in the liquid within the pores of the particles. However, for a linear isotherm, assumptions of homogeneous-solid diffusion or only pore surface ditfusion instead of pore diffusion give the same dimensionless equations and thus the same solution. The equations are

ac (1-4)aqA

__+)&~ at+ E

(1)

with the initial and boundary conditions

a(X)

(3)

P(r,x,O)=Pi(r,x)

(4)

S’(r, x, 0) = S:(r, x)

(5)

C(X,O)=

c (0, t) = co c(x, t)
x+f=

= kf[c(x,

t>o

(6)

t>o

(7)

t)-P(ro,x,

t)l.

I , 0

II4 L

112L

columns are thought to be lirst upstream parts of one column with semi-infinite length. Although this is not true for a real multicolumn arrangement, it is usually thought that the error in the resulting solution is negligible. A check upon this has been made with the exact boundary condition (i.e. (ac/ax)(L, t)=O) at the end of a finite column subdivided in four parts. General solutions can be obtained by making the equa- ’ tions dimensionless. In this work the time scale is related to the particles. The dimensionless retention time for those is

This is a suitable definition for comparisons between periodic operation in one column, multicolumn periodic countercurrent operation, and continuous countercurrent operation. The parameters used are the dimensionless groups that arise when the dimensionless variables are introduced. A numerical factor is included only in the Biot number parameter, Bi = kf2ro/(esDpf). The dimensionless variables are

(9) c, =

E*Dpf at=(k,*ES)

1

(10)

2 84 (r 5 >

where q is an “effective” concentration in the particle and {ssDpj/(k, + e,)} corresponds to an “effective” diiusivity. The principal column arrangement is shown for four columns in series in Fig. 1. The concentration profiles, a(x) and qi(r, x), at the beginning of one cycle are not known in advance. Part of these are the remaining profiles in the column(s) that already has been in operation one cycle or more. The boundary condition (6) is exact only when there is no axial dispersion. The condition (7) imply that the

(kq +4e)co

R=; x=;

Equation (2) can then be reformulated to

a4

L

Fig. 1. Principal arrangement of four columns in periodic countercurrent operation. (A total of five colunms in the system, since one is always under regeneration.)

Local equilibrium between the solid phase and the fluid phase is assumed with the following linear adsorption isotherm: S’ = kqP.

3/4L

esDp+ ’ = (kcq+ 4 )ro and the parameters are 8=&f r. D,(w

7

Bi = g PeL = 2

. -Ul-e) V

(kq

I P

(bed length parameter)

t l s) (distribution ratio)

(Biot number) (Peclet number).

Numerical solution of multicolumnadsorption processes

347

with those presented by Rosen[7]. The accuracy in calculated concentration values is estimated to about 1%. The computations have been made for two and four columns in series, eight breakthrough concentration values (0.01,0.02, 0.05,0.10,0.20,0.30,0.50,0.75), four bed length parameter values and three film resistances. In order to contine the computation time and the number of parameters, all computations presented in this paper have ac, 1 a %=~a (12) been made with the assumption of negligible axial dispersion usual for such adsorption systems. Because the with the boundary conditions column length to particle diameter ratio is large (> 1000) the lowest possible value of a Peclet number based on the column length is about 500[ 181.Computations using this C&O,7) = 1 (13) value have shown that the axial dispersion effect is G(X,T)<~ for X+m (14) negligible. The distribution ratio, D, has everywhere been set much larger than the bed length parameter to assure no influence by this parameter. The value D = 8000/3 has been used which value is sufficiently large to make the capacity of the fluid negligible in comparison with the capacity of the adsorbent. The degree of adsorbent utilization, Y, is defined as the ratio between the utilized NUhiERICAL COMF’UTATION capacity and the stoichiometric capacity. When the disNo analytical solution for the actual model applied to tribution ratio is large the difference between the degree semi-continuous countercurrent operation is available. of utilization based on the particles and that based on the Moreover, the numerical evaluation of the existing analyt- whole column including the bed fluid capacity is negligiical solutions for one periodic column is for some of the ble. In this work the degree of adsorbent utilization has been calculated in two ways. The time mean concentracommon parameter combinations very time-consuming. tion value in the effluent can be used to give v = An implicit finite difference method of the Crank(r/&(1 - ckdout).However, a numerically more accurate Nicolson type has been used for the numerical computaway to get the degree of utilization is to use the volume tions in this work. Methods of this type have previously been proved to be efficient for these kinds of averaged concentration value in the particles leaving for equations[lO, 14, IS]. Implicit schemes are used for both regeneration. The analytical solution obtained by Kasten and eqns (11) and (12). The resulting set of coupled systems of algebraic equations have previously been solved by some Amundson [ 1l] for a continuous countercurrent column iterative technique. In this work, bowever, a modification has been used to calculate the curves for that case, of the Thomas’ method[ 161has been used. In this modifi- equivalent to an infinite number of columns in series with cation, the tridiagonal coefficient matrices are factorized the same total length in periodic countercurrent operation. separately starting with the matrix for the concentrations RESULTS inside the particles; the equations can then be decoupled and solved separately starting with the concentrations in ID Figs. 2-4 the breakthrough concentration values in a the streaming fluid. This method is direct, making the commonly interesting range are plotted against the total computations rapid. It is constructed to be usable also for retention time for the particles in different column arcases with nonlinear isotherms. The method is described rangements. Curves are shown for no film resistance, elsewhere [17]. (Bi + m, Fig. 2) moderate 6lm resistance (Bi = 40, Fig. 3) Each computation is started from an initial state with all and high film resistance (Bi = 10, Fig. 4). The curves for columns in the multicolumn system completely regener- one column are just the usual breakthrough curves. For design purposes it is useful to have diagrams which ated. When a predetermined breakthrough concentration value is reached in the effluent (i.e. break point) one cycle include the time mean values of the efluent concentrais finished, and the computation is halted. The movement tions and the degree of adsorbent utilization. Such figures (Figs. 4-7) are presented for each film resistance for the of the columns upstream is simulated and the concentration in the added regenerated column is set zero (i.e. the commonly interesting range of concentration values. In regeneration is assumed to be complete) and the computa- Appendix 2 an example of using the diagrams is shown. Dosages of activated carbon for treatment of municipal tion is then continued for the next cycle. After each cycle the mean concentrations in the fluid water in different apparatus arrangements are calculated. phase and in the particles of the bed are computed. The DJSCUSSION time mean concentration in the effluent over the fmished cycle is also determined. In a single column operated periodically the concentraThe convergence of the numerical method has been tion profiles never stabilize for a linear isotherm. Howchecked by successive reduction of the steplengths. Cal- ever, in a multicolumn system or in a pulsed bed system culated values for one column show very good agreement under periodic countercurrent operation steady periodic In Appendix 1 the relation between some of these parameters and those used by Rosen[7l and Vermeulen[l3] is given. The equations in dimensionless form are

348

G. SVEDBERG 4 _

L

5:2/3 Bi-m

.05

.l

.5

1

2

.l

.5

1

5

.5

1

5

-I

_

6:5/3 Bi-00

-I .l

Fig. 2. Dimensionless concentration in effluent vs dimensionless retention time for the particles with number of columns as parameter. 5:2/3 Bi:40

, G+t ,7_

5:1/3 Bi:LO

.6.5-

Fig. 3. Dimensionless concentration in effluent vs dimensionless retention time for the particles with number of columns as parameter.

.l

.5

1

T

2

.l

Bi=lO

s= 513

.5

1

5

Fig. 4. Dimensionless concentration in etlluent vs dimensionless retention time for the particles with number of columns as parameter.

.Li5

.6-

Bi=lO

,7_ s=113

Bi :lO

6.213

S.116

0

.Ol, 0

.05-

IO-

.Ol,

.O! >-

.lC I-

, .l

, 1 I

, .2

.2

I

, .3

.3

, .4

I

.1

, .5

I

.5

, .6

I

.6

, .7

I1 .7

.6

I

.9

I

1

P

J

.4

.4

.5

.7

2

6m

.5

.6

.6

.7

.6

.6

.9

.9

1

4 L#_ n:l

ft

Fig. 5. Time mean value of dimensionless concentration in effluent vs utilization of the adsorbent with number of columns as parameter.

I

_ _ Bi-m

350

G. SVEDBERG

.Ol 0

.l

.2

.3

A

.5

.6

.7

.6

.9

1

0

.l

.2

.3

.L

.5

.6

.7

.6

.9

1

.3

.4

.6

.6

.7

.6

.9

1

Fig. 6. Tie mean value of dimensionless concentration in efiluent vs utilization of the adsorbent with number of columns as parameter.

profiles (or cyclic steady state) have been found to be established after three to five cycles. The breakpoint concentration values in the effluent from one periodic column and two or four columns in series in periodic countercurrent operation can in Fig. 24 be compared with the e&tent concentrations from one column with the same total length but in continuous countercurrent operation. The relative differences in retention time between the various numbers of columns for a given breakthrough concentration decrease when the total bed length increases. Also these differences decrease when the film resistance decreases. The conclusions fit the general rule that the choice of arrangement of the apparatus becomes more important the lower the separation capacity is. The relative differences in retention time between the continuous countercurrent mode of operation and other modes of operation become larger for higher breakthrough concentrations. For a column with a total length equal to the longest bed in this study (S = 5/3) the ratios between the retention times for periodic countercurrent operation with one, two and four column subdivisions and continuous countercurrent operation are about 0.60, 0.83 and 0.86, respectively, for a breakthrough concentration of 0.10. For a breakthrough concentration of 0.25 the corresponding ratios are 0.60, 0.77 and 0.81, respectively (no tilm resistance).

The bed length parameter values chosen in this study are thought to cover the commonly interesting range of degree of adsorbent utilization. Naturally, an increase in the total bed length gives better utilization. An increase in the mean value of the effluent concentration also increases the utilization. A change in the total length has a more pronounced effect the greater the flhn resistance. For a fixed total length the relative differences between the curves for one, two, four and an infinite number of columns become greater the larger the film resistance. In Table 1 approximate ranges for the ratio between the degree of adsorbent utilization for finite numbers of columns and infinite number of columns are shown for four different values of the time mean concentration in the effluent. The lowest ratios correspond to the shortest total bed length and the largest tihn resistance; analogously the Table 1. Approximate upper and lower bounds for the ratio \y./% (i.e. degree of adsorbent utilization for n columns in periodic countercurrent operation through degree of utilization for a column in continuous countercurrent operation at constant total bed length) ~&out 1 column 2 COlunLnS 4 collmns a,25

0.10 - 0,97

a,93 - > 0999

0.97 - > 0.99

0,io

0.55 - 0.90

0.65 -

0,96

0,94 - ' 0399

0.05

0,50 - 0,80

09'15 -

0997

0,92 - ' 0,99

0.01

0.35 - O,TO

0.70 -

0,9T

0,9O -

0999

.l

.2

.3

.4

.5

.6

.?

_

5:213

_

Bi:lO

.6

.3

.4

.5

.6

.7

.6

.9

1

Fig. 7. Time mean value of dimensionless concentration in efauent vs utilization of the adsorbent with number of columns as parameter. highest ratios are found for the longest bed and no film

resistance. Subdivision of one column into two columns (or operation of one column as a pulsed bed) gives an appreciable increase in the degree of adsorbent utilization. With four subdivisions the degree of utilization is for all studied cases greater than 90% of that reached with a column in continuous countercurrent operation.

Acknowledgements-The author wishes to thank Jvars Neretnieks and Richard C. Aiken for very helpful discussions during the preparation of the manuscript. NOTATION

CONCLUSIONS

Subdividing a periodic column into two or more columns and operating them periodically in a countercurrent mode or operating one column as a pulsed bed gives a high increase in the utilization of the adsorbent. The profiles in the columns, which, for a linear adsorption isotherm, never stabilize in a column of infinite length, reach a cyclic steady state within three to five cycles. One periodic column will give more than 35% and in the usual applications about 60% of the degree of adsorbent utilization reached for a column with the same total length in continuous countercurrent operation. Two columns give more than 70%, usually about 85%, and four columns more than 90% and often 95% of the utilization reached for continuous operation. Thus periodic countercurrent operation in a multicolumn system or in a pulsed bed system often is a very attractive mode of operation from an economic point of view.

D

DL

4 Qf ID

k k

L P PeL 4 r R” Re S’

Blot number = (kf2ro/csDpf) concentration in fluid phase concentration in fluid phase at inlet dimensionless concentration in fltid phase = c/c0 dimensionless concentration in particle = 4 I(co(k4 + 6 )) distribution ratio = [(l - c)/e]($ + 4) axial dispersion coefficient diflusivity in ordinary fluid phase Musivity in fluid phase in particle j-factor for mass transfer = Sh /(Re SC Ia) equilibrium constant fluid-particle mass transfer coefficient total length of the column arrangement concentration in fluid phase in particle Peclet number = (Lu/cDL) effective concentration in particle = S’ + E.P radial coordinate for particle radius of particle dimensionless radial coordinate for particle = r/r, Reynold number = (pu2r0/17) concentration in solid phase

352

G.

SVEDBERG

SC Schmidt number = (v/pDf) Sh Sherwood number = (kf2rO/Df) t retention time for the adsorbent particles in the column arrangement 0 superficial fluid velocity bed length coordinate dimensionless bed length coordinate = x/L

Greek symbols length dimensionless bed parameter = [e,D,fL(l - l)/ro*ol void fraction of bed void fraction of particle viscosity density of fluid dimensionless retention time for the particles = [cDptt KL, + es)roZl degree of adsorbent utilization Superscripts A volume averaged variable - time averaged variable 1 maximum value of variable Subscripts i initial state n number of columns out efauent REFERENCES [l] Parkhurst J.D., Dryden F. D., McDermott G. N. and English J., J. W.P.C.F. 1967No. 10 R70. [2] Gulp R. L. and Gulp G., Advanced Wastewater Treatment, Van Nostrand, New York 1971. [3] Kaempf H. .I. and Winkler H. E., Wasser, Lufi und Betrieb 197216 No. 9 1. [4] Helfferich F., Ion Exchange McGraw-Hill,New York 1%2. [5] Fornwalt H. J. and Hutchins R. A., Chem. Engng May 9,1966 155. [6] Chen I. W., Cunningham F. L. and Buege J. A., Ind. Engng Chem. Process Des. Develop 197211 No. 3 430. [7j Rosen J. B., Ind. Engng Chem. 195446 1590. @I Babcock R. E., Green D. W. and Perry R. H., A.I.Ch.E. J. 1%5 12 922. [91Pellett G. L., Tappi 196649 75. PI Fleck R. D. Jr, Kirwan D. J. and Hall K. R., Ind. Engng Chem. Fundam. 197312 95. Ull Kasten P. R. and Amundson N. R., Ind. Engng Chem. 195244 1704. WI Neretnieks I., Swedish Paper J. 1974II 407. 1131Vermeulen T., Ado. Chem. Engng 19582 147. D41 McGreavy C., Nussey C. and Cresswell 0. L., I. Chem. E. Symp. Ser. 1%7 23 111. WI Garg P. R. and Ruthven D. M., Chem. Enana Sci. 197328 791. Ml Lapidus L., Digital Computation for Chemical Engineers McGraw-Hill, New York 1%2. WI Svedberg G.,.Ph.D. Thesis, Royal Institute of Technology, Stockholm 1975. WI Perry J. H. and Chilton C. H., Chemical Engineers Handbook, 5th Edn, McGraw-Hill, New York 1973. t191 Westermark M., .I. W.P.C.F. 197547 704. WI Wilson E. J. and Geankoplis C. J., Ind Engng Chem. Fundam. 19665 9.

definitions have been chosen to give a convenient presentation for a special process or a certain mode of operation. The most common parameters are those used by Rosen[7j and Vermeulen[l3]. Here a table is presented where the relation between the parameters used by different authors is shown.

This

work Film resistance parameter Bed length parameter Dimensionless time parameter Flow ratio

Rosen [7]

Bi

Y =2/Bi

s

x=38

Vermeulen[l3]

N,/N, = Bill0 N, = 156

T

y=2(~--s/D)

;

f=+l/D)

0, =15(7-6/D)

2=&?-l/D

The time required to put in a volume of feed stoichiometrically equivalent to the capacity of the particles in a column is L(le)(k, t c,,)/u. The group e8DJ((k, t e.)rO*) has the dimension inverse time and gives the time scale for the intraparticie diffusion. Thus the bed length parameter, 8 = (L(1 - e)/o)(cDJr,Z), may be thought of as a ratio of one time specifying the streaming fluid and another time specifying the diffusion in particles. The dimensionless retention time for the particles, 7, differs from Rosen’s dimensionless contact time parameter, y, only by a numerical factor if the distribution ratio, D, is large (or more precisely if S/D Q T). In that case the relation to the throughput parameter, Z, defined by Vermeulen is simply Z = T/8. SUF’PLE=ARY NOTATION number of transfer units for film diffusion number of transfer units for solid-phase internal diffusion dimensionless bed length parameter dimensionless time parameter throughput ratio dimensionless film resistance parameter time modulus APPENDIX 2 Example of using the diagrams for designing purposes

Activated carbon can be used for adsorption of organic compounds from municipal wastewater. The most important factors in such an adsorption process from an economic point of view are the carbon dosage, the bed length and the arrangement of the apparatus. Calculations of the carbon dosages are made for one bed length with the different arrangements of the apparatus that have been treated in this paper (one periodic column, columns with two and four subdivisions in periodic countercurrent operation and one continuous countercurrent column). The reduction of adsorbable organic compounds has been chosen to 90%. The data for the activated carbon have been taken from Westermark[l9]. Data for the activated carbon: Type of carbon Void fraction of carbon particles Void fraction of bed Equilibrium constant Diiusivity in fluid in particles DitTusivityin fluid

WV-G

es = 0.50m”/m’ e = 0.40m3/m-’ k., = 9200 kg COD/m3particle [ kg COD/m3liquid I Dpf = 4.68x lo-” m*/s Df = 2.50x lo-” m2/s

(The high value of D,,f compared to Df is thought to be due to surface migration and a slightly nonlinear isotherm.) Density of water at 10°C Viscosity of water at 10°C

APPENDIX 1

p=l.OOXl@kgjm’ 7)= 1.30x 10e3Ns/m*

Relations between parameters

Assumed data

A large number of parameters has been used in the literature to describe general solutions to fixed bed processes. Often the

Radius of carbon particles r, = 5.0~ 10e4m. Water velocity based on empty cohmm u = 4.0 x lo-’ m/s.

353

Numerical solution of multicolumn adsorption processes 1.30x 10-3 SC= loa . 2.5oxlo-,o= 5.20x ld

Cafcufations Calculations are made for a bed length parameter value of S = l/3. This is about the value found by Westermark[l9] by an economic ontimization of a continuous countercurrent adsorber. The detinitibn of 8 gives the corresponding real bed length as

and then

.

B1 =

L = l/3 . (5.0x 10-9’. 4.0 x lo-” = 2 4 m 050*468x10-‘“~~l-o.4n~ . . . .

Results are presented here for Bi = 40 and for Bi -+a. Values for the actual I%ot number can then be achieved by interpolation. The carbon dosage is here calculated as L(1 - e)/& which is the amount of carbon particles needed for treatment of one unit volume of water. The degrees of utilization for diierent types of column arrangements have been taken from Figs. 5 and 6. The following mass balance relation

The film resistance can be calculated via an empiric relation given by Wilson and Geankoplis[20] j,

_

“09

2.50x lo-” = 73.4. +1.09(3.08.5.20x 1O3)“3 4.68x lo+. 0.50 0.40

Re-l,’

E

L(l-E)(ILq tE$)Y= tu(l-E@J

valid for 1.6~ lo-” < Re < 55, 0.35< E < 0.75. This gives Sh = l.O9/c(ReSc)“” where the Sherwood number is based on the diiusivity in the fictive film surrounding the particles. The Biot number is based on the dilTusivityin the pores in the particles and thus

can then be used to get the carbon dosage as L(l-E)_(l-L)~

to

Bi=fiSh. *, s

w,

For one periodic column and cohunns with two and four subdivisions in periodic countercurrent operation the minimum reductions have been found in Figs. 2 and 3 via calculation of

From the given data one can get

to

Re=103~4.0x10-3~2~5.0x10-4=308 1.3x lo-’

Number of columns

The

Degree of utilization Y

1 2 4 m

+4

Y

~=L(l-e)(k,+e~)=(l-c,,.,). results are:

m’ carbon m” L(l-e)

to

Minimum reduction

(1-LA.

100%

Bi=40

Bi+m

Bi=40 (X 10-3

Bi-tm (X 10-3

Bi=40

Bi+m

0.44 0.60 0.65 0.67

0.54 0.70 0.75 0.77

2.2 1.6 1.5 1.5

1.8 1.4 1.3 1.3

73 78 83 90

68 75 80 90