Znd. Eng. Chem. Res. 1994,33, 1593-1599
1593
A Rectangular Capillary Suction Apparatus D. J. Lee’ Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 10617, Republic of China
Y. H. Hsu Department of Chemical Engineering, Yuan-Ze Institute of Technology, Taoyuan, Taiwan 32026, Republic of China
Fluid flow and cake formation in a rectangular capillary suction apparatus (RCSA) are investigated experimentally and theoretically. Water, methanol, ethanol, and ethylene glycol are used to study the effects of liquid properties, and CaC03, kaolin, and bentonite slurries are employed for studying the effects of cake formation on capillary suction time (CST). A theory based on a diffusion-like approach is developed. The liquid saturation under the inner cell will approach a constant value when the wet front distance is large. A method based on this experimental finding for estimating the cake specific resistance is proposed. The agreement between experiments and calculations is close. The RCSA is superior to the cylindrical CSA when treating liquids with small diffusivities or slurries with high solid concentration and/or with high averaged specific resistance.
Introduction
A cylindrical capillary suction apparatus (CCSA) is composed of two plastic plates, a stainless steel cylindrical column,several electrodes servingas the sensors, Whatman No. 17 chromatography paper, and a timer. When sludge is poured into the column, the time the filtrate needs for travelling between two concentric circles is called the capillary suction time (CST). Usually, a long CST means a large cake specificresistance (Baskervilleand Gale, 1968). Since the CCSA experiment is much simpler to perform and more timesaving than the conventional filtration tests, the device has been widely used in various fields since first developed by Gale and Baskerville in 1967 (Baskerville and Gale, 1968; Karr and Keinath, 1978; Kavanagh, 1980; Nguyen, 1980; Leu, 1981; Ju, 1982; Konechnik et al., 1987; Wilcox et al., 1987; Dohanyos et al., 1988; Stephenson, 1988; Vesilind, 1988; Vesilind and Davis, 1988; Lu et al., 1989; Agerkvist and Enfors, 1990; King and Forster, 1990; Hayashi et al., 1990; Lee and Hsu, 1992,1993,1994; Meeten and Lebreton, 1993). Though a CCSA is very simple to use, there exist several shortcomings which make the CCSA only a qualitative not a quantitative tool in determining the cake characteristics (Ju, 1982). One among others is the anisotropic property of the machine-made filter paper which causes the filtrate to move faster along the grain than across. The resulting wet area shape will be elliptical rather than circular. Gale and Baskerville (1967) found that the difference between the two principal radii is about 20 % . Leu (1981) proposed the concept of rectangular CSA (RCSA) to overcome this difficulty. The RCSA makes use of only one direction of the filter paper and leads to a unidirectional flow field. The device is also discussed further in Unno et al. (1983) and in Tiller et al. (1990). Experimental findings show that the wet front distance grows with t1I2,and the CSTs obtained in a CCSA will be larger than those obtained in a RCSA, which implies that the RCSA might be the proper tool in treating slurries with high solid concentration and/or high averaged specific resistance. However, the existing RCSA analyses in the literature are all based on the piston-like approach.
There are two ways of describing the liquid movement in a capillary suction apparatus (Nguyen, 1980): the pistonlike approach, which treats the liquid flow in the filter paper as a displacement process (Nguyen, 1980; Leu, 1981; Ju, 1982; Unno et al., 1983; Vesilind, 1988; Tiller et al., 1990), and the diffusion-like approach, which treats the process as a diffusion process with the diffusivity a concentration-dependent property (Lee and Hsu, 1992, 1993).
Lee and Hsu (1992) studied the fluid flow in a CCSA with the diffusion-like approach since it is more realistic than the piston-like approach. By assuming a power law dependence of diffusivity, D = D s n , effects of various liquids and the parameters of the CCSA on the CST are investigated. A pseudo-steady-state model is proposed. On the basis of diffusion-like approach, Lee and Hsu (1993) utilized CCSA for estimating the averaged specific resistance of a cake. The capillary suction pressure versus saturation relation is constructed. Various slurries are tested, and the predicted specific resistance agrees well with the experiment. It is also mentioned that the use of the piston-like approach might introduce serious errors in estimating the liquid invasion volume (and also the superficial velocity) and the averaged specific resistance of the cake. There exists no RCSA theory based on the diffusionlike approach in the open literature to the authors’ best knowledge. To extend the works in a CCSA to a RCSA, in this report, various liquids and slurries are tested to study the effects of fluid properties and cake formation on the CST. The method for estimating the averaged specific resistance of the cake is proposed and compared with experiments.
Analysis Fluid Flow. The process is shown schematically in Figure 1. For the two-phase system neglecting gravitational force, the following“diffusion-like”equation may be written from Darcy’s law and the continuity equation (Chang, 1988; Lee and Hsu, 1992):
~~~~
* To whom correspondence should be addressed. E-mail:
[email protected]: 886-2-362-3040. 0888-588519412633-1593$04.5OlQ
0 1994 American Chemical Society
1594
Ind. Eng. Chem. Res., Vol. 33, No. 6, 1994
m
An integral method is utilized for analyzing this problem (Lee and Hsu, 1992). Integrating eq 2 with respect to z+, the result is
slurry
wet area
(5)
'h
An expression of s as function of z+ is needed to complete the analysis. An observation leads to the conclusion that the form of eq 1 is basically the same as that for a transient heat conduction problem with temperature-dependent conductivity. Landau and Lifshitz (1959) had treated such a problem and demonstrated that, for a small time interval, the independent variable W near the moving front would vanish according to W a IxJl/",where x was the distance from the moving front. Therefore, following a similar argument in Lee and Hsu (1992), the liquid saturation profile can be taken as
and
Substituting eqs 6a and 6b into eq 5, the result is
I
y2
+ CTSO" = 0
(7)
where y=--
and
c=--2(n + 1) n"
Clearly the dimensionless invasion distance y will move with t1/2,which is consistent with the experimental findings. The slope is equal to 2(n + l)D$n2Z02,which is a function of liquid properties and the cell thickness, and is independent of the cell width. For checking purpose, the liquid invasion volume is
where
V = &'&Bs dz
and
where In a RCSA with no cake formation, the liquid saturation under the rectangular cell will remain as unity, since liquid always exists above the filter paper during operations. A clear boundary between the moving front and the remaining paper can be observed in all experiments, which suggests that the liquid saturation should be zero in the region outside the front area. The boundary conditions can therefore be written as
-
Z + l l
(4a)
V = BZGt
z+ = z+= Z(7)/Z0
(4b)
which is the result for the piston-like approach (Leu, 1981). Cake Formation. When fluid is flowing steadily through a porous media (the cake), the pressure drop can be stated as
s=so
s =0
When index n approaches infinity, the liquid invasion volume derived from the diffusion-like approach will converge to that obtained from the piston-like approach (Lee and Hsu, 1992). Taking the n m limit, eq 10 becomes:
where Z ( r ) is the front distance at dimensionless time r , and SO is unity.
(12)
Ind. Eng. Chem. Res., Vol. 33, No. 6, 1994 1595 1
= ~paavcOvu
aPcake
where aav,V, p, Co, and u are the averaged specificresistance for the cake, the filtrate volume, the liquid viscosity, the solid concentration, and the superficial velocity, respectively. When it is assumed that the liquid saturation under the inner cell will approach a constant value when y is large (as will be shown later), then the capillary suction pressure under the cell will also remain unchanged. When the liquid head in the cell is neglected, the pressure drop across the cake will be constant and is provided by the capillary suction pressure of the filter paper under the inner cell (Lee and Hsu, 1993). That is, hPcake
= '~('0)
(14)
Under this condition, the process is similar to a constant pressure filtration process. The Pcversus SO relation has been proposed by Lee and Hsu (1993) and will be used in this work. For a constant pressure filtration process, if the specific resistance is a function of the applied pressure difference only, the v2's will be a linear function of time (McCabe et al., 1985). In such a case, if CO does not change significantly during experiments, eq 13 can be integrated as t=
Coaavp
v2+
1 - rectangular cell filter paper transparent plate electrodes data acquisition system 6 - personal computer
2345-
c,
Figure 2. (a, top) Experimental setup. (b, bottom) Top view of the upper plate. Numerical values are in cm. Table 1. Physical Properties and Calculated Reference Effective Diffusivity for Various Liquids. T = 28 OC. * denotes along the grain; denotes across the grain. Diffusivities listed in the last column are data from cylindrical CSA tests
2A2aPcake
where C1 is an integration constant. The specificresistance can therefore be obtained from a linear regression of the t versus v2 data. By combining eqs 14 and 15, the following expression can be obtained:
The subscript m indicates that the condition is under the constant liquid saturation region. If the liquid saturation SO,- is known, eq 16 can be rearranged with the help of eq 10 as
Equation 16 can be used in the cases when a direct reading of the liquid invasion volume is possible, and eq 17 is suitable for the cases when the liquid invasion volume data are unavailable. The standard procedure for estimating the specific resistance is as follows. If the liquid invasion volume and the wet front dynamic data are available with the CST, the liquid saturation SO at fixed time intervals can be calculated from eq 10. After the saturation has reached the constant region, the slope of t versus VZ data can be obtained from a linear regression analysis. The specific resistance can be then calculated via eq 16, if the specific resistance is a function of the pressure difference only. On the other hand, if accurate liquid invasion volume data are not available, eq 17 can be used instead if the liquid saturation SO is known.
Experimental Section The experimental setup is shown in Figure 2. Figure 2a shows the device. Two transparent plates, a rectangular
liquid watera waterC watera waterC MeOHa MeOHc MeOHa MeOHC EtOHa EtOHc EtOHa EtOHc EGa EGC EGa EGC
ZO,
IL,
10%
cm
Pas
N/m
Ddexp), 10"m2/s
0.4 0.4 0.8 0.8 0.4 0.4 0.8 0.8 0.4 0.4 0.8 0.8 0.4 0.4 0.8 0.8
0.80 0.80 0.80 0.80 0.60 0.60 0.60 0.60 1.0 1.0 1.0 1.0 13.0 13.0 13.0 13.0
72.0 72.0 72.0 72.0 22.0 22.0 22.0 22.0 22.0 22.0 22.0 22.0 47.2 47.2 47.2 47.2
7.35 6.57 7.27 6.60 5.98 5.06 6.02 4.98 3.14 2.71 3.02 2.65 0.41 0.35 0.41 0.34
Ddcal), DdCCSA), 10-5m2/s 1W m2/s
5.79 4.85 5.72 4.81 2.62 2.20 2.60 2.19 0.43 0.37 0.43 0.37
7.01 7.01 7.01 7.01 5.46 5.46 5.46 5.46 3.07 3.07 3.07 3.07 0.41 0.41 0.41 0.41
cell into which the liquid is poured, and a Whatman No. 17 filter paper form the main body. The data acquisition system and the experimental procedures can be found elsewhere (Lee and Hsu, 1992). Figure 2b shows the top view of the upper plate. Small pin holes are drilled along the central line, and electrodes are installed at these locations to serve as sensors. Pure fluid tests with fluid flowing along and across the grain are conducted. Methanol, water, ethanol, and ethylene glycol are employed as the testing liquids, and the physical properties of these substances are listed in Table 1. Rectangular cells with inner thickness 0.4 and 0.8 cm are used. The width of the filter paper and also the cell are fixed as 6.9 cm. A CCSA with an inner cylinder radius of 0.535 cm is used in comparison tests. Kaolin, CaC03, and bentonite slurries are employed in slurry tests. A vacuum filtration system equipped with an electronic balance is installed for measuring the averaged specific resistance of the cake. The weight of the filtrate can be recorded continuously and sent to a personal computer automatically. The pressure difference for filtration is fixed as 62 cmHg. The characteristics of
1596 Ind. Eng. Chem. Res., Vol. 33, No. 6, 1994 Table 2. Characteristics of the Slurries Used in This Study slurry
%, w/w
kg/m3
103w, Pas
dew), 10llm/kg
adcal), 10llm/kg
CaC03 CaC03 CaC03 CaC03 kaolin kaolin kaolin kaolin bentonite bentonite
2.0 6.0 10.0 20.0 2.0 6.0 10.0 20.0 2.0 6.0
20.8 67.6 116.1 276.1 20.8 68.8 120.4 361.3 21.1 70.9
0.94 1.03 1.01 1.00 1.04 0.98 1.00 1.03 1.05 1.05
0.408 0.383 0.471 0.470 2.18 2.49 3.62 2.93 3070 3530
0.2-0.4 0.41 0.44 0.43 2.4 2.3 3.2 2.7 4300 4100
CO,
these slurries are listed in Table 2. The inner cell for slurry tests is fixed as 0.9 cm, and only tests with fluid flowing along the grain are conducted. The experimental procedure is as follows. After the RCSA is cleaned and dried, the filter paper, the rectangular cell, and the electrodes are positioned. By injecting a fixed amount of liquid into the cell, the contact of the liquid with the filter paper will trigger the timer in a personal computer. The wet front invasion distance can then be traced by recording the time the liquid arrives at each electrode. The experiment finishes when all electrodes receive the signal of liquid arrival. The measurement error of the time is less than 0.01 s, and the error for position measurement is less than 0.5 mm. The CST data recorded are then plotted against the electrode position. The invasion distance of the wet front at fixed time intervals can also be obtained by interpolation. At least three runs for each experimental condition are conducted to check the reproducibility. Liquid invasion volume data can be directly read out in experiments. It is noted, however, that the accuracy and reproducibility of the liquid invasion volume measurement is highly dependent on the operator's skill. Much care has to be taken in measurements. The amount of slurry employed in this study is 6 mL. With this amount of slurry, the liquid head is very small when compared with the capillary suction head, and the slurry will not exhaust during experiments. At least three runs for each experimental condition are conducted to check the reproducibility. The effects of particle sedimentation on the CST had been proven to be only minor in a CSA test (Lee et al., 1994) and are therefore neglected in this study. Results and Discussion
Fluid Flow. Experimental results for pure liquids are shown in Figure 3. It is clear that each group of data can be fitted by a straight line with slope of 1/2 on a log-log scale when y is large. For a fixed sampling distance, the CST increases as the column width decreases, and decreases when the liquid viscosity decreases andlor the liquid surface tension increases. It is also noted that the wet front moves faster along the grain than across. The difference between the two principal radii is about 20%, which is consistent with the investigations in a CCSA (Gale and Baskerville, 1967). Since the cell width/thickness ratio is large, the liquid might not distribute uniformly in the cell at the moment of injection. It is observed that the shape of the wet front is slightly curved at the beginning of the experiments and is dependent on the method of injection. Under such conditions, the fluid flow field might not be unidirectional and the basic assumptions of the theory fails. In Figure 3, clear deviation exists between the theory and the
Zo Awatera .4 Liquid
1R
bwater: . 4 4 Water .8 Water: . E -MeOH 4 D M ew' : 4 =MeOHa 8 a ~ a : 0EtOH 4 9 EtOH' :4 bEtOHa 8 0 E t p C 18 nEG .4 v
.5
-
-
:-~ -
-
P EG' .4 E G ~ .8 -
,
0
I/
0
"
D
'
,'
1
I 2
E G ~ .e
I
I
3
4
5
log CST
Figure 3. log y vs log CST. Pure liquid tests. 20 is in cm. All dashed lines are lines with slope 1/2. MeOH, methanol; EtOH, ethanol; EG, ethylene glycol; a, along the grain; c, across the grain.
Liquid
Zn -
Watera .4 3 Water' .4 4 Watera .E V WaterC .8 D MeOHa 4 D M ~ O H:4 ~ Mal? 8 3 MeOHC ' 8 0 E tOHa '4 is EtOH' '4 0 EtWa 18 0 E t p c .8 A
0
U
EG
0 EG' 0 D
-0
1
2
EG" EG'
3
.4
.4
.a .a 4
log 7
Figure 4. Comparison between experimentaldata and eq 7. 20 is in cm. MeOH, methanol; EtOH, ethanol; EG, ethylene glycol; a, along the grain; c, across the grain.
experimental data when the invasion distance is small. The use of the data obtained from electrodes located along the central line of the upper plate to represent the overall fluid flow behavior might therefore introduce serious errors when y is small. When the invasion distance is large, however, usually for y larger than about 3-4, a rather straight wet front boundary can be obtained. The effective diffusivity of each liquid test can be obtained from a linear regression analysis. The best-fitted results are listed in Table 1. Usually, the correlation coefficient is as high as 0.999. As suspected, the diffusivity for water and ethylene glycol is respectively the largest and the smallest one. It is clear that the diffusivity of each liquid is independent of the cell thickness but is larger when the flow direction is along the grain than across. The difference between the diffusivities for flow along two principal radii is about 20 % ,which is close to the difference observed in invasion distance measurements. Experimental results are compared with eq 7 in Figure 4. The theory agrees well with the experiments when n
Ind. Eng. Chem. Res., Vol. 33, No. 6, 1994 1597 1.0I
I
I
I
8
1
-
I
rJ.
Q
~ C a 2
A
kao
cal
Ca2
o Ca3
I
II
A
v c D 0 @
'-
Ca4 kaol kao2 kao3 kao4 benl ben2
3
4
3
I
el'
9
1 0
"
1
2
3
4
log CST
5
10
Y
Figure 5. logy vs log CST. Slurry tests. 20= 0.9 cm. All dashed lines are lines with slope 112. Along the grain. Cal, 2.0% w/w CaC03 slurry; Caz, 6.0% w/w CaC03 slurry; Ca3,10.0% w/w CaC03 slurry; i ,4, 20.0% w/w CaC03 slurry; kaol, 2.0% w/w kaolin slurry; kao2, 6 . ~ w/w kaolin slurry; kao3,10.0% w/w kaolin slurry; kao4,20.0% w/w kaolin slurry; Benl, 2.0% w/w bentonite slurry; Ben2,6.0% w/w bentonite slurry.
Figure 6. V/BZo& vs y. Dot and dashed line, calculation result for the piston-like approach (eq 22); solid lines, calculation results for the diffusion-like approach (eq 10) for various values of SO. Cal, 2.0% w/w CaC03 slurry; Ca2, 6.0% w/w CaC03 slurry; Ca3, 10.0% w/w CaC03 slurry; Ca4, 20.0% wlw CaC03 slurry; kaol, 2.0% w/w kaolin slurry; kao2,6.0% w/w kaolin slurry; kao3,10.0% w/wkaolin slurry; ka04, 20.0% w/w kaolin slurry; Benl, 2.0% w/w bentonite slurry; Ben2, 6.0% w/w bentonite slurry.
is between 3 and 4, which is consistent with the results for the CCSA tests (3.6) (Lee and Hsu, 1992). It is suggested by Lee and Hsu (1992) that the effective diffusivity for various liquids is directly proportional to the liquid surface tension and is inversely proportional to the liquid viscosity. Therefore, based on some reference fluid, the diffusivities for various liquids can be estimated. When water is taken as the reference substance, the calculated diffusivities are also listed in Table 1. I t is clear that the calculated results agree fairly well with the diffusivities obtained from the linear regression analysis. To examine the relation between the diffusivities obtained from RCSA tests and from CCSA tests, pure liquid tests in a CCSA are conducted and the diffusivities estimated by curve fitting are listed in the last column of Table 1. The values are different from those shown in Lee and Hsu (1992),which are caused from an inadvertent error in their regression program. I t is noted that the diffusivity from the CCSA test falls between the diffusivities obtained from RCSA tests along and across the grain and is close to the geometric mean of the two values. This result is quite natural since the equivalent radius, which is the square root of the product of the long and short principle radii, is used in CCSA tests for overcoming the anisotropic problem. Cake Formation. Experimental results for slurry tests are shown in Figure 5 . The data for pure water tests are also shown in the figure for comparison. All data can be fit by straight lines with a slope of 112 on a log-log scale. It is clear that the formation of cake increases the CST significantly. The time the wet front needs to travel a fixed distance increases with the solid concentration and the averaged specificresistance. Take y = 5 as an example. The CST for pure water, CaC03 slurry (6.0% w/w), kaolin slurry (670 w/w), and bentonite slurry (6% w/w) are 41, 56,60, and 2500 s, respectively. The fact that the test for the 2% wlw CaC03 slurry almost coincides with the pure water test suggests that the use of a RCSA is inadequate in treating slurries with low specific resistance and/or low solid concentrations (Coa,, < 10l2m-2). It is also clear that for the slurries with a specific resistance larger than
about 1015m/kg,the CST will be longer than several hours. Since one of the advantages of CSA tests is the short experimental time required when compared with the conventional filtration tests, a long CST might cause the CSA to lose its major superiority. To check the validity of the diffusion-like approach, the liquid invasion volume and the corresponding y data are plotted in Figure 6. The dot and dash line represents the calculation results for the piston-like approach (eq 12), and the solid lines represent the calculation results for the diffusion-like approach (eq 7) with various SO'S. It is clear that the results from the piston-like approach seriously overestimated the liquid invasion volume (and also the superficial velocity) in RCSA tests. In some cases, the deviation can be as high as 300%. Take y = 5 as an example again. The liquid invasion volume from the piston-like approach is 3.73 mL, the experimental results for pure water, CaC03 (6.0% w/w), kaolin (6.0% w/w), and bentonite slurry (6.0% w/w) are about 3.00,2.92,2.67, and 1.25 mL, respectively. It is clear that the degree of overestimation increases with the solid concentration and the specific resistance of the cake. The diffusion-like approach can, however, describe the fluid flow behavior in a RCSA with cake formation well. The data for the pure water test fit well with the line of SO = 1.0, which is the upper bound of the liquid invasion volume for fixed wet front distance. The data for slurry tests can also be fit well by eq 10 if the SO value (less than unity) is properly chosen. From the figure, the liquid saturation approaches a constant when y is large, which is similar to the experimental findings for a CCSA (Lee and Hsu, 1993). Under such circumstances, the requirement in the analysis for a constant SO in estimating the averaged specific resistance is fulfilled. The SO,- values for CaC03, kaolin, and bentonite slurries are about 0.91.0, 0.8-0.95, and 0.40-0.50, respectively, which is a function of the product of the solid concentration and the averaged specific resistance. These values are all larger than those obtained in CCSA experiments. Since the bentonite slurry is the most difficult one to filter among the slurries used in this work, it is reasonable to find that
1598 Ind. Eng. Chem. Res., Vol. 33, No. 6, 1994 2.8
I
,
I
I
I
4
I
I
I
i
't-
1
5
0
9
Y
Figure 7. Comparison between CCSA and RCSA tests. Bentonite slurry, 10% wiw. 20= 0.9 cm. Ro = 0.535 cm. the liquid saturation for the bentonite slurry is also the lowest one among these slurries. It is therefore concluded that the diffusion-likeapproach is adequate in describing the flow process in a RCSA. The analysis based on a piston-like approach might overestimate the pressure drop across the cake, and the condition becomes more serious when the slurry concentration and/ or the cake specific resistance increases. Since when the wet front distance is large the liquid saturation so is a constant during experiments, the RCSA process can be viewed as a constant pressure filtration process. From eq 17 and the capillary suction pressureliquid saturation relation (eq 14 in Lee and Hsu, 19931, the averaged specific resistance for slurries are calculated and listed in the last column of Table 2. The agreement is satisfactory. However, for CaC03 slurries with low solid concentrations, a larger uncertainty exists in determining the liquid saturation from experimental data, since the results are very close to the pure water tests and are hard to differentiate. A test with a 109% w/w bentonite slurry performed both in a CCSA and in a RCSA is shown in Figure 7. It is clear that the dimensionless time for the CCSA test might be 20 times larger than that for the RCSA test. This result further supports the conclusion that when treating a slurry with a high solid concentration and/or a high specific resistance of the cake, a RCSA is superior to the cylindrical CSA. However, as discussed above, a cylindrical CSA will be a more proper choice when treating slurries with low concentrations and a low specific resistance.
Conclusions The capillary suction time is a strong function of the inner cell thickness, the sampling locations, and the liquid properties but is independent of the cell width. For a fixed sampling distance, the CST increases as the column width decreases, and decreases when the liquid viscosity decreases and/or the liquid surface tension increases. A model based on a diffusion-likeapproach is developed. The dimensionless invasion distance is proportional to t1I2, and the slope is a function of the liquid effective diffusivity and the cell thickness. The agreement between the experimental data and the theory is satisfactory. The capillary suction time will increase when cake forms in a RCSA. For fixed sampling locations, the CST
increases with the solid concentration and the specific resistance of the cake. The liquid invasion volume obtained from a piston-like approach is overestimated, and the deviation can be as high as 300% for bentonite slurry tests. On the other hand, the liquid invasion volume can be well described by a diffusion-like approach with the liquid saturation under the inner cell less than unity. The SO value will approach a constant value during experiments. In the region with constant SO, the fluid flow across the cake is similar to a constant pressure filtration process. The procedures for estimating the specific resistance of the cake are proposed. Calculation results agree well with the experiments. The RCSA is superior to the cylindrical CSA, especially when treating slurries with high solid concentrations and a high averaged specific resistance, since it is more timesaving and avoids the anisotropic problem. However, the cylindrical CSA will be the proper choice when the solid concentration or the cake resistance is low.
Acknowledgment The authors are grateful to National Science Council, R.O.C., for supporting the project NSC82-0402-E002-346.
Nomenclature A = area, m2 E = cell width, m C = coefficient defined in eq 9 Co = solid concentration, kg/m3 C1 = integration constant D = diffusivity, m2/s D, = effective diffusivity, m2/s G ( y ) = function defined in eq 11 n = index M a k e = pressure drop across the cake, Pa P, = capillary suction pressure, Pa R = wet front radius in a CCSA, m Ro = inner cylinder radius, m s = liquid saturation so = liquid saturation under the cell t = time, s V = liquid invasion volume, m3 z = distance from the front, m y = 2/20- 1 for a RCSA, or =R/Ro - 1 for a CCSA 2 = wet front distance, m 20 = cell thickness, m Greek Letters cy,,
= averaged specific resistance, m/kg
= wet paper porosity 6 = wet paper thickness, m t
w = liquid viscosity, Pa s CT = liquid surface tension, N/m T = dimensionless time
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Received for review August 26, 1993 Revised manuscript received March 11, 1994 Accepted April 1, 1994' Abstract published in Advance ACSAbstracts, May 1,1994.
