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Residence time distribution analysis in controllable flow conditions: case of rotating disk reactor
Chemical En@neerC,,g Sdenre Vol. 39, No. 5, pp. 813-819, hined in Grzat Bntain.
1984 8
CKW-2509/X4 S3.00+ .OO 1984 Pergamon Press Ltd.
RESIDENCE TIME DISTRIBUTION ANALYSIS IN CONTROLLABLE FLOW CONDITIONS: CASE OF ROTATING DISK REACTOR MYUNG Department
JIN KIM, YOUNG
SUNG GHIM and HO NAM CHANG*
of chemical Engineering, Korea Advanced Institute of science and Technology, P-0. Box 150 Chongyang, Seoul, Korea
(Received
24 March
1983; in revised form 21 June 1983; accepted 31 August
1983)
Abstract--A four-parameter multistage model was used to illustrate the mixing phenomenon in a rotating disk reactor. The parameters were. estimated by the nonlinear regression from the residence time distribution data of a tracer. The dead space portion was negligibly small due to the sticient mixing of the liquid, but the size of the mixing regions, cross flow rate and number of stages varied with design and operation factors. Variations of the parameters were closely related to the expected fluid motions in the reactor. A dimensionlessgroup combiningthe Reynolds numberand the Ekman number was proposed for the analysis of mixing in ihe reactor.INTRODUCITON
Residence time distribution (RTD) analysis has been a valuable tool in obtaining information on the fluid motion and mixing in a continuous flow system[l, 21. The most common approach is to choose an appropriate model and to fit the model output to the experimental output. Many types of mixing models have already been proposed and can be divided into two groups: a dispersion model which has been used mainly for empty tubes and packed beds; and a combined model in which complete mixing and plug flow regions are located in series and parallel arrangements. As the number of mixing stages increases, that is, as the scale of fluid motion causing the mixing effect decreases, the combined model approaches the dispersion model. After examining the literature in chemical engineering, however, one can readily see that the dispersion model has been more widely used. The major disadvantage of the combined model is that the involved parameters may not be closely related to the physical characteristics in the system. Because of this limitation, several workers[l-31 cautioned against the danger of empirically fitting the experimental data, thereby discouraging use of the combined model. We believe that this particular defect of the combined model is not intrinsic, but it arises from conditions in the accompanying systems. The mixing patterns in the stirred tanks and fluidized beds are too complex to be explained by parametered deterministic models. However, in the rotating disk reactors whose schematic diagram is shown in Fig. 3, the fluid motion can be characterized by the axial direction of the bulk flow, the transverse direction of the rotating flow and the mixing results from interactions of the two. The dominance of these flows and their interactions are easily related to the mixing cells and
*Author
to
whom correspondence should be addressed.
CES Vol. 39. No. -
813
streams in the combined model. The interconnection between the model parameters and the real fluid motions will be emphasized throughout the work. The prototype of the rotating disk reactor employed in the present work can be found in biological wastewater treatment [4] and blood oxygenation [S]. The main feature of this reactor is its regular contact of the solid phase with alternating gas and liquid phases. Gas is easily transferred through the thin liquid film entrained along the disk surfaces, and the liquid is spontaneously agitated by rotating disks[6]. Though the rotating disk reactor is not widely used and reported data is rare, it is promising as a general three phase contact system for the future.
FLUID MOTZON AND MIXING MODEL
(1) Fluid motion due to the rotation of disks When the disks steadily rotate in the same direction with a sufficiently high speed, a boundary layer will develop on each disk. The situation is quite complex, however, and a large number of multiple steady states has been reported [7J Two well-known solutions are introduced here because they provide enough qualitative information to devise a model. One solution is described by Batchelor [8] who assumed that one disk behaves as a centrifuge with an inflow in the axial direction while the fluid is sucked at a neighboring disk. In this case, the fluid between the boundary layers will rotate with a constant angular velocity (Fig. la). But, Stewartson[9] suggested that both disks behave as centrifuges. The fluid rotates only in the boundary layer while a radial and axial flow results outside the boundary layer (Fig. lb). In practice, though, the finite disks rotate in a vessel and are partially immersed in a liquid phase with a free surface. Three factors namely, the disk edge, trough wall and free surface, should also be considered. The disk edge will induce a radial jet Bow with swirling around the edge and a vortex near the rim center[ lo]. These flows will be confined by the
M. J. KIMet
814
al.
mixing regions, the crossflow between them and a dead region. The upper mixing region of the fractional volume ccis arranged with consideration to the mixing by the suction and pumping flow between the disks. The lower mixing region with a fractional volume m indicates that the turbulence near the disk edge is caused by the radial jets and bulk flow in the axial direction. A dead region d is added for the relatively minor fluid motion stagnant near the disk center and a multistage model is used to represent the degree of axial mixing. (b)
Fig. 1. Streamlinesof the flow between two rotating disks. (a) Batchelor’s solution; (b) Stewartson’ssolution. trough wall and interact with the bulk stream flow in the axial direction. It is difficult to access the effect of the free surface, and only an indefinite impression could be obtained from flow visualization experiments. The existence of a free surface. on the bulk fluid and of a thin liquid 6lm in the gas phase obstructed the development of rotational inertia in the liquid phase. While following the movement of the die, we were able to observe a stagnant region below the free surface near the disk center. (2) Liquid mixing model The combined model will contain several adjustable parameters. A larger number of parameters may facilitate the fitting of the mode1 to the experimental data, but the estimated parameters are much more likely to have less physical significance. Wolf and Resnick[l I] tried to characterize a multiparameter model with two parameters. Levich et a/.[121 proposed a three parameter mode1 for the dispersion in porous media, and EMham and Gibilaro [13] obtained the analytical solution of the frequency response in the time domain. Raghuraman and Varma[14] used a four parameter model incorporating the crossflow between active and dead regions to show that the mode1 accommodated the experimental data of fluidized beds. A four parameter model is also used in this work, as shown in Fig. 2. The model consists of two perfect
EXPERIMENTAL
CONDITIONS
(1) Materials and apparatus Figure 3 shows the schematic diagram of the experimental setup for measuring the RTD of a tracer. The trough and disks were made of acrylic resin and their specifications are given in Table 1. Ring type spacers of 0.47 cm thickness were inserted on the axis to adjust the distance between the disks. The inlet and outlet flow rates were kept constant and equal by the two identical peristaltic pump heads on the same motor generator (Cole-Parmer). Another motor generator (Ultramaster Flex and Servodyne System, ColeParmer) was used to obtain the constant rotational speed of the disks at even low speeds. Tap water was used as a carrier fluid and a 2 N KC1 solution as a tracer. The tracer concentration at the effluent was measured by its electrical conductivity. The electrical circuit and probe were constructed according to the method of Khang and Fitzgerald[lS], which showed good linearity in the experimental range.
Fig. 3. Schematic diagram of experimental apparatus: 1, motor; 2, influent; 3, disks; 4, trough; 5, probe; 6, effluent; 7, conductivity meter; 8, recorder.
Table I. SpeoiCzation of rotating disk reactor Inner
Diameter
Length
of
Thickness
21.8
cm
Trough
28.0
cm
of Disk
0.27
cm
Diameter
of Disk
Material
of Trough
Shape
Fig. 2. Liquid mixing model.
of Trough
of Trough
14, and
Disk
18.
Acrylic
20.
21 cm
Resin
Half-cylindrical
Residence time distribution
analysis in controllable flow conditions
(2) Method Clean tap water was fed for a period of 3-4 times longer than the mean holding time so that any remaining tracer from the previous run would be completely washed out. At steady state flow conditions, 0.5 or 1.0 cm3 of 2 N KC1 tracer was injected at the inlet. The exit concentration was continuously monitored on the recorder until it dropped nearly to zero. The studies were performed with varying rotational speeds, flow rates, distances between disks, sizes of disks, and fractional areas of the submerged portion of disks. (3) Flaw visualizarion A mixture of water and glycerol (50 : 50) was used as a working fluid, and ink was used as a dye[l6]. The relation of shear stress to the shear rate confirmed that the mixture was a Newtonian fluid with viscosity of 9.7 CP at 15°C. The glycerol seems to retard the dispersion and settling of the ink dye. DATA ANALYSIS
815
integration n-i
I
E(t)
=
brie”+
“‘i=,-_k;o(f
l)“-‘-’
(&-i-k-l)! (a, + a)“-* X(1,-~11,)2”~‘~~{~--_).,(#-~--_)!k(
(Zn - i -k (&+aY X(l,-~l,)“-‘-‘(n-k)!(n-~-k)!k! L _ I-
- l)!
(g)
-(a+b+c)+,/(a+b+c)i-dab F. L
a = -(a 2
+6
+c)+,/(a
+b
+c)2-4ab
(9)
2
(2) RTD density function from experimental data The material balance for the injected tracer is given as
(1) Analytical expression of the RTD density function The material balances of a tracer for each compartment in Fig. 2 are
s*
M=
0
QWW.
(10)
Rewriting eqn (10) in terms of the amount lost in the efflmnt until t = f, and that remaining in the reactor will be ma
2
=fQC,
+QCi_
I
-
(f + l)QC,
(2)
M=
” QC.(t)dt
+ Vc.(t,).
(11)
I0 for the ith stage with initial conditions at 1 = 0, Ci = C; = 0. Application fer function
of the Laplace transform for the ith stage as
G(s) z(,_=
s2+(a
b(s + a) +b+c)s
(3)
The amount remaining in the reactor after t = 3 - 40 was less than 7% of the total amount for each experiment. Thus,
gives the transM=
fffb
(4)
where
Since Q in eqn (12) is constant and the electrical conductivity is a linear function of C,(t) in the experimental range,
E(t)
I
=- GO)
Ml&
crp
The overall transfer by m=m=
G(s1
function
F(s) can be represented
b(s + a) s*+(a+b+c)s+ab
E(t)=$$=L-‘{F(s)] As shown in similar works[l3, 141, the analytical inversion of eqn (6) can be obtained by contour
cxp
=
C.(t)
s I’
CnO)d~
0
1“-(6)
When M units of a tracer is injected as a pulse at the inlet, the exit age distribution function is
I
where R,, denotes tivity meter.
%O)
_
the output
(13)
voltage
of the conduc-
(3) Normalized RTD density function The normalized RTD density function related to E(t) as follows E(t)
= ELzq).
E(r)
is
(14)
816
M. J. ICIMet al.
QO
10
OS
15
20
25
30
TIME,T
Fig. 4. Normalized
residence time distribution curves. 5 rpm, ---; 10 rpm, -‘-; 100 rpm, -. . ‘-; complete mixing, -.
to eqn (13)
According
,yr) 0s
*
E(T)l,, = g
0
(15)
%OW
Figure 4 shows E(r) curves for various rotational speeds of the disk from 5 to IOOrpm. As was expected, the response curve was close to that resulting from complete mixing especially at a higher speed of rotation. The response curves in all experiments were closer to those from complete mixing rather than those from plug flow. This shows that the combined model with various mixing regions in Fig. 2 was more appropriate than the dispersion model. (4) Parameter estimation Independent parameters of the model a, m, f and n were estimated by curve fitting in the time domain since the analytical expression of the RTD density function was possible. The objective function to be minimized is given as
FO =
: [E(tj)lmcdel j=l
E(tj11exp12
(16)
Table 2. Experimental V/Q
Disk
Diameter
Rotational
(cm1
(rpm)
22.5
20
5. 10, 25, 50,
22.5
14. 18.
12. -
22.5,
5. 25, 100
20. 21
22.5
20
22.5
12.
22.5,
55.5
21
25
Submerged
Portion
(X)
17.5
7.1. 14.2,
7.1.
14.2,
17.5, 25
14, 18,
20.
Disk
for RTD analysis Distance Disks
between
Figure
(cm)
Number
4.7
4
2.35
5
2.35
6
2.35
7
2.35
a
17.5, 18.8
14. 18.
55.5
RESULTS AND DISCUSSION
Figures 5-8 show the effect of design and operation factors on the model parameters and their experimental conditions are given in Table 2. The most important factors are the rotational speed and the disk size, shown in Fig. 5. As the disk size increases, the fraction of the lower mixing region m increases and the effect is more pronounced at low rotational speeds. This indicates that when the disks are surrounded by a closely fitting cylindrical wall, rotational inertia cannot develop over the bulk stream in the axial direction. With a high rotational speed some compatible effect of rotational inertia is observed, but only the turbulence caused by interactions between
100
20. 21
5,
25,
21
100
18.8
9.7,
17.5,
27..
39.5
7.1,
14.2,
17.5, 25
.-;
where the first term in the r.h.s. of the equation is given in eqn (8) and the second term in eqn (13). Nonlinear optimization was carried out by a modified Gauss-Newton method proposed by Marquardt[l7]. Since stage n must be an integer, optimization was achieved by increasing n by one, starting from one. In all the computational cases, the average error at each point was less than 5% of the experimental value.
conditions
Speed
(min)
25 rpm, -a a-; 50 rpm, -.
18.8
18.8
817
Residence time distribution analysis in controllable flow conditions
“or----l
“21----1 10-
08-
0,01 10
20
SUBMERGED
00106’
*
0.9
08 SIZE,
Fig. 7. Effect of degree of submergence on the parameters
1D
4 m, 4 f.
R Rt
Fig. 5. Effect of the size and rotational speed of disk on the parametersm,4f.a=l-m-d;5rpm,-;25rpm, --‘f lOOrpm, --.
the bulk stream
AREA, %
+Ao,,
07 DISK
mixing
40
30 DISK
and the disk edge contributes
as the rotational
speed
to the
decreases.
An increase of flow rate should accentuate the effect of the disk size because it emphasized the flow in the axial direction. We can confirm this relationship in Fig. 6 where the fraction of the lower mixing region is the greatest when the fluid rapidly flows in the axial direction through the narrow gaps between the disk edge and trough wall. But, the effect of the degree of submergence is not clearly defined as was described in the earlier section. Only a retardation of the rotational inertia due to presence of a free surface can be surmised because the fraction of the upper mixing region increases with the degree of submergence (Fig. 7). In addition, we
cannot discern the effect of the distance between disks[6], because neither the rotational inertia nor the turbulence by the bulk stream can grow when the disks are arranged at long intervals, and vice versa. Back mixing is affected by the molecular and eddy diffusions, but the latter is clearly dominant. Ghim and Chang[lS] demonstrated that the total effect of axial dispersion combined with the bulk flow decreases with the flow rate in spite of an increase in the axial dispersion. The same phenomenon is observed in the right side of Fig. 8 because the degree of back mixing is inversely proportional to the number of stages. Figure 8 also shows that back mixing increases with the rotational speed but decreases with the disk size. Considering that the fraction of the upper mixing region a increases with the rotational speed and that the region is not located in the main streamline (see Fig. 2), we can easily understand the effect of rotational speed. The effect of the disk size can be explained as the inverse effect of the rotationat
“2r----
‘Or---ce-
%-
z z:: 6-
z 5 !2 2
0.01’
0.6
*
:
0.8
07 DISK
SIZE.
:::
0.9
4,.
I.0
R h
Fig. 6. Effect of the disk size and flow rate on the parameters m, d, f. a= 1 -m - d; flow rate 36 ml/min, ---; 88 ml/min, -‘-; 164 ml/min, -.
Y-
/
o/
4-
.p,_
/: J
I--._*--_-_0
,’
I&-____---_a__---*
2-
0.6
0.7 DISK
0.9
0.8 SIZE.
i
1.0
j$
Fig. 8. Effect of disk size, rotational speed and flow rate on the parameter n. Rotational speed 5 rpm, -. 25 rem, --‘--; IOOrpm, ---; flow rate 36ml/min, I; 8i ml/min, 0; 164ml/min, 0.
818
M. J. Km
speed, but it can also be understood by viewing the increase in the interstitial velocity as a result of the gap between the disk edge and the trough wall decreasing with the disk size. Up to now we have examined the relationship between the model parameters and real fluid motions in the reactor. We have tried to quantitize this relationship through a dimensionless group x, defined by the product of the Reynolds number based on the bulk stream flow and the Ekman number modified for finite disks:
II
et al.
quite well: 1 -=10+31X*-@,a+,=1 a
! = 5.7 f
The modified Ekman number is exactly the same as the reciprocal of the Taylor number, which is widely established as the criterion of stability for Taylor vortices in the annulus between the two concentric cylinders when the inner cylinder rotates[l9]. We decided to separate the Taylor number into the Ekman number and the correction factor for the finite disks surrounded by the trough wall because the latter is more suitable for the present system. As can be seen in eqn (17) the increase in IF corresponds to the increase in the flow rate and disk size, and the decrease in the rotational speed and degree of submergence. It not only contains all the significant factors, but also represents their effects on the model parameters. Figure 9 shows the dependence of each parameter on the dimensionless group x. In spite of limited experimental information and the adoption of a simple model, the results fit
1
#.41
(18)
n=1.3+4.2~~.~ Acknowledgements-The authors are indebted to the Korea Science and Engineering Foundation for partial support of this research.
=Re,.Ek*=[-g-].[(-&(~)-q. 07)
.
NOTATION
C
d
E(t) E(r) Ek’
tracer concentration, g mole/liter volume fraction of dead region exit age distribution density function, min- ’ normalized exit age distribution density function modified Ekman number, (v/wR’) _ [(R, - R),‘R]-3’2
f F(s) 6 n m
cross flow rate factor between two perfect mixing regions overall transfer function in eqn (4) objective function in eqn (16) height of liquid level, cm volume fraction of lower perfect mixing region amount of tracer added as a pulse, g mole number of stages volumetric liquid flow rate, l/min radius of disk, cm radius of trough, cm output voltage of conductivity meter, mV Reynolds number, Q/vH variables in Laplace domain, min-’ time, min liquid phase volume of each stage, 1 total liquid phase volume, 1
Greek symbols 0: volume
fraction of upper perfect mixing region B overall mean holding time (V/Q), mm Y kinematic viscosity, cm’/s IC dimensionless group defined in eqn (17) r normalized time, c/O o angular velocity of disk, min-’
-1n-5.0 -40
1 -30
-29
-1.0
0.0
1.0
2.0
Ln Tc
Fig. 9. Relationship between the model parameters K, f, n and dimensionless group 1~.
REFERENCES [I] Himmelblau D. M. and BischoB K. B., Process Analysis and Simulation. Chap. 4, Wiley. New York 1974. [2] Seinfeld J. H. and Lapidus L., Mathematical Methods in Chemical Engineering, Vol. 3. Prentice-Hall, Englewood Cliffs, New Jersey 1974. [3] Froment G. F., A.1.Ch.E.J. 1975 21 1041. [4] Smith E. D., Miller R. D. and Wu Y. C. (Eds.), Proc. 1st Nor. Symp./Workshop on Rotating Biological Conracror Techrwlogy. University Pittsburg 1980. [5] Lightfoot E. N., Transport Phenomena and Living Systems, pp. 380-387. Wiley, New York 1974. [6] Kim M. J., MS. Thesis, KAIST, Seoul, Korea 1983.
Residence
time distribution
analysis in controllable
[7l Holodniok M., Kubicek M. and Hlavacek V., J. Fluid Mech. 1981 108 227. [(I] Batchelor G. K., @art. J. Appl. Math. 1951 4 29. 191 Stewartson K., Proc. Cattab. Phil. Sot. 1953 49 333. [IO] Richardson P. D., Int. J. Heat Mass Transfer. 1976 19 1189. [ll] Wolf D. and Resnick W., Ind. Engng Chem. Fundk 1963 2 287. [12] Levich V. G., Markin V. S. and Chismadzhev Y. A., Chem. Engng Sci. 1967 22 1357. [13] Buffham B. A. and Gibilaro L. G., Chem. Engng Sri. 1968 23 1399.
flow conditions
819
[14] Raghuraman J. and Varma Y. B. G., Chem. Engng Sci. 1973 28 585. [15] Khan8 S. J. and Fitzgerald T. J., Znd. Eng?tg Chem. Fundls 1975 14 208. [16] Lee Y. C. and Talhnadge J. A., A.K.Ch.E.J. 1972 18 858. [17j Kuester J. L. and Mize J. H., Oprimizarion Technique with Fortran, p. 240. McGraw-Hill, New York 1973. [IS] Ghim Y. S. and Chang H. N., Erd. Engng Chem. Fundls 1982 21 369. [19] Schlichting H., Eolara!ary Iqer Theory, 6th Bdn, P. 500. McGraw-Hill, New York 1979.
1984 8
CKW-2509/X4 S3.00+ .OO 1984 Pergamon Press Ltd.
RESIDENCE TIME DISTRIBUTION ANALYSIS IN CONTROLLABLE FLOW CONDITIONS: CASE OF ROTATING DISK REACTOR MYUNG Department
JIN KIM, YOUNG
SUNG GHIM and HO NAM CHANG*
of chemical Engineering, Korea Advanced Institute of science and Technology, P-0. Box 150 Chongyang, Seoul, Korea
(Received
24 March
1983; in revised form 21 June 1983; accepted 31 August
1983)
Abstract--A four-parameter multistage model was used to illustrate the mixing phenomenon in a rotating disk reactor. The parameters were. estimated by the nonlinear regression from the residence time distribution data of a tracer. The dead space portion was negligibly small due to the sticient mixing of the liquid, but the size of the mixing regions, cross flow rate and number of stages varied with design and operation factors. Variations of the parameters were closely related to the expected fluid motions in the reactor. A dimensionlessgroup combiningthe Reynolds numberand the Ekman number was proposed for the analysis of mixing in ihe reactor.INTRODUCITON
Residence time distribution (RTD) analysis has been a valuable tool in obtaining information on the fluid motion and mixing in a continuous flow system[l, 21. The most common approach is to choose an appropriate model and to fit the model output to the experimental output. Many types of mixing models have already been proposed and can be divided into two groups: a dispersion model which has been used mainly for empty tubes and packed beds; and a combined model in which complete mixing and plug flow regions are located in series and parallel arrangements. As the number of mixing stages increases, that is, as the scale of fluid motion causing the mixing effect decreases, the combined model approaches the dispersion model. After examining the literature in chemical engineering, however, one can readily see that the dispersion model has been more widely used. The major disadvantage of the combined model is that the involved parameters may not be closely related to the physical characteristics in the system. Because of this limitation, several workers[l-31 cautioned against the danger of empirically fitting the experimental data, thereby discouraging use of the combined model. We believe that this particular defect of the combined model is not intrinsic, but it arises from conditions in the accompanying systems. The mixing patterns in the stirred tanks and fluidized beds are too complex to be explained by parametered deterministic models. However, in the rotating disk reactors whose schematic diagram is shown in Fig. 3, the fluid motion can be characterized by the axial direction of the bulk flow, the transverse direction of the rotating flow and the mixing results from interactions of the two. The dominance of these flows and their interactions are easily related to the mixing cells and
*Author
to
whom correspondence should be addressed.
CES Vol. 39. No. -
813
streams in the combined model. The interconnection between the model parameters and the real fluid motions will be emphasized throughout the work. The prototype of the rotating disk reactor employed in the present work can be found in biological wastewater treatment [4] and blood oxygenation [S]. The main feature of this reactor is its regular contact of the solid phase with alternating gas and liquid phases. Gas is easily transferred through the thin liquid film entrained along the disk surfaces, and the liquid is spontaneously agitated by rotating disks[6]. Though the rotating disk reactor is not widely used and reported data is rare, it is promising as a general three phase contact system for the future.
FLUID MOTZON AND MIXING MODEL
(1) Fluid motion due to the rotation of disks When the disks steadily rotate in the same direction with a sufficiently high speed, a boundary layer will develop on each disk. The situation is quite complex, however, and a large number of multiple steady states has been reported [7J Two well-known solutions are introduced here because they provide enough qualitative information to devise a model. One solution is described by Batchelor [8] who assumed that one disk behaves as a centrifuge with an inflow in the axial direction while the fluid is sucked at a neighboring disk. In this case, the fluid between the boundary layers will rotate with a constant angular velocity (Fig. la). But, Stewartson[9] suggested that both disks behave as centrifuges. The fluid rotates only in the boundary layer while a radial and axial flow results outside the boundary layer (Fig. lb). In practice, though, the finite disks rotate in a vessel and are partially immersed in a liquid phase with a free surface. Three factors namely, the disk edge, trough wall and free surface, should also be considered. The disk edge will induce a radial jet Bow with swirling around the edge and a vortex near the rim center[ lo]. These flows will be confined by the
M. J. KIMet
814
al.
mixing regions, the crossflow between them and a dead region. The upper mixing region of the fractional volume ccis arranged with consideration to the mixing by the suction and pumping flow between the disks. The lower mixing region with a fractional volume m indicates that the turbulence near the disk edge is caused by the radial jets and bulk flow in the axial direction. A dead region d is added for the relatively minor fluid motion stagnant near the disk center and a multistage model is used to represent the degree of axial mixing. (b)
Fig. 1. Streamlinesof the flow between two rotating disks. (a) Batchelor’s solution; (b) Stewartson’ssolution. trough wall and interact with the bulk stream flow in the axial direction. It is difficult to access the effect of the free surface, and only an indefinite impression could be obtained from flow visualization experiments. The existence of a free surface. on the bulk fluid and of a thin liquid 6lm in the gas phase obstructed the development of rotational inertia in the liquid phase. While following the movement of the die, we were able to observe a stagnant region below the free surface near the disk center. (2) Liquid mixing model The combined model will contain several adjustable parameters. A larger number of parameters may facilitate the fitting of the mode1 to the experimental data, but the estimated parameters are much more likely to have less physical significance. Wolf and Resnick[l I] tried to characterize a multiparameter model with two parameters. Levich et a/.[121 proposed a three parameter mode1 for the dispersion in porous media, and EMham and Gibilaro [13] obtained the analytical solution of the frequency response in the time domain. Raghuraman and Varma[14] used a four parameter model incorporating the crossflow between active and dead regions to show that the mode1 accommodated the experimental data of fluidized beds. A four parameter model is also used in this work, as shown in Fig. 2. The model consists of two perfect
EXPERIMENTAL
CONDITIONS
(1) Materials and apparatus Figure 3 shows the schematic diagram of the experimental setup for measuring the RTD of a tracer. The trough and disks were made of acrylic resin and their specifications are given in Table 1. Ring type spacers of 0.47 cm thickness were inserted on the axis to adjust the distance between the disks. The inlet and outlet flow rates were kept constant and equal by the two identical peristaltic pump heads on the same motor generator (Cole-Parmer). Another motor generator (Ultramaster Flex and Servodyne System, ColeParmer) was used to obtain the constant rotational speed of the disks at even low speeds. Tap water was used as a carrier fluid and a 2 N KC1 solution as a tracer. The tracer concentration at the effluent was measured by its electrical conductivity. The electrical circuit and probe were constructed according to the method of Khang and Fitzgerald[lS], which showed good linearity in the experimental range.
Fig. 3. Schematic diagram of experimental apparatus: 1, motor; 2, influent; 3, disks; 4, trough; 5, probe; 6, effluent; 7, conductivity meter; 8, recorder.
Table I. SpeoiCzation of rotating disk reactor Inner
Diameter
Length
of
Thickness
21.8
cm
Trough
28.0
cm
of Disk
0.27
cm
Diameter
of Disk
Material
of Trough
Shape
Fig. 2. Liquid mixing model.
of Trough
of Trough
14, and
Disk
18.
Acrylic
20.
21 cm
Resin
Half-cylindrical
Residence time distribution
analysis in controllable flow conditions
(2) Method Clean tap water was fed for a period of 3-4 times longer than the mean holding time so that any remaining tracer from the previous run would be completely washed out. At steady state flow conditions, 0.5 or 1.0 cm3 of 2 N KC1 tracer was injected at the inlet. The exit concentration was continuously monitored on the recorder until it dropped nearly to zero. The studies were performed with varying rotational speeds, flow rates, distances between disks, sizes of disks, and fractional areas of the submerged portion of disks. (3) Flaw visualizarion A mixture of water and glycerol (50 : 50) was used as a working fluid, and ink was used as a dye[l6]. The relation of shear stress to the shear rate confirmed that the mixture was a Newtonian fluid with viscosity of 9.7 CP at 15°C. The glycerol seems to retard the dispersion and settling of the ink dye. DATA ANALYSIS
815
integration n-i
I
E(t)
=
brie”+
“‘i=,-_k;o(f
l)“-‘-’
(&-i-k-l)! (a, + a)“-* X(1,-~11,)2”~‘~~{~--_).,(#-~--_)!k(
(Zn - i -k (&+aY X(l,-~l,)“-‘-‘(n-k)!(n-~-k)!k! L _ I-
- l)!
(g)
-(a+b+c)+,/(a+b+c)i-dab F. L
a = -(a 2
+6
+c)+,/(a
+b
+c)2-4ab
(9)
2
(2) RTD density function from experimental data The material balance for the injected tracer is given as
(1) Analytical expression of the RTD density function The material balances of a tracer for each compartment in Fig. 2 are
s*
M=
0
QWW.
(10)
Rewriting eqn (10) in terms of the amount lost in the efflmnt until t = f, and that remaining in the reactor will be ma
2
=fQC,
+QCi_
I
-
(f + l)QC,
(2)
M=
” QC.(t)dt
+ Vc.(t,).
(11)
I0 for the ith stage with initial conditions at 1 = 0, Ci = C; = 0. Application fer function
of the Laplace transform for the ith stage as
G(s) z(,_=
s2+(a
b(s + a) +b+c)s
(3)
The amount remaining in the reactor after t = 3 - 40 was less than 7% of the total amount for each experiment. Thus,
gives the transM=
fffb
(4)
where
Since Q in eqn (12) is constant and the electrical conductivity is a linear function of C,(t) in the experimental range,
E(t)
I
=- GO)
Ml&
crp
The overall transfer by m=m=
G(s1
function
F(s) can be represented
b(s + a) s*+(a+b+c)s+ab
E(t)=$$=L-‘{F(s)] As shown in similar works[l3, 141, the analytical inversion of eqn (6) can be obtained by contour
cxp
=
C.(t)
s I’
CnO)d~
0
1“-(6)
When M units of a tracer is injected as a pulse at the inlet, the exit age distribution function is
I
where R,, denotes tivity meter.
%O)
_
the output
(13)
voltage
of the conduc-
(3) Normalized RTD density function The normalized RTD density function related to E(t) as follows E(t)
= ELzq).
E(r)
is
(14)
816
M. J. ICIMet al.
QO
10
OS
15
20
25
30
TIME,T
Fig. 4. Normalized
residence time distribution curves. 5 rpm, ---; 10 rpm, -‘-; 100 rpm, -. . ‘-; complete mixing, -.
to eqn (13)
According
,yr) 0s
*
E(T)l,, = g
0
(15)
%OW
Figure 4 shows E(r) curves for various rotational speeds of the disk from 5 to IOOrpm. As was expected, the response curve was close to that resulting from complete mixing especially at a higher speed of rotation. The response curves in all experiments were closer to those from complete mixing rather than those from plug flow. This shows that the combined model with various mixing regions in Fig. 2 was more appropriate than the dispersion model. (4) Parameter estimation Independent parameters of the model a, m, f and n were estimated by curve fitting in the time domain since the analytical expression of the RTD density function was possible. The objective function to be minimized is given as
FO =
: [E(tj)lmcdel j=l
E(tj11exp12
(16)
Table 2. Experimental V/Q
Disk
Diameter
Rotational
(cm1
(rpm)
22.5
20
5. 10, 25, 50,
22.5
14. 18.
12. -
22.5,
5. 25, 100
20. 21
22.5
20
22.5
12.
22.5,
55.5
21
25
Submerged
Portion
(X)
17.5
7.1. 14.2,
7.1.
14.2,
17.5, 25
14, 18,
20.
Disk
for RTD analysis Distance Disks
between
Figure
(cm)
Number
4.7
4
2.35
5
2.35
6
2.35
7
2.35
a
17.5, 18.8
14. 18.
55.5
RESULTS AND DISCUSSION
Figures 5-8 show the effect of design and operation factors on the model parameters and their experimental conditions are given in Table 2. The most important factors are the rotational speed and the disk size, shown in Fig. 5. As the disk size increases, the fraction of the lower mixing region m increases and the effect is more pronounced at low rotational speeds. This indicates that when the disks are surrounded by a closely fitting cylindrical wall, rotational inertia cannot develop over the bulk stream in the axial direction. With a high rotational speed some compatible effect of rotational inertia is observed, but only the turbulence caused by interactions between
100
20. 21
5,
25,
21
100
18.8
9.7,
17.5,
27..
39.5
7.1,
14.2,
17.5, 25
.-;
where the first term in the r.h.s. of the equation is given in eqn (8) and the second term in eqn (13). Nonlinear optimization was carried out by a modified Gauss-Newton method proposed by Marquardt[l7]. Since stage n must be an integer, optimization was achieved by increasing n by one, starting from one. In all the computational cases, the average error at each point was less than 5% of the experimental value.
conditions
Speed
(min)
25 rpm, -a a-; 50 rpm, -.
18.8
18.8
817
Residence time distribution analysis in controllable flow conditions
“or----l
“21----1 10-
08-
0,01 10
20
SUBMERGED
00106’
*
0.9
08 SIZE,
Fig. 7. Effect of degree of submergence on the parameters
1D
4 m, 4 f.
R Rt
Fig. 5. Effect of the size and rotational speed of disk on the parametersm,4f.a=l-m-d;5rpm,-;25rpm, --‘f lOOrpm, --.
the bulk stream
AREA, %
+Ao,,
07 DISK
mixing
40
30 DISK
and the disk edge contributes
as the rotational
speed
to the
decreases.
An increase of flow rate should accentuate the effect of the disk size because it emphasized the flow in the axial direction. We can confirm this relationship in Fig. 6 where the fraction of the lower mixing region is the greatest when the fluid rapidly flows in the axial direction through the narrow gaps between the disk edge and trough wall. But, the effect of the degree of submergence is not clearly defined as was described in the earlier section. Only a retardation of the rotational inertia due to presence of a free surface can be surmised because the fraction of the upper mixing region increases with the degree of submergence (Fig. 7). In addition, we
cannot discern the effect of the distance between disks[6], because neither the rotational inertia nor the turbulence by the bulk stream can grow when the disks are arranged at long intervals, and vice versa. Back mixing is affected by the molecular and eddy diffusions, but the latter is clearly dominant. Ghim and Chang[lS] demonstrated that the total effect of axial dispersion combined with the bulk flow decreases with the flow rate in spite of an increase in the axial dispersion. The same phenomenon is observed in the right side of Fig. 8 because the degree of back mixing is inversely proportional to the number of stages. Figure 8 also shows that back mixing increases with the rotational speed but decreases with the disk size. Considering that the fraction of the upper mixing region a increases with the rotational speed and that the region is not located in the main streamline (see Fig. 2), we can easily understand the effect of rotational speed. The effect of the disk size can be explained as the inverse effect of the rotationat
“2r----
‘Or---ce-
%-
z z:: 6-
z 5 !2 2
0.01’
0.6
*
:
0.8
07 DISK
SIZE.
:::
0.9
4,.
I.0
R h
Fig. 6. Effect of the disk size and flow rate on the parameters m, d, f. a= 1 -m - d; flow rate 36 ml/min, ---; 88 ml/min, -‘-; 164 ml/min, -.
Y-
/
o/
4-
.p,_
/: J
I--._*--_-_0
,’
I&-____---_a__---*
2-
0.6
0.7 DISK
0.9
0.8 SIZE.
i
1.0
j$
Fig. 8. Effect of disk size, rotational speed and flow rate on the parameter n. Rotational speed 5 rpm, -. 25 rem, --‘--; IOOrpm, ---; flow rate 36ml/min, I; 8i ml/min, 0; 164ml/min, 0.
818
M. J. Km
speed, but it can also be understood by viewing the increase in the interstitial velocity as a result of the gap between the disk edge and the trough wall decreasing with the disk size. Up to now we have examined the relationship between the model parameters and real fluid motions in the reactor. We have tried to quantitize this relationship through a dimensionless group x, defined by the product of the Reynolds number based on the bulk stream flow and the Ekman number modified for finite disks:
II
et al.
quite well: 1 -=10+31X*-@,a+,=1 a
! = 5.7 f
The modified Ekman number is exactly the same as the reciprocal of the Taylor number, which is widely established as the criterion of stability for Taylor vortices in the annulus between the two concentric cylinders when the inner cylinder rotates[l9]. We decided to separate the Taylor number into the Ekman number and the correction factor for the finite disks surrounded by the trough wall because the latter is more suitable for the present system. As can be seen in eqn (17) the increase in IF corresponds to the increase in the flow rate and disk size, and the decrease in the rotational speed and degree of submergence. It not only contains all the significant factors, but also represents their effects on the model parameters. Figure 9 shows the dependence of each parameter on the dimensionless group x. In spite of limited experimental information and the adoption of a simple model, the results fit
1
#.41
(18)
n=1.3+4.2~~.~ Acknowledgements-The authors are indebted to the Korea Science and Engineering Foundation for partial support of this research.
=Re,.Ek*=[-g-].[(-&(~)-q. 07)
.
NOTATION
C
d
E(t) E(r) Ek’
tracer concentration, g mole/liter volume fraction of dead region exit age distribution density function, min- ’ normalized exit age distribution density function modified Ekman number, (v/wR’) _ [(R, - R),‘R]-3’2
f F(s) 6 n m
cross flow rate factor between two perfect mixing regions overall transfer function in eqn (4) objective function in eqn (16) height of liquid level, cm volume fraction of lower perfect mixing region amount of tracer added as a pulse, g mole number of stages volumetric liquid flow rate, l/min radius of disk, cm radius of trough, cm output voltage of conductivity meter, mV Reynolds number, Q/vH variables in Laplace domain, min-’ time, min liquid phase volume of each stage, 1 total liquid phase volume, 1
Greek symbols 0: volume
fraction of upper perfect mixing region B overall mean holding time (V/Q), mm Y kinematic viscosity, cm’/s IC dimensionless group defined in eqn (17) r normalized time, c/O o angular velocity of disk, min-’
-1n-5.0 -40
1 -30
-29
-1.0
0.0
1.0
2.0
Ln Tc
Fig. 9. Relationship between the model parameters K, f, n and dimensionless group 1~.
REFERENCES [I] Himmelblau D. M. and BischoB K. B., Process Analysis and Simulation. Chap. 4, Wiley. New York 1974. [2] Seinfeld J. H. and Lapidus L., Mathematical Methods in Chemical Engineering, Vol. 3. Prentice-Hall, Englewood Cliffs, New Jersey 1974. [3] Froment G. F., A.1.Ch.E.J. 1975 21 1041. [4] Smith E. D., Miller R. D. and Wu Y. C. (Eds.), Proc. 1st Nor. Symp./Workshop on Rotating Biological Conracror Techrwlogy. University Pittsburg 1980. [5] Lightfoot E. N., Transport Phenomena and Living Systems, pp. 380-387. Wiley, New York 1974. [6] Kim M. J., MS. Thesis, KAIST, Seoul, Korea 1983.
Residence
time distribution
analysis in controllable
[7l Holodniok M., Kubicek M. and Hlavacek V., J. Fluid Mech. 1981 108 227. [(I] Batchelor G. K., @art. J. Appl. Math. 1951 4 29. 191 Stewartson K., Proc. Cattab. Phil. Sot. 1953 49 333. [IO] Richardson P. D., Int. J. Heat Mass Transfer. 1976 19 1189. [ll] Wolf D. and Resnick W., Ind. Engng Chem. Fundk 1963 2 287. [12] Levich V. G., Markin V. S. and Chismadzhev Y. A., Chem. Engng Sci. 1967 22 1357. [13] Buffham B. A. and Gibilaro L. G., Chem. Engng Sri. 1968 23 1399.
flow conditions
819
[14] Raghuraman J. and Varma Y. B. G., Chem. Engng Sci. 1973 28 585. [15] Khan8 S. J. and Fitzgerald T. J., Znd. Eng?tg Chem. Fundls 1975 14 208. [16] Lee Y. C. and Talhnadge J. A., A.K.Ch.E.J. 1972 18 858. [17j Kuester J. L. and Mize J. H., Oprimizarion Technique with Fortran, p. 240. McGraw-Hill, New York 1973. [IS] Ghim Y. S. and Chang H. N., Erd. Engng Chem. Fundls 1982 21 369. [19] Schlichting H., Eolara!ary Iqer Theory, 6th Bdn, P. 500. McGraw-Hill, New York 1979.