I - 0 L’STRLAI, S S D EAYGINEERINGCHEMISTRY
April, 1926
34 1
Some Practical Applications of the Lewis Filtration Equation’ By Harold C. Weber a n d R o b e r t L. Hershey I I E P B R T M E N T OF C H E M I C A L ENGINEERING,
I
l l l A S S A C H U S E T T S I N S T I T U T E OF
S RECENT years students a t the Boston Station of the
School of Chemical Engineering Practice have regularly run tests on the Sweetland filter presses of the Revere Sugar Refinery in Charlestown, Mass. The data gathered in these tests have been used to examine the validity of the filtration equations given by Walker, Lewis, and McAdams in their “Principles of Chemical Engineering.” The present paper is an exposition and a summary of the results of these examinations together with the conclusions to be drawn from this experience.
TECHNOLOGY, CAMBRIDGE,
MASS.
pressure or constant velocity conditions. B u t for nonrigid and compressible sludges it immediately breaks down, due t o the fact t h a t the case is not so simple as it appears. I n the filtration of slimy precipitates, as the pressure is increased the rate fails to increase anywhere near in proportion; in constant velocity runs the pressure starts low and remains such for a considerable period, finally building up t o a maximum pressure. The difficulty is t h a t alpha is not a constant. I n order t o overcome this difficulty, the assumption is made t h a t alpha is a power function of both pressure and velocity. In making this assumption, the following limitations are to be noted:
(1) That although alpha is a power function of the pressure, i t is independent of The derivation a n d i n t e g r a t i o n of the filtration the duration of that pressure. A l t h o u g h consideration The symbol used throughout e q u a t i o n s given by Walker, Lewis, a n d M c A d a m s in of the theoretical basis of this paper for the pressure ex“Principles of Chemical Engineering” is given. A n ponent is s. the above-mentioned equae x p e r i m e n t a l m e t h o d of e x a m i n i n g t h e s e equations o n ( 2 ) That t h e v e l o c i t y tions is without the scope c o m m e r c i a l a p p a r a t u s filtering defecated s u g a r solueffect is caused by scouring of this paper, it has seemed of small particles off from t i o n s is described, a n d representative r e s u l t s of a series advisable, since the equalarger aggregates t o plug up of e x p e r i m e n t s a r e t a b u l a t e d a n d discussed. I t i s small channels; once scoured, tions and their derivation concluded that, w i t h i n the accuracy of p l a n t m e a s u r e a cake will stay scoured; it is have not before been prements, t h e e q u a t i o n is generally valid, and that i t has therefore irreversible only for sented in any periodical, to mixed precipitates, such as its chief practical u s e i n t h i s t y p e of filtration in d e t e r give the best known derivak i e s e 1 g u h r and aluminium mining w h e t h e r a proper a m o u n t of filter cell i s being hydroxide, or the colloidal tion. It is that of Tattersused. carbon-granular carbon mixfield? and is as folloivs: tures encountered in the filtration of tars. With homoIn considering the flow of geneous sludges, such as carefluids through f i e s l u d g e s , the openings between particles are so small t h a t straight-line flow fully prepared aluminium hydroxide or ferric hydroxide, there holds, except possibly for extremely high velocity; even then the . should be no scouring effect. The symbol used here for this rate of flow is usually roughly proportional to the pressure or driv- power function is t . ing force. The general expression for rate of flow may then be The over-all resistance R’then becomes applied for the rate through a filter cake:
Derivation a n d I n t e g r a t i o n of E q u a t i o n s
n’here
P
= pressure or driving force
volume of filtrate over-all resistance presented t o flow 8 = time T; =
R
=
=
$ (similar -~
to I
=
”> R-
P
+
- R’ R” where K ’ = resistance of cake R“ = resistance of press; neglect temporarily CYL R‘ should equal -, where a is the coefficient of resistance, L is A the thickness of the cake a t any time, and A is the actual area of cloth through which filtration takes place. A , L , and P should all employ the same units; in this work square feet, feet, and pounds per square foot are used. Let v = the volume of sludge as laid on the filter per volume of filtrate; o is constant for any one suspension. Then L =
vv A
The expression for the general case of filtration then becomes d I’ P A2-t
This may be considered to be the fundamental basic general equation for filtration under any conditions. The term r is the constant of resistance or the specific resistance of any one particular sludge.
This equation will apply reasonably well t o rigid, noncompressible sludges. It will hold fairly well for either constant
KO claim can be made that this derivation is above criticism. What are perhaps the most exhaustive mathematical and theoretical studies of filtration have been made by Drew,3 and TT700dward and Edmunds.4 Drew shows that the basic assumption of the applicability of Poiseuille’s law is entirely valid. It is stated that v is a constant for any particular sludge; but this is not strictly true, for if the
Received February 6, 1926. Quantitative Study of the Filtration of a Compressible Sludge.” Master’s thesis, M. I . T., 1922.
“The Theory of Filtration,” Master’s thesis, M. I. T., 1923. “Quantitative Study of Factors Affecting Resistance of Cake,” Master’s thesis, M. I. T., 1922. ’
and
1
* “A
R’ =
LY
vv 7 A
8
Filter
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sludge is compressible, u must decrease with increasing pressure. Experiments6 indicate the compressive effect to be dependent not only on the pressure but also on the length of time the pressure is exerted. If, however, one includes this time effect in the mathematical sta-tement of the problem, a practicable solution becomes difficult if not impossible. It is important to note that the term P represents the total pressure drop through the cake-i. e., from the surface of the cloth next the cake to the outside of the cake. It should be obvious that a direct measurement of this quantity is difficult. If, on the other hand, the total pressure on the press be used, the mathematical analysis is greatly complicated; although the difficulties would not be insurmountable if the correction for press resistance were constant. Unfortunately this is not so, for, as Tattersfield remarks, the first filtration seems to be “in and not on the cloth.” The best method of attack seems to be to measure the total pressure drop and then to correct this value by the method to be outlined. I n this way a satisfactory value for P, the drop through the cake, can be obtained. To define the pressure term in the equation as the total pressure on the press promises a result of a forbidding complexity. The problem is thus reduced to one of experimental procedure. Integration must be performed on this general equation, if it is to be of practical value. This is generally done for two sets of conditions-i. e., for those of constant pressure and those of constant rate. These integrations are given by Tattersfield as follows: Case A : Constant Rate Conditions = constant
Case B: Constant Pressure Conditions Rewriting the basic equation,
Integrating,
When V = 0, 8
=
:.
0
A little algebraic manipulation will serve to show Equations l and 2 to be of the following forms, respectively: . t’ = K.pl-5 (la) By taking logarithms of both sides, (la)and (2a) become log 1’
log
=
log K
v = log K’
+ (1 l+t + (-J
S)
log P
(3)
e
(4
log
Similarly, a plot of values of log V versus log 0 from a constant pressure run should give a straight line of slope Also, t cannot ordinarily have values of less than 2 + t zero; for this would mean that scouring-i. e., plugging of the cake-would decrease with increasing flow, a condition conceivable in only one case. As this case is precisely that encountered in the filtrations here described, it will be discussed fully later. Experimental
It will suffice for a preliminary examination of the validity of these equations to determine whether the above-mentioned straight lines can be obtained and whether the values of their slopes are within the field of possibility. The actual experimentation must then be concerned with the measurement of V , P, and 8. It is important to understand clearly the exact meanings of these symbols. V is the volume of the filtrate which has passed through the filter since the very beginning of the filtration. P is the pressure drop through the cake itself, and must be carefully distinguished from the total pressure drop through the press. 8 is the time of filtration measured from the instant a t which V = 0-that is, the moment when the first drop of filtrate emerges from the filter leaves. The actual measurement of these quantities on a commercial filter press is extremely difficult. The tests here described were performed on Sweetland filters, filtering defecated sugar sirups. Filter aids of the kieselguhr type were used. I n some instances a “precoat” was laid on the filter leaves; that is, one pressful of sludge with a high filter aid content was run through the press preceding the actual filtration. This was a matter of plant procedure, and not a special feature of test runs. The ordinary cycle in practice is operation neither a t constant rate nor at constant pressure. During- the first part of the cycle the ,so pressure is increasing, care being taken not to increaseit so rapidly 130 that the filtrate is not clear. Thus the pres- ,,o sure is increased to the maximum pressure $ capability of the pumps 9 0 ( u s u a l l y a b o u t 60 u pounds per square inch gage) and for the rest of the run the pressure is about constant. The usual cycle lasts about 11/* hours. The first difficulties are therefore those of press operation. For a rigid examination of the equations, the data should be collected for a run made either at a constant rate or a t a constant pressure throughout its entirety. It then becomes a matter for nice judgment to choose the pressures or rates a t which to operate. I n constant-pressure runs too high a pressure causes cloudy fiItrate during the first of the runs; too low requires an excessively long time to complete the filtration. I n constant-rate runs a high initial rate soon requires a pressure beyond the capabilities of the pumps for its maintenance; too low a rate is wasteful of time and difficult to measure accurately. Of course, the limitations of time are imposed only by the necessity for uninterrupted
;
constant is 0
which are equations of straight lines of slopes (1 - s) and + t , respectively. 2-tt If these equations are valid, then, a plot of log V versus log P of the values obtained in a constant-rate run should give a straight line of slope (1 - s). The value of s should never be negative, for this would mean that a decrease in cake resistance due to compression of the cake would be occasioned by an increase of the pressure. This seems absurd. The value of (1 - s) cannot, therefore, be more than unity. 8 Munning, “A Study of the Filtration of Suspensions,” Undergraduate thesis, M. I. T., 1921.
Vol. 18, KO.4
April, 1926
INDUSTRIAL A N D ENGINEERING CHEMISTRY
production from the filter. I n any case the desired constant conditions cannot be attained from the very beginning of the run. The most to be hoped for is the achievement of these conditions as soon after the beginning of the run as possible. The simplification of the equations to terms including V , P, and 8 is contingent upon the constancy of A , the filter area, a condition not always existent in commercial presses. Constant observations must be taken to insure against invalidation of the data by plugging of filter leaves. Of the three quantities V , P, and 8, V is undoubtedly the most easily and accurately measured. A weir, especially designed and calibrated against water, has been used in these tests. This method, of course, gives instantaneous values of d V / d 0 , which may be plotted against 8.and the area under the curve is, of course, 1’. d check measurement may be made by catching the filtrate in a tank, which may be gaged. Usually, however, it is inconvenient to use a single tank as a r e c e i v e r a n d this method is rarely employed. The direct measurement of P is practically impossible. The best that can be done is to measure t h e t o t a l pressure on the press, which includes not only P , but a pressure due to the resistance of the 0 20 40 60 80 100 TIME IN MINUTES p r e s s i t s e l f . It thus becomes necessary to estimate P from the measured value of the total pressure. For runs a t constant pressure it is only necessary to know that P is constant. I n these runs it has been rustomary to maintain the total pressure a t a constant value, a procedure that assumes constancy of the pressure due to press resistance. This assumption is known to be erroneous, particularly during the early part of the filtration when, as before mentioned, the filtration seems to be in and not on the cloth. However, it seems probable that by the time the total pressure has been adjusted to a constant value, changes in the pressure due to press resistance will have become negligible, and that for most of the run a constant total pressure will indicate a constant value of P. For constant-rate runs the authors have employed the following scheme for correcting the total pressure. It will be remembered that V can be measured with a reasonably high degree of accuracy, so that a plot of the t m e values of V against the total pressure can be made. Figure 1 shows the data of four representative constant-rate runs so plotted. Such a curve may be extrapolated to 77 = 0; the corresponding pressure may be considered as the pressure that would have existed (on the press) a t the moment of appearance of the first drop of filtrate had the filtrate been at constant rate from the beginning. Of course, it is permissible to plot only such data as are taken after the filtration has been adjusted to the constant rate. The value of P may be determined by subtracting this hypothetical initial pressure from the observed pressures. This assumes the correction to remain constant, an assumption based on the considerations discussed in the preceding paragraph. Ordinary Bourdon gages are used to measure pressures, a low-range gage being used for low pressures on constant-rate runs.
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8 is of interest only in constant-pressure runs. It cannot be measured directly, the moment when 8 = 0 being obscured by the facts that the run cannot be made a t constant pressure from the very start and that the filtrate cannot be observed until some time after it emerges from the leaves. However, by arbitrarily assuming 8 = 0 a t some designated moment, say, that of the appearance of the filtrate a t the outlet, 0 may be plotted against V which is accurately known and the curve extrapolated to V = 0. Figure 2 shows such a plot for four representative pressure runs. Only data taken after the constant pressure has been attained may be plotted, and the extrapolation indicates the moment a t which 8 would have been 0 had the filtration been a t constant pressure from the beginning. It is apparent from the slope of the curves in Figure 2 that the condition of constant pressure must be realized as soon as possible in order that points near the 0 axis may be plotted. Treatment of Data
By use of such curves as are shown in Figure 1 the values of P may be estimated simply by subtracting from the observed pressure values the value of the intercept on the prYsure,axis. The logarithm of the values of P thus obtained may be plotted against the logarithms of the corresponding values of V . This has been done in Figure 3. The curves correspond to the similarly numbered curves in Figure 1. As would be expected if the equations were d log V valid, these curves are straight lines. The slope of these straight lines is equal to (1 - s) and a measurement of the slope enables the calculation of s to be made.
(drp
I n a similar manner the proper values of 0 may be obtained from such curves as are shown in Figure 2 , by subtracting the intercept on the time axis from the observed values of time. The logarithms of 8 may then be plotted against the logarithms of V , as has been done in Figure 4. The curves correspond to the like numbered ones in Figure 2. They are straight lines and their slopes
equal to 2C* tion of t .
+
’
dm 7are
A measurement of the slope allows the calcula-
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344
Results and Discussions
The Compression Coejicient The curves plotted in Figure 3 are representative of their type. It will be noted that they are all good straight lines, indicating the validity of the general equation. Table I gives the values of s obtained for these four runs. Table I-Values
of
a
Run
S
2
1
0.36 0.04
3 4
0.00 0.00
All four values of s are equal to or greater than zbro, as is to be expected. A zero value for s is often obtained in these tests and indicates that sufficient filter aid is being added. If the filter aid is considered as furnishing a surface on which the slimy, gelatinous precipitate may stick, i t follows that until enough filter aid has been added to take care of all the precipitate a filtration test will show a compressible cake (s > 0). When sufficient filter aid has been added, however, compression of the cake will no longer be obtained, the filter aid itself not being a compressible substance. When there is no cake compression, s = 0. To obtain the greatest assistance from the filter aid just enough (or very little more) should be added to furnish surface for all the precipitate; if less is added the unattached precipitate greatly hinders the filtration; if more is added, the excess filter aid only increases the burden on the press without assisting in the filtration. While a determination of the value of s will indicate whether sufficient filter aid is being used or not, it will not show whether too much is being used.
The Scouring Coeflcient The curves in Figure 4 are also representative of their type. They are good straight lines, and the values of t , as shown in Table 11, show this to be the special case where t may be negative. Table 11-Values of t Run 1 1 -0.57 2 -0.28 3 0.22 4 0.16
Such a value of t has already been interpreted to mean that increasing flow causes a decrease in plugging of the cake. The explanation for this apparently paradoxical phenomenon, as suggested by W. K. Lewis, is as follows: I n the first place, the phenomenon will be found only when a sludge containing slime and a granular substance is being filtered. Here, specifically, the filter aid is granular and the precipitate resulting from defecation slimy. It is reasonable to assume the cake to contain many small holes, either in or between the filter-aid particles and a smaller number of larger holes or chambers, these latter existing mainly between the particles. The slimy precipitate is loosely attached to the surface of the filter-aid particles. It is conceivable that with a small or moderate rate of flow the smaller holes would become plugged with slime, which, as the flow increases, is blown out in tendril-like formation without becoming detached from the walls of the small holes, into the adjacent larger chambers, thus reducing the plugging of the cake. This is exactly the effect predicted-i. e., a decrease in plugging with an increase in rate of flow. It should be remarked that positive values of t are obtained about as often as negative values. This is probably an indication that in such cases so much filter aid has been used that the slime layer on the particles is so thin as to make the
Vol. 18, No.
above-mentioned phenomenon impossible. Unfortunately, in plant operation, the time of addition of the filter aid to the solutions is such that conclusively confirmatory data on this matter are as yet unavailable. It has not been found possible, in these tests, to run both constant-rate and constant-pressure runs on the same sludge. Indeed, many runs have been invalidated by the failure of a sludge supply of constant quality before sufficient time had elapsed to insure a satisfactory experiment. It
I
2
3
6
8 IO 20 30 60 8=TIME IN MINUTES
100
200
is an interesting suggestion, and one to be tested rather in the laboratory than in the plant, whether a sludge of this type giving a positive value of s will not also give a negative value of t; and whether one giving a zero value of s will not give a positive or zero value of t. Conclusions
These experiments indicate that the filtration equation here used is sufficiently sound to justify its use in plant tests of similar filtrations. I n such tests its greatest value will be found in estimating the proper amount of filter aid, by means of constant-rate runs. For this it has proved extremely reliable. It seems highly probable, though not so certain, that constant-pressure runs can also be interpreted to indicate the proper ratio of precipitate to filter aid. The complexity of the filtering process and the large number of uncontrollable and inestimable variables make the use of the equation in filter design a matter of small promise indeed. The many experimental difficulties encountered by the authors and their students in early experiments indicate that the utmost care must be exercised to measure all values as accurately as possible and to maintain the closest watch that the supposedly constant quantities do not, by some vagary of plant operation, experience a change during the run. Acknowledgment
The authors are grateful to the Revere Sugar Refinery and its employees for their hearty cooperation in making the experiments, to the students in the School of Chemical Engineering Practice whose work has furnished data for this article, and to W. K. Lewis. whose criticisms and suggestions were of much help in interpreting the results. I -
