Accurate predicting the viscosity of biodiesels and blends using soft computing models

Accepted Manuscript Accurate predicting the viscosity of biodiesels and blends using soft computing models

Ali Aminian, Bahman ZareNezhad PII:

S0960-1481(17)31242-9

DOI:

10.1016/j.renene.2017.12.038

Reference:

RENE 9535

To appear in:

Renewable Energy

Received Date:

04 April 2017

Revised Date:

01 November 2017

Accepted Date:

09 December 2017

Please cite this article as: Ali Aminian, Bahman ZareNezhad, Accurate predicting the viscosity of biodiesels and blends using soft computing models, Renewable Energy (2017), doi: 10.1016/j. renene.2017.12.038

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ACCEPTED MANUSCRIPT Highlights:

Soft computing models for the viscosities of the biodiesels and blends of biodiesels. The presented model is superior to the well-known theoretical models. The results of estimations compared for eighteen types of biodiesels. The GA and SA optimized model has high accuracy for the new viscosity data.

ACCEPTED MANUSCRIPT 1

Accurate predicting the viscosity of biodiesels and blends using soft

2

computing models

3

Ali Aminian, Bahman ZareNezhad*

4

Faculty of Chemical, Petroleum and Gas Engineering, Semnan University, PO Box

5

35195-63, Semnan, Iran

6

*Corresponding author: Bahman ZareNezhad, E-mail: [email protected]

7

Abstract

8

While the viscosity is an important factor influencing the atomization and combustion

9

behavior of biodiesels, the viscosity prediction of biodiesels, blend of biodiesels, and

10

blends of biodiesel-diesel fuels can be utilized for the replacement of conventional

11

diesel fuels by the biodiesels from environmental pollution and renewability stand

12

points. Therefore, a Support Vector Machine (SVM), an Adaptive Neuro Fuzzy

13

Inference System (ANFIS), and feedforward neural network model trained by Genetic

14

Algorithm (GA), Simulated Annealing (SA), and Levenberg-Marquardt (LM) are

15

proposed for accurate prediction of the viscosity of various biodiesels based on a high

16

number of experimental viscosity data. The performances of the developed models are

17

compared to choose the one with the highest accuracy, which in turn led to pick up

18

ANFIS model. Also, the neural network model trained by the stochastic optimization

19

algorithms is provided better performance compared to other soft computing models

20

while took into account new data. Also, the comparisons between the proposed model

21

and the most well-known biodiesel viscosity models proofing the superiority of the

22

developed model for predicting the viscosity of eighteen types of biodiesels with the

23

correlation of determination of 0 .9964 and ARD of 2.51%.

24

Keywords: Soft computing; Biodiesel; Viscosity; Blend; Stochastic optimization 1

ACCEPTED MANUSCRIPT 1

1. Introduction

2

Viscosity is one of the key features of biodiesels, which can be used to assess the

3

efficiency of a biodiesel for substituting the existing petrodiesel fuels. In order to

4

control the excessive use of traditional diesel fuel due to its renewability problem,

5

global warming from CO2 emissions and the content of sulfur and aromatic

6

compounds, biodiesel has been increasingly received attention mainly due to its

7

outstanding properties compared to those of diesel fuel. However, the higher viscosity

8

of biodiesel compared to that of diesel fuel is a potential problem from atomization

9

and equipment design standpoints. The higher viscosity will results in droplets with

10

larger size and the more complicated combustion chamber including the pump and

11

injector elements. On the other hand, a fuel with low viscosity may not meet the

12

criteria for adequate lubrication that leads to leakage or increased wear. Thus, any

13

biodiesel fuel does need to meet the kinematic viscosity specifications (determinations

14

at 40 C) in biodiesel standards which are 1.9-6.0 mm2/s in the American standard

15

ASTM D6751 [1] and 3.5-5.0 mm2/s in the European standard EN 14214 [2]. Also, the

16

higher production costs are limiting biodiesel replacement for diesel fuel [3].

17

In recent years, researchers are trying to accelerate the biodiesel replacement for diesel

18

fuel by blending different biodiesels, use of biodiesel-diesel blends and/or improving

19

its problematic issues like the viscosity, especially in cold climate conditions.

20

Therefore, the thermodynamic properties of reformulated biodiesels can meet the

21

existing regulatory standards concerning the quality of the biodiesel.

22

There are many mathematical models for determining the viscosity of diesel fuels [4-

23

6], but reliable models for calculating the viscosity of biodiesels are scarce. Several

24

researchers reported experimental data and modeling for biodiesels and blends of

25

biodiesels [7-12]. Borges et al. [13] developed a new technique for estimation of the

2

ACCEPTED MANUSCRIPT 1

content of FAME in biodiesels from viscosity data. Since the reaction extent in the

2

process of transesterification of vegetable oil into biodiesel is a crucial and important

3

task, monitoring the reaction yield from FAME content in a biodiesel from viscosity

4

data can be utilized to produce biodiesel from economical point of view in the

5

industrial scale. Therefore, finding the relationship between viscosity and the FAME

6

content is an aim of this work.

7

Also, the principle of corresponding state using one- and two-reference fluids

8

implemented to predict the biodiesel viscosity [14]. They used a total of 193

9

experimental data with average relative deviation of 6.66% in which a two-reference

10

fluids model revealed better predictions. Developing accurate models to predict the

11

viscosity of biodiesel blends and mixture of biodiesel-diesel fuels is valuable to

12

properly simulate atomization and combustion behavior for such fuels. For instance,

13

Barabas and Todorut [15] presented experimental and modeling methods for the

14

temperature dependent viscosity of biodiesel-diesel-bioethanol blends. They used the

15

principal rule of Key for prediction purposes while their models outperform other

16

alternatives.

17

Also, the Quantitative structure-property relationship (QSPR) has been used as a base

18

assumption for relating the structure and property of a compound via group

19

contribution method to predict the viscosity of biodiesels [16]. The total number of

20

data points was 330 while the main drawback of the QSPR-based model was different

21

biodiesels require different set of numeric coefficients. Geacai et al. [17] reported

22

experimental data and modeling for two biodiesel blends and a biodiesel-diesel blend

23

of total 45 data points with different coefficients for the model associated with each

24

blend.

3

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Recently, empirical correlations developed for the viscosity of ethanol and butanol

2

blends with diesel and biodiesel fuels, however, interaction coefficient correlations in

3

alcohol-diesel and alcohol biodiesel fuels used for each system in order to increase the

4

accuracy of the empirical correlations. The three-parameter Grunberg-Nissan equation

5

with rigorous four-body interaction model of McAllister used to model the viscosities

6

of ethanol and butanol blended with diesel and biodiesel fuels [18]. Moreover, the

7

existing viscosity models examined to predict the blended viscosity of mixtures of

8

oils/biodiesel. Twelve mixing rules, for example modified Shu and Barrufet,

9

Setiadarma and Grunberg and Nissan’s mixing rules, employed for estimation

10

purposes [19]. Gülüm and Bilgin measured and modeled hazelnut biodiesel blended

11

with Ultra Force Euro diesel fuel at the volume ratios of 5, 10, 15, 20, 50 and 75%.

12

They developed one equation for the temperature dependency of the viscosities and

13

another for mass fraction dependency of the viscosities, which are quite difficult in

14

order to find the viscosities as function of both temperature and mass fraction [20].

15

Also, the viscosity of jojoba oil blends with biodiesel and/or petroleum diesel as

16

function of temperature and mixture composition presented and predicted by using

17

four viscosity-temperature correlations and then the mixture viscosities obtained via

18

existing mixing-rules equations for biodiesel/diesel fuels [21, 22]. Corach et al. [23]

19

fitted the kinematic viscosity of soybean biodiesel blended with diesel fuel into an

20

Arrhenius-type equation as a function of temperature and composition while biodiesel

21

composition estimated as a function of temperature and permittivity. However, the

22

aforementioned equation requires the knowledge about the permittivity of

23

biodiesel/diesel blend [23]. In addition, the Martin’s rule of free energy additivity used

24

to cover the kinematic viscosity of saturated and unsaturated FAME and further

25

extended it to estimate the kinematic viscosity of pure and blended biodiesels [24].

4

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However, R2 and ARD at different temperatures for 191 data points of pure and

2

blended biodiesels reported as 0.9818 and 5.38%, respectively, via the equation

3

proposed using Martin’s rule of free energy additivity.

4

As seen, semi-empirical models for viscosity of biodiesel blends do need mixing rules

5

based on the viscosity of their pure biodiesel, while models developed using Martin’s

6

rule of free energy additivity are not accurate enough when they applied to blends of

7

biodiesels. Furthermore, the high cost of experimentations associated with measuring

8

the viscosity of pure and blends of biodiesels makes valuable the use of a general

9

predictive model that has high accuracy. Although a number of approaches have been

10

developed for the viscosity of biodiesels, there is a need for a more comprehensive

11

model taking into accounts the effect of temperature and composition tested against a

12

large amount of experimental data. In addition, the use of theoretical approaches for

13

estimating the viscosity of biodiesel systems is of great practical interest, but there are

14

still significant errors in those predictions. As a result, the thermodynamic and semi-

15

empirical approaches have less viscosity prediction accuracy when applied to various

16

biodiesels, blends of biodiesels and mixture of biodiesels-diesels.

17

In this paper, a SVM model, an ANFIS model and feedforward neural network (FNN)

18

model trained by GA, SA and LM optimization algorithms are developed in order to

19

accurately predict the viscosity of biodiesels compared to the most important biodiesel

20

viscosity models concerning pollution emissions and renewability of diesel fuels.

21

2. Modeling by soft computing models

22

2.1. Determining the most influencing variables

23

The intrinsic molecular structures are accounted for the properties of chemical

24

compounds by considering the additivity contribution of each similar group of atoms

25

of a compound to the whole molecule. Therefore, the sum of Gibbs free energy of all

5

ACCEPTED MANUSCRIPT 1

the contribution groups leads to the overall Gibbs free energy of the whole molecule

2

[16]: 𝑧

3

∆𝐺 =

∑ ∆𝐺

(1)

𝑖

𝑖=1

4

where i stands for the i-th group. If the Andrade Gibbs free energy equation for the

5

viscosity is substituted into Eq. (1) [25, 26],

- ∆𝐺/𝑅𝑇

(∑𝑧𝑖= 1 ‒ ∆𝐺𝑖)/𝑅𝑇

6

𝜂 = 𝐴𝑒

7

where A is a constant, R denotes universal gas constant, T is temperature and η

8

represents viscosity. For biodiesels comprising saturated and unsaturated fatty acid

9

methyl esters, Eq.(2) can be rewritten in term of fatty acid methyl esters [24], ∆𝑆𝑓 𝑅

= 𝐴𝑒

∆𝐻𝑓

+ 𝑙𝑛 𝐴

𝑙𝑛 𝜂 =

11

By reformulating Eq. (3),

𝑠0

‒

∆𝐻𝑖

∆𝑆𝑑

∆𝐻𝑑

𝑛𝑑

𝑅 𝑅 𝑅 1 𝑅 𝑧 𝑅 ⏟ + 𝑧 ⏟ ‒ ⏟ + 𝑛𝑑 ⏟ ‒ ⏟ 𝑇 ℎ0 𝑇 ℎ1 𝑇 ℎ2 𝑠1 𝑠2

10

⏟

∆𝑆𝑖

(2)

(3)

ℎ0 + ℎ1𝑧 + ℎ2𝑛

𝑑

12

𝑙𝑛 𝜂 = 𝑠0 + 𝑠1𝑧 + 𝑠2𝑛𝑑 ‒

13

where ΔS and ΔH stand for entropy and enthalpy, z is number of carbon atoms, nd

14

represents number of double bonds, i and f denote i-th atom group and f-th fatty acid

15

group with zero carbon atom. Therefore, based on the thermodynamic formulation for

16

the viscosity of FAMEs, the viscosity can be corrollated against the temperature,

17

number of carbon atoms and double bounds in FAMEs. Since biodiesel is comprised

𝑇

6

(4)

ACCEPTED MANUSCRIPT 1

of FAMEs of different compositions, the average numbers of carbon atoms and double

2

bonds in FAMEs are calculated for prediction purposes, as follow: 𝑛

3

𝑧=

∑𝑥 𝑧

(5)

𝑖 𝑖

𝑖=1

𝑛

4

𝑛𝑑 =

∑𝑥 𝑛

(6)

𝑖 𝑑

𝑖=1

5

where xi is the weight fraction of i-th componen. It can be concluded that the most

6

influencing variables for prediction of viscosity of biodiesels are temperature, the

7

average numbers of carbon atoms and double bonds in FAMEs. Since theoretical and

8

semi-empirical equations are usually underestimate or overestimate the viscosity of

9

biodiesels, a neuromorphic model has been developed for precise predicting the

10

viscosity of different biodisels, biodiesels blend and biodiesels-diesels blend over a

11

wide ranges of influencing variables, namely, a temperature range of 0 to100 C.

12

2.2. The theory of the soft computing models

13

Soft computing methodologies are adaptive models that can capture highly nonlinear

14

relationships exist between dependent and independent variables. Soft Computing

15

technique can be integrated into an optimization algorithm in order to find suitable

16

solutions by modifying model parameters [27].

17

Combining neural networks and the use of efficient optimization algorithms appear to

18

be a very helpful approach in order to avoid trapping neural network models into local

19

minima solutions. Therefore, many researchers are trying to simply integrate

20

evolutionary learning algorithms with neural network models that can offer the best

21

solution when applied to unseen data. The architecture of the model for the sets of

22

input-output data can be examined by the process of learning in which different 7

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topologies are being examined by evaluating different optimization procedures [28,

2

29].

3

In this study, the well-known soft computing techniques namely, support vector

4

machine, adaptive neuro-fuzzy inference system, three-layered feed forward neural

5

network model trained by Levenberg-Marquardt, genetic and simulated annealing

6

optimization algorithms are implemented in order to explore the performance of each

7

model for accurate predicting the viscosities, while take into account the effect of fatty

8

acids concentration of biodiesels for reliable estimations.

9

The performance of each network architecture is accomplished by calculating the

10

mean squared error (MSE) and the correlation coefficient (R2). The MSE and R2

11

equations are as follow:

12

1 𝑀𝑆𝐸 = 𝑁

𝑁

∑ (𝑡 ‒ 𝑦 ) 𝑖

2

(7)

𝑖

𝑖=1

𝑁

∑ (𝑡 ‒ 𝑦 ) 𝑖

13

𝑖=1

2

𝑅 =1‒

2

𝑖

𝑁

(

𝑁

∑ 𝑡 ‒ ∑ 𝑡 /𝑁 𝑖=1

𝑖

𝑖

𝑖=1

)

(8)

2

14

where N is the total number of data points, ti and yi are the target and calculated

15

values, respectively.

16

By examining different network architectures in terms of the number of hidden layers

17

and nodes for a FNN model, the optimal architecture can be selected once the network

18

trained and tested well for a defined shape of transfer function and the learning

19

algorithm. The relationship between the overall network output and the input variables

20

for a feedforward neural network is [28]:

8

ACCEPTED MANUSCRIPT

[∑ [∑ 𝑚

] ]

1

𝑛 ℎ 𝐼 𝑦𝑘 = 𝑓𝑜 𝑤𝑘𝑗𝑓ℎ 𝑤𝑗𝑖𝑥𝑖 + 𝑏ℎ𝑗 𝑗=1 𝑖=1

2

where activation or transfer functions fo and fh are related to the output and hidden

3

layers, respectively, xi denotes the ith input, wji, wkj, bok and bhj are the network

4

parameters to be optimized. The Levenberg-Marquardt [27, 28], genetic [30] and

5

simulated annealing [31, 32] optimization algorithms are used for network training.

6

For the Levenberg-Marquardt algorithm, the objective function is defined as:

7

F(w)=eT(w)×e(w)

8

where e is the error vector in terms of the network parameters w. In each iteration, the

9

set of network parameters are updated via the following recurrence formula:

+ 𝑏𝑜𝑘

(9)

(10)

10

ws+1=ws−(JsTJs+λI)−1×JsT×es

11

where λ and J denote the learning rate and the Jacobian matrix, respectively.

12

Also, genetic algorithm is used for optimizing the parameters of the three-layered

13

FNN. Genetic algorithms are inspired from natural genes and evolutionary

14

computations such as probabilistic population and mutation [33]. GA is a type of

15

stochastic search method that its simplicity without the use of derivatives makes the

16

genetic algorithm useful for the discontinuous and non-differentiable problems.

17

In genetic algorithm, strings of random population created and the crossover and

18

mutation probability functions specified for a maximum of generations. After random

19

population, the fitness of each individual evaluated followed by selecting pairs of

20

individuals based on their performance. The FNN model and average absolute relative

21

deviation objective function are employed to evaluate the fitness of each individual.

22

After the reproduction step, the selected individuals moved to the mating pool where

23

crossover and mutation with their corresponding probability performed to produce

24

offsprings. Several offsprings with high performance generated from the selected

(11)

9

ACCEPTED MANUSCRIPT 1

individuals to improve the fitness value. These steps continue until maximum number

2

of generations exceeded or condition of convergence satisfied.

3

The fitness function defined as

4

𝐹(𝑥) =

5

where F(x) and f(x) are the fitness and objective functions, respectively.

6

Moreover, simulated annealing algorithm is used to compare its performance against

7

the genetic and Levenberg-Marquardt optimization algorithms. Simulated annealing is

8

also an evolutionary optimization algorithm analogous to the thermodynamic process

9

in which the lowest energy state of a substance formed by slowly cooling (heating) the

10

substance [34]. The Boltzman probability distribution function in terms of the

11

objective function and control parameters is commonly used to place the random

12

points near global solution:

13

P(E) =exp(-E ⁄kBT)

14

where P is the probability distribution, kB denotes the Bolzmann constant and T is

15

temperature. Starting by a random point l, the probability of this point at initial high

16

temperature evaluated. The second point created and the probability calculated at this

17

point using Eq. (13). The difference between the function values of these points

18

determined, if the difference was equal or smaller than zero the second point accepted;

19

if not, a random point r in the range of 0 to 1 created and the point accepted with the

20

following criterion,

21

r≤exp(-ΔE ⁄kBT)

22

where ΔE=E(l+1)-E(l). Setting new temperature as T=T×C (0
23

should be continued until T approached to a small value. Table 1 shows the prediction

24

performance of the FNN model trained by LM, GA and SA optimization algorithms.

25

From Table 1, it can be seen that a three-layered feedforward network would be

1 1 + 𝑓(𝑥)

(12)

(13)

(14)

10

ACCEPTED MANUSCRIPT 1

adequate for predicting the viscosities of biodiesels. The generality analysis performed

2

in order to avoid the overfitting problem.

3

In addition, a SVM model developed to compare its result with those of the

4

aforementioned model. SVM is a supervised learning method useful for classification

5

purposes [27]. A linear function is implemented to approximate the nonlinear

6

relationships among input-output datasets by using a separable hyperplane that

7

perfectly maximizes the margin among different classes,

8

y=wx+b=0

9

where w and b are adjusting parameters. To find w and b, the non-linear convex

(15)

10

programming problem defined below can be implemented,

11

Max w,b

2 w

subject to y i wx i  b   1  0

(16)

12

The above equation is used to maximize the margin. In order to consider the non-

13

separate ability of datasets, slack variables ξi ≥ 0; i =1,...., m introduced and the

14

classification problem to be solved becomes,

15

Max w,b

m 2  C i w i 1

subject to y i wx i  b   1   i  0

(17)

16

where C accounts for points with misclassification. In the case of linearly

17

nonseparable datasets, a kernel function K(xi, xj) can be used to project the input data

18

set xi into a high-dimensional space such that x→ϕ(x). Thus, the classification

19

function expressed as,

20

m  f x   sgn   α i y i K X i , X   b *     i 1 

11

(18)

ACCEPTED MANUSCRIPT 1

where αi is the Lagrange multiplier used in an optimization algorithm, such as

2

quadratic programming algorithm, to determine the values of w and b in Eq.(15).

3

The radial basis distribution function is employed for the kernel function while penalty

4

parameter C and the parameter of the kernel function need to be optimized to have a

5

model with high accuracy. The correlation of coefficient (R2) and MSE of the

6

developed SVM model for predicting the viscosities of biodiesels are obtained as

7

0.9801 and 6.42×10-4, respectively. Fig.1 illustrates classification using SVM model.

8

Finally, ANFIS is also a supervised learning algorithm based on fuzzy sets and theory.

9

It has five layers where fuzzification, firing strength, implication and defuzzification

10

performed to map the desired output(s) from input datasets [35]. The parameters of the

11

input and output membership functions can be optimized by a hybrid method in which

12

a gradient vector with respect to the parameters continued to work until the criterion

13

for convergence met [36]. The validation data sets can be used to enhance the

14

predictive-ability of the ANFIS model when the parameters of the membership

15

functions are being optimized. The following steps implemented for modeling through

16

ANFIS:

17

- Fuzzifying the inputs

18

- Applying fuzzy operators

19

- Implication

20

- Aggregation

21

- Defuzzifying

22

In the first step, fuzzification via a membership function performed to find the degree

23

of belonging of each input to the fuzzy sets. In the second layer, the fuzzy operators 12

ACCEPTED MANUSCRIPT 1

AND and OR applied for a multi-inputs system to produce a single value for the

2

incoming signals. In the next step, the weight of all rules, a value between 0 and 1,

3

obtained. Each rule comprises antecedent and consequent parts in which antecedents

4

defined the weights of every rule and the consequent part is a fuzzy set. In implication,

5

the weighted values from antecedent passed through implication process to form fuzzy

6

sets by truncating or scaling the output fuzzy sets. In the fourth step, the reshaped

7

fuzzy sets from the previous step combined together to produce single fuzzy set for

8

each desired output. Finally, defuzzifying applied for each fuzzy set obtained in the

9

fourth step to produce a single number. The schematic diagram showing a fuzzy

10

inference system is depicted in Fig.2. The ANFIS model has R2-value of 0.9964 and

11

MSE of 1.17×10-4 for 81 rules.

12

Table 2 provides comparisons among the developed models for predicting the

13

viscosities of biodiesels. As shown in Table 2, ANFIS model is superior to the other

14

soft computing techniques in terms of MSE and R2-value of 1.17×10-4 and 0.9964,

15

respectively, for the training and testing datasets used during the model development.

16

2.3. Training result

17

In order to predict the viscosity of biodiesels, blends of biodiesels-diesels and blend of

18

biodiesels, the required data points are taken from different resources [7, 8, 27-41].

19

The biodiesel mass fraction, temperature, the average numbers of carbon atoms and

20

double bonds in FAMEs constituting the studied biodiesels are chosen to be the input

21

variables for the viscosity. The total number of data points is 527 for the viscosity of

22

biodiesels, blends of biodiesels-diesels and blend of biodiesels. Following the

23

procedure mentioned in the previous section, the experimental datasets are divided

24

into training and testing sets; 70% for training and the remaining 30% for testing. For

25

FNN model, the minimum error for the testing datasets can be utilized to recognize the 13

ACCEPTED MANUSCRIPT 1

number of hidden layers and hidden neurons for a set of activation functions. Testing

2

datasets are fully kept unseen to the ANN model to test the generalize-ability of the

3

model. The learning phase is a trial and error task in which the network performance,

4

in terms of training and testing errors, should be monitored for different activation

5

functions, number of hidden layers and hidden neurons. In this study, the Levenberg-

6

Marquardt, GA and SA algorithms are implemented, while the tan-sigmoid is found to

7

be the proper activation function for one hidden layer of nine neurons (Table 1). The

8

feedforward network architecture for predicting the viscosity of biodiesels, blends of

9

biodiesels-diesels and blend of biodiesels as functions of state variables, namely, the

10

biodiesel mass fraction, temperature, the average numbers of carbon atoms and double

11

bonds is illustrated in Fig.3.

12

From Table 2, ANFIS performs better compared to three-layered feedforward neural

13

network of nine hidden nodes trained by LM, GA and SA algorithms and SVM model;

14

therefore, the developed ANFIS model is selected for prediction purposes. The parity

15

plots for training and testing datasets of the ANFIS model are displayed in Figs. 4 & 5.

16

The correlation coefficients for training and testing calculations are 0.9980 and

17

0.9932.

18

3. Results and discussion

19

For a complex liquid fuel like biodiesel, evaluation of the presented soft computing

20

model for viscosity are performed based on the data from different references [7, 8,

21

37-41]. Since the most commercially available biodiesels are blended with petroleum

22

diesels, the blends of biodiesels are recognized by their mass fractions; therefore, the

23

mass fraction of biodiesel is added as an input variable. The compositions of the

24

biodiesels studied in this work are given in Table 3. The viscosity of biodiesels @ 40

25

C as a property to meet the international standards for fuels is provided in Table 4 for

14

ACCEPTED MANUSCRIPT 1

the studied biodiesels. The ANFIS model predictions are compared with Ceriani's

2

model [42], Yuan's model [37], revised Yuan's model [8] and Krisnangkura's model

3

[39], which are shown in Table 5. Figs. 6-11 reveal the viscosity predictions of the

4

proposed model in comparison with the experimental data and the most important

5

biodiesel viscosity models. As shown in Figs. 6-11, the viscosity of biodiesels

6

decreases as temperature increases from 0 °C to 100 °C for all types of biodiesels and

7

blends.

8

The presented ANFIS model is compared with Ceriani, Yuan, revised Yuan and

9

Krisnangkura models regarding the viscosity prediction of Cotton seed oil biodiesel

10

[41] over the temperature range of 293-373 K in Fig. 6. As can be seen in Fig. 6, the

11

current viscosity models exhibit prediction errors. The underpredictions and

12

overpredictions of the viscosity models may lead to poor combustion associated with

13

the high viscosity of biodiesels. In contrast, utilizing the ANFIS model provides much

14

more accurate predictions of the biodiesel viscosity. The accuracies of the proposed

15

and the theoretical viscosity models are calculated by using the percent average

16

relative deviation (ARD%):

17

1 𝐴𝑅𝐷% = 𝑁

𝑁

∑

|(𝜂)𝑖,𝑝𝑟𝑒𝑑 ‒ (𝜂)𝑖,𝑒𝑥𝑝|

𝑖=1

(𝜂)𝑖,𝑒𝑥𝑝

× 100

(19)

18

The ARD for the Ceriani [42], Yuan [37], revised Yuan [8], Krisnangkura [39] and

19

ANFIS models regarding the viscosity prediction of Cotton seed oil biodiesel are 9.06,

20

5.35, 4.42, 3.53 and 2.43%, respectively, showing the superiority of the ANFIS model.

21

In addition, the calculated and experimental viscosity of Coconut biodiesel [39] is

22

compared in Fig. 7. As shown, Ceriani, Yuan and revised Yuan models overpredict the

23

viscosity of Coconut biodiesel while Krisnangkura's model underpredicts the viscosity

15

ACCEPTED MANUSCRIPT 1

over the temperature range of 298-323 K with the ARD of 5.88% for the latter model.

2

However, the proposed model performs well with the ARD of 3.03%.

3

The predicted viscosities by using the proposed model are compared with

4

experimental data for a 75% soybean oil methyl ester (SMEA) blended with No. 2

5

diesel [38] at different temperatures, as depicted in Fig. 8. The predicted viscosity of

6

SMEA 75 biodiesel blended with No. 2 diesel are in good agreement with the

7

measured data as shown in Fig. 8. The overall ARDs of different models regarding the

8

prediction of the viscosity of SMEA (25, 50 and 75% mass fraction) blended with

9

No.2 diesel are obtained as 10.2, 9.56, 9.9, 11.45 and 4.08%, respectively.

10

Also, the comparisons among different models and experimental data for predicting

11

the viscosity of a 75% genetically modified soybean oil methyl ester (GMSME)

12

blended with No. 2 diesel [38] over the temperature range of 293-373 K is illustrated

13

in Fig. 9. The overall ARDs of Ceriani, Yuan, revised Yuan, Krisnangkura and ANFIS

14

models regarding the prediction of the viscosity of GMSME (25, 50 and 75% mass

15

fraction) blended with No.2 diesel are 9.4, 6.47, 5.49, 8.26 and 2.87%, respectively.

16

It may be seen that the viscosity of the biodiesel reduces by raising temperature.

17

Therefore, accurate prediction of the viscosity of different biodiesels and blends at

18

various temperatures can reduce the problems related to the poor atomization behavior

19

of biodiesels, especially when temperature decreases.

20

Furthermore, Fig. 10 shows the viscosity of Coconut biodiesel from [37] at different

21

temperatures. It can be seen that the ANFIS model outperforms the other alternatives

22

with the ARD of 10.93, 9.1, 7.14, 13.61 and 1.92% for Ceriani, Yuan, revised Yuan,

23

Krisnangkura and ANFIS models, respectively. The predicted viscosities are in

24

excellent agreement with experimental data for Coconut biodiesel [37]. As shown,

25

Ceriani and Krisnangkura models predict the viscosities with larger errors showing the

16

ACCEPTED MANUSCRIPT 1

less suitability of these models for prediction purposes. Generally, at higher

2

temperatures Ceriani's model shows less accuracy than does at lower temperatures,

3

and vice versa, Krisnangkura's model performs better at higher temperatures. Also, the

4

revised Yuan's model provides lower prediction errors than Yuan's model.

5

Fig. 11 shows the variation of viscosity versus temperature for 90% mass fraction

6

methyl linoleate (ML) in low-sulfur petrodiesel [7]. Methyl linoleate (C18:2) is a neat

7

fatty ester blended with a petrodiesel enriched in C8-C25 straight chain alkanes. It is

8

well known that the higher viscosity of ML compared to other biodiesels attributed to

9

its longer chain. The figure demonstrates that Krisnangkura's model deviates

10

considerably from measured viscosity data. Although it has an equation for

11

unsaturated FAME, for example C18:2, but a blend of an unsaturated FAME with a

12

diesel fuel may contribute to this behavior because of the strong interactions exist

13

between two liquids. On the other hand, the proposed ANFIS model accurately

14

predicts the viscosity of 10-90% ML in diesel fuel with the overall ARD of 1.86%

15

compared to those from Ceriani, Yuan, revised Yuan and Krisnangkura models having

16

ARD of 7.05, 3.8, 3.75 and 7.78%, respectively. The lower ARD% of the ANFIS

17

model proves the accuracy of the proposed model for predicting the viscosity of neat

18

fatty esters blended with diesel fuels over the temperature range of 273-313 K.

19

According to Table 5, Krisnangkura's model is developed for pure fatty esters than for

20

biodiesel blends. This is an Arrhenius type temperature dependent formula which is

21

suffers from the density measurements to convert kinematic viscosities into dynamic

22

one and some compositions not presented in those equations. In the case of Ceriani's

23

model, the higher temperatures influenced the accuracy of the group contribution-

24

based equation in which Ceriani's model can only be applied for the viscosity of

25

biodiesels at low and/or intermediate temperatures. The Ceriani's model should be

17

ACCEPTED MANUSCRIPT 1

fitted for each fatty compound via rigorous calculations with poor temperature-

2

dependent parameters at temperatures above 335 K.

3

Furthermore, the viscosity of biodiesel fuels can be correlated by the Andrade

4

equation modified by Tat and Van Gerpen [9], as shown below:

5

𝑙𝑛(𝜂) = 𝐴 +

6

where η is the viscosity, T is the absolute temperature and A, B, and C are the

7

correlation constants. Krisnangkura et al. [43] proposed a correlation to calculate the

8

viscosity of mixtures of biodiesel at different temperatures:

9

𝐵 𝐶 + 𝑇 𝑇2

𝑙𝑛(𝜂) = 𝑙𝑛(𝐴) + ∅.𝑉 +

(20)

𝐵 𝐶.𝑉 + 𝑇 𝑇

(21)

10

where Φ and V are adjustable parameter and the volume fraction of biodiesel.

11

However, the Φ .V term in Eq. (12) contributes significantly less than the other terms,

12

therefore, the following three-terms equation is used for blends:

13

𝑙𝑛(𝜂) = 𝑙𝑛(𝐴) +

𝐵 𝐶.𝑉 + 𝑇 𝑇

(22)

14

The values of parameters for the studied biodiesels and blends along with the ARD%

15

and R2-valus for each correlation are given in Table 6.

16

None of the theoretical models are suited for various biodiesel-diesel blends because

17

they intrinsically developed for pure fatty compounds and biodiesels. The molecular

18

interactions at atomistic scale impede theoretical models from deep understanding

19

phenomena defining these interactions; such that group contribution methods are not

20

providing successful results. The correct modifications of group contribution methods

21

are needed to calculate the interactions between individual components covering a

22

wide range of operating conditions; however, the presented neuromorphic models can

23

effectively capture the nonlinear interactions among different variables. The overall 18

ACCEPTED MANUSCRIPT 1

ARD in viscosity prediction of the biodiesels and the blends via theoretical models of

2

Ceriani, Yuan, revised Yuan, Krisnangkura and Andrade are about 6.83, 5.51, 4.87,

3

7.41 and 6.33%, respectively, while the ARD for the ANFIS model is 2.51%

4

indicating significant improvement in biodiesel viscosity predictions. Fig. 12 shows

5

the ARDs for the biodiesels while Fig. 13 is for the biodiesel blends with diesels.

6

In order to test the generalize-ability of each model developed in this work, new

7

viscosities from biodiesels and biodiesel-diesel fuel and biodiesel-biodiesel blends are

8

gathered. As shown in Table 7, the FNN trained by GA and SA provided much more

9

accurate results than SVM, ANFIS and the FNN trained by LM. It can be explained

10

that stochastic optimization techniques are approaching global minima solutions, while

11

SVM, ANFIS and the FNN trained by LM models showed less accurate results for the

12

new datasets. From Table 7, it is found that genetic and simulated annealing

13

algorithms can successfully provide accurate results for new inputs not included in the

14

range of the datasets used for the models development.

15

4. Conclusions

16

A new neural network-based model is presented for viscosity of biodiesels and blends.

17

The viscosity of biodiesels is influencing the complete combustion and proper

18

atomization process in diesel engines. Therefore accurate estimation of the viscosity of

19

biodiesels is crucial. The effects of the most important influencing variables extracted

20

from group contribution theory and Gibbs free energy analysis on the viscosity of

21

biodiesels are took into accounts through SVM, ANFIS, and FNN model trained by

22

GA, SA, and LM. The proposed model optimized by GA an SA algorithms is found to

23

be accurate enough for predicting the viscosity of several well-known biodiesels and

24

new datasets not included in the range of operating conditions used during models

25

development. The inputs to the model are mass fraction of biodiesel, temperature, the

19

ACCEPTED MANUSCRIPT 1

average numbers of carbon atoms and double bonds in FAME constituting the

2

biodiesels while the output is the viscosity. The results of the proposed models for the

3

new datasets proof the generality of the models for accurate predicting the viscosity of

4

biodiesels.

5 6 7 8 9

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10

stock for distillate fuels. ASTM, West Conshohocken, PA.

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[30] Deb K. Multi-objective optimization using evolutionary algorithms. John Wiley

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fuzzy systems by supervised learning algorithms. IEEE Int Conf Fuzzy Systems

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(FUZZ-IEEE) 2003; 2: 226-31.

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biodiesel fuels. Fuel 2009; 88: 1120-26.

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Viscosity of Selected Biodiesel Fuels and Blends with Diesel Fuel. J Am Oil Chem

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Soc 2005; 82: 195-9.

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biodiesel viscosity at various temperatures. Fuel 2006; 85: 107-113.

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Sant’Ana HB. Viscosities and densities of binary mixtures of coconut + colza and

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coconut + soybean biodiesel at various temperatures. J Chem Eng Data 2010;55:3909-

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14.

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3

Densities and viscosities of binary mixtures of babassu biodiesel + cotton seed or

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soybean biodiesel at different temperatures. J Chem Eng Data 2010;55:5305-10.

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[42] Ceriani R, Gonçalves CB, Rabelo J, Caruso M, Cunha ACC, Cavaleri FW, et al.

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Group contribution model for predicting viscosity of fatty compounds. J Chem Eng

7

Data 2007;52:1846-53.

8

[43] Krisnangkura K, Sansa-ard C, Aryusuk K, Lilitchan S, Kittiratanapiboon K. An

9

empirical approach for predicting kinematic viscosities of biodiesel blends.

10

Fuel 2010; 89: 2775-80.

11

[44] Esteban B, Riba J-R, Baquero G, Rius A, Puig Rita. Temperature dependence of

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density and viscosity of vegetable oils. Biomass Bioenergy 2012; 42: 164-71.

13

[45] Nogueira CA, Nogueira VM, Santiago DF, Machado FA, Fernandes FAN,

14

Santiago-Aguiar RS, De Sant’Ana HB. Density and Viscosity of Binary Systems

15

Containing (Linseed or Corn) Oil, (Linseed or Corn) Biodiesel and Diesel. J Chem

16

Eng Data 2015; 60: 3120-31.

17

[46] Parente RC, Nogueira CA, Carmo FR, Lima LP, Fernandes FAN, Santiago-

18

Aguiar RS, Sant'Ana HB. Excess Volumes and Deviations of Viscosities of Binary

19

Blends of Sunflower Biodiesel + Diesel and Fish Oil Biodiesel + Diesel at Various

20

Temperatures. J Chem Eng Data 2011; 56: 3061-67.

21

Nomenclature

22

A, B and C constants of the Yuan's model

23

A1k , A2k , … Ceriani's parameters to be optimized

24

bj

25

C

bias penalty parameter 24

ACCEPTED MANUSCRIPT 1

f

activation function

2

f0, f1 parameters

3

f(x)

predicted value

4

K

kernel function

5

m

number of hidden nodes

6

M

molecular weight

7

n

number of input nodes

8

nd

number of double bonds

9

N

total number of data fed to the model

10

Nk

number of groups k in the molecule

11

Nc

total number of carbon atoms

12

Ncs

number of carbons of the substitute fraction

13

R

universal gas constant

14

R2

correlation coefficient

15

s0, s1 parameters

16

T

temperature (K)

17

ti

target value

18

wji weight value

19

xi

i-th input to the network; weight fraction

20

y

predicted output

21

z

number of carbon atoms

22

Greeks

23

α, β, γ, δ parameters

24

∆𝐺

overall Gibbs free energy

25

∆𝑆

Entropy change

25

ACCEPTED MANUSCRIPT 1

∆𝐻

Enthalpy change

2

η

viscosity

3

Subscript

4

exp

experimental value

5

pred

predicted value

6

Superscript

7

I

input layer

8

h

hidden layer

9 10 11

26

ACCEPTED MANUSCRIPT Table 1. Comparison of the performance of optimization algorithms for training and testing the three-layered FNN model. Levenberg-Marquardt

Genetic algorithm

Simulated annealing

No. of node 3

MSE

R2

MSE

R2

MSE

R2

3.03E-04

0.9894

1.90E-03

0.9662

1.60E-03

0.9698

4

2.90E-04

0.9912

5.00E-03

0.9275

3.50E-03

0.9449

5

2.12E-04

0.9928

4.75E-03

0.9302

5.30E-03

0.9206

6

1.93E-04

0.9924

4.20E-03

0.9385

3.50E-03

0.9438

7

1.65E-04

0.9944

6.40E-03

0.902

4.70E-03

0.9204

8

1.79E-04

0.9940

7.40E-03

0.8983

1.90E-03

0.9574

9

1.29E-04

0.9944

4.19E-04

0.9866

5.50E-03

0.9237

10

1.33E-04

0.9950

4.20E-04

0.9860

6.50E-03

0.9080

11

1.94E-04

0.9948

4.47E-04

0.9864

4.47E-04

0.9864

12

1.93E-04

0.9936

4.81E-04

0.9852

9.98E-04

0.9704

27

ACCEPTED MANUSCRIPT

Table 2. Performance of the soft computing techniques considered in this work regarding training and test data sets. Model

Training data

Testing data

MSE

R2

MSE

R2

FNN-LM

9.07×10-5

0.9961

2.46×10-4

0.9931

FNN-GA

1.69×10-4

0.9905

8.85×10-4

0.9751

FNN-SA

8.52×10-4

0.9557

8.45×10-3

0.9012

SVM

3.33×10-4

0.9890

1.40×10-3

0.9624

ANFIS

6.17×10-5

0.9980

2.47×10-4

0.9932

28

ACCEPTED MANUSCRIPT Table 3. Compositions of the biodiesels. Coconut [37]

YGME [37]

Coconut [39]

Soy A [8]

C8:0

0.092

0.048

C10:0 C12:0 C14:0 C16:0 C16:1 C18:0 C18:1 C18:2 C18:3

0.064 0.487 0.17 0.077

0.062 0.527 0.175 0.074

0.1618

0.024 0.076 0.014

0.0382 0.288 0.5046

0.022 0.054 0.022

0.0170 0.1947 0.1438 0.5467 0.0796 0.0069

Soy B [8]

B1 [8]

Sunflower [8]

Palm [8]

Rapeseed [8]

0.0007 0.1078 0.0007 0.0395 0.2302 0.5366 0.0703

0.018 0.047 0.047 0.019 0.7113 0.0989

0.0002 0.0007 0.0641 0.0009 0.0423 0.2393 0.6425 0.0012

0.0003 0.0025 0.0057 0.4252 0.0013 0.0403 0.4199 0.0981 0.0009

0.0001 0.0004 0.0007 0.0526 0.002 0.0163 0.6249 0.2094 0.0699

0.0589

0.0036

0.006

C20:0

0.0025

0.0038

C20:1

0.0052

0.0023

0.0003

0.0015

0.0123

C22:0

0.0021

0.008

0.0077

0.0009

0.0135

C22:1

0.0008

0.0019

C24:0

Cont. B+Petroleum

SMEA

SMEB

GMSME

YGME

GP

Coconut

[7]

[38]

[38]

[38]

[38]

[8]

[40]

Babassu [41]

C8:0

0.0408

C10:0

0.0365

0.051

0.0002

0.3535

0.2811

C12:0 C14:0 C16:0

0.1079

C16:1

0.0008

0

0

0.0127

0.0013

0.1984

0.2556

0.1049

0.1081

0.0397

0.1744

0.1057

0.1383

0.1541

0.0012

0.0011

0.0013

0.0203

0.0013

C18:0

0.0421

0.0427

0.0454

0.0299

0.1238

0.0266

0.0394

0.0504

C18:1

0.2441

0.242

0.2496

0.8254

0.5467

0.4105

0.143

0.2079

C18:2

0.5338

0.5136

0.5066

0.0498

0.0796

0.3667

0.0473

C18:3

0.0721

0.0748

0.0727

0.037

0.0069

0.071

C20:0

0.0036

0.0037

0.003

0.0025

0.0044

C20:1

0.0028

0.0032

0.005

0.0052

0.0067

C22:0

0.004

0.0042

0.0036

0.0021

0.0045

C22:1

0.0007

0

0

C24:0

0.0014

0.0012

0.0012

0.0012

29

Cotton seed [41]

0.0062 0.2409

0.0256 0.1574 0.5699

ACCEPTED MANUSCRIPT Table 4. Properties of Biodiesels Biodiesel

viscosity @ 40 C

Ref.

sunflower

3.636 (mPa s)

[8]

soy B

3.548(mPa s)

[8]

B palm

3.961(mPa s)

[8]

rapeseed commercial biodiesel SMEA

3.942(mPa s)

[8]

4.15 (mm2/s)

[7]

3.67 (mPa s)

[37]

Palm

3.87 (mPa s)

[37]

Coconut

2.32 (mPa s)

[37]

Canola

3.7 (mPa s)

[37]

SMEB GMSME YGME

4.41

(mm2/s)

[38]

4.87

(mm2/s)

[38]

5.02

(mm2/s)

[38]

Coconut

2.15 (cP)

[39]

Coconut

2.45 (mm2/s)

[40]

Babassu

3.18 (mm2/s)

[41]

(mm2/s)

[41]

Cotton seed

3.99

30

ACCEPTED MANUSCRIPT Table 5. The most important viscosity models for biodiesels. Biodiesel viscosity model Ceriani 𝑙𝑛(𝜂𝑖) 𝐵1𝑘 𝐵2𝑘 = 𝑁𝑘 𝐴1𝑘 + ‒ 𝐶1𝑘𝑙𝑛 𝑇 ‒ 𝐷1𝑘𝑇 + 𝑀𝑖 𝑁𝑘 𝐴2𝑘 + ‒ 𝐶2𝑘𝑙𝑛 𝑇 ‒ 𝐷2𝑘𝑇 + 𝑄 𝑇 𝑇

∑ 𝑘

(

𝑄 = 𝜉1𝑞 + 𝜉2 𝛽 𝑞 = 𝛼 ‒ ‒ 𝛾𝑙𝑛 𝑇 ‒ 𝛿𝑇 𝑇 𝜉1 = 𝑓0 ‒ 𝑁𝑐 𝑓1 𝜉2 = 𝑠0 ‒ 𝑁𝑐𝑠 𝑠1

)

(

∑ 𝑘

Reference

)

[42]

η in mPa.s Yuan

[37]

𝐵 𝑙𝑛(𝜂) = 𝐴 + 𝑇+𝐶 η in mPa.s Revised Yuan 𝐵 𝑇+𝐶 For parameters fitted against new and accurate data 𝑙𝑛(𝜂) = 𝐴 +

[8]

Krisnangkura 492.12 108.35𝑧 + 𝑓𝑜𝑟 (𝐶6:0 ‒ 𝐶12:0) 𝑇 𝑇 403.66 109.77𝑧 𝑙𝑛(𝜂) =‒ 2.177 ‒ 0.202𝑧 + + 𝑓𝑜𝑟 (𝐶12:0 ‒ 𝐶18:0) 𝑇 𝑇 2051.5 𝑙𝑛(𝜂) =‒ 5.03 + 𝑓𝑜𝑟 𝐶18:1 𝑇 1822.5 𝑙𝑛(𝜂) =‒ 4.51 + 𝑓𝑜𝑟 𝐶18:2 𝑇 1685.5 𝑙𝑛(𝜂) =‒ 4.18 + 𝑓𝑜𝑟 𝐶18:3 𝑇 2326.2 𝑙𝑛(𝜂) =‒ 5.42 + 𝑓𝑜𝑟 𝐶22:1 𝑇 η in mm2/s 𝑙𝑛(𝜂) =‒ 2.915 ‒ 0.158𝑧 +

31

[39]

ACCEPTED MANUSCRIPT Table 6. Results of correlation with Eqs. (20 & 22) Fuel type

Temperature range

No. of Experimental data

Babassu

[293-373]

5

Cotton seed

[293-373]

5

Coconut

[293-373]

5

Biodiesel+ Diesel

[273-313]

81

[283-353]

15

[278-363]

18

B1

[283-353]

15

Sunflower

[283-363]

17

Rapeseed

[278-363]

18

Palm

[288-363]

16

GP

[278-363]

18

Coconut

[298-323]

3

[293-373]

20

[293-373]

20

[293-373]

20

[293-373]

20

YGME

[293-373]

5

Coconut

[293-373]

5

[293-373]

35

𝑙𝑛(𝜂) = 𝑙𝑛(0.074) +

1029.8 154.7𝑉 + 𝑇 𝑇

17.3

0.9095

[41]

[273-313]

81

𝑙𝑛(𝜂) = 𝑙𝑛(0.121) +

1044.3 131.4𝑉 + 𝑇 𝑇

15.79

0.9122

[7]

[273-313]

81

𝑙𝑛(𝜂) = 𝑙𝑛(0.16) +

964.8 87.4𝑉 + 𝑇 𝑇

15.23

0.9280

[7]

Soy A Soy A

SMEA+ Diesel SMEB+ Diesel GMSME+ Diesel YGME+ Diesel

Cotton Seed Biodiesel + Babassu Biodiesel methyl oleate +low-sulfur petrodiesel methyl linoleate+lowsulfur petrodiesel

Andrade Equation 1922.8 550.3 + 2 𝑇 𝑇 2018.9 1752.8 𝑙𝑛(𝜂) = ‒ 5.09 + + 2 𝑇 𝑇 61.5 314530.4 𝑙𝑛(𝜂) = ‒ 2.1 ‒ + 2 𝑇 𝑇 1170.7 108.7𝑉 𝑙𝑛(𝜂) = 𝑙𝑛(0.079) + + 𝑇 𝑇 2056.6 78.1 𝑙𝑛(𝜂) = ‒ 5.23 + ‒ 2 𝑇 𝑇 91.4 337170.9 𝑙𝑛(𝜂) = ‒ 1.86 ‒ + 2 𝑇 𝑇 1100.6 144041.3 𝑙𝑛(𝜂) = ‒ 3.68 + + 2 𝑇 𝑇 689.7 117322.2 𝑙𝑛(𝜂) = ‒ 2.2 + + 2 𝑇 𝑇 807.4 148440.4 𝑙𝑛(𝜂) = ‒ 2.7 + + 2 𝑇 𝑇 2142.5 1584.3 𝑙𝑛(𝜂) = ‒ 5.4 + ‒ 2 𝑇 𝑇 843.1 161416.1 𝑙𝑛(𝜂) = ‒ 3.05 + + 2 𝑇 𝑇 1950.9 2207.9 𝑙𝑛(𝜂) = ‒ 5.1 + ‒ 2 𝑇 𝑇 1409.1 128.3𝑉 𝑙𝑛(𝜂) = 𝑙𝑛(0.028) + + 𝑇 𝑇 1603.6 198.5𝑉 𝑙𝑛(𝜂) = 𝑙𝑛(0.015) + + 𝑇 𝑇 1262.4 158.6𝑉 𝑙𝑛(𝜂) = 𝑙𝑛(0.044) + + 𝑇 𝑇 1366.4 182.1𝑉 𝑙𝑛(𝜂) = 𝑙𝑛(0.032) + + 𝑇 𝑇 2143.6 2791.7 𝑙𝑛(𝜂) = ‒ 5.3 + ‒ 2 𝑇 𝑇 1795.4 17834.2 𝑙𝑛(𝜂) = ‒ 5.07 + + 2 𝑇 𝑇 𝑙𝑛(𝜂) = ‒ 5.1 +

ARD%

R2

Ref.

2.20

0.9979

[41]

4.55

0.9982

[41]

0.565

0.9999

[40]

12.67

0.9390

[7]

1.56

0.9977

[8]

0.714

0.9997

[8]

12.11

0.9974

[8]

11.70

0.9904

[8]

8.65

0.9934

[8]

2.17

0.9969

[8]

4.15

0.9963

[8]

2.83

0.9994

[39]

7.27

0.9874

[38]

4.72 11.05 9.74 4.65 1.19

0.9753 0.9849 0.9816 0.9988 0.9998

[38] [38] [38] [37] [37]

Table 7. The generalize-ability analysis of the models for the new datasets not used during the models development. Ref.

𝑛

𝑧

T (K)

Exp.

FNN-GA

32

FNN-SA

FNN-LM

SVM

ANFIS

ACCEPTED MANUSCRIPT

Pure biodiesel [44]

w Linseed Biodiesel+ (1-w)Diesel w=0.516 [45]

w Corn Biodiesel+ (1-w)Diesel w=0.707 [45]

w Coconut Biodiesel + (1-w) Colza Biodiesel w=0.397 [40]

Fish [46]

Sunflower [46]

w Sunflower Biodiesel+ (1-w)Diesel w=0.616 [46] w Fish Biodiesel+ (1-w)Diesel w=0.419

1.242

17.582

283

9

8.482928

7.727363

8.565492

8.491521

8.218474

1.242

17.582

293

1.242

17.582

303

6.78

6.44063

6.139977

5.3

4.948701

4.898533

6.220207

6.66415

6.191424

4.710804

5.122183

4.8934

1.242

17.582

313

4.26

3.868261

3.934179

3.733437

3.864886

4.004777

1.242 1.242

17.582

323

3.51

3.0911

17.582

333

2.94

2.535195

3.188826

3.06247

2.891519

3.300699

2.614847

2.569791

2.201338

2.730824

1.242

17.582

343

2.51

2.139529

2.173991

2.19676

1.793596

2.336974

1.242

17.582

353

1.242

17.582

363

2.16

1.859298

1.835985

1.918916

1.667543

2.088642

1.9

1.661896

1.577137

1.720893

1.822425

1.862798

1.242

17.582

373

1.69

1.523746

1.379043

1.58672

2.257484

1.58079

1.242 1.242

17.582

383

1.51

1.427865

1.227491

1.500159

2.97196

1.239224

17.582

393

1.36

1.362061

1.11155

1.44741

3.96509

0.859743

1.242

17.582

403

1.23

1.317595

1.022832

1.418576

5.236106

0.46059

1.242

17.582

413

1.13

1.288212

0.954919

1.407714

6.784235

0.052233

1.492

17.914

293.15

4.2551

5.542549

5.226299

5.381001

5.548413

6.331283

1.492

17.914

303.15

3.3437

4.297136

4.178493

4.211428

4.131532

4.998473

1.492

17.914

313.15

2.703

3.393415

3.368972

3.407208

3.000527

4.118199

1.492

17.914

323.15

2.2399

2.741281

2.74638

2.82571

2.154682

3.413005

1.492

17.914

333.15

1.8905

2.272965

2.269139

2.392767

1.593283

2.753168

1.365

17.74

293.15

5.0913

5.864933

5.543614

5.814417

6.122832

7.369508

1.365

17.74

313.15

3.1749

3.568182

3.565597

3.749153

3.496534

4.550276

1.365

17.74

333.15

2.1864

2.375759

2.389696

2.735622

2.009668

3.063239

1.365

17.74

353.15

1.6078

1.769015

1.699083

2.049106

1.656405

2.312518

1.365

17.74

373.15

1.2332

1.466635

1.295836

1.440266

2.430874

1.852588

0.2376

14.0584

293.15

5.6734

3.718474

4.453408

4.222233

6.650801

6.887521

0.2376

14.0584

313.15

3.5026

2.183974

2.849596

2.887709

4.442965

5.132890

0.2376

14.0584

333.15

2.3938

1.4116

1.902291

1.798958

3.42771

3.329761

0.2376

14.0584

353.15

1.7494

1.033748

1.341948

0.83299

3.59919

2.974310

0.2376

14.0584

373.15

1.3351

0.856489

1.002071

0.438402

4.951504

2.341865

0.9586

17.1746

293.15

6.2801

6.399041

6.103666

6.666859

6.698226

8.280182

0.9586

17.1746

313.15

3.8107

3.851032

3.930853

3.897027

3.859681

5.252918

0.9586

17.1746

333.15

2.5688

2.534641

2.625356

2.519383

2.162983

3.531692

0.9586

17.1746

353.15

1.858

1.869397

1.849571

1.858726

1.602163

2.577382

0.9586

17.1746

373.15

1.4044

1.540222

1.390583

1.565881

2.171201

1.89915

1.536

17.858

293.15

6.0298

6.33262

6.205729

5.992084

6.420175

5.906141

1.536

17.858

313.15

3.7244

3.795361

3.952986

3.713519

3.706905

3.768702

1.536

17.858

333.15

2.5439

2.475294

2.611012

2.579556

2.124341

2.500645

1.536

17.858

353.15

1.8558

1.802936

1.823953

1.913985

1.666556

1.901861

1.536

17.858

373.15

1.4128

1.467684

1.366081

1.561631

2.327556

1.410975

1.536

17.858

293.15

4.9981

5.656939

5.427142

5.583719

5.81542

5.511

1.536

17.858

313.15

3.1035

3.450159

3.483597

3.572266

3.277399

3.620204

1.536

17.858

333.15

2.1126

2.298692

2.332643

2.562836

1.878332

2.320019

1.536

17.858

353.15

1.5492

1.709266

1.659962

1.919768

1.612454

1.209196

1.536

17.858

373.15

1.1945

1.413725

1.269417

1.340939

2.473955

0.607715

0.9586

17.1746

293.15

4.6096

5.271387

4.972845

5.119213

5.74398

1.052873

0.9586

17.1746

313.15

2.8469

3.236233

3.244503

3.217079

3.128906

1.159247

0.9586

17.1746

333.15

1.949

2.188546

2.214236

2.253608

1.667754

1.062311

33

ACCEPTED MANUSCRIPT [46]

0.9586

17.1746

353.15

1.4255

1.660285

1.605181

1.714239

1.354687

0.9748431

0.9586

17.1746

373.15

1.0896

1.399862

1.2463

1.341897

2.183827

0.7549627

34

ACCEPTED MANUSCRIPT

two-dimensional feature space separating hyper-plane

mapping ϕ

original one-dimensional space

Fig. 1. One-dimensional space projecting example into two dimensional space using SVM.

35

ACCEPTED MANUSCRIPT

Rule 1

X1

Rule 2 y

∑ X2

Rule 3

Rule 4

Fuzzifying crisp inputs

Evaluation of all rules (Obtaining antecedent and consequent parts of each rule)

Aggregation

Fig. 2. Schematic diagram of ANFIS structure.

36

Defuzzifying

ACCEPTED MANUSCRIPT 1

T, K X1 2 mass% X2

Viscosity

𝒛 3

X3

𝒏𝒅

X4 n

bn

Fig. 3. Feedforward neural network model architecture.

37

ACCEPTED MANUSCRIPT

Predicted viscosity

14

R2=0.9980

12 10 8 6 4 2 0 0

2

4

6 8 10 Experimental viscosity

12

14

Fig. 4. ANFIS training result for the viscosity of biodiesels. 70% of experimental data used to learn the nonlinear relations between variables.

38

ACCEPTED MANUSCRIPT

Predicted viscosity

14

R2=0.9932

12 10 8 6 4 2 0 0

2

4

6 8 10 Experimental viscosity

12

14

Fig. 5. ANFIS testing result for the viscosity of biodiesels. 30% of experimental data used to test the model against the completely unseen data.

39

ACCEPTED MANUSCRIPT 7 Viscosity, mPa.s

Exp., Cotton seed [41]

6

ANFIS model Ceriani

5

Yuan Revised Yuan

4

Krisnangkura

3

Andrade

2 1 290

300

310

320

330

340

350

360

370

Temperature, K Fig. 6. Comparison between the proposed model and experimental data of Cotton seed biodiesel [41] and the results of the theoretical models.

40

380

Viscosity, cSt

ACCEPTED MANUSCRIPT 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

Exp., coconut [39] ANFIS model Ceriani Yuan Revised Yuan Krisnangkura Andrade

295

300

305

310 315 Temperature, K

320

325

Fig. 7. Viscosity variation with temperature for Coconut biodiesel [39] and prediction accuracy of different models.

41

ACCEPTED MANUSCRIPT

Kinematic viscosity (mm2/s)

8

Exp., SMEA 75 [38]

7

ANFIS model Ceriani

6

Yuan Revised Yuan

5

Krisnangkura Andrade

4 3 2 1 290

300

310

320

330

340

350

360

370

380

Temperature, K

Fig. 8. Comparison between the predicted kinematic viscosity and experimental data for SMEA 75 biodiesel blended with No. 2 diesel [38] and the prediction results of four theoretical models.

42

ACCEPTED MANUSCRIPT

Kinematic viscosity (mm2/s)

8 Exp., GMSME 75 [38]

7

ANFIS model

6

Ceriani

5

Revised Yuan

Yuan Krisnangkura

4

Andrade

3 2 1 0 290

300

310

320

330

340

350

360

370

380

Temperature, K Fig. 9. Comparison between the predicted kinematic viscosity and experimental data for GMSME 75 biodiesel blended with No. 2 diesel [38] and the prediction results of four theoretical models.

43

Kinematic viscosity (mPa.s)

ACCEPTED MANUSCRIPT 4.5

Exp., Coconut [37]

4

ANFIS model

3.5

Ceriani

3

Yuan Revised Yuan

2.5

Krisnangkura

2

Andrade

1.5 1 0.5 290

300

310

320

330 340 350 Temperature, K

360

370

380

Fig. 10. Comparison between the predicted dynamic viscosity and measured data for Coconut biodiesel [37] with the prediction results of theoretical models.

44

ACCEPTED MANUSCRIPT

Kinematic viscosity (mm2/s)

13

Exp., ML 90 [7] ANFIS model Ceriani Yuan Andrade Revised Yuan Krisnangkura

12 11 10 9 8 7 6 5 4 3 272

277

282

287 292 297 Temperature, K

302

307

Fig. 11. Comparison between experimental and predicted values of different models regarding kinematic viscosity of 90 % methyl linoleate (ML) blended with diesel [7].

45

312

ACCEPTED MANUSCRIPT

Ceriani

Yuan

Revised Yuan

Krisnangkura

ANFIS

18 16 ARD%

14 12 10 8 6 4 2

Fig.12. Comparison of the performance of different models regarding prediction of the viscosity of biodiesels and a biodiesels blend.

46

] [3 3

su

[3 3

]

]

ab

as

se ed +B

n se ed

ot to C

ot

to

n

C

Ba

ba

ss

u

[3 2

[3 3

]

] ut

[6 d

[6 on oc C

R

ap

es

ee

G P

]

] [6 lm

Pa

w

er

[6 ]

[6 ] B1

nf lo Su

] [6 B

y So

A

[6 ]

] y

on oc C

So

ut

[3 1

[3 E

M G

Y

C

oc

on

ut

[2 9

]

0]

0

ACCEPTED MANUSCRIPT Ceriani

Yuan

Revised Yuan

Krisnangkura

ANFIS

14 12

ARD%

10 8 6 4 2

] 30 )[ d an

an

50

50

5,

5,

(2

(2 Y

G

M

E

E SM M G

% 75

75 d

75 d an 50 5, (2

EB SM

] % )[

30 %

)[

)[ % 75 an d 50 5,

(2 EA SM

30

]

] 30

[5 ] eu m ol et r

L+ P M

+P et O M

B+ P

et

ro

ro le um

le um

[5

[5

]

]

0

Fig.13. Comparison of the performance of different models regarding prediction of the viscosity of biodiesel-diesel blends.

47