A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive
ESTIMATION OF SLUDGE SEDIMENTATION
ESTIMATION OF SLUDGE SEDIMENTATIONPARAMETERS FROM SINGLE BATCH SETTLING CURVESALEXIS VANDERHASSELT 1M and PETER A.VANROLLEGHEM 2*M1Laboratory of Microbial Ecology,University Gent,Coupure Links 653,B-9000Gent,Belgium and2BIOMATH Department,University Gent,Coupure Links 653,B-9000Gent,Belgium(First received 1December 1998;accepted in revised form 1April 1999)Abstract ÐIn this paper a comparison is made between two means of obtaining the parameters for the settling velocity models that are at the core of the solid ¯ux theory:(i)the traditional approach using zone settling velocity (V ZS )data obtained from a dilution experiment and (ii)a new direct parameter estimation method relying on a single batch settling curve (SBSC).For four distinct sludges,settling curves were recorded at di erent sludge concentrations (X )and the Vesilind parameters were calculated in the traditional way.The value of the resulting model was evaluated by cross-validating it on its ability to describe complete SBSC's.Provided the dynamics of the sludge blanket descent were moderate a settler model incorporating these traditional Vesilind parameters could reasonably match the experimental batch settling curves.However,when the dynamics of the sludge blanket descent were fast,the Vesilind model failed.It was tried whether other settling velocity models could result in better ®ts to the SBSC.Here,the Cho model turned out to be the most e ective one.When cross-validating the Cho model on the dilution experiment V ZS -data,however,it was found to be less performing than the Vesilind model in describing the relationship between V ZS and X .The fact that the Vesilind model is superior to the Cho model in describing such relationships,while the Cho model is better in describing complete settling curves,clearly points out that current settling models are still very empirical in nature.An important ®nding was that with all settling velocity models tested,practical identi®ability problems appeared,indicating the need for better experimental designs.Finally,the ¯ux curves associated with the SBSC-estimated Cho parameters were compared with the traditional Vesilind ¯ux curves.Although both types of ¯ux curves generally showed similar trends,the reliability of SBSC-based ¯ux curve predictions is at present insu cient to warrant replacement of the traditional estimation of settling characteristics.#1999Elsevier Science Ltd.All rights reservedKey words Ðactivated sludge,batch settling curves,¯ux curves,parameter estimation,sedimentation,settling modelsNOMENCLATUREA surface of the clari®er (m 2)V 0Vesilind maximum settling velocity (m/h)k 'Cho parameter (kg/m 2h)n Vesilind parameter (m 3/kg)n 'Cho parameter (m 3/kg)r h Takacs settling parameter associated with the hindered settling component of the settling velocity equation (m 3/kg)r p Takacs settling parameter associated with the low concentration and slowly settling component of the suspension (m 3/kg)SBSC Single Batch Settling CurveV 0'Takacs maximum theoretical settling velocity (m/h)V ZS stirred zone settling velocity (m/h)X sludge concentration (kg/m 3)X min minimum attainable sludge concentration (kg/m 3)X t threshold sludge concentration for activation of the minimal ¯ux restriction Q uunder¯ow ¯ow rate (m 3/h)INTRODUCTIONThe design and operation of secondary clari®ers is commonly based on the solid ¯ux theory (Ong,1992;Daigger,1995).The basic data required for the application of this theory can be obtained from multiple batch tests by which the stirred zone settling velocities (V ZS )over a range of sludge con-centrations (X )are measured (dilution experiment).The relationship between X and V ZS is most often characterised by the Vesilind equation [Eq.(1)]which contains two parameters:V 0and n .The key underlying assumption is that the settling velocity is only dependent on the local sludge concentration.The Vesilind parameters can then be used to con-Wat.Res.Vol.34,No.2,pp.395±406,2000#1999Elsevier Science Ltd.All rights reservedPrinted in Great Britain0043-1354/99/$-see front matter395/locate/watresPII:S0043-1354(99)00158-X*Author to whom all correspondence should be addressed.Tel.:+32-9-264-5932;fax:+32-9-223-4941;e-mail:peter.vanrolleghem@rug.ac.bestruct the ¯ux curve which is the key input of thesolid ¯ux theory.Obtaining the Vesilind parameters by a dilution experiment is unfortunately time con-suming and labour intensive and can result in scat-tered data (Ekama et al.,1997).An alternative approach consists of linking sludge volume indices as SVI,DSVI and SSVI to V 0and n through empirical functions (Daigger,1995;Ozinsky and Ekama,1995).This linking is done on the basis of extensive historical data sets on sludge volume indi-ces and Vesilind parameters.These relationships have to be used with care because the V ZS is in¯u-enced by factors not incorporated in the sludge con-centration and sludge volume indices (Ozinsky and Ekama,1995).A more structural analysis of the di erent SVI indices allowed Bye and Dold (1998)to seriously question the validity of any correlation between Vesilind parameters and sludge volume in-dices.This leaves operators and researchers at pre-sent with the choice between a labour intensive and a non-unconditional approach.Cacossa and Vaccari (1994)and Vanrolleghem et al.(1996)successfully ®tted one-dimensional settling models to single batch settling curves (SBSC).Hence,batch settling curve based parameter esti-mation appears to have the potential of being a good means of e ciently obtaining information on sludge settling characteristics for use in the solid ¯ux theory.However,each of the above studies was restricted to one speci®c kind of sludge and in neither of the mentioned papers ¯uxes calculated with the obtained parameter estimates were com-pared against ¯uxes obtained with the traditional approach using dilution experiment V ZS 's.Consequently,the work that resulted in this paper focused on the ability of SBSC-based charac-terisation of settling velocity models for use in solid ¯ux theory instead of the method based on dilution experiment V ZS -data.Attention was particularly paid to the practical identi®ability of the parameters from such SBSC data sets and to the comparison between the resulting ¯ux curves and the ¯ux curves obtained from traditional interpretation of dilution experiment V ZS -data sets.METHODSExperimental layoutThe recently developed Settlometer (Vanrolleghem et al.,1996;commercially available from Applitek nv,Deinze,Belgium)allows to automatically record complete batch sludge settling curves,and quantify the initialsettling velocity (V ZS )and the stirred sludge volume (SSV).Sludge settles in batch mode in a settling column that has a height of 70cm and a diameter of 14cm,and is equiped with a stirrer (0.3rpm).Four di erent kinds of sludge (Table 1)were brought to the laboratory where they were aerated over night before use in an experiment.In this way sludge with stable sedi-mentation characteristics was obtained (Vanderhasselt and Verstraete,1999).The storage and settling experiments were performed at room temperature.Each sludge was thickened for 1h and subsequently pumped into the Settlometer where it was allowed to settle for 35min.Next,the sludge was rehomogenised by shortly mixing with air.To set up an experiment at a lower sludge concentration,part of the sludge in the device was sub-sequently replaced by supernatant and a new settling curve was recorded.The amount of sludge to be removed at each dilution step was calculated in such a way that the obtained sludge concentrations were more or less equidi-stant.This procedure was repeated until six settling curves were recorded.Further,care was taken to ensure that at the highest sludge concentration applied still a substantial descent of the sludge blanket occurred,while at the lowest concentration,still a sharp sludge blanket/water interface could be detected.The suspended solids concentration was determined by centrifugation (NBN 366.01,1956).From the dilution experiment described above,the Vesilind par-ameters were obtained by linear least-squares regression of the log(V ZS )against the sludge concentration.Parameters obtained with this approach will be referred to as the tra-ditional Vesilind parameters.Modelling layoutIn the past,quite a number of settling velocity models have been presented in literature (Vesilind,1968;Dick and Young,1972;Takacs et al.,1991;Otterpohl and Freund,1992;Cho et al.,1993;Watts et al.,1996).On lab settler data Grijspeerdt et al.(1995)identi®ed the Takacs et al.(1991)model [Eq.(2)]to be the best.Watts et al.(1996)stated that for the prediction of sludge blanket dynamics the Takacs model [Eq.(2)]can be simpli®ed to the Vesilind model [Eq.(1)].The latter model clearly is the most frequently used (Ong,1992;Ozinsky and Ekama,1995;Bye and Dold,1998).Cho et al.(1993)also advised that the number of parameters in a model should be lim-ited to two in order to be simple enough for application in practice.In view of these re¯ections the Vesilind model was identi®ed as the most interesting to start ter in the study the two-parameter Cho settling velocity model [Eq.(3)]was applied as well.V s V 0e ÀnX1 V s V H 0 e Àr h X ÀX minÀe Àr p X ÀX min2V s kH eÀn H XXX 3Batch settling experiments were thought to be an inter-esting information source for parameter estimation as the data are the result of only the physical properties of theTable 1.``Traditional ``Vesilind parameters for the di erent sludgesSludge Concentration range (kg/m 3)V ZS range (m/h)V 0(m/h)n (m 3/kg)Correlation coe cientB 15.6±3.00.5±3.610.30.190.9693D 4.1±0.50.4±4.7 5.50.640.9712O 14±2.20.8±9.112.20.210.9677P6.6±1.00.7±3.26.90.360.9854Alexis Vanderhasselt and Peter A.Vanrolleghem396measuring device and the settling properties of the sludge.The settling device can be assumed to have constant prop-erties.There are no bulk ¯ows in the batch-wise operating device and,consequently,no associated momentums that in¯uence the movement of the sludge blanket.Additionally,model calibration problems such as the lo-cation of the in¯uent layer (Watts et al.,1996;Ekama et al.,1997)or the determination of short circuit ¯ows from the feedlayer to the sludge recycle (Dupont and Dahl,1995)are avoided.The settling velocity models were incorporated in a set-tler model discretisised into 49horizontal layers of equal volume.This discretisation can be seen as a ®nite di er-ence approximation of the underlying partial di erential equation and was introduced by Stenstrom (1975).The large number of layers was chosen to resolve the detailed behaviour of the settling process as recommended by Jeppsson and Diehl (1996).Further,this number of layers is required to obtain a resolution that is similar to the ac-curacy of the measured settling curve.The layer structure was combined with the minimal ¯ux restriction that restrains the gravitational ¯ux from a certain layer to the layer below to the minimal gravitational ¯ux of both layers (Stenstrom,1975).Vitasovic (1989)activated this restriction only when the sludge concentration reached a certain minimum concentration (X t ),i.e.3kg/m 3.Ekama et al.(1997)identify this restriction as being necessary for numerical stability of the simulation.However,there is also physical ground to this concept:a layer can not unconditionally discharge sludge to a layer in which there is already a substantial amount of sludge.To avoid that X t would in¯uence the result without being actually esti-mated,it was decided to activate the minimal ¯ux restric-tion throughout the whole concentration region (X t =0kg/m 3)for all sludges considered.This does not pose a pro-blem for the description of batch settling data because the sludge concentrations that determine the height of the sludge blanket are relatively high and the sludge concen-tration increases continuously from top to bottom.Further,for Vesilind and Cho based settler models these sludge concentrations are located in the descending part of the ¯ux curve,so the minimal ¯ux restriction will always be active regardless the fact whether X t is equal to 0or 3kg/m 3.The height of the sludge blanket was de®ned as the lo-cation of the uppermost layer with a sludge concentration higher than a certain threshold.Vitasovic (1986)and Vanrolleghem et al.(1996)used 3kg/m 3as the threshold.From a numerical simulation with di erent sludge blanket thresholds it was found that the de®nition of this threshold is not critical.Evidently,this statement only holds as long as the threshold sludge concentration is lower than the sludge concentration at the start of the ex-periment.For practical reasons this sludge blanket threshold was therefore set to 1kg/m 3,being lower than the initial sludge concentration of all modelled exper-iments.When a settling curve is recorded (Fig.1)one observes that some start-up phenomena take place before the des-cent of the blanket starts.To incorporate also these phenomena in the batch settling model,the settling vel-ocity model was slightly changed.This was done by multi-plying the settling velocity model with an on±o term.This term is 0when the simulation time (t )is smaller than the start-up time (T st )and 1when it is larger than T st .In this way the start-up phenomena are incorporated in the model without a ecting the settling part itself.Moreover,this method has the advantage that the settling velocity model is activated as soon as the blanket starts to des-cend.RESULTSBefore starting the parameter estimation,the clari®er model was evaluated from a theoretical point of view.Analysis of the concentrationpro®lesFig.1.Settling curves from the dilution experiment with the B sludge.Estimation of sludge sedimentation parameters 397showed that the evolution of the sludge blanket in the zone settling phase is determined only by the in-itial sludge concentration.Only when the settling curve starts to bend (i.e.when the settling velocity decreases)higher concentrations have a signi®cant impact on the evolution of the sludge blanket.Hence,only from that time instant onwards,the data contain information on more than one sludge concentration.Therefore,it is postulated to be an essential experimental condition that the obtained settling curve shows su cient amount of bending before reliable parameter estimation can be per-formed on a SBSC.This statement can be sup-ported from a mathematical point of view:thereisFig.2.Data and model output from the Vesilind based model for the O sludge at 11.0kg/m 3.(A)Traditional calibration on set of V ZS .(B)Optimisation result V 0=13.1m/h,n =0.219m 3/kg,T st =6.70E À02h.Alexis Vanderhasselt and Peter A.Vanrolleghem398only one additional parameter needed to describe a straight line once one point (the initial height at the starting of the sedimentation phase)of the curve is known.Settling curves at di erent sludge concentrations were recorded for each sludge.As an example the resulting settling curves for the B sludge are depicted in Fig.1.For the di erent sludges,the tra-ditional Vesilind parameters resulting from such di-lution experiment are summarised in Table 1.Looking at the parameter values one can conclude that sludges with good settling properties (B,O)as well as poor settling sludges (D,P)were included in the experimental plan.Cross-validation 1:traditional Vesilind parameters to single batch settling curveFor each experimental data set,the batch settler model was solved using the traditional Vesilind par-ameters.For T st ,a value was used that gave a mini-mal squared error (SSE)between the simulation result and the data.Provided that the dynamics in the sludge blanket were rather slow,reasonable agreement (and,thus,cross-validation)could be obtained between the data and the simulation results e.g.Fig.2(A).Still,when the zone settling phase is ending and the settling curve starts to bend [in Fig.2(A)between t =0.2and t =0.5h]asmallFig.3.Data and model output for the B sludge at 5.0kg/m 3.(A)For the Vesilind based model with traditional calibration on set of V ZS .(B)Optimisation results of the Cho based model k '=29.6kg/h m 2,n '=0.12m 3/kg,T st =1.10E À02.Estimation of sludge sedimentation parameters 399discrepancy appears between the data and the simu-lation result.For settling curves with faster dynamics (higher V ZS ),e.g.Fig.3(A),it becomes obvious that the discrepancy becomes bigger for this transition zone.Still,although the model (with Vesilind settling velocity model and parameters obtained from a dilution experiment)is not able to predict a solids pro®le that allows a smooth tran-sition between the zone settling phase and the com-paction phase,it is able to accurately describe the zone settling phase and the ®nal height of the blan-ket.Similar trends could be observed with the other sludges.It can be concluded that the model allows to adequately describe slow dynamic batch settling curves when it is calibrated with parameters from a dilution experiment.Parameter estimation from a slow dynamic single batch settling curveFor the estimation from a SBSC approach,a direction set algorithm (Brent,1973)was used to ®nd the parameters (V 0,n and T st )that would result in a minimal SSE between the observed data and the simulation results.For the data of Fig.2(A)this resulted in the ®t depicted in Fig.2(B).As can be seen for this slow dynamic SBSC,the obtained ®t is rather good.The ®t in Fig.2(B)is better than the ®t of Fig.2(A).This is as expected because the simulated curve in Fig.2(A)is not the result of a parameter estimation exercise,but the result of a cross-validation of parameters obtained from the dilution experiment.Further,the estimated Vesilind parameters (V 0=13.1;n =0.22)can be considered rather close to the ones obtained from the traditional determination (V 0=10.3;n =0.19).Although the above is important for the quality assurance of lab experiments,it is the ¯ux curves in which practitioners are interested.In order to evalu-ate this,the ¯ux curves for both parameter sets (tra-ditional vs SBSC estimation)were plotted against each other (Fig.4).As can be seen from this ®gure,both curves are rather close to each other,although in the upper concentration region di erences run up to 17%.Fig.4can also be considered as a cross-validation from the SBSC approach to the dilution approach.In the present case,the cross-validation result is reasonable.However,in other cases (data not shown)the results were less.Evaluation of practical identi®abilityWhen estimating parameters it is important to check whether the available data are informative enough to give unique values to the model's par-ameters.When the information content of a data set is too low,it is possible that di erent parameter sets can be found giving equal ®t.This practical identi®ability phenomenon surfaces when di erent estimates are found when the parameter search al-gorithm is started from di erent initial parameter guesses.In order to assess eventual practical identi-®cation problems in this application,the sum of squared errors was calculated for a large number of parameter sets in a parameter subspace.In the resulting Fig.5one observes no sharp minima but rather broad valleys.This clearly points towards identi®cation problems,i.e.it is not straightforward to ®nd the lowest sum of squared errors,leading to the best parameter set.This identi®ability problem was further con®rmed by tests that were conducted in which the search algorithm was startedfromFig.4.Flux curves for the traditional and the SBSC (X =11.0kg/m 3)estimated Vesilind parametersfor the O sludge.Alexis Vanderhasselt and Peter A.Vanrolleghem400di erent initial parameter sets,resulting in di erent optimisation results.As the SBSC approach was not successful in ®nd-ing unique parameter values,it was evaluated whether a multiple response ®tting method as used by De heyder et al.(1997)could provide a solution.In this approach,parameters are estimated using di erent settling curves obtained with di erent con-centrations of one kind of sludge.The identi®cation is then based on the fact that each settling curve contains di erent information due to the di erent initial sludge concentration and the fact that for all curves the same Vesilind parameters must hold.Note,however,that T st is dependent on V ZS (Vanderhasselt et al.,1999)and hence,it is di erent for each curve.Therefore,an optimal T st value was determined for each settling curve.Subsequently,for each settling curve the sum of squared errors was calculated in the V 0/n parameter space.The resulting error surfaces were quite similar inshapeFig.5.Simulation error as function of di erent parameter combinations for O sludge at 11.0kg/m 3.Top:n =0.219m 3/kg.Bottom:T st =0.067h.Estimation of sludge sedimentation parameters 401as the one at the bottom bining these surfaces did not result in a graph with a sharp minimum.Consequently,multiple response ®tting was not successful in solving the identi®ability pro-blem.Parameter estimation from a fast dynamic single batch settling curveFor sedimentation curves with faster dynamics it was not possible to obtain Vesilind parameters which were able to describe the curve adequately.In particular,no parameter set could be found that was able to describe the transition between the zone settling phase and the compression phase substan-tially better than the traditional Vesilind parameters [Fig.3(A)].Because this Vesilind approach was not fully suc-cessful,alternative settling velocity models as pro-posed by Takacs et al.(1991)and Watts et al.(1996)were evaluated.Neither of these models was able to describe the data substantially better than the Vesilind model.A hybrid Vesilind settling velocity model [Eq.(3)]that was proposed by Cho et al.(1993)to describe the relation between V ZS and X in dilution exper-iments,was tested subsequently.It is important to realise that special attention must be directed to the lower concentration region when this Cho model is used to calculate settling velocities:when the sludge concentration reaches zero,the settling velocity goes to in®nity,which is not physically possible.In order to prevent this,a threshold concentration below which the settling velocity is set to zero was incorporated.Arbitrarily this constant was set at 0.1kg/m 3.The precise value of this threshold was identi®ed not to be critical for this particular con-centration.One should note the correspondence between this threshold concentration and the non-settleable fraction X min in the Takacs settling vel-ocity model.Among all models tested,the Cho model gave the best ®ts to the settling curves under consider-ation,i.e.it gave the lowest SSE.The better ability of the Cho model to describe complete batch settling curves is obvious if one compares the ®t of the Vesilind model [Fig.3(A)]with the one of the Cho model [Fig.3(B)].While the Cho model pro-vided better ®ts indicating its superior structural ¯exibility,still identi®ability problems similar to the problems observed with the Vesilind model were encountered.In Fig.6the Cho ¯ux curves calculated with par-ameters obtained from di erent SBCS at di erent concentrations are depicted.Next to the Cho ¯ux curves the traditional Vesilind ¯ux curve is depicted.Considering Fig.6,it is clear that the ¯uxes obtained with the SBSC-estimated parameters should be used with precaution as there can be some di erence between the ¯ux curves obtained from parameters estimated from SBSC's at di erent sludge concentrations.For the D sludge as well as for the other sludges the highest ¯ux curves were obtained with parameters estimated from the settling curve with the highest initial sludge concen-tration.In the higher sludge concentration region the Cho ¯ux curves with parameters obtainedfromFig.6.Flux curves for D sludge,calculated with the Vesilind parameters traditionally calibrated on aset of V ZS 's and Cho parameters obtained from di erent SBSC's.Alexis Vanderhasselt and Peter A.Vanrolleghem402the SBSC approach were situated below the tra-ditional Vesilind ¯ux curve.For each sludge the ¯ux curves with the par-ameters obtained from the middle of the three selected sludge concentrations are plotted in Fig.7.It becomes clear that the ¯uxes resulting from the parameter estimation follow the same trend as observed with the Vesilind ¯ux curves obtained from dilution experiments.Sludges that have hightraditional Vesilind ¯ux curves like B and O also have high Cho-from-SBSC ¯ux curves.Sludges with low traditional ¯ux curves like D and P have corre-sponding low Cho ¯ux curves.Cross-validation 2:Cho SBSC-estimated parameters to dilution experiment V ZSIn order to further investigate the validity of the Cho model,it was evaluated whether the Chopar-Fig.7.Flux curves for the di erent sludges according to the Vesilind model traditionally calibrated on a set of V ZS 's (closed symbols)and the Cho model obtained from the middle concentration SBSC's(opensymbols).Fig.8.Natural logarithm of the V ZS in function of the P sludge concentration.Estimation of sludge sedimentation parameters 403ameters resulting from the SBSC approach could describe the X/V ZS relationships as observed in the dilution experiment(cross-validation).To this end the natural logarithm of the observed and par-ameter predicted V ZS were plotted as function of the sludge concentration,e.g.Fig.8.The Cho par-ameters resulting from the SBSC approach described the observed V ZS/X data less accurate than the Vesilind parameters from the dilution ex-periment.Also the observed correspondence was even lower for the poorly settling sludges as D and P compared to the well settling sludges.DISCUSSIONThe Vesilind parameters calculated from a di-lution experiment V ZS's allow a49layer sedimen-tation model to roughly describe complete experimental batch settling curves.However,when the settling dynamics observed in the experiment increase,the discrepancy between the simulation result and the data enlarges,especially at the tran-sition between zone settling and compression phases.On this observation di erent comments can be made:.When the sludge blanket reaches the bottom of the batch settler,compression phenomena start (or might start)to determine the descent of the sludge blanket in a direct way.In the com-pression phase sludge particles lean on each other.This is a fundamentally di erent situation than in the zone settling phase which is the only phenomenon characterised in the dilution exper-iment(Ekama et al.,1997).Here,the descent of the blanket is depending only on the equilibrium between gravitational and hydraulic friction forces..When the sludge blanket is low,it is also close to the conical bottom of the settling column.Some boundary processes related to the conical bottom and scraper could in¯uence the observed pro-cesses.A hampering of the blanket movement at the bottom could be substantiated by the®nding that in nearly all cases the¯ux curves obtained from parameter estimation on a SBSC are situ-ated below the traditional Vesilind¯ux curves at the high concentration end.The Cho model[Eq.(3)]was found to be the most e ective in®tting the observed settling curves. Grijspeerdt et al.(1995)identi®ed the Takacs model [Eq.(2)]to be the best,although they didn't include the Cho model in their comparison.Here,it has to be mentioned that the values Grijspeerdt et al. (1995)found for the Takacs r p and r h parameters are rather close to each other indicating that both exponential terms interfere with each other over quite a broad range.This is in contradiction with the mechanistic explanation Takacs et al.(1991) give to their model.Further,Grijspeerdt et al.(1995),like Takacs et al.(1991)and Watts et al. (1996),used not a settling curve but a discrete solids pro®le as experimental data for the parameter estimation.Considering the empirical nature of the settling models it would not be surprising that the optimal model structure is depending on the nature of the experiments to be described.In this respect cross-validation of parameters between the two types of data sets as performed by Cacossa and Vaccari(1994),is certainly a task for further research.Because data on concentration pro®les were lack-ing,the Cho model[Eq.(3)]was cross-validated in an alternative way:it was evaluated on its ability to describe the evolution of the V ZS as a function of the suspended solids concentration(dilution exper-iment).For the good settling sludges like the O and B sludge,the model was able to reasonably describe the observed behaviour.However,for the poorly settling sludges like P and D,the Cho model[Eq.(3)]was not able to describe the trends as well as the Vesilind model[Eq.(1)].Hence,the Vesilind model[Eq.(1)]appears to be better in describing the V ZS obtained from dilution experiments, whereas the Cho model[Eq.(3)]is better in describ-ing complete single batch settling curves.This points clearly towards the empirical nature of the used settling velocity models and on the fact that not all processes involved in the batch settling pro-cess are understood and taken into account.In the past,practical identi®cation problems as-sociated with the estimation of parameters from a single settling curve have only been addressed to a minor extent.Cacossa and Vaccari(1994)(vaguely) mentioned that the parameters estimated by their algorithm were``somewhat sensitive to the initial guesses used''.These authors stated further that reasonable parameter estimation was only possible if the sludge volume was less than half the initial volume after60min of sedimentation in their120-cm tall stirred column.However,no detailed analy-sis was reported.In the current study identi®ability problems were still encountered with curves that had a sludge volume less than a quarter of the in-itial one after30min of settling in a70-cm tall col-umn.Vanrolleghem et al.(1996)stated that no more than three parameters in a Takacs based batch settler model could be identi®ed from SBSC's recorded with the Settlometer.Despite these practi-cal identi®ability problems encountered while using SBSC's,the determination of Vesilind parameters through the more labour intensive,traditional di-lution experiment can also be troublesome as Ekama et al.(1997)report that scattered data can be obtained.From slow dynamic SBSC's one can obtain¯ux curves which are quite close to the ones obtained by a traditional dilution experiment.However,this is not always the case.For fast dynamic SBSC's the traditional Vesilind model is not able to describeAlexis Vanderhasselt and Peter A.Vanrolleghem 404。
diffusion模型解读
diffusion模型解读## Diffusion Models Explained.Diffusion models are a type of generative model that has gained popularity in recent years for their ability to generate high-quality images, text, and other data from scratch. They work by gradually "denoising" a random input, adding details and structure until a coherent output is produced.Diffusion models are based on the idea of diffusion, a process that spreads out a substance over time. In the context of generative modeling, diffusion is used to spread out the noise in a random input, making it gradually more uniform. By reversing the diffusion process, the model can then learn to generate data that is increasingly structured and detailed.The training process for a diffusion model involves two main steps:1. Forward diffusion: In this step, a random input is gradually "denoised" by adding noise to it over multiple iterations. This process makes the input increasingly uniform and unstructured.2. Reverse diffusion: In this step, the model learns to reverse the diffusion process, generating data that is increasingly structured and detailed. The model is trained on a dataset of real data, and it learns to generate data that matches the distribution of the real data.Diffusion models have several advantages over other generative models. First, they can generate high-quality data that is visually appealing and realistic. Second, they are relatively easy to train, and they can be used to generate a wide variety of data types. Third, they are computationally efficient, and they can be used to generate large amounts of data quickly.However, diffusion models also have some limitations. First, they can be slow to generate data, especially forlarge datasets. Second, they can be difficult to control, and it can be difficult to generate data with specific properties.Despite these limitations, diffusion models are a promising new approach to generative modeling. They havethe potential to revolutionize the way we generate data,and they could be used to create new applications in a wide range of fields.## 中文回答：扩散模型通俗解释。
机械专业毕业论文中英文翻译--在全接触条件下,盘式制动器摩擦激发瞬态热弹性不稳定的研究
Frictionally excited thermoelastic instability in disc brakes—Transientproblem in the full contact regimeAbstractExceeding the critical sliding velocity in disc brakes can cause unwanted forming of hot spots, non-uniform distribution of contact pressure, vibration, and also, in many cases, permanent damage of the disc. Consequently, in the last decade, a great deal of consideration has been given to modeling methods of thermo elastic instability (TEI), which leads to these effects. Models based on the finite element method are also being developed in addition to the analytical approach. The analytical model of TEI development described in the paper by Lee and Barber [Frictionally excited thermo elastic instability in automotive disk brakes. ASME Journal of Tribology 1993;115:607–14] has been expanded in the presented work. Specific attention was given to the modification of their model, to catch the fact that the arc length of pads is less than the circumference of the disc, and to the development of temperature perturbation amplitude in the early stage of breaking, when pads are in the full contact with the disc. A way is proposed how to take into account both of the initial non-flatness of the disc friction surface and change of the perturbation shape inside the disc in the course of braking.Keywords: Thermo elastic instability; TEI; Disc brake; Hot spots1. IntroductionFormation of hot spots as well as non-uniform distribution of the contact pressure is an unwanted effect emerging in disc brakes in the course of braking or during engagement of a transmission clutch. If the sliding velocity is high enough, this effect can become unstable and can result in disc material damage, frictional vibration, wear, etc. Therefore, a lot of experimental effort is being spent to understand better this effect (cf. Refs.) or to model it in the most feasible fashion. Barber described the thermo elastic instability (TEI)as the cause of the phenomenon. Later Dow and Burton and Burton et al.introduced a mathematical model to establish critical sliding velocity for instability, where two thermo elastic half-planes are considered in contact along their common interface. It is in a work by Lee and Barber that the effect of the thickness was considered and that a model applicable for disc brakes was proposed. Lee and Barber’s model is made up with a metallic layer sliding between twohalf-planes of frictional material. Only recently a parametric analysis of TEI in disc brakes was made or TEI in multi-disc clutches and brakes was modeled. The evolution of hot spots amplitudes has been addressed in Refs. Using analytical approach or the effect of intermittent contact was considered. Finally, the finite element method was also applied to render the onset of TEI (see Ref.).The analysis of nonlinear transient behavior in the mode, when separated contact regions occur, is even accomplished in Ref. As in the case of other engineering problems of instability, it turns out that a more accurate prediction by mathematical modeling is often questionable. This is mainly imparted by neglecting various imperfections and random fluctuations or by the impossibility to describe all possible influences appropriately. Therefore, some effort aroused to interpret results of certain experiments in addition to classical TEI (see, e.g.Ref).This paper is related to the work by Lee and Barber [7].Using an analytical approach, it treats the inception of TEI and the development of hot spots during the full contact regime in the disc brakes. The model proposed in Section 2 enables to cover finite thickness of both friction pads and the ribbed portion of the disc. Section 3 is devoted to the problems of modeling of partial disc surface contact with the pads. Section 4 introduces the term of ‘‘thermal capacity of perturbation’’ emphasizing its association with the value of growth rate, or the sliding velocity magnitude. An analysis of the disc friction surfaces non-flatness and its influence on initial amplitude of perturbations is put forward in the Section 5. Finally, the Section 6 offers a model of temperature perturbation development initiated by the mentioned initial discnon-flatness in the course of braking. The model being in use here comes from a differential equation that covers the variation of the‘‘thermal capacity’’ during the full contact regime of the braking.2. Elaboration of Lee and Barber modelThe brake disc is represented by three layers. The middle one of thickness 2a3 stands for the ribbed portion of the disc with full sidewalls of thickness a2 connected to it. The pads are represented by layers of thickness a1, which are immovable and pressed to each other by a uniform pressure p. The brake disc slips in between these pads at a constant velocity V.We will investigate the conditions under which a spatially sinusoidal perturbation in the temperature and stress fields can grow exponentially with respect to the time in a similar manner to that adopted by Lee and Barber. It is evidenced in their work [7] that it is sufficient to handle only the antisymmetric problem. The perturbations that are symmetric with respect to the midplane of the disc can grow at a velocity well above the sliding velocity V thus being made uninteresting.Let us introduce a coordinate system （x1; y1）fixed to one of the pads (see Fig. 1) thepoints of contact surface between the pad and disc having y1 = 0. Furthermore, let acoordinate system （x2; y2）be fixed to the disc with y2=0 for the points of the midplane. We suppose the perturbation to have a relative velocity ci with respect to the layer i, and the coordinate system （x; y）to move together with the perturbated field. Then we can writeV = c1 -c2; c2 = c3; x = x1 -c1t = x2 -c2t,x2 = x3; y = y2 =y3 =y1 + a2 + a3.We will search the perturbation of the uniform temperature field in the formand the perturbation of the contact pressure in the formwhere t is the time, b denotes a growth rate, subscript I refers to a layer in the model, and j =-1½is the imaginary unit. The parameter m=m（n）=2pin/cir =2pi/L, where n is the number of hot spots on the circumference of the disc cir and L is wavelength of perturbations. The symbols T0m and p0m in the above formulae denote the amplitudes of initial non-uniformities (e.g. fluctuations). Both perturbations (2) and (3) will be searched as complex functions their real part describing the actual perturbation of temperature or pressure field.Obviously, if the growth rate b<0, the initial fluctuations are damped. On the other hand, instability develops ifB〉0.2.1. Temperature field perturbationHeat flux in the direction of the x-axis is zero when the ribbed portion of the disc is considered. Next, let us denote ki = Ki/Qicpi coefficient of the layer i temperature diffusion. Parameters Ki, Qi, cpi are, respectively, the thermal conductivity, density and specific heat of the material for i =1,2. They have been re-calculated to the entire volume of the layer (i = 3) when the ribbed portion of the disc is considered. The perturbation of the temperature field is the solution of the equationsWith and it will meet the following conditions:1，The layers 1 and 2 will have the same temperature at the contact surface2，The layers 2 and 3 will reach the same temperature and the same heat flux in the direction y,3，Antisymmetric condition at the midplaneThe perturbations will be zero at the external surface of a friction pad(If, instead, zero heat flux through external surface has been specified, we obtain practically identical numerical solution for current pads).If we write the temperature development in individual layers in a suitable formwe obtainwhereand2.2. Thermo elastic stresses and displacementsFor the sake of simplicity, let us consider the ribbed portion of the disc to be isotropic environment with corrected modulus of elasticity though, actually, the stiffness of this layer in the direction x differs from that in the direction y. Such simplification is, however, admissible as the yielding central layer 3 practically does not take effect on the disc flexural rigidity unlike full sidewalls (layer 2). Given a thermal field perturbation, we can express the stress state and displacements caused by this perturbation for any layer. The thermo elastic problem can be solved by superimposing a particular solution on the general isothermal solution. We look for the particular solution of a layer in form of a strain potential. The general isothermal solution is given by means of the harmonic potentials after Green and Zerna (see Ref.[18]) and contains four coefficients A, B, C, D for every layer. The relateddisplacement and stress field components are written out in the Appendix A.在全接触条件下，盘式制动器摩擦激发瞬态热弹性不稳定的研究摘要超过临界滑动盘式制动器速度可能会导致形成局部过热，不统一的接触压力，振动分布，而且，在多数情况下，会造成盘式制动闸永久性损坏。
my model的英语作文
When writing an essay in English about My Model,its important to consider the context in which the term model is being used.Here are a few different approaches you might take,depending on the specific meaning of model in your essay:1.A Role Model:Begin by introducing who your role model is and why they are important to you. Discuss the qualities and achievements of your role model that you admire. Explain how their actions or life story has influenced your own life or goals.Example Paragraph:My role model is Malala Yousafzai,a Pakistani activist for female education and the youngest Nobel Prize laureate.Her courage and determination to fight for girls education rights in the face of adversity have deeply inspired me.Malalas story has taught me the importance of standing up for what I believe in,even when it is difficult.2.A Fashion Model:Describe the physical attributes and style of the model.Discuss the impact they have had on the fashion industry or their unique contributions to it.Explain why you find their work or presence in the industry notable.Example Paragraph:Kendall Jenner is a fashion model who has made a significant impact on the industry with her unique style and presence.Her tall and slender physique,combined with her ability to carry off diverse looks,has made her a favorite among designers and fashion enthusiasts alike.I admire her for her versatility and the way she uses her platform to promote body positivity.3.A Model in Science or Technology:Introduce the model as a theoretical framework or a practical tool used in a specific field.Explain the principles behind the model and how it is applied.Discuss the benefits or limitations of the model and its implications in the real world.Example Paragraph:The Standard Model in physics is a theoretical framework that describes three of the four known fundamental forces excluding gravity and classifies all known elementary particles.It has been instrumental in understanding the behavior of subatomic particles and predicting the existence of new particles,such as the Higgs boson.However,the models inability to incorporate gravity or dark matter has led to ongoing research for amore comprehensive theory.4.A Model in Business or Economics:Introduce the business or economic model and its purpose.Explain how the model works and the strategies it employs.Discuss the success or challenges associated with the model and its potential for future growth.Example Paragraph:The subscriptionbased business model has become increasingly popular in recent years, particularly in the software panies like Adobe have transitioned from selling packaged software to offering services on a subscription basis,allowing for continuous revenue streams and a more predictable income.This model has been successful in fostering customer loyalty and providing a steady income,although it requires ongoing innovation to maintain customer interest.5.A Model in Art or Design:Describe the aesthetic or functional qualities of the model.Discuss the creative process or design principles that inform the model.Explain the cultural or historical significance of the model and its influence on contemporary art or design.Example Paragraph:The Eames Lounge Chair,designed by Charles and Ray Eames,is a model of modern furniture that has become an icon of midcentury design.Its elegant form,made from molded plywood and leather,exemplifies the designers commitment to blending comfort with aesthetics.The chairs timeless appeal has made it a staple in both residential and commercial settings,influencing countless furniture designs that followed. Remember to structure your essay with a clear introduction,body paragraphs that develop your points,and a conclusion that summarizes your main e specific examples and evidence to support your claims,and ensure your writing is clear,concise, and engaging.。
spatial theory
在佐伦的理论体系之中，场景（scene）是构
成空间复合体的基本单位。与地形层面相关 的场景是地点（place）。地点是空间中可以 被度量的一点，如：房子、城市、山林或河 流等。地点之间往往有清晰的分界。与时空 体层面相关的空间单位是行动幅度（zone of action）。行动幅度依据事件发生的区域，而 不是空间的连续性和清晰的地质疆界来划分 。对于空间单位的最重要的研究在于视角场 （field of vision），即一个空间单位是从地
The Critique of everyday life
One of Lefebvre's most important contributions to social thought is the idea of the "critique of everyday life," which he pioneered in the 1930s. This work was influential in French theory, particularly for the Situationists, as well as in politics (e.g. for the May 1968 student revolts). While the theme presented itself in many works, it was most notably outlined in his eponymous 3 volume study, which came out in individual installments, decades apart, in 1947, 1961, and 1981.
语言的选择会对文本空间建构的效果产生影
大模型diffusion原理
大模型diffusion原理英文回答：Diffusion is a fundamental principle in large-scale modeling that describes the process of how substances or information spread through a system. It is commonly used in various fields such as physics, chemistry, biology, and social sciences to understand how different entities interact and propagate.In physics and chemistry, diffusion refers to the movement of particles from an area of high concentration to an area of low concentration. This is driven by the random motion of particles, which leads to a net flow of particles down the concentration gradient. For example, if you open a bottle of perfume in one corner of a room, the scent molecules will diffuse and spread throughout the entire room over time.In biology, diffusion is essential for variousbiological processes. For instance, oxygen and carbondioxide are exchanged in the lungs through the process of diffusion. Oxygen molecules move from the air sacs in the lungs to the bloodstream, where they are transported to different parts of the body. Similarly, carbon dioxide molecules produced as waste in cells diffuse from the bloodstream to the lungs, where they are exhaled.In social sciences, diffusion is used to study the spread of ideas, innovations, or behaviors within a population. It is often referred to as "cultural diffusion" or "social diffusion." For example, the adoption of new technologies, such as smartphones or social media platforms, can spread rapidly through a population through social networks and word-of-mouth. The diffusion of these innovations can be influenced by factors such as social influence, communication channels, and individual characteristics.Large-scale modeling of diffusion involves representing the system as a network of interconnected nodes. Each node represents a location, individual, or entity, and theconnections between nodes represent the pathways through which diffusion occurs. Mathematical models, such as the diffusion equation or network models, are used to simulate and analyze the spread of substances or information in the system.In conclusion, diffusion is a fundamental principle in large-scale modeling that describes how substances or information spread through a system. It is a universal phenomenon that can be observed in various fields, from physics and chemistry to biology and social sciences. Understanding diffusion is crucial for predicting and managing the spread of substances, ideas, or behaviors in complex systems.中文回答：扩散是大规模建模中的一个基本原理，描述了物质或信息在系统中的传播过程。
第4章 扩散
4.1.1 菲克第一定律 (Fick’s first law)
1858年，菲克（Fick）参照了 傅里叶（Fourier）于1822年建 立的导热方程，获得了描述物质 从高浓度区向低浓度区迁移的定 量公式。
假设有一单相固溶体，横截面积 为A，浓度C不均匀，在dt时间 内，沿x方向通过某处截面所迁 移的物质的量与该处的浓度梯度 成正比，这就是菲克第一定律。
而焊接面浓度Cs＝C2/2。 在任意时刻，浓度曲线都相对于x＝0，Cs＝（C1﹢C2）/2为 中心对称。随着时间的延长，浓度曲线逐渐变得平缓，当 t→∞时，扩散偶各点浓度均达到均匀浓度（C1﹢C2）/2。
② 扩散的抛物线规律：由关系式看出， 如果要求距焊接面为x处的浓度达到C， 则所需要的扩散时间可由下式计算
根据上式可以估算达到一定渗碳层深度所需要的时间 。
由 x K Dt 可知，规定浓度的渗层厚度正 比于渗碳时间的平方根。显然，同一条件 下要使渗层厚度增大一倍，则渗碳时间将 为原来的四倍。 另外，除了化学热处理之外，金属的真空 除气、钢铁材料在高温下的表面脱碳也是 半无限长扩散的例子，只不过对于后者来 说，表面浓度始终为零。
第一节 表象理论 第二节 扩散的热力学分析 第三节 扩散的原子理论 第四节 扩散激活能 第五节 无规则行走与扩散距离 第六节 影响扩散的因素
扩散(diffusion):物质在一定的温度下，处于晶格平衡位 置上的原子进行快速热振动，当获得足够大能量时，部分 原子可脱离晶格的束缚而发生微观意义上的跃迁并因此而 导致一种宏观上的物质流动现象。
Diffusion is a process of mass transport that involoves the movement of one atomic species into another.
WIGNER-POISSON AND NONLOCAL DRIFT-DIFFUSION MODEL EQUATIONS FOR SEMICONDUCTOR SUPERLATTICES
1. Introduction Nonlinear charge transport in semiconductor superlattices has been widely studied in the last decade 1,2 . A superlattice (SL) is a convenient approximation to a quasi-one-dimensional crystal that was originally proposed by Esaki and Tsu to observe Bloch oscillations, i.e., the periodic coherent motion of electrons in a miniband in the presence of an applied electric field. When the materials were grown, many interesting nonlinear phenomena were observed, including self-oscillations of the current through the SL due to motion of electric field pulses, multistability of stationary solutions, and so on 1,2 . In the effort to describe quantitatively transport in these materials, a large number of quantum and semiclassical equations, hydrodynamic and drift-diffusion models have been proposed. However, systematic derivations of reduced balance equations (hydrodynamic or drift-diffusion equations) are scarce. Hydrodynamic equations have been derived by Lei and coworkers 3,4 and their numerical solutions describe self-oscillations of the current in different regimes 5 . They write operator Heisenberg equations, and average them to obtain