A new model to predict the densities of nano fluids using statistical mechanics and artificial intelligent plus principal component analysis

F.Youse fi*,Z.Amoozandeh

Department of Chemistry,Yasouj University,Yasouj,75914-353,Iran

1.Introduction

Nano fluids are two phase mixtures consisting of solid nanoparticles with sizes varying generally from 1 to 100 nm dispersed within heat transfer liquids such as water,ethylene glycol,propylene glycol,and light oils.Thermal properties and stability of nano fluids have been a hotly discussed topic during the last two decades due to their potential for application in heat transfer[1].

Nano fluids, firstly described by Choi[2]and other researchers,discover a new type of heat transfer fluid superior to conventional microparticle fluids in terms of thermophysical properties[2].Moreover,enhancing heat transfer eliminates problems arising from microparticle fluids.Therefore,nano fluids have valuable applications in practical heat transfer processes due to their high potential for enhancement of heat transfer.The thermophysical properties of nano fluids,such as thermal conductivity,viscosity,and density,are important in heat transfer application involving heat transfer fluid for thermal engineering[3].

Among the various thermophysical properties of nano fluids,most attention has been dedicated to thermal conductivity and viscosity[4–8],while less attention has been paid to other properties such as density(ρ),specific heat capacity(Cp),and isentropic compressibility(Ks).Thermal conductivity and viscosity are not adequate to calculate theoretical heat transfer coefficient.Density,specific heat,and isentropic compressibility are also very significant for heat transfer computations.Therefore,these properties should perfectly determine because of their in fluence on nano fluid flow and heat transfer characteristics[9].On the other hand,experimental evaluation of nano fluids suffers from limitations such as complexities in preparing monodisperse suspensions;and methodical problems in measuring particle size,concentration,and the homogeneity of its solution.As a consequence,the ranges of the considered variables are also limited.Hence a few density measurements have been presented for various nano fluids at different situations[10],while suggesting the reported theoretical correlation for the evaluation of nano fluids density is based on simple model.

Earlier,Ihm–Song–Mason equation of state(ISM EOS)[11]and Tao–Mason equation of state(TM EOS)[12]were extended to fluid and fluid mixtures[13–21].In addition,the applications of equation of state and artificial neural networks approaches[19,22]were studied to approximate the properties ofpure polymers.Generally,the artificialneuralnetwork(ANN)is a powerful and successful method for complex nonlinear systems due to unique advantages such as high speed,simplicity,and large capacity which reduce engineering attempt.In recent years,ANN modelling has been successfully used for predicting of thermophysical properties of pure and mixture fluids[22–26].

This research is focused on the capability of both TM EOS(with the Pak and Cho equation)and the ANN to estimate thermodynamic properties of nano fluids in different conditions.As a final point,the efficiency of these approaches is compared with experimental data and an experimental Pak and Cho equation.

2.Theory

2.1.Tao–Mason equation of state

The common equations of state are based on the van der Waals family of cubic equations,the extended family of virial equations,or equations based more strongly on the results from statistical mechanics and computer simulations[27–29].The TM EOS belongs to the latter category.Tao and Mason expressed a perturbed correction term that has an effect on the attractive forces,and combined it with the Ihm–Song–Mason(ISM)equation of state[11]to present an advanced equation ofstate(TMEOS)[12].This equation ofstate for pure materials is as follows:

where,ω,λ,ρ,Tc,andKTare the Pitzer acentric factor,an adjustable parameter,number density,critical temperature,and usual meaning.Also,B2,α andbare the second virial coefficient,the scaling parameter,and the effective van der Waals co-volume.

The TM EOS requires usage of the second virial coefficient(B2)in company with the parameters α,andb.It should be mentioned that knowledge of experimental second virial coefficient data is adequate to calculate values of the other two temperature-dependent parameters,if the intermolecular potential is not accessible[12].In this case,there are numerous correlation schemes,typically based on the corresponding state principal that leads to the computation of the second virial coefficient.

Tao and Mason formulated α,and b in terms of the Boyle temperature(TB)and the Boyle volume(vB)[12].However,theB2values can be calculated from the Tsonopolous correlation[30]in the absence of sufficient experimental data.

In this project,to attain higher accuracy,a corresponding state correlation was examined in order that TM EOS might be applied to nano fluids.In this respect,the following correlation equation forB2using a new scaling parameter with minimum input(such as molar density at the room temperature)has been extended.This correlation for second virial coefficient is presented as follows:

where ρris density at room temperature.

Tao and Mason's observation shows that the dimensionless quantities α/υBandb/υBas almost common functions of the reduced temperature(T/TB)can be calculated from the exponentialformulas based on a LJ(12–6)potential[12].At this point,the scaling factors(TBand υB)are the Boyle temperature and Boyle volume,which can be stated based on the room parameters.The empiricalequations given in[12]forα/υBandb/υBas,a function ofT/TBcan be rescaled by 298.15 K and ρr,temperature and density in room point,instead ofTBand υBas Eslami[31].Therefore,the inputparameter decreased,and this parameter easily obtains in contrast to Boyle parameters or boiling parameters.

where the constantsa1,a2,c1,andc2are?0.0860,2.3988,0.5624,and 1.4267,respectively.In the previous study,TM EOS was extended to the refrigerant mixtures with following equation[19].

TheGijterm is the pair distribution function and was calculated by Ihmet al.[11]:

whereTcmixis the traditional pseudocritical temperature.

In the present method,the second virial coefficient and the other two temperature-dependent parameters evaluation can be extended to mixtures using simple geometric mean of liquid density at the room temperature;i.e.,

The adjustable parameter(λ)obtains from volumetric properties at high temperature and pressure and authorizes the whole procedure as self-correcting.

3.ANN Modelling

The artificial neural network(ANN)is the nonlinear mathematical method thatattracts the greatestinterestdue to its simplicity, flexibility,and availability for various training algorithms,as well as its large modelling capacity[32–34].An artificial neural network,which is derived based on the activity process of the human brain,has hitherto been employed for modelling by many scientific disciplines[33–36].Several applications and good descriptions of the artificial neural network(ANN)were presented in previous publications[37,38].

The ANN forms the input layer(independent variables),the output layer(dependent variables),and one or more neuron layers called hidden layers can be located between them.The structure of an ANN is described by the number of its layers,the number of neurons in each layer,and the nature of learning algorithms and neurons transfer functions.The important elements of a neural network are the neurons,which are organized in input and output layers of hidden layers(Fig.1).

3.1.Principal component analysis

Principal component analysis(PCA)is a statistical method that changes an orthogonal transformation,to convert a set of observations ofpossibly correlated variables into a setofvalues oflinearly uncorrelated variables called principal components.It is a way of recognizing patterns in data,and expressing the data in such a way as to highlight their similarities and dissimilarities.While patterns in data can be hard to find in data of high dimension,the PCA is a great tool for analysing data.The other key bene fit of the PCA is that once you have found these patterns in the data,you can compress the data by means of reducing the number of dimensions without much loss of information.Principal components are guaranteed to be independent if the data set is jointly normally distributed.The PCA is sensitive to the relative scaling of the original variables[39].

Finally,the PCA was used to make a classifier system more effective and it is based on the assumption that most information about classes is contained in the directions,including which the variations are the largest.The most common derivation of the PCA is in terms of a standardized linear projection,which maximizes the variance in the projected space[40].The details of the PCA are stated in[41].

3.2.Network training and selection of the best network architecture

The most regular neural network technique for solving difficulties is multilayerperceptrons(MLP).The MLP discovers the data pattern using algorithms known as“training”,these algorithms modify weights of the neurons according to the error between the values of actual output and target output that provide nonlinear regression between inputs and outputs variables and are extremely useful for recognizing patterns in complex data.An example of training algorithms is the backpropagation algorithm that is widely used to train ANN in various applications.

Two transfer functions that are used in the hidden layer and output layer are “Tansig”and “Purelin”,respectively.The back propagation algorithm of Levenberg–Marquardt(trainlm)was applied to determine optimalnetstructure.The trialand error approach is mostbasic method of training a neural network.The number of hidden layers is to be selected depending on the complexity of the problem but,generally,one hidden layer is satisfactory for modelling most of the problems.In this method, firstly the number of hidden layers considered one and in the next step,in the trial and error approach,the number of neurons in the hidden layer is varied one by one to attain the desired objective function outputs.This trend is continued to find the number of neurons thatleadsto the lowesterrorforthe testing subsetand isreported asthe optimum number of neurons in the hidden layer.

The mathematical de finition of the error criteria,including average deviation percentage(AAD)and correlation coefficient(R2)values,were given as below:

Fig.1.The topology of a three layer MLP.

4.Results and Discussion

In this project,the density,specific heat capacity,and isentropic compressibility(Ks)of some nano fluids such as Sb2O5,SnO2/(EG+H2O),ZnO/(EG+H2O),Al2O3/(EG+H2O),ZnO/(PEG+H2O),ZnO/PEG,and TiO2/EG were estimated from the extended TM EOS at various temperatures and volume fractions.In addition,the artificial neural network with PCA technique has been utilized to compute the densities of the aforementioned nano fluids as a function of the temperature,volume fraction ofnanoparticle,diameter ofnanoparticle,and the densities of base fluids and ranges of input–output variables for each nano fluid are given in Table 1.

Table 1Summary of the input–output dataset characterization

Experimental densities for Sb2O5,SnO2/(EG +H2O),ZnO/(EG+H2O),and Al2O3/(EG+H2O)were obtained from Vajjha and Mahagaonkar[42].Furthermore,the experimental densities for ZnO/(PEG+H2O)and ZnO/PEGwere attained from[43]and the experimental densities for TiO2/EG were determined from[44].

The equation for the density of two-phase mixtures for particles of micrometre size is available in the literature on slurry flows[45].Pak and Cho[46]adopted the same equation for nanometre sized particles,which is expressed by the formula

wherengis the density of the nano fluid,ρpis the density of the particle,φ is the particle volume concentration,and ρbfis the density of the base fluid.Pak and Cho conducted the experiment at only one temperature(298 K)for γ-Al2O3and TiO2nano fluids up to 4.5%volume concentration to verify Eq.(4).

As above mentioned,the experimentalevaluation ofnano fluids have many limitations,such as complexities in preparing mono-disperse suspensions,methodological problems in measuring particle size,concentration,and the homogeneity of its solution.Hence,several density measurements have been presented for various nano fluids in different situations[10].Therefore,novel approaches such as equation of state and artificial neural networks are efficient in predicting the density and other thermodynamic properties of nano fluids,based on available data.

In this investigation,the TMequation of state was used to determine the density of base fluids.Then we replaced the density of the base fluid in the Pak and Cho[46]equation and introduced a new equation ofstate to determine the density of nano fluids for the first time.

Extension of statistical,mechanically-based equation of state for pure and mixed liquids requires preliminary modifications of the TM EOS.In the beginning,the second virial coefficient was developed using the density of liquid at room temperature that can simply be measured in contrast to the critical parameters.Consequently,the number of input parameters in the Tsonopolous'correlation[30](critical temperature,critical pressure,and acetric factor)are reduced to one parameter,including density at room temperature(ρr).

Besides,kin Eq.(2)isa weak function ofthe acentric factorso thatkis approximated to 1.093 andA2was estimated to 1.64[47].Then,the parameters α andbwere correlated by means of Eqs.(8)and(9),respectively.In both equations,the inputparameters(the molardensity at room temperature)are more available than the Boyle temperature and volume,and these modifications lead to a decrease in the number of input parameters of B2,α andbfrom five(including critical temperature,critical pressure,acentric factor,Boyle temperature,and Boyle volume)to one(molar density at room temperature).

In addition,the parameter of λ for pure base fluid was adjusted by the nonlinear regression method as follow:

where the parameters that were used for calculation of λ are listed in Table 2.After that,with these modifications using Pak and Cho's[46]equation,the densities of nano fluids can be computed.

Table 2Coefficients in Eq.(19)

The MLP is trained,validated,and tested at random with 70%(199 data points),15%(43 data points),and 15%(43 data points),respectively.

At first,a set of observations are converted into a set of values of linearly uncorrelated variables with a statistical method:namely,principal component analysis(PCA).The number of principal components variable in this work is equal to the number of original variables but the variables values of linearly uncorrelated.

In the second step ofthe training procedure,alldata points produced from PCA were scaled to the range of[0,1]as follows:

Fig.2.Effect of the number of hidden layer neurons on AAD.

The possibility of over-training is a problem in the ANN,and can be overcome by suitable selection of the number of neurons in the hidden layer.The numbers of hidden neurons were determined by trial and error and were begun with two neurons in the hidden layer,with the number of neurons increased regularly.The performance of the network in training phase should increase with the increase in the numberofneurons;atthe same time as the performance ofthe network in testing data phase leads to the optimumvalue ofhidden neurons(see Fig.2).In this project,the mean square error(MSE)was chosen as a compute of the performance of the neural network.The net with one hidden layer(20 neurons)and with a mean square error of 1.12 × 10–5leads to the most excellent prediction in Fig.2.Fig.3 shows the progress of training,validation,and test errors as a function ofthe numberoftraining epochs(based on an early stopping approach).

Fig.3.Evolution of training,validation,and test errors as a function of the number of training epochs during ANN training.

Fig.4.Modelling ability of the optimized ANN to predict the effective densities of all nano fluids suspension:(R2=0.999,AAD=0.48%).

Fig.5.Mean absolute relative error for the train,test and validation of densities of all nano fluids with the experimental data.

Fig.6.Deviation plot for the calculated density of TiO2/EG from TM EOS and ANN at different temperatures,pressures and two volume fractions(Fig.6a.(?=1.75%)and(Fig.6b).?=5%),compared with the experiment[44].

Fig.8.Deviation plot for the calculated density of ZnO/PEG+H2O from TM EOS and ANN atdifferenttemperatures,volume fractions and two mole fraction of PEG(a:X PEG=0.062 and b:X PEG=0.15)compared with the experiment[43].

Based on testing data,the optimized neural network model was used to predict the densities of nano fluids;the evaluation between predictive values and experimental values is carried out and shown in Fig.4.The results of Fig.4 demonstrate good agreement between the predicted and the experimental values of densities of mentioned nano fluids with an absolute average error,AAD%=0.48%,and high correlation coefficients,R2=0.999.

The error analysis oftrained netover temperature variation for train,test,and validation point are presented in Fig.5.

TM EOS was combined with Pak and Cho's equation to calculate densities of TiO2/EG nano fluids and compare with experimental data[44]at different temperatures,pressures,and two volume fractions(1.75%and 5%)(Fig.6).The results show that these models have a good level of agreementwith experimentaldata[45].The AADof densities of TiO2/EG from new EOS and the ANN with experimental data at volume fraction 1.75%are 0.58%and 0.48%,respectively.In addition,the AAD at volume fraction 5%are 1.67%and 0.58%,respectively.Fig.7 shows the deviation plot for the calculated density of ZnO/PEG from new EOS and ANNat differenttemperatures and volume fractions compared with the experiment[43],and the AAD of these models from the literature are 0.67%and 0.11%.

The deviation plot for the calculated density of ZnO/PEG+H2O from TM EOS and ANN at different temperatures,volume fractions,and two mole fractions of PEG(0.062 and 0.15)compared with the experiment[43].The absolute average deviations ofthese models from experimental data at mole fraction of 0.062 are 0.59%and 0.14%,respectively(see Fig.8a).Also,the AAD at mole fraction of 0.15 are 1.62%and 1.05%,respectively(see Fig.8b).

Fig.9.Deviation plot for the calculated density of Al2O3/(EG+H2O)from TM EOS,ANN and Pak&Cho at different temperatures and two volume fractions(a:?=1%and b:?=8%),compared with the experiment[42].

Fig.10.Deviation plotfor the calculated density of Sb2O5,SnO2/(EG+H2O)from TM EOS,ANN and Pak&Cho at differenttemperatures and two volume fractions(a:?=1%and b:?=5.8%),compared with the experiment[42].

The absolute average deviation from experimental data[42]for the predicted densities of the three systems(Al2O3/(EG+H2O),Sb2O5,SnO2/(EG+H2O);and ZnO/(EG+H2O))are calculated from new EOS,ANN,and experimental Pak and Cho,and are presented in Figs.9–11.Itshould be stated thatin the experimentalPak and Cho,densities ofbase fluids are obtained from experimental data,while in the first model the densities of base fluids are computed from TM EOS.These figures show that the ANN model is in good agreement with experimental data and is superior to others.Also,the new EOS is better than the experimental Pak and Cho.

In this research,some thermodynamic properties such as heat capacity and isentropic compressibility of nano fluids can are predicted using the equations below:

Fig.11.Deviation plot for the calculated density of ZnO/(EG+H2O)from TM EOS,ANN and Pak&Cho at different temperatures and two volume fractions(a:?=2%and b:?=5%),compared with the experiment[42].

Fig.12.Deviation plotfor the calculated heatcapacity ofZnO/PEGfrom TMEOSatdifferent temperatures and volume fractions,compared with the experiment[43].

wheredanduare density and speed of sound of nano fluids,respectively.

The deviation plot for the calculated heat capacity of ZnO/PEG at different temperatures and volume fractions is compared with the experiment[43]in Fig.12.It can be seen from this figure that the calculated heat capacity has good agreement with literature with AAD=1.04%.

In addition,the plot of deviation of heat capacity of ZnO/(PEG+H2O)at different temperatures,volume fractions,and mole fraction of PEG(0.06 and 0.15)is presented in Fig.13,and the absolute average deviations for these systems are 1.07%and 0.23%,respectively.

As a final point,the calculated isentropic compressibility of ZnO/PEG from Eq.(22)at different temperatures and volume fractions,is compared with the experiment[43]and presented in Fig.14.The AAD of this system is 0.65%.

The calculated isentropic compressibility of ZnO/PEG+H2O at different temperatures,volume fractions,and mole fraction of PEG(0.06 and 0.15)is also compared with the literature[43](see Fig.15)and the AADs are 0.70%and 0.93%,respectively.Since the values of AAD can establish the fact that the calculated values are more or less close to the experimental data,it can be claimed that the new EOS can predict the experimental density and thermodynamic properties of nano fluids with a high degree of accuracy.

Fig.13.Deviation plotforthe calculated heatcapacity ofZnO/(PEG+H2O)from TMEOSat differenttemperatures,volume fractions and two mole fraction ofPEG(a:X PEG=0.06 and b:X PEG=0.15),compared with the experiment[43].

Fig.14.AAD plot of the calculated isentropic compressibility of ZnO/PEG from TM EOS at different temperatures and volume fractions,compared with the experiment[43].

Table 3 contains the AAD of the calculated molar density of all nano fluids using the EOS and ANN at various temperatures,pressures,and volume fractions.Asisclearfrom Table 3,in mostcasesthe accuracy of the calculated densities using the ANN are superior to new EOS but also that both of them are in agreement with experimental data.The AADs of new EOS and ANN are 1.11%and 0.48%,respectively.

5.Conclusions

In this work,the density,specific heat capacity,and isentropic compressibility(Ks)of some nano fluids such as Sb2O5,SnO2/(EG+H2O),ZnO/(EG+H2O),Al2O3/(EG+H2O),ZnO/(PEG+H2O),ZnO/PEG,and TiO2/EG were estimated from the extended TM EOS together with the Pak and Cho equation at various temperatures,pressures,and volume fractions.The TM EOS was performed using minimum input(density of room temperature).Also,the artificial neural network+PCA technique(with 20 neuron in hidden layer)was performed over the whole range of available conditions.The AADs of the calculated molar density of all nano fluids using the new EOS and ANN at various temperature,pressure,and volume fractions are 1.11%and 0.48%,respectively.In addition,the heat capacity and isentropic compressibility of the aforementioned nano fluids were predicted using obtained densities of new EOS.The results showed that these properties are in agreement with the literature.

Fig.15.AADplotofthe calculated isentropic compressibility ofZnO/(PEG+H2O)from TM EOS at different temperatures,volume fractions and two mole fraction of PEG(a:X PEG=0.06 and b:X PEG=0.15),compared with the experiment[43].

Table 3Result of density prediction for all mentioned nano fluids

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