-Studies inFiltration111. Derivation of General Filtration Equations B. F. RUTH University of Minnesota, Rlinneapolis, Rlinn.
I+
FEIXCFILTER (3-FOOT DIbMETER, 2-FOOT FACE) FOR LIMESLUDGE
N THE first two papers of this series (6, 7 ) evidence was presented which indicated that a simple parabolic equation of the type, (V C)p = K (0 O0), was able to describe the constant-pressure filtration behavior of a wide range of materials, independently of their degree of compressibility, homogeneity, concentration, etc., with an accuracy sufficient for engineering purposes. Upon the basis of this evidence mas formulated an empirical generalization termed the “fundamental axiom of constant-pressure filtration.” In a strict sense it would perhaps have been better t o have described this generalization as a maxim rather than an axiom, since the latter word refers to a self-evident truth, incapable of, or requiring no proof. For this reason it will hereafter be referred to as the “fundamental maxim of filtration,” using maxim in the sense that it refers to a rule or precept sanctioned by experience. The principal contribution of this maxim to filtration theory lies, not in the fact that it recognizes the approximately parabolic shape of all time-volume discharge curves secured in constant-pressure filtration, but rather in the fact that it postulates a perfect parabolic behavior for all such tests. That the filtration data of some workers approaches perfect parabolic behavior quite closely has long been recognized as a phenomenon encountered in the filtration of granular noncompressible precipitates, under conditions usually associated with negligible septum resistance (12). The failure to recognize that perfect parabolic behavior is a universal property of all time-volume discharge curves has been largely due t o two factors:
+
curve t o be expected for actual substances which are usually more or less compressible in nature. The significance of Sperry’s axiom (9) to the effect that the origin of the timevolume coordinate system may be taken at any point along the curve was not recognized, and as a result niany experimenters continued t o employ analytical methods which were valid only when septum resistance was negligible. The result of this procedure in most instances was to strengthen the belief that compressibility was a phenomenon which caused the time-volume discharge of actual substances to depart more or less from the perfect parabolic form expected for ideal materials. For this reason the principal task of the filtration theorist has heretofore been the reconciliation of the approximate with the perfect parabolic data. The resulting com~~
This paper establishes the mathematical background of a treatment of constantrate and constant-pressure filtration that has been used with complete satisfaction for several years. Although the experimental verification of its various phases must necessarily extend through a series of papers, this fairly complete discussion at the outset is essential to familiarize the reader with the notation and equations to be used throughout. Thereafter it should serve as a valuable compendium for future reference. Mathematical simplicity in relation to the engineering requirements of analysis and design has been the prime objective of this development, yet neither accuracy nor correctness of concept has been sacrificed in its attainment.
1. Inadequate means of analyzing a parabola ahen the origin of the coordinate axes is displaced from the vertex (1. e., septum resistance is no longer negligible). 2 . The presence in the literature of a large amount of data which, because of experimental error, actually departs considerably from parabolic form
Although Sperry (8, 9) as early as 1917 had shown both from theoretical considerations and actual tests that the form of the time-volume discharge curve for an ideal noncompressibie substance was that of a perfect parabola with the coijrdinate axes displaced from the vertex, there existed no means of predicting the shape of the time-volume discharge 708
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J l : 3 l < , 1935
io9
plexity of such treatments a r e d o u b t l e s s responsible for t h e w i d e s p r e a d idea that the ineclianisrn of filtration is one of such extreme variability t,hat the e n g i n e e r niay perhaps never hope to find law and o r d e r i n it.s iperation. fluaminatiun in thesc .aboratories of the tinievolume discharge curves ri a wide variety of maierials has shown t.liat, i n d e r p r o p e r l y conrolled conditions, these :urvcs always exhibit, :ur pr is in g ly perfect )araholic form, and that, vheii a n y c o n s i s t e n t leparture from this fonn s observed, it can usually be traced to one of it n u m i ~ rof ixperiniental errors. These errors have charactcrist,ic effects ipon the sensitive analytical plot of &,'dVarid V , so that the ,xperienced operator can tell by inspefition what type of 'mor has acted, irrespective of the sonree of the data. Illitsrations of the typical action of soriie of these factors will lie :iven at a later time. The fundamental maxim tltils possesses a twofold signifiance: It postulates not only that the time-volume discharge urve is parabolic, biit further that any cniisisteiit,departure :om this form is to be considered as indicative of t,he undesirble aot.ion of some extraneous factor, the ident,ity of which lay be established through elimination by trial and error. Ince attention is t,tius directed t o such factors, it is easy to fentif?.and, in most cases, to eliininatc them. The fundaniental rnasim may, in emsequence, be condeled as n criterion enabling us t.o judge whether or riot n ltration h a s been properly performed. flaving estalrlishod s validity, it becomes possible io dismiss from consideralion
The fundaniental tnarirri of filtration with its essential pustulate that filter septum resistance may lie exprosseed i o terms d filtrate volumes h a s served as the basis ftrr the treatment of constant-pressuret: filtration. In contrast, i t is found just as essential to the sirnple treatrnerrt of constant-rate filtraiioii that. this c x ~ i cept be discarded and sepiurn rcsistariee: be expressed as a pressure drop. General fdtratiem equations appliceble to all pre61t eoneent.rationsare fbrmufated for a nuniber of typcs of filtrations, and various factors influencing their accuracy are discussed. I t is shown that from theoreiical considerations w-e may expect to find that the true shape of the time-volume discharge curve will depart slightly from a perfect parabola in the initial stages of filtration. -
_I_~~___~_._~
-
any unsatisfactory data after the cause has once been ideniified. A great s i m p l i f i c a t i o n in the niat.hernatical trtlatnient of filtration d a t a beeoisies t i t m i x ? pussiirle. A considerable portion of this siniiriifimtinn is dlle to t l i c f a c t t,Il&t, since t h e o b s e r v e d stages of filtratioii are ai:airately rqiresent,ed by an cqutition ,of the feptum resistance as a filtrate volume C. The analysis of the data is easily made. The plot of V against P is easily extrapolated t o a definite value of C. Resistance a is then given by
a
at any desired value of P along the curve.
Constant Pressure-Volume Gradient The form of the equation for constant pressure-volume gradient mill depend upon the assumptions made. If we assume that the filtration is started by the instantaneous application of some low pressure, P I , of such a value that PI = G‘C and that thereafter P = G’ (V C), the derivation leads to a simple equation. Substituting in Equation 1:
+
P
P r1 +
-Z-dV =E= A de r
Wa
r* +
721
A
+
G’(V C) (V C)psa A ( l - ms)
+
(43) Plotting the values of a as obtained from Equation 43 against P , we should be able to determine a as a function of P.
Equation for Constant Pressure-Time Gradient If pressure is increased by small and equal increments a t either equal intervals of time or volume, the result is to compound a large number of short constant-pressure filtrations into a single test in which the individual short sections of time-volume parabolas merge into a single curve as the size of the pressure increment is decreased. In the limit, a linear relation between P and 0 or P and V is obtained. Expressing this relation as P = GO for the first case and substituting into Equation 1:
Separating the variables and integrating:
For a noncompressible prefilt, the plot of P against B is a straight line cutting the P axis a t a positive value of PI; which, extrapolated to the time axis, yields a negative intercept equivalent to 80. The plot of V against 0 is a straight line passing through the origin. For compressible sludges the plot of P or V against 0 is a curve concave t o the time axis. In practice it is quite difficult to begin the filtration by an application of the correct initial pressure, PI, such that the ratio (P - P1)/V is equal to the ratio P/(V C). Moreover, for a compressible prefilt, if cloth resistance remains constant, the equivalent value of C will decrease as P increases. Hence a more valid and general derivation for a constant pressurevolume gradient filtration will follow from the assumption of constant septum resistance and a constant ratio P/V = G’. If the cloth resistance is taken as T I , the pressure Z dV drop through it a t any time is ; i3 , Substituting into Equation 1:
+
Z d-V -P A do r
Simplifying: (44)
Simplifying:
(45)
Integrating :
Pz = r2
Z dV P-rl-A d8 Vpsa A ( l - ms)
Z dV
- G ’ V - r , -A- do Vpsa A(1 - ms)
For 0 me may substitute P/G:
For a noncompressible preflt the plot of V against P is a straight line. Because of the septum resistance present z dV for a short period a t the start of the test when V = 0, will not increase as rapidly as it will later, so that the ex-
Z psaV -A 2 (1 - ms)
+ rlZ - In V = G‘8 + constant A
If the experimental limits V = 0,O = 0, and V = V , 0 = 8, are substituted, it is seen that an infinite period of time must.
INDUSTRIAL A S D ENGINEERING CHEhIISTRY
122
apparently elapse before V can attain any finite value. This is quite understandable because, with the resistance r1 already present a t the start of filtration, an infinitesimal pressure, d p , will require an infinite period of time to filter the required volume dV. Experimentally the conditions a t the start of the filtration never fulfill the theoretical conditions; hence, the above equation should be integrated between limits over which the conditions are satisfied. If we integrate between the limits VI to I- and O1 to 8:
z z 2
psa
(V - VI)
V = G’(e - el) + rlZ In VI
Theoretically, the time O1 t o filter volume VIis infinite; practically, owing t o the necessity of applying a finite small value of P a t the start, it is negligibly small. If VI is taken a t unit volume:
where O0 is the experimentally observed time to filter unit volume. The only field t o which this and the previous type of filtration have practical application is to that of compressible prefilts, and for these materials, because of their high resistance, r1 In V is negligible above low pressures. a is then given by - ms) A9GG’z(6- &)(I - ms) a = -A 2 G , (6 - &)(l (49) Z (V - 1)ps ’Or Z(P - G ) ~ s and may be expressed in the form given by Equation 17 by plotting (a - ao)against the pressure logarithmically.
Choice of Units to Be Used in Filtration Calculations The author has found neither homogeneous English nor C. g. s. units to be desirable in filtration calculations. Pure English units are undesirable because they yield inconveniently large values for specific resistance. Moreover, the advantageous relationships possessed by the metric system are lost, thereby increasing the work of calculation and the chance for error many fold. On the other hand, the C. g. s. system is undesirable because it also yields large values for the constants and requires one t o think in terms of square and cubic centimeters when working with areas and volumes perhaps a million times greater. If one views the problem from the proper angle, it becomes apparent that it makes no difference what hybrid mixture of units may be chosen, as long as they facilitate laboratory measurement and calculation, and help in the easy visualization of the processes involved. Any question as to the dimensions of quantities involved always reduces to one of the dimensions of specific resistance. If the dimensions of this quantity are thoroughly understood, the trouble a t once disappears. The filtration engineer desiring to apply the foregoing equations may feel quite free t o define l / a in any fashion he pleases, such as rate of filtrate flow in cubic feet per second or per hour, of a liquid of one English unit viscosity, through one pound of solids deposited as a layer in a cylinder one square foot in area, produced per unit of pressure drop, when the total pressure drop across the layer is P pounds per square foot. The dimensions of a may be most clearly perceived by writing Equation 10 in the form:
- z $ T + = W =T 2g (50)
-1 = _ Z d_V ( V + C ) P ~ A d e A ( l - ms) P a
If a is to be converted from one system of units to another, the form
VOL. 27, S O . 6
may be conveniently utilized. By ( I / ~ ) P is indicated the fact that l l a is dependent upon pressure drop P through the layer of solids. Throughout this work the units minutes, liters, kilograms, square decimeters, and pounds per square inch have been used. Minutes as the unit of time seemed a logical choice because filtrations are rarely accomplished in less than thiq time, and are usually completed n-ithin a reasonable number of minutes. Since pressures are almost universally measured in pounds per square inch in this country, any other unit would involve a conversion factor. Although pounds and cubic feet are the standard units of weight and volume, it is just these quantities that are most advantageouqly expressed in metric units. In the metric system the centimeter is inconveniently small, and the meter inconveniently large. Therefore the square decimeter of area, the cubic decimeter or liter of volume, and the kilogram of weight are chosen. With this system of units the specific permeability, l / a , of a filtered material is defined as the rate of flow in liters per minute of filtrate having a viscosity of one centipoise (viscosity relative to water at 20” C.) through a weight of one kilogram of solids enclosed in a cylinder of one square decimeter area, produced per pound per square inch of pressure drop, when the total pressure drop across this layer is P pounds per square inch. With this system of units no very large or very small quantities occur. Moreover, the specific Tesistances are of reasonable magnitude. Thus the resistance of a filter cloth will vary from 0.5 to 50 units, meaning that from 2 to 0.02 liter per minute will flow through 1 sq. dm. under a pressure head of 1 pound per square inch. The specific resistance of chemical precipitates of medium compressibility will vary from 100 to 10,000 units, depending upon particle size. The resistance of extremely compressible precipitates will vary from 5000 to 500,000 units, depending upon the pressure. In following papers data will often be presented in terms of liters volume and seconds time of filtration. When this is true it should be borne in mind that the value of K obtained from a plot of dB/dV in seconds per liter against V in liters must always be multiplied by 60 before calculating the value of a.
Conversion of Units of Specific Resistance Although the above system of units has been found very convenient, and industrial application would probably benefit froin the use of the meter stick, the necessity of converting specific resistances from one system of units to another. whether homogeneous or not, is bound to arise. Because workers in these laboratories have experienced some difficulty a t times in performing such transformations, several typical cases mill be given to illustrate a procedure which has been found fairly reliable. 1 ZdVW1 K F , the symbols refer to In an equation such as = quantities expressed in the definite units of minutes, relative viscosity, liters, kilograms, etc., which for lack of a better name will be termed “convenient.” Bearing in mind the conversion factor for each unit, me may convert to the English system:
-1ff Eng. units
=
2 X 0.000672 9/9.29
d6 X 60
w$922?4
-1
1SDCSTNI.iL AND ENGINEERING CHEMISTRY
JUNE, 1935
z
dl‘_ 1.I’ = - _ -1_ _ = _ _ -1 1,922,000 d d8 9 P
:.
Q
convenient units 1,922,000 a in Eng. units = 1,922,000 X CI in “convenient filtration
units” The crmversion t o metric units
iq
similar :
723
PO = back-pressure generated by flowmeter or recording device
PA = hydrostatic pressure of filtrate at any point in the cake, varying from P, in the frame to 0 a t filtrate exit Pd = pressure stress exerted a t deposition zone by impact of fluid as solids are brought to rest -4 = filtering area, sq. dm. B = filtering time, minutes, t o secure volume
1 C . g. 5. units =
v or
v
CI
z x 0.01 -4 x
loo
I
x 1000 1 w x 1000 de x 60 1 A x loo
-I
dT‘
e, p x 453 1 X (2.54)2
(52)
-~1
.‘
.
Z d- V _ r-n=1 - convenient units 4220 2 d8 A P 4220 a a in C. g. s. units = 4200 X a in filtration units
The inetric factor is seen to be l j G 3 of the English factor because the unit weight of solids is taken as one gram instead of one pound.
Conclusion A survey of the notation and equations to be used throughout subsequent papers has heen presented in a single unit, in the belief t h a t t’hisplan best, serves the interests of the reader who desires to become familiar with the subject. I n presenting a t this time t h e general equations to be used iIi treatment of filtration data, a number of important questions have been ignored. Cnt,il such questions as the validity of applying Poiseuille’s law to fluid flow through clean or thinly covered filter septa, the actual behavior of the initial stage of flltration, etc., are settled, the presentation of a complete mathematical treatment must admittedly be premature. The author believes, however, t h a t as the papers to fo’ilow appear from time to time these objections will eventually be withdrawn. Meanwhile, the difficulties of both exposition and interpretation are diminished by the availabilit’yof bhis paper. Nomenclature ‘I’ = filtrate volume, liters
v
= prefilt volume, liters
C = filtrate volume equivalent to resist,ance a t point V = 0 C b = filtrate volume equivalent to filter base resistance C L = filtrate volume retained by leads
vc
=
prefilt volume equivalent to resistance a t point
v
=
o
B = weight of prefilt, kg. B, = weight of prefilt equivalent to d o t h resistance p
= density of filtrate, kg./cu. dm.
of prefilt d = sp. gr. of a precipitat>e 6’ = density of wet filter cake, kg. of solid liquid/cu. dm. 6 = density of wet filter cake, kg. of dry solids/cu. dm. of wet cake V- = weight of solids on a filter septum or in prefilt, kg. W b = weight of solids equivalent to volume of filtrate Cb,kg. s = weight fraction of solids in a solute-free prefilt S, = grams total dried matter/gram of volatile solvent S , = grams of soluble solids/gram of volatile solvent S = grams of resistance-forming solids/gram of volatile solvent = SI - S2 S’ = grams of precipitate/gram of volatile solvent S“ = grams of iilter aid/gram of volatile solvent m = ratio of weight of a wet, solute-free filter cake to its dry weight *If = ratio of weight of a met filter cake containing soluble solids to its dry weight T I ‘ , r l ” , T * , . . . = resistance per unit area of cloth, throttling layer, filter cake, etc. = effective resistance of septum during flitration PJ = total filtration pressure, pounds/square inch P = (PJ- Ph), pressure drop t,hrough filter ca,ke, varying from 0 a t deposition zone t.o PI a t filtrate exit (identical with pressure stress) PI = pressure drop through septum, pounds/square inch Pn = pressure drop through cake, pounds/square inch u = density
+
= hypothetical filtering time to secure volume C or
vc
90 = calculated tinie correction t o observed atering time Z = relative viscosity, centipoises a = average specific resistance of solids through which a pressure drop, P , exists aOl y, 8, n = coefficients in expression of a as a. function of pressure stress ao‘, y ’ , $’, n’ = coefficients in expression of a, as a function of pressure when obtained from a differential plot a,, = specific resistance at any point in a filter cake under uniform pressure stress P v = pore volume, cc./gram, or cu. dm./kg. K , J, H = symbols representing a group of factors whose product is constant G, G‘ = constant gradients of pressure-time and pressurevolume rate = constant gradient of volume-time
Acknowledgment The author wishes to thank G. H. Montillon and R. E. Nontonna for their interest and help during the development of the material presented in this paper.
Literature Cited (1) Donald, >I. B., and Hunneman, R. D., Trans. Inst. Chem. Engrs. (London), 1, 97 (1923). (2) Gilse, J. P. 11. van, Ginneken, P. J. H. van, and Waterman, H.L., J . SOC.Chem. Ind., 49, 4441‘ (1930). (3) Pickard, J. A., Ind. Chemist, 4, 186 (1928). (4) Ruth, B. F., and Kempe, L. H., unpublished work on “Theory of Filtration in Continuous Rotary Filter Presses.” ( 5 ) Ruth, B. F., and Meyette, C. L., unpublished work on “Flow of Fluids through Porous Media.” (6) Ruth, B. F., Montillon, G. H.. and Montonna, R. E., IND.Esct. CHEM.,25, 76 (1933). (7) Ihid., 25, 153 (1933). (8) Sperry, D. R., Chem. & M e t . Eng., 15, 198 (1916). (9) Ihid., 17, 161 (1917) (10) Tattersfield, Master’s Thesis, hlass. Inst. Tech., 1922. (11) Underwood, A. J. V., J. SOC.Chem. I d . , 47, 325T (1928). (12) Underwood, A. J. V., Trans. Inst. Chem. Engrs. (London), 4, 19 (1926). (13) Univ. Minnesota, unpublished work in these laboratories on “Compressibility of Chemical Precipitates and Granular Solids.” (14) Walker, W. H., Lewis, W. K., and Mcrldams, W. H., ‘‘Principles of Chemical Engineering,” New York, McGraw-HillBook Co., 1923 and 1927. (15) Webber, H. C., and Hershey, R. L., IND.ENG.CHEM.,18, 341 (1926). of an oversight, the symbol p has been utilized in the nomenclature of both the limited and general cases as representing density of filtrate, whereas in the latter case it is p ’ , density of volatile solute-free solvent that should have been used. If p is retained in the equations as filtrate density, a new symbol representing the ratio p ’ / p (ratio of weight of solvent evaporated to original weight of filtrate when a sample of filtrate is evaporated to dryness) must be introduced. The product f ’ p should then be substituted for p wherever this latter symbol is used in the nomenclature of the general case. In the limited case dealing with solutefree suspensions,f‘ is unity, so that f’p, p’, and p are all identical in value. . ~ U T H O R ’ SCORRECTION. Because
r,
RECEIVED June 14, 1934. A portion of this paper was abstracted from part of a thesis submitted by B. F. R u t h in December, 1931, t o the faculty of the Graduate School, University of Minnesota, in partial fulfilment of the requirements for the degree oi doctor of philosophy.
