Decanter Theory

CHAPTER 4 Decanter Theory The decanter centrifuge lends itself to a wide range of theoretical treatments, both process and mechanical. Reif and Stahl [1] observed that the decanter incorporates three "extensive fundamental problems", dewatering of the solids, clarification of the liquor, and conveyance of the cake produced. These three major subjects will be covered here, together with separate sections to deal with specific decanter processes, such as thickening, classification, threephase separation and the latest technology of "dry solids" operation. Separate sections concerning allied topics, such as particle size technology and fluid flow will also be included briefly. A few important mechanical aspects will also be covered, such as resonance, maximum bowl speed, and bearing and gearbox life. Firstly some basic decanter theories will be expounded.

148

Decanter Theory

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d2S

dt 2

= --c02 r.cos(O)

= -co2r

(4.5)

(4.6)

Thus, anything moving in a circle of radius r, at an angular velocity ~, will experience an acceleration towards the centre of the circle of ~2r. In the centrifuge, it is the liquid that moves round in a circle, and the particles in suspension are free to move relative to the liquid. Thus, relative to the liquid, the suspended particles experience an acceleration, ~2r, radially outwards. Thus, the gravitational force, F, on a particle of mass m, is the product of its mass and acceleration, where: F ~- m r ~ 2 (4.7) In centrifuge parlance the term "g" (or g-level) is often used. This is the number of times the acceleration in the centrifuge is greater than that due to gravity alone. To differentiate between "g", which is dimensionless, and the acceleration due to gravity, having dimensions of LT -2, the ratio of the two accelerations "g" will be denoted here by gc, and that due to gravity simply by g. Thus: gr =

ad2t '

g

(4.8)

Note that the g-level within a centrifuge will thus vary, proportional to the radius, t h r o u g h o u t the depth of the liquid, and is proportional to rotational bowl speed squared. (To calculate gc using a simple expression, use rpm 2 x diameter in m m / 1 . 7 8 9 x 106; for example, rotating at 3000 rpm at 450 mm diameter, gc would be 32 x 4 5 0 / 1 . 7 8 9 - 2264. 4.1.2 Differential

The difference in rotational speed between the bowl and the conveyor is commonly referred to as the conveyor differential speed, N. Conveyor differential speed is calculated from a knowledge of the rotational bowl speed, S, the gearbox pinion speed, Sp, and the gearbox ratio, RGB: N = ( S - Sp) RGB

(4.9)

When an epicyclic gearbox is used, the conveyor rotates at a speed less than the bowl speed, while with a Cyclo gearbox the conveyor rotates at a speed above the bowl speed. This fact can have an effect on process performance

Decanter Theory

151

with short bowls, if the feed is not fully accelerated to bowl speed on entry [2]. With the conveyor rotating faster t h a n the bowl, the liquor has to get up to speed to find its way to the liquor discharge hub. With conveyors rotating slower than the bowl, the liquor could wind its way a r o u n d the helix of the conveyor to the centrate discharge ports, w i t h o u t ever getting up to bowl speed. Note, to effect scrolling in the required direction, the flight helix on a conveyor using an epicyclic gearbox would be left-hand, and with a Cyclo gearbox it would be r i g h t - h a n d (assuming conventional equipment and operation). With modern technology, the speeds of the bowl and gearbox pinion can be continuously measured with t a c h o m e t e r s or proximity probes, and their signals fed to a simple PLC to work out, and even control, the differential. The PLC would need the gearbox ratio p r o g r a m m e d in to execute this duty.

4.1.3 Conveyor torque Conveyor torque, T, is a m e a s u r e of the force exerted by the conveyor in moving the separated solids t h r o u g h the bowl, up the beach and out of the decanter. It equals the pinion torque, T~,,times the gearbox ratio: T = Rc~ x Tp

(4.10)

It is not easy to measure conveyor torque directly, whereas pinion torque can usually be obtained from i n s t r u m e n t a t i o n on the pinion braking system. This signal is often given to a PLC for control purposes. Conveyor torque is a vital measure in the control of modern decanter systems.

4.1.4 Process performance calculations Consider the two-phase decanter separation system in Figure 4.2. Input is the feed at a rate of Qf, of density pf, with a solids fraction xf, and an additive, often a flocculant, of density pp, at a rate of Qp, with a solids fraction Xp. There are two products, cake at a volumetric tlow rate Q,~, at density p.~, and solids fraction xs, and a centrate at flow rate 01, at density pl, with a solids fraction of xl. From m e a s u r e m e n t s of some of the eight process parameters mentioned, it is required to assess the performance of the decanter. It is normal to monitor the feed rate, Qf, and the additive rate, Qp, with flow meters. Periodically, gravimetric analyses are conducted on samples of feed, cake, centrate and, if necessary, the additive. Performance is judged by how high is the solids recovery, R, and how low is the flocculant dose, PD, w h e n this is used. Recovery is the percentage of solids in the feed that reports to the cake discharge. Flocculant dose, sometime referred to as polymer dose, is the a m o u n t of dry polymer used per unit dry solids in the feed, usually expressed as kg/t db (kilograms per tonne dry basis).

152

Qf x,.

Basic Theories

qp

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Xs

X! Figure 4.2. Decanter Mass Balance.

As an intermediate parameter in the calculations, it is necessary to calculate the centrate rate, Ol, by conducting a total and a solids mass balance across the decanter. Total mass balance: @PS + GPp = O~p~ + O~pJ

(4.11)

Qfpyxf + QpppXp = QspsXs + Olp~xj

(4.12)

Solids mass balance:

Eliminating Qsps from equations (4.11) and (4.12): OlPl - OfPf (Xs - xf ) + QPPP (Xs -- Xp)

(xs-

(Xs-

(4.13)

Recovery of solids is calculated by subtracting the percentage loss of solids in the centrate from 100. Thus:

R=loo

1

Qlx,p,~

(4.14)

Polymer dosage is given by: PD = GXpPp

(4.1 5)

Polymer dose levels are frequently expressed in kg/t db, with the dry basis measure applying to both solids rate and polymer rate. During these calculations, one must take care with the units used. Volumetric flow rates are invariably measured and the density terms are often ignored as they are usually close to unity. However, the density terms must be used when density values are significantly above unity. The gravimetric analyses of the samples should all be total solids (i.e. samples are evaporated to dryness, and thus measure suspended and dissolved solids), which all except

Decanter Theory

153

the centrate generally are. However, centrates are generally analysed in terms of suspended solids. Any dissolved solids in the centrate c a n n o t be considered a measure of a decanter's inefficiency, as it is suspended solids w h i c h it separates, and any dissolved solids attached to the cake by virtue of its moisture content represents a bonus. Dissolved solids in the centrate are usually low and can be ignored, but w h e n they are not, then they should be included in the equations w h e n calculating centrate rate.

4.2 Particle Size Distribution Very few process slurries contain particles of uniform size. A large proportion of slurries, processed by decanters, contain solids which have a particle size distribution which conforms closely to a logarithmic probability distribution. The logarithmic probability equation was derived by Hatch and Choate [3] in 1929: dz

d{ln(d) }

exp[

Z =

~

In(a,,)

.

-

{ln(d) - ln(d~) }2] , 2 {ln(a.)}-

(4.16)

where Z is the total number of particles: d is the particle diameter: ag is the geometric standard deviation: dg is the geometric mean diameter: and z is the number of particles less than diameter d. Integrating this equation gives the formula for a cumulative number distribution: C,, - 5 + 5 erf x/21n(~rg)

(4.17)

where Cn is the cumulative fraction of the number of particles below size d: and erfis a tabulated integral from - 1 to + 1. It can be shown, by using equation (4.16), that the equation for the cumulative weight distribution is similar:

C,,. = ~ + ~erf x/2 In(;.) -

x/2

(4.18)

where Cw is the cumulative weight or volume of particles below size d. Inverting and simplifying equation (4.18): In(d) - al + a2erf -1 (2C,,. - 1) where al and a2 are constants, functions ofdg and ag.

(4.19)

Decanter Theory 155

Since Hatch and Choate first published their equation, special graph paper has been developed and printed whereby plotting the cumulative percent of particles by n u m b e r or weight, oversize or undersize, against particle size, results in a straight line. The mathematics of the distribution are such that one can readily transfer between weight and number distributions, and even area and diameter distributions [4]. The diameter at the 50% line on the graph gives the geometric m e a n diameter for the type of distribution plotted, be it number, weight or whatever. The geometric standard deviation, which is the same for all types of distribution, is given by the size for w h i c h 8 4 . 1 3 % of the n u m b e r , weight, etc., is smaller, divided by the geometric mean size: d84,13 erq -- d5o

dso -

- - - dl 5.87

(4.20)

The relationship between the various means is given by" dgu, - dge3(in ~ 4 , - d.qe2On~

(4.21)

)2

d ql - d~e(In (70)2

(4.22)

(4.2 3)

where dgw is the geometric mean for a weight distribution: dgs is the geometric mean for an area distribution" dg] is the geometric mean for a diameter distribution: and dg is the geometric mean for a number distribution. Figure 4.3 shows a typical distribution plotted as a number, area, and weight distribution on the specially scaled graph paper. From a grapb such as this, the two basic parameters, ~rg and dg, can readily be obtained and from these more pertinent information can also be obtained. For example, the total surface area. Av, of the solids ir~ the slurry can be calculated: 6

At -- dT,, .e} ~M~ ~-~

(4.24)

This parameter is useful for paints and pigments, giving the covering power of the solids. It is also useful in assessing relative flocculant demand, as this is proportional to the surface area of the particles. In general, the decanter removes the coarsest or densest particles from the slurry, leaving a n l y the finest or least dense in the centrate. Knowing the required percentage solids removal from the slurry, the recovery, one can read the desired cut point from the distribution graph. This then gives an appreciation of the feasibility of the desired separation. Experience will tell whether the decanter would be capable of achieving the required cut point. For instance, a cut point of 2 - 5 Bm would be feasible on a decanter with most

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slurries, but, say, less than O.1 pm would probably be impossible, however high the density of the particles. The cut point size is the smallest particle size that has to be settled in the decanter. Technically 50% of particles of that size settle and 50% are lost in the centrate: above that size the separational efficiency increases and below it vice versa. In consequence, the size distributions in both the cake and the centrate will also exhibit logarithmic probability distributions. Figure 4.4 depicts examples of weight frequency distributions for feed, cake and centrate. Note that the centrate and cake lines intersect at the cut point size, and at a frequency level half that of the feed at that size. Hence, there is a 50% split between cake and centrate at the cut point. These frequency distributions are plotted as cumulative weight distributions in Figure 4. S.

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4.3 Clarification The separation of solid particles and agglomerates from the suspending liquors, within a decanter centrifuge, has invited n u m e r o u s theories. Few of these theories have provided exact results, which has given the opportunity for m a n y more. This is no reflection on those providing the theories, as the system, let alone the processes used, are quite complex. In a decanter, liquid and solids flow in a helical path within a cylindrical vessel with a conical end. The rotational velocity can vary from the bowl wall to the pond surface. Many theories start by assuming discrete spherical particles that settle in a l a m i n a r flow regime, when often this is far from w h a t actually happens. The expression of liquid from the cake can be as a result of one or more mechanisms. It could be by filtration, compaction, or simple drainage against the scrolling action. Nevertheless over the years a n u m b e r of theories have been proposed, which allow a reasonable fit with m u c h of the data, or specific categories of data. This chapter gives some of the more c o m m o n and usable theories.

4.3.1 Sigma theory For more t h a n 40 years the clarification of liquor in decanter centrifuges, in fact in most sedimentation-type centrifuges, has been characterised by the Sigma theory of Ambler [ 5]. This theory assumes that spherical particles settle in a laminar flow regime according to Stokes' law [6], in a cylindrical bowl rotating at constant angular velocity. Since the first publication by Ambler, several variations have been used which are approximations, or developments, of the original. Refer to Figure 4.6. Consider the smallest particle in the feed sludge that has to be separated, the cut point size, de. This particle has a density, Ps, and settles in a liquor of density, PL, and viscosity, r/L. The feed slurry enters the decanter at a rate of Qf, at a pond radius, rl, at point X at time t=0. By the time the particle traverses the clarifying length of the centrifuge, L, in time t=te, the particle m u s t settle to a radius r 2, at point Y, the bowl internal radius, if it is to be collected by the conveyor. The centrifuge rotates at a constant a n g u l a r

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Decanter Theory

163

= 7rLg:.(r2 + rl)Vs

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(4.46)

where Ar is the pond depth: and V is the pond volume. For this derivation of clarification capacity, it is readily deduced that" E = 7rL

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In the graph of Figure 4.7 the various formulae for Sigma developed so far (equations (4.32), (4.39), (4.40) and (4.47)) are compared for various ratios ofrl/r2. The common factor 7rLw2/g is removed and r2 is taken as unity, for the graphical comparison. By means of Figure 4.7, a number of observations may be made. The expansion of the logarithmic term to give an easier formula for Ambler's Sigma is a very acceptable approximation. The even simpler formula last developed above is also acceptable for shallow ponds (radii ratio greater than 0.75). However, there is a significant difference for the formula used for deep ponds. Notice that with zero pond radius, the shallow pond versions of Sigma have finite Sigma values while the deep pond version has a zero value. This is because a particle starting at the centre line will experience no g to initiate its fall, while those which by definition start half way into the pond, or are subjected to a mean g throughout the pond will always have a finite settling rate. However for practical designs the radius ratio will always be appreciably over zero, generally in the range 0.4-0.8. It will be seen from the various Sigma formulae that increasing the length of the bowl increases Sigma pro rata. Thus, in this respect Sigma is additive. Some like to include the Sigma value of the beach in their formula [8], especially when feeding on the beach. For this, using equations (4.46) and (4.40): E = 2 7 r w 2 Lc

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where Lk is the wetted beach axial length" and Lc is the cylindrical length of the bowl.

164

Clarification

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Considering the assumptions used in the derivation, and the approximations used, one could question whether the use of the simpler equation (4.47) would not suffice, for use with shallower pond machines at least. It is the ratio of Sigma values which is used when scaling from one decanter size to another. Using equation (4.48), the ratio will be little affected as the extra term will increase by approximately the same ratio with geometrically similar machines. Another expression, in place of Sigma, uses an empirical formula taking a nominal bowl radius, the ~ bowl radius, r~ (which equals three quarters of r2). This expression [2 ] is termed the "area equivalent", Ae~ and is determined by"

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166

Clarification

of the same geometry. However, it m u s t be noted t h a t this only applies to process materials with solids exhibiting a skew Gaussian (logarithmic probability) distribution. W h e n scaling from one m a c h i n e to another: Qf2 = ~Z2

(4.55)

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(4.56)

4.3.2 Sigma enhancement The use of conical discs, or angled vanes, on the conveyor will theoretically e n h a n c e the Sigma value, the clarification capacity, of the centrifuge. To estimate the Sigma value of a stack of conical discs, the formula used for disc stack centrifuges m a y be employed [ 11 ]: ED =

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3

g

.(r~ -- r~). cot O

(4.5 7)

where ED is the Sigma value for the disc stack; nD is the n u m b e r of discs; r ~ is the outside radius of the discs; and 0 is the half included angle of the discs. The total Sigma value for the centrifuge is obtained by adding ED to the Sigma value calculated for the conveyor section between the feed zone and the discs. There is little published on the effect of longitudinal angled vanes, but the equation is derived in a similar fashion to t h a t used for the disc stack centrifuge: nvLvw 2

~

7rLvw 2

Ev = 2g. c o t , (r~ - r~) -~- ~ r ~ g

(4.58)

where Ev is the Sigma value for the vanes; Lv is the length of the vanes: nv is the n u m b e r of vanes; and ~bis the angle between the vane and a radius. If the vanes do not extend the full length between the feed zone and the centrate discharge, t h e n the Sigma of the plain section needs to be added to Ev to obtain the total Sigma value for the centrifuge. Caution is needed in using these extended Sigma values, particularly for the angled vanes. This is because flow t h r o u g h the vanes or discs can channel, to

Decanter Theory 167

take the easiest path. This will reduce the effectiveness of the devices. Good designs, therefore, will endeavour to ensure even distribution of the flow across the vane and disc openings. Even then as liquor flows from the outer edge of vanes or discs, towards the centrifuge axis for discharge at the weir lips, considerable changes in kinetic energy occur. This can cause very complex flow patterns, turbulence and Coriolis effects.

4.3.3 Flocculant requirement Chapter 5 is devoted to flocculants. However, it is appropriate at this j u n c t u r e to mention the need for polymers in some process applications, particularly effluent applications which are a large m a r k e t for the decanter. In these applications, without a polymer flocculant, it would not be possible to employ a decanter. It is clear from equation (4.28), Stokes' law, that the settling velocity of a particle, Vs, is proportional to the square of its diameter. Thus doubling the particle diameter will increase vs by a factor of four. This results in greater separation efficiency. The objective of flocculants is to c h a n g e the electrochemical forces on the surface of the particles, so as to bind t h e m together such that they act as one large particle. Once flocculated, these particles must be handled carefully so as not to break them up mechanically. This is especially true w h e n processing them in a decanter. In most applications, the a m o u n t of polymer used is just sufficient to flocculate a sample of the feed. The a m o u n t necessary, as assessed in the laboratory, is generally the a m o u n t used in practice on the centrifuge, plus or minus a small fraction. However, recently there has been considerable development in decanters and their use in obtaining extra-dry cakes from compressible sludges, particularly effluents. In these instances, the consumption of polymer has increased considerably. The a m o u n t of flocculant needed increases as the extent of dryness required in the cake increases, and it increases exponentially. The a m o u n t of flocculant required also increases with the feed rate to the centrifuge [ 12 ]. In practice, on a "dry solids" application, the polymer used will be two to three times t h a t which would be used on a standard application with the same process material. There has not been a theoretical formula proposed to quantify polymer demand. However, the available data suggest a format similar to equation (4.59): P D - kl + k2.e (~:'-k~

(4.59)

where kl, k2, and k3 are constants. Practical data can be very erratic, as it is easy to overdose w h e n striving for extra dryness. When assessing the m i n i m u m polymer requirement, it is necessary carefully to adjust all operating parameters, to ensure performance is at the limit, without contingency levels added.

4.4 Classification Classification, the fractionation or separation of particles by size, could be considered as merely inefficient clarification. The cut, or desired classification, is adjusted by altering the centrifuge's efficiency. This is most easily done by altering the feed rate or bowl speed. However, adjustment of pond depth or differential may, in certain circumstances, be used. In a thick suspension hindered settling occurs, when there is a tendency for the larger particles, which should settle, to get held up by the dense concentration of the smaller particles. In these circumstances higher differentials could be used, to agitate the suspension and so release the heavier particles. The disadvantage of this is that the cake or "heavy fraction" tends to be wetter, as a result of the higher differential, and thus entrains larger quantities of the smaller particles. To correct this, a shallow pond is selected to allow release of liquid containing the smaller particles on the dry beach. In some classification applications, the required cut point is very sharp and the rheology of both separated phases is such that they remain quite fluid. In this type of application the pond used would be relatively deep, and separation would be akin to a liquid/liquid separation, using a hydraulic balance under some form of baffle. Very occasionally there will be found a classification application where it is required to separate two distinctly different particles, such as in the refining of minerals. In these cases the two different substances to be separated may have markedly different densities. This is particularly acceptable and quite advantageous when the denser material comprises the larger-sized particles. However, if this is not so, one must consider a combination of density and particle size for the cut point of each of the two substances, in relation to Stokes' law. One could visualise the situation of a large, low-density particle settling faster than a high-density, small particle. Thus for such a process to be feasible: 2 dc2h(Psh - Pf) > dc,(PslPl)

(4.60)

where dch iS the required cut point size of the heavy fraction; dd is the required cut point size of the light fraction; Psh is the density of the heavy solids; and P~l is the density of the light solids.

Decanter Theory 169

Each of the two solid constituents will have their own size distribution from which a cut point size can be chosen to give the desired purity of product and yield. Poor efficiencies can occur in some classification applications, due to n a t u r a l agglomeration of particles. In these applications, the use of dispersants is quite common. Dispersants have the opposite effect to flocculants, and can be equally powerful.

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Decanter Theor!!

171

172

Three-Phase Separation

will become necessary. Back-pressure from a centripetal pump or skimmer pipe will require recalculation of weir settings. Working with three-phase separation will require revision of the formulae used for performance evaluation. Not only will there be interest in solids recovery and clarity (absence of solids), for both liquid phases, there will be an avid interest in what has happened to the oil. How free of water is the oil? What is the recovery of the oil? How much oil is left in the cake and the water phase? In some of the three-phase applications, water is added to dislodge the oil from the solids. With water addition there are two input streams, feed and water, and three outlet streams, light liquid phase (oil), heavy liquid phase (water) and cake (solids). Each stream is analysed for the three elements, oil, water and solids. Four of the five streams are monitored for flow rate. The cake rate would be difficult to measure accurately. To analyse performance, a mass balance across the centrifuge is performed for the three separate elements and the total mass, after which the cake rate is eliminated. Formulae are then developed for pertinent recoveries and purities. It is not necessary to develop these here as the pertinent formulae will depend on the application, and in any case the development is similar to that already shown for two-phase separation (see Section 4.1 ).

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174

Thickening

scrolling rate, Ss, of a particular decanter would be proportional to the differential, N. In a thickening process one might expect the scrolling rate to be an inverse function of cake solids content, because as the "cake" becomes thicker it gets more viscous and the scrolling efficiency reduces. Thus: N

s~-

Xs

(4.63)

Dividing the scrolling rate by the solids feed rate gives an empirical "thickening factor", ~, where: N = (4.64)

O.sxsx~

As the factor 9 is increased, cake dryness decreases and solids recoveries increase, and vice versa. Very good correlations can be found between cake dryness and ~, and with solids recovery and 9 for fixed pond depths and fixed polymer dosages.

4.7 Conveying In the past, the vast majority of decanter applications were limited by the clarification capacity of the centrifuge. With the development of the decanter, better designs, higher g-levels, and better knowledge of the technology, today many applications are governed by what happens at the other end of the machine. Performance is now often limited by how efficiently the solids can be dewatered, and how efficiently and at what rate the solids can be discharged. 4.7.1 The Beta theory

The decanter capacity, related to a solids conveying limitation, is indicated by" Oj p f 3 z f O( N P n l r 2 ( r 2 psXs

-

rl)

(4.65)

where xf is the solids concentration in the feed; Xs is the solids concentration in the cake; pr is the density of the feed; Ps is the density of the cake; N is the differential between conveyor and bowl; P is the pitch of the conveyor: and nl is the n u m b e r of leads or flights on the conveyor. This is k n o w n as the Beta theory, derived by Vesilind [ 13 ], where: fl -- 2 7 r N P n , r2 (r2 -

rl)

(4.66)

Thus for solids conveying limitation, scaling from one machine to another:

Qf2

=

32

(4.67)

where the subscripts 1 and 2 refer to the centrifuges 1 and 2, respectively. The Beta value calculated by the above formula does not take into account scrolling efficiency, nor the fact that the depth of solids between flights reduces to a m i n i m u m at the point of discharge. However, if scaling is between machines of similar geometry, efficiencies will be similar and dimensions of

176 Conveying the flight area at discharge will be proportional to that in the cylindrical section. Thus, equation (4.65) will be valid. One of the basic assumptions for equation (4.65) is that the cake fills the space between the flights, from the scrolling surface to the pond surface, throughout the length where capacity is being considered. With non-cohesive cakes, those that tend to be fluid or creamy, a head of cake touching the conveyor hub at discharge will not be possible and here the more appropriate formula would be, instead of equation (4.65): Of pfxf cx NPnlr2 PsXs

(4.68)

When thickening is involved, equation (4.68) is certainly more appropriate. If liquor levels are close to or above the solids discharge level, some fluid mechanics technology may need to be invoked with some attention to cresting heights. 4.7.2 Conveying on the beach

It is important in decanter work to have a good understanding of the factors that affect the mechanism and efficiency of scrolling [10], particularly up the beach. Consider the general case of a single particle, mass m, being pushed up the beach, angle a, by a flight of the conveyor. Consider the forces on that particle (in Figure 4.9). Constructing a vector diagram of these forces, there is the normal force, FN, perpendicular to the face of the flight, the scroll friction at right-angles to this, the weight of the particle resolved down the beach parallel to the bowl axis, and finally the friction from the beach. The direction of the beach friction is indicative of the direction of travel of the particle up the beach, at angle 0 to the axis. It is worth considering what factors minimise 0 and thus maximise scrolling efficiency. Reducing scroll friction by polishing and smoothing is one factor. Maximising beach friction is another: this is generally done by ribbing or grooving, which effectively polarises the friction, to stop slippage when cake rotates with the conveyor. This means that it is easier for the cake to slide up along the grooves, which are in line with the centrifuge axis, than to shear over itself at right-angles to the grooves. Reducing the acceleration force on the particle, the weight resolved down the beach, by reducing bowl speed or decreasing beach angle will improve scrolling efficiency but will have adverse effects on other features of the process. The same goes for reducing conveyor pitch. It will be noticed that the weight resolved down the beach has a buoyancy component. When the particle leaves the pond for the dry beach this buoyancy effect is lost. Instantaneously the weight resolved down the beach

Decanter Theory

o ~ ,\

177

Direction # of Motion r ,, . ,, .m g .sin (x Fs

m ' g = ~d___~(p_ pt)gr

m'g .s"

1 S w _._..Jl ~s - 1 + tan~-tanO - P

S.~

Best ~s = p = co,,,2~

Figure 4.9. Force Vector Diagram.

markedly increases. It can increase so much with some materials, that scrolling ceases. This is where the use of a baffle disc feature can be effective. A baffle disc is fitted onto the hub of the conveyor at the foot of the beach to restrict excessive cake flow, and the pond level is raised above the solids discharge level. Therefore, theoretically the cake is below the pond surface right up to the point of discharge, thus m a i n t a i n i n g any beneficial buoyancy effects. Scrolling efficiency is also maintained, and enhanced by means of the differential hydraulic pressure across the disc.

4.7.3 Dry solids conveying When a decanter is operated to obtain the driest cake from a compressible sludge, the decanter bowl will be virtually full of cake [9], from front to rear, with next to no volume of clear s u p e r n a t a n t . The dryness of the cake, as will be seen in Section 4.9.3, with a constant torque, is inversely proportional to the volume of the cake. In a dry solids decanter, there is generally a restriction, for example a baffle disc, against which the conveyor compresses the cake. Sometimes the cake discharge aperture forms the restriction or acts as an extra restriction in series

178

Conveying

with the baffle. The maximum possible t h r o u g h p u t of the decanter is proportional to a combination of these restriction areas, to the conveyor differential and also to the cake dryness. It is proportional to cake dryness because the dryer the cake, the greater is its density and thus the mass per unit volume being conveyed. Thus: QfR pfxf = PNAr PsXs

(4.69)

where R is solids recovery: xf is feed solids content; Xs is cake solids content: Ar is a function of the area of the conveyor restriction; and P is conveyor pitch. It is generally found that, with a properly operated dry solids decanter, the scrolling capacity at the point of maximum restriction, the smallest area, is greater than that calculated using equation (4.69), but less than that predicted by the equation at any other restriction in series.

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Decanter Theory

181

The more cohesive organic materials, such as clay-like substances and municipal effluents, can drain by d e c a n t a t i o n with the squeezing action mentioned. Dewatering will c o m m e n c e u n d e r the liquor surface. Thus, it will be appreciated that there c a n n o t be a generalised equation for dewatering as there is for clarification. As with clarification though, special designs and devices can be incorporated, especially with respect to the conveyor, to e n h a n c e the dewatering capability of the decanter.

4.9.2 Washing In some decanter applications it is required to remove, from the solids, some dissolved impurities in the liquid held in the cake. This is achieved by spraying rinse liquid onto the solids as they are conveyed up the beach. Admitting the rinse too far up the beach can cause problems, by w a s h i n g the cake back into the pond, or producing too wet a cake. W a s h i n g too far down the beach risks poor w a s h i n g efficiency, w h e n rinse by-passes the cake, by streaming over the surface of the supernatant. To maximise rinsing efficiency, it is necessary to keep the cake flooded with rinse liquor, but not to add excess unless it flows t h r o u g h the cake, rather t h a n over it. The ideal location for admitting the rinse, therefore, is at the junction between the wet and the dry beach. For o p t i m u m location of the rinse nozzle(s) it is necessary to have a good appreciation of the cake profile around the wet and dry beach junction. Consider the idealised system depicted in Figure 4.1(). Feed enters the system at rate Of with suspended solids content xf and dissolved impurity content yr. Cake is discharged at the right at rate 0,~ with solids content x's and impurities y~.

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x,:y, Figure 4.10. Rinsing.

182

Dewatering and Washing

Rinse flows countercurrently at rate Qw with solids content Xw = 0 and impurities Yw = 0. The centrate, which includes the spent rinse, flows out at the left at rate Ql with solids content xl (which will be assumed equal to zero) and impurities content Yl. In the system considered, the cake remains completely flooded without excess liquor above it. Impurity level in the feed is If where: If = lO0.y/%wb = l o 0 . Y f % d b xl

(4.72)

Similarly, in the cake: (4.73)

Is = l o 0 . YS%db Xs

Volumetric dry solids flow is QsDwhere: (4.74) Ps

where ps is the solids density and ps is the cake density. The voidage flow in the cake at discharge is therefore Qv where (4.75)

O,, = Os - OsD

Assuming that the solids are impervious, and without surface adsorption, then Ow must equal or exceed Qv to remove all the impurity. If Qw is less than Qv then the impurity level of the solids as they emerge from the pond, Ie, will be proportional to the difference between these two figures, assuming plug flow: le --

lO0. Yf. Q Qv" ~--' xs 0,,

~0

(4.76)

After emergence from the pond, further dewatering takes place on the dry beach, assuming that there is a dry beach. Thus, the impurity level of the discharged cake will reduce to Is where Ix = Ie o-Ss Ps(1 - xs) 9 " O,, "Pl

(4.77)

In practice solids are not impervious, and diffusion has to be relied upon to reduce impurity levels. Consider a modification of Figure 4.10, as in Figure 4.11, to include diffusion.

Decanter Theory

183

Ow

Of

c , = y,, = 0

el

................................._~ Diffusion Q,

I

C 3 = C~

O. C2

Figure 4.11. Rinsing with diffusion.

The c o n c e n t r a t i o n s cl to c4 are the impurity c o n c e n t r a t i o n s in the liquor, as s h o w n in Figure 4.11. Thus: c3 - (1 - xl)

(4.78)

,l]s c2 =~1~ - x ~ )

(4.79)

Yf t'~ = (1 -- Xf)

(4.80)

c4 -

y,,, -

0

(4.81)

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-

(cl

-

c2)(l

-

x.~).Os

(4.82)

If the diffusion process is 100% efficient: c3 - cl Substituting equations rearranging"

(4.81)

and

(] r162

(4.83)

(4.83)

-

x~)G

into

-

(1 - x~)O~

equation

(4.82)

and

0,,,

(4.84)

However, the diffusion process is seldom, if ever, 100% efficient. The mass transfer factor, JD, for this type of situation is given by [ 14]"

184

Dewateringand Washing

]D = -hD - --(Sc) 0"67

(4.85)

l/c

where Sc is the Schmidt n u m b e r , a dimensionless group: Sc = n pD

(4.86)

and h D iS the mass transfer coefficient; Uc is the superficial velocity of the rinse; 7/is the rinse viscosity; D is the diffusivity of the impurity: and p is the density of the rinse. The mass transfer factor, JD. is a function of the modified Reynolds number, Rein, where: uc4P Re,, = (1 - e)r/

(4.87)

where d p is the characteristic size of the bed, i.e. the typical m e a n pore size; and e is the cake voidage. To estimate the total mass transfer of impurity, Na, the following basic equation is used: Na = hDA,,Ac

(4.88)

where Ac is the surface area of the cake bed: and Ac is the mean concentration difference between the cake particle surface and the rinse. From equation (4.88) it is seen t h a t mass transfer will decrease as pond level increases, because the surface area of the bed, Ac, decreases. Washing efficiency should be unaffected by g-level, if all other parameter values are held constant, unless a higher g-level enables a lower residual moisture level in the cake, and thus a proportionally lower level of impurities. As differential is increased, the layer of cake becomes thinner, and therefore the superficial velocity of rinse liquor proportionally increases. However, the Reynolds n u m b e r remains essentially constant as the characteristic size of the bed decreases, which in t u r n means t h a t the mass transfer factor remains constant. Thus, from equation (4.85), the mass transfer coefficient will be proportional to superficial velocity. This means that washing efficiency should improve with differential as is found in practice. The theory would suggest that w a s h i n g efficiency should be unaffected by feed rate. However, there comes a point, w h e n feed rate is increased, at which the thickness of the cake on the beach is such that the rinse c a n n o t flood the bed because of the high g field. This then invalidates the theory. To m a i n t a i n a constant bed thickness as feed rate increases would require a pro r a t a increase

Decanter Theory

185

in bowl diameter. This suggests that the capacity of a decanter, limited by washing efficiency, would be proportional to its diameter. 4.9.3 Solids compaction

Many centrifuge manufacturers have expended significant development effort over the past 10 years or more to improve the cake product dryness from decanters when employed on compressible sludges, particularly effluent sludges. The efforts have not been wasted in that several suppliers now offer special ranges of decanters for "dry solids". It has been found [15] that, in "dry solids" operation, the cake dryness produced is proportional to the torque developed by the conveyor or vice versa. It has been shown [12] that:

where xs is cake dryness; T is cake yield stress: T is conveyor torque; and V is pond volume in bowl. To improve dryness in a dry solids decanter, conveyor differential is reduced, and thus throughput has to be reduced, as a result of which the cake compacts and gets dryer, resulting in increased torque. Pushing this reduction too far will result in overspill of the solids into the centrate. The question for the centrifuge specialist is " w h a t is the limit of dryness achievable, assuming that the practical torque limit of the centrifuge is not reached?" The maximum dryness achievable, without producing dirty centrate, will improve with bowl speed and pond depth and with reduced feed rate.

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Decanter Theory 187

where Z' is the Sigma scale-up value for the compaction process of "dry solids". Note from equations (4.90) and (4.93): ~P_/ = { 1 -- ( r l / r 2 ) 2 } ~

(4.94)

An alternative approach to this version of compaction will now be made, which is based on conventional compaction theory [18] where processing capacity is found to be a function of volume rather than area. Consider the compaction of a concentrated compressible sludge in a cylinder in a field of 1 g, as illustrated in Figure 4.12. After time t the interface which develops between the settling sludge and the clear supernatant will be at a height H above the base of the cylinder. This interface will have a velocity v. After an infinite time, settlement will cease when v will be zero, and the height of the interface will be at height H~. At this point the weight of the cake is no longer sufficient to express any more liquid from between the pores of the cake. The rate of sedimentation is given [ 18] approximately by the expression: dH v----~(H-H~) dt

(495)

In this system, it is the compressive forces of the weight of the solids that are forcing the liquid at an ever-reducing rate, up through the ever-reducing spaces between the particles. As in most fluid flow systems, this flow rate will be proportional to g, as will be seen in the laboratory bottle spin centrifuge. The total volumetric flow of supernatant, O, which will equal the volumetric shrinkage of the cake, will be proportional to the cross-sectional area, A, of the cylinder. Introducing this area, and the proportionality constant, into equation (4.9 5), the following expression results: Q=ksA(H-H~)

where ks will have units ofT- ~ and is a constant for a given sludge.

\

Figure 4.12. Solids compaction in a 1gfield.

(4.96)

188

Dry Solids Operation

The height components of equation (4.96), H and H:~. when multiplied by a density are equivalent to pressure heads. Consider now compaction in the centrifuge. In the decanter centrifuge the geometry is quite different, as illustrated in Figure 4.13. Nevertheless, the equivalent pressure heads may be derived using equation (4.61 ) for pressure, Pr, developed at radius r within a centrifuge. In equation (4.61) an expression is given for the pressure developed at radius r with liquor above to a level of radius rl. In the present case, it is required to know the pressure of the head of cake, at the bowl wall radius, r2, with the surface of the cake at radius, !"1. Thus, r needs to be replaced by r 2, and rl by tin equation (4.61 ). Thus, the term/-/in the 1 g mode will be equivalent in the centrifuge to: ~2

2g (r~ - !"2)

(4.97)

and H~ will be equivalent to: ~,2

2--g (r~ - r~)

(4.98)

where r~ is the radius to which the sludge would settle in infinite time. Substituting equations (4.97) and (4.98)into (4.96) the following equation is obtained: ~d2

(21 = k6A-~fl [(,'~ - r 2) - (r 2 -/'2,~)]

(4.99)

where Q~ is the centrate rate that equals the cake volume reduction rate: and k~ is a constant for the sludge and system. It is k n o w n that w h e n a decanter is operated to its limit, it will be full of solids, so: r = rl (4.100) and thus: A = 2 7rrl L where L is the length of the bowl where compaction takes place.

Figure 4.13. Compaction in the decanter.

(4.101 )

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190 Dry Solids Operation For acceptable performance: R~ 1

(4.110)

Substituting equations (4.11 O) and (4.109) into equation (4.108): (4.111)

Substituting equation (4.111 ) into (4.10 7 ): (4.112)

Thus, the highest possible feed rate, Qr, is proportional to g-level multiplied by volume times a function of xs, x~ and xr. In any single system, x~ and Xr will be constants. If xs is plotted as ordinate against Of/glV, then a line cutting the ordinate axis at a dryness of x~: will result, with a negative slope. Over the range of drynesses generally tested the line will be close to a straight line. The term gl V is referred to as "g-volume", or as g-Vol in equations, in the scalingup calculations of Chapter 7. With centrifuges of similar type and geometry, it should be possible to scale performance from one machine to another. Where geometries are dissimilar, the parameter x~: is liable to vary, as it will be a function of the g developed within the centrifuge and the depth of pond used. Shallow pond machines are not able to produce as m u c h pressure at the bowl wall as deep pond machines and thus the ultimate dryness achievable will be less. Some fundamental work [19] has indicated the relationship between a compressible cake's dryness and its yield stress. As a moist cake is subjected to a stress, or pressure, there is an equilibrium level of moisture for each load value. The graphical results of this, for one particular sludge, are shown in Figure 4.14. It cannot be said how closely this relationship is translatable to any other effluent, but it is anticipated that similar sludges will behave similarly, with perhaps some adjustments to the constants of the equation for the line. The slope of this line is 0.2 6. A figure of 0.2 5 will be taken here to formulate firstorder assessments of performance. Thus approximately: x~ cx ~

(4.113)

where Pc is the pressure to which the cake is subjected. This pressure will be proportional to density difference between solids and liquor, and also to cake voidage. Nevertheless, this pressure will be proportional to the total pressure, and so using total liquid pressures in comparing two systems will be valid, where the solids occupy the entire pond volume of the decanter.

Decanter Theory

191

2.00 1.80 1.60

f

A

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o

0

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Log10 (Yield Pressure Pa) Figure 4.14. Cake !lield stress.

Thus, when comparing or scaling capacities of two decanters, the pressures at the bowl walls should be the same or the value of x~ should be adjusted in the light of the different pressures. Not only will the ordinate intercept change, but so will the slope of the line. The slope of the line could also be affected by changes in k6. which is constant for the sludge and system. If the geometry of the centrifuge changes, or the g-level, then this could affect the compaction e n h a n c e m e n t effect of the conveyor enumerated by k6. Equation (4.104) can be differentiated with respect to r] to find the optimum ratio of r ] / r 2 for maximising feed rate. The value of this optimum ratio is 1 / x/~ -- O. 5 8. When polymer is used, as invariably is the case, a slight adjustment to equation (4.1 1 2 ) is needed. The subject of the equation then becomes the total flow to the decanter, Qt, where: Ot = Q / + Op

(4.114)

where Qp is the polymer feed rate. The feed solids Xf needs to be replaced by x'f where" 9,

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194 FluidDynamics device to do this, viscous drag of the pond is the only means by which these actions can be accomplished. If the viscosity is low, considerable turbulence can occur, affecting cresting, interface location, stability, and sedimentation and re-entrainment of settled solids.

4.11.2 Moving layer In a pond of an operating decanter centrifuge, there often tend to be two distinct liquid layers. The upper or surface layer, the moving layer, moves rapidly and turbulently towards the discharge weirs. Under this moving layer, the pond is quiescent, allowing solids to settle under a laminar flow regime, and then to compact. This is a simplistic picture, as the shape of the conveyor and its movement adds to the complexity. It is sometimes useful to estimate the depth of the moving layer to know w h e n it is liable to disturb and re-entrain sedimented solids. It will be appreciated that the thickness of the moving layer will depend upon whether the flow is axial or helical. Research has shown that for axial flow: h,,, ~x

4q;

(4.126)

where hmis the thickness of the moving layer. It will be seen that the formula is independent of path length, the clarifying length. It has also been found that moving layer thickness closely follows cresting height (see Section 4.11.3). Thus, the shape, size and number of weir plates used can affect the moving layer thickness. The moving layer thicknesses found in helical flow are greater than those calculated using equation (4.12 6).

4.11.3 Cresting The level difference between centrate and cake discharges can be quite critical when optimising process performance. When the level difference is small, the degree and consistency of the centrate cresting can play an important part in the process performance optimising. The crest height, the pond surface level above the actual weir height, is a function of the centrate rate, the total weir width and centrifuge g-level, as well as physical constants of the liquor such as viscosity and density. Crest height is hc, given by: (4.127)

Decanter Theory

195

where co is a constant and generally approximately 0.415; and B is the total length of weirs. This equation is derived from the Francis formula [20]. Experimental data show that, due to the interrupted n a t u r e of the discharge weir, the calculated value of crest height needs to be increased by 35% for axial flow and 90% for helical flow. With a 360 ~ internal weir, B would be the full circumference, and thus would cause the least cresting. 4.11.4 Feed zone acceleration

Feed zones are designed to accept the m a x i m u m possible feed rate, and bring it up to bowl speed with the m i n i m u m of splashing and rejection. Bringing the feed up to the angular velocity of the bowl is not necessarily enough. As the process material flows out of the feed zone to the pond, it has a constant linear velocity fixed by its a n g u l a r velocity at its point of exit from the feed zone. To m a i n t a i n its a n g u l a r velocity extra linear velocity is required as the radius increases [21 ]. The power required to bring the feed material up to bowl speed at the pond surface is Pp, where:

PP - Or p.r~2 ,~

(4.12 8 )

Power available in the feed stream at the pond surface is PA. where: 1

P a -- -~ Of Pf~X r-~

(4.129)

The power lost on entry is thus the difference between equations (4.124) and (4.125), and this is dissipated in heat and turbulence on entry. Thus to minimise turbulence and power loss, it is necessary to design the decanter with the pond surface as close as practicably possible to the centre line. Nevertheless, other process considerations may require the taking of a different view.

4.12 Power Consumption The total power input required by a decanter centrifuge comprises a number of separate power components: PT -- PP + PwI: + Ps + PB

(4.130)

where PT is the total power required by the decanter: Pp is the power required to accelerate the process material to the bowl speed at the discharge radius; PWF is the power to overcome windage and friction: Ps is the power required for conveying: and PBis the power for braking. From equation (4.128): Pp - O.rpfw2,'d

(4.131)

where rd is the process discharge radius. Naturally, if cake and centrate are discharged at different radii, then these two power components have to be calculated separately. The windage and friction component is given by: PWF = k7 + k8w + k,~,,'2

(4.132)

where k7, ks and k9 are constants. Pwv can be calculated with difficulty but is more generally derived practically in the factory by measuring the power absorbed for different bowl speeds. The conveying component is given by: Ps = N T

(4.133)

where N is the conveyor differential; and T is the conveyor torque. Similarly, the braking component is: PB

--

SpTp

(4.134)

where Sp and Tp are the pinion speed and torque, respectively. In some types of backdrive the braking power can be regenerated, so that the total power used is reduced.

Decanter Theory 197

4.12.1 Main motor sizing The p o w e r of the m a i n m o t o r will be based on the c a l c u l a t i o n of PT from e q u a t i o n ( 4 . 1 3 0 ) , while its physical size will be influenced by its s t a r t i n g r e q u i r e m e n t s . Motor m a n u f a c t u r e r s rate their m o t o r s o n the basis of the m a x i m u m p o w e r delivered at the m o t o r shaft, PM. This h a s to be g r e a t e r t h a n PT to cater for frictional losses in the drive belts a n d fluid coupling, if used. Thus, the m o t o r p o w e r is PM: PM .r

= PT

(4.135)

w h e r e (D is the fluid drive efficiency; and (B is the efficiency of the drive belts. The p o w e r used, h o w e v e r , will be g r e a t e r t h a n PM, d u e to losses in the m o t o r itself and losses in some c o n t r o l gear w h e n used, s u c h as an inverter. P o w e r is lost w i t h i n a m o t o r due to a n u m b e r of factors, w h i c h include: 9 9 9 9 9

iron losses in the m a g n e t i s i n g material, p r o d u c i n g h e a t in the m o t o r rotor and stator; friction in the rotor bearings; e n e r g y needed to drive a cooling fan, internally a t t a c h e d to the m o t o r shaft: w i n d a g e losses; and copper losses (the p o w e r lost due to the resistance of the w i n d i n g s , s o m e t i m e s referred to as the I-'R losses).

These five factors combine to give a m o t o r efficiency. CM, of less t h a n unity. Extra p o w e r is also necessarily supplied to the motor, w h e n the p o w e r factor is less t h a n unity. The p o w e r factor will n e v e r be unity, a n d is a m e a s u r e of how m u c h the c u r r e n t lags or leads the applied voltage. It is m e a s u r e d as the cosine of the p h a s e angle b e t w e e n c u r r e n t a n d voltage. W h e n an i n d u c t i o n m o t o r is c o n n e c t e d to an AC electrical supply, w h e t h e r the m o t o r does useful work or not. a c u r r e n t is d r a w n to excite the m o t o r . This c u r r e n t , i n s t a n t a n e o u s l y on start-up, lags 9 0 ~ out of phase with the voltage, a n d is reactive c u r r e n t , or so-called idle or wattless c u r r e n t . The power factor increases as the m o t o r accelerates. W h e n the m o t o r is put to work. it will take in addition to its excitation current, a c u r r e n t according to the a m o u n t of work to be done. The p o w e r factor will increase and will be m a x i m u m w h e n the m o t o r works at its full power rating. T h u s the power t a k e n from the m a i n s supply will be Pc w h e r e : FpPc.4~r = P~I w h e r e Fp is the p o w e r factor.

(4.136)

198 PowerConsumption To combat the anomaly of a low power factor, the installation of a capacitance bank, ideally directly across the motor windings, causes the motor current to reach its m a x i m u m value closer to w h e n the voltage does in the alternating cycle. Therefore, a suitably designed capacitor added to an induction motor will reduce the lag of current, by any desired a m o u n t . Generally, in industry, because the cost of small capacitors is high, it is more economical and expedient to install large banks of capacitors at the supply source, and automatically switch in and out various sets of capacitors as the d e m a n d fluctuates. Moreover, a leading current, w h i c h is possible if the capacitor is too large, increases wattless current as m u c h as a lagging current. Motor m a n u f a c t u r e r s supply motors in standard increments of power. Thus, after power demand for the decanter is calculated, the next larger size is specified. Motor m a n u f a c t u r e r s can supply tables of efficiency and power factors for ranges of loading. Also available are performance curves for their motors, giving output torque against rotational speed. The selected size of motor, for economic reasons, needs to be as near as possible to the power d e m a n d e d by the centrifuge. Details of the installation need to be considered in the motor specification. These factors would include the ambient temperature, w h e t h e r the installation is indoors or out, and w h e t h e r any hazards exist, such as flammable materials in use, and w h e t h e r the motor will need to be hosed down. The installed electrical services need to be assessed to ensure that they are adequate for the method of starting contemplated. It is important that supply cables be adequately sized to minimise voltage drops before reaching the main motor. The power supplied to the motor reduces proportionally to the square of any voltage drop. Nevertheless c u r r e n t will increase to compensate for the drop in voltage, increasing the heating and losses. Moreover regulations restrict voltage drops to a total of 4%. If reduced voltage starting is used, it is important that the reduced starting torque is never less than the sum of the friction and windage torque. Since the torque available to accelerate the bowl is equal to the difference between the motor torque and friction and windage torque, the motor may not reach full speed in a reasonable time, unless care is taken. 4.12.2 Main motor acceleration

Most decanter rotating assemblies have high inertias, which can require several minutes' acceleration [22], or run-up time, ta. If the run-up time is too short, drive belts will slip and wear out prematurely, or even break. If the runup time is too long, then the motor could overheat and burn out. The r u n - u p time is:

ta=

~M(IM + lP) ( T a - Ti)

(4.13 7)

Decanter Theory

199

where ]M is the inertia of the motor; lp is the inertia of the decanter at the motor; 02M is the motor speed; Ta is the m o t o r torque; and Tl is the reactive torque of the decanter. The decanter inertia is given by" 022

lp - -~M lo

(4.138)

where ]D is the inertia of the rotating assembly. Both Ta and Tl vary with speed, and not linearly. Examples of motor and decanter torque/speed curves are shown in Figure 4.15. To use equation (4.13 7), Ta and Tl are averaged over the speed range from zero to full motor speed. Given the inertia at the motor shaft, the equations in this section are used to determine w h e t h e r the torque of the chosen motor is sufficient to accelerate the decanter bowl smoothly to speed, w i t h o u t slippage of the belts. The t h e r m a l limits and the torque limit of the n u m b e r of drive belts used and their cross-sections have to be checked, with the pre-set diameter of the smallest pulley taken into account. Causing the belts to slip will end in their failure, while producing copious a m o u n t s of dust in the belt guard, which could be an explosion hazard.

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40 Speed

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Figure 4.15. Motor and decanWr torque~speed curves.

100

4.13 Mechanical Design The design of a good and reliable decanter centrifuge requires a thorough knowledge of most mechanical engineering disciplines such as machine dynamics, strength of materials, bearing design, and gearbox design. Some of the more important fundamental aspects of the mechanical design of decanter centrifuges will now be discussed. The need for a careful mechanical design can be illustrated by examining the energy accumulated in a decanter centrifuge in operation. The rotational energy in a medium size decanter with rotational inertia of 50 kg m 2. rotating at 3 6 0 0 rpm, will be 3.55 M]. This energy corresponds to the kinetic energy of a vehicle weighing 9.2 tons travelling at 100 km/h. Furthermore, it can be shown that the rotational energy of a decanter centrifuge will increase with the fourth power of the diameter, when the centrifugal force at the bowl wall, go, and the length/diameter ratio, )~, are kept constant. The ratio of the diameters of the largest to the smallest industrial bowl is over 10. Thus, the ratio between the rotational energy of the largest and the smallest decanter centrifuge on the market is over 104. With the high energy involved, failure of one of the major rotating components of a decanter centrifuge can cause severe damage, both to the decanter and its surroundings. For all decanter designs a risk analysis, evaluating all possible failure modes, must be carried out. A European standard [23] deals with the foreseen risks for centrifuges in normal operating environments. The standard gives requirements for design, verification, and installation of centrifuges. The high risk of failure of a decanter requires a high quality, both of design and production, as well as periodic inspection during use, to ensure that unanticipated deterioration of materials of construction has not occurred. From the energy comparison above, it is seen that the risk increases with size, and a design which is adequate for a small laboratoryscale decanter, may be extremely dangerous on a large industrial scale.

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withstand this force with a sufficient safety margin. Components of the decanter may be subjected to several other design-dependent static loads, which must be considered by the designer. One example is the axial load, acting on the conveyor, caused by conveying the solids. The decanter will also be subjected to cyclic loads, which can cause m e c h a n i c a l fatigue damage on both the rotating assembly and on the stationary parts. Among the cyclic loads which must be considered by the designer are the bending forces on the shafts, caused by the weight of the rotor, the loads from belt drives, unbalance forces from the rotor, and cyclic loads from frequent starts and stops, or intermittent loading with process material. On complicated hub geometries, often it will be necessary to make a finite element calculation of the stresses to make a proper fatigue evaluation. The notch sensitivity and ductility of the material m u s t be considered. Quality assurance procedures during m a n u f a c t u r i n g , such as X-raying of critical welds and die penetrant testing of castings, must be maintained.

4.13.2

Critical

speeds

The n a t u r a l frequencies and critical speeds of a decanter will depend on its actual configuration. A conventional decanter centrifuge consists of a flame holding the double rotor - - conveyor and bowl - - in rigid bearings. The m a i n motor can be attached to the rotor flame either by a rigid connection or flexibly through vibration isolators. Further the motor can be attached to a sub-flame or to a n o t h e r part of the supporting structure. These factors are more fully described in Section 2.1.

Decanter Theory

203

At speeds below operating speed the main flame and the rotor can be considered as one rigid body. If the decanter flame is m o u n t e d on soft vibration isolators the decanter assembly will have six natural frequencies and associated vibration modes below the operating speed. The n a t u r a l frequencies are determined by the spring stiffness of the vibration isolators and the mass and inertia of the total system. When the main motor also is supported on the decanter flame by vibration isolators, it will have six additional natural frequencies below the operating speed. The important critical speed for a decanter is the lowest speed at which there is significant flexible deformation of the rotor. This speed is called the first rotor critical speed. Decanters will always have a certain unbalance, both due to the handling of solids from the process and due to wear on the rotor. Operating the decanter close to, or just above, the flexible critical speed of the rotor will result in high vibration levels and very high stresses in the rotor components. The critical value of the rotor speed will therefore be an upper limit for the operating speed, and the decanter must be operated below this speed with a safe margin. The first rotor critical speed will mainly be a function of conveyor geometry, bowl geometry, gearbox weight, main bearing stiffness and conveyor bearing stiffness. The first rotor critical speed will decrease with the length of the decanter. The critical speed of a decanter can be calculated by using a finite element method and verified by measurements. It is normal practice to test decanters at a speed 15-20~ above operating speed, to verify the design integrity, and such an over-speed test can also reveal if the operating speed is close to a critical speed. These factors influencing the first rotor critical speed have been more fully covered by Madsen [24].

4.13.3 Liquid instability problems Often very large vertical and horizontal vibrations are seen in some speed intervals on decanters when they are started and stopped with liquid inside the bowl. The vibration frequencies in the instability intervals correspond to rigid body natural frequencies of the decanter, but the vibrations are not caused by unbalances. Rather, they are due to interaction between the liquid inside the bowl and the decanter. The vibrations which occur in some instability speed ranges are subsynchronous, i.e. the vibration frequency is a fraction (normally about 0.7) of the actual operating speed. If, for example, the decanter is vibrating at an operating speed of 1000 rpm, the vibration frequency will be around 700 rpm. The vibrations are usually harmless, but as very large forces could be acting on the foundations of the decanter, the manufacturer must supply information on the magnitude of these forces, and the foundation must be designed to withstand these forces. By having a constant flow of water to the decanter during starting and stopping, the instability vibrations can be suppressed.

204

MechanicalDesign

The complicated d y n a m i c p h e n o m e n o n , w h i c h is related to all rotating cylinders with an internal a n n u l u s of liquid, has been dealt with in a n u m b e r of publications [25, 26].

4.13.4 Length/diameter ratio In general the bowl strength, the first rotor critical speed, and the m a x i m u m permissible speed of the m a i n bearings control the m a x i m u m speed at which a decanter can be operated. It is argued [24] t h a t a long, slender decanter centrifuge will give advantages with respect to overall economy, power c o n s u m p t i o n and process performance. For a centrifuge with the L/D ratio above 3, the critical speed will often be the m a i n factor controlling the m a x i m u m obtainable speed and it can therefore be desirable to increase the critical speed in order to obtain a high L/D ratio w i t h o u t sacrificing the m a x i m u m operating speed. W h e n a decanter bowl, for calculation purposes, is approximated to a beam, its n a t u r a l frequency is inversely proportional to the square of its length. In that g-force is proportional to the square of bowl speed, and it is necessary to keep resonance frequency above bowl speed, the m a x i m u m bowl speed is proportional to its length to the fourth power. To obtain g-forces in the range 20()0-3()O0, generally required for commercial decanters, the m a x i m u m length-to-diameter ratio, for the most frequently used designs, has to be restricted to a little over 4.0 [24]. In order to increase the critical speed of the rotor a n u m b e r of different modifications can be made to the rotor system. By supporting one or both main bearings in a flexible pillow block the first critical speed of the rotor can be turned into a low speed rigid-body motion for the rotor. It can then be operated supercritically with respect to this critical speed. Other modifications are the floating conveyor and the separately supported gearbox. These sorts of modifications have been utilised by Alfa Laval in producing a decanter with an L/D of over 5 which can operate with up to 10 0 0 0 g. How these modifications extend the possible L/D ratio and clarification capacity was graphed by Madsen, for 2 5 0 m m diameter bowls, and reproduced in Figure 4.17.

4.13.5 Bearing life One of the most frequent reasons for breakdown of decanters is failure of one of the m a i n bearings. The operating conditions of decanters are often very arduous, and there can be a high load on the bearings. The failure of a main bearing on a properly designed decanter will not lead to a dangerous situation, but it can cause d a m a g e to other parts of the decanter, and expensive downtime.

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MechanicalDesign

This simple formula was developed around 1950 and was based on data for bearing fatigue failure. Since the formula was published, considerable progress has been made both in the knowledge of bearing failure, in bearing manufacturing, and in lubrication, which is not reflected by this formula. The formula is based on the assumption of fatigue failure, although it is in fact not the most frequent failure mode for bearings. More realistic methods for calculation of bearing life have since been developed, which both account for the improved quality of bearings, for the bearing environment and for the lubrication conditions. In 1977 the same ISO standard [27] introduced an adjusted life-rating Lloah formula:

Lloah--b,.b2.b3.(~--~)'"

(4.144)

b2

where bl is a constant accounting for reliability; is a constant accounting for the material used; and b3 is a constant accounting for environmental conditions. The key to avoiding failure of bearings is proper maintenance and lubrication. By monitoring and analysing vibrations measured with sensors directly on the bearing housings of a rotating machine, bearing faults can often be detected before they lead to failure. Several systems for detection of bearing faults, by continuous vibration monitoring, are available, and some decanter manufacturers offer their own specialised systems. For critical installations, and installations with several decanters, such monitoring systems can be a good investment, to avoid inconvenient bearing failures, damage to the machine, and unnecessary downtime. 4.13.6 Gearbox life

The decanter manufacturer will often quote the expected life of the gearbox. This will be based on the fatigue life of the gear teeth, which is proportional to the ninth power of the torque encountered. Thus, one has to be extremely careful not to overload the gearbox above its torque rating. An 8~ increase of torque over its rating will halve the expected life of the gearbox. 4.13.7 Feed tube

Each component of the decanter has its own natural frequency, even the stationary components, which could resonate sympathetically if this frequency is close to the bowl speed. The feed tube is a good example, being a long, thin tube. Apart from the inverse square relationship with length, resonance frequency is also proportional to the fourth power of diameter in its simplest form. Unless care is taken the feed tube can be caused to resonate like a tuning fork. The design engineer thus endeavours to maximise diameter and

Decanter Theory 207

minimise the length of the feed tube. Other t e c h n i q u e s employed include tapering the feed tube and making it of lighter materials. Of course the double concentric tube used, when flocculant is added, helps to increase the natural frequency.

4.14 Nomenclature

Symbol

Description

tl 1

Constant Constant Cross-sectional area Total surface area of bowl and beach Surface area of cake bed Dry beach area Centrifuge "area equivalent" Function of restriction area in the conveyor Wet beach area Constant Constant Constant Total length of weirs Expected bearing life in revolutions Weir discharge coefficient Impurity concentration in feed mother liquor Impurity concentration in cake moisture Impurity concentration in centrate Impurity concentration in wash liquor Dynamic load capacity of bearing Equivalent dynamic load of bearing Cumulative fraction by number of particles below sized Cumulative weight or volume of particles below size d Particle diameter Cut point size Required cut point size of the heavy fraction Required cut point size of the light fraction Geometric mean diameter (number basis)

A AB

A~ AD

Ae~_ 4 A~ Aw bl b, b3 B BI() r C1 C C3 C4

C CE Cn C Vr

d dc dch

dcl dg

Dimensions

L2 L2 L2 L2

L2 L2 L2

L

MLT -2 MLT -2

Decanter Theory 209

dgl dgs dgw dm dp d15.87 d so d84.13 D DAy Dp e F FB FN Fs Fp Fx g g~ gc I gl

g-Vol hD hm H If 1D ]D JM kav kl

kl k2 ks k4 ks k6 k7 ks k9

Geometric m e a n diameter (length basis) Geometric m e a n diameter (area basis) Geometric m e a n diameter (weight/volume basis) Hydraulic m e a n diameter Characteristic size of cake bed 15.87% of all particles are below this diameter Median diameter 8 4 . 1 3 % of all particles are below this diameter Diffusivity Mean diameter of pond Pipe diameter Cake voidage Force Beach friction Normal force from conveyor flight Scroll friction Power factor M a x i m u m axial force Acceleration due to gravity Acceleration n u m b e r of times greater t h a n gravity Mean value ofgc in pond Centrifuge g-level at the pond surface g-volume, product of g l and V Mass transfer coefficient Thickness of moving layer Height of interface at time t Height of interface after infinite settling time Impurity concentration % in cake at exit from pond Impurity concentration % in feed Impurity concentration % in cake Mass transfer factor Inertia of the rotating assembly Inertia of the main motor Average cake permeability Constant Constant Constant Constant Constant Constant Constant Constant Constant Constant

L L L L L L L L L2T-1 L L MLT -2 MLT -2 MLT -2 MLT -2 MLT -2 LT-2

L~ LT-2 L L L

ML 2 ML 2 L2

Z-1 T-1 ML2T-3 ML2T-2 ML2T-1

210

Nomenclature

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Na P P Pc PA PB Pc PD PM Pp Ps PT Pwr P2 P2m

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Ol (2p (L QsD

(2, Qw V t"d t"x t"1 F2

Product c o n s t a n t for compaction Clarifying length Cylindrical length of the bowl Wetted beach length Length of vanes on a conveyor Expected bearing life in revolutions Expected bearing life in h o u r s Expected bearing life in h o u r s - adjusted formula Mass Revolutions per minute of bearing Number of discs in a disc stack Number of vanes in a vane stack Number of leads or flights on the conveyor Differential Mass transfer rate Perimeter length Conveyor pitch Pressure in the cake Power available in the feed stream in the bowl Power required for braking Power taken from mains supply Polymer dose Total power absorbed by the main m o t o r Power required to accelerate the process stream Power required for conveying Total power absorbed by the decanter Power to overcome windage and friction Pressure at bowl wall Maximum pressure at bowl wall Volumetric flow rate of feed Volumetric flow rate of feed to decanter 1 Volumetric flow rate of feed to decanter 2 Volumetric flow rate of centrate Volumetric flow rate of flocculant Volumetric flow rate of cake Volumetric flow rate of solids in cake Total volumetric flow rate of flocculant and feed Volumetric flow rate of cake voidage Volumetric flow rate of rinse Radius Radius of discharge of process stream Radius used in Ambler Sigma derivation Pond radius Bowl inside radius

LT-a L L L L T T

M T-1

T-1

MT-a L L ML-1T-2 ML2T-3 ML2T-3 ML2T- 3 ML2T - 3 ML2T -3 ML2T-3 ML2T - 3 ML2T- ~ ML-1T-2 ML-IT-2 L3T-1 L3T-1 L3T-1 L3T-1 L~T-1 L3T-1 L~T-1 L3T-a L3T-1 L3T-1 L L L L L

Decanter Theory

r3 too

R

Re Rein ROB S Sl 82

S Sc Sp Ss t

te tw

T Ta Tl Tp To u Uc v Va Vc Vr Vs

V W Xf X ft Xl .Yp ~s ~Coo

Yr Yl Ys Yw Z

Z

Ouside radius of a disc or vane stack Three quarters bowl radius Radius to which the sludge would settle in infinite time Solids recovery Reynolds n u m b e r Modified Reynolds number Gearbox ratio Distance Distance Distance Bowl speed Schmidt number Pinion speed Volumetric scrolling rate Time Time for particle to traverse decanter Thickness of bowl shell Conveyor torque Motor torque Reactive torque of decanter Pinion torque Heel torque Velocity Superficial velocity Settling velocity of interface Axial velocity Tangential velocity Radial settling velocity Stokes settling velocity Pond volume Constant Solids fraction in feed Solids fraction in total feed and flocculant Solids fraction in centrate Solids fraction in flocculant Solids fraction in cake Solids fraction in cake after infinite compaction time Impurity fraction in feed Impurity fraction in centrate Impurity fraction in cake Impurity fraction in rinse Number of particles less t h a n diameter d Total number of particles

L L L T-1 7-1

LST-1 T T L ML2T-2 ML2T-2 ML2T-2 ML2T-2 ML2T -2 LT-1 LT-1 LT-1 LT-1 LT-1 LT-1 LT-1 L3

211

212

/31 /32

-7 Ac Ar

CB ~D ~s C, q r/L 0 2 71"

P Pb Pr Pl PM Pp Ps PS Psh Psi O'g (7t

s s

Y]2 T

0

~,0M

Nomenclature

Beach angle Beta value: the scrolling capacity Beta value; the scrolling capacity of centrifuge 1 Beta value; the scrolling capacity of centrifuge 2 Angle rotated after time t Mean concentration difference Pond depth Bowl separational efficiency Efficiency of the drive belts Fluid drive efficiency Scrolling efficiency Separational efficiency of decanter 1 Separational efficiency of decanter 2 Viscosity Viscosity of supernatant The half included angle of a disc stack Ratio of bowl length to diameter Pitch angle Universal constant Density Density of bowl material Density of feed Density of centrate Density of supernatant Maximum bulk density of process material Density of flocculant Density of cake Density of solids Density of the heavy solids Density of the light solids Geometric standard deviation Average tangential stress in bowl shell Sigma; equivalent settling area of centrifuge Sigma value for a disc stack Sigma value for a vane stack Modified Sigma for a compaction process Sigma for bowl 1 Sigma for bowl 2 Cake yield stress Cake path angle up the beach Angle between a vane and the radius Thickening factor Angular velocity Motor speed

o

L3T-a L3T-1 L3T-1 o

L

ML-1T-a ML-1T-a o

o

ML-3 ML- 3 ML-3 MLML-3 ML-3 ML-3 ML-3 ML- 3 ML-3 ML-3 ML-1T-2 L2 L2 L2 L2 L2 L2 ML-aT-2 o o

L-3 T-1 T-1

4.15 References 1

F Reif, W Stahl. Transportation of moist solids in decanter centrifuges. Chem

Eng Prog 85(11) (1989) 57-67 2

B Madsen. Flow and sedimentation in decanter centrifuges. IChemE

Symposium Series 113 (1989) 301-17 3

4 5 6 7 8 9

l0 11 12 13 14 15 16 17 18

T Hatch, S P Choate. Description of the size properties of non-uniform particulate substances. Harvard Engineering School, Publ. No. 35 (1928-29) 369--87 F A Records. The Performance of a 4" micronizer. AWRE series 0 reports Number O41/61, Feb. 1962 C M Ambler. The evaluation of centrifuge performance. Chem Eng Prog 48(3) (1952) 150-8 G G Stokes. On the effect of the internal friction of fluids on the motion pendulum. Trans Cam Phil Sot" 9 (1851) 8 G A Frampton. Evaluating the performance of industrial centrifuges. Chem Proc Eng 44(8)(1963) 402-12 C M Ambler. Theory of scaling up laboratory data for the sedimentationtype centrifuge. J Biochem MicrobiolTechnol Eng I (1959) 185-205 S Yano. Experimental studies of separational efficiencies in centrifugal sedimenters. Proceedings of the first China - J a p a n joint international conference on filtration and separation, China, Nov. 1991. Chinese Mechanical Engineering Society & Society of Chemical Engineering, Japan FA Records. Recent advances in sludge processing. Aqua Enviro, University of Leeds, 19 Nov. 1991 A Lavanchy, F W Keith. Centrifugal separation. Kirk Othmer Encyclo Chem Technol & Engng 2nd Edn,Vol. 4, p. 719 N Corner-Walker, FA Records. Filtration+Separation 37(8) (2000) PA Vesilind. Scale-up of solid bowl centrifuge performance. ] Envirmlln Eng Division, ASCE, April 1974 Coulson, Richardson. Chem Eng I (1962) 254 W W-F Leung. Torque requirement for high-solids centrifugal sludge dewatering. Filtration+Separation 35 (1998) 883 (Figure lb) NCorner-Walker. Filtration+Separation 37 (2000) 28-32 E A Relter, 1~ Schilp. Solid-bowl centrifuges for wastewater sludge treatment. Filtration+Separation 31 (5) (1994) Coulson, Richardson. Chem Eng 2 (1956) 515

214 References

19 20 21 22 23 24 25 26

27

] Eiken, B Madsen, ] Oppelstrup. Private communication Perry. Chemical Engineers' Handbook. 3 rd edition 409 W W-F Leung, A H Shapiro. Improved design of conical accelerators for decanter and pusher centrifuges. Filtration+Separation 33 (1996) 735 Electro Courier IV (1976), No. 2 European Standard EN12547. Centrifuges- common safety requirements, CEN 1999 N F Madsen. Slender decanter centrifuges. I Chem E Symposium Series 113 (1989) 281-99 J A Wolf Jr.Whirl dynamics of a rotor partially filled with liquid. ASME ]App Mech December (1968) 678-82 F Ehric. A state-of-the-art survey in rotor dynamics - nonlinear and selfexcited vibration phenomena. Proceedings 2nd International Symposium on Transport Phenomena, Dynamics and Design of Rotating Machinery, Hemisphere Publishing Corporation (1989) ISO 281, Rolling bearings- dynamic load and life ratings