FILTRATION EQUIPMENT
THEORY Flow rate o f filtrate into a cake is not equal to the flow rate out. Ignoring this fact m a y cause significant errors, especially when dealing with thick slurries by F. M. TILLER and C . J. HUANG, University of Houston, Houston, Tex.
SINGLE
AND MULTIPLE PHASE fluid flow through particulate beds occurs on a universal scale in both nature and mandirected operations. I n spite of its basic importance in such widely divergent fields as filtration, determination of specific surface area, flow of underground water, petroleum reservoirs, mud flow in drilling of wells, cake washing, aerosol filtration involving particulate clouds, particle agglomeration, percolation, two-phase flow in packed towers, flow through ceramics and porous sintered metals, thickening and clarification, flocculation of colloids, and drainage and drying of packed beds, relatively little emphasis has been placed on experimental and theoretical investigations in comparison to such fields as mass and heat transfer. Several obstacles have impeded progress in the filtration and flow through porous media field. First, the complexity of even the most simple model discourages investigators from attempting solutions. Second, the nature of particles and precipitates forming filter beds is such that reproducibility becomes a serious problem. The researcher finds that he must not only deal with the difficult flow problem but must also contend with the vagaries of crystal formation and flocculation. Third, young men entering the teaching field have tended to follow along more developed paths in such fields as thermodynamics, heat transmission, and mass transfer. I n spite of the relatively little emphasis that has been placed on the field, fairly good progress has been made in the past few years. The development of the compression-permeability cell in 1946 by Ruth (10) and the A.1.Ch.E. Filtration Symposium in 1952 mark turning points in the development of theoretical and experimental studies. A small but important body of knowledge created in the past 7 to 8 years can serve as a springboard for more substantial progress in basic research and industrial filter design.
Many basic scientific areas contribute to the developing, but still incomplete, theory of liquid and gas flow through beds of solids. Fluid dynamics of very slow motion, basic ideas of friction, electrokinetic phenomena, statistical studies of beds of particles, flocculation, capillarity, and compressibility all command substantial roles in studies of laws governing flow through porous media. Although empiricism still plays a preponderant part in formulation of useful analytical expressions, nevertheless, theoretical advances are beginning to make headway in elucidation of basic phenomena. Approaches to the theory of liquid flow through compressible, porous media have evolved from varying viewpoints. Differential equations involving empirical resistance and compressibility coefficients have been the main source of formulas for practical, computational purposes. Equations have been developed using both the Darcy permeability coefficient and the Ruth specific filtration resistance. I n either case, the coefficients must be related experimentally to the solid compressive pressure. Although flow is almost always viscous in flow through filter beds, no reliable theoretical relation between permeability and porosity as determined
Last month I&EC presented a group of three articles on Centrifugation Equipment. The article, “Theory of Filtration Equipment,” begins the second set of three from the ACS St. Louis Meeting on Selection and Evaluation of Chemical Equipment. Next month, I &EC will conclude this presentation with a series on Crystallization Equipment. Each group covers theory, design, and applications of the equipment discussed.
by compressive pressure has been developed. An open field remains in which a fundamental breakthrough would be highly desirable. Fundamental to flow through filter beds is the insufficiently studied phenomenon of compressibility. The nature of the cake, void distribution, flow rates, and pressure drop are all intimately related to cake compression. Little more than a meager accumulation of experimental data is available to the researcher. Since compressibility is a complex process involving basic ideas of friction, transmission of forces, migration of fines, and orientation and shape of particles, progress along theoretical lines is expected to be rather slow. Both permeability and compressibility are closely related to the nature and arrangement of the particles within the bed. I n attempts to develop a mechanistic model, flow has been visualized as occurring in idealized beds of rigid spherical particles. Unfortunately this line of attack has not been followed with sufficient vigor to yield important results. Numerical solution of the Navier-Stokes equation subject to complex boundary conditions would lead to improved understanding of the relation between porosity and permeability. Treatment of flow through the complex network of flow channels can be undertaken from a statistical viewpoint; and although a statistical approach ( 7 I) has been pursued at some length, the results have not as yet received widespread attention. Researchers in flow through compressible porous media principally aim at discovery of relations between volume of throughput, over-all pressure differential, rate of flow, average bed porosity, mass of solids deposited, and time. I n formulating expressions relating the average and integrated values of the variables, the investigator inevitably must concern himself with interior and local phenomena. He is led to the development of models (9) of particulate VOL. 53, NO. 7
JULY 1961
529
SUBSCRIPT
I
/
--
beds and accompanying patterns of flow of fluids through the interstices of the solids. Determination of laws governing interaction between particles and frictional flow in causing compaction is basic to any fundamental investigation. While the investigator aims at theoretical analysis and basic understanding of the flow process, empiricism and descriptive analysis must continue as important components of investigations in the field of flow through porous media at the present time. Frictional Drag and Solid Compressive Pressure
When suspended solids are deposited during cake filtration, liquid flows through the interstices of the com-
Fs
SUBSCRIPT i
pressible bed in the direction of decreasing hydraulic pressure gradient. The solids are retained by a screen, cloth, porous metal, or other solid bed known as the septum or filter medium. The solids forming the cake are compact and dry at the medium whereas the surface layer is in a wet and soupy condition. The porosity is a minimum at the point of contact between the cake and medium where x = 0, Figure 1, and a maximum a t the surface ( x = L ) where the liquid enters. The frictional drag of the liquid passing through the voids is responsible for the decreasing porosity. I n Figure 1 the particles forming the filter bed are illustrated in a manner which indicates that the porosity decreases as the liquid passes through the solids and approaches the septum. I n
dF dF dF dF
Figure 2 A . the liquid is shown as it passes frictionally along the particles in the porous media. The drag on each particle is communicated to the next particle; and, consequently, the net solid compressive pressure increases as the medium is approached, thereby accounting for the decreasinq porosity. As flow in most filtrations is viscous: theoretically the drag and hydraulic pressure drop could be calculated by proper use of the Navier-Stokes equations. I n practice, knowledge of the geometrical configuration of the particles within the bed is insufficient to permit calculation based upon solution of the basic differential equations. Nevertheless, it is possible to relate the forces of compression to decrease of the liquid pressure. As a first basic postulate, it is assumed that the particles are in point contact (Figure 2 ) and that the liquid completely bathes each particle and communicates the liquid pressure uniformly in a direction along a plane perpendicular to the direction of flow. Under this assumption, the hydraulic pressure p z is effective over the entire cross section of the cake as the area of contact is negligible. The net force on the total mass within the differential distance, dx, is given by Net force
=
F,
+ dF, + A($* + d j Z ) Fs - A d P ,
(1)
This force equals the product of rhe mass within dx and the acceleration. The differential mass includes both the mass of liquid e,pA dx and the mass of solid (1 - e.)psA dx. Although the solid actually moves in the cake toward the medium, the acceleration is negligible. The average velocity of the liquid is generally less than 0.004 foot per second (1 g.p.m. per square foot with a porosity of 0.5), and the acceleration is so small as to be negligible. Consequently Equation 1 can be written as dF,
+ A dp,
= 0
(2)
A pseudo-solid compressive pressure is defined as p . = F,/A yielding dP,
+ dPs
=
0
(3)
Integration gives
where p is the applied filtration pressure at the surface of the cake. I n actual cakes, there is a small area of contact A, = cA between particles, and solid pressure could be defined as FJA,. This definition serves no useful purpose excepting as it relates to deformation of the particles. However, for area rather than point contact, Equation 2 would have to be modified to the form Figure 2.
530
Compressive force due to frictional drag
INDUSTRIAL AND ENGINEERING
CHEMISTRY
dF,
4-( A - A,)d$,
= 0
(5)
O n integration, this equation would yield
1.0
which reduces to Equation 4 when A,/A = 0. At the present stage of filtration theory, it appears that assumption of A , / A = 0 is justified.
08
Porosity US a Function of Solid Pressure
The porosity of a porous bed can be determined as a function of applied pressure in a consolidometer (75) or a compression-permeability cell (10). Data are available for numerous substances (1,3,6,8,12, 13,16). Unfortunately, the same chemical substance can give widely different E, us. pa relations dependent upon particle size and shape. In general, the science of filtration has not advanced to the point where the porosity of a cake can be correlated with physical properties of the substance. Consequently, the investigator must obtain his own data for each new situation. For moderately compressible solids (72, 73, 75, 76) in the range of up to 100 p.s.i., the porosity can be represented by the power function ex =
e0pa-x
P* > P i
0.6
P
0.4
0.2 0
0.2
0.4
Hydraulic Pressure and Porosity Variation in a Cake
The hydraulic pressure in a filter cake is affected by the flow rate q,, filtration resistance a,, and porosity e, in addition to other parameters. In Figure 3, the variation of p,/p as a function of x / L is illustrated for a number of substances (79). Tiller and Cooper (78) showed that the following relation approximates the hydraulic pressure variation
0.8
I .o
x L
(7)
where pi is some low pressure in the approximate range of 0.1 to 1.0 p.s.i. Below pi the porosity reaches a limiting value ei which represents the porosity of a cake laid down under virtually zero pressure. The exponent X generally varies from 0-0.05. The majority of porosities lie in the range of 0.4-0.9. In order to relate the porosity in a filter cake to the porosity in the compaction cell, as a second basic postulate, it is assumed that the porosities in the cake and cell are equal when the solid compressive pressure in the filter, as indirectly calculated by 16, = p - p., equals the compaction pressure in the cell. While this highly important assumption has never been directly confirmed, it is supported by indirect evidence and calculations (72, 73).
0.6
Figure 3.
Hydraulic pressure distribution
where n is the compressibility coefficient and j3 equals the exponent in the relation 1 -
€2
=
BPd3
pII > pi
(9)
The exponent ,8 generally varies from 0 to 0.25. Because of the relatively small change in E, for most substances in the 0 to 100 p.s.i. range, it is possible to rcpresent both e, as in Equation 7 and (1 - e,) as power functions of pa. The average porosity eav of a filter bed is important as it determines the liquid content of the discharged cake, Based on Equation 7, it was shown (8) that the average porosity could be approximated by 1 - n
are sufficient for evaluation of eavo, po, and p in Equation 11. For thick slurries and solids which have large changes in porosity with pressure, Equation 11 must be used with caution. The porosity variation in a compressible bed can be determined indirectly from experimental values of the hydraulic pressure (72, 73, 79) or by calculation based on permeability-compression cell data (72, 20). For a wide variety of circumstances, the porosity is a function of x / L alone. Combination of Equations 7 and 8 yields the following power function relation
The exponent in Equation 12 can vary quite widely. Variation of Flow Rate with Bed Thickness
where eavo is an experimeutally determined value of the average porosity a t pressure De. Equation 11 is useful in that minimal data are needed for its application. One average porosity from an actual filtration plus B. us. p8 data obtained from a consolidometer
For many years, it was assumed that the flow rate q1 = dv/d@ was constant throughout a filter cake. It was argued that the rate of flow into the surface of the cake must equal the rate of flow out a t the medium corrected for the liquid deposited in the cake. Generally the VOL. 53, NO. 7
JULY 1961
531
A
E
c
0
D
X
Figure 4. Illustration of relationship of ratio of q2 qi to q1 - qi. Area
-
represented by
Led
xdez i s proportional
to qx - qi. Area ABC is proportional to
q1
-
4_i = s2-
qd.
mass of dry solids per unit area w and the filtrate volume per unit area u were related by Tiller (75). Ze,=-
ps
1 - ms
the cake, there would be a uniform porosity equal to E%, the value in an infinitesimal surface layer where p a = 0. The area ABDO in Figure 4 represents the volume, E ~ Lof, liquid per unit area which would remain in the cake if there were no compression. The area CBDO under the e z us. x curve, equal to f ezdx, represents the volume of liquid retained in the pores of the cak?; and, consequently, the area ABC to the left of the porosity curve, equal to Jxde,, corresponds to the liquid which has been squeezed from the cake. The ratio of the flow rate qt a t the surface of the cake to the flow rate q1 at the medium is given by Tiller and Cooper (77). q1
1
- ms
= 1
- ( m , - m)s 1 - ms
(14)
where m is the average value of the ratio of the mass of wet to mass of dry cake and m, is the value in a n infinitesimal surface layer. The quantity m is related to the porosity by
where ealz is the average porosity between € 1 snd e,. The values of ei and mi are fixed, while ear and m depend upon the total pressure and the slurry concentration. Therefore, q z / q l is a function of total pressure and slurry concentration as well as x / L . I n Figure 5, a plot of qJq1 for cement material (12, 79) is shown as a function of w,/w and s at a pressure of 71.5 p.s.i. The limiting curve marked ( q ~ q J L i m is equivalent to a plot of (qz - q i ) / (41 qi) as obtained from Equat:on 16. When s = l/mi,qi = 0 ; and the ratio (4% - qi)/(ql qi) is identical with q 2 / q l . As s decreases, the variation of the flow rate with distance becomes less pronounced. T h e area under q 2 / q 1 us. w,/w curves equals J , which is defined by Equation 23.
-
-
Local Filtralion Resiskaeaee
The average filtration resistance is generally defined by the equation
LY
v
The average ratio m of mass of wet to mass of dry cake changes rapidly during the first minute of filtration-insufficient data are available to indicate if there may be examples in which the variation is prolonged-and then becomes essentially constant. When m is constant, dwld8 is directly proportional to dv/d8. However, during short rotary filtration, assumption of constant m is not warranted ; and conventional methods are highly questionable. In Figure 4 the porosity variation as a function of x and 0 is shown. As time increases, the porosity a t each point in the solid decreases; and, consequently, liquid is squeezed out of each element of volume. Since flow is additive, the flow rate necessarily increases as the medium is approached. If there were no compressive action within
10
As s increases and (1 - mds) approaches zero, the ratio qJq1 dccreases and approaches 7ero. When s is zero, the ratio is unity and the flow rate is constant throughout the cake. The explanation of the effect of slurry concentration on flow rate variation is related to the ratio of the rate a t which liquid is squeezed from the cake to the flow rate a t the medium. If the slurry is dilute, a relatively large amount of liquid flows through the cake compared to the amount which is squeezed out by compression. O n the other hand, when the slurry is concentrated, the cake thickness increases rapidly ; and the liquid squeezed from the cake may be large compared with the flow rate in the surface layer. I n the limiting case, where the slurry has a concentration s = l / m t , a solid is being pumped rather than a slurry; and the flow of free liquid qi becomes zero, thereby leading to a ratio of zero for q d / q l . I t has been shown (79) that the flow rates were related by the expression
and the local resistance a , is defined by
See Figure 1 for a diagram of the cake. On the basis of experimental data, it is assumed that a z is a function of p a alone (it may be affected by flocculation and slurry concentration) and is related to the hydraulic pressure in a cake in a manner similar to the local porosity 8,. Values of a2 are obtained from a compression-permeability cell (70) or are indirectly calculated from average values ( 7 4 ) . I t is a fundamental postulate that aZ has identical values in a cake and permeability cell when p p , in the cake equals ,b, of the cell. Grace ( 3 ) and Shirato and Okamura (73) have presented the most extensive data for filtration resistances in the literature. For moderately compressible materials (13, 16) up to 100 p.s.i., the point filtration resistance can be related empirically to the compressive pressure by
-
08 = 06
ff*
qx -
q,
04
02
'0
02
04
06
08
IO
%/W
Figure 5. Effect of slurry concentration s on ratio of q z / q i for cement material (1 2) at a pressure of 71.4 p.s.i.
532
where the integral in the denominator equals the total area to the left of the e2 us. x / L curve, and the integral in the numerator equals the area between E, and e;. Substituting for qi/ql in Equation 16 permits solution for qJq1
e = 1 - [%2] (i)]-e[ 1 -
INDUSTRIAL AND ENGINEERING CHEMISTRY
J
E'
- 6av (17)
ffopa"
P s > Pi
= ffi= ffopi.
ps < P i
(20)
(21)
where it is assumed that 01% approaches a limiting value ai at some low pressure p$. Accurate determination of ad is beset with the same difficulties as accurate determination of ed. Virtually no reliable data are available for values of porosity and filtration resistance of highly compressible material a t very low pressures. Since some calculations are quite sensitive to the values of ad and ed, considerable ,caution must frequently be used (2).
The value of a. (73, 15) decreases with increasing slurry concentration, possibly owing to flocculation or changing cake structure.
Average Filtration Resistance
Equation 19 can be rearranged into the form
where g& = p q l R m represents the pressure required to overcome the resistance of the medium. The factor J (79) is defined by
tices can be deleterious if a break-up of large particles is brought about by excessive shear action. One of the largest impediments to successful use of filtration theory lies in the vulnerability of a to varying process conditions. Frequently in the past, the variation of a with pl has been neglected. However, it has been demonstrated (77) that as long as the filter media offers an appreciable portion of the total re-
sistance, a will vary with q1 and will be less than its limiting value during the initial stage of filtration. Calculations for a dilute slurry (s = 0.003) of talc filtered a t a constant pressure of 40 p.s.i. indicated that approximately 7 minutes were required for a to reach 90% of its limiting value. I n this example, the ratio of the medium resistance to ultimate filtration resistance was about 0.03. Grace ( 5 ) indicated
,Types of Filtration Processes For purposes of calculation, filtration processes may be classified
(74)according to the relation of the pressure and flow rate to time. Generally, the pumping mechanism determines flow characteristics and serves as a basis for division into the following categories :
The term q x / q l can be obtained from Equation 17. Since q x / q l is less than 1, J wil! also be less than unity. If qJq1 is a unique function of w,/w or x / L , the value of J will be independent of time. Generally, for longer filtrations in which pl is small compared to p , J will be a function of p and s in accord with Tiller and Cooper’s findings (77). The following definition of a is valid when E, p x / p , and q 2 / q l are functions of x / L alone.
where aR is the resistance with s = 0 and J = 1. This definition differs from those used in the past (3, 8, 70, 75) by inclusion of the factor J . Since J depends upon the slurry concentration and p l depends upon the exit rate of flow q 1 = du/d8, the filtration resistance can be written in the general functional form 01
= f ( p , s, dvldo)
(25)
In addition to the variables indicated in Equation 25, a may be affected by flocculation, possible changes in cake structure due to the action of concentrated slurries, and, in short filtrations, the failure to reach equilibrium values equal to those obtained in compression-permeability cells. Generally, these effects would be expected to be a maximum in short rotary filtrations where slurries are quite concentrated. Flocculation and particle size can be affected enormously by the history of the pre-filt. Changes in temperature, pH, agitation, and reaction time can cause variations in the nature of the precipitate and the difficulty of filterability. Pumping prac-
Constant pressure filtration Constant rate filtration Variable rate-variable pressure filtration Centrifugal pump Constant rate pump with bypass control 0 Stepped pressure by manual methods Flow rate us. pressure characteristics for the four types of filtration are illustrated in Figure 6. Arrows on the curves point in the direction of increasing time. The constant pressure and constant rate curves are represented by vertical and horizontal lines, respectively. When a centrifugal pump is used, the rate us. pressure curve follows a downward trend dependent upon the pump characteristics. When a bypass is added to a constant rate pump, two possibilities occur. If a pressure relief valve is placed in the bypass, the filtration will follow a constant rate line until a predetermined pressure is reached, and then it will continue at constant pressure. If an ordinary valve is placed in the bypass, there will be a continuous bleed from the filter. As the pressure builds up, the recycle will increase; and the filtration will follow a variable pressure-variable rate operation. The maximum pressure will depend upon the resistance placed in the bypass and can be determined by allowing the full capacity of the pump to flow through the bypass. I n laboratory testing, stepped pressure operation is desirable in that any desired relation of pressure to rate or time can be obtained by a skillful operator. By manually adjusting air pressure in a small press, it is possible to simulate various pumping conditions.
5 CONSTANT
-
RATE
.- - - - - -- - - - ----_
PRESSURE
i
+ARROWS POINT IN DIRECTION OF INCREASING TINE
\I1
PRESSURE
I
Figure 6. Relation of flow rate to pressure for different methods of operation
VOL. 53, NO. 7
JULY 1961
533
Variable Pressure-Variable Rate Filtration Example 1. Talc is to be filtered in a press with a centrifugal pump having the characteristics (74)given in the first two columns of the following tabulation.
I Rate of Disrharge, G.P.M./Sq.Ft.
Pressure, P.S.I.
Pressure at Medium, J.1 P.S.I.
+P.S.I. -
0 0.975 1.95 2.725 3.9 4.a75 5.0 5.55
43.5
43.5 42.6 38.4 32.3 24.1 10.3 5.0 0
0 0.1 0.2 0.3 0.4 0.5 0.54 0.57
I
+,
+I,
41.6
36.4 29.6 20.2 5.4 0
-
Other quantities necessary to the solution are 0.001 lb. mass/(ft.) (sec.) 62.4 lb. mass/cu. ft. = 0.003 fraction solids in slurry
p
=
p
=
s
R , = (2.0)1010 ft.-' = 2.5 a, = 8.66(10'a)$,o.606
m
Since s is quite small, J will be close to unity. Changing v to gal./sq. ft., up; q1 to gal./(sq. ft.) (min.), qo and @ to p.s.i., $ in Equation 32 leads to
I
- 0.0075)
(32.2) (1
(144) ($ -
(0.001) (0.003) (62.4) [(7.484:(60)]
(8.66)
I
(1 - 0.506)
PRESSURE DROP ACROSS CAKE
V-p,
(34)
IThe pressure at the medium is related to
I
1J1)o.~~~
*'
0001 2-
qg
32.2 (7.48) (60)
qe by
((2) 144)(10'0)
= 9.79 qg
Figure 7. Graph for solution of Examples 1 and 2 (35)
Correcting the discharge pressure for $1 yields the final column in the table containing the pump discharge characteristics. Placing Equation 34 in logarithmic form, there results
In Figure 7, the pump characteristics are plotted on logarithmic paper in the form of log qg us. log ($ $1). A series of straight lines with slope 0.494 corresponding to Equation 36
-
LBS./ (SQ.IN.)
are drawn in Figure 7, and their intersections with the pumI curve give pressure drop us. rate of flow values. Total volumq can be obtained from the intercepts along line $ - $1 = which equal 1.9O/vQ. For example, with # - $1 = 30 p.s.i. the pump pressure is found to be 32.9 p.s.i.; and the intercep is 0.05 giving a value of up = 37.5 gal./sq. ft. Other value of vu can be found and the time calculated by using Equatioi 29.
~~
that the ratio R,/a should in general be less than 0.1. The ratio R m / a may be replaced by a mass w, of dry solids per square foot equivalent to the resistance of the medium. The equivalent thickness of the medium is given by R, mRm wo = OrP*(l
-
E-
OcPav
€*")
+
P,(l
-
e*")
(26)
where psv = pea" p e (1 Since a increases and e." decreases with time, the equivalent thickness decreases with time. An order of magnitude of the
5 34
minimum equivalent thickness of cake can be calculated with R,/a = 0.03, ps = 150, and E," = 0.5. Using these values yields Le, = 0.005 inches. Maximum values of Le, near the start of filtration may easily approach 1 inch or more. In general, design should be aimed at the lowest possible values of Leq. However, if Le, is too small at the start of filtration, the medium may be too open, and cloudy filtrates may result. Good design should yield values of Le, in the range of 0.01 to 0.06 inches (4).
INDUSTRIAL AND ENGINEERING CHEMISTRY
Applying Filtration Theory
The design and operation of filters involve many aspects including mechanical features, optimization of cost or production rate, selection of media, precoating and addition of fiiter aids, washing and drying, and pumping equipment. The concentration, nature of the slurry, and product desired dictate the type of equipment which should be employed. Basic to the design of most filtrations is the relation of the pressure drop, filtrate volume, and time. This section will deal with this aspect of filtration.
FE/THEORY Equation 22 can be rearranged to include the factor J in the following manner : qzdw, = J q l w =
In any filtration operation, it is necessary to obtain the integral of dpb/crZ either by numerical methods or by integration of empirical expressions for aZ. If crz is related to p s by Equations 20 and 21, integration yields
P
Solving for q1 and taking logarithms yields log 41 = (1
- n)
log ( P - P I )
+
Combined with pump characteristics in the form of log q us. log p , Equation 33 leads to a rapid solution for q 1 us. p and u. The use of the method will be illustrated in following example 1,
Equation 27 relates the total mass of dry cake w to the rate of flow q1 and the pressure p . I n the solution of Equation 27, either q 1 must be replaced in terms of dw/dO or w must be replaced by Equation 8. Following the latter procedure yields
Integrating and combining terms yields Constant Pressure Filtration
If the pressure is constant and q1 = dv/dO, Equation 28 can be placed in the form
where q1 = dv/dO. I t is assumed m is constant in the derivation of equation 28, and, consequently, short rotary filtrations are not amenable to the methods which will be developed. I n order to solve this equation for a general case, it is necessary to relate J to p and s and q 1 to p. The caIcuIation of J has been discussed by Tiller and Shirato (20). The relationship between q1 and p will depend upon the discharge characteristics of the pumping mechanism. In general, it is desired to determine the relationship between the volume of filtrate, time, and pressure. The first step in finding the desired values requires that the volume be found as a function of q 1 and p using Equation 28. Once the v us. q1 curve has been determined, the time may be obtained from the following integral
If n is 0.5 or less, the term in pi can be neglected for total pressures above 10 p d . However, if n is as large as 0.7, the pi term must be included. When n becomes large, the power function approximation is less accurate, and numerical methods should be employed for the integration.
where g& = pR,dv/dO. The factor J will be constant since both p and s are constant. Substituting for the value of the integral as given in Equation 31 yields dv
2
-
go(1 - ms) ( p - ~pcr~( 1 n)
- p i ) ’ - ” - nPi’-n Jv
(38)
Variable Pressure-Variable Rate Filtration
where p must be used in pounds per square foot. The general solution of u us. 6 can be obtained in a similar manner to the method employed for variable pressure-variable rate filtration if pi is neglected. The pump curve of Figure 7 is replaced by a curve of pressure drop us. rate or p - P I = p - (pR,/gc)du/d6 us. dv/dO. Then lines of slope (1 - n) are constructed, and volume and time are obtained in an identical manner. The procedure illustrated in the
Jahreis (7) developed an ingenious solution for variable pressure-variable rate filtration in which he took advantage of the power function relation between cr and p. If p r is neglected in Equation 31 and the result is substituted in Equation 28, one obtains
,Constant Pressure Filtration Example 2. The material of Example 1 is to be filtered at constant pressure of 43.5 p.s.i. A new tabulation of qg = iup/dO,I), and I)’, is made, thus B
Rate of Filtration, G.P.M. (Sq. Ft.)
4.6 4.0 3.0 2.0 1 .o 0.5 0.3 0.1
Pressure at PBSSV?t?
Medium
&I,
J., P.S.I.
P.S.I.
43.5 43.5 43.5 43.5 43.5 43.5 43.5 43.5
43.5 39.0 29.3 19.5 9.8 4.9 2.9 1 .o
*-
*I,
P.S.I.
0 4.4 14.2 24.0 33.7 38.6 40.6 42.5
+
A plot of dv,/dO = qs us. - +I has been placed on Figure 7 . The total pressure is constant a t 43.5 p.s.i., and the pressure at the medium is determined by Equation 35. A procedure identical with the one developed in Example 1 may be employed for obtaining u p and the time. When the pressure drop is 10 p.s.i. across the cake (43.5- 10 = 33.5 p.s.i. across the medium), conditions of filtration are represented by line A . The volume of filtrate is obtained from the intercept and is found to be 0.14 gal./sq. ft. The rate of filtration drops from 4.6 to 3.8 gal./(sq. ft.) (min.) during the filtration of this volume of liquid. The time is roughly computed to be about 0.04 minute. When line B is reached, the rate drops to 0.08 gal./(sq. ft.) (min.); and the pressure drop across the cake reaches a value of 34 p.s.i. The volume filtered is 1 .O gal./sq. ft., and the total time approximates 5 minutes. The filtration rapidly passes through the region involving the top portion of the curve of rate us. pressure drop across the cake.
+
VOL. 53, NO. 7
JULY 1961
535
same doubling of pressure for a material like talc with n = 0.506 would result in a reduction of time of 30% rather than 50%. Constant Rate Filtration
If the rate is constant, Equation 29 becomes
Figure 8. Effect of compressibility on pressure vs. time curves for constant rate filtration
( p - pi)l-n - npil-m gdl
-Concentrated
where p must be used in pounds per square foot. Compressible materials with large values of n undergo less increase of rate with increasing pressure than relatively incompressible materials because of the effect of the power 1 - n. Doubling the pressure for an incompressible material ( n = 0) filtered in accord with Equation 39 would require half as long for the same volume of filtrate. The
p
= 0.001 Ib. mass/(ft.)
s
= = =
p
n
a. =
p = B = pa =
(sec.)
0.4 62.4 0.332 1.01 X 10'2 (using p.s.i.) 0.095 0.32 166 lb. mass/cu. ft.
In order to use Equation 39, it is necessary to obtain J from Figure 9 and m from Equation 1 5 . Calculating from Equation 10 gives I
-B
(1
'" - ms) a 0 ( l - n)Jq12e
(41)
Slurries
Example 3. A 40y0 slurry of Hong Kong pink kaolin is to be filtered at a constant pressure of 50 p.s.i. in a filter with negligible medium resistance. Find the relation between time and volume if the parameters have the following values:
Ea>. =
=
where p must be used in pounds per square foot. This equation is valid for p > p,. In many filtrations, the term np,(l-.) can be neglected. The factor J is not, in general, constant and should be included in calculations relating pressure to time. In Figure 8, the effect of compressibility is illustrated with a plot of pressure against a relative time scale based on varying values of n. For n < 0.1, a straight line can be used to approximate the data over a considerable pressure range. When n increases to 0.3, it is difficult to represent the pressure us. time as a linear function beyond 5 to 10 p.s.i. When n reaches values near
RELATIVE TIME
example is of value principally for short filtrations where p l remains a substantial fraction of the total pressure. Caution must be observed as boih m and J may vary, particularly if the slurry is concentrated. If p l is small and the filtration is lengthy, Equation 38 can be integrated to give
(40)
v = qle
Combined with Equations 28 and 31, there results
-
"->
I - n
(sq. ft.) (sec.). Find the relation between pressure and time. Letting p1 and pi be zero in Equation 41 yields
where p has been replaced by to emphasize the units as pounds per square inch. Both m and J will vary with time since pressure is increasing. Since Equation 49 was developed on the assumption that (1 - ms) was constant, it is only an approximation for the present example. For a more exact procedure, formulas developed in a previous article (17) can be employed. Substituting in Equation 49
\--,
pP
= 1 - 0.32
(
1
0 095 -: X 0.668)
5Oo.0Q' = 0.602
where 8, is in minutes.
Rearranging
(45)
Calculating m m = 1
(0.602) = + (62.4) -(166) (0.398)
The value of J is 0.902. VI
=
(46)
(47)
(0.001) (0.4) (62.4) (1.01) (1012) (0.902) (0.668) (60)em
1.86(10-4)8,
Example 4. The Hong Kong pink kaolin of the previous example is to be filtered a t a constant rate of l . 2 ( l O - 5 ) cu.ft./
536
Tabu-
Substituting in Equation 39
2gc(l - ms) 144+b1-n pspa,J 1 -n e
- _ _ _2(32.2) _ ~( 1 - 0.612) (500.6@)(144) =
Values for m and J must be obtained as function of p . lating values yields
INDUSTRIAL AND ENGINEERING CHEMISTRY
* 20 40
60 80 100
fSV
0.635 0.610 0.595 0.584 0.575
m
(l-0.4m)
1.655 1.589 1.553 1.527 1.508
0.328 0.364 0.379 0.389 0.397
J
0.888 0.894 0.899 0.902 0.904
*0."8
7.40 11.86 15.41 18.67 21.68
8,
91.5 149.4 201.0 249.0 294.6
FE/THEORY
S
Figure 9. I vs. S at constant pressure for Hong Kong pink kaolin
0.7, the curves are characterized by an initial flat portion followed by a rapidly increasing slope. I n the range for n > 0.7, the curves have only qualitative value. Above n = 1, an infinite pressure is reached a t finite time (74) according to Equation 41. Practically, the equations indicate that for very compressible materials filtered a t a constant rate, there is a finite time a t which the pressure rises rapidly to large values. I n constant pressure or variable pressure-variable rate filtration, there is a corresponding point at which the rate drops to a very small value, and “blinding” is said to have occurred. Concentrated Slurries
It is necessary to know how J is related to p and s when concentrated slurries are involved. Shirato and Tiller (20) have demonstrated how J can be calculated in a rigorous manner when porosity and filtration resistances are available. A plot illustrating J as a function of p and s is shown for Hong Kong pink kaolin (73) in Figure 9. The correction factor decreases as s increases and is less affected by pressure. I n order to carry out calculations for this type of kaolin, it is necessary to have the relationship of 0 1 ~and E $ to p8’ As an approximation of the experimental data, the following relations may be used: =
1.01(1012)~,o.a3*
e2 =
1-
E=
0.6951,b.0.06*6 0.32~,0~og5
(42)
(43) (44)
The limiting values as #8 approaches zero are = 0.72, 014 = 0.85 (10l2)ft./lb. mass, and #< = 0.6 p.s.i. The valueof a. in Equation 42 may be affected by slurry concentration. Acknowledgment
The authors wish to thank the National Science Foundation for a grant which made preparation of this manuscript possible.
Nomenclature
P
= exponent in Equation 9, dimen-
A A,
€I
= porosity in infinitesimal surface
E:.
= porosity
€o
=
€1
=
ESP
=
= cross-sectional area, sq. ft. = area of contact between par-
ticles, sq. ft. €3 = constant defined in Equation 10, dimensions meaningless F, = solid compressive force, lb. force g, = conversion factor, poundal/ pound force, (lb. mass)(ft.)/ (lb. force) (sec.2) J = correction out factor for filtration resistance, ratio of average flow rate to rate a t medium, qav/ql, dimensionless L = cake thickness, feet Le, = thickness of cake equivalent to resistance of medium, feet = ratio of mass of wet to mass of dry cake, dimensionless = value of m in infinitesimal surface layer of cake = compressibility coefficient, Equation 20, dimensions meaningless = applied filtration pressure, lb. force/sq. ft. = low pressure below which az and eo are constant, lb. force/sq. ft. = solid compressive pressure a t distance x from medium, also total compressive pressure, lb. force/sq. ft. = hydraulic pressure a t distance x from medium, lb. force/sq. ft. = arbitrary pressure, lb. force/ sq. ft. = pressure a t interface of medium and cake, lb. force/sq. ft. = rate of flow a t interface of medium and cake, gal./(sq. ft.) (min.) = value of q2 in infinitesimal surface layer cake, cu. ft./(sq. ft.) (sec.) = rate of flow of liquid in cake a t distance x from medium, cu. ft./ (sq. ft .)(sec.) = value of qz a t interface of medium and cake, cu. ft./(sq. ft.) (sec.) = average valve of qz = medium resistance, l/ft. = fraction solids in slurry, dimensionless = volume of filtrate, cu. ft./sq. ft. = volume offiltrate, gal./sq. ft. = distance from medium, ft. = total mass of dry solids per unit area, lb. mass/sq. ft. = mass of solids equivalent to resistance of medium, lb. mass/ sq. ft. = mass of solids Der unit area in distance x frbm medium, lb. mass/sq. ft.
Greek Letters average specific resistance, ft./ lb. mass average value of a uncorrected for variation of flow rate value of specific resistance a t distance x from medium where solid compressive pressure is pB,ft./lb. mass constant in Equation 20, dimensions meaningless, pa in lb. force/sq. ft.
sions meaningless
€avo
e,vz
=
e
=
0,
=
h
=
$
= =
$1
layer, dimensionless at distance x from medium, dimensionless constant in Equation 7, dimensions meaningless porosity in layer adjacent to medium, dimension!ess average porosity, dimensionless value of E , , at pressure bo, dimensionless average porosity for cake lying between medium and distance x , dimensionless time. seconds time, minutes exponent in Equation 7 , dimensions meaningless applied pressure, lb. force/sq. in. pressure at interface of medium and cake, lb. force/sq. in.
Literature Cited (1) Coimbra, A. L., Ph.D., thesis, Vander-
bilt University, Nashville, Tenn., 1949. (2) Cooper, Harrison, M.S. thesis, University of Houston, Houston, Tex., 1958. (3) Grace, H. P., Chem. Eng. Progr. 49, 303, 367 (1953). (4) Grace. H. P.. “Art and Science of . ,Liquid ‘ Filtration,” 50th Anniversary Meeting of A.I.Ch.E., 1958. (5) Grace, H. P., A.l.Ch.E. Journal 2, 307, 316 (1956). (6) Hutto, F. B., Jr., Chem. Eng. Progr. 53, 328 (1957). (7) Jahreis,’Carl, “Role of Pumping Equipment in Liquid Filtration,” Atlanta Meeting, A.I.Ch.E., 1960. (8) Kottwitz, Frank, Boylan, D. R., A.1.Ch.E. Journal 4, 175 (1958). (9) Kozeny, J., Sitzer-Akad. Wiss. W e n , Math. naturw. Klasse 136, lla, 271 (1927). (10) Ruth, B. G., IND.ENC.CHEM.38, 564 (1946). (11) Scheidegger, A. E., ‘(Physics of FlowThrough Porous Media,” MacMillan, New York, 1957. (12) Shirato, Mompei, D. Eng. thesis, Nagoya University, Nagoya, Japan, 1960. (13) Shirato, Mompei, Okamura, S . , Chem. Eng. (Jafian) 19, 104,111, (1955); 20, 98, 678 (1956); 23, 11, 226 (1959). (14) Tiller, F. M., A.2.Ch.E. Journal 4, 170, (1958). (15’) Tiiler, F. M., Chem. Eng. Progr. 49, 467 (1953). (16 Zbid., 51, 282 (1955). (171 Tiller F. M., Cooper, Harrison, A.1.Ch.E: Journal 6, 595 (1960). (18) Tiller. F. M.. Coouer. Harrison, “The Role of Por’ositv ih Filtration V: Porosity Variation ik Filter Cakes,’; A.1.Ch.E. Journal, to be published. (19) Tiller, F. M., Shirato, Mompei, “The Role of Porosity in Filtration VI, New Definition of Filtration Resistance,” A.1.Ch.E. Journal, to be published. (20) Tiller, F. M., Shirato, Mompei, “The Role of Porosity in Filtration VII, Calculation of Flow Rate Variation in Filter Cakes,” A.1.Ch.E. Journal, to be published. RECEIVED for review May 10, 1961 ACCEPTED May 10, 1961 Divisioq of Industrial and Engineering Chemistry, 139th Meeting, ACS, St. Louis Mo., March 1961. VOL. 53. NO. 7
JULY 1961
537
