Deposits in Thermal Power Plant Condensers

Deposits in Thermal Power Plant electrical condensersAbstractUnexpected fouling in condensers has ceaselessly been one and only(a) of the main operational concerns in caloric might upstanding kit and caboodles. This report card describes an approach to predict fouling deposits in thermal antecedent plantcondensers by means of support sender machines (SVMs). The hebdomadary fouling formation carry out and relief fouling phenomenon be analyzed. To improve the inductive lawsuiting transaction of SVMs, animproved derivative ontogenesis algorithm is introduced to hone the SVMs parameters. The presage theoretical account establish on optimized SVMs is utilise in a incident study of 300MW thermal agency station. The experimentresult says that the proposed approach has more accurate forecasting results and better changingself-adaptive ability to the condenser operating(a) conditions change than asymptotic pretending and T-S blear-eyed amaze.Keywords Fouling pro phecy Condensers Support transmitter machines Differential ontogeny1. IntroductionCondenser is one of key equipments in thermal power plant thermodynamic cycle, and its thermalperformance directly impacts the economic and safe operation of the general plant 1. Fouling of steamcondenser tubes is one of the most important factors touch their thermal performance, which reduces rough-and-readyness and come alive conveyancing capability with condemnation 2, 3. It is piece that the utmost decrease in long suit due to fouling is about 55 and 78% for the evaporative coolers and condensers, respectively2. As a consequence, the formation of fouling in condenser of thermal power plants has special economicsignifi bedce 4-6. Furthermore, it re awards the concerns of modem federation in respect of conservation oflimited resources, for the environment and the natural world, and for the progress of indus rill workingconditions 6, 7.The fouling of catch fireing system exchangers is a e ncompassing ranging topic coveting galore(postnominal) aspects of technology, thedesigning and operating of condenser mustiness contemplate and try the fouling ohmic subway system to the heattransfer. The knowledge of the progression and mechanisms of formation of fouling will allow a design of* Manuscriptan appropriate fouling mitigation strategy such as optimal cleanup spot schedule to be made. The most common apply sit arounds for fouling adhesion ar the thermal resistance regularity and heat transfer coefficient method6-10. However, the residual fouling of periodic fouling alluviation process and the dynamic changes ofheat exchanger operating condition are not considered in these imitates. Consequently, the estimation errorof those methods is very large.Artificial Neural Ne bothrks (ANNs) are capable of efficiently dealing with many indus mental test troublesthat corporationnot be handled with the same accuracy by other techniques. To eliminate most of the diffi culties oftraditional methods, ANNs are utilise to estimate and catch the fouling of heat exchanger in recent years.Prieto et al 11 presented a archetype that uses non-fully connected feedforward artificial neural vanes forthe gaugeing of a sea weewee-refrigerated power plant condenser performance. Radhakrishnan et al 12developed a neural network establish fouling puzzle employ historical plant operating selective information. Teruela et al 13described a systematic approach to predict ash deposits in coal-fired boilers by means of artificial neuralnetworks. To minimize the boiler cipher and efficiency damagees, Romeo and Gareta illustrated a hybridsystem that combines neural networks and fuzzy logic expert systems to control boiler fouling and optimizeboiler performance in 14. Fan and Wang proposed diagonal re electric current neural network 15 and multipleRBF neural network 16 base models for measuring fouling in thermal power plant condenser.Although the technique of AN Ns is able to estimate the fouling ontogenesis of heat exchanger withsatisfaction, on that point are whatever businesss. The option of structures and types of ANNs dependents onexperience greatly, and the dressing of ANNs are based on empirical risk minimization (ERM) principle18, which aims at minimizing the train errors. ANNs therefore face just about disadvantages such asover-fitting, topical anesthetic optimal and bad generalization ability. Support sender machines (SVMs) are a newmachine t all(prenominal)ing method deriving from statistical learning theory 18, 19. Since later 1990s, SVMs arebecoming more and more habitual and have been successfully employ to many areas such as writtendigit recognition, speaker identification, puzzle out approximation, chaotic time series forecasting, nonlinearcontrol and so on 20-24. Established on the theory of geomorphologic risk minimization (SRM) 19 principle,compared with ANNs, SVMs have some distinct advantages such as glo bally optimal, small sample-size,good generalization ability and resistant to the over-fitting occupation 18-20. In this paper, the use of SVMsmodel is developed for the predicting of a thermal power plant condenser. The prediction model was utilisein a case study of 300MW thermal power station. The experiment result shows that the prediction modelbased on SVMs is more precise than thermal resistance model and other methods, such as T-S fuzzy model17. Moreover, to improve the generalization performance of SVMs, an improved contraryial evolutionalgorithm is introduced to optimize the parameters of SVMs.2. fortnightly fouling process in condenserThe accumulation of unwanted deposits on the proves of heat exchangers is usually referred to asfouling. In thermal power station condensers, fouling is chiefly formed inside the condenser tubes, reducingheat transfer amongst the hot fluid (steam that condenses in the external surface of the tubes) and the cold wet flowing through the tu bes. The presence of the fouling represents a resistance to the transfer of heatand therefore reduces the efficiency of the condenser. In order to maintain or affect efficiency it is oftennecessary to clean condensers. The Taprogge system has found wide application in the power industry forthe maintenance of condenser efficiency, which is one of on-line cleaning systems 6. When the foulingaccumulation in condensers r individuallyed a threshold, the laver guard balls cleaning system is activated,slightly oversized loaf rubberise balls constantly passed through the tubes of the condenser by the waterflow, and the fouling in the condenser is cut back or eliminated. The progresses of fouling accumulating andcleaning continue alternatively with time. Therefore, the fouling evolution in power plant condensers isperiodic.However, the sponge rubber ball system is only effective of preventing the accumulation ofwaterborne mud, biofilm formation, scale and corrosion product deposition 6. As for some of inorganicmaterials strongly attached on the inside surface of tubes, e.g. calcium and atomic number 12 salts, back not beeffectively decreased by this technique. As a result, at the end of every sponge rubber ball cleaning period,there still exist a lot of residual fouling in the condensers, and the residual fouling will be stash awaycontinuously with the time. Where, the fouling bed be cleaned by the Taprogge system is called soft fouling,and those passel not be cleaned residual fouling is called embarrassing fouling. When the residual fouling accumulatedto some degree, the cleaning techniques that can eliminate them, such as chemistry cleaning method,should be used.Generally, the foul degree of heat exchanger is expressed as fouling thermal resistance, delimitate as the rest amidst rates of deposition and removal 6. In this paper, the equal fouling thermalresistance of soft fouling and disfranchised fouling expressed as Rfs and Rfh, respectively. wher efore, the condenserfouling thermal resistance Rf in any time is the sum of soft fouling thermal resistance and hard foulingthermal resistance, expressed as Eq. (1).( ) ( ) ( ) ( ) ( ) ( ) 0 0 0 R t R t R t R t R t t R t t f fs fh f fs fh ? ? ? ? ? ? ? (1)where ( ) 0 R t f is the initial fouling.Fig. 1 periodic fouling evolution in power plant condensersFig. 1 demonstrates the periodic evolution process of fouling in power plant condensers. In fact, theevolution process of fouling in a condenser is very complex, which is related to a great number of variables,such as condenser ram, cooling water callosity, the pep pill of the circulating water and thecorresponding respite and outlet temperatures, the non-condensing gases present in the condenser, and so on.The Rfs(t) and Rfh(t) expressed a very complex material and chemical process, their accurate mathematicmodels are very hard to be obtained. Hence, measure and prediction of fouling development is a verydifficult task. Since t he fouling evolution process is a very complex nonlinear dynamic system, thetraditional techniques based on mathematic analysis, i.e. asymptotic fouling model, are not efficient todescribe it 11. SVMs, as a small sample method to deal with the highly nonlinear classification andregression problems based on statistic learning theory, is expected to be able to reproduce the nonlinearbehavior of the system.3. SVMs regression and parameters3.1 SVMs regressionSVMs are a group of supervised learning methods that can be applied to classification or regression.SVMs represent an extension to nonlinear models of the generalized portrayal algorithm developed byVladimir Vapnik 18. The SVMs algorithm is based on the statistical learning theory and theVapnik-Chervonenkis (VC) dimension introduced by Vladimir Vapnik and Alexey Chervonenkis 19. Here,the SVMs regression is applied to forecast the fouling in power plant condensers. allow the habituated training information sets represented as ?( , ), ( , ), , ( , )? 1 1 2 2 n n D ? x y x y ? ? ? x y , where di x ? R isan input vector, y R i ? is its corresponding desired payoff, and n is the number of training data. In SVMs,the original input space is mapped into a high dimensional space called feature space by a nonlinear represent x ? g(x) . Let f (x) be the SVM outputs corresponding to input vector x. In the feature space,a linear function is constructedf (x) ? wT g(x) ? b (2)where w is a coefficient vector, b is a threshold.The learning of SVMs can be obtained by minimization of the empirical risk on the training data.Where, ? -intensive loss function is used to minimize the empirical risk. The loss function is defined asL? (x, y, f ) ? y ? f (x) ? max(0, y ? f (x) ) e(3)where ? is a positive parameter to allow approximation errors smaller than ? , the empirical risk is?niemp i i L x y fnR w1( , , )1( ) ? (4)Besides using ? -intensive loss, SVMs tries to reduce model complexity by minimizing 2 w , whichcan be describ ed by shrink from variables. Introduce variables i? and i , then SVMs regression is obtained asthe side by side(p) optimization problemmin ??? ?nii i w C12 ( ? )21 ? ? (5)s.t. i i i y ? f (x ) ? ? , i i i f (x ) ? y ? ? , i? , i ? 0where C is a positive constant to be regulated. By using the Lagrange multiplier method 18, theminimization of (5) becomes the problem of maximizing the following dual optimization problemmax ( ? )( ? ) ( , )21( ? ) ( ? )1 1 , 1j j i jni ji inii i i inii ? y ? ? ? ? ? ? ? ? ? K x x? ? ?(6)s.t. ( ? ) 01? ? ??nii i ? ? ,C = i , i? =0where i and i? are Lagrange multipliers, and union ( , ) i j K x x is a symmetric function which isequivalent to the dot product in the feature space. The center of attention ( , ) i j K x x is defined as the following.( , ) ( ) ( ) jTi j i K x x ? g x g x (7)There are some kernels, i.e. polynomial kernel K(x, y) ? (x ? y ? 1) d and hyperbolic tangentkernel ( , ) tanh( ( ) ) 1 2 K x y ? c x ? y ? c can be used. Where t he Gaussian function is used as the kernel.)2( , ) exp( 22?x yK x y?? ? (8)Replacing i i i ? ? ? ? ? ? and sexual intercourse 0 ? ? ? ?i , then the optimization of (6) is rewritten asmax ( , )211 1 , 1j i jni jinii inii ? y ? ? ? ? ? K X X? ? ?? ? (9)s.t. 01? ??nii ?,C ? i? ? ? CThe learning results for training data set D can be derived from equivalence (9). Note that only some ofcoefficients i? are not zeros and the corresponding vectors x are called support vectors (SV). That is, onlythose vectors whose corresponding coefficients i i are not zero are SV. whence the regression function isexpressed as equation (10).f x K x x b i jpii i ? ? ?( ) ( ? ) ( , )1? ? (10)It should be noted that p is the numbers of SV, and the constant b is expressed as? ? ? ???? ?? ? ? ? ???? ?? ? ? ? ?? ?pii i i ipii i i i b y K x x y K x x1 1min ( ? ) ( , ) max ( ? ) ( , )21 ? ? ? ? (11)3.2 SVM parametersThe quality of SVMs models strongly depends on a proper setting of parameters and SVMsapproxim ation performance is sensitive to parameters 25, 26. Parameters to be regulated includehyper-parameters C, ? and kernel parameter? , if the Gaussian kernel is used 25. The values of C,? and ? are relate to the actual object model and there are not fixed for different data set. So the problemof parameter selection is complicated.The values of parameter C, ? and ? affect model complexity in a different way. The parameter Cdetermines the trade-off among model complexity and the tolerance degree of deviations larger than ? .The parameter? controls the largeness of the ? -insensitive zone and can affect the numbers of SV inoptimization problem. The kernel parameter? determines the kernel width and relates to the input range ofthe training data set. Here, parameters selection is regarded as compound optimization problem and animproved differential evolution algorithm is proposed to select suitable parameters value.4. Improved Differential growingDifferential evolution (DE) algorithm is a simple but goodly state-based stochastic hunttechnique for solving global optimization problems 27. DE has 3 operations genetic genetic mutation, crossover voter andselection. The crucial idea behind DE is a plan for generating trial vectors. Mutation and crossover areused to generate trial vectors, and selection then determines which of the vectors will survive into the next extension. The original DE algorithm is described in the following briefly.4.1 Basic differential evolutionLet S ? Rn be the search space of the problem on a lower floor consideration. Then, the DE algorithm utilizesNP, n-dimensional vectors X x x xt S i NPintititi ( , , , ) , 1,2, , 1 2 ? ? ? ? ? as a population for eachgeneration of the algorithm. t denotes one generation. The initial population is generated hit-or-missly andshould veil the whole parameter space. In each population, two manipulators, namely mutation and crossover,are applied on each individual to yield a trial vector for each ta rget vector. Then, a selection human body takesplace to determine the trial vector enters the population of the next generation or not.For each target individual ti X , a edition vector 1 , , 11?1 ? ? t ?nt ti V v ? v is laid by thefollowing equation.( ) 1 2 31 trtrtrti V ? ? X ? F ? X ? X (12)Where F ? 0 is a real parameter, called mutation constant, which controls the amplification of thedifference vector ( ) 2 3trtr X ? X to avoid search stagnation. fit in to Storn and Price 27, the F is setin (0, 2. 1 r , 2 r , 3 r are indexes, ergodicly selected from the set 1,2,, NP . Note that indexes mustbe different from each other and from the running index i so that NP must be a least four. pursuit the mutation phase, the crossover (recombination) operator is applied on the population.For each mutant vector t ?1i V , a trial vector 1 , , 11?1 ? ? t ?nt ti U u ? u is generated, using the followingdodge. ?? ?? ????, ( ) ( )1 , ( ) ( )1x rand j CR and j randn iv rand j CR or j randn i utijtt ijij (13)Where j=1, 2, ?, n. rand( j) is the jth evaluation of a uniform random number generator within 0, 1.CR is a crossover probability constant in the range 0, 1, which has to be heady previously by the user.randn(i) ? (1,2,,n) is a randomly chosen index which ensures that t ?1i U gets at least one fractionfrom t ?1i V . Otherwise, no new mention vector would be produced and the population would not alter.To decide whether the trial vector t ?1i U should be a instalment of the population comprising the nextgeneration, it is compared to the corresponding target vector ti X , and the greedy selection strategy isadopted in DE. The selection operator is as following. ???? ??, otherwise1 , ( 1 ) ( )1titititt ii XU f U f XX . (14)4.2 Modification of MutationFrom the mutation Eq. (12) we can check off that in the original DE iii vectors are chosen at random formutation and the base vector is then chosen at random within the three, which has an exploratory effect butit slow s down the convergence of DE. In order to accelerate the convergence speed, a modified mutationscheme is adopted.The randomly selected three vectors for mutation are sorted by rising slope in terms of the fittingnessfunction value. The tournament best vector is ttb x , the better vector is ttm x and the worst vector is ttw x .For speeding up convergence, the base vector in the mutation equation should select ttb x , and the directionof difference vector should direct to ttm x , that is to choose ( t )twttm x ? x as the difference vector. Then thenew modified mutation strategy is as following Eq. (15).1 ( t )twttmttbti v ? ? x ? F ? x ? x . (15) after(prenominal) such modification, this process explores the region round each ttb x in the direction of ( t )twttm x ? xfor each mutated point. The mutation operator is not random search any more, but a classical search.However, the vectors for mutation are selected randomly in the population space, so in the wholeevolutionary process it is still a random search, which can ensure the global optimization performance ofthe algorithm 28.5 optimisation procedures of IDE for SVMs5.1 Objective functionThe objective of SVMs parameters optimization is to minimize deviations between the outputs oftraining data and the outputs of SVMs. Where, the mean square error (MSE) is used as the performancecriterion.211( ( , ))21 ????? ? ??Kkk k y f x wKObj (16)Where K is the number of training data, k y is the output of the kth training data, and f (x ,w) k is theoutput of SVMs correspond to input k x . Then the objective of the IDE is to search optimal parameter C,? and ? to minimize Objmin F(C,? ,? ) ? minObj (17)Generally, the search range of these parameters is C? 1, 1000, ? ? (0, 1, ? ? (0, 0.5. Forspecial problem, the search range is changeable.5.2 Optimization proceduresThe probing procedures of the improved differential evolution (IDE) for optimization of SVMsparameters are shown as below.Step1 input signal the training d ata and test data, select the Gaussian kernel function.Step2 find the number of population NP, the difference vector scale factor F, the crossoverprobability constant CR, and the maximum number of generations T. Initialize randomly the individuals, i.e.C, ? and ? , of the population and the trial vector in the given searching space. Set the currentgeneration t=0.Step3 Use each individual as the control parameters of SVMs, train the SVMs using training data.Step4 Calculate the fitness value of each individual in the population using the objective function givenby equation (17).Step5 Compare each individual?s fitness value and get the best fitness and best individual.Step6 Generate a mutant vector according to equation (15) for each individual.Step7 According to equation (13), do the crossover operation and yield a trial vector.Step8 accomplish the selection operation in terms of equation (14) and generate a new population.Step9 t=t+1, return to Step3 until to the maximum number of generations.6 Case study6.1 Fouling prediction schemeThe formation and development of fouling in condensers is influenced not only by cooling waterhardness and turbidness but also by working conditions of condensers, such as velocity of the cooling waterand the corresponding inlet and outlet temperatures, the fecundation temperature of steam under entrancepressure of condenser, the non-condensing gases present in the condenser, and so on. According to theprevious analysis of periodic fouling process of power plant condensers, the fouling can be classified assoft fouling and hard fouling. Therefore, two SVMs models are developed to forecast thermal resistance ofsoft and hard fouling, respectively. Then, the whole prediction fouling thermal resistance ( f R? ) incondenser is the sum of output of soft fouling prediction model ( fs R? ) and output of hard fouling predictionmodel ( fh R? ).Generally, the evolution of soft fouling is determined by the velocity (v), turbidity (d), inlet ( Ti) andoutlet temperatures (To) of cooling water, saturation temperature of steam under entrance pressure ofcondenser (Ts), and prediction time range (Tp) (the running time in a sponge rubber ball cleaning period).Therefore, these variables are chosen as inputs of the soft fouling thermal resistance predictive model. Asfor hard fouling of the class of calcium and magnesium salts, it is related to the residual fouling at thebeginning and the end of previous sponge rubber ball cleaning period (corresponding thermal resistance isRfb,n-1, Rfe,n-1, respectively), hardness of cooling water (s), saturation temperature of steam under entrancepressure of condenser (Ts), and the accumulating running time of condenser (Ta). Hence, those variables arechosen as the inputs of hard fouling thermal resistance prediction model. The soft and hard foulingprediction model based on SVMs illustrated in Fig. 2 and Fig. 3, respectively.( , ) 1 K x x( , ) 2 K x x( , ) p?1 K x x( , ) p K x x1 1 2 2 1 1 ? ? ? ? p p ? ?p p ( , ) 1 K x x( , ) 2 K x x( , ) p?1 K x x( , ) p K x xSbTs1 1 2 2 1 1 ? ? ? ? p p ? ?p p Ta RfhRfb,n-1R fe,n-1?Fig. 2 Soft fouling prediction model Fig. 3 Hard fouling prediction modelThe parameters of the two prediction models are optimized by the IDE algorithm. Fig. 4 illustrates thefouling prediction model using SVMs optimized by IDE.?Fig. 4 fouling prediction model based on SVMs optimized by IDE6.2 Experiment resultsIn this section, experiments on N-3500-2 condenser (300MW) in Xiangtan thermal power plant arecarried out to prove the effectiveness of the proposed approach. The cooling water of this plant is riverwater that pumped from the Xiangjiang river. The Taprogge systems are installed in the plant to on-lineclean the condensers. At present, the condenser is cleaned every two days using the Taprogge system, andevery cleaning time is about 6 hours. Obviously, the fitted cleaning period is not optimal, because thefouling accumulating process is dynamic chan ging with the operating conditions changing.The experiment system consists of sensors for operating condition parameters measuring, dataacquisition system, PC-type computer, etc. A set of 1362 real-time running data in different operatingconditions in 84 cleaning periods is collected to train and optimize the SVMs model for fouling prediction,another set of 300 data is chosen for model verification. The proposed IDE is used to optimize the SVMsparameters. The control parameters of IDE are the following. The number of population is 30, the crossoverprobability constant CR is 0.5, the mutation factor F is 0.5, and the maximum number of generations is 100.The selection of above parameters is based on the literature 27 and 28. After application of IDE, theoptimal SVMs parameters of soft fouling prediction model are C=848, ? =0.513, ? =0.0117, the optimalSVMs parameters of hard fouling prediction model are C=509, ? =0.732, ? =0.0075.The velocity, turbidity, and inlet temperature of cool ing water is different in summer and winter, theevolution of fouling in condensers is also different in the two seasons. In the experiments, four spongerubber ball cleaning periods in different seasons are investigated. Among them, three periods, i.e. the first,18th and 40th period, are in summer, and the other period is in winter. The hardness and turbidity of coolingwater is 56mg/L and 17mg/L in summer, and is 56mg/L and 29mg/L in winter.To demonstrate the effectiveness of the proposed approach, the comparison between the SVMs model,T-S fuzzy logic model 17 and asymptotic model is considered. The asymptotic model is obtained byprobability analysis method, and the corresponding expression is the following 17.( ) ? 41.3?1? ?(t ?1.204) /14.57 f R t e (17)Table 1 and Table 2 show the fouling thermal resistance prediction results of the above three models inthe first and the 18th cleaning periods, respectively. From the Table 1 and Table 2, we can see that comparedwith tradition asympt otic model and T-S fuzzy logic model, the SVMs based prediction model has higherprediction precision. Fig. 5 and Fig. 6 show the predicted fouling thermal resistance evolution based on theoptimized SVMs model and asymptotic model. Fig.6 clearly shows that the asymptotic model is not able toforecast the fouling evolution process at the beginning stage of the 18th cleaning period, the reason is thatthe residual fouling in the periodic fouling formation process is not considered in the asymptotic models.Table 1 fouling thermal resistance prediction results in the first cleaning periodRunningtime Tpa(hour)Operating conditions measurevalues Rf(K.m2/kW)Prediction values (K.m2/kW) coition errorv(m/s) Ti(?) Ts(?)SVMsmodelT-SmodelAsymptoticmodelSVMsmodelT-SmodelAsymptoticmodel0 2.0 19.1 33.2 0.0258 0.0260 0.0258 0.62 0 5 2.0 18.5 33.3 0.0995 0.0992 0.1018 0.0947 0.26 2.31 4.8210 2.0 15.6 31.9 0.2028 0.2037 0.2007 0.1872 0.45 1.04 7.6915 2.0 14.3 31.6 0.2501 0.2494 0.2411 0.2528 0.27 3.6 1. 0820 2.0 15.5 33.5 0.2865 0.2864 0.2830 0.2993 0.03 1.22 4.4825 2.0 15.5 34.0 0.3174 0.3172 0.3123 0.3323 0.06 1.61 4.6930 2.0 16.1 34.8 0.3420 0.3393 0.3321 0.3558 0.79 2.89 4.0435 2.0 14.4 34.6 0.3567 0.3562 0.3497 0.3724 0.14 1.96 4.4040 2.0 14.2 34.9 0.3722 0.3736 0.3600 0.3842 0.37 3.28 3.22Table 2 fouling thermal resistance prediction results of the 18th cleaning periodRunningtime Ta(hour)Operating conditions Measuringvalues Rf(K.m2/kW)Prediction values (K.m2/kW) Relative errorv(m/s) Ti(?) Ts(?)SVMsmodelT-SmodelAsymptoticmodelSVMsmodelT-SmodelAsymptoticmodel632 2.0 14.0 29.8 0.0774 0.0791 0.074 2.26 0 637 2.0 14.2 30.9 0.1772 0.1773 0.1850 0.0947 0.06 4.40 46.56642 2.0 12.5 30.4 0.2474 0.2479 0.2438 0.1872 0.21 1.46 24.33647 2.0 11.9 30.4 0.2898 0.2908 0.2955 0.2528 0.36 1.97 12.77652 2.0 10.6 30.1 0.3230 0.3222 0.3354 0.2993 0.25 3.84 7.34657 2.0 11.4 31.5 0.3447 0.3437 0.3525 0.3323 0.28 2.26 3.60662 2.0 10.2 31.2 0.3655 0.3652 0.3648 0.3558 0.08 0.19 2.65667 2.0 10.7 32.0 0.3831 0.3815 0.3767 0.3724 0.42 1.67 2.79672 2.0 11.8 33.5 0.3985 0.3978 0.3912 0.3842 0.18 1.83 3.59To eliminate the influence of residual fouling and improve the prediction precision, an improvedasymptotic models are i