Computer-aided measurement evaluation of residence-time-distribution curves

Com/wr.

them. Engng, Vol. Great Britain

12, No.

t/3,

pp.

159-163,

1988

0098-1354/88

Printed in

$3.00 + 0.00

Pergamon Press plc

COMPUTER-AIDED MEASUREMENT AND OF RESIDENCE-TIME-DISTRIBUTION

EVALUATION CURVES

J. HUNEK, J. SAWINSKY, A. B~LINT and L. PoDMANIczKyt Department

of Chemical Engineering, Technical University of Budapest, H 1521 Budapest, Hungary (Received 8 July

1987)

Abstract-Utilizing the sampling system of a SAM85 process control microprocessor which was connected to a TPA 1140 fast-computer continuous-phase backmixing in a new RDC extractor type has been studied by residence-time-distribution analysis. Evaluation of weighing functions has been done on the basis of the 1-D diffusion model using different optimization and moment methods. Explicit functions have been derived by Laplace transformation from the model for truncated moments. The correlation of the results has shown that perforation of stator rings has not increased axial mixing.

INTRODUCTION

cL t?*

In this work, backmixing in the continuous phase of a pilot-scale RDC extractor with perforated stator rings have been studied. The results themselves have a chemical engineering interest too, but the computer-aided experimental technique as well as the comparison of the different mathematical methods for evaluation have much more to show even for a further aim of computer-aided process control. With the ever increasing sophistication of mathematical models proposed for flow systems there is an increasing need for methods of evaluating the parameters with greater accuracy. Computer-aided sampling excluding human errors can improve experimental techniques, while the fast evaluation gives the possibility of process control sometimes even with more complicated mathematical methods. For the representation of axial mixing phenomena the 1-D diffusion model has been widely used (Mecklenburgh and Hartland, 1975; Himmelblau and Bischoff, 1968; Danckwerts, 1953; Sawinsky et al., 1980; Ingham, 1971): I 8% ac ac ----=-, Pe dz2

Bz

as

1)

Fig.

I

truncated signal (Fig. 1) to those of the model correspondingly truncated. The appropriate values of the k th truncated moments can be derived as: 98 Mf= Qk.g(9) d9, (2) s0 where 9* is the dimensionless truncation time and g(S) is the weighting function. The Laplace transform of g(9) [the G(s) transfer function] can bc derived from the model and the boundary conditions (Levenspiel and Smith, 1957). We have derived the kth truncated moments as: a= a*

(1)

18=0

Tracer impulse techniques are the most common for determining its parameter, the P&let number. While optimization methods either in the time or in the frequency domain are usually the most accurate, renewed interest is being shown in the new modifications of the faster and more direct moments method. The main problem with the ordinary moments (Levenspiel, 1962; Sawinsky et al., 1979) is that the higher ones are unreliable due to magnification of small errors in the tail. A modification suggested by Kafarov et al. (1968) is to select the parameters by comparing the moments of the

(a)

(b)

_

(3)

(c)

Fig. 2. (a) Open (infinite); (b) semi-open (semi-infinite); and (c) closed (finite) systems for diffusion.

TDepartment of Chemical Technology. 159

160

J. HUNEK er al.

For process control purposes the real time of a response function must not be too long. This can be achieved by choosing the injection and measurement points close enough to each other. In this case the system can be regarded as “infinite” and “open for diffusion” at least one end (Fig. 2). For systems open for diffusion at both ends (Fig. 2a) the transfer functon is (Levenspiel and Smith, 1957):

formation as: M:=F,(~+~)-F~(~--~)-~F,-!-,

(6)

where F, = erfc [K(l - .!J*)],

(7)

F2 = e”.erfc

[K(l + 9*)],

(8)

exp[-KK2(1 - G*)*]

(9)

___

F3 = and

where a=

li

1 +;.s.

(10)

(5)

From (3) and (4) the Ist, 2nd and 3rd truncated moments have been derived by inverse Laplace trans-

M:=F,

-F2

;+&+; (

)

$-if; (

)

(11)

-26(&+3 M*=F 3

_?+E+d+! ’ ( Pe’ Pe* -F2

Pe

2

$$+;-f (

-F,($-$-%)-expe-$-.)

xF,.erfc.K

+F,,

(12)

where F4 = 2.5 + 7

and

+ 9*

- ( E+Pe

F =9*.e-” p.

5 J;r

9*(5KZ + 2K’)

) K*-l.25Pe 9*

$1

+(E+Pe)(f

1

(13)

-4K2-2K4)

+K)].

(14)

However, the functions are rather complicated, they can be easily used in a computer program to calculate Pe from experimental truncated moments if 9* is given. With the aid of Figs 3-5. The appropriate truncation time can be chosen. Sometimes the tracer injection or the measurement has to be made at the inlet or at the outlet respectively (Fig. 2b). In this case the system is open for diffusion at only one end. Here the numerical integration of the weighing function (Strand et al., 1962): g(9)=Eexp[-F(&-fiy]-$

P%

Figs 3-S. Truncated moments for open system (see Fig. 2a).

x exp(Pe)erfc[fi(-$

+ fi)],

(15)

Evaluation 1.0

1.6

E

0.6

0.4

hl:

0.6 0.2

0

10

’

30

-

=yf+pe Pe 4 ctgy,A__,

Pe

40

(18)

(7)

1.6

Pe 4yj

(19)

Following the same method as in the case of an open system, i.e. starting from the Laplace transform of equation (16) we have arrived at functions which were too complicated for inverse Laplace transformation but we could find another exact way to get the truncated moments: fortunately equation (16) consists of a series of uniform convergence, and this fact allows us to change the places of the integral and the infinite summa:

50

Pe

1.2

161

and y, values are the roots of the following characteristic equation:

1.2

20

curves

u

(61

1.5

s*=

of residence-time-distribution

* M2

Mf(9)

= 2.expe):,

RjI:

Sk.exp(-ejg)d% (20)

According to equation (20) the first three truncated moments for the closed system have been derived as:

I 0

10

20

30

40

50

Pe I.4

-

1.2

M:=Z.exp

03)

1.8

F 0

$ s[l J--1 I

-(I

+u,$*)exp(-a,9*)],

(21)

1.5

MI

0.6

I?*

x[2-(2+2a,9*+aj29*2)exp(-uj~*)].

= 1.2

0.6

iUT=2,exp

P 0

(22)

t $[6-(6+60,9* I-1 /

0.4 q°F

u

0.2

0

+3a?9*2-uaj9*3)exp(-uj9*)]. / 1

I

10

20

I

I

I

30

40

50

Pe

Figs 6-8. Truncated moments for semi-open system (see Fig. 2b). has proved simpler. Results can be seen on Figs 6-g. The approaching functions were polynomials for each previously fixed trunction time. If the investigated system is a small one (usually laboratory size) the tracer injection can be done at the inlet and the measurement at the outlet. The system can be regarded as finite and closed for diffusion at both ends (Fig. 2~). For this case, the analytical expression of the g(9) weighing function is (Miyauchi, 1953; Nagata, 1975):

g(9) = 2.exp

3 J-I,ER,exp(-aj9), 0

(16)

where: R

_

/-

(- l)j+lyf 2

Yj+%+Pe

1

(17)

(23)

The appropriate u-unction time can be chosen on the basis of Figs 9-11. In computer aided evalution of RTD curves it is advantageous if there is a possibility to choose the injection and measurement points according to Fig. 2a, because that ensures the least computational time. The possible range of the values of the P&clet number itself determines the advisable length of the investigated part (L), because if Pe > 20 + 50 the sensitivity of all methods are rather small. For the first two cases (Fig. 2a and b) our evaluation method has been compared with the method of ordinary moments choosing the usual (Gyilman ef al., 1966): 2 fJ2=-+fPe

A Pe*

(24)

expression for the second central moment where A = 8 for the open and A = 3 for semi-open system. From the optimization methods in the time domain we used the Hooke-Jeeves (Hooke and Jeeves, 1961) optimization for curve fitting as well as an algorithm worked out on the basis of the linearization

J.

162

HUNEK et al. n ce-‘t

76-

*0

5-

0 + .

My

9.36 10.77

+/’

0

12.20 13.63 15.00

0.6 34-

+**

0.6

/

d -51 mm s -37mm

1 21f&yfy,,

Pe

, 200

400

,

600

,

600

, 1000

,

, 1200

d.n 1.2

”

(IO)

r

Fig.

l9*=1.0

E

12.

Correlationof date evaluatedby the methodof truncated moment.

2nd

1.5

0.9

0.6

1.0 0.4

1.2

5

10

15

20

0.6

Figs 9-11. Truncated moments for the closed system.

memory of SAM85 for each response curve. The evaluation according to the previously described methods have been done by a TPA 1140 fast computer which was connected online to the SAM85 Nonchanging experimental data (flowrate, sampling time interval, geometrical data, etc.) were previously given in a data-file. The transfer of 300 data between the two computers was not fast (2.5 min), however, for all but the Hooke-Jeeves’ method, the computation speed on the TPA was approximately that of printing the results. The Hooke-Jeeves’ optimization was about 100 times slower and it always needed a good starting value for another method. Averaged squared deviations of points of measured weighing functions from the theoretical values are compared in the first two columns of Table 1 for 25 measurements. Results were correlated according to Stemerding et al. (1963) and Ingham (1971):

De*

method in probability-coordinates of Westerterp and Landsman (1962). Experimental sodium chloride tracer impulse response signals on a 2 m x 4 100 mm RDC extractor column have been measured by conductivity cells at two points positioned according to Figs 2a and b. The real time sampling system of a SAM85 process control microprocessor was fitted to the conductivity meters so that 25 + 250 digital data were stored in the

TH=B+CT

d.n

S ;

0

of results

IO’.squared deviations of

HooktJeeves (1961) Westerterp (1962) Methodof Moments

2 centr. 2 trunc. 3 trunc.

1.7 1.9 2.1 1.8 1.9

13 16 20 19 15

_

(25)

The results for C-values are tabulated in the 5th and 6th column of Table 1. All but one agree well with Stemerding’s C = 0.012 slope for nonperforated stator rings. The cause of the one exception is possibly the error of the method discussed in Westerterp and Landsman (1962) for Pe ~2. According to Ingham (1971) experimental data for the B-intercept on the ordinate, do not coincide.

Table I. Comparison

Calculated and measured weighing functions 2b 2a

2

Experimental data from equation (25) 2a 2b 12 47 26 15 15

2a = for system open for diffusion at both ends (Fig. 2a) 2b = for system open for diffusion at one end (Fig. 2b)

22 23 103 64 22

Slope of equation (25) c x 10’ k 2b 1.14 0.80 1.16 1.21 1.11

1.01 I .05 1.19 1.16 1.01

Evaluation

of residence-time-distribution

The most important conclusion from the correlation is that, however, the perforated stator ring (S = 37 mm with 24 x 4 8 mm perforations) has 2.5 times the cross-sectional area of the nonperforated one with the same diamater, axial mixing has not changed. Stemerding’s linear correlation [equation (25)] enabled us to compare the variance of these results, too (see Table 1, 3rd and 4th columns). The fit for the evalution from A4: is shown on Fig. 12 for the “open system” (see Fig. 2a). Tabulated data and experiences on computation times show the adantages of the truncated moments method. The utilization of SAM85 in the system combined with this effective and fast mathematical evaluation method opens the possibilities of computer aided process control as well. Acknowledgement-The authors wish to thank Mr Csaba Fritz for his experimental contribution to this publication during his undergraduate years. NOMENCLATURE a = See equation

A, B C c D,, d F,. FZ, F>. F4, Fs g(9) G(s) H K L I MI n Pe R

(18) Constants Slope of Stemerding correlation Concentration Effective diffusivity (cm2 s-‘) Rotor diameter (cm) Functions, see equations (7-9, 13, 14) respectively = Weighing function = Transfer function = Height of one compartment in RDC extractor (cm) = Function see equation (10) = Characteristic length (see Fig. 2) (cm) = Distance from injection point (cm) = kth truncated moment = Rotor speed (s-l) = u.L/D,,, Peclet number = See equation (17)

= = = = = =

curves

163

S = Diameter of stator ring (cm) s = Variable of Laplace transformation 1,T= Time, mean residence time (s) u = Linear velocity (cm S-’ ) z = I/L dimensionless length 9-l = Sign of inverse Laplace transformation Greek symbols y = c* = 9 = 9* =

See equation (19) Second central moment r/T dimensionless time Dimensionless truncation

time

Subscript j=Serialnumberj=l,2,3... REFERENCES

Danckwerts P. V., Chem. Engng Sci. 2, 1 (1953). Gyilman V. V., M. B. Ajzenbud and E. 2. Sulc, Khim. Prom. 42, 123 (1966). Himmelblau D. M. and K. B. Bischoff, Process Analysis and Simulation. Wiley, New York (1968). Hooke R. and T. A. Jeeves, J. Ass. comput. Mach. 8, 212 (1961). Ingham J., Recent Advances in Liquid Liquid Extraction (C. Hanson, Ed.), Chap. 8, p. 227. Pergamon Press, Oxford (1971). Kafarov V. V., V. G. Vygon and L. S. Gordeev, Theor. Found. Chem. Engng 2, 288 (1968). Levenspiel O., Chemical Reaction Engineering, Chap. 9. Wiley, New York (1962). Levensuiel 0. and W. K. Smith. Chem. Enana. __ Sci. 6. 227 (i95i).

Mecklenburgh 1. C. and S. Hartland, The Theory of Backmixing, Chap. 2. Wiley, New York (1975). Miyau& T. ??ngaku k&ku 17, 382 (1953). Sawinsky J., J. Hunek and L. Podmaniczky, ht. them. Engng Process In& 20, 681 (1980). Sawinsky J., J. Hunek and B. SimPndi, Hung J. Znd. Chem. 7, 69 (1979). Stemerding S. Jr., E. C. Lumb and J. Lips, Chem Ing Tech&. 35, 844 (1963). Strand C. P., R. B. Olney and G. H. Ackerman, AIChE JI 8, 252 (1962). Westerterp K. R. and P. Landsman, Chem. Engng Sci. 17, 363 (1962).