Ind. Eng. Chem. Res. 1992,31, 2379-2385
6 = angle made by flow direction with horizontal plane
6 = angle of incidence 4 = rate of turbulent energy dissipation per unit mas, m2 s3 t = solidity coefficient VH = hydraulic efficiency, (eq 4) = pumping effectiveness (eq 8) 0 = tangential coordinate cc = molecular viscosity, Pes cct = eddy viscosity, Pes cceff = total effective viscosity, Pes p4 = effective viscosity for the variable 4 p = liquid density, kg m-3 Qk
= Prandtl number for turbulent kinetic energy
= Prandtl number for turbulent energy dissipation rate 4 = generalized notation for transport variable T, = wall shear stress, N m-2 u,
Literature Cited Deshpande, G. B. Fluid Mechanics of Agitated Reactors. M. Chem. Eng. Thesis, University of Bombay,, 1988. Duncan, W. J.; Thom, A. S.; Young, A. D. Mechanics of Fluids; Arnold: London, 1970. Fort, I. Mixing: Theory and Practice; Uhl, V. W., Gray, J. B., Eds.; Academic Press: New York, 1986;Vol. 111. Joshi, J. B.; Pandit, A. B.; Sharma, M. M. Mechanically Agitated Gas-Liquid Reactors. Chem. Eng. Sci. 1982,37,813-844. Kaminoyama, M.; Saito, F.; Kamiwano, M. Numerical Analysis of 3D Flow of Pseudo-Plastic Liquid in a Stirred Vessel with a Turbine Impeller. Znt. Chem. Eng. 1990,30,720-728. Mann, R. Gas-Liquid Contacting in Mixing Vessels; Inst. Chem. Eng.: Rugby, 1983. Mishra, V. P. Unpublished results, 1992. Nagata, S. Mixing: Principles and Applications; Kodanska Ltd.1 Wiley: Tokyo, New York, 1975.
2379
Oldshue, J. Y.Fluid Mixing Technology; McGraw-Hik New York, 1984. Pericleous, K. A.; Patel, M. K. The Modelling of Tangential and Axial Agitators in Chemical Reactors. PCH, Physicochem. Hydrodyn. 1987,8,105-123. Placek, J.; Tavlarides, L. L. Turbulent Flow in Stirred Tanks-I Turbulent Flow in the Turbine Impeller Region. AIChE J. 1985, 31, 1113-1120. Ranade, V. V. Design of Multiphase Reactors. Ph.D. Thesis, University of Bombay, India, 1988. Ranade, V. V.; Joshi, J. B. Flow Generated by Pitched Blade Turbines-I Experimental. Chem. Eng. Commun. 1989,81, 197-224. Ranade, V. V.;Joshi, J. B. Flow Generated by a Disc Turbine-I Experimental. Chem. Eng. Res. Des. 1990a,68,19-33. Ranade, V. V.; Joshi, J. B. Flow Generated by a Disc Turbine-11: Mathematical Model and Comparison with Experimental Data. Chem. Eng. Res. Des. 1990b,68, 34-50. Ranade, V. V.; Joshi, J. B.; Marathe, A. G. Flow Generated by Pitched Blade Turbines-11 Mathematical Model and Comparison with Experimental Data. Chem. Eng. Commun. 1989,81, 225-247. Ranade, V. V.; Bourne, J. R.; Joshi, J. B. Fluid Mechanics and Blending in Agitated Tanks. Chem. Erg. Sci. 1991,46,1883-1893. Rewatkar, V. B.; Joshi, J. B. Effect of Impeller Design on Liquid Phase Mixing in Mechanically Agitated Reactors. Chem. Eng. Commun. 1991,91,322-353. Saraph, V. S.Fluid Mechanics of Agitated Reactors. M. Chem. Eng. Thesis, University of Bombay, 1989. Tatterson, G. B. Fluid Mixing and Gas Dispersion in Agitated Tanks; McGraw-Hilk New York, 1991. Wallis, R. A. Axial Flow Fans: Design and Practice; Newnes: London, 1961.
Received for review January 9, 1992 Revised manuscript received June 10, 1992 Accepted June 29, 1992
Fluid Flow in Capillary Suction Apparatus D. J. Lee**+ and Y. H. Hsu Department of Chemical Engineering, Yuan-Ze Institute of Technology, Taoyuan, Taiwan, 32026, R.O.C.
The fluid flow through porous media in capillary suction apparatus (CSA)was investigated experimentally and theoretidy. Water, with and without surfactant additives, methanol, and ethylene glycol were used to study the effects of liquid on capillary suction time of apparatus with various column radii and sampling locations. A model based on the saturation-diffusion mechanism was developed. The results showed satisfactory agreement with the experimental data. A modified capillary suction time was proposed based on the model which incorporated only the liquid effect and was independent of the apparatus used. Experiments with kaolin slurry showed that the model could be used to describe the fluid flow behavior in CSA with cake formation, if the liquid saturation under the inner cylinder was taken as less than unity.
Introduction Capillary suction apparatus (CSA) has been widely used since first developed by Gale and Baskerville in 1967 (Wilcox et al., 1987; Dohanyos et al., 1988; Vesilind and Davis, 1988; Lu et al., 1989; King and Forster, 1990). A CSA is composed of two plastic plates, Whatman No. 17 chromatugraphypaper, a stainless steel cylindrical column, several electrodes serving as sensors, and a timer. Sludge is poured into the column and the time the filtrate needs to travel between two concentric circles is called the capillary suction time (CST). Since the filter paper is ma-
* T o whom correspondence should be addressed.
Present address: Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan, 10764,ROC.
chine-made,the filtrate would move faster along the grain than across. Gale and Baskerville (1967) found that the difference is about 20%. Baskerville and Gale (1968) obtained a correlation between the CST and the specific resistance under filtration of sludge. It was shown that a sludge with a long CST indicated a sludge that when filtrated would form a cake with high specific resistance. Karr and Keinath (1978) investigated the factors influencing the sludge dewaterability. In their work, a CSA with columns of two inner radii was used. It was clear that the CST was a strong function of the column radius. Nguyen (1980) developed a theory for modeling the CSA by assuming the liquid moving in the filter paper as a piston-like process. An equivalent radius defined as the square root of the product of the long and short principle
0888-5885f 92f 2631-2379$Q3.oO/Q 0 1992 American Chemical Society
2380 Ind. Eng. Chem. Res., Vol. 31, No. 10, 1992
radii was suggested to overcome the difficulty raised from the anisotropic permeation inherent to the filter paper. Leu (1981)extended and corrected some errors in the work of Nguyen and proposed the concept of a rectangular CSA which was also discussed further in Unno et al. (1983). Ju (1982) studied the effects of particle sedimentation on the CST. An expression based on a two-dimensional analysis was proposed to relate the CST and the specificresistance of the filtration cake. Vesilind (1988) used CST 88 a measure of sludge dewaterability. A simple model was developed which showed that the diameter of the watered area depended on t'lz. From the works in the literature, the parameters of the CSA, such as inner radius of the cylindrical column, the distance between two concentric cycles for sampling CST, and the direction of the sampling locations, vary from case to case. The data from various sources were hard to compare (Vesilind and Davis, 1988). An appropriate definition of CST should only include the properties of the sludge and should not be influenced by the CSA used. There exists two ways of describingthe water movement in a porous media (Nguyen, 1980): the piston-like approach, which treats the liquid flow in the filter paper as a displacement process; and the diffusion-like approach, which treats the process as a diffusion process with the diffusivity to be a saturation-dependent property. The models developed from piston-like approach usually fitted well with the experimental data. However, when these models are applied to a specific system, permeation, Kp, and capillary pressure, P,, of the liquid/substrate combination must be known in a priori. Besides, from the physical point of view, it was uncomfortable to accept the assumption that the liquid saturation would change discontinuously from unity to zero at the moving front, without a transition zone between these two extremes. The theoretical study using the diffusion-like approach to treat the fluid flow in CSA had never appeared in the literature to the authors' knowledge, although it seemed more realistic than the piston-like approach. In this report, fluid flow in CSA was studied experimentally and theoretically. Effects of various liquids and the parameters of CSA were investigated. A model based on a saturationdependent diffusion process was developed. An expression of a modified CST based on the model was propoaed which included only the properties of the fluid and could be used to link the data from various investigations. Finally, experiments with kaolin slurry were conducted to show the feasibility of extending the results obtained from pure liquid tests to the cases with cake formation.
Experimental Section Figure la shows the device. Two transparent plates with Whatman No. 17 chromatographypaper formed the main body. A stainless steel cylinder with a radius of 0.95,0.535, or 0.20 cm was located at the center of the upper plate into which the testing liquid is poured. Water, ethylene glycol (EG),and methanol were used as the testing fluids. In some testa, water with surfactant additive (sodium dodecyl sulfate, SDS) was used to study the effects of surface tension, since a small amount surfactant additives affects surface tension but not other physical properties of the solution (Yang, 1983). Figure l b shows the top view of the upper plate. Small pin holes were drilled along and a c r w the grain direction. Electrodes were installed at these locations to serve as sensors. A multichannel data acquisition system was employed to detect the signals of the arrival of the liquid at various electrodm. A personal computer was used to access and analyze these data.
,,k \ : \ nI:
v32,2i,
1
I
j
I
1
1
. . . . ... ...:;.....
*
4 1 -Filter Paper
5- Electrodes
2-Cylindrical Column
6-Data Acquisition System
3-Upper Plate
7-Personal Computer
4-Lower Plate
Figure 1. (a, top) Experimental setup. (b,bottom) Top view of the upper plate. Numerical values are in centimeters.
The procedure for each experiment was as follows. The device was first cleaned, dried, and assembled. The cylindrical column with desired radius was positioned, the sampling locations of interest were chosen, and the data acquisition system and the computer were connected. Both the directions along and across the grain were of interest. Five or six electrodes were located along each of these two directions. The distance between the center and various sampling positions were measured with an error less than 0.5 mm. The amount of the liquid added was fixed to be 3 mL, which could guarantee that the liquid would not be exhausted during experiments. The experiments started with injecting liquid into the cylindrical column. At the time of the injection, the sensor installed in the column would activate the timer in the computer. The data acquisition system would then monitor every electrode continuously and record the time needed for the liquid to travel from the column to each electrode. Experiments were finished when all installed electrodes had received the signal of the liquid arrival. All data were then sent to the personal computer for further analysis. The CST data recorded along each direction were then plotted against the radius of the corresponding sampling positions. The measurement error of the timer was esti-
Ind. Eng. Chem. Res., Vol. 31, No. 10, 1992 2381 Table I. Averaged Thickness and Averaged Porosity of Wet (Water-Saturated)Whatman No. 17 Filter Paper 1U-l 6, mm 6 1 0.130 0.72 2 0.131 0.71 3 0.130 0.72
mated to be less than 0.01 8. The radii of the moving front at regular time intervals were then interpolated from these curves. At least three runs for each experimental condition were conducted to check the reproducibility. The maximum relative error was about 10% and was caused mainly from the individual quality variation between filter papers. The error caused by experiments of small column radius would be greater than that of large column radius. The averaged porosity c and averaged thickness 6 of the filter paper were measured following the method in Leu (1981). In each test, several pieces of filter paper were weighted and then dipped in water. After the papers were saturated with water, the mass and the thickness of the papers were measured and averaged. The data are listed in Table I. It was shown that the averaged thickness obtained (0.130 mm) was very close to that reported in Leu (1981)(0.137 mm), but the averaged porosity (0.72) showed some deviations from the data in Leu's thesis (0.60). The CST experimentswith fluids containing solids were conducted using kaolin slurry as the testing substance. The slurry was prepared by mixing certain amounta of kaolin particles with deionized water stirred vigorously with a magnetic stirrer. To prevent serious sedimentation in the inner cylinder which might distort the experimental results (Ju, 1982),the slurry was filtered with a coarse filter plate to remove the large particles. The weight percent of the solid content in the resulting filtrate was then determined by sampling and drying. By labeling the inner cylinder with careful calibrations, the liquid volume penetrated into the filter paper could be recorded simultaneously with the CST. The error in determining the liquid volume was estimated to be less than 3%. In the experimenta with slurry, the inner cylinder radius was fixed to be 0.535 cm.
Analysis In this section, the fluid flow in CSA was modeled by the diffusion-like process. The concept of equivalent radius proposed by Nguyen (1980) was used. Since in some previous works (Leu, 1981; Ju, 1982) the hydrostatic head of the liquid in the inner cylinder was shown to be very small when compared with the capillary suction pressure, the gravitation force could be neglected safely in this work. For a two-phase system neglecting gravitation force, the following equation may be written from Darcy's law and the continuity equation:
where K and K,, where the permeability and the relative permeability, respectively. By defining the effective diffusivity as
The effective diffusivity defined in eq 2 included both the characteristics of liquid and the medium and was a complicated function of saturation, wetting properties, paper pore size distribution, and other terms which were not easily determined (Corney, 1977). Various forms of D had been proposed for many liquid/porous media systems for which eq 3 could be solved. In thia work, a simple power law was used for the first approximation as D = D,s" (4) since the effective diffusivity would change from 0 to some value when saturation increased from 0 to 1. For circular geometry, the dimensionless form of eq 1 would be
where
In CSA with no cake formation, the liquid saturation within the inner radius of the column would remain unity, since liquid always existed above the filter paper during operations. A clear boundary between the moving front and the remaining paper could be observed in all experiments, which suggested that the liquid saturation should be zero in the region outside the front area. The boundary conditions could be written as s = so:
r+ 5 1
(6a)
s = 0:
r+ = R+ = R(r)/Ro
(6b)
where R(r) was the front radius at dimensionless time r , and so was unity in pure liquid testa. For most cases, the velocity of the moving front was not large. It was thus interesting to use the integral method to analyze the problem. Integration of eq 5 with respect to r+ would give the resulting equation
zlt=w+ d d ~ =+
*I
(7) dr+ r+=1 An approximation function of s as function of r+ was needed to complete the analysis. Landau and Lifshitz (1959) analyzed a transient heat conduction problem with temperature-dependentthermal conductivity which was basically the same as that treated by Chang (1988). It was shown that for a very small time interval, the dependent variable, W, near the moving front would vanish as sn
rt
W
a Ilel'/"
(8)
where x was the distance from the moving front. Equation 8 and the boundary conditions eqs 6a and 6b suggested a approximation function as
eq 1could be rearranged as a "diffusion-like" form (Chang,
1988): a s p t = v.(Dv~) where s is the liquid saturation.
(3)
The saturation profile of eq 9 is shown in Figure 2 for various n values. It can be seen that the fluid flow could be taken as piston-like when the index n is large. When the index is not so large, the saturation distribution de-
2382 Ind. Eng. Chem, Res., Vol. 31, No, 10, 1992
t
0
R(T)
RO
r
"I
Figure 2. Liquid saturation profile for various n values. 1.0
Liquid
Water .95 Water Water
d 0
+
2.0
1.0
MeOH SDSl SDSl SDSl SDS2 .95 SDS2 ,535 SDS2 , 2 0 EG .95 EO ,535 EG . 2 0 3.0
Figure 3. log (R&) va log CST. SDS1,0.04 M SDS aqueous solution; SDS2, 0.02 M SDS aqueous solution.
viates from the piston-like profile. Substituting eq 9 into eq 7 results
+ c2y2 + co = 0
(10)
where c3
2n3 3(n + 1)(2n + 1) n2 cz = 2(n + 1)
=
co = -?Son
0
1.0
2.0
log(CST)
Figure 4. Effects of water added amount on log (R&) va log CST. Ro = 0.95 cm.
could not be explained by the work of Vesilind (1988). The result based on the piston-like approach for the case with no cake formation was (Leu, 1981)
where
l o g (CST)
c3y3
0
(104
y=R+-1 (10d) Since C3 > 0, Cz> 0, and Co< 0, there exists one and only one valid solution. The solution of y is a universal function of T if the integer n is known.
Results and Discussion The experimental results are shown in Figure 3. The ordinate is the logarithm of the product of the two principle front radii, R, and R,, at fixed CST. Therefore, the ordinate shown in the figure is the square of the equivalent radius, and is a measure of the wetted area. From Figure 3, it was clear that the time for liquid to travel between fixed concentric circles would be longer when the column radius was smaller. It was also shown that the CST increased when liquid viscosity increased and/or the surface tension decreased. Each data group could be fitted by a straight line on a log-log scale. The slopes of these lines varied from 0.51 (EG, Ro = 0.95 cm) to 1.00 (water + SDS, Ro = 0.2 cm) and was a function of column radius and liquid properties. Therefore, the result
The calculation results of water are shown as dashed lines on Figure 3. The values of permeation and capillary suction pressure for water/Whatman No. 17 paper were taken from Leu (1981). The mean relative error of the watered area was estimated to be about 20'70, and was satisfactory in most applications. The results of other liquids could not be compared here, since the permeation and capillary suction pressure data for those liquid/paper combinations were not available. To compare the experimental results with the present theory (eq lo), the index n and the reference effective diffusivity D,were needed. These are discussed as follow. Index n. Kissa (1981) studied the drop penetration into the fibrous assemblies and divided the process into two phases: (I) the drop sinking and spreading period with some liquid remaining on the surface; (11) the spreading liquid all contained within the substrate, with no liquid existing on the substrate surface. Chang (1988) developed a kinetic model to describe ink-drop spreading in phase 11. The similarity solution for the cylindrical coordinate was A ( t ) a t'/("+') (12) where A was the spreading area. Equation 12 could be used to find the index n of the filter paper if the liquid was made to exhaust during experiments. The effects of the amount of water added were teated, and the results with the 0.95-cm column radius are shown in Figure 4. When more than 3 mL of water was added, all results coincided with the data shown in Figure 3. When the added amount was less, the curve obtained showed transition. Taking the 1-mL case as an example, the curve A-C was the experimental result. For curve A-B, the, data coincided with those obtained with sufficient water. When the liquid front advanced to point B,the water penetrated into the filter paper was estimated to be 1.02 mL, and the slope of the curve decrased, Le., the front moving velocity suddenly slowed down. The curve B-C
Ind. Eng. Chem. Res., Vol. 31, No. 10,1992 2383 Table 11. Physical Properties and Calculated Reference Effective Diffusivitu for Various Solutions liquid LL, Pa8 u, N/m D,, lo4 mz/s T, O C water 0.80 72.0 1.26 30 1.05 72.0 0.96 20 water 0.74 30 0.60 22.0 MeOH 1.70 39.7 0.48 20 MeOH (60 vol %) 1.22 27.2 20 0.46 MeOH (80 vol %) 30 0.05 13.0 47.2 EG 30 0.64 0.80 37.1 SDS (0.04 M) 30 0.65 0.80 37.8 SDS (0.02 M) 20 1.01 1.13 55.5 acetone (5 wt %) 20 1.25 52.2 0.85 acetone (10 wt %) 1.15 20 1.29 72.9 glycerol (10 wt %) 20 0.85 1.74 72.4 glycerol (20 wt %) 20 0.53 2.79 71.7 glycerol (33 wt %) 20 0.34 4.22 70.7 glycerol (42 wt %) 20 0.03 37.4 67.1 glycerol (75 w t %)
might be viewed as the phase I1 of the drop penetration process in Kissa (1981). Taking all data on curve B-C, the best fitted index n would be about 3.6. This was the characteristic of Whatman No. 17 chromatographic paper. The n value obtained indicated that the piston-like assumption used in various studies to describe the fluid flow in CSA was questioned. Determination of D,. From eq 2, it was shown that the reference diffusivity was inversely proportional to liquid viscosity. Leverett gave a correlation that the capillary pressure in porous media was proportional to the liquid surface tension (Scheidegger, 1960). It was thus naturally to suggest that P, a u cos 8, where 8 was the contact angle. Therefore, for the first approximation,the effective diffusivity should be proportional to the surface tension, the cosine of the contact angle, and inversely proportional to the liquid viscosity. If the diffusivity of some reference liquid was known, the effective diffusivity of other liquids could then be found from
Experimental results of water showed that when D, = 1.26 X m2/s, the data were best matched with the theory. Taking water as the reference liquid, the effective diffusivity of other liquids could be calculated from eq 13. The range of contact angle of water was shown to be around 30-60' for ordinary metal substrate and was enlarged to 100' when the substrate was a coated PTFE film (Diesselhorstet al., 1977). Usually, the contact angle for other organic liquids was less than that of water when on a nonmetallic substrate. In this study, the contact angle of water was taken as 70°,and for all other pure liquids and mixture Solutions, the contact angle was taken as 60'. Since the moving front velocity was relatively small, the contact angle was taken as a constant throughout the test and the dynamic contact angle effect was negligible. The results of D, thus obtained were shown in Table 11. Comparison. The comparison of the predictions using eq 10 and the experimental data is shown in Figure 5. Good agreement is obtained when the index n was taken to be between 3 and 4,which is consistent with the value, 3.6,obtained in the previous analysis. The data of Nguyen (1980)are also shown in the fgure. The physical data and the calculated reference effective diffusivity for various mixture solutions are also listed in Table 11. The theory agreed well with all Nguyen's results for nine mixture solutions with methanol, glycerol, and acetone. For cases with columns of very small radii, some deviation might occur when the front radius becomes large.
- -1
0
1
2
3
4
5
log T
Figure 5. Comparison between experimental data and eq 10. The data for water mixtures with methanol, acetone, and glycerol were from Nguyen (1980). SDS1, 0.04 M SDS solution; SDS2, 0.02 M SDS solution: Acel, 5 vol % acetone; Ace2, 10 vol % acetone; MeOH1,60 vol % methanol; MeOH2,80 vol % methanol; Glyl, 10 wt % glycerol; Gly2,20 wt % glycerol; Gly3,33 wt % glycerol; Gly4, 42 wt % glycerol; Gly5, 75 wt % glycerol.
This might come from the assumption that the approximation function of the saturation profile assumed in eq 9 could be applied to the whole region of interest, which should be valid only within the region near the front. However, since columns with very small radii were seldom used in applications,and since the relative error would be large in such experiments, use of eq 10 to describe the fluid flow in CSA with no cake formation was still satisfactory in most applications. Capillary Suction Time. From eq 10,the capillary suction time for filtrate moved between two concentric circles, R1 and R2, is CST = tz - ti
From eq 14,a modified CST was proposed to incorporate the effects of various parameters as
From this definition, the modified CST was the inverse of the product of the effective diffusivity and the liquid saturation in the filter paper under the column, and the CST data for a given fluid obtained by various CSA would be the same. A Test with Cake Formation. In previous discusions, the liquid saturation so with no cake formed on the filter paper was assumed to be unity without proof. When cake formed on the filter paper, excess pressure drop was needed to drive the liquid flowing through the cake. Since the head of the liquid in the cylinder was small, the pressure drop should be provided by the capillary suction pressure of the filter paper under the cylinder. Therefore, it was speculated that so should be less than unity and vary during experiment. Experiments with kaolin slurry were conducted to illustrate the feasibilityof using eq 10 to describe the fluid flow in CSA with slurry. The results are shown in Figure 6. It is clear that the formation of cake increased the CST significantly. The more solid particles added, the longer
2384
Ind. Eng.Chem. Res., Vol. 31, No, 10, 1992
t
Q
t
Water .95
Q Water ,535 0 A
Water .20 Kaol ,535
t
-1
o Kao2 ,535 Kao3 ,535
-1
0
1
2
I 3
I
J
4
5
0
1
log T
Figure 6. log y venue log r in CSA with pure water and kaolin slurry. Kaol, 2.1 w t % kaolin slurry; Kao2,4.3 w t % kaolin slurry, h 0 3 , 6.1 wt % kaolin slurry.
the time needed for the wet front to travel between two fixed points. The solid linea in Figure 6 are the calculation results of eq 10 with various so values. For the cases of 2.1,4.3, and 6.1 w t %, the y versus 7 curves fitted well with the lines with so = 0.71, 0.68, and 0.65, respectively. To confirm this point, the liquid saturation so was calculated directly from the liquid penetration volume and the wet front dynamic data. Since the saturation profile was described by eqs 6a and 9, the liquid volume that penetrated into the paper could be found as
V = JR6f2urs dr
(16) Therefore, so could be calculated with the help of eq 16 if V and y at various time intervals were known. F’igure 7 summarim the results. The 80)s obtained with kaolin slurry first decreased slightly and then approached a constant value. After the so attained the constant region, the capillary suction pressure of the filter paper under the inner cylinder was the same and the condition was very similar to the traditional constant pressure filtration process. The data for pure water are also shown in Figure 7 for comparison. Within the experimental error, the so could be taken as unity, which confirmed the assumption made previously. Therefore, eq 10 was adequate to describe the fluid flow behavior in CSA with pure liquids or slurry, if the liquid saturation under the inner cylinder was equal to or less than unity. Further analysis for the CSA with cake formation will be treated in a forthcoming paper. Conclusions The fluid flow procese in the capillary suction apparatus was studied experimentally and theoretically. The following conclusions were obtained in this study: 1. Capillary suction time was a strong function of the column radius, the location of measurement, and the liquid properties. For fixed sampling locations, CST increased as the column radius decreaeed and decreased when liquid viscosity increased and/or the surface tension decreased. 2. A model based on the diffusion-like approach was developed to model the fluid flow in CSA with the effective dBusivity approximated by a simple power law D = Osz6. The reference diffueivityof liquih could be evaluated from the liquid propeties. The agreement between the exper-
2
3
4
5
6
Y
Figure 7. so versus y. Kaolin slurries.
R,,= 0.535 cm.
imental data and the theory was satisfactory. 3. A modified CST based on the model was proposed to be a measure of the easiness of the fluid flow in the filter paper which was independent of the CSA used. It could link the works of various investigators. 4. Experiments with kaolin slurry showed that the results from the pure liquid case could be extended to describe the fluid flow behavior in CSA with cake formation, if the saturation under the inner cylinder was less than unity. Acknowledgment
We are grateful to the Environment Technologies Research Center, Yuan-Ze Institute of Technology, for the support of Project ETRC-79005. Nomenclature CB,C2, Co = coefficients in eq 10 D = effective diffusivity, mz/s D,= diffusivity at s = 1,m2/s K = permeability, m2 Kp = effective permeability, m2 n = index in eq 2 P, = capillary suction pressure, N/m2 Ro = cylinder inner radius, m R = equivalent radius of the front, m R, = long principle radius of front, m R, = short principle radius of front, m s = liquid saturation so = liquid saturation under the inner cylinder t = time, s V = liquid volume, m3 x: = distance from the front, m y=r+-1
Greek Letters 6 = wet filter paper t
thickness, m
= wet paper porosity
19 = contact angle, rad cc = liquid viscosity, Pa.8 p = liquid density, kg/m3 u = surface tension, N/m r = dimensionless time
Literature Cited Baskerville, R. C.; Gale, R. S.A Simple Automatic Instrument for Determining the Filterability of the Sewage Sludge. Water Polkct. Control 1968,67,233-241. Chang, L. S . Spreading of Ink Drops in Paper: a Kinetic Model. Society of Photographic Sci. Engineers Conference,New Orleans; 1988.
Ind. Eng. Chem. Res. 1992,31,238&2388 Corney, A. T. Mechanics of Heterogeneous Fluids in Porous Media. Water Resources: 1977. Dieseelhorst, T.; Grigull, U.; Hahne, E. Hydrodynamic and Surface Effects on the Peak Heat Flux in Pool Boiling. In Heat Trcmnsfer in Boiling; Hahne, E.; Grigull, U., Eds.; Academic: New York, 1977; Chapter 6. Dohanyos, M.; Grau, P.; Sedlacek, M. Interpretation of Dewaterability Measurements by Capillary Suction Time (CST). Water Sci. Technol. 1988,20, 265. Gale, R.S.; Baskerville, R. C. Capillary Suction Method for Determination of the Filtration Properties of a Solid-Liquid Suspension. Chem. Znd. 1967, 9, 355-356. Ju, S. J. A Study on the Batch Gravitational Filtration. Master’s Thesis, National Taiwan University, Taipei, Taiwan, 1982. Karr, P. R.;Keinath, T. M. Influence of Particle Size on Sludge Dewaterability. J. Water Pollut. Control Fed. 1978, 50, 1911-1930.
King,R.0.; Forster, C. F. Effects of Sonication on Activated Sludge. Enzyme Microb. Technol. 1990,12,109. Kissa, E. Capillary Sorption in Fibrous Assemblies. J. Colloid Interface Sci. 1981, 83, 265. Landau, L. D.; Lifshitz, E. M. Fluid Mechanics; Pergamon Press: Oxford, 1959. Leu, W. F. Cake Filtration. Ph.D. Dissertation, University of Houston, Houston, TX, 1981. Lu, W. M.; Ju, S. F.; Chang, U. J. Using CST to Measure Particle
2386
Characteristic Diameter and the Specific Reeitance. Proceedings of the Symposium on Tronuport Phenomena and Applications; Chinese Institute of Chemical Engineers: Taipei, Taiwan, 1989. Nguyen, C. T. A Model for the Capillary Suction Apparatus. Master’s Thesis, University of Houston, Houston, TX,1980. Scheidegger, A. E. The Physics of Flow Through Porous Media; Macmillan: New York, 1960. Unno, H.; Muraiso, H.; Akehata, T. Theoretical and Experimental Study of Factors affecting Capillary Suction Time (CST). Water Res. 1983, 17, 149. Vesilind, P. A. Capillary Suction Time as a Fundamental Measure of Sludge Dewaterability. J. Water Pollut. Control Fed. 1988,60, 215.
Veailind, P. A.; Davis, H. A Using the Capillary Suction Time Device for Characterizing Sludge Dewaterability. Water Sci. Technol. 1988, 20, 203.
Wilcox, R. D.; Fisk, Jr., J. V.; Corbett, G. E. Filtration Method Characterizes Dispersive Properties of Shales. SPE Drill. Eng. 1987, 2, 149.
Yang, Y. M. Surface Effects and Boiling Heat Transfer. Ph.D. Dissertation, National Cheng Kung University, Tainan, Taiwan, 1983.
Received for review January 21, 1992 Revised manuscript receiued July 17, 1992 Accepted July 29,1992
Power Consumption for a Scraped-Surface Heat Exchanger with a “No-Leak”Reactor Mehmet R.Altiokka Chemical Engineering Department, Faculty of Engineering and Architecture, Anadolu University, Eskigehir, Turkey
In this work, the power consumption for a scraped-surface heat exchanger with a “no-leak” reactor was investigated to scale up a pilot plant reactor for industrial use. The experimental results obtained at various speeds of rotation and fluid viscosities were analyzed by nonlinear regression computation using the Nelder algorithm. Using the results of this analysis, the power consumption expression given for a “with-leak” reactor in a previous work was modified to give that of the reactor with “no leak”. That is, PL = [251D,1~79nB0%B/(D,- Di)0*31]2v1*93p0.s1.
Introduction Scraped-surface heat exchanger type reactors are extensively used in parafii-wax planta and in petrochemical plants for crystallization. They are also suitable for chemical reactions OcCuTLing in a thick Viscous media. This type of reactor generally consists of two concentric cylinders sealed at both ends. The inner cylinder is connected to a variable-speed unit and has blades attached to the outer rim. As the inner cylinder rotates, the blades scrape the inner surface of the outer cylinder thereby providing good heat transfer. Without the scraping action of the blades, a thick viscous liquid layer would build up on the inner surface of the outer cylinder causing poor heat transfer between the reactor contents in the annulus and the cooling/or heating media. Because there is no clearance between the blades and the inner surface of the outer cylinder, the reactor used in this work is called a scraped-surface heat exchanger with “no leak”; otherwise a scraped-surface heat exchanger “with leak” would be a better name. In scaling up a pilot plant reactor for industrial use, the power consumption of the impeller and other mechanisms is one of the most important considerations. Therefore, the objective of this work is to find an expression to predict the power requirement for “no-leak” reactors and generalize the relation.
The factors causing the power consumption in such a reactor are the shear stress created by the liquid, the scraping action of the blades against the wall, friction in the bearhgs, and the rotation of mass of the inner cylinder and blades themselves. Only the power consumption due to the shear stress created by the liquid can be calculated theoretically when the reactor is operated within the laminar flow regime since the velocity profile is known (Bird et al., 1960; Brewster and Nissan, 1958; Atiokka, 1991). Unfortunately, it is not possible to develop a mathematical expression to distinguish between the power contribution due to the other factors mentioned above. However, it is acceptable to suggest that the power requirements to overcome all frictions can be combined into one term which is proportional to the speed of rotation of the inner cylinder. Therefore, the total power consumption can be given by
KlIP/1+ K a (1) where the first term represents the power dissipated in liquid when the reactor is operated within the laminarflow regime while the second is considered to be friction term. K1and K2are constants, and K1can be calculated theoretically from the reactor dimensions (Altiokka, 1983). However, Trommelen and Boerema (1966) showed that even in the laminar flow regime there is no reliable PT
0888-5885f 92f 2631-238Ei$03.O0 f 0 Q 1992 American Chemical Society
