Stochastic Model for a Vortex Depinning in Random Media

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Stoichio-kinetic modeling and optimization of chemical synthesis Application to aldolic condensation

Available online at Chemical Engineering and Processing47(2008)349–362Stoichio-kinetic modeling and optimization of chemical synthesis:Application to the aldolic condensation of furfural on acetoneNadim Fakhfakh a,Patrick Cognet a,∗,Michel Cabassud a,Yolande Lucchese a,Manuel D´ıas de Los R´ıos ba Laboratoire de G´e nie Chimique,UMR5503CNRS,INPT(ENSIACET),UPS,5rue Paulin Talabot,BP1301,31106Toulouse Cedex1,Franceb Cuban Research Institute for Sugar Cane Byproducts(ICIDCA),P.O.Box4026,Havana City,CubaAvailable online16January2007AbstractThe condensation reaction of furfural(F)on acetone(Ac)gives a high added value product,the4-(2-furyl)-3-buten-2-one(FAc),used as aroma in alcohol free drinks,ice,candies,gelatines and other products of current life.This synthesis valorises the residues of sugar cane treatment since furfural is obtained by hydrolysis of sugar cane bagasse followed by vapor training extraction.In the face of numerous and complex reactions involved in this synthesis,it is very complicated to deﬁne the kinetic laws from exact stoichiometry.A solution allowing to cope the problem consists in identifying an appropriate stoichiometric model.It does not attempt to represent exactly all the reaction mechanisms,but proposes a mathematical support to integrate available knowledge on the transformation.The aim of this work is the determination of stoichiometric and kinetic models of the condensation reaction of furfural on acetone.Concentrations of reagents and products are determined by gas and liquid chromatography.Concentration proﬁles obtained at different temperatures are used to identify kinetic parameters.The model is then used for the optimization of the production of FAc.The interest of such tool is also shown for the scale up of laboratory reactor to a large scale.The anticipation of the reaction behaviour in large scale is crucial especially when the reactor presents important limitations of thermal exchange capacity.©2007Elsevier B.V.All rights reserved.Keywords:Furfural;Acetone;Chromatography;Aldolic condensation;Batch reactor;Stoichio-kinetic modeling1.IntroductionChemical industries of industrialized countries turn increas-ingly to products with high added value,especially in the sector ofﬁne chemistry(pharmaceutical products,cosmetics,etc.). This type of industry[1–3]is different from traditional chem-ical industry.Theﬁne chemical industry like in France is well known to be a strategic area which needs a lot of investments not onlyﬁnancial but in high level scientiﬁc knowledge for the R&D.The syntheses of such products are generally complex and involve secondary reactions which are to be minimized.In thisﬁeld,the faster development of processes is essen-tial in order to answer the rapid evolutions of the market.The detailed studies of the mechanisms and kinetics of reactions are generally not carried out for reasons of duration and cost.Nev-ertheless,for optimization and advanced control of processes∗Corresponding author.Tel.:+33534615260;fax:+33534615253.E-mail address:Patrick.Cognet@ensiacet.fr(P.Cognet).ofﬁne chemistry,it is necessary to obtain a reliable model of the system.This problem is often circumvented by the use of linear or quadratic polynomial models which parameters can quickly be identiﬁed by the installation of experimental plan-ning[4].Nevertheless,these models rapidlyﬁnd their limits in a restrictedﬁeld of validity and a difﬁculty of accounting for the dynamics of the syntheses.Sinceﬁne chemical reactions are usually complex,theirs kinetics are poorly known.The real problem is the fast devel-opment of realistic and safety stoichio-kinetic models of the synthesis[5–8].However,due to high purity requirements, environmental regulation and competitive pressure on the new products,the development of dynamic models has become an important objective.The approach proposed does not depend on a detailed and predictive model of the process and at the same time it does not ignore what we already know about the process,such as material balances,heat and mass transfer characteristics,etc. Nevertheless,a stoichiometric model should describe the differ-ent stages of the synthesis,or the most important tendencies.It0255-2701/$–see front matter©2007Elsevier B.V.All rights reserved. doi:10.1016/j.cep.2007.01.015350N.Fakhfakh et al./Chemical Engineering and Processing47(2008)349–362can be obtained by creating pseudo-reaction(roundup of several reactions)or pseudo-compound(roundup of several compounds or addition of losses)[5].The identiﬁed model can be used to calculate the kinetic of the different reactions determining thus the rate of the chemical transformation.This technique has been used with success by several researchers[9–15]for the model-ing of chemical syntheses like the epoxidation of oleic acid[14], or the polymerization of acrylonitrile-styrene[10]carried out in batch reactors.In several cases,the optimization problems of discontinuous reactor are formulated with two kinds of objectives:maximum conversion problem,the operative time isﬁxed a priori,or min-imum time problem,the conversion rate of wished product is ﬁxed a priori.Garcia et al.[16]considered theﬁrst case.They converted the problem of optimal control into a non-linear problem solved by a reduced gradient algorithm(GRG)coupled with the golden search method.This tool allows to optimize simultaneously different variables(temperature,feedﬂow rate and amount of reactant,operation time,etc.)and to take into account bounds and linear and/or non-linear constraints on the variables.The use of constraints allows to reach a solution witch is not only a numerical solution but witch is closer to the experimental reality.Aziz and Mujtaba[17]are interested to the consecutive reac-tion optimization in batch reactors.The optimization problems are formulated with environmental and operational constraints and solved by the control vector parameterisation(CVP)tech-nique.Two different models are presented:a shortcut model, consisting of only mass balance and reaction kinetics,allows determination of the optimal reactor temperature proﬁles to achieve a desired performance.The optimal temperature proﬁles can then be used as a basis for the detailed design of the reactor (i.e.reactor volume,heating/cooling conﬁguration,etc.).The detailed model,consisting of mass and energy balances,reaction kinetics and cooling/heating conﬁguration,allows determi-nation of the best operating conditions of already designed reactors.In this work,the methodology is illustrated through its application to a complex chemical transformation:aldolic con-densation of furfural(F)on acetone(Ac),which allows mainly two products noted(FAc)and(F2Ac)to be obtained.This synthesis valorises the residues of sugar cane treatment since furfural is obtained by hydrolysis of the sugar cane bagasse then extracted by vapor training.The(FAc)is used as aroma in sev-eral types of food industries.The study of this synthesis has been made with the collaboration of Cuban Research Institute for Sugar Byproducts(ICIDCA).2.Theoretical part2.1.Identiﬁcation of a stoichiometric modelThe stoichiometry of chemical transformation determines the proportions according to which the different constituents react or are formed.In general,these proportions are integer or semi-integer.The stoichiometry of a reaction system involving NC species A j(j=1,NC)and NR reactions R i(i=1,NR),can be written: NCj=1νij A j=0(1)whereνij is the stoichiometric coefﬁcient of A j in the reaction R i.•Ifνij>0then A j is a product in the reaction i;•Ifνij<0then A j is a reactant in the reaction i;•Ifνij=0then A j is not involved in the reaction i.For a batch reactor and a data base of NE experiments(k=1, NE),the mole number of the compound A j in the chemical transformation,represented by several reactions R i,is given byn jk=n0jk+n0NRi=1νij X ik(2)where n jk is the mole number of A j in the experiment k at the instant t,n0jk the initial mole number of A j in the experiment k, X ik the extent of the reaction R i in the experiment k,and n0is the normalizing factor equal to the sum of the initial reactants mole numbers.n0=NCj=1n0jk,k=1,NE(3) For simpliﬁcation reasons we note:Y jk=y jk−y0jk=n jk−n0jkn0(4) Eq.(2)becomes:Y jk=NRi=1νij X ik(5)The equation system representing the whole set of equations can be put under the following matrix form:[Y jk]=[νij]T[X ik](6) Or more simply:Y=νT X(7) Several techniques have been developed[5]allowing the iden-tiﬁcation of the stoichiometry of chemical syntheses.Theﬁrst method proceeds with iterative way by constructing reaction by reaction a more and more complex system to improve the representativeness of the studied synthesis.The second method treats the problem in a more global manner and determines a stoichiometric matrix in only one stage:it is the singular val-ues decomposition(SVD)method[18,19].An approach called “target factor analysis”(TFA)[20]enables to know whether a postulated stoichiometry from a priori information is compatible with the abstract factors.N.Fakhfakh et al./Chemical Engineering and Processing47(2008)349–362351 2.2.Identiﬁcation of kinetic modelThe molar balance for a reactor operating in batch or semi-batch mode gives:d n jd t=Fe j+R j V r(8)with n j is the mole number of A j at instant t,Fe j the feed rate ofthe compound A j(j=1,NC),and V r is the reactor volume andR j=NRi=1νij r i(9)R j is the production rate if it is positive and consumption rate if it is negative and r i is the rate of reaction i.Eq.(8)may be written with extent of reaction,we obtain:d X i d t =V rn0r i(10)In this work,the transformation is supposed to be a pseudo-homogeneous one and the kinetic law is written as a classical Arrhenius’s law.It is important to emphasize that the form of the kinetic law and its degree of complexity depend on the user and the desired accuracy of the tendency model.So,we haver i=k0i e−E a i/RT NCj=1C a ij j(11)where k0i is the pre-exponential factor for reaction i,E ai the acti-vation energy for reaction i,C j the concentration of constituent j,and a ij is the order of constituent j in the reaction i.According to(11),Eq.(10)may be written:d X i d t =Vn0k0i e−E a i/RTNCj=1C a ij j(12)The orders are assumed to be part of the data of the problem and are chosen a priori to be equal to the absolute value of the stoichiometric coefﬁcients of every reactant.The identiﬁ-cation of kinetic parameters(pre-exponential factor,activation energy)is determined by minimizing the difference between the experimental concentrations and those computed with the iden-tiﬁed parameters for the different constituents according to the following criterion:J=NEk=1NCj=1C0C01k(C f jk id−C f jk exp)2(13)withC00=NEk=1C01k(14)and C01k is the concentration of a key reactant in experiment k.The whole procedure has been implemented on software, Batchmod[21].The correlation coefﬁcient(r)is used to measure the“good-ness ofﬁt”.It is deﬁned asr=Ni=1(x i−¯x)(y i−¯y)Ni=1(x i−¯x)2Ni=1(y i−¯y)2(15)where x i means data points and y i means model points.The average of the data points(¯x)and the model points(¯y) are simply given by¯x=1NNi=1x i and¯y=1NNi=1y i(16)As the model better describes the data,the correlation coef-ﬁcient will approach unity.For a perfectﬁt,the correlation coefﬁcient will approach r=1.2.3.Optimization of chemical synthesisThe general procedure of optimization is formulated as the following[22]:min f(x),x∈ n;g i(x)=0,i=1,m e;g j(x)≤0,j=m e+1,m;x l≤x≤x u(17) where f is the objective function to minimize,g i the equality con-straints,g j the inequality constraints,m e the number of equality constraints,m the total number of constraints,x l the low limit of x variable,and x u is the up limit of x variable.The goal of the problem is to minimize a function f that depends on several variables.These variables are limited and submitted to equality and inequality constraints.In general,the function f is not linear and is not given under explicit shape of variables.The optimization of a chemical synthesis is the determina-tion of the working conditions(temperature,feed-rate,operative time),that maximize a synthesis criterion(output,concentration, etc.)under some constraints.The resolution of the problem requires the discretisation of temperature proﬁles and feed-rates intoﬁnite intervals inside the interval of operation[t0,t f],where t0represents the initial time of operation and t f theﬁnal time of operation.The interval[t0,t f] is discretised into aﬁnite number(n int)of intervals.A function is deﬁned to represent the evolution of the control variable v(t) in every time interval:v(t)=Φ(t,z j),t∈t j−1,t j,j=0,n int(18) where t is the commutation time and z j is the temperature and feed-rate values in bounds of each interval.In order to avoid complex temperature and feed proﬁles,Φfunction is assimilated to a simple function:•A linear function for the temperature:v(t)=z j−1+(t−t j−1)z j−z j−1j j−1,j∈[1,n int](19)352N.Fakhfakh et al./Chemical Engineering and Processing 47(2008)349–362The last temperature of interval j is supposed to be equal to the initial temperature of interval j +1.•A constant function for the feed-rate:v (t )=a j ,j ∈[1,n int ](20)The program determines the mass ﬂow in every interval and supposes that it remains constant in this interval.The feed-rate is thus a succession of landings.The resolution of the optimization problem returns to the determination of (n int +1)temperatures and n int values of feed-rate.This optimization method allows to scale up a chemical reac-tion in batch reactor with safety constraints [23,24].2.4.Energy balance—thermal ﬂux modelingA classical Semenov-type analysis [25]is used to describe the exothermic reaction.The rate of heat production is proportional to the reaction speed,which means it is an exponential function of temperature.It is given by Eq.(21):Q released =V r H k 0exp −E aRTC initial(21)where Q released is the heat ﬂux released by the reaction,V r the reacting volume, H the heat of reaction,k 0the pre-exponential factor of reaction,E a the activation energy,and C initial is the initial concentration.The thermal ﬂux evacuated out of the reactor is expressed by Eq.(22).It is proportional to a temperature difference between reacting solution and coolant ﬂuid,exchange area and global heat transfer coefﬁcient.A little variation of coolant ﬂuid tem-perature induces a linear variation of thermal ﬂux evacuated from the reactor:Q evacuated =UA (T cf −T reactor )(22)where Q evacuated is the thermal ﬂux evacuated with jacket reac-tor,U the global heat transfer coefﬁcient,A the exchange surface reactor,T cf the cooling ﬂuid temperature,and T reactor is the reactor temperature.3.Experimental part3.1.Aldolic condensation of furfural on acetoneThe aldolic condensation [26–29]of furfural (F)on acetone (Ac)takes place in alkaline medium.It implies the generation of a carbanion obtained from abstraction of a proton in alpha of acetone carbonyl function and leads to the 4-(2-furyl)-3-buten-2-one (FAc).Because of the symmetry of the acetone molecule,a second attack of the furfural can happen which then leads to the di-adding product,the 1,4-pentadien-3-one,1,5-di-2-furanyl (F 2Ac).The different steps for the formation of (FAc)molecule can be written:(a)Extraction of a proton on acetone and formation of thecarbanion:(b)Condensation of the carbanion on the carbon of furfuralcarbonylfunction:(c)Fixation of H +on theoxy-anion:(d)Regeneration of hydroxide ion base and dehydration in basicmedium:Finally,the reactions of formation of (FAc)and (F 2Ac)are:(23)(24)The reversibility of reactions (23)and (24)are negligi-ble.Besides these two reactions,others may happen.Amongst the known reactions,furfural can react with itself in an oxydo-reduction reaction (Cannizaro reaction)to give a higherN.Fakhfakh et al./Chemical Engineering and Processing 47(2008)349–362353oxidation product,the 2-furoic acid (Furo),and a lower oxida-tion product,the furfuryl alcohol (Furfu)[30].This reaction can take place in highly alkaline medium.On the other hand,acetone can react on itself to give the 4-hydroxy-4-methylpentan-2-one (Ox 1),which after dehydration leads to the mesityl oxide (Ox 2)[30].(25)(26)All used chemicals have analytical grade.The 1,4-pentadien-3-one,1,5-di-2-furanyl is not commercialized,therefore,it has been prepared and puriﬁed in the laboratory.3.2.Conditions of the reactionThe reactions are achieved in discontinuous mode in a jack-eted glass reactor of 250mL capacity (Fig.1).The acetone and the furfural are charged in the reactor with equi-molar quantities.The solvent used is an equi-molar mixture of water and ethanol.The presence of ethanol in the medium favours the dissolution of FAc and F 2Ac which are not soluble enough in water.An aque-ous solution of sodium hydroxide (0.03mol L −1)is injected to trigger the reaction.The temperature of the medium can be main-tained constant thanks to a heating-cooling system.The reaction volume is constant and equal to 98mL.The initial compositions are chosen according to the sug-gestions of the Cuban Research Institute for Sugar Byproducts (ICIDCA).It was not varied because of industrial restrictions.3.3.Analytical procedureThe concentrations of reagents and products are determined by gas and liquid chromatography.Only the acetone is measured by gas chromatography,the formed products (FAc and F 2Ac)being heat-sensitives.For the two techniques,ethanol is used as an internal standard in addition to its solventrole.Fig.1.Experimental equipment used for the chemical synthesis.354N.Fakhfakh et al./Chemical Engineering and Processing47(2008)349–362Table1Wavelengths corresponding to a maximal UV absorptionProductsλmax(nm)Acetone265Furfural274FAc322F2Ac3822-Furoic acid228Furfuryl alcohol220Mesityl oxide2324-Hydroxy-4-methylpentan-2-one232For gas chromatography,the injections are realized in split mode.Aﬂame ionization detector(FID)is used.A polar column (FFAP,25m×0.32mm i.d.)with aﬁlm thickness of0.25␮m separates the solvent and the matrix of components.The injector is maintained at a temperature of250◦C.Theﬂame ionization detector is heated until300◦C.The initial temperature of the oven is50◦C.After1.5min,the temperature increases quickly at the rate of50◦C/min until240◦C then it remains constant during5min.The total time of analysis is10.3min.Helium is used as the carrier-gas.The products(F),(FAc)and(F2Ac)are measured by high performance liquid chromatography(HPLC)[31–36].The sys-tem is equipped with UV–vis detector and an automatic injector of25␮L in full loop.An ODS Hypersyl C185␮m column (125mm×4mm i.d.)is used for the separation of the products.Chemical compounds have different maximum UV absorp-tion at different wavelengths.Table1shows the wavelength at which the maximum UV absorption is observed for the com-pounds of the synthesis.The use of three different wavelengths (265,322and382nm)allows to measure with precision the quantity of different compounds.The eluent is a mixture of water and methanol.Fig.2sum-marizes the procedure of HPLC analysis adopted to follow the concentrations of the different compounds in the aldolic con-densation of furfural on acetone.4.Results and discussion4.1.Identiﬁcation of a stoichiometric matrixSeveral reactions have been carried out at temperature of24, 29,34and40◦C and atmospheric pressure.The study has not exceeded40◦C because the acetone ebullition temperature is 56◦C.The mole number variation of the compounds at24◦C (Fig.3)shows the disappearance of acetone and furfuralandFig.3.Mole number evolution of the compounds at24◦C.the apparition of(FAc)and(F2Ac).The conversion of furfural approaches95%while that of acetone is near62%.The pro-duction of(F2Ac),more important than that of(FAc),explains the best conversion of the furfural.Furthermore,the analysis has not revealed the presence of either2-furoic acid or furfuryl alco-hol.Also neither(Ox1)nor(Ox2)was detected(Eq.(26)).Then suggested reactions for the synthesis are only reactions(23)and (24).In order to verify the accuracy of the supposed stoichiometric matrix,it is necessary to verify that molar and mass balances are correct.Assuming that only reactions(23)and(24)occur,the molar balance can be written asn0F=n F+n FAc+2n F2Ac(27)n0Ac=n Ac+n FAc+n F2Ac(28)n H2O=n FAc+2n F2Ac(29) It is necessary to determine the number of moles of water formed during the synthesis to be able to establish the molar and mass balances.The formed water is not determined by chromatogra-phy but calculated according to Eq.(29).However,water present at the beginning in the medium does not appear in the balance and was taken into account in the concentration of the solvent.The mass balance shows a deﬁcit which increases with time and then decreases at the end of the synthesis(Fig.4).The fur-fural and the acetone were more consumed than the prediction with the supposed reaction scheme.Supplementary reactions must therefore be added to the stoichiometric model.The Can-nizaro reaction being rejected,the possibility of polymerization reactions was studied.The main reaction of polymerizationmen-Fig.2.Change of wavelength and eluent composition for the analysis by HPLC.N.Fakhfakh et al./Chemical Engineering and Processing 47(2008)349–362355Fig.4.Mass balance for the experiment at 24◦C (H 2O calculated from Eq.(21)).Fig. 5.Example of HPLC chromatogram (experiment at 40◦C,reaction time =10min).tioned in the literature [37]is the formation of (FAc)n from the(FAc)in basic medium,n being superior or equal to 2.In the HPLC chromatograms obtained,an unidentiﬁed peak can be noticed.It is present during the ﬁrst minutes of the syn-thesis and then it disappears.It has a retention time near that of (FAc).The product (P)which corresponds to this peak is sensitive to the wavelength of 322nm (Fig.5).To identify the product (P),we used the HPLC analysis cou-pled with mass spectroscopy (Fig.6).This analysis revealed that this product has an important molar mass near 350g mol −1.The developed formula of (P)shown in Fig.6is one of possible formulas deduced from the fragments identiﬁed on the spectrum.Since product (P)appears at the same time as (FAc),then it cannot be a polymer of the later.It is rather formed directly from the furfural and the acetone (Eq.(30)).This equation is only a representation and does not account for a chemical mechanism.It would then imply a ﬁve reactants encounter,which is not realistic.The product (P)disappears during the reaction,this disappearance could be represented by Eq.(31).HPLC and GC analyses has not reveal the presence of other products aside furfural,acetone FAc,F 2Ac and P,that is why the decomposition of P to FAc and F 2Ac is the most probable and not other products.Finally,it was suggested that this synthesis could be represented by the four reactions (23),(24),(30)and (31).(30)(31)The molar balances associated to the reactions (23),(24),(30)and (31)become:n 0F =n F +n FAc +2n F 2Ac +3n P (32)n 0Ac =n Ac +n FAc +n F 2Ac +2n P (33)n H 2O =n FAc +2n F 2Ac +3n P(34)With the new reactions system proposed for the synthesis,the concentrations of (P)and water can be calculated by two ways,from the balance on acetone (Eqs.(33)and (34)),or from the balance on furfural (Eqs.(32)and (34)).The two methods have been tested and the values obtained were very close.Contrarily to the previous case (system of two reactions)the representation of the synthesis by four reactions allows to get a correct mass balance.The maximal error observed is 6.1%(Fig.7).The experiment carried out at temperatures of 24,29,34and 40◦C allow to generate a range of input/outputconcentrationsFig.6.Spectrum of mass spectroscopy (the peak at 240.3amu is visible on the spectra of all analyzed samples).356N.Fakhfakh et al./Chemical Engineering and Processing 47(2008)349–362Fig.7.Mass balance for the experiment at 24◦C (H 2O calculated from Eq.(26)).for the six components of the synthesis and therefore to feed the algorithm of calculation.The SVD method provides the follow-ing singularvalues:Knowing that the most important values indicate the number of necessary reactions for the description of the synthesis,it can be assumed that the chemical transformation can be described adequately by a system of four or ﬁve reactions.According to the available knowledge about the transforma-tion,it was postulated a stoichiometry with four reactions:•The ﬁrst reaction represents the formation of (FAc)from fur-fural and acetone.•The second reaction represents the formation of (F 2Ac)from (FAc)and furfural.•The third reaction represents the formation of (P)from three moles of furfural and two moles of acetone.•The fourth reaction represents the consumption of (P).One mole of (P)gives one mole of (FAc)and one mole of (F 2Ac).Table 2contains the postulated stoichiometry and Table 3contains the calculated stoichiometry.Eq.(35)allows the error between the target matrix and the calculated matrix for every reaction to be calculated:error(i )=NC j =1νtarget ij−νcalculated ij 2 NC j =1 νtarget ij2(35)Table 2Target stoichiometric matrixFAc FAc F 2Ac P H 2O Reaction 1−1−11001Reaction 2−10−1101Reaction 3−3−20013Reaction 411−1Table 3Calculated stoichiometric matrixFAc FAc F 2Ac P H 2O Reaction 1−0.983−1.0120.9850.048−0.0170.980Reaction 2−1.0370.043−0.9850.9940.034 1.044Reaction 3−2.988−2.022−0.0430.074 1.021 2.987Reaction 4−0.0150.0251.0390.916−1.0340.016Table 4Kinetic parameter values (reaction rate in mol L −1s −1)Activation energy,E a (J mol −1)Pre-exponential factor,k 0Reaction 174581.559 3.7979×108Reaction 235959.731 2.2094×102Reaction 358914.459 4.4197×104Reaction 4117919.1919.5420×1015An error of 3.02%was obtained for the main reaction,4.05%for the second reaction,1.93%for the third reaction and 6.03%for the fourth reaction.Results show that the postulated stoi-chiometry is compatible with the abstract factors.The model can therefore be adopted for the identiﬁcation of kinetic parameters.4.2.Determination of the kinetic parametersThe determination of the stoichiometry of the chemical trans-formation allows to ﬁnd the proportions according to which the components react.The next step for modeling consists in identi-fying the kinetic parameters for each reaction.In this work,the transformation is supposed to be a pseudo-homogeneous one and the kinetic law is written as a classical Arrhenius’s law.The orders of compounds are taken equal to absolute values of stoichiometric coefﬁcients for the reactants and zero for the products.Table 4gives the values of activation energy (E a )and pre-exponential factor (k 0)identiﬁed according to Eq.(11).The ﬁrst reaction is sensitive to temperature;the production of (FAc)increases as temperature rises.The second reaction is less sensitive to temperature than the ﬁrst.The third reaction which corresponds to the formation of (P)is far less activated by temperature than the fourth reaction which corresponds to the consumption of (P).The disappearance of (P)is so more rapid at 40◦C than at 24◦C.From the kinetic parameters,the time-dependence of theoret-ical concentrations can be illustrated.The comparison between the time-dependence concentrations obtained from the experi-ment and from the model is given in Figs.8–11.A statistical analysis is provided regarding the signiﬁcance of the ﬁtted kinetic parameters.Table 5shows the correlation coefﬁcientsTable 5Correlation coefﬁcientsFAc FAc F 2Ac P H 2O Reaction at 24◦C 0.99950.99840.89150.99740.92670.9995Reaction at 29◦C 0.99730.99860.90690.99860.91560.9973Reaction at 34◦C 0.99530.99560.90410.99940.90510.9953Reaction at 40◦C0.99720.99910.92520.99700.96880.9972。

SIMPACK风电仿真分析案例

Chair of Machine Elements, Dresden University of TechnologyDetermining reliable load Assumptions in Wind turbines using SIMPACKthe intensified efforts to provide alternative and renewable sources of energy led to a substantial worldwide growth in the wind turbine industry over the last few years. the advantages of this clean energy source and the recent successes in increasing power output rates for on- and offshore wind turbines is extremely encouraging. However the need for more durable wind turbines must still be addressed. the operation of anchored flexible light weight constructions under high dynamic stochastic loads was a relatively new challenge that arose with the wind turbine. these challenges are being met with the aid of the multi-body system (MBS) software SIMPACK.IntroDuCtIonThe Chair of Machine Elements at Dresden University of Technology does research which, for the past several decades, has focused on machine elements like shafts, gearings and bearings. Since 2001, MBS software has played an integral role in the development of guidelines, standards and verification of the dynamic behavior of drivetrains. MBS software has been used to develop modeling strat-egies that realistically represent the dynamic behavior of machine elements and have a high correlation with measured data. The main focus of research has been the investigation of the dynamic behavior of large drivetrains which can be found in roller mills, compressors, ships, fans, shearers, cranes and wind turbines.MotIVAtIonIn comparison to the wind turbine design software used by wind turbine manufacturers, multi-body system simulation software allows for a more precise modeling of the drivetrain components. Instead of a detailed rotor model and a simple 3-mass torsional vibration model for the gear box and the generator, all components can be represented with up to six degrees of freedom and with coupling stiffnesses. The resulting simulation model of the wind turbine consists of the substructure's rotor, drivetrain, coupling, generator, supporting structure, tower and foundation (Fig. 1). It allows the determination of the natural frequencies of the complete structure, and additionally, shows the mode shapes of all drivetrain components. To detect possible ranges of resonances, the effects of the rotor, the gear meshing frequencies, and rotation speeds of the components can be compared to the calculated natural frequen-cies. Simulation in the time domain enables the determination of torques, forces, displacements, velocities and accelerations for the modeled components and degrees of freedom. The resulting values at the different load conditions can be used for the design of the components, bearings and gearings, as well as the gear box housing and supporting structure.BASICS oF DrIVetrAIn SIMulAtIonThe analysis of drivetrains operating under high dynamic loads pre-supposes the assembly of a detailed simulation model which is ableFig. 1: Flexible multi-body system model of a wind turbine drivetrainFig. 2: Gear box substructuresto represent the dynamic behavior of the drivetrain in the frequency and time domain. Even if high performance computers are available the level of detail of the simulation model has to correspond to the formulated question to ensure a feasible calculation effort. Based on the available data and the experience of the engineer, a discrete simulation model can be assembled. A suc-cessive and modular assembly of fully param-eterized simulation models allows a clear and reproducible modeling process. The modular concept requires decomposition of the drivetrain into its substructures. Using this approach, a simulation model of a com-mon wind turbine consists of the following substructures: rotor, main shaft, coupling, generator and an additional subdivided gear box. For the gear box, a combination of helical/spur and planetary gear stage substructures is necessary (Fig. 2). Each substructure consists of model components which can be subdivided into shafts, gear stages, bearings and supporting structures. This combination of single substructures leads to the complete simulation model ofthe wind turbine.Fig. 3: Modeling of shafts by discretisation, SiMBeaM model and FeM-method Fig. 5: Variable gearing stiffness over the contact path using Fourier coefficientsFig. 4: Gear box of a 3 MW wind turbine“The degrees of freedom of each substructure can be adjusted toaccommodate varying degrees of detail of the overall model.”The degrees of freedom of each substructure can be adjusted to accommodate varying degrees of detail for the overall model. Sub-modeling enables easy verification of the appropriate level of model detail.MoDelInG oF DrIVetrAIn CoMPonentSThe dynamic behavior of a drivetrain results from the gear ratio and the distribution of the mass, mass of inertia and stiffness. The de-termination of mass parameters is possible with three-dimensional CAD models or by using simple analytical approaches. Great effort is required to accurately calculate the various stiffness parameters for all of the drivetrain components.The torsional stiffness of the drivetrain is mainly characterized bythe stiffness of the shafts. Special consideration must be given to slender shafts whose elastic properties need to be accounted for.Additionally, the bending stiffness of such shafts may have consid-erable influence on the dynamic behavior as well as the resultingdisplacements. The required simulation model can be assembled in three ways:a) by using the method of discretization, b) via the beam approach or c) by implementing modally reduced finite-element models (Fig. 3).For models that include the axial and ra-dial motion of the shafts, the properties ofbearings must be accounted for. Essentially, the modeling of the bearings is realized by a force element which introduces the reaction forces in the axial and radial directions as well as the reaction moments, if necessary. The bearing properties can be described by average bearing stiffness, characteristic curves or complex models imported as DLLs.The transmission ratio between the rotor and the high speed gen-erator can be realized by a gear box consisting of a set of planetary and helical gear stages (Fig. 4). The changing speeds and torques in a gear box as well as the varying gearing stiffness resulting from the total overlap ratio have an important influence on the dynamic behavior of the drivetrain and must be considered in the simulation model. SIMPACK offers the special force element Gear Pair (FE 225)Chair of Machine Elements, Dresden University of Technologywhich enables a comfortable modeling of gearing. An alternative modeling approach offers the mathematical description of the re-sulting forces in the gearing by user routines. Based on the calcula-tion of the tooth normal force in the ideal pitch point, the complete tooth contact is simplified and described in one point. The tooth normal force consists of stiffness and damping dependent parts. Information about the displacements and velocities in tangential, radial and axial directions resulting from the relative position of the gears can be determined from the joint states and the corresponding trigonometric relationships. The gearing stiffness can be considered as average contact stiffness according to DIN 3990 and variable gearing stiffness over the path of contact using Fourier coefficients (Fig. 5).In order to improve the model even further, modeling of the cou-pling and generator is necessary. The most important influence on the dynamic behavior results from the rotorblades. A simplified approach for modeling of the rotorblade stiffness can be done by splitting the blades into mass segments coupled by spring damper elements to represent the bending stiffness (Fig. 6). Therefore, the information of the mass and stiffness distribution as well as the natural frequencies in edge- and flapwise direction are sufficient. In addition, if the information of every profile section is available, the rotorblade generation in SIMPACK offers an automated procedure.A detailed modeling of the shafts,bearings, gearings, coupling, generatorand rotor allows the representation ofthe dynamic interaction between thecomponents and the determination ofdisplacements, velocities, accelerations,forces and torques. Even if spring-damper elements are used to sup-port the components, the surrounding structure, e.g. the gear box housing or the main frame, is assumed to have a stiff coupling to the global reference system. Thereby the influences resulting from the flexible structure of a wind turbine are neglected. The enhance-ment of the stiff multi-body system model using modally reduced finite-element structures allows the consideration of these effects. The implementation of a flexible structure in SIMPACK is based on a meshed finite-element model of the component geometry and the definition of the material properties. The connection points to the Fig. 6: Discretisation of a rotorblade supportingspring-damperelements canbe modeled bymeans of multi-point constraints(MPC). Due to theresulting complex-ity and degreesof freedom of theFEM-models a reduc-tion of the structure isrequired. The applicationof the approach accordingto Craig-Bampton requiresthe definition of the connec-tion points between the flexiblestructure and the rigid bodies.The mode shapes of the reducedmodel are used to determine thedeformation under load. The number ofnatural frequencies chosen for the modalreduction defines the valid frequency rangeand the accuracy of themodel, which is alsoinfluenced by the choiceof frequency responsemodes in the SIMPACKAdd-On Module FEMBS.The implementation of flexible structuresallows representation of the flexibility of thesupporting structure as well as the consider-ation of the stiffness of the drivetrain com-ponents, e.g. shafts and planetary carriers,with a higher degree of accuracy.AnAlYSIS oF nAturAl FreQuenCIeSAnD eXCItAtIonSThe resulting flexible multi-body systemmodel allows the determination of the natu-ral frequencies and can take into accountvarious degrees of freedom. The resultingfrequencies can be compared to the excita-tion frequencies to determine possible reso-nances. Relevant excitations are the first,second, third, sixth (ninth, twelfth) order ofthe rotor rotation frequency. Additionally,the first and second orders of the rotationfrequencies of all drivetrain components, aswell as the meshing frequencies of the gearstages, have to be considered. The compari-son of natural frequencies and excitationsby means of a Campbell diagram revealspossible resonances. The analysis of thecorresponding mode shapes allows furtherstatements as to whether the excitation ofa natural frequency can cause critical opera-tion points or not.AnAlYSeS In tHe tIMe DoMAInThe detailed flexible multi-body systemmodel also offers the possibility of determin-Fig. 7: Windloads on flexiblerotorblades“SIMPACK offers the special force element Gear Pair (FE 225) which enables a comfortable modeling of gearing.”ing resulting displacements, deformations, velocities, accelera-tions, forces and torques under the dynamic loads resulting from the stochastic wind field. To model a realistic wind field, different approaches are available. The common wind turbine design software tools like Bladed, Flex5 and AeroDyn are mainly used to define the design loads. An in-terface between AeroDyn and SIMPACK is available (FE 237: Wind AeroDyn*)which offers the possibility of determining wind loads based on the rotorblades, tower and wind turbine parameters. If all the required information is not given, different simplified approaches based on measurements can be used. A wind model based on the information of the wind speed, power coefficients and assumed load distribu-tions over the blade length and the height can also be used. Using nine segments for each blade and by superimposing a stochastic wind field, the tower shadow can be calculated. An enhancement of this approach replaces the as-sumptions for the wind field and load distribu-tion by measurement results captured as torque and bending moments at the main shaft. An algorithm adjusts the single blade forces to achieve the measured results on the main shaft in the simulation model so that the measured states of the wind turbine can be represented by the model. The changing forces at the flex-ible modeled rotorblades are shown in Fig. 7 as arrows and scaled deformations.The operation of the wind turbine under dif-ferent load conditions and extreme load cases like emergency stops can also be calculated. The resulting speeds, torques and forces in the gearing of the first planetary gear stage are shown in Fig. 8. In addition, the information of displacements of the main shaft, gear box and gear box components like the sun shaft (Fig. 9) can be obtained. This data can lead to a deeper understanding of the dynamic behavior of the system and the resultant loads under different load conditions.Fig. 8: emergency stop simulation, speed, pitch angle and forceFig. 9: Displacement of the sunConCluSIonWind turbines are anchored flexible complex drivetrains which oper-ate under highly dynamic stochastic loads. For onshore wind tur-bines, and especially for offshore wind turbines, the request for high reliability requires comprehensive knowledge of dynamic behavior already in the design phase. This includes information about possible excitations and natural frequencies which can cause resonances in the operational speed range. Additionally, the displacements, defor-mations and resulting forces in the drivetrain, as well as the influ-ence of wind turbine control under maximum loads during normal operation and emergency cases, has to be analyzed. The multi-body method offers the ability to realistically model a wind turbine while considering all relevant components and degrees of freedom. This approach enables the required knowledge to be obtained in order to fully understand the dynamics of wind turbines.。

基于群体平衡的汽轮机动叶表面盐析颗粒分布特性

2018年第37卷第2期 CHEMICAL INDUSTRY AND ENGINEERING PROGRESS·437·化工进展基于群体平衡的汽轮机动叶表面盐析颗粒分布特性胡鹏飞，李勇，曹丽华，吴雪菲（东北电力大学能源与动力工程学院，吉林吉林 132012）摘要：为深入了解汽轮机动叶内盐析颗粒的微观行为，本文以某超临界汽轮机高压级动叶为研究对象，应用计算流体力学与群体平衡模型耦合方法，对汽轮机动叶内盐析颗粒在流场中的分布进行数值模拟研究，获得了盐析颗粒在动叶内的粒径分布及不同负荷时叶片尾缘处盐析颗粒数量密度分布规律。

模拟结果表明：在汽轮机动叶吸力面附近的盐析颗粒粒径较压力面附近盐析颗粒粒径小，且叶根处颗粒粒径小于叶顶处；动叶压力面的颗粒数量密度呈前缘点尾缘点处大、中间段小的分布规律，并且盐析颗粒在叶片上的数量密度分布最大值并不出现在组分数及粒径最大处，而是出现在平均粒径为110～150μm 的盐析颗粒沉积位置处；当汽轮机30%负荷运行时，粒径40μm 盐析颗粒的数量密度是其在汽轮机额定负荷运行时的1.5倍，而粒径140μm 盐析颗粒的数量密度仅为汽轮机额定负荷运行时的80%。

关键词：汽轮机动叶；盐析颗粒；群体平衡模型；两相流中图分类号：TK26 文献标志码：A 文章编号：1000–6613（2018）02–0437–07 DOI ：10.16085/j.issn.1000-6613.2017-1765Distribution characteristics of salting-out particles on the surface of steamrotor blade based on population balance model （PBM ）HU Pengfei ，LI Yong ，CAO Lihua ，WU Xuefei（School of Energy and Power Engineering ，Northeast Electric Power University ，Jilin 132012，Jilin ，China ）Abstract ：In order to get a better understanding of microscopic behavior of salting-out particles in a steam turbine ，a high-pressure grade rotor blade was employed in a supercritical steam turbine as a research object and the distribution of salting-out particles in the flow field from a steam turbine rotor blade was simulated using CFD-PBM method. The diameter distribution of salting-out particles in a rotor blade and the number density distribution of salting-out particles in the tailed-edge area of rotor blade with different load situations were obtained. The simulation results showed that the salting-out particle diameter near the suction side was smaller than that near the pressure side in a steam turbine rotor blade ，and the salting-out particle diameter at the blade bottom was smaller than that at the blade tip. The particle number density distribution law at the pressure side of rotor blade was presented that the particle number density was larger both at the leading edge and at the tailed-edge of rotor blade while the particle number density was smaller in the middle parts of rotor blade ，and the maximum value of salting-out particle number density distribution did not appear in the position having the maximum component number and particle diameter in the rotor blade ，but it appeared in the positionwhere salting-out particles with the average diameter 110—150μm deposit. When steam turbine was under 30% load operation ，the number density of salting-out particles with 40μm diameter was 1.5第一作者及通讯作者：胡鹏飞（1985—），男，博士研究生，讲师，主要研究方向为汽轮机节能技术与优化运行。

Fluid-Structure Interaction

Fluid-Structure Interaction Fluid-Structure Interaction (FSI) is a complex and interdisciplinary fieldthat involves the interaction between a fluid flow and a solid structure. This interaction can have significant effects on the behavior and performance of both the fluid and the structure, making it a crucial consideration in various engineering applications. The study of FSI is essential in understanding phenomena such as flutter in aircraft wings, vortex-induced vibrations in offshore structures, and blood flow in arteries. One of the key challenges in FSI is accurately modeling and simulating the interaction between the fluid and the structure. This requires a deep understanding of both fluid dynamics andstructural mechanics, as well as sophisticated computational tools to solve the coupled equations governing the FSI problem. The development of numerical methods for FSI simulations has been a major focus of research in recent years, with the goal of improving the accuracy and efficiency of FSI models. In addition to numerical simulations, experimental techniques play a crucial role in studying FSI phenomena. Physical testing allows researchers to validate and calibrate their numerical models, as well as to investigate complex FSI behaviors that aredifficult to capture through simulations alone. Experimental facilities such as wind tunnels, water tanks, and biomedical labs provide valuable data for understanding the dynamics of FSI and for developing new design guidelines for engineering structures. The study of FSI has wide-ranging applications across various industries, including aerospace, civil engineering, biomechanics, and marine engineering. In aerospace, FSI is essential for predicting the aerodynamic performance and structural integrity of aircraft components, such as wings and fuselage. In civil engineering, FSI is used to analyze the behavior of buildings and bridges under wind and earthquake loads, as well as to optimize the design of offshore structures for oil and gas production. From a biomedical perspective, FSI is critical for understanding the flow of blood in arteries and theinteraction between blood vessels and surrounding tissues. This knowledge is essential for diagnosing and treating cardiovascular diseases, as well as for designing medical devices such as stents and artificial heart valves. In marine engineering, FSI is used to study the response of ships and offshore platforms towaves and currents, as well as to optimize the design of marine structures for improved performance and safety. Overall, the study of Fluid-StructureInteraction is a fascinating and challenging field that requires amultidisciplinary approach to address complex engineering problems. By combining theoretical analysis, numerical simulations, and experimental testing, researchers can gain valuable insights into the behavior of fluid-structure systems and develop innovative solutions for a wide range of practical applications. The continued advancement of FSI research will lead to new discoveries and technologies that will shape the future of engineering and science.。

Ornstein–Uhlenbeck process - Wikipedia, the f

Ornstein–Uhlenbeck process - Wikipedia,the f...Ornstein–Uhlenbeck process undefinedundefinedFrom Wikipedia, the free encyclopediaJump to: navigation, searchNot to be confused with Ornstein–Uhlenbeck operator.In mathematics, the Ornstein–Uhlenbeck process (named after LeonardOrnstein and George Eugene Uhlenbeck), is a stochastic process that, roughly speaking, describes the velocity of a massive Brownian particle under the influence of friction. The process is stationary, Gaussian, and Markov, and is the only nontrivial process that satisfies these three conditions, up to allowing linear transformations of the space and time variables.[1] Over time, the process tends to drift towards its long-term mean: such a process is called mean-reverting.The process x t satisfies the following stochastic differential equation:where θ> 0, μ and σ> 0 are parameters and W t denotes the Wiener process. Contents[hide]1 Application in physical sciences2 Application in financialmathematics3 Mathematical properties4 Solution5 Alternative representation6 Scaling limit interpretation7 Fokker–Planck equationrepresentation8 Generalizations9 See also10 References11 External links[edit] Application in physical sciencesThe Ornstein–Uhlenbeck process is a prototype of a noisy relaxation process. Consider for example a Hookean spring with spring constant k whose dynamics is highly overdamped with friction coefficient γ. In the presence of thermal fluctuations with temperature T, the length x(t) of the spring will fluctuate stochastically around the spring rest length x0; its stochastic dynamic is described by an Ornstein–Uhlenbeck process with:where σ is derived from the Stokes-Einstein equation D = σ2 / 2 = k B T / γ for theeffective diffusion constant.In physical sciences, the stochastic differential equation of an Ornstein–Uhlenbeck process is rewritten as a Langevin equationwhere ξ(t) is white Gaussian noise with .At equilibrium, the spring stores an averageenergy in accordance with the equipartition theorem.[edit] Application in financial mathematicsThe Ornstein–Uhlenbeck process is one of several approaches used to model (with modifications) interest rates, currency exchange rates, and commodity prices stochastically. The parameter μ represents the equilibrium or mean value supported by fundamentals; σ the degree of volatility around it caused by shocks, and θ the rate by which these shocks dissipate and the variable reverts towards the mean. One application of the process is a trading strategy pairs trade.[2][3][edit] Mathematical propertiesThe Ornstein–Uhlenbeck process is an example of a Gaussian process that has a bounded variance and admits a stationary probability distribution, in contrast tothe Wiener process; the difference between the two is in their "drift" term. For the Wiener process the drift term is constant, whereas for the Ornstein–Uhlenbeck process it is dependent on the current value of the process: if the current value of the process is less than the (long-term) mean, the drift will be positive; if the current valueof the process is greater than the (long-term) mean, the drift will be negative. In other words, the mean acts as an equilibrium level for the process. This gives the process its informative name, "mean-reverting." The stationary (long-term) variance is given byThe Ornstein–Uhlenbeck process is the continuous-time analogue ofthe discrete-time AR(1) process.three sample paths of different OU-processes with θ = 1, μ = 1.2, σ = 0.3:blue: initial value a = 0 (a.s.)green: initial value a = 2 (a.s.)red: initial value normally distributed so that the process has invariant measure [edit] SolutionThis equation is solved by variation of parameters. Apply Itō–Doeblin's formula to thefunctionto getIntegrating from 0 to t we getwhereupon we seeThus, the first moment is given by (assuming that x0 is a constant)We can use the Itōisometry to calculate the covariance function byThus if s < t (so that min(s, t) = s), then we have[edit] Alternative representationIt is also possible (and often convenient) to represent x t (unconditionally, i.e.as ) as a scaled time-transformed Wiener process:or conditionally (given x0) asThe time integral of this process can be used to generate noise with a 1/ƒpower spectrum.[edit] Scaling limit interpretationThe Ornstein–Uhlenbeck process can be interpreted as a scaling limit of a discrete process, in the same way that Brownian motion is a scaling limit of random walks. Consider an urn containing n blue and yellow balls. At each step a ball is chosen at random and replaced by a ball of the opposite colour. Let X n be the number of blueballs in the urn after n steps. Then converges to a Ornstein–Uhlenbeck process as n tends to infinity.[edit] Fokker–Planck equation representationThe probability density function ƒ(x, t) of the Ornstein–Uhlenbeck process satisfies the Fokker–Planck equationThe stationary solution of this equation is a Gaussian distribution with mean μ and variance σ2 / (2θ)[edit ] GeneralizationsIt is possible to extend the OU processes to processes where the background driving process is a L évy process . These processes are widely studied by OleBarndorff-Nielsen and Neil Shephard and others.In addition, processes are used in finance where the volatility increases for larger values of X . In particular, the CKLS (Chan-Karolyi-Longstaff-Sanders) process [4] with the volatility term replaced by can be solved in closed form for γ = 1 / 2 or 1, as well as for γ = 0, which corresponds to the conventional OU process.[edit ] See alsoThe Vasicek model of interest rates is an example of an Ornstein –Uhlenbeck process.Short rate model – contains more examples.This article includes a list of references , but its sources remain unclear because it has insufficient inline citations .Please help to improve this article by introducing more precise citations where appropriate . (January 2011)[edit ] References^ Doob 1942^ Advantages of Pair Trading: Market Neutrality^ An Ornstein-Uhlenbeck Framework for Pairs Trading ^ Chan et al. (1992)G.E.Uhlenbeck and L.S.Ornstein: "On the theory of Brownian Motion", Phys.Rev.36:823–41, 1930. doi:10.1103/PhysRev.36.823D.T.Gillespie: "Exact numerical simulation of the Ornstein–Uhlenbeck process and its integral", Phys.Rev.E 54:2084–91, 1996. PMID9965289doi:10.1103/PhysRevE.54.2084H. Risken: "The Fokker–Planck Equation: Method of Solution and Applications", Springer-Verlag, New York, 1989E. Bibbona, G. Panfilo and P. Tavella: "The Ornstein-Uhlenbeck process as a model of a low pass filtered white noise", Metrologia 45:S117-S126,2008 doi:10.1088/0026-1394/45/6/S17Chan. K. C., Karolyi, G. A., Longstaff, F. A. & Sanders, A. B.: "An empirical comparison of alternative models of the short-term interest rate", Journal of Finance 52:1209–27, 1992.Doob, J.L. (1942), "The Brownian movement and stochastic equations", Ann. of Math.43: 351–369.[edit] External linksA Stochastic Processes Toolkit for Risk Management, Damiano Brigo, Antonio Dalessandro, Matthias Neugebauer and Fares TrikiSimulating and Calibrating the Ornstein–Uhlenbeck process, M.A. van den Berg Calibrating the Ornstein-Uhlenbeck model, M.A. van den BergMaximum likelihood estimation of mean reverting processes, Jose Carlos Garcia FrancoRetrieved from ""。

ornstein-uhlenbeck noise 原理 -回复

ornstein-uhlenbeck noise 原理 -回复Ornstein-Uhlenbeck noise (OU noise) is a stochastic process that is widely used in various fields, including physics, biology, and finance. This noise process is named after physicists Leonard Ornstein and George Eugene Uhlenbeck, who first introduced it in 1930 as a model for the velocity of a Brownian particle subject to friction.To understand the principles behind Ornstein-Uhlenbeck noise, let us first explore the basics of stochastic processes. In mathematics and st atistics, astochastic process is a collection of random variables indexed by a timeparameter. These random variables can represent the behavior of a system over time, where the value of the variable at a given time is uncertain.One common example of a stochastic process is Brownian motion. Brownian motion describes the random movement of particles suspended in a fluid, such as the movement of pollen grains in water. The particle's position changes randomly over time due to the continuous bombardment of water molecules.Ornstein-Uhlenbeck noise, on the other hand, models a system with some level of mean reversion or tendency to return to a central value. This characteristic isoften observed in real-world phenomena, such as the behavior of stock prices or the motion of a pendulum subject to damping.The Ornstein-Uhlenbeck process can be mathematically defined as a stochastic differential equation (SDE). The SDE representing Ornstein-Uhlenbeck noise is as follows:dX(t) = θ(μ- X(t))dt + σdW(t)In this equation, dX(t) represents the infinitesimal change in the value of the variable X at time t. The first term θ(μ- X(t))dt models the mean reversion behavior, where θrepresents the strength of mean reversion and μis themean or central value that the pr ocess tends to move towards. The second term σdW(t) captures the instantaneous random fluctuations, where dW(t) is an increment of standard Wiener process or Brownian motion, and σrepresents the intensity of these fluctuations.By solving the SDE, we can generate sample paths of the Ornstein-Uhlenbeck process. These sample paths exhibit a characteristic behavior where the process tends to revert to its mean value while still allowing for random fluctuations around it. The rate of mean reversion and the in tensity of fluctuations can be adjusted by changing the parameters θand σ.In practical applications,Ornstein-Uhlenbeck noise is often used as a tool for generating synthetic time series data that resemble certainreal-world phenomena. For example, in finance, the motion of stock prices is often modeled as a stochastic process with mean reversion to capture the observed tendency of prices to revert to their long-term average. Ornstein-Uhlenbeck noise provides a useful framework for simulating such proces ses. Similarly, in physics, this noise process has been used to model the motion of charged particles subject to random forces.Furthermore, Ornstein-Uhlenbeck noiseis used as a regularization technique in machine learning and optimization problems. By a dding this noise to the gradient updates during training, it introduces a form of randomness that can help prevent the model from getting stuck in local optima and improve its generalization performance.。

Mathematical Model for Fluid Flow and Heat Transfer in the Cooling Shaft of

Cpf Cp~ dp dh F ks K g h~f m n nx ny p qw q~ r r R0 RI S t t, ty Tf T~ uf Us ut vf vs v~ V Vf
general transport equation fluid specific heat, (J/(kg K)) solid specific heat, (J/(kg K)) particle diameter, (m) hydraulic diameter of the passage of gas flow, (m) inertial coefficient, dimensionless the coefficient of solid flow potential permeability, (mz) gravitational acceleration, -9.81, (m2/s) fluid-solid heat transfer coefficient, (W/(m 2 K)) mass flux through a control volume face unit outward normal vector the x -direction component of n the y -direction component of n fluid pressure, (Pa) heat flux at the wall, (W/m 2) source term the position vector radial coordinate, (m) the diameter of prechamber, (m) the diameter of the cooling chamber, (m) the surface area of cell face, (m2) unit vector tangential to the boundary. the x -direction component of t the y -direction component of t fluid temperature, (K) solid temperature, (K) Darcy velocity in x direction, (m/s) the coke descending velocity in x direction, (m/s) velocity vector component in t -direction Darcy velocity in r direction, (m/s) the coke descending velocity in r direction, (m/s) velocity vector component in n -direction, (m/s) velocity vector, (m/s) gas velocity vector, (m/s)

Deterministic and Stochastic Models

Irina Overeem Community Surface Dynamics Modeling SystemUniversity of Colorado at BoulderSeptember 20081Course outline 1•Lectures by Irina Overeem:•Introduction and overview•Deterministic and geometric models•Sedimentary process models I•Sedimentary process models II•Uncertainty in modeling•Lecture by Overeem & Teyukhina :•Synthetic migrated data2Geological Modeling: different tracks Static Reservoir ModelReservoir DataSeismic, borehole and wirelogsSedimentary Process Model Stochastic Model Deterministic ModelData-driven modeling Process modelingFlow ModelUpscalingDeterministic and Stochastic Models •Deterministic model-A mathematical model which contains no random components; consequently, each component and input is determined exactly.•Stochastic model -A mathematical model that includes some sort of random forcing.•In many cases, stochastic models are used to simulate deterministic systems that include smaller-scale phenomena that cannot be accurately observed or modeled. A good stochasticmodel manages to represent the average effect of unresolvedphenomena on larger-scale phenomena in terms of a randomforcing.4Deterministic geometric models•Two classes:–Faults (planes)–Sediment bodies (volumes)•Geometric models conditioned to seismic•QC from geological knowledge5Direct mapping of faults and sedimentaryunits from seismic data•Good quality 3D seismic data allows recognition of subtle faults and sedimentary structures directly.•Even more so, if (post-migration) specific seismic volume attributes are calculated.•Geophysics Group at DUT worked on methodology to extract 3-D geometrical signal characteristics directly from the data.6L08 Block, Southern North SeaSeismic volume attribute analysis ofthe Cenozoic succession in the L08block, Southern North Sea. Steeghs,Overeem, Tigrek, 2000. Global andPlanetary Change, 27, 245–262.07 January 20227Fault modelling Fault surfaces•from retrodeformation (geometries ofrestored depositional surfaces)10More faultmodellingin Petrel•Check plausibility of impliedstress and strain fields11FanFan Feederchannel Delta ForesetsGas-filled meandering channelDeterministic sedimentary model fromseismic attributesObject-based Stochastic Models •Point process: spatial distribution of points (object centroids) in space according to some probability law•Marked point process: a point process attached to (marked with) random processes defining type, shape, and size of objects •Marked point processes are used to supply inter-well object distributions in sedimentary environments with clearly definedobjects:–sand bodies encased in mud–shales encased in sand16Ingredients of marked point process •Spatial distribution (degree ofclustering, trends)•Object properties (size, shape,orientation)17An example: fluvial channel-fill sands •Geometries have become more sophisticated, but conceptual basis has not changed: attempt to capture geological knowledgeof spatial lithology distribution by probability laws18•Examples of shape characterisation:–Channel dimensions (L, W) and orientation–Overbank deposits–Crevasse channels–LeveesExploring uncertainty of object properties(channel width)•W = 100 m•W = 800 m•Major step forward:object-based model ofchannel belt generated byrandom avulsion at fixedpoint•Series of realisationsconditioned to wells(equiprobable)Stochastic Model constrained by multipleanalogue data •Extract as much information as possible from logs and cores (Tilje Fm. Haltenbanken area, offshore Norway).•Use outcrop or modern analogue data sets for facies comparison and definition of geometries •Only then ‘Stochastic modeling’will begin22Lithofacies types from core Example: Holocene Holland Tidal Basin 07 January 202223SELECTED WINDOW FOR STUDYModern Ganges tidal delta, India distanc e5kmChannel widthTidal channels Conceptual model of tidal basin (aerial photos, detailed maps)Quantify the analogue data into relevantproperties for reservoir model •Channel width vs distance to shorelineThe resulting stochastical model……28Some final remarks onstochastic/deterministic models•Stochastic Modeling should be data-driven modeling•Both outcrop and modern systems play an important role in aiding this kind of modeling.•Deterministic models are driven by seismic data.•The better the seismic data acquisition techniques become, the more accurate the resulting model.References •Steeghs, P., Overeem, I., Tigrek, S., 2000. Seismic Volume Attribute Analysis of the Cenozoic Succession in the L08 Block (Southern North Sea). Global and Planetary Change 27, 245-262.• C.R. Geel, M.E. Donselaar. 2007. Reservoir modelling of heterolithic tidal deposits: sensitivity analysis of an object-based29 stochastic model, Netherlands Journal of Geosciences, 86,4.。

Stochastic Modeling of Power Demand Due to EVs Using Copula

Stochastic Modeling of Power Demand Due toEVs Using CopulaAlicja Lojowska ,Student Member,IEEE ,Dorota Kurowicka,Georgios Papaefthymiou ,Member,IEEE ,andLou van der Sluis ,Senior Member,IEEEAbstract—The driving patterns characterizing electric vehicles (EVs)are stochastic and,as a consequence,the electrical load due to EVs inherits their randomness.This paper presents a Monte Carlo procedure for the derivation of load due to EVs based on a fully stochastic method for modeling transportation patterns.Under the uncontrolled domestic charging scenario three variables are found to be crucial:the time a vehicle leaves home,the time a vehicle arrives home,and the distance traveled in between.A detailed transportation dataset is used to derive marginal cumu-lative distribution the variables of interest.Since the variables are a joint distribution function is built using a copula Subsequently,simulated EV trips are combined with a typical charging pro ﬁle so that the energy con-tribution to the system is computed.The procedure is applied to analyze the effect of the EV load on the national power demand of The Netherlands under different market penetration levels and day/night electricity tariff scenarios.Index Terms—Copula,dependence structure,domestic charging,EVs load,price incentives,transportation dataset,uncontrolled charging.I.I NTRODUCTIONTHE development of electric vehicles (EVs)is currently driven by the need to decrease reliance on foreign oil sup-plies and to reduce CO ernmental plans in many countries support a signi ﬁcant increase in the number of EVs on the roads by the next decade.A respective scenario for The Netherlands assumes there will be one million EVs on the road between the years 2020and 2025[1].The future deployment of EVs leads to an increase in elec-tricity demand.To assess the effects on the power system,an estimation of EVs power demand is required.In the related lit-erature,a number of methods have been proposed for the pre-diction of the load of future EVs.These methods are usuallyManuscript received July 13,2011;revised December 01,2011;accepted March 03,2012.Date of publication May 01,2012;date of current version Oc-tober 17,2012.This work was supported by SenterNovem,an agency of the Dutch Ministry of Economical Affairs,under Project IOP EMVT 08103.Paper no.TPWRS-00940-2011.A.Lojowska and L.van der Sluis are with the Electric Power Systems Group,Delft University of Technology,Delft 2628CD,The Netherlands (e-mail:a.lo-jowska@tudelft.nl;l.vanderSluis@tudelft.nl).G.Papaefthymiou is with the Ecofys Germany GmbH and with the Electric Power Systems Group,Delft University of Technology,Delft 2628CD,The Netherlands (e-mail:g.papaefthymiou@tudelft.nl).D.Kurowicka is with the Institute for Applied Mathematics,Delft University of Technology,Delft 2628CD,The Netherlands (e-mail:d.kurowicka@tudelft.nl).Color versions of one or more of the ﬁgures in this paper are available online at .Digital Object Identi ﬁer 10.1109/TPWRS.2012.2192139based on driving patterns that are taken into account by the di-rect use of data of real commuting habits (single realizations).For example,in [3]EVs power demand is derived based on rep-resentative driving patterns obtained through clustering of data from the Danish National Travel Survey.The same dataset is also utilized in [4]to create average plug-in patterns that are input for studying the optimal con ﬁguration of an integrated power and transport system.The work in [5]presents a method-ology for modeling EVs that uses a full-year input time series of driving patterns obtained from an analogous travel survey for Finland.Axsen et al.[10]derive PHEV energy consumption by employing data collected through a web-based survey of new vehicle buyers in California.The simulation studies carried out by NREL [8]and,under the REHEV project [7],were based on driving habits of the St.Louis population recorded using GPS technology [9].Other approaches utilize national statistics [6],e.g.,daily average distance or even hypothetical transportation demand [16].In [17],a Monte Carlo technique is used to gen-erate a set of future EVs’electricity demand pro ﬁles based on information about an expected future EVs’load and an expected amount of uncertainty involved.However,the literature lacks methods that allow the modeling of the inherent stochasticity of the driving patterns,as elaborated in [11].Contrary to the direct use of data,stochastic simulation methods rely on transportation data (typically travel surveys)in order to capture the uncertainty of the main variables describing the behavior of EVs.This paper presents the setup of such a stochastic method;the EV’s power demand can be simulated using three random variables:the time of arrival at charging area,time of departure from charging area,and distance trav-eled in between.The variables are found to be correlated and to have nonstandard distribution functions.In such a situation,to perform a stochastic modeling a copula function has to be used in creating a joint distribution function.The stochastic simulation setup is used for the generation of synthetic datasets that comply with the uncertainty of the inputs.These synthetic datasets capture the whole uncertainty of the behavior of EVs (contrary to single realizations)and are scal-able to different EV populations (allowing uncertainty reduction effects in large populations).Moreover,such stochastic setups can be used for the design of systems when transportation data are not readily available or future changes in transportation be-havior are expected 1.1Inthe latter case,expert judgement techniques can be used to extract infor-mation on probability distributions of the variables characterizing commuting habits of the future population [19].0885-8950/$31.00©2012IEEEThis paper is structured as follows.Section II covers informa-tion regarding the transportation dataset.In Section III,the vari-ables representing the driving patters are studied together with the statistical relationships between them.A modeling method-ology is proposed.In Section IV,a Monte Carlo simulation is presented and is followed by Section V,where the results are discussed.Conclusions and future work are given in Section VI.II.T RANSPORTATION D ATASETStatistical information on the variables representing driving patterns are extracted from the transportation data of the year 2008provided by the Dutch Ministry of Transportation [2].Per-sons taking part in the transportation survey were eligible to give details regarding trips they made in The Netherlands during the elicitation period.The dataset includes information about com-muting activities like time of departure,address of departure’s place,main purpose of the commuting,mean of transport,ad-dress of the place a person arrived and time of arrival,the dis-tance of the trip,and so on.A.h2h TripsThe paper focuses on domestic charging,i.e.,the case when charging infrastructure is organized in the way that vehicle users have recharge access only at their home.From the system point of view,the timing and EVs’power demand are of interest.Recharging an EV’s battery can take place in the time interval which starts at the moment a driver arrives home and ﬁnishes upon departure from home.The battery state of charge (SoC)of an EV is strongly related to the distance traveled since the last charging.Therefore,in modeling driving patterns for the load computation we are interested in the following variables:the time a vehicle leaves home,the time a vehicle arrives home and the overall distance traveled since the last visit at home.The sequence of trips which started at home one day and ﬁnished the same or next day is referred throughout the paper as an h2h trip,e.g.,a sequence of the following trips:home work shop home.According to the dataset,75%of the respondents had a single h2h trip in a day whereas 21%a double h2h trip and the rest had three or more h2h trips.The paper presents the modeling of single as well as double trips since they comprise the signi ﬁcant majority of trips.2Respectively,other charging infrastructures can be consid-ered as well.If one focuses on charging at home and at work,the set of variables would be extended by the time a vehicle ar-rives at work and the time a vehicle departures from work.B.Potential UsersThe possibility to charge the battery only at home makes the EVs suitable for people whose typical distance of an h2h trip does not exceed the maximum driving range of the vehicle’s battery.Potential EV users are therefore chosen according to the maximum driving range of a vehicle’s battery.In the simulation2Thehigher order trips could be modeled as well,but they require more vari-ables to consider.A bigger transportation dataset would have to be used in order to provide good quality estimates of the respective model parameters.Fig.1.Charging pro ﬁle of the Nissan Altra battery.studies,the case of Nissan Altra EV with lithium-ion battery 3is used as example [6],[18].The battery-charging pro ﬁle is shown in Fig.1.The capacity of the battery is equal to 29.07kWh.The maximum driving range depends on several factors,including climate control,speed,and driving style,and,consequently,dif-fers for urban and freeway area.We assume a maximum driving range of 130km [6],[18].In order to prevent a so-called range anxiety,the subset of potential users are chosen according to the maximum range of 110km so that at least about 10%of energy would be left after an h2h trip.III.S IMULATION A PPROACHModeling single as well as double h2h trips requires capturing the stochastic behavior of the random variables representing them.Modeling of single h2h trips requires using three vari-ables:start-time ,end-time of the h2h trip,and trav-eled distance ,whereas a double h2h trip needs six vari-ables:start-time,end-time,and traveled distance of the ﬁrst h2h trip and the second h2h trip .This section presents the data analysis of an h2h trip in three steps.In the ﬁrst step,the one-dimensional (1-D)marginal dis-tribution of the variables representing h2h trips are derived.The second step involves examining the stochastic dependence be-tween the variables.In the last step,a stochastic simulation ap-proach is presented which enables the modeling of the variables together with their dependence structure.A.Marginal The probability the start times of h2h trips are shown in Fig.2.We can see that the start-time of a single h2h trip and the time of the ﬁrst departure from home in a double h2h trip are described by similar distributions.In both cases,a high percentage of drivers tend to leave their home between 7and 9am,most likely to commute to work.After this peak,the probabilities gradually decrease and,for a single h2h trip,exhibit some less signi ﬁcant peaks around 1and 7pm.It should be noted that,since a vehicle user is going to make a second trip in the case of a double h2h trip,the probability distribution function (pdf)of is less concentrated in the evening hours than the pdf of .The pdf of (the start-time of the second3Themethod can be applied with any type of batteries,also those of plug in hybrid EVs (PHEV).In the case of PHEVs,the choice of potential users would have to take into account the maximum distance using both an electric motor and an internal combustion engine.Fig.2.Distributions of start-times of a single and double h2htrip.Fig.3.Distributions of end-times of a single and double h2h trip.The timescale is extended to the next day.h2h trip in a double h2h trip)reaches two peaks at 1pm and at 7pm,where it is the highest.Fig.3presents the probability distributions of arrival times.The peak for a single h2h trip occurs between 4and 6pm,when people tend to return from work.The histograms of arrival times for double h2h trips have peaks in the same interval but also during late morning and late evening .The hours higher than 24on the -axis indicate that an h2h trip started one day and ﬁnished the next rmation on end-time of h2h trips are crucial in computing the starting time of recharging vehicles batteries.Under the assumption that there are no price incentives,Fig.3shows that the recharging can take place at any hour of a day.The pdfs of traveled distance between home and home are shown in Fig.4.For all types of h2h trips,short distances are the most common,i.e.,up to 20km.Distances of double h2h trips (and )are characterized by a higher occurrence of low values than the respective distances of a single h2h trip,which is a consequence of makingmultiple h2h trips per day instead of one.No distance exceeds 110km due to the as-sumptions discussed in Section II-B.Fig.4.Distributions of traveled distance of a single and double h2h trip.B.DependenceStructureIn order to verify if there is dependence among the above-mentioned variables,the measure of rank correlation [21]is ap-plied.The rank correlation measures a monotonic relationship between variables [20],[21].If tends to increase when in-creases,the rank correlation coef ﬁcient is positive and vice versa .The rank correlation coef ﬁcient can take the values between 1and 1.Rank correlation derived for the variables representing a single h2h trip:,,and are given by(1)The rank correlation for a double h2h trip is computed based on the variables ,,,,,and asfollows:(2)Testing the hypothesis of no correlation against the alterna-tive that there is a nonzero correlation was applied to the ele-ments of the correlation matrices.All coef ﬁcients in the ma-trix are found to be statistically signi ﬁcant assuming a 0.05level of signi ﬁcance.In the matrix,all coef ﬁcients are signi ﬁcant except from ,,and .This suggests no monotonic dependence between the starting time of the ﬁrst trip,and all of the variables representing the second h2h trip.It can be observed from the correlation matrices that the h2h distance is negatively correlated with the departure time,which indicates that the later aperson leaves home,the shorter distance she/he travels.The relation of the h2h distance with the arrival time is opposite:the longer the distance thatis traveled the later she/he comes back home.The relations between departure and arrival times seem to be different fordifferent trip types.For the single h2h trip,the dependence coef ﬁcient is equal to 0.35,whereas in the ﬁrst trip in the double h2h trip is close to zero,indicating a lack of monotonic dependence between the variables.The rank correlation coef ﬁcient between departure and arrival times of the second h2h trip is 0.8,which is relatively high.The rank coef ﬁcient of arrival time of the ﬁrst trip and the departure time of the second trip also have a relatively high value.C.Modeling Using Copula FunctionsAs shown in Section III-B,dependencies exist between the variables representing both types of trips.Therefore,it is cru-cial to model the correlation structure between them in order to obtain simulations which re ﬂect the features of the input data.All of the individual variables of interest have different marginal distributions.In this case,a multivariate joint distribution func-tion can be created using a copula function [20],[21],which represents the dependence structure between the variables.A copula is a function that joins univariate distribution functions to form multivariate distribution functions.Speci ﬁcally,in accor-dance with Sklar’s theorem [22],the continuous random vari-ables with cumulative distribution functions (CDFs),respectively,are joined by copula if their jointdistribution function can be expressed asDenoting for,where are realizations of uniform variables ,respectively,theabove formula can be rewritten asFor modeling Dutch driving patterns,two joint multivariatedistribution functions are needed:the ﬁrst for a single tripand the second for a double trip.A normal/Gaussian copula is a common choice for modeling the dependence structure between variables.In order to ﬁnd out if the normal copula function is suitable for modeling de-pendency between the variables representing driving patterns,scatter diagrams are considered.In general,it is dif ﬁcult to ex-plore the character of the dependence structure using scatters between the variables in their natural scale due to the in ﬂuence of their marginal distributions [14],[20].Isolating the effects of the marginal distributions can be achieved by ranking the data or,in other words,by transforming the random variables to uni-form distributions.In this new uniform domain,the character of the relationship can be revealed.Fig.5(a)presents the scatter di-agram of the ranked variables and .It can be observed that the points are distributed quite uniformly with a slightly higher density on the diagonal intersecting zero point which conforms to a low respective correlation coef ﬁcient in the matrix.In this case,lack of any signi ﬁcant tail dependence makes the ap-plication of this copula function appropriate.The granular na-ture of the plotted data can be explained by a tendency of re-spondents to round the distance traveled as well as the arrival time.Fig.5.Scatter plots between variables in uniform domain.(a)versus.(b)versus .Fig.6.(a)Scatter plot between simulatedand simulated .(b)Q-Qplot of simulated and originalvariable .However,because the and variables satisfy the relation,their scatter diagram,shown in Fig.5(b),reveals anonstandard dependence structure.Such a nonstandard depen-dence occurs also in the case of other pairs of variables satis-fying an inequality relation:and ;and ;and .To the best of our knowledge,there is no parametric copula function available in the literature for the modeling of such a dependence structure.In order to overcome this problem,the normal copula function with the so-called conditional sam-pling is applied.This procedure is based on selecting samples during the simulation process that satisfy the required relations.For example,the simulation of a single trip would require the following simulation code:while doend whilewhere is a coef ﬁcient matrix of a normal copula function for single trips (more details are given in the rest of the paper).Fig.6(a)presents the scatter diagram of the ranked simulated variables and .Although the scatters of the ranked originaland ranked simulated data look alike,the latter might not sat-isfy the requirements of the copula function,i.e.,the uniformly distributed margins.As a consequence there is a risk that the distribution of the simulated and original variables might not fully comply.This side effect can be investigated by exploring the quantile–quantile (Q-Q)plots of the simulated and original data.Fig.6(b)presents the Q-Q plot for the original and simu-lated variable ,which is found to have the most severe devi-ations among all of the considered variables representing single as well as double trips.We can see that the difference between distributions is rather insigni ﬁcant.Denoting CDFs of ,respectively,the normal copula is de ﬁned aswhereare realizations of the uniform variableson the interval ,respec-tively,denotes standard normal cumulative distribution andthe coef ﬁcient matrix of the copula function,which is the product moment correlation matrix between standard normal variables .When modeling dependent random variables using a normal copula function,the following steps of the joint normal trans-form methodology [20],[21]should be followed.Step 1)Transform random variables ,to uniforms using their CDFs:Step 2)Transform uniform variables tonormal variables using inverse standard normal distribution :Step 3)Estimate product moment correlation be-tween normal variables .Simulatefrom multivariate standard normaldistribution with correlation .Step 4)Transform the simulated values back to the originaldomain by applying standard normal CDF and the inverse of the respective CDF as follows:.IV .EV S IMULATIONHere,the models representing driving patterns are combined with the battery characteristics to create a Monte Carlo simula-tion of the electricity demand due to electric cars.In order to perform the simulation of commuting behavior,the two joint multivariate distribution functions and are built up using normal copula functions and ,respec-tively,and the marginal empirical For this,parameters of the copula functions and are estimated by means of the maximum-likelihood method.These coef ﬁcient matrices have similar values to the rank correlations computed in the previous section.The copula coef ﬁcient is given as(3)Simulating a person’s commuting plans involves ﬁrst the deci-sion if he/she will travel during a day and,second,how manyh2h trips will he/she have.The probability that a person does not travel during one day can be computed from the transporta-tion dataset,as respondents were obliged to report lack of com-muting as well.According to the survey,17%of the drivers do not commute at all during a day.We assume that a person who commutes makes either a single or a double h2h trip.Based on the ﬁgures presented in Section II-A,there is 78%probability for a single h2h trip and 22%for a double h2h trip.Thus,the use of distribution functions and will be determined by these probabilities.The state of charge of a battery (SoC)is an available battery capacity at a given moment expressed as a percentage of the total capacity.A linear relationship between SoC at the begin-ning of charging and the distance traveled since the last charging time is applied,as in [6].Since,in this procedure,the idea of fully charged battery at the beginning of charging is not adopted,the formula for the initial SoC is(4)where ,denote the start time and end time of a trip,respec-tively,is the distance traveled in between,and denotes an EV’s driving range.The SoC of a battery at the beginning of a trip might be lower than 100%if a person starts a trip before the vehicle’s battery became fully charged after the last trip.In this way,the initial SoC depends not only on the dis-tance traveled since the last charging but also on the charging history.4The simulations are performed for two consecutive days.During the ﬁrst day,it is assumed that a car starts its h2h trip with a fully charged battery.Only the results from the second day are used to calculate the EV’s load.This removes the effect of the initial assumption about the initial state of charge,leaving only the desired random distributions.At the beginning of the simulation procedure for an EV ,the decision is made whether the driver will commute during the ﬁrst day and the second day.The driving patterns are simulated using distributions and based on the probability of oc-currence of their respective h2h trips,i.e.,78%and 22%.These driving patterns have to satisfy the logical requirements (see Section III-C)so the start-time of a trip cannot exceed end-time of a trip.The initial SoC is computed at every home-arrival-time using (4).Then,by applying the charging pro ﬁle given in Fig.1,the electricity demand is derived per each hour.In order to com-pute total power demand due to a number of EVs,the simulation procedure is run for each EV ,and the individual hourly power demands are added up.V .S IMULATION R ESULTSIn order to analyze the scenario of one million EVs,the Monte Carlo simulation algorithm is run for 1000EVs and the resulted load is scaled up accordingly.In Section V-A,we elaborate on the daily load power curve of the EVs for different levels of range anxiety.Sections V-B and V-C present one of the possible4Inreal life,the SoC would also depend on a vehicle speed and on a driving mode (urban/highway)but,due to the lack of data,they could not be taken into account.Fig.7.Power demand due to one million EVs under the scenario of uncon-trolled domestic charging.The solid line together with the uncertainty is for the case when commuters plug in the vehicles just after arrival at home.The line marked with stars is for the case assuming that commuters would charge the ve-hicle if the SoC is less than50%or if there is not sufﬁcient energy in the battery for the next trip.applications of the methodology:examining the effect of the EV load on the national electricity demand.A.Load Due to EVsA worst case scenario referred to in the literature[6]is that of uncontrolled domestic charging(no price incentives for off-peak charging).In this scenario,people would tend to start charging their vehicles as soon as they arrive home,i.e.,during peak hours for the system.As shown in Fig.3,drivers arrive at different times of the day,thereby their electricity demand is going to be present throughout the entire day instead of a few particular evening hours.The load due to one million EVs is presented in Fig.7for two cases:commuters plug in the vehicles just after arrival at home and the case assuming that commuters would charge the vehicle if the SoC is less than50% or if there is not enough energy in the battery for the next trip. The simulation procedure is run100times in order to show the uncertainty in the EVs’power demand.These100simulations can be considered as the simulation of the load for100different days.We can see that under the basic principle of Monte Carlo simulation,uncertainty propagation,the resulted shape of the power curve takes after the distribution of the arrival time of a h2h single trip(see Fig.3).Furthermore,the uncertainty range varies depending on the hour.The highest uncertainty occurs when the load has peaks:around7pm,1pm,as well as around midnight.The maximum average load of0.84GW is reached at hour19. It could seem that this is a quite low value taking into account the fact that a relatively high percentage of people arrive at home around6pm and they mostly charge at6.5kW power level. Nevertheless,the power demand at a given hour depends not only on the fraction of cars that arrived before that hour but also on their initial SoC,related to the distance traveled,which de-termines the length of time a car stays plugged in to the system. In Fig.4,we see that most people tend to have short trips,and, as a consequence,the charging takes relatively little time.The average distance traveled for single as well as double trips is ing(4)and the charging proﬁle of the Nissa Altra(see Fig.1),this leads to an average charging time of1.16h to the full capacity.5The so-called range anxiety is considered as one of the major factors that could limit the wide-scale adoption of electric vehi-cles.The solid line in Fig.7represents the load power curve for the highest level of the range anxiety,so electric drivers plug in their vehicles every time they arrive home no matter how high the initial SoC is.Assuming a maximum driving range of 130km,the average distance of26km traveled by EVs users is equivalent to only20%of a battery discharge.Therefore,a fully charged battery of the Nissan Altra EV can serve much more than the energy needed to satisfy the usual driving habits of the potential EVs’users.Because frequent charging impacts the battery lifespan,commuters might prefer to charge the bat-teries only if the SoC gets below a certain threshold.In Fig.7, we can see the load power curve for the case when the batteries get charged if the initial SoC is lower than the threshold of50% or if the battery SoC is too low for the next planned trip.Al-though,in this situation,about66%fewer cars will contribute to the electricity demand,the load due to EVs does not decrease signiﬁcantly.For example,the peak load attains0.71GW—only 15%less than the base case.Attention should be drawn to the increased frequency of the long charging times.The average charging is equal now to3.1h,which is almost three times more than previously.In consequence,this increases the electricity demand per vehicle and also explains the power curve shifting.B.Results for Different Market Penetration LevelsOne million EVs will correspond to approximately10% market penetration in2020in The Netherlands[15].For the purpose of examining the cumulative effect of the load due to EVs on the system,the computed EVs’load is added to a predicted national demand proﬁle for the year2020.Since the worst case study is usually of the highest interest[7],the load forecasts are computed by applying an annual average growth rate of2%[12]to the daily load curve[13]with the highest hourly peak recorded in the year2010in The Netherlands, which occurred on December1st.Fig.8presents the results for10%as well as for20%penetration levels corresponding to two million EVs.It appears from theﬁgure that the winter load peak increases by3.4%and6.8%in the case of10%and 20%penetration levels,respectively.Moreover,the highest percentage increase takes place at hour19and is equal to 3.9%and7.8%for the respective penetration levels.The90% conﬁdence interval for the peak load value in the case of10% is(23.09,23.28)GW and(23.76,24.14)GW for the20%. The results in the related literature vary depending on the data for driving patterns,battery characteristics,market penetration levels,total system load shape,and so on[7],[8][6].For ex-ample,a similar study for the U.K.system[7]reveals a rela-tively lower increase of the evening peak electricity demand be-cause of an almost two times lower charge rate and the driving patterns utilized[9].In studies considering distribution systems, a higher load increase may be reported due to the higher pene-tration levels for the speciﬁc part of the system but also by the 5To obtain these average values,it was assumed that EVs start a trip with a full battery.。

Part3-StochasticFrontierModel

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Stochastic FrontierModels Stochastic Frontier Model

AdventuresInStochasticProcessesSolutionManual

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Fluid-Structure Interaction and Dynamics

Fluid-Structure Interaction and Dynamics Fluid-structure interaction (FSI) is a complex and fascinating field that involves the interaction between fluid flow and solid structures. This interaction can have significant impacts on the behavior and performance of various engineering systems, ranging from aircraft wings to cardiovascular stents. Understanding and predicting the dynamics of FSI is crucial for designingefficient and reliable systems. One of the key challenges in FSI is accurately modeling the behavior of both the fluid and the structure. Fluid flow is typically described by the Navier-Stokes equations, which govern the motion of viscous fluids. On the other hand, the structure is often modeled using finite element analysis, which represents the solid deformations under external forces. Combining these two models to simulate the interaction between the fluid and the structure requires sophisticated numerical techniques and computational tools. The dynamics of FSI can be influenced by a variety of factors, such as the geometry of the structure, the properties of the fluid, and the boundary conditions. For example, the shape and flexibility of an aircraft wing can affect its aerodynamic performance, while the viscosity and density of the fluid can impact the flow patterns around the wing. Additionally, the boundary conditions at the interface between the fluid and the structure play a crucial role in determining the overall behavior of the system. In the context of biomedical engineering, FSI plays a critical role in understanding the behavior of blood flow in arteries and veins. Cardiovascular diseases, such as atherosclerosis, can alter the flow patterns in blood vessels, leading to potentially life-threatening conditions. By simulating the FSI dynamics in blood vessels, researchers can gain insights into the underlying mechanisms of these diseases and develop new treatment strategies. Despite the challenges and complexities associated with FSI, advancements in computational fluid dynamics (CFD) and structural analysis have enabled researchers to make significant progress in this field. High-performance computing resources and advanced simulation techniques have made it possible to simulate complex FSI problems with a high degree of accuracy and reliability. These simulations can provide valuable insights into the behavior of engineering systems and help optimize their design and performance. In conclusion, fluid-structureinteraction and dynamics are essential components of many engineering systems, with applications ranging from aerospace to biomedical engineering. By understanding and predicting the complex interactions between fluids and structures, researchers and engineers can design more efficient and reliable systems. Continued advancements in computational tools and simulation techniques will further enhance our ability to study and optimize FSI dynamics, leading to innovative solutions and breakthroughs in various industries.。

stochastic resist model evolution -回复

stochastic resist model evolution -回复[Stochastic Resist Model Evolution]Stochastic resist model, also known as random field resist model, is a mathematical model used to predict and analyze the behavior of materials with random resistivity distributions. This model has significantly evolved over the years to better understand and describe the complex behavior of various materials, such as semiconductors, thin films, and composite materials. In this article, we will explore the step-by-step evolution of the stochastic resist model and its applications in different fields.Before delving into the details of the model's evolution, let's first understand its basic principles. The stochastic resist model is based on the assumption that the resistivity of a material is not constant but instead a function of random variables. This randomness can be attributed to numerous factors, including impurities, defects, and variations introduced during the manufacturing process.In its early stages, the stochastic resist model was primarily employed to understand the electrical transport properties ofdisordered systems, such as amorphous semiconductors. These systems often exhibit significant spatial inhomogeneities in their resistivity distribution, making it challenging to accurately model their behavior using conventional approaches.The first step in the evolution of the stochastic resist model was the development of theoretical frameworks to describe the statistical properties of resistivity distributions. Researchers introduced concepts from statistical physics and applied them to study the behavior of disordered materials. These concepts include the correlation length, which represents the characteristic scale over which the resistivity fluctuates, and the spatial correlation function, which quantifies the correlation between resistivity values at different locations.The next significant advancement in the stochastic resist model came with the introduction of numerical simulation techniques. These techniques allowed researchers to generate random resistivity distributions with specific statistical properties and study their impact on the transport properties of materials. Monte Carlo simulations, in particular, became widely adopted in the field, as they provided a powerful tool to analyze the stochastic behaviorof resistive materials.As computational power increased over the years, more sophisticated numerical techniques were developed to simulate the stochastic resistivity distributions. One such technique is the random walk simulation approach, where electrons hop from one lattice point to another, taking into account the resistivity values at each location. This approach provides insight into the macroscopic transport properties of materials and has been applied to various systems, including composite materials and thin films.In recent years, advancements in experimental techniques have greatly contributed to the evolution of the stochastic resist model. Researchers now have access to high-resolution imaging techniques that allow them to directly observe and quantitatively analyze the resistivity distribution within materials. This experimental data, combined with computational simulations, have led to a better understanding of the relationship between the microstructure and the resistivity properties of materials.Moreover, the stochastic resist model has found applicationsbeyond the realm of electrical transport properties. It has been extended to analyze other physical phenomena, such as heat flow and diffusion in random media. By considering the random resistivity distribution, scientists can gain insight into how these different quantities propagate through materials with spatially varying properties.In conclusion, the stochastic resist model has undergone significant evolution over the years to better understand and describe the behavior of materials with random resistivity distributions. From the development of theoretical frameworks to the advancement of numerical simulation techniques and experimental observations, the model has become a powerful tool in various fields of research. By incorporating the stochastic nature of resistivity into material analyses, scientists can gain a deeper understanding of the underlying physics and improve the design and development of materials for various applications.。

一种旋转失重模拟试验装置设计

第 39 卷第 2 期航天器环境工程Vol. 39, No. 2 2022 年 4 月SPACECRAFT ENVIRONMENT ENGINEERING187 E-mail: ***************Tel: (010)68116407, 68116408, 68116544一种旋转失重模拟试验装置设计黄强1，荆江2，王佳南1，胡鑫1，高越1（1. 北京强度环境研究所，北京 100076； 2. 北京航天长征飞行器研究所，北京 100076）摘要：为了验证某些航天产品在同时满足失重和自旋条件下的工作性能，设计了一种旋转失重模拟试验装置，由上下独立的单端承载旋转机构和转动平稳释放机构装配而成。

文章给出装置的设计原理、转动应力参数仿真和平稳释放实现方法等，以及采用的关键技术。

国内首例高速自旋失重试验的验证结果表明，旋转失重试验装置的使用克服了传统试验方法无法同时提供旋转环境和失重环境的局限性，在有效避免共振的前提下，实现产品在3.5 r/s转速下平稳下落，倾斜角度未超过1°，试验结果满足设计要求。

关键词：旋转失重；机构设计；参数仿真；试验验证中图分类号：V416.8文献标志码：A文章编号：1673-1379(2022)02-0187-06 DOI: 10.12126/see.2022.02.011A design of rotating and weightlessness simulation test deviceHUANG Qiang1, JING Jiang2, WANG Jia’nan1, HU Xin1, GAO Yue1(1. Beijing Institute of Structure and Environment Engineering, Beijing 100076, China;2. Beijing Institute of Space Long March Vehicle, Beijing 100076, China)Abstract: In order to verify the working performance of some aerospace products under simultaneous weightlessness and self-rotation conditions, a simulation mechanism is designed by assembling two independent mechanisms, namely, the upper single-end bearing rotary mechanism and the lower stable rotation release mechanism. This paper presents the design principle, the simulation of the rotary stress parameters, and the method and the steps for implementing the stable rotating release. The results of the high-speed spin and weightlessness test, as first carried out in China, show that with the designed device, better than the traditional test method, both the rotating environment and the weightlessness environment can be provided, and, under the premise of effectively avoiding the resonance, the test piece may fall smoothly at a speed of 3.5 r/s with a tilt angle not more than 1°, that well meets the design requirements.Keywords: rotation and weightlessness; mechanical design; simulation of parameters; test validation收稿日期：2021-10-15；修回日期：2022-02-08引用格式：黄强, 荆江, 王佳南, 等. 一种旋转失重模拟试验装置设计[J]. 航天器环境工程, 2022, 39(2): 187-192HUANG Q, JING J, WANG J N, et al. A design of rotating and weightlessness simulation test device[J]. Spacecraft Environment Engineering, 2022, 39(2): 187-1920 引言在部分航天产品的地面性能参数检测中，需要检验其在自旋失重状态下的性能参数是否满足设计需求。

船舶SCR脱硝尿素喷射分解及氨气分布均匀性的优化

CHEMICAL INDUSTRY AND ENGINEERING PROGRESS 2017年第36卷第2期·742·化工进展船舶SCR脱硝尿素喷射分解及氨气分布均匀性的优化王铮1，刘道银1，刘猛1，吴勤瑞2，檀净2，陈晓平1（1东南大学能源热转换及其过程测控教育部重点实验室，江苏南京 210096；2南京中船绿洲环保有限公司，江苏南京 210039）摘要：以某船型柴油机尾气选择性催化还原法（SCR）脱硝工艺为对象，采用数值模拟研究喷射管内氨气（NH3）生成过程及分布均匀性。

模型包括尿素液滴的雾化、蒸发、分解以及与流场的相间作用等。

结果表明，模型可详细预测烟道内气体速度场、温度场、NH3等气体组分场、液滴轨迹等，NH3转化率及分布均匀性受到烟气温度、入口湍流强度、扰流器的影响。

提高烟气温度、提高烟气入口湍流强度、增加扰流器，均促进NH3生成并提高分布均匀性，但是，扰流器的旋片角度过大，会导致部分液滴冲击到管道壁面，不利于NH3生成。

对比扰流器旋片角度为15°、30°和45°的情况，发现旋片角度30°时对提高NH3生成及分布最为有利，其出口NH3分布不均匀性从无绕流器条件下的39.57% 降低到9.36%。

关键词：选择性催化还原；脱硝；氨分布均匀性；数值模拟中图分类号：TK09 文献标志码：A 文章编号：1000–6613（2017）02–0742–08DOI：10.16085/j.issn.1000-6613.2017.02.047Optimization of uniformity of NH3 distribution and thermolysis of urea inthe SCR marine processWANG Zheng 1，LIU Daoyin1，LIU Meng1，WU Qinrui2，TAN Jing2，CHEN Xiaoping1（1Key Laboratory of Energy Thermal Conversion and Control of Ministry of Education，Southeast University，Nanjing 210096，Jiangsu，China；2CSSC Nanjing Environment Protection Co.，Ltd.，Nanjing 210039，Jiangsu，China）Abstract：The distribution process of NH3 in the SCR marine mixing chamber was investigated by numerical simulation.The simulation model considered atomization，evaporation，and thermolysis of urea，and its interaction with gas turbulence. This model provides the detailed information of velocity field，temperature field，component field，and trajectories of the droplets. Production and distribution of NH3 was affected by flue gas temperature，inlet turbulent intensity，and swirl mixing. It is found that，by increasing the flue gas temperature，inlet turbulent intensity or adding a swirl mixer，the evaporation and thermolysis of urea solution accelerated，and the distribution uniformity increased.However，if the swirl angle was too large，some drops would impact the wall of the pipe，which would decrease the NH3 production. Comparing the cases of swirl angle with 15°，30°and 45°，it is found that the 30°swirl mixer yielded the best performance，which decreased NH3 distribution non-uniformity from 39.57% to 9.36%.Key words：SCR；de-NO x；NH3 distribution uniformity；numerical simulation近年，国际海事组织（IMO）修订了船舶排放尾气氮氧化物（NO x）排放控制标准，加强对船舶尾气排放的控制。

永川地区深层页岩气储层不同尺度裂缝精细建模

大庆石油地质与开发Petroleum Geology ＆ Oilfield Development in Daqing2023 年 12 月第 42 卷第 6 期Dec. ，2023Vol. 42 No. 6DOI ：10.19597/J.ISSN.1000-3754.202301027永川地区深层页岩气储层不同尺度裂缝精细建模葛勋1，2，3 郭彤楼4 黎茂稳1，2，3赵培荣5 范宏娟6王鹏6 李王鹏1，2，3钟城7（1.页岩油气富集机理与高效开发国家重点实验室，北京102206；2.中国石化页岩油气勘探开发重点实验室，北京102206；3.中国石化石油勘探开发研究院，北京102206；4.中国石化西南油气分公司，四川成都610041；5.中国石油化工集团有限公司，北京100728；6.中国石化西南油气分公司勘探开发研究院，四川成都610041；7.南京大学地球科学与工程学院，江苏南京210033）摘要：目前已有关于裂缝的研究主要是地震单属性预测，裂缝预测的精度较低。

为了进一步提高压裂改造水平与页岩气井产能，需对页岩储层不同尺度的裂缝进行预测，尤其是对小尺度裂缝进行精细预测。

利用相干、曲率、蚂蚁体3种地震属性对永川地区龙马溪组一段的天然裂缝进行定量预测，并针对不同尺度裂缝进行多属性裂缝综合建模。

结果表明：永川地区裂缝分为3个等级，断距大于100 m 的大尺度裂缝、断距50~100 m 的中尺度裂缝和断距小于50 m 的小尺度裂缝，相干、曲率、蚂蚁体属性预测结果大体一致，整体上断层走向以NE —SW 为主，大尺度裂缝主要发育在新店子背斜附近，蚂蚁体可以更清晰地刻画出小尺度裂缝。

裂缝综合建模显示，永川地区深层页岩气储层总体裂缝发育程度由好到差依次为新店子背斜主体、南部向斜、北区向斜，南部向斜微裂缝最发育，有利于压裂改造，因此将永川南部作为首要滚动产建目标区，气井试采效果好，测试产量整体高于永川中、北区。

StochasticModelling

Stochastic ModellingUnit1:Markov chain modelsRussell Gerrard and Douglas WrightCass Business School,City University,LondonJune2004Contents of Unit11Stochastic Processes2Markov Chains3Poisson Processes4Markov Jump Processes5Martingales1Stochastic processes1.1The Stochastic ProcessA stochastic process is just a sequence of random variables.It involves a random element and a time element.Any collection{Xα:α∈A}of random variables may be considered as a stochastic process. Examples:∗rainfall since midnight∗the number of cars passing a traﬃc census∗stock market index,observed day by day(or minute by minute)∗a football team’s points score,match by match∗the size of the hole in the ozone layerAn unpredictable(random)observable process may be modelled to predict its future be-haviour:∗short-term movements(exchange rates)∗medium-term extrema(storm damage claims,derivatives)∗long-term asymptotics(steady-state costs,eg health insurance)Usefulness for modelling is determined by the nature of the dependence of X n(or X(t))on preceding X values.The set of all values X can ever take is the state space.1.2Classiﬁcation of stochastic processes•Based on time parameter:discrete-time or continuous-time.We write{X1,X2,...}in discrete time,{X(t):t≥0}in continuous•Based on state space:discrete or continuous.A counting process is discrete,taking only integer values;a no-claims bonus scheme is another example of a discrete state space;the size of an insurance claim is treated as continuous.1.3The HistoryH t—the history of X up until time t—is the collection of answers to all questions about the behaviour of X in[0,t].We can write H Xt to avoid confusion.In formal notation,H Xt=σ({X s:0≤s≤t}).I E(X t|H t)=X t,but I E(X t+s|H t)=X t+s.{H Xt:t≥0}is theﬁltration generated by X.Aﬁltration may contain the histories of many processes at once.(The same things are true in discrete time.)1.4StationarityA stochastic process X is stationary if•the distribution of X t is the same for all t•for any k the distribution of the vector(X t,X t+1,...,X t+k)is the same for all t. Example:we might assume that the process of interest rates is stationary.1.5Stationary,independent incrementsAn increment of a process X is a quantity of the form X t+s−X t.X has stationary increments if the distribution of X t+s−X t is the same as that of X s−X0 for all s,t.X has independent increments if X t+s−X t and X u+v−X u are independent r.v.s whenever the intervals(t,t+s)and(u,u+v)do not overlap.1.6The Markov propertyThis is about memorylessness.‘The future is independent of the past,given the present’Formally,X has the Markov property if the distribution of X t+s given H t is the same as the distribution of X t+s given X t.1.7The Martingale property(Named after a gambling system,in turn named after a horse harness.)On average,the process stays where it is.Like the no-arbitrage principle.Formally,I E(X t+s|H t)=X t.(H is usually,but does not have to be,H X.)2Markov Chains2.1IntroductionA discrete-time process{X n:n≥0}with a discrete state space is a Markov Chain if it possesses the Markov property:‘The future is independent of the past,given the present’Example:Weather.State space{rainy,sunny}.Example:Random walk,binomial lattice model.Example:A no-claims discount scheme with four levels:Level1no discountLevel210%discountLevel325%discountLevel440%discountAfter a year without making a claim,move up to the next discount level(unless already in 4).After a year with one claim,move down to the next discount level(unless already in1).Example:An actor’s career.The actor is employed,unemployed or doing temporary work.A month of employment is followed by another such month,with prob.0.8,or by unemploy-ment otherwise.After a month of unemployment,the actorﬁnds temporary work with probability0.4,em-ployment as an actor with probability0.2.After a month of temporary work the probability ofﬁnding employment is0.2,or the tem-porary work continues with probability0.7;otherwise unemployment looms.2.2Transition probabilitiesDeﬁne the transition probability from i to j asp ij=P(X n+1=j|X n=i),independent of events before time n by the Markov property.We assemble the p ij into a matrix P,the transition matrix.Note:entries≥0,row sums=1.Example(actor)P=0.80.200.20.40.40.20.10.7.2.3Chapman-Kolmogorov EquationsThe k-step transition probability p(k)ijis deﬁned as P(X n+k=j|X n=i).Chapman-Kolmogorov:P(k)=P k,the k th power of matrix p.2.4Long-term behaviourIn many cases we observe that for large n the distribution of X n converges to a limitπ,ie.P(X n=j|X n=i)→πj,the limit being the same regardless of the starting point.We canﬁndπby solvingπP=πwith the constraint thatπj=1.2.5Fitting a Markov Chain modelObserve the process for an extended period,or several copies of the process(one actor for several years,or several actors for one year).n i=number of times the process is in state i,n ij the number of transitions from i to j.Then estimate p ij by p ij=n ij/n i.2.6Time-inhomogeneous Markov ChainsTransition probabilities may change with time.Young drivers and very old drivers may have more accidents than middle-aged drivers,for example.P would therefore depend on n,giving P n.The analogous form of the Chapman-Kolmogorov Equations continues to hold,but there is no long-term limit.Even if individual behaviour is not time-homogeneous,the behaviour of the population as a whole could be treated as time-homogeneous because of statistical equilibrium.3Poisson Processes3.1The standard Poisson ProcessEvents can happen at any time.The probability of an event in any time interval(t,t+dt) isλdt+o(dt),independently of other intervals.Suppose T1,T1+T2,T1+T2+T3,...are the times of theﬁrst,second,third,...events.The T i are independent of one another,and all have exponential distribution with rateλ. Therefore X(t),the number of events in(0,t),is a Poisson random variable with meanλt. X is a stochastic process in continuous time(t>0).3.2Time-dependent Poisson ProcessThe underlying rate at which events occur changes in some deterministic way.For example, more domestic storm damage claims occur in autumn than in summer.X(t)is still a Poisson r.v.,but the mean is now more complicated.3.3Age-dependent Poisson ProcessesThe transition rate depends on the time since the last transition(‘age’).If h(t)is the transition rate at age t,we haveS(t+dt)=S(t){1−h(t)dt}, where S is the survivor function,S(t)=P(time to next transition>t).The solution isS(t)=exp−th(u)du.X(t)is no longer a Poisson r.v.:instead,X is a simple renewal process.4Markov Jump Processes4.1The Markov propertySame as in discrete time:P(X(t+s)=j|H t)depends only on X(t),j,s and possibly t. Initially we only consider cases where this probability is independent of t(time-homogeneous cases);use the notationp ij(s)=P(X(t+s)=j|X(t)=i).Later there will be variations involving time-inhomogeneity or age-dependence. Example:reversionary annuity:states are{both partners alive,only husband alive,only wife alive,neither alive}Example:marriage model:states{never married,married,divorced,remarried,widowed} Example:long-term care model:states{healthy,short-term sick,long-term sick}4.2The Behaviour of Markov ProcessesThe length of time spent in the current state x must have a memoryless(ie,exponential) distribution,with rate parameter(λx)which depends only on the state itself.This implies that the mean time spent in state x on any one visit is1/λx.Once the jump occurs,the probability that it takes the chain to state y is r xy,regardless of duration in x.The matrix R is a discrete-time transition matrix.The associated chain is the jump chain of X.A state x for whichλx=0is absorbing:if the process hits the state,it will never leave.4.3The Chapman-Kolmogorov EquationsAs in discrete time the Chapman-Kolmogorov Equations areP(s+t)=P(s)P(t)Notice that P(0)=I,the identity matrix.4.4The Kolmogorov Diﬀerential EquationsSetting t=ds in the Chapman-Kolmogorov equations givesP(s+ds)=P(s)P(ds)orP(s+ds)−P(s)=P(s){P(ds)−I},which givesP (s)=P(s)Q,where Q=P (0),the generator matrix of the chain.This is called the Kolmogorov Forward Equation.By putting s=dt in the Chapman-Kolmogorov equations,we obtain the Kolmogorov Back-ward Equation,P (t)=Q P(t).The formal solution of the Kolmogorov DEs is P(t)=exp(tQ),but this exponential may be hard to evaluate.Example:Linear birth-and-death process.The rate of transitions from x to x+1is the birth rate xβ;from x to x−1the death rate xδ.Thereforeq x,x−1=xδ,q x,x+1=xβ,q x,x=−x(β+δ)with all other entries in the Q-matrix being zero.Thus the Q-matrix isQ=0000...δ−(β+δ)β0... 02δ−2(β+δ)2β... 003δ−3(β+δ)... ...............4.5Long-term behaviourAs for the discrete-time version,a time-homogeneous Markov jump process converges(under certain conditions),in the sense thatP(X(t)=j)→πj as t→∞regardless of the starting position,whereπis the solution toπQ=0.We can therefore calculate the long-run proportion of time spent in the states.4.6Fitting a Markov jump modelThe elements of R,the transition matrix of the jump chain,are estimated as in discrete time. Estimates for the parametersλx are based on average durations in state x.Testing goodness ofﬁt can take many forms,such as∗is exponential a good distribution toﬁt?∗are destinations independent of durations?∗do2nd and subsequent visits have diﬀerent durations?4.7Time-inhomogeneous Markov modelsUsed when there is an underlying reason for transition rates to change with time. Example:care model:rates of falling sick are higher in winter,recovery rates lower.Also used for a single individual with age-varying transition rates.The transition rates are:P(X(t+dt)=j|X(t)=i)=q ij(t)dt+o(dt).The formal solution P(t)=exp(tQ)no longer holds.The only possibility is to attempt to solve the DEs by hand.Example:Poisson process with rateλ(t)=2λt/(1+t2):d dt p0,j(t)=−2λt1+t2p0,j(t)+2λt1+t2p0,j−1(t).Even in this case it is not obvious that there is an explicit solution.In more complicated cases numerical approximation is required.It is unlikely that X converges to a limiting distribution4.8Semi-Markov modelsTwo restrictions imposed by the Markov format are relaxed:∗the durations need not be exponential∗the destinations may depend on the durationsMuch moreﬂexible.But diﬃcult toﬁt because there are so many parameters.Only practical when there is a prior idea of the distribution of the durations.Long-run proportion of time spent in the various states may be found as above.5Martingales5.1Basic propertiesFormally,I E(X t+s|H t)=X t.(H is usually,but does not have to be,H X.)Example:random walk with zero-mean increments,binomial lattice,even a(non-Markov) process with increased volatility after a jump.It follows from the deﬁnition that I E X t=I E X0.5.2Martingale convergenceThe usefulness of martingales arises from convergence theorems:A martingale bounded above or below converges,X t→X∞.If X is bounded above and below,or has bounded variance,then I E X∞=I E X0.5.3Stopping timesA random time T is a stopping time if you can always tell whether or not it has happened yet.Example:theﬁrst time something occurs,but not the last time something occurs. Formally:the answer to the question“Is T≤t?”must lie in H t.5.4Stopped MartingalesIf X is a martingale and T a stopping time,then Z,deﬁned byZ t=X t if T>tX T if T≤tis a stopped martingale.A stopped martingale is also a martingale.5.5Optional StoppingSuppose X is a martingale,certain to hit0or a eventually,and T is theﬁrst time of hitting one of them.Let q be the probability that X hits0ﬁrst.Then Z is a bounded martingale,so converges,and Z∞=0(with prob.q)or a(with prob. 1−q),so I E Z∞=a(1−q).But I E Z∞=I E Z0=X0.Hence q=1−X0/a.5.6Practical issues“Firstﬁnd your martingale”.Commonly the process being studied is not a martingale.But transformations can help(such as Y t=h X t).。

土壤压力计型号TRI-MOD-S 说明书

PRESSUREMETERModel TRI-MOD-SAPPLICATIONSThe TRI-MOD-S pressuremeter is a reliable and effective tool used to measure in-situ the strength and stress-strain properties of all types of soils and soft rocks. It quickly and economically provides a large volume of data encompassing the variability of the geotechnical conditions on a site.A well-proven method used to analyze the pressure-meter test data gives realistic values of:• Bearing capacity of shallow or deep foundations • Settlement of all types of foundations• Deformation of laterally loaded piles or sheet piles • At-rest pressure coefﬁ cient (Ko)DESCRIPTIONThe TRI-MOD-S pressuremeter is a device used to run in situ loading tests in boreholes at various depths. It is comprised of the following components:• The probe: a single cylindrical cell hydraulically inﬂ ated with 6 strain gauges and cantilevered arms, and ﬁ tted with an inﬂ atable metallic sheath • Hydraulic manual pumps• Pneumatical pump or cylinder for deﬂ ating the probe • The datalogger• The tubing and electrical cable• Two calibration tubesFEATURES• Inﬂ ated hydraulically up to 20 000 kPa • Direct reading of diametric changes with six electrical strain followers • Test in “N” size borehole • Very simple to operate• Readout displays diametrical expansion in hundredth of mmSPECIFICATIONSPROBEDiameter (min.) 73 mmDiameter (max.) 76.2 mm at 20 000 kPa82.0 mm at 10 000 kPa Working pressure (max.) 20 000 kPa (3000 psi)Length of in atable sheath 460 mm Typical sheath inertia 75 kPa PRESSURE GAUGE Range 20 000 kPa Accuracy1% F .S.DIGITAL READOUTP-3 Vishay strain indicator with selector switch Resolution of diametrical change 0.01 mmProducts and speci cations are subject to change without notice. E50059-060510© Roctest Limited, 2005.TEST PROCEDURE AND RESULTSThe probe is placed at the test depth in a pre-drilled borehole obtained by a method adapted to the soil conditions: augering, rotation with drag bit and bentonite slurry, shelby tube driving, etc.Stress-control i s u sed t o r un t he t est. E qual i ncrements of pressure are applied to the probe and held cons-tant for one minute. The diametric changes are log-ged 30 and 60 seconds after each pressure step is rea ched.In situ stress-strain curves are obtained by plotting the changes in each of the 3 instrumented diameters or their average against pressure.The limit pressure P L is the pressure corresponding to the doubling of the volume of the initial cavity and is a direct re ection of the soil's bearing capacity:Q a = (C/F) P Lwhere: Q a = Allowable bearing capacityC = Shape factorF = Safety factorThe modulus of deformation E used to calculate set-tlement is given by:E = (1 + ) (∆P/∆R) R where: = Poisson’s ratio ∆P= Increase in pressure∆R / R = Relative change of radiusWith special limitations, the “at rest” pressure of thesoil can be estimated along with the reaction factor used in the design of laterally loaded foundations.Example of pressuremeter test results。

Vortex tube models for turbulent dynamo action

(September 14, 1998)
Abstract
The possibility of dynamo action resulting from a pair of elongated vortex structures immersed in an electrically conducting uid is investigated. For elongated vortex structures the critical magnetic Reynolds number for dynamo action is about half that for the spherical rotors which have been studied previously. When applied to Kolmogorov turbulence with vortex structures of scale comparable to the dissipation length, this model can explain dynamo action only when the magnetic Prandtl number ( = kinematic viscosity / magnetic di usivity) exceeds a critical value that is larger than unity. It is argued that in astrophysical bodies where this condition is not satis ed (in stellar convection zones, for example), dynamo action must instead result from motions on all scales up to the size of the region.

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the velocity is zero for F < Fc, and increases for F > Fc as
v ∼ (F − Fc)θ,
(2)
where the exponent θ is called the velocity exponent.
The continuum equation for the dynamics of interfaces in random media may be written simply as [9]
PACS numbers: 68.35.Fx, 05.40.+j, 64.60.Ht
In the past few years, there has been an explosion of studies in the ﬁeld of dynamics of ﬂuctuating interfaces due to theoretical interests in the classiﬁcation of universality for stochastic models and also due to applications to various physical phenomena such as crystal growth, vapor deposition, electroplating, biological growth, etc. A number of discrete models and continuum equations for interface dynamics have been introduced and studied [1-3]. An interesting feature of nonequilibrium interface dynamics is the nontrivial dynamic scaling behavior [4] of the interface ﬂuctuation width, i.e.,
of
the
pin-
Hale Waihona Puke ning strength. Thus the equation is replaced by
∂h(x, t) = ν∇2h + λ (∇h)2 + F + η(x, h),
(4)
∂t
2
which is called the quenched Kardar Parisi Zhang (QKPZ) equation. The QKPZ equation leads to a diﬀerent universality class from the QEW equation. Recently several stochastic models in the QKPZ universality class have been introduced [9,11]. From the models, it has been naturally concluded that the surface at the threshold of the PD transition Fc can be described by the directed percolation (DP) cluster spanned perpendicularly to the surface growth direction in 1+1 dimensions. The roughness exponent α of the interface is given as the ratio of the correlation length exponents ν⊥ and ν of the DP cluster in the transversal and the longitudinal direction that is α = ν⊥/ν ≈ 0.63.
Kon-Kuk University, Seoul 143-701, Korea
We present a self-organized stochastic model for the dynamics of a single ﬂux line in random media. The dynamics for the ﬂux line in the longitudinal and the transversal direction to an averaged moving direction are coupled to each other. The roughness exponents of the ﬂux line are measured for each direction, which are close to α ≈ 0.63 for the longitudinal and α⊥ ≈ 0.5 for the transversal direction, respectively. The dynamic exponents are obtained as z ≈ 1 for both directions. We discuss the classiﬁcation of universality for the stochastic model.
systems. More generally, recently a new continuum equa-
tion was introduced [10], which includes a nonlinear term
λ 2
(∇h)2
induced
from
the
anisotropic
property
W (L, t) =
1 Ld′
(h(x, t) − h¯(t))2 1/2 ∼ Lαf (t/Lz),
x
(1)
where h(x, t) is the height of site x on substrate at time t. h¯, L, and d′ denote the mean height, system size, and substrate dimension, respectively. The angular brackets stand for statistical average. The scaling function behaves as f (x) → const for x ≫ 1, and f (x) ∼ xβ for x ≪ 1 with z = α/β. The exponents α, β and z
arXiv:cond-mat/9707084v1 [cond-mat.stat-mech] 9 Jul 1997
Stochastic Model for a Vortex Depinning in Random Media
Byungnam Kahng, Kwangho Park and Jinhee Park Department of Physics and Center for Advanced Materials and Devices,
are called the roughness, the growth, and the dynamic exponents, respectively.
Recently the problem of the pinning-depinning (PD) transition of interfaces in random media has also attracted interests in association with the problem of dynamics of ﬂuctuating interfaces. Examples include the dynamics of domain boundaries of random Ising spin systems after being quenched below the critical temperature [6], wetting immiscible displacement of one ﬂuid by another in a porous medium [5], pinning ﬂux lines in type-II superconductors [7], ﬂuid imbibition in paper [8], etc. In the problem of the PD transition, interface is pinned when external driving force F is weaker than pinning strength induced by random media, while it moves with a certain velocity v when the force F is greater than the pinning strength. Thus there exists a threshold of external applying force Fc across which the PD transition occurs. The role of the order parameter is played by the mean velocity, v = x ∂h(x, t)/∂t /Ld′. Accordingly,
smoothening eﬀect of surface tension, the second term
the uniform driving force, and the third a random force
with short range correlations, satisfying η(x, h) = 0 and η(x, h)η(x′, h′) = 2Dδ(x − x′)δ(h − h′) with noise