Viscosity equations of pure fluids in an innovative extended corresponding states framework

Fluid Phase Equilibria 199 (2002) 281–294

Viscosity equations of pure fluids in an innovative extended corresponding states framework II. Application to four fluids G. Scalabrin a,∗ , G. Cristofoli a , D. Richon b b

a Dipartimento di Fisica Tecnica, Università di Padova, via Venezia 1, I-35131 Padova, Italy Laboratoire de Thermodynamique, Ecole Nationale Supérieure des Mines de Paris 35, rue St Honoré, 77305 Fontainebleau, France

Abstract This paper represents the second part of a work devoted to an innovative version of the historical extended corresponding states (ECS) technique for the development of viscosity equations on the whole ηTρ surface for individual fluids. In the first part [1], the theoretical aspects of modelling the ECS transport properties modelling have been discussed, and the fundamental characteristics of the proposed method have been tested, using viscosity values generated from conventional dedicated viscosity equations. The very promising results suggest a move from generated to experimental data correlation to determine dedicated viscosity equations for a number of fluids for which conventional equations are available in literature. This is done in this second part, where the fluids studied are the alkane ethane and the haloalkane refrigerants R123, R134a and R152a. The absolute average deviations (AADs) obtained with primary data are, respectively, 1.04, 1.13, 0.92 and 0.71% with a significant improvement with respect to the conventional equations. Considering that the expected experimental accuracy for good viscosity data is generally in the range 1–2%, the obtained results seem to be very promising. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Extended corresponding states; Neural networks; Viscosity dedicated equation; Ethane; R123; R134a; R152a; Refrigerants

1. Introduction In the first part of this work, an innovative extended corresponding states (ECS) model for the development of viscosity equations on the whole ηTρ surface of interest fluids is proposed [1]. The scope of the method is to obtain the viscosity of a target fluid distorting the viscosity equation independent ∗

Corresponding author. Tel.: +39-49-827-6875; fax: +39-49-827-6896. E-mail address: [email protected] (G. Scalabrin). 0378-3812/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 ( 0 2 ) 0 0 0 0 9 - 2

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variables of a reference fluid through shapes functions. As an alternative to the historical ECS method for transport properties, based on the generation of shape functions in the thermodynamic domain and their import into the transport properties modelling, the present method avoids the thermodynamic domain and makes it possible to obtain a single shape function exclusively based on viscosity data of the target fluid. A detailed analysis was performed in order to verify the most effective method for the independent variables distortion, showing that a single shape function based on a neural network technique is fully effective as function approximator of a viscosity surface. The very promising results suggest a move from generated to experimental data correlation to determine dedicated viscosity equations for a number of fluids for which conventional equations are available in literature. This also allows a comparison between the present and the conventional techniques in terms of the final representation accuracy. A synthetic description of the multilayer feedforward neural network (MLFN) architecture and formalism assumed for the shape function θ(Tr , ρr ) representation is presented in the following section to allow the new viscosity equations to be easily applied. 2. The ECS-NN model In the proposed ECS-neural networks (ECS-NN) model, the distortion of the reduced temperature and density is obtained with a single shape function: ηrx (Tr , ρr ) − ηr0 (Tr θ, ρr θ ) = 0

(1)

where the superscript x indicates the fluid of interest, superscript 0 the reference fluid, and the subscript r the reduced variables. Such a shape function θ (Tr , ρr ) has then to be generated from experimental data and its functional form is based on a neural network. The core of the problem is to fit the ECS model with the shape function θ(Tr , ρr ) on the available viscosity data. Among different neural network architectures, the MLFN with only one hidden layer seems to be the most effective as a universal approximator of continuous functions in a compact domain [2–4]. In this architecture, there are several neuron layers (multilayer) and the information goes in only one direction, from input to output (feedforward). Reference is here made to Fig. 4 of the first part of this work [1]. The I − 1 inputs Ui enter the I − 1 neurons of the input layer. The inputs Ui represent the independent variables of the problem, which in our case are the reduced temperature and density. The last neuron, at the number I, receives the Bias1. The J neurons of the hidden layer receive the weighted sum of signals from the input layer. A non-linear transfer function is applied to this sum. The neuron number J + 1 receives only the Bias2 value. If Hj is the output of the j hidden layer, this is 
I  Hj = f (2) wij Ui , 1 ≤ j ≤ J i=1

HJ +1 = Bias2

(3)

where f is the transfer function and wij are the weighting factors. Finally, the K neurons of the output layer receive the weighted sum of signals from the hidden layer and, once again, apply a non-linear transfer

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function to the sum. In our case K = 1 and the output Sk represent the dependent variable of the problem, which is the shape function θ (Tr , ρr ). Because Sk is the output of the output layer, i.e. the final output of the MLFN, it is   J +1  Sk = f  wjk Hj  , 1 ≤ k ≤ K (4) j =1

When J and f have been chosen, the weighting factors wij [I ×J ] and wij [(J +1)×K] can be fitted on the experimental viscosity datasets of the target fluid. This regression step is called the training step. Bias1 and Bias2 are two constants able to make the convergence easier and faster during fitting. The transfer function used in the present work has a sigmoid form: f (x) = α

1 1 + e−2βx

(5)

As a consequence of the choice of the transfer function, Eq. (5), Sk ≤ α for every k = 1, . . . , K. In addition, the training step is faster if all the inputs are of similar magnitude and for this aim filtering functions compressing and expending the variables are introduced. Indicating with V1 = T r , V2 = ρ r and W1 = ηr the present filtering equations are: ui =

Amax − Amin , Vmax,i − Vmin,i

i = 1, 2

(6)

sk =

Amax − Amin , Wmax,k − Wmin,k

k=1

(7)

Wk = sk (Sk − Wmin,k ) + Amin ,

k=1

(8)

Ui = ui (Vi − Vmin,i ) + Amin ,

i = 1, 2

(9)

U3 = Bias1

(10)

where J is the number of neurons in the hidden layer, Amin and Amax are the allowable range limits of the compressed input variables, Vmin,i and Vmax,i are the limits of the independent input variables for the training set, and Wmin,k and Wmax,k are the limit values of the output functions. Due to the characteristics of the present problem, the MLFN parameters are set here to the following values: I = 3, K = 1,

Bias1 = 1.0, Bias2 = 1.0,

Amin = 0.05, Amax = 0.95,

α = 1.0 β = 0.005

In this way the input variables and the output function have both been compressed in the range 0.05–0.95. To complete the MLFN definition, the following parameters have to be calculated for each target fluid through the training step: J, Vmin,i , Vmax,i , Wmin,k , Wmax,k , w ij , w j k . Since the MLFN architecture is always the same, its connotative contents are in general the number of hidden layers, the number of nodes, and the matrixes of weighting factors wij and w j k . It has been established that a single hidden layer suffices for representing a continuous function. The number J of neurons in the hidden layer has to be found by subsequent trials; this number has to minimise the residual error during the training procedure. In addition, for each number of hidden layer nodes, two matrixes wij and wj k of coefficients have to be

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found. The determination of the optimum number of hidden nodes and the fitting of the two matrixes wij , wj k are part of the training procedure. The weighting factors are found by minimising the following objective function by means of an optimisation procedure: 2 NPT  1  ηr0 [Tr,i θ (w), ¯ ρr,i θ(w)] ¯ − ηrx [Tr,i , ρr,i ] FOB = (11) NPT i=1 ηrx [Tr,i , ρr,i ] where NPT is the number of viscosity points, ηr0 the reduced viscosity of the interest fluid, ηrx the reduced viscosity of the target fluid, and θ (w) ¯ the shape function approximator. 3. Application of the ECS-NN model The good capabilities of the new method suggest that it should be applied directly to experimental data correlation for the development of dedicated viscosity equations. This procedure was then applied to a number of fluids for which conventional viscosity equations are available in the literature, allowing a comparison between the present and the traditional techniques in terms of the final representation accuracy. The fluids studied are ethane and the haloalkane refrigerants R123, R134a and R152a. For each of these compounds, multiparameter equations of state and viscosity equations are available, with references given in Table 1, together with the fundamental parameters needed for the models. Propane was selected as reference fluid and its parameters and references are indicated in Table 1 as well. Propane has been chosen due to the ranges of validity of the viscosity dedicated equation, which are wider than those of the fluids considered as target. The equations of state are not needed for the model itself, but for the conversion T , P → T , ρ in the independent variables of the experimental data. For the reference fluid propane, the equation-of-state is not required. Because the ECS-NN model relies on experimental data, the ranges which they cover give directly the ranges of validity of the individual viscosity equations obtained through the model. On this occasion, all the data available for each fluid were considered for the correlation of the shape function. We tested all available experimental viscosity data versus the conventional viscosity equation and included all data with deviations of less than 6%. The data screened in this way were considered as primitive. Using these data, a first neural network version was regressed for the shape function. After this preliminary screening, Table 1 Parameters for the studied fluids Fluid

PM (kg kmol−1 )

Tc (K)

Pc (MPa)

ρ c (kg m−3 )

ηc (␮Pa s)

Ref.a

Ref.b

R134a R152a R123 Ethane Propane

102.032 66.051 152.93 30.069 44.098

374.18 386.41 456.831 305.33 369.85

4.05629 4.5157 3.6618 4.8718 4.24766

508.0 368.0 550.0 206.57 220.49

25.179 21.644 27.851 15.976 17.102

[5] [7] [9] [11] −

[6] [8] [10] [12] [13]

a b

Thermodynamic equation-of-state. Viscosity dedicated equation.

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Table 2 Viscosity correlation by the ECS-NN equation for the whole surface and of the Krauss et al. conventional equation [6] for R134aa Phase

Range T (K)

Primary data l 273–333 sl 273–333 v 274–333 sv 273–333 v 308–403 sl 223–353 sl 241–350 l 258–361 v 297–438 v 298–423 sc 398–423 v 294–371 sl 238–343 l 293–343 sc 423 l 213–373

Range P (MPa)

0.95–14.6 ∗

0.14–1.47 ∗

0.1 ∗ ∗

0.32–6.44 0.03–0.32 0.1–3.93 4.18–5.63 0.28–0.59 ∗

0.66–51 10.1–29.8 1.58–30.2

Overall primary Secondary data l 200–300 sl 175–320 sl 233–333 sl 273–343 sl 251–343 v 233–600 sl 203–373 sl 250–306 sl 198–298 l 248–298 v 303–424 sc 423–424 sv 247–351 sv 253–353 v 253–483 sc 383–483 v 382–405 sc 382–405 Total Overall

0.62–33.7 ∗ ∗ ∗ ∗

0.1 ∗ ∗ ∗

0.73–100.1 0.1–4 4.47–6.42 ∗ ∗

0.1–4 4.5–11 0.66–3.41 4.25–6.23

Conventional equation from Krauss et al. [6]

Present model

AAD (%)

AAD (%)

Bias (%)

NPT

Ref.

[14] [14] [15] [15] [16] [17] [18] [19] [20] [21] [21] [22] [23] [23] [24] [24]

Bias (%)

Max (%)

Max (%)

2.17 2.82 2.72 2.72 2.99 2.76 1.88 1.11 1.29 1.47 0.88 2.16 3.65 0.94 0.76 4.90

−2.17 −2.82 2.72 2.72 2.99 −2.11 −1.76 −0.01 1.29 1.32 −0.58 2.16 −3.65 −0.86 0.73 −3.56

3.36 3.06 4.30 5.49 5.46 9.43 7.05 3.05 1.94 4.12 1.85 3.41 −5.09 3.44 2.39 19.24

1.56 2.37 0.74 1.18 0.57 0.82 0.99 0.95 0.45 0.65 0.80 0.36 1.20 0.88 1.03 1.66

−1.56 −2.37 0.29 0.31 −0.57 0.56 −0.29 0.88 −0.45 0.30 0.78 −0.17 −0.56 −0.88 1.03 0.93

2.71 3.11 2.18 3.73 1.10 2.27 2.92 3.15 0.65 2.44 1.70 1.29 3.56 3.56 1.49 5.02

32 7 22 7 6 14 91 23 71 113 13 37 17 43 8 73

2.14

−0.38

19.24

0.92

−0.04

5.02

577

11.13 16.10 3.80 5.09 9.67 2.02 4.07 4.69 15.95 4.86 2.96 12.24 2.45 2.38 2.36 11.26 2.26 2.15

−11.13 −16.10 −3.80 −5.09 −9.67 −0.86 −1.39 −4.69 −15.95 −4.81 −2.23 −12.24 −0.26 −0.22 −2.18 −11.26 2.00 −2.15

25.32 43.44 7.37 11.89 24.57 5.32 14.27 6.93 36.18 9.96 11.38 13.48 8.13 7.86 10.70 14.86 4.34 4.35

5.40 8.34 2.37 4.70 8.32 2.06 2.60 2.94 9.53 2.20 3.43 10.3 2.58 2.43 3.00 9.04 1.67 1.92

−5.40 −8.34 −1.51 −4.70 −8.32 −1.93 2.48 −2.94 −9.53 −2.18 −3.43 −10.33 −2.46 −2.39 −3.00 −9.04 1.67 −1.92

10.49 20.69 7.43 11.99 24.66 4.65 3.81 6.23 21.02 4.68 10.88 11.87 8.40 7.91 37.95 14.55 3.54 4.96

63 30 21 8 12 11 18 14 5 37 35 6 11 11 313 65 6 5 671

3.85

−2.86

43.44

2.67

−2.16

37.95

1248

[25] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [34] [35] [35] [35] [35] [36] [36]

a Error: AAD: average absolute deviation (%), Bias: bias (%), Max: maximum absolute error (%); NPT: number of experimental points. Phase: sl: saturated liquid, l: liquid, sv: saturated vapour, v: vapour, sc: super critical, zd: vapour at zero density limit.

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the first ECS-NN model was tested versus the primitive data. Avoiding the experimental points with deviations higher than 2%, a finer screening was done to identify the primary data, over which the final model was fitted. For each of the four fluids studied, the validation results of the individually obtained models are presented in Tables 2–5. The MLFN parameters of the shape function θ(Tr , ρr ) are reported for each fluid in Appendix A, together with the residual error of the training step.

Table 3 Viscosity correlation by the ECS-NN equation for the whole surface and of the Krauss et al. conventional equation [8] for R152aa Phase

Range T (K)

Primary data sl 243–373 sv 294–386 v 273–303 v 298–423 sc 398–423 sv 273–333 v 275–333 sl 273–333 l 273–333

Range P (MPa)

∗ ∗

0.1–0.68 0.1–4.45 4.52–5.28 ∗

0.19–1.24 ∗

1.45–17.7

Overall primary Secondary data sv 243–373 sv 243–373 sl 243–373 v 233–600 sl 113–383 sl 255–323 sl 200–318 v 298–323 sl 243–333 sv 249–386 sv 253–363 v 253–493 sc 393–493 sl 273–343 sl 273–343 sl 223–333 sl 242–351 Total Overall a

∗ ∗ ∗

0.1 ∗ ∗ ∗

0.1 ∗ ∗ ∗

0.1–4.5 5–12 ∗ ∗ ∗ ∗

Conventional equation from Krauss et al. [8]

Present model

AAD (%)

AAD (%)

Bias (%)

Max (%)

Bias (%)

Max (%)

NPT

Ref.

[37] [38] [39] [40] [40] [15] [15] [41] [41]

0.91 3.15 0.58 0.77 0.72 1.54 1.30 0.66 0.90

−0.35 3.15 0.41 0.0397 −0.11 1.54 1.30 −0.66 0.49

2.32 6.22 1.87 2.21 1.37 2.77 2.03 1.97 2.71

0.85 1.86 0.49 0.39 1.44 0.44 0.45 0.92 1.02

−0.07 1.63 −0.42 −0.04 −1.44 0.38 0.42 −0.60 0.17

2.80 4.97 1.02 1.15 2.33 1.70 1.44 2.55 2.34

35 19 49 99 14 7 21 7 32

0.98

0.42

6.22

0.71

−0.01

4.97

283

7.18 10.69 0.98 0.71 42.60 2.48 58.53 0.64 27.18 4.88 1.60 1.41 9.86 2.54 1.74 16.25 3.26

−3.02 −9.80 −0.71 0.26 −29.9 2.48 −45.9 −0.64 −27.2 −4.56 −1.26 −1.26 −9.86 −1.36 −1.68 16.25 −2.02

21.04 32.05 2.58 1.80 232.4 4.06 350.0 1.15 38.73 26.26 11.48 13.32 19.00 8.25 5.95 136.6 9.20

6.81 11.23 0.85 1.08 48.08 2.98 10.49 1.23 27.56 5.69 2.32 1.79 9.73 3.13 2.35 4.82 3.09

−3.95 −10.66 −0.40 1.08 48.02 2.98 5.58 −1.23 −27.56 −5.55 −2.18 −0.31 −9.73 −1.32 −1.64 4.66 −2.19

20.58 31.66 3.03 4.15 443.3 5.65 30.68 1.74 39.09 28.31 12.35 14.58 20.17 8.42 6.13 15.93 8.92

36 34 36 11 26 11 9 2 8 12 12 332 63 8 8 12 12 632

4.65

−2.94

350.0

4.28

−0.22

443.3

915

[37] [37] [42] [29] [30] [31] [43] [44] [45] [35] [35] [35] [35] [46] [27] [17] [47]

Error: AAD: average absolute deviation (%), Bias: bias (%), Max: maximum absolute error (%); NPT: number of experimental points. Phase: sl: saturated liquid, l: liquid, sv: saturated vapour, v: vapour, sc: super critical, zd: vapour at zero density limit.

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Table 4 Viscosity correlation by the ECS-NN equation for the whole surface and of the Tanaka and Sotani conventional equation [10] for R123a Phase

Range T (K)

Range P (MPa)

Primary data v 323–423 sl 170–320 v 308–363 sl 273–353 l 233–418 l 200–300

0.1–1.96 ∗

0.1 ∗

1.06–20.7 3.23–33.6

Overall primary Secondary data sc 473–523 v 303–523 sv 303–443 sv 301–433 v 303–423 l 293 Total

4–7 0.1–3.5 ∗ ∗

0.1–2.03 0.1

Overall

Conventional equation from Tanaka and Sotani [10]

Present model

NPT

Ref.

AAD (%)

Bias (%)

Max (%)

AAD (%)

0.25 5.87 0.69 1.68 1.89 6.09

−0.09 −5.85 0.47 −1.68 −0.31 −6.05

0.82 25.21 1.77 4.42 5.59 14.47

0.66 1.04 0.50 0.93 1.09 2.14

−0.04 −0.14 0.27 −0.30 0.88 −1.82

2.04 3.64 1.36 2.26 3.55 4.67

42 23 4 9 62 29

[48] [25] [16] [27] [49] [25]

2.70

−2.05

25.21

1.13

−0.03

4.67

169

4.63 3.05 2.38 2.76 1.51 2.25

−4.63 −3.05 −2.37 −2.76 −1.51 2.25

6.65 7.17 7.23 5.61 7.49 2.25

11.6 3.11 1.95 2.06 1.38 3.16

−11.6 −3.11 −1.89 −2.03 −1.38 3.16

14.67 9.25 6.87 5.09 6.23 3.16

25 244 15 15 31 1 331

2.89

−2.66

25.21

2.69

−2.30

14.67

500

Bias (%)

Max (%)

[35] [35] [35] [35] [34] [50]

a

Error: AAD: average absolute deviation (%), Bias: bias (%), Max: maximum absolute error (%); NPT: number of experimental points. Phase: sl: saturated liquid, l: liquid, sv: saturated vapour, v: vapour, sc: super critical, zd: vapour at zero density limit.

Table 5 Viscosity correlation by the ECS-NN equation for the whole surface and of the Hendl et al. conventional equation [12] for ethanea Phase

Range T (K)

Primary data l 193–303 l 298 v 319–499 sc 319–499 sc 305–477 l 299 v 324–408 sc 324–408 zd 258–523 zd 301–476 zd 296–303 zd 298–348

Range P (MPa)

0.17–4.79 4.48–55.16 1.71–4.55 5.16–54.89 4.90–36.03 4.38–35.81 0.69–3.45 6.89–55.16 ∗ ∗ ∗

0.10

Conventional equation from Hendl et al. [12]

Present model

AAD (%)

AAD (%)

1.42 0.42 1.88 1.49 0.72 0.38 0.90 1.90 0.36 0.70 0.10 0.23

Bias (%) 0.54 0.37 −0.18 −0.55 −0.51 0.07 0.16 −0.39 0.16 −0.65 0.10 0.21

Max (%) 3.99 0.98 2.97 5.86 3.13 1.26 1.40 4.51 1.06 1.05 0.12 0.51

1.65 0.43 1.87 1.34 0.98 0.73 1.09 1.53 0.40 0.70 0.29 0.47

Bias (%) 0.05 −0.07 0.21 −0.44 −0.05 −0.25 0.79 0.57 0.40 −0.57 0.29 0.47

NPT

Ref.

13 16 7 56 135 26 8 32 10 5 2 16

[51] [52] [53] [53] [55] [55] [56] [56] [57] [58] [59] [60]

Max (%) 3.92 1.23 3.64 4.29 3.92 1.21 1.76 4.94 0.92 0.99 0.33 0.66

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Table 5 (Continued ) Phase

zd zd v sc sl sc

Range T (K)

293–633 298–468 298–444 310–444 95–300 320

Range P (MPa) ∗

0.10 0.93–4.14 5.51–55.16 ∗

5.25–31.03

Overall primary Secondary data sc 305–322 v 288–473 sc 313–473 l 288–303 v 273–523 sc 312–523 l 258–304 v 298–348 sc 305–348 l 101–167 l 172–288 l 111–164 l 294 l 100–290 v 299–477 zd 212–393 v 290–320 zd 194–273 zd 290–523 zd 293–393 zd 303 zd 195–273 zd 273 zd 308–351 zd 296 zd 305–408 zd 250–473 zd 339 zd 373–673 zd 293–373 Total Overall

4.88–6.57 0.69–4.14 5.17–34.47 4.15–34.47 2.03–4.053 5.07–81.06 2.03–81.06 0.20–4.59 4.87–12.94 0.10 0.10–3.46 0.20 7.64–51.91 0.30–32.11 0.10–4.86 ∗

0.60–4.76 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗

Conventional equation from Hendl et al. [12]

Present model

AAD (%)

AAD (%)

Bias (%)

Max (%)

Bias (%)

NPT

Ref.

[62] [63] [52] [52] [54] [54]

Max (%)

0.09 0.35 2.07 0.69 0.73 1.30

0.09 −0.32 −0.90 −0.20 −0.18 −0.80

0.19 0.65 5.24 3.61 2.43 2.16

0.25 0.34 1.43 1.09 1.13 0.77

−0.06 −0.23 −0.45 0.73 −0.38 −0.49

0.73 0.56 3.80 3.02 2.28 1.51

12 5 24 41 42 19

0.95

−0.32

5.86

1.04

−0.02

4.94

469

7.77 3.43 10.06 8.38 3.60 6.12 5.53 2.18 1.44 1.53 10.47 26.79 0.60 1.48 1.41 0.58 3.77 3.95 0.50 1.93 2.20 3.35 3.20 0.76 0.80 1.12 1.36 1.15 1.36 1.95

7.77 3.00 10.04 8.38 −0.45 5.94 5.53 2.04 0.42 −0.27 4.43 26.79 −0.45 −1.06 −1.25 −0.15 −3.51 −3.95 0.50 −1.93 −2.20 −3.35 −3.20 −0.76 −0.80 −1.12 −1.36 −1.15 −1.36 −1.95

13.50 10.51 22.17 10.85 14.59 16.43 7.34 7.14 5.23 4.24 18.87 56.41 1.15 73.84 4.88 1.01 5.98 7.16 1.23 2.29 2.20 6.59 3.20 1.21 0.80 3.05 2.33 1.15 1.92 2.26

7.89 3.95 10.83 7.78 4.11 6.01 4.95 1.86 1.56 1.90 10.02 34.10 1.09 2.31 1.57 0.64 3.30 2.53 0.55 1.82 2.03 2.24 2.82 0.65 0.57 1.07 1.22 1.07 1.83 1.84

7.89 3.54 10.76 7.78 0.01 5.83 4.95 1.86 0.79 −0.07 3.77 34.10 −1.09 −0.95 −1.50 0.64 −3.30 −2.53 0.55 −1.82 −2.03 −1.96 −2.82 −0.65 −0.57 −1.03 −1.22 −1.07 −1.83 −1.84

14.13 11.60 23.61 10.65 14.48 13.59 7.49 4.87 4.83 5.48 18.84 74.22 1.56 74.42 5.91 0.98 5.16 4.72 1.07 2.22 2.03 4.19 2.82 1.14 0.57 2.99 2.02 1.07 3.23 2.19

18 63 56 23 24 82 62 145 256 13 11 9 8 83 65 7 20 2 6 6 1 2 1 4 1 13 19 1 3 9 1013

2.66

1.42

73.84

2.79

1.62

74.42

1482

[64] [65] [65] [65] [57] [57] [57] [60] [60] [66] [67] [67] [53] [54] [55] [61] [54] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80]

a Error: AAD: average absolute deviation (%), Bias: bias (%), Max: maximum absolute error (%); NPT: number of experimental points. Phase: sl: saturated liquid, l: liquid, sv: saturated vapour, v: vapour, sc: super critical, zd: vapour at zero density limit.

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As already mentioned, the present selection of the primary data for the three refrigerants R134a, R152a and R123 has led to the same composition of the datasets used for the developments of the conventional viscosity equations. Even if the separation of the points of a single data source into primary and secondary classes would be possible using the present correlative model, for them it was decided to keep integral each data source. For ethane the data set screening, performed with the present model, has led to a selection of the primary data sources slightly different from that of the conventional equation. The validation of the conventional equation was performed following this new selection. The residual error of the four ECS-NN dedicated viscosity equations, in terms of AAD%, is in general much lower that of the conventional equations. It is particularly interesting to note that the primary data Bias%, for each of the four new equations, is very close to zero, demonstrating that the viscosity surface is perfectly centred with respect to the error noise of the primary data. The same cannot be said for the conventional equations. The obtained AAD and Bias values can be considered as evidence of the great flexibility of the present correlative model.

4. Conclusions The present work demonstrates that the original ECS modelling for transport properties can be fundamentally modified through the proposed ECS-NN method with the evident advantages of: • • • •

avoiding the thermodynamic shape functions calculation; reducing the ECS model to a single shape function; obtaining a totally correlative method based only on experimental data; reaching a higher accuracy and flexibility in the viscosity surface representation.

Table 6 Comparison of the results obtained by the conventional equations and the present ECS-NN model for the four studied fluidsa Fluid

Range T (K)

Range P (MPa)

Primary data Ethane R123 R134a R152a

193–633 170–523 175–600 113–600

Overall data Ethane R123 R134a R152a

193–633 170–523 175–600 113–600

a

Conventional equation AAD (%)

Bias (%)

0.01–81.0 0.1–33 0.02–100 0.1–17.7

0.95 2.70 2.14 0.98

−0.32 −2.05 −0.38 0.42

0.01–81.0 0.1–33 0.02–100 0.1–17.7

2.66 2.89 3.85 4.65

1.42 −2.66 −2.86 −2.94

Present model Max (%)

NPT

AAD (%)

Bias (%)

Max (%)

5.86 25.21 19.24 6.22

1.04 1.13 0.92 0.71

−0.02 −0.03 −0.04 −0.012

4.94 4.67 5.02 4.97

469 169 577 283

73.84 25.21 43.44 350.0

2.79 2.69 2.67 4.28

1.62 −2.30 −2.16 −0.22

74.42 14.67 37.95 443.3

1482 500 1248 915

Error: AAD: average absolute deviation (%), Bias: bias (%), Max: maximum absolute error (%); NPT: number of experimental points.

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The proposed model appears to be a suitable method for developing viscosity equations of high accuracy, totally based on experimental data. In addition, it is also shown [1] that a few tens of data points, regularly distributed on the ηTρ surface of the target fluid, are sufficient to draw a very precise equation, with evident saving of experimental efforts. The non-theoretical and completely heuristic nature of the model also allows its application to the statistical screening of the experimental data. Furthermore, the variables conversion T , P → T , ρ does not necessarily require an equation-of-state, as is the case for the ECS transport property model, and this can be carried out by a density model like the one recently presented by Scalabrin and co-workers [81,82] for the refrigerants family of fluids. The overall results for the four studied fluids are summarised in Table 6, where the comparison with the conventional equations is also reported. In the same Table, the ranges of availability of the primary data and then of validity of the ECS-NN equations are reported. A total of 4145 point have been processed for the four fluids, with very promising results, in particular if compared with the conventionally accepted experimental accuracies for this property. List of symbols Amin , Amax f(x) FOB Hj NPT P R Sk T Ui Vmin,i , Vmax,i w ij , wj k Wmin,k , Wmax,k

allowable range limits of the compressed input variables transfer function objective function output of j neuron in the hidden layer number of data points pressure universal gas constant output of the output layer temperature inputs limits of the independent input variables for the training set weighting factors limit values of the output dependent variable for training

Greek letters φ η θ ρ

shape factor viscosity shape factor density

Superscripts 0 x

reference fluid fluid of interest

Subscripts c r

critical reduced quantity

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