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equations on continuous thickeners

Continuous Thickening TheoryIn a continuous thickener the downward solids velocity with respect to the thickener wall is the sum of the settling velocity of the solids by gravity alone, Vi, plus the downward velocity caused by removal of the thickener underflow, U. The rate at which solids move downward by gravity alone is expressed as the product of the concentration of the solids in a layer, Ci, and the settling The continuousflow gravity thickener: Steady state behavior The equations governi

Description of equations on continuous thickeners

on the method of thickener area calculation based on the data jan 01, 1986 · when applying batchsettling results to continuous thickening, it is implicitly assumed that the thickener wili exhibit ideal thickener properties. shannon and tory (1966) have defined an ideal thickener as one in which there is no radial variation in either the solids concentration or in the downward component of suspension velocity. a critical review of thickener design methods tfig. 3 variables in a continuous thickener. kona no.ll (1993) 2.1 mishler 39;s method. the first equation to predict the capacity of a thickener was developed by mishler6l in 1912 and corresponds to a simple macroscopic mass balance in the equipment. consider a thickener working at steady state, as shown in fig. 4. chapter 3 sludge thickening marmara Üniversitesiat some time, at some level quot;iquotin the thickener, the total flux g i is the sum of these two fluxes, i.e. the underflow flux and the batch settling flux: or if this equation is plotted the graph below would be obtained (figure 6). figure 6. total flux curve [3]. counter current washing and leaching calculationscounter current decantation (ccd) with thickeners is one of the methods generally used. the process engineer needs a quick and accurate ccd calculation method. several theoretical 112 ccd calculation methods are available in the literature . the simultaneous 1, 2 equation method for a complex circuit as used would take too long to solve the the continuousflow gravity thickener: steady state behavior the equations governing consolidation in a continuousflow gravity thickener are developed based on the assumption that a flocculated suspension possesses a compressive yield stress py() that is a function of local volume fraction only. these equations are used to model the steady state operation of a thickener. the bed height required to achieve a given underflow concentration is found to (pdf) thickener design, control and developmentpilotscale continuous thickener te stwork can be conducted using larger r aked thickenersthese are typically about 1 metre diameter w ith a sidewall of about 4 metr es. these can confirm the sm a model of continuous sedimentation of flocculated jul 31, 2006 · the chief purpose of this paper is to formulate and partly analyze a new mathematical model for continuous sedimentationconsolidation processes of flocculated suspensions in clarifierthickener units. this model appears in two variants for cylindrical and variable crosssectional area units, respectively (models 1 and 2). continuous thickening theoryin a continuous thickener the downward solids velocity with respect to the thickener wall is the sum of the settling velocity of the solids by gravity alone, vi, plus the downward velocity caused by removal of the thickener underflow, u. the rate at which solids move downward by gravity alone is expressed as the product of the concentration of the solids in a layer, ci, and the settling thickener design and theory problemscoe and clevenger believed that throughout free settling the settling velocity u was a function of c only. in mathematical terms, u = u(c). about compression they made the following statement: the water liberated by compression finds its way out of zone d (their compression zone) through tubes or channels which form drainage systems upwards through the zone. thus they conceived channeling to be symptomatic of compression. if you saw channeling, you knew you had reached compression. recently it has become apparent that channels must form, either permanently or intermittently, in a concentration range which is not yet in compression, at least as we have conceived compression in the past. and in this range u is not a function of c only. this divides coe and clevenger free settling into two regimes, giving three altogether: 1. zone settling coe and clevenger free settling in which u = u(c) 2. phase settling free settling with channeling, u u(c) 3. compression floc stru see full list on 911metallurgist kynch started from the postulate that u = u(c). as noted above, this was assumed true by coe and clevenger for their free settling, and we have used it as the defining relationship for the zone settling regime. using a continuity approach, kynch showed that: 1. a concentration discontinuity would propagate with a velocity: u = g/c 2. a locus of constant concentration in a concentration gradient will propagate with a velocity: u = dg/dc see full list on 911metallurgist the lower conjugate concentration poses an unresolved problem. continuity would demand that this concentration exist between the feed point and whatever higher concentration zones exist below. but there is no necessity to assume continuity. and many thickeners show a mud line far belowthe feed point, with clear supernatant above, at least near the periphery where sample lines are commonly provided, so a lower conjugate concentration does not have to exist. yet scott reports finding concentrations apparently corresponding to upper conjugate in some test operations. if feed is more concentrated than a lower conjugate concentration, there is a large body of literature in the civil engineering field which says it will plunge to the bottom of such a zone like a submerged waterfall as a density current. it will spread out at the level of hydrostatic equilibrium. and it can be shown mathematically that by the time it has spread out over the entire thickener area its load of solids will see full list on 911metallurgist comings long ago showed solids profiles in a continuous thickener. concentration varies with depth, and such behavior was common industrial knowledge. theory based on the postulate that u = u(c) does not permit this. therefore the idea that zone settling could explain all thickening behavior was never logically tenable. classically, pulps not in zone settling were assumed to be in compression. coe and clevenger defined a compression zone as that portion of the pulp where the flocs considered as integral bodies, have settled to a point where they rest directly one upon another. after pulp enters zone d, further separation of liquid must come through liquid pressed out of the flocs and out of the interstitial spaces between the flocs. see full list on 911metallurgist classically it was believed that channeling accompanied compression. this has always been hard to rationalize, and leaves several questions. first, what hs channels open, when solids pressure necessarily exceeds the yield value of the solids structure to cause compaction? if it is hydraulic pressure, what pressure gradients then remain to push water out of the floc structure into the channels? and why should channels open up to begin with, when the structure is being crushed together overall? as far as we know, nobody has thought this through and come up with a completely plausible explanation. on the other hand, there can be no doubt that channeling does occur in sedimenting pulps. in the first place, you can often see it. particularly with metallurgical pulps it visibly takes the form described by coe and clevenger, great drainage systems upward through the pulp. at their mouths these rivers of fluid may be of the order of a millimeter across, and they erupt through the surface see full list on 911metallurgist in batch settling, the compression point has been generally conceived as that at which solids just below the pulp supernatant interface pass into the compression regime. it is identified with a discontinuity in the slope of a batch settling curve of h vs t. from a to b the subsidence plot is linear, corresponding to initial concentration. at b higher concentration or kynch zones start appearing at the interface, giving a first slope discontinuity. at some point c there is a second discontinuity in slope, which is taken as the compression point. the slope discontinuities are usually more easily spotted on a log log plot of h vs t. experience shows, however, that in many loosely flocculant pulps there is no discontinuity apparent to identify as the compression point. this gives rise to considerable uncertainty in the coe and clevenger procedure, and also to the idea that if the results were wrong, it was because the compression point had not been properly identified. see full list on 911metallurgist from the foregoing it will be apparent that there are many unresolved problems in the theory of thickening. but the practicing engineer cannot cease specifying thickener size until theoretical types get them straightened out. he must do the best he can with the information available. fortunately things are not quite as bad practically as this paper makes them sound. the announced idea of the paper was to emphasize what we do not know. in most metallurgical applications the coe and clevenger procedure will continue to yield thickeners of adequate design, as it has for the past fifty s. however, the talmage and fitch procedure should be preferred for determining thickener area. it is simpler, and even though it has been shown to be theoretically not quite valid in the vicinity of the compression point, empirically it has been shown to predict area more accurately, (and more rapidly) than the coe and clevenger procedure. see full list on 911metallurgist autoprecipitation modelling in a thickenertransient behavior of flocculated suspensions in an ideal batch or continuous thickener. international journal of mineral processing, 55 (november 1999), 267282. 12. r. bürger and f. concha, mathematical model and numerical simulation of the settling of flocculated suspensions. international journal of multiphase flow, 24 (may 1998), 1005 3

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Advantages of equations on continuous thickeners

on the method of thickener area calculation based on the data jan 01, 1986 · when applying batchsettling results to continuous thickening, it is implicitly assumed that the thickener wili exhibit ideal thickener properties. shannon and tory (1966) have defined an ideal thickener as one in which there is no radial variation in either the solids concentration or in the downward component of suspension velocity. a critical review of thickener design methods tfig. 3 variables in a continuous thickener. kona no.ll (1993) 2.1 mishler 39;s method. the first equation to predict the capacity of a thickener was developed by mishler6l in 1912 and corresponds to a simple macroscopic mass balance in the equipment. consider a thickener working at steady state, as shown in fig. 4. chapter 3 sludge thickening marmara Üniversitesiat some time, at some level quot;iquotin the thickener, the total flux g i is the sum of these two fluxes, i.e. the underflow flux and the batch settling flux: or if this equation is plotted the graph below would be obtained (figure 6). figure 6. total flux curve [3]. counter current washing and leaching calculationscounter current decantation (ccd) with thickeners is one of the methods generally used. the process engineer needs a quick and accurate ccd calculation method. several theoretical 112 ccd calculation methods are available in the literature . the simultaneous 1, 2 equation method for a complex circuit as used would take too long to solve the the continuousflow gravity thickener: steady state behavior the equations governing consolidation in a continuousflow gravity thickener are developed based on the assumption that a flocculated suspension possesses a compressive yield stress py() that is a function of local volume fraction only. these equations are used to model the steady state operation of a thickener. the bed height required to achieve a given underflow concentration is found to (pdf) thickener design, control and developmentpilotscale continuous thickener te stwork can be conducted using larger r aked thickenersthese are typically about 1 metre diameter w ith a sidewall of about 4 metr es. these can confirm the sm a model of continuous sedimentation of flocculated jul 31, 2006 · the chief purpose of this paper is to formulate and partly analyze a new mathematical model for continuous sedimentationconsolidation processes of flocculated suspensions in clarifierthickener units. this model appears in two variants for cylindrical and variable crosssectional area units, respectively (models 1 and 2). continuous thickening theoryin a continuous thickener the downward solids velocity with respect to the thickener wall is the sum of the settling velocity of the solids by gravity alone, vi, plus the downward velocity caused by removal of the thickener underflow, u. the rate at which solids move downward by gravity alone is expressed as the product of the concentration of the solids in a layer, ci, and the settling thickener design and theory problemscoe and clevenger believed that throughout free settling the settling velocity u was a function of c only. in mathematical terms, u = u(c). about compression they made the following statement: the water liberated by compression finds its way out of zone d (their compression zone) through tubes or channels which form drainage systems upwards through the zone. thus they conceived channeling to be symptomatic of compression. if you saw channeling, you knew you had reached compression. recently it has become apparent that channels must form, either permanently or intermittently, in a concentration range which is not yet in compression, at least as we have conceived compression in the past. and in this range u is not a function of c only. this divides coe and clevenger free settling into two regimes, giving three altogether: 1. zone settling coe and clevenger free settling in which u = u(c) 2. phase settling free settling with channeling, u u(c) 3. compression floc stru see full list on 911metallurgist kynch started from the postulate that u = u(c). as noted above, this was assumed true by coe and clevenger for their free settling, and we have used it as the defining relationship for the zone settling regime. using a continuity approach, kynch showed that: 1. a concentration discontinuity would propagate with a velocity: u = g/c 2. a locus of constant concentration in a concentration gradient will propagate with a velocity: u = dg/dc see full list on 911metallurgist the lower conjugate concentration poses an unresolved problem. continuity would demand that this concentration exist between the feed point and whatever higher concentration zones exist below. but there is no necessity to assume continuity. and many thickeners show a mud line far belowthe feed point, with clear supernatant above, at least near the periphery where sample lines are commonly provided, so a lower conjugate concentration does not have to exist. yet scott reports finding concentrations apparently corresponding to upper conjugate in some test operations. if feed is more concentrated than a lower conjugate concentration, there is a large body of literature in the civil engineering field which says it will plunge to the bottom of such a zone like a submerged waterfall as a density current. it will spread out at the level of hydrostatic equilibrium. and it can be shown mathematically that by the time it has spread out over the entire thickener area its load of solids will see full list on 911metallurgist comings long ago showed solids profiles in a continuous thickener. concentration varies with depth, and such behavior was common industrial knowledge. theory based on the postulate that u = u(c) does not permit this. therefore the idea that zone settling could explain all thickening behavior was never logically tenable. classically, pulps not in zone settling were assumed to be in compression. coe and clevenger defined a compression zone as that portion of the pulp where the flocs considered as integral bodies, have settled to a point where they rest directly one upon another. after pulp enters zone d, further separation of liquid must come through liquid pressed out of the flocs and out of the interstitial spaces between the flocs. see full list on 911metallurgist classically it was believed that channeling accompanied compression. this has always been hard to rationalize, and leaves several questions. first, what hs channels open, when solids pressure necessarily exceeds the yield value of the solids structure to cause compaction? if it is hydraulic pressure, what pressure gradients then remain to push water out of the floc structure into the channels? and why should channels open up to begin with, when the structure is being crushed together overall? as far as we know, nobody has thought this through and come up with a completely plausible explanation. on the other hand, there can be no doubt that channeling does occur in sedimenting pulps. in the first place, you can often see it. particularly with metallurgical pulps it visibly takes the form described by coe and clevenger, great drainage systems upward through the pulp. at their mouths these rivers of fluid may be of the order of a millimeter across, and they erupt through the surface see full list on 911metallurgist in batch settling, the compression point has been generally conceived as that at which solids just below the pulp supernatant interface pass into the compression regime. it is identified with a discontinuity in the slope of a batch settling curve of h vs t. from a to b the subsidence plot is linear, corresponding to initial concentration. at b higher concentration or kynch zones start appearing at the interface, giving a first slope discontinuity. at some point c there is a second discontinuity in slope, which is taken as the compression point. the slope discontinuities are usually more easily spotted on a log log plot of h vs t. experience shows, however, that in many loosely flocculant pulps there is no discontinuity apparent to identify as the compression point. this gives rise to considerable uncertainty in the coe and clevenger procedure, and also to the idea that if the results were wrong, it was because the compression point had not been properly identified. see full list on 911metallurgist from the foregoing it will be apparent that there are many unresolved problems in the theory of thickening. but the practicing engineer cannot cease specifying thickener size until theoretical types get them straightened out. he must do the best he can with the information available. fortunately things are not quite as bad practically as this paper makes them sound. the announced idea of the paper was to emphasize what we do not know. in most metallurgical applications the coe and clevenger procedure will continue to yield thickeners of adequate design, as it has for the past fifty s. however, the talmage and fitch procedure should be preferred for determining thickener area. it is simpler, and even though it has been shown to be theoretically not quite valid in the vicinity of the compression point, empirically it has been shown to predict area more accurately, (and more rapidly) than the coe and clevenger procedure. see full list on 911metallurgist autoprecipitation modelling in a thickenertransient behavior of flocculated suspensions in an ideal batch or continuous thickener. international journal of mineral processing, 55 (november 1999), 267282. 12. r. bürger and f. concha, mathematical model and numerical simulation of the settling of flocculated suspensions. international journal of multiphase flow, 24 (may 1998), 1005 3

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The Project of equations on continuous thickeners

theory and application of thickener designby equation 3. the underflow operating line will betangential to the flux curve ifthe feed flux gp is greater than gu as is the case in figure 1. by performing a continuous thickening experiment under nonsteady state conditions (gpgt;gu) at a particular underflow velocity, and determining the final underflow 643kb a.g. waters, k.p. galvin 8 1991chapter 3 sludge thickening marmara Üniversitesiat some time, at some level quot;iquotin the thickener, the total flux g i is the sum of these two fluxes, i.e. the underflow flux and the batch settling flux: or if this equation is plotted the graph below would be obtained (figure 6). figure 6. total flux curve [3]. thickening springerlinkdec 20, 2013 · the theory of sedimentationconsolidation is deduced from the equations for a particulate system and constitutive equations for the solidfluid interaction force and sediment compressibility are postulated. batch and continuous sedimentation are analyzed and simulations are compared to data from the literature. simulation of batch and continuous thickeners sciencedirectjan 01, 1993 · continuous thickeners: runs c (constant under flow solids concentration). 2047 simulation of batch and continuous thickeners x distance down column or thickener, m r distance to sediment surface, m greek letters ap difference between the solids density and the fluid density, kg/m3 s, volume fraction of solids in sediment value of e, at cake continuous countercurrent decantation calculationsa total of (12~). coming into this same thickener is the overflow of thickener z, plus the solution after precipitation which carries 39a total value of 8, (400 tons of solution, precipitated to two cents). the only other source of values coming to thickener y is the underflow of thickener x. the value of its dynamics of continuous thickeningthickening this equation is subject to the boundary con ditions: 1. flux (at z = 0) = flux imposed on the thickener whenever a dilute solids concentra tion exists within the thickener, and 2. d(cv8)/dz = 0 at the bottom of the thickener. solution of equation 3 requires definition of the velocity of displaced fluid at any position steadystate, control, and capacity calculations for diffusion equation, with the discontinuous flux appearing in the recently analyzed clarifierthickener (ct) setup. this setup includes both clarification and thickening zones of clarifierthickener units, while the earlier ideal continuous thickener (ict) concept explicitly models the thickening zone only. (pdf) kynch theory of sedimentationnov 18, · the traditional equation to calculate unit areas for continuous thickeners based on kynch 39;s theory is not correctly interpreted in the literature. a different explanation is given here. view fulltext autoprecipitation modelling in a thickenertransient behavior of flocculated suspensions in an ideal batch or continuous thickener. international journal of mineral processing, 55 (november 1999), 267282. 12. r. bürger and f. concha, mathematical model and numerical simulation of the settling of flocculated suspensions. international journal of multiphase flow, 24 (may 1998), 1005

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