FLOW THROUGH POROUS MEDIA
Moisture Content of a Fine-Coal Filter Cake EFFECT OF VISCOSITY AND SURFACE TENSION C H A R L E S E. SILVERBLATT' A N D DONALD A. DAHLSTROM' NORTHWESTERN UNIVERSITY. EVANSTON, ILL.
T h i s work was initiated t o study t h e relationship of several filtration variables as they affect t h e moisture content of a fine-coal filter cake. Although t h e effects of viscosity and surface tension were of primary concern, t h e other variables discussed include air rate, pressure drop across t h e filter cake, and wetting agent concentration. Moisture content was primarily a function of viscosity for dilute concentrations of wetting agent. Relatively high concentrations of wetting agent had a profound effect on both moisture content and air rate. A general relationship between moisture content and a multiple correlating factor was suggested.
I"
MOST filter operations the question of cake moisture is a very important one. In the past many investigators have noted that an increase in slurry temperature resulted in a decrease in cake moisture. However, little work has been done to isolate the effects of the properties that are actually changed by a variation in temperature; surface tension and viscosity. If it mag be assumed that a filter cake is composed of many capillaries distributed in a random fashion it is reasonable to assume that surface tension will determine the lower limit of cake moisture (1, 2). Dimensional analysis yields the dimensionless group AP') __K(-d g , d cos
CY
which is called the capillary number. Experimental data for thin beds, as deep as 2 inches in thickness relate residual saturation, S,, to the capillary number
Residual saturation decreases with a decrease in surface tension but only t o the 0.264 power. Viscosity is considered to have no influence on residual saturation. If during the removal of filtrate from the cake, the flow of the filtrate is laminar, the rate of removal is inversely proportional to the filtration constant, Ct Ct =
ML'Xd2 I
h AP
This constant appears in the equation
which expresses the time required to reduce cake moisture to any value as a function of the physical properties of liquid and cake solids. Thus, a decrease in viscosity would cause an increase in the rate of filtrate removal and a subsequent decrease in cake moisture. Should the flow of filtrate be turbulent, no viscosity terms appear in the rate equation; however, the rate is related t o factors that are proportional to viscosity. Accordingly, 1
Present address, The Eimco Corp., Palatine, Ill.
June 1954
viscosjty is very significant in influencing the rate of approach to residual saturation. This investigation was undertaken to determine quantitatively the effects of surface tension and viscosity upon the moisture content of a cake formed from a slurry of fine-coal and water. This system was chosen because of the importance of cake moisture in the preparation of coal. The conclusions drawn from this work may be applied to other slurries also. APPARATUS A N D P R O C E D U R E
The filter used in this investigation was a water-jacketed batch vacuum filter using a 4-inch square screen or filter media made of Bixby-Zimmer stainless steel wedge wire. A cross-section sketch of the filter is given in Figure 1. The wedge wires of triangular cross section were welded to two stainless steel rods so as to form a flat filtering surface. The usable filter area was 0.0955 square feet and the average distance between the wedge wires was 0.015 inches. The vacuum source for the filter consisted of four evacuated tanka connected to the filter by a I-inch inside diameter hose. This constant volume vacuum system permitted both control of the vacuum level and accurate measurement of the amount of air passed through the cake. The total volume of the tanks and connecting lines and hose was 43.68 cubic feet. The system pressure was measured by a mercury msnometer connected to the first tank. A schematic drawing of the experimental layout is given in Figure 2. The feed coal used in this work was a sample of filter cake obtained from the Ceredo preparation plant of the Truax-Traer Coal Co. a t Ceredo, W. Va. This material was thoroughly mixed to ensure a uniform size-consist throughout. The screen analysis for the feed coal is given in Table I. A 60% slurry containing varying amounts of wetting agents, Aerosol OT and Tergitol CFV, was used for all runs. Before making a run the pressure in the vacuum system was lowered to the desired level and the temperatures of the slurry and filter water jacket were properly adjusted. The thoroughly mixed slurry was poured on the filter deck, and as soon as the deck was covered with slurry, the valve to the vacuum system was opened. Timing was started after the disappearance of water from the surface of the cake, and the vacuum system was shut off after a predetermined amount of air had passed through the cake. The cake thickness was measured, and then the total cake was used for a moisture determination by drying overnight a t 210" F. Surface tension and viscosity determinations were made on the filtrates from slurries prepared in the same ratios as those used in the various runs. Surface tension was measured using a du Nouy ring tensiometer, and viscosity was determined with an Ostwald viscometer. For the wetting agents used, the viscosity of the liquid was a function of temperature only.
INDUSTRIAL AND ENGINEERING CHEMISTRY
1201
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT Table I. W e t Screen Analysisof Coal Used in Experimental Filtration Studies (Dorothy Seam, Truax-Traer Coal Co., Ceredo, W. Ya.) Specific Gravity of Coal = 1.363
G. S. Std. Mesh
wt. %
-8 8 12 12 X 16 16 X 20 20 X 30 30 X 40 40 X 50 20 X 70 70 X 100 100 X 140 140 X 200 -200
2.99 2.50 3.15 5.76 8.65 13.32 16.59 15.61 13.81 7.07 3.81 6.74
x
AP = 10 AP = 20
i-7*--------1
Figure 1.
check reproducibility and in each case the duplicates wcre averaged and treated as a single observation. One convenient a a y of summarizing the data is shown in Table 111. The averages a t the ends of the rows and below the columns indicate the effects of viscosity and surface tension upon cake moisture regardless of the variations of the other two variables. A further examination of the data of Table I11 shows that, the trends in the dat,a are quite consistent. This is a very good indication that, there is little or no interaction between viscosity and surface tension. The averages for a.ll tehe data a t each of the two pressure levels are found in Table I1 and are Average of Sine Moisture Determinations 18 30 17 51
The difference betxeen these tn-o averages is highly rignific:ari( as will be shown later in the analysis of variance. Table IV gives the averages of three moisture determinatioiis for each of the levels of the variables chosen for the experimental plan given earlier. Again, the method of presentat,ion makes it readily apparent that the trends in the data are quite consistent. This consistency indicates t,hat there is little or no interaction between pressure level and viscosity or pressure level and surface tension. The variation characteristic of t,hese laboratory determillations provides an objective standard for testing the significancv of differences among the group averages used in the previous discussion. An analysis of variance was used to summarize the data and provide tests of significance. When the variance for two factor interactions were tcsicd against the variance from three factor int,eractjons and e u o r , the I: values were found to be in the vicinit,y of 1.0 and none of the interactions were significant. Therefore, the sums of squarcs and degrees of freedom for all interactions were pooled t o increase thc wnsitivity of following tests. The resulting variance valuc? v a s used to test the significance of surface tension, viscosity, and pressure drop across the filter cake.
Basic Design of Test Filter
RESULTS
AA
Analysis of Variance. In older to determine the significance of TEST FlLTeR the variables involved and the magnitude of any interactions that might exist, the experimental plan given in Table I1 was VACUUM T A N F S set up. Data were taken at two levels of vacuum ( p i ~ ~ s udrop ie across the cake) and three levels each of surface tension and visFigure 2. Schematic Drawing of Filter Vacuum System cosity. Preliminary runs indicated the choice of surface tension levels, and since viscosity was effected only by changes in temperature, the values chosen represent three convenient temTable I I. Experimental Plan and Results perature levels: 50°, 80°, and 120" F. The (Total cake moisture a t controlled levels of vacuum, viscosity, and surface tension) second level of vacuum was used not only to AP = 20 AI-' = 10 give a wider range of data but also to check a Average 70 32 26 Average 0 70 32 pe possible general relationship for filtration vari22.08 21.68 15.77 20 83 1H 84 16 72 23.20 22.57 p = 1.31 ables found by Piros, Brusenback, and Dahl20.15 19,40 12,62 17 39 18 34 13.90 20.88 20.26 p = 0.861 strom (5, 6). 1 5 30 17 67 10.87 17.37 15.75 17.14 11.69 18.41 p 0.560 The data obtained from the experimental 19.58 17 51 13,09 19.87 18 30 19.99 14.10 20 82 Average d a n are tabulated in Table 11. There were duplicate determinations for three points to 3
1202
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 46, No. 6
FLOW THROUGH POROUS MEDIA AP
Table I I I. 70 23.20 22.08 20.85 20.15 18.41 17.37 20.34
U
p
= 1.31
p
= 0.861
p =
0.560
Average
Table I V .
32 22,57 21.68 20.26 19.40 17.14 17.67 19.79
20.34 17.86 15.52
Moisture Determination Averages for Experimental Plan
p =
= = u = c =
P
p
u
-
1.31 0.861 0.560 70 32 26
Averages of Three Moisture Determinations AP = 10 AP = 20 20.83 19.84 17.39 18.34 l5,30 15.75 20.82 19.87 19.99 19.58 14.10 13,09
= 2 inches
Air quantity = 25 CF/sq. ft. Aerosol OT concn. = 0 t o 0.02 wt. % U = 31.2 to 77.1 dynes/cm. = 0.65 to 1.79, (cp.)-' 1 /P Air rate = 13.10 to 20.21 CFM/sq. ft. Cake moisture = 17.14 to 23.20 wt. %
Average
26 16.72 15.77 13.90 12.62 11.69 10.87 13.59
= 10 inches Hg
L
Summary of Data
A tabulation of the data is found in Table VI. For convenience, straight line relationships were assumed between all combinations of the four factors; cake moisture, reciprocal of viscosity, surface tension, and air rate. Multiple regression procedures based on the method of least squares were used t o obtain Cake moisture = 27.4708
- 3.8936 (l/p)
+ 0.0147 ( u ) 0.1853 (air rate)
This type of approach was necessary because of variations in the data. Although these data were within the limits of random variations, the effects of surface tension and air rate could not be graphically separated without the aid of the above equation. A tabulation of the calculated and experimental cake moistures may be found in Table VI. For 11 of the 12 points, the average absolute deviation was 1.29%. For all 12 points, the average absolute deviation was 1.59%. With the exception of one point the deviations in cake moistures were 0.42 percentage points or less; the exception was 0.94 percentage points. The arithmetic sum of the deviations was +0.02. The square of the sum of the deviations has the following relationship to the multiple correlation coefficient, R, and the total sum of squares, SS: (Sum of deviations)2 = (1
- R*)(SS)
, 0
1.6
0.8
1l P ,
W.)-
Figure 3. Effect of Viscosity on Cake Moisture Parameters of Surface Tension
The difference between the averages for surface tensions of 32 and 26 dynes per cm. (6.20 percentage points) was conspicuously greater than between the averages for surface tensions of 70 and 32 dynes per cm. (0.55 percentage points) (Table 111). Therefore, the 2" of freedom for surface tension were divided into 1" of freedom for the contrast between surface tensions of 70 and 32 dynes per cm. and 1O of freedom for the contrast between surface tensions of 70 and 32 versus 26 dynes per cm. While the mean squares for both were significant, the difference between surface tensions of 70 and 32 dynes per cm. was relatively unimportant compared t o the contrast between surface tensions of 70 and 32 versus 26 dynes per cm. The analysis of variance is shown in Table V. The difference between the averages a t the two pressure levels was found to be highly significant. However, due to the relatively small decrease in cake moisture that accompanied a twofold increase in pressure drop across the cake, the extra expenditure that would be required t o obtain the higher pressure drop may not be worth while. The mean square for viscosity was also found to be highly significant. This verifies the general conclusions that could be drawn from the tabulations of Tables I11 and IV. General Equation. Whenever possible it is desirable to allow the factors to vary and then to reduce the results to equational form. Obviously this procedure allows the data to cover a wider range than would be possible with the same number of determinations using the classical method of varying one factor a t a time. A number of runs were made holding certain factors constant while others were allowed to vary as follows: June 1954
16' 0
I
I
I
0.8 1/ P ,
w.1-
I
I 1.6
'
Figure 4. Effect of Viscosity on Cake Moisture Parameters of Air Rate
Table V. Analysis of Variance of Data of Table I I Source of Total -. C
Variation
u = 70vs. 32 u =
70 & 32 us. 26
P
AP
Two Factor Interaction (C),
(PI
(AP) (c) ( A P ) : (PI ( A P ) , (4, ( P ) , and error Pooled interactions and error a
6
F Required Degrees at Sum of of Mean Squares Freedom Square F 5% 1% 242.2073 17 ... 168.3733 2 ... 0.9296 1 d.'si96 7.3' 4:75 9 : 3 3 1 167,4437 1309.2b 4 . 7 5 9 . 3 3 167.4437 69.4747 2 34.7374 271.6b 3 . 8 8 6 . 9 3 1 2.8243 2 2 . 1 6 4 75 9 . 3 3 2 8243
...
0,4334 0.3374 0,2750
4 2 2
0,1084 0.1687 0.1375
. ,.
..
.
..
.. ..
0.4892
4
0.1223
...
..
..
1.5350
12
0 1279
.
..
..
..
..
..
..
Significant a t 5% level. Significant a t 1% level.
INDUSTRIAL AND ENGINEERING CHEMISTRY
1203
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
22
I
I
10
0.8
0
Figure 5.
0 AP A
1.6
'
I/@, (CP.1-
Cake Moisture as Function of Reciprocal Viscosity
= 10 inches H g
AP = 20 inches Hg
Upper group: concentration of Aerosol OT, 0 t o 0.02 w t .
%
Lower group: concentration of Aerosol OT, 0.10 w t . except as noted by small numbers t o r i g h t of points
%
The R 2 is a measure of the effectiveness wit,h u-hich the equation predict,s t,he experimental cake moist,ure. For the equation given above, R Z = 0.967. This is an extremely high value when compared to the required value of 0.860 at, the 1% significance level. Some idea of the relative importance of the three variables, viscosity, surface tension, and air rate, may be obtained by noting the cont,ribut,ion of each t o the square of the multiple correlation coefficient, R Viscosity = 0,728 Surface tension = 0.046 Air rate = 0.193 R2 = 0 967
Thus it is seen that viscosit'y effects account for the major portion 01" the variat,ion in moisture content,, Lyhile the contribution of surface tension effects is relatively unimportant. This is in accord wit,h the theory presented earli~rin the introduction.
Table VI. Cake moisture, wt.
Data Used t o Derive Equation
70 = 27.4705
- rate) 3.8936
(l/p)
+ 0.0147
Cake R-loisture,
Run
l/p
u
2 13,18 15, 17 4 5,22 21 20
1.16 1.16 1.1G 1.46 1.79 1.79 1.79 0.89 0.76 0.76 0.76 0.65
70.8 47.9 31.8 69.0 68.3 46.0 31.2 73.E
25 24 6,8
7 5 , ~
51.5 34.2 77.1
Air Rare CFLf/Sq. Ft. 19.23 18.46 18.29 18.62 18.72 19.13 20.21 13.82 15.32 13 10 14.78 13.42
~ mt. yo _
Actual 20.85 19.30
20.20 19.18 18 4 1 17.53 17.18 22.22 23.20 23.07 22.57 23.18
A4veragedeviation = 1.59%. Average deviation for 11 values = 1.25%.
1204
Calcd.
20.43 20.24 20.03 19.35 18.04 17.63 17.21 22.53 22 78 22.84 22.28 23.59
(u)
- 0.1833 (air
Deviation _ Actual 7c - 0 42 2.01 0.94 4 87 - 0 23 1.14 0.17 0.85 -0.37 2 01 0.08 0.46 0.41 0.07 0.31 1.40 - 0 42 1.81 -0.23 1.00 -0.29 1.28 0 41 1.77
Figures 3 and 4, which are plot,s of cake moisture versus the reciprocal of viscosity with parameters of surface tension and air rate, respectively, are graphical representations of the effects of these variables upon cake moisture. These curves m r e obtained by using the equation present,ed abore. The variation of the paramet,ers is necessarily linear because linear rclationships betn-een all variables yl'ere assumed in deriving the equation. Again cake moisture decreases with a decrease in viscosit in surface tension and an increase in air rate. The results indicat,e the quant,itative influence of viscor-it,g and surface tension on final nioijture content. The actual moi,:ture content is also a function of size distribution, orientation, physical characteristics of t,he solids as well as other operat,ing conditions for each material filtered. Hoa r, the relative influerice of surface tension and viscosity on inert solids should be similar t,o that encountered in this study. Effect of Conc e n t r a t i o n of W e t t i n g Agent. For wetting agenh in general, the curve of surface tension as a function of wetting agent concentration starts v-it,li a large negativc slope a n d t h e n aaymptoticall y approaches some value of surface tension. T h e r e fore, for conc(wtrations g r e a t (lr 0 0.2 0.4 than a particular W T . yo AEROSOL OT IN 8OLUTlOh ADDED 10 FEED COAL value the further Figure 6. Cake Moisture as Funcreduction in surtion of Concentration of Aerosol OT face tension of the solut#ionis negligible. When AeroI I I I sol O T was employed at coiiceiitrations somewhat greater than 0.02 wt'. %, there n-as a sharp drop in cake moisture but only a siiiall decrease i n s u r face tension. This e f f e c t w a s first not,ed in t h e analysis of vari0 0.2 0.4 V O L . yc TEROITOL CW I N SOLUTION A D D E D T O CEED COAL ance, Table V, w h i c h shov-s a Figure 7. Cake Moisture as Funcvery large mean tion of Concentration of Tergitol C W square value for the comparison of cake moktures at surface teiisioii o€ 70 and :32 versus 26 dyncs per cin. and a relatively small nieaii square for the comparison at ,surface tensions of 70 and 32 dj-nes per em. Figure 5 , which is a plot of cnlre moist,ure versus rceiprocal viscosity, includes dat,a taken over a wide range of -ierosol OT concentrations. The data fall into t,xo definite groups; Ihc u p p c ~which includes data for which relatively low concentmiations of Aerosol OT were used, aiid the lover for which higher concentrations were used. Since the effect is the same at both pressure levels, it is unlikely the result of a change in surface tension, but rather it is caused by some kind of surface reaction involving the wetting agent.
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
Vo1. 46,No. 6
FLOW THROUGH POROUS MEDIA For the present work it was necessary to alter their factor as follows. The air rate, C F M / s q . ft., was employed a t standard conditions of 29.92 inches Hg and 32’ F. Unless the square root of AP were used, separate parameters for each pressure level were obtained. This one-half power of AP can be justified only if the air flovi is in the fully turbulent state. Apparently this was the case although it has been generally conceded that fluid flow through filter cakes is usually in the laminar or transition zones. The solids used in the present work contained fewer slimes than those used by Piros et al., along with a coarser size distribution. Since the earlier work was
U v)
>
ft‘
\
5 wl
14
5
2
0 WT,
yo AEROSOL
0.9
qnTEROITOL
014
0.2
0 VOL.
CW I N SOLUTION ADDED TO FEED COAL
0.4
OT I N SOLUTION ADDED TO FEED COAL
Figure 8. Air Rate through Filter Cake as Function of Concentration of Aerosol OT
This apparent function of the concentration of the wetting agent is clearly seen in Figures 6 and 7 , which relate cake moisture to the concentrations of Aerosol OT and Tergitol CW. These curves also indicate that this special effect is not peculiar t o Aerosol OT but may be common to wetting agents in general. The air rate through the filter cake is also affected by an increase in the concentrations of the wetting agent. Figures 8 and 9, which relate air rate to the concentrations of Aerosol OT and Tergitol CW, show that as the concentration of wetting agent is increased, the air rate passes through a definite minimum. Therefore, for each wetting agent there is a definite narrow concentration range that will yield the minimum cake moisture and the minimum air rate. This high concentration of wetting agent possibly causes a regular progression of the air-water interface through the cake. In the oil production field it has long been known that if the more permeable zones are allowed to denude too rapidly along with a drop in reservoir pressure, dissolved gas will come out of solution and occupy the majority of void volume in these sand pores. Such sands have a very low permeability to oil, and consequently, it is possible to block off the lower permeability oil sands from production. If the latter sands must deliver oil t o the more permeable but gas filled sands in order t o reach the well bore, the rate of production may be far below efficient operation and with a much higher loss in reservoir energy in the form of increased gas rates. It is possible that when low or zero concentrations of wetting agent are used, the air channels through the larger openings in the cake and isolates “pockets” of moisture. Subsequent passage of air through the cake removes very little of this moisture, because the permeability of the air filled pores is very low with respect to water. This hypothesis is strengthened by the sharp drop in air rate a t the critical wetting agent concentration that first produces the low final moisture content. General Relationship. Piros, Brusenback, and Dahlstrom (6) obtained a general relationship between total cake moisture and the multiple factor
multiple factor, l / p , was included. Surface tension was not included as it was shown earlier t o have very little effect,. The result,ing new correlating factor was ( 6 )
All 46 runs were included in Figure 10 which shows the relationship between total cake moisture and the new correlating factor. As would be expected from previous results the data are divided into two parameters. The higher moisture content parameter includes eoncentrations of 0 to 0.02 wt. % ’ Aerosol OT and 0 to 0.1 vol. % Tergitol CW, while the lower moisture content parameter includes the higher concentrations of wetting agent. Applications of various simplifications of this correlating factor t o industrial problems have proved to be very useful. INDUSTRIAL APPLICATIONS
Coal, as mined, contains varying quantities of gangue material. At the coal preparation plant this heavy gangue material is separated from the coal by suitable beneficiation methode. As most methods utilize a hydraulic medium, the operator is faced with the problem of removing the water from the coal product.
g e4
$
d3 eo ‘16
Y
4 -I
Q:
c
le
0
c
8
0
100
Figure I O .
This factor is purely empirical and was derived by a mtionalization of the variables involved. The equations presented in the introduction are intended to help justify the use of these variables. June 1954
200
300
Cake Moisture as Function of General Correlating Factor
0 AP =
10 inches H g A AP = 20 Inches H g Upper parameter. Low wetting agent concn. Lower parameter. High wetting agent concn.
INDUSTRIAL AND ENGINEERING CHEMISTRY
1205
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
February March April May June ?July
4
August September
-
The results obtained with coal should also apply in a general sense t o any other insoluble solid. It may also be possible to lower cake moistures effectively by washing the partially dried cake with a relatively concentrated solution of wetting agent. This solution n-ould be collected separately and recycled.
r
-
-
SUMMARY
-m E 8
The effects of suiface tension and viscosity upon the moisture content of he-coal filter cakes were investigated using a laboratory batch vacuum filter Data were presented to show that, when using aka-o. . I April sol OT as a wetting agent, the effect of surface tenMaya W June* sion for concentrations of wetting agent not greater 9 July 1 X than 0.02 vvt. % n-as statistically significant but relaAugust 1 0 September tively minor in magnitude. I t is doubtful that the October November use of small amounts of wetting agent for cake moisDecember6 i ture control would be economical. However, fol relatively high concentrations of wetting agents o February (Aerosol OT and Tergitol CW) the cake moisture March L April dropped sharply and the air rate passed through a $Mayc 1 I I /minimum. Viscosity was considerably more effective than surface tension in reducing the moisture content of a finecoal filter cake when using relatively low concentraFigure 11. Total Moisture as Function of M o n t h of Year tions of wetting~-agent. An equation for final cakt, a D a t a o n J u n e a n d J u l y n o t sufficient t o bestatistically significant. moisture as a function of reciprocal viscosity, surface b New t h e r m a l dryer started December 1952. tension, and air rate obtained from multiple regresN e w t h e r m a l dryer s h u t d o w n M a y 1,1953. sion procedures on the experimental data serves to quantitatively point up the much greater significance There are two major reasons for maximum economic water of viscosity over surface tension. removal from the coal. First, the consumer buys B.t.u.'s and A general relationship is presented that relates filter cake moiaan increase in moisture not only reduces boiler efficiency but also ture of the material used t o a multiple factor. This type of i n c r e a s e s c o s t s p e r B.t.u. Secondly, during the winter months the temperature of the Table V I I . Data Tabulaltion plant circulating water may T e t t i n g Agent drop to 40" F., and thus, if a Cake Slurry C F / C F M ,' X i i l t i p l e Moist, 9s. Factor Run Temp. Concn., sq. car of cold wet coal is shipped XO. F. AP AP 0 L u Wt. % Ft. Ft. x 10-3 at. % I.r in below f r e e z i n g w e a t h e r , 1.55 22.90 25.7 12.98 6 40 ... ... fi5 . 8 serious freezing of the coal 1.55 23,45 25.4 13.86 05.3 8 40 5.42 15.98 23 40 AeroioiOT olio 1.55 25.3 64 3 may easily occur. 1.31 23.20 25.3 15.32 77.1 50 36 1 . 3 1 23.07 24.1 25 Aerosoi OT 0 : 065 13.10 73.6 50 By maintaining the plant 1.31 25.4 14.78 24 50 Aerosol OT 0.02 26.3 22.57 circulating water a t summer 6.96 (6.6 1.31 25.6 16.72 26 Aerosol OT 0.10 50 1.12 8H (1 2 4 . 8 12.16 21.63 7 60 .. temperatures or higher, the 8fi 6 1.12 2 4 . 8 16.48 9 22.80 60 .. 0.861 24.2 19.23 20.85 111.8 2 80 moisture content of all the 0.861 llO.ti 24.6 17.22 19.47 80 Aeroboi OT 0 :665 13 19.70 0.861 110.0 24.4 19.12 coal would be maintained at a 18 80 Aerosol OT 0,005 110.1 0.861 24.8 17.94 20.63 80 Aerosol O T 0.02 15 lower consistent level. The 115.4 0.861 25.9 18.64 19.88 80 Aerosol OT 0.02 17 117 1 0,861 25.7 1 0 . 5 8 14.82 80 Aerosol OT 0.06 37 sensible heat added t o the coal 110.6 11.32 24.2 0,861 15.43 80 Aerosol OT 0.09 39 111.8 during t h e winter m o n t h s 0.861 8.65 13.97 24.9 80 AerosolOT 0.10 14 109.6 0.861 8.93 13.84 24.6 80 Aerosol OT 0.10 16 should greatly alleviate the 114.6 0.861 12.33 24.9 11.22 80 Aerosol OT 0.15 40 113.6 0.861 12.78 24.6 15.54 80 Aerosol OT 0 , 4 0 38 freezing problem. 141,2 19.18 2 4 . 8 18.62 0.685 4 100 ... ... 172 3 0.560 18.19 24.6 18.36 Evidence of increased drain120 5 lii8 0 0.560 18.62 24.0 19.08 120 22 age in the larger sizes of coal 0.560 171.7 21 17.55 24.7 19.13 120 0,560 170.2 17.14 24.6 20.21 120 20 due t o an increase in water 0,560 175 7 6.84 26.0 120 11.69 19 1.31 7.79 102 6 24.0 22.08 temperature is given in Figure 50 28 1.31 6.28 100 1 24.0 21.68 50 Aeroboi O T 0 :02 29 11. which relates the moisture 1.31 4.34 99.1 23.4 15.77 50 Aerosol OT 0.10 27 153.9 23.2 19.20 0.861 20.15 80 30 content of one product coal t o 145.7 23.1 14.37 19.40 0.861 80 Aerosoi O T 0 : 02 32 7.43 153.4 24.2 12.62 0.861 80 $erosol OT 0.10 the months of the year (6). 31 244,3 0.560 2 4 . 1 21 .90 17.37 120 33 In this case the plant circulat230.3 0.560 23.5 22.62 17.67 120 .4ero&i OT 0 ;02 35 233.5 24.1 10.48 0,560 10.87 120 Serosol OT 0 , l O 34 ing water was taken from the 0.861 166.4 36.4 21.94 18.96 1 80 ... .., 36.28 277.8 46.8 0.861 18.30 10 ... 80 Ohio River which is known t o 0,861 214.8 47.6 21.82 19.13 11 80 ... 324.4 17.52 0.861 71.5 25.81 12 have a temperature varying 80 0.685 8 . 3 15.17 49.1 22.57 3 100 with the months of the year. 2 3 . 6 22.25 112.1 21.22 0.861 80 41 111.4 0.861 23.3 24.32 21.88 42 80 This same application should 111.1 2 3 . 3 17.40 0.861 20.52 80 43 121.1 0.861 18.00 24 4 13.12 80 44 be of interest in all minerals 113.6 24.5 7.26 0.861 14.80 45 80 14.43 113.4 0.861 24 3 9.39 80 beneficiation fields as a liquid 46 medium is g e n e r a l l y u s e d . K%er December
1-
s
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E
i
r-'
I
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.
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1206
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 46, No. 6
FLOW THROUGH POROUS MEDIA correlation should aid in predicting the most economical combination of filt,ration variables,
S
= void saturation or fraction of cake voids filled with wet-
S,
= effective saturation = voids containing wetting fluid in
S, X
=
u
=
u’
= =
ting fluid
ACKNOWLEDGMENT
The authors wish to gratefully thank R. L. Sutherland of the Truax-Traer Coal Co. for his helpful suggestions and for supplying the coal used in this investigation. A special thanks is given to Marjorie L. Sutherland, statistical consultant, whose help and guidance made possible a thorough analysis of the data.
e
p p’
= =
=
active flow divided by voids containing both fluids in active flow residual saturation; limiting value of S cake porosity, fraction voids time, minutes surface tension, 1iq.-air interface, dynes/cm. surface tension, liq.-air interface, lb.//ft. liquid viscosity, cp. liquid viscosity, lb.,/(ft.)(min.) or lb.m/(ft.)(sec.)
NOMENCLATURE
B I BL IOG RAPHY
Cake moisture = wt. yo total moisture .4ir rate = CFM/sq. ft. CF/sq. It. = cubic ft. of air/sq. ft. of filter area drawn through cake and measured a t std. conditions of 1 atm. and 32” F. CFMlsa. ft. = cubic ft. of air Der min./sa. ft. of filter area drawn thrbugh cake and measured a t std. conditions ( C F M l s q . ft.),,. = cubic ft. of air/min./sq. ft. of filter area a t av. vake prpssure and 32” F. Ct = filtration constant d = cake thickness, ft. lb., ft. = conversion factor = 32.2 gc 1b.f sec.2 lb., cu. ft. K = permeability, lb.1 set.* L = cake thickness, inches AP = pressure drop across filter cake, inches Hg. AP‘ = pressure drop across filter cake, lb.j/sq. ft. A p = difference between final and initial pressures in the vacuum system after correcting for leakage, inches Hg. R = multiple coirelation coefficient
(1) Brown and Associates, “Unit Operations,” pp. 21C-55, New York,
John Wiley & Sons, 1950.
(2) Brownell, L. E., and Katz, D. L., Chem. Eng. Progr., 43, 601
(1947). (3) Brusenback, R. A., “Relationship of Filtration Variables to Filter Cake Properties with Particular Reference to Cake Moisture Content,” Master’s thesis, Northwestern University, Evanston, Ill., 1952. (4) Gore, W. L., “Statistical Methods for Chemical Experimentation, New York, Interscience Publishers, 1952.‘ (5) Piros, R. J., Brusenback, R. A , , and Dahlstrom, D. 9.,Mining Engineering, 4, 1236-44, December 1952. ( 6 ) Silverblatt, C. E., Dahlstrom, D. A., “Economic Dewatering of Coal,” Annual Joint Fuels Conference, A.I.1W.E.-A.S.M.E. Chicago, Ill., October 29-30, 1953. (7) Snedecor, G. W., Statistical Methods, p. 220, Ames, Iowa, Iowa State College Press, 1946. (8) Sutherland, Dr. Marjorie L., statistical consultant, Chicago, Ill., private communication, 1953. (9) Sutherland, R. L., chief combustion engineer, Truax-Traer Coal Co., Chicago, Ill., private communications, 1952, 1953. RECEIVED for review November 23, 1953.
ACCEPTEDMarch 26, 1954.
Residual Equilibrium Saturation of Porous Media H.S . D O M B R O W S K l l
AND
L. E. B R O W N E L L
U N I V E R S I T Y OF M I C H I G A N , A N N A R B O R . M I C H .
A general correlation is presented t o predict t h e capillary retention of wetting fluids by porous media. T h e fraction of voids filled with t h e wetting fluid under equilibrium conditions is termed t h e residual equilibrium saturation. T h e correlation takes into account bed permeability and depth, liquid density, surface tension, and contact angle and t h e desaturating driving forces of gravity, centrifugal force, and pressure gradient of air as a displacing fluid. Static and dynamic end effects are incorporated i n t h e correlation.
1
N EARLIER studies ( 4 ) a general correlation was presented for the simultaneous flow of two fluid phases through porous media. Each fluid was treated as a single phase with modificatjlons for the effect of one fluid on the other. One fluid normally wets the solid (flows adjacent to the solid) and prevents contact of the other fluid with the solid. The second fluid flows through the remaining void space and is contacted by the first fluid rather than the solid. If the voids are completely filled with the first or “wetting” fluid, the porous medium is said to be “saturated” with the first fluid. If the voids are only partly filled with the wetting fluid, the fraction of voids filled with wetting fluid is termed the saturation. If, in the simultaneous flow of two fluid phases the flow of the 1
Present address, E. I. du P o n t de Semours & Co., Kewport, Del.
June 1954
wetting fluid to a porous medium is stopped, the flow of this fluid from this medium continues until an equilibrium value of saturation is reached; this is termed the residual equilibrium saturation. 811 the wetting fluid does not flow from the medium because capillary forces retain the wetting fluid in the smaller interstices of the porous medium. The determination of this residual equilibrium saturation is important as it can be used to predict the oil held in porous oil sands, the residual moisture in cakes from filters and centrifuges, and the holdup in packed towers. It can also be used to predict the relationships for the simultaneous flow of two homogeneous phases through porous media. This article describes the different variables and their influence on residual equilibrium saturation. Static and dynamic end effects are taken into consideration in addition to end effect-free porous beds.
INDUSTRIAL AND ENGINEERING CHEMISTRY
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