ISSN 0036-0244, Russian Journal of Physical Chemistry A, 2020, Vol. 94, No. 1, pp. 30–40. © Pleiades Publishing, Ltd., 2020.
CHEMICAL THERMODYNAMICS
AND THERMOCHEMISTRY
Hyperbolic Correlation Between the Viscosity Arrhenius Parameters
at Liquid Phase of Some Pure Newtonian Fluids and their Normal
Boiling Temperature
E. Mlikia,b, T. K. Srinivasac,*, A. Messaâdid, N. O. Alzamela,b, Z. H. A. Alsunaidia, and N. Ouerfellia,b
aDepartments
of Mathematics and Chemistry, College of Science, Imam Abdulrahman Bin Faisal University,
P.O. Box 1982, Dammam, 31441 Saudi Arabia
b
Basic and Applied Scientific Research Center, Imam Abdulrahman Bin Faisal University,
P.O. Box 1982, Dammam, 31441 Saudi Arabia
cDepartment of Physics, A.S.N Women’s Engineering College Tenali, Andhra Pradesh, 522 201 India
dUniversité de Tunis El Manar, Laboratoire de Biophysique et Technologies Médicales,
LR13ES07, Institut Supérieur des Technologies Médicales de Tunis, Tunis, 1006 Tunisia
*e-mail: sritadikonda@gmail.com
Received April 29, 2019; revised April 29, 2019; accepted May 14, 2019
Abstract—Viscosity is the most important characteristic of hydraulic fluid and which is influenced by pressure and temperature. Using statistical methods for correlation analysis and regression, eventual causal relationship between parameters of the Arrhenius-type equation and principally the boiling point of some classical Newtonian liquids is attempted. Empirical validation utilizing data on viscositabty of pure Newtonian liquids studied at atmospheric pressure and at different ranges of temperature gives excellent statistical results.
Indeed, we found a significant strong correlation between the Arrhenius activation energy (Ea), the Arrhenius
temperature (TA) and the boiling point (Tb). Consequently, an original hyperbole-type equation modeling
this relationship is proposed which allows the prediction of the boiling temperature and the type of isobaric
liquid-vapor diagram through information on viscosity Arrhenius parameters values, only in liquid phase,
and will be thus very useful in the handling of engineering data especially for the study of hydraulic components and systems efficiency.
Keywords: Viscosity, Arrhenius behavior, Boiling temperature, Newtonian liquids, Correlation
DOI: 10.1134/S0036024420010239
Effectively, this will be very gainful for hydraulic
fluid properties simulation in the design and optimization of several industrial processes, such as in Food
Industry, Chemical Industry, Pharmaceuticals, Cosmetics and Hydraulic-Mechanics, etc.
INTRODUCTION
Liquids viscosity is one key transport property
embroiled in chemical engineering, process design
and development [1–6]. Numerous empirical and
semi-theoretical equations of liquid viscosity have
been suggested based generally on three principal theories: the theory of reaction rate [7–9], the molecular
dynamic approach [10] and the distribution function
theory [11]. Several semi-empirical expressions have
been proposed in literature [12–17] and investigated in
previous works to discuss the viscosity-temperature
dependence [18–22].
The present study focuses on analyzing the existence of any eventual causal association between the
Arrhenius-type equation parameters and boiling
points of some classical Newtonian liquids. This relationship may allow estimating these phase change
temperatures through the study of variation of viscosity versus temperature of a pure solvent only in its liquid phase.
LITERATURE REVIEW ON THE LIQUID
VISCOSITY-TEMPERATURE DEPENDENCE
Explicit Dependence and Newtonian Behavior
As the temperature increases, the thermal agitation
increase and consequently the mean free path
decreases. Hence, numerous expressions, generally
classified under three types: two-, three- and multiconstant equations, have been suggested in the literature for describing the liquid shear viscosity (η) versus
temperature (T) through available experimental data
of literature for interpolation methods.
30
HYPERBOLIC CORRELATION BETWEEN
The simplest form of representation of Newtonian
fluid viscosity versus temperature is a relationship with
1 two parameters where the neperian logarithm form
becomes the most popular and it is the so called
Arrhenius type-equation which may be linearly formulated as follows [18–22]:
()
Ea 1
(1)
,
R T
where R is the perfect gas constant, As and Ea, the preexponential factor and the Arrhenius activation energy
for the pure fluid system respectively.
ln η = ln As +
For the Newtonian fluids not obeying to the viscosity Arrhenius behavior, the most popular nonlinear
equation is suggested firstly by Vogel and lately known
as Vogel-Fulcher-Tammann-type model [18–22]:
B ,
T −C
where A, B and C are constants.
ln η = A −
(2)
Notice that in this work we focus on the Arrhenius
type-equation for studying any eventual relationship
between its parameters and boiling points (Tb) of some
classical Newtonian liquids in the hope to predict its
value through information on viscosity Arrhenius
parameters only in liquid phase.
Implicit Dependence and Derived Temperatures
From thermodynamic quantities and similar
amounts, certain specific temperatures can be determined by manipulating certain derivatives as well as
those of the Maxwell equations. Noting that there are
certain common points with the definition of the
enthalpy of vaporization and the enthalpy of activation
viscous flow (ΔH*) as well as the viscosity activation
energy (Ea) [23–37]. This leads us to think that the
specific temperatures derived from the previous quantities are causally correlated with the boiling temperature of the fluid that we studied its dynamic viscosity.
It is in this context that we will present certain correlations and try to reduce a way of predicting the boiling
temperature.
Among the derived temperatures we cite the Arrhenius temperature (TA) of a pure liquid (Eqs.1 and 3),
the Arrhenius mean temperature (TAm) of a liquid
binary mixture (Eq. 4), the current temperature of
Arrhenius (TAc) deduced from partial molar magnitudes (Eq. 5) and the thermodynamic temperature
(T#) deduced from Maxwell’s equations (Eq. 11).
TA = −
Ea
.
R ln As
(3)
quasi-linear behavior where the slop gives the Arrhenius mean temperature (TAm) expressed as follows:
Ea
(4)
=TAm ln As + B.
R
Where B is the intercept of the ordinate when ln(As) is
mathematically null and it is thoroughly related to the
shear viscosity of the binary fluid system at boiling
point; we remark that the Tam value is close to the
boiling temperature Tb of one of the two pure constituting components of the binary mixture [23–37].
Nonetheless, if we eliminate the hypothesis of
Eq. (4) that the Arrhenius mean temperature (TAm)
nothing is more constant over the whole domain of
molar composition, we can review it as variable Arrhenius’ current temperatures (TAci ) related to the pure
component (i). It can be determinate from the following partial derivatives at selected molar composition
(x1):
−
⎛ ∂(Eai ) ⎞
(5)
TAci = ⎜
⎟ .
⎝ ∂(−R ln( Asi )) ⎠P
Whereas in several cases of previously studied binary
liquid mixtures, we have observed a strong correlation
existing between the limiting Arrhenius’ current temperatures (TAci ) and the vaporization temperature of
the isobaric liquid vapor equilibrium of the corresponding binary system [23–37]. The following equation asserts all these observations,
⎛ ∂(Eai ) ⎞
⎜
⎟
∂xi ⎟
(6)
lim ⎜
≈ −RΔTbi ,
xi →1 ⎜ ∂ ( ln A ) ⎟
si
⎜ ∂x ⎟
⎝
⎠
i
and from which we have estimated in previous works
with a good approximation the boiling point (Tbi) of
some pure liquid constituents (i).
In addition, the free energy (ΔG*) of activation of
viscous flow are related as follows [7–9, 38]:
⎛ η( x ,T , P )V ( x1,T , P ) ⎞
ΔG *( x1,T , P ) = RT ln ⎜ 1
⎟ , (7)
hN A
⎝
⎠
where R, η, h, NA and V are the perfect gas constant,
dynamic viscosity of binary fluid system, Plank’s constant, Avogadro’s number and molar volume of fluid
system at molar composition (x1), absolute temperature T and pressure P respectively, and:
(8)
ΔG * = ΔH * − T ΔS *.
Where ΔH* and ΔS* are the enthalpy and the entropy
of activation of viscous flow of binary fluid system at
molar composition (x1) and can be calculated in the
general case as follows:
⎛ ∂(ΔG */T ) ⎞
ΔH * = ⎜
⎟ ,
⎝ ∂(1/T ) ⎠P
Generally for binary mixtures the plot (Ea) as a function of (ln As) at different compositions exhibit a
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A
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MLIKI et al.
100
1000
T*
Tm
TA
Ti/K
Tb
100
1000
Fig. 1. (Color online) Classification of different mean temperatures (Ti ) used in this statistical investigation. Great vertical bar (|):
Average value and the small vertical bar (l): delimitation of the Confidence Interval (CI).
⎛ ∂(ΔG *) ⎞
ΔS * = ⎜
⎟ .
⎝ ∂(T ) ⎠P
(10)
Moreover, considering the partial derivatives functions of Maxwell equations and the Gibbs free energy
expression (Eq. (7)), we can determinate the thermodynamic temperature (T#) at constant pressure by the
partial derivative (ΔH*) with respect to (ΔS*) as follows:
⎛ ∂(ΔH *) ⎞
#
⎜
⎟ =T .
⎝ ∂(ΔS *) ⎠P
(11)
RESULTS AND DISCUSSION
Arrhenius and Temperature Parameters
A sample of 103 experimental data provided previous works and from the literature [14–19, 23–37] on
viscosity of pure Newtonian liquids studied at atmospheric pressure and at different temperature ranges is
used in this paper in order to analyze the existence of
any eventual correlation between parameters of the
Arrhenius type-equation, such as the activation energy
(Ea) and the entropic factor (ln As), and the melting
temperature (Tm) and boiling temperature (Tb).
Table 1 presents the experimental values of these
parameters. Also, the table provides experimental values of additional temperature parameters which are
the Arrhenius temperature (TA) and the Arrhenius
activation temperature (T*= Ea/R) as introduced by
Ouerfelli et al. [23–35]. We note that the existence of
some samples’ repetitions, that’s because the data are
provided from different references of the literature and
it are realized at different ranges of temperature, which
it’s statistically considered as different and independent observations and it will well enrich the investigated set of samples, and lead to reliable statistical dispersion.
Table 2 reports the principle descriptive statistics of
temperature parameters for the 103 studied observations, such as the Arithmetic mean (Ti ), the Confidence Interval (CI) of the mean and the coefficient of
variation (CV) from which we can admit the following
classification, also show by Fig. 1 which confirms that
there is no net intersection between any consecutive
(CI).
TA < Tm < Tb < T *.
Moreover, according to the (CV) values, the (T*)parameter is the most dispersed variable, inversely to
the boiling temperature (Tb) which is the most
2
homogenous.
Analysis of Correlation
Firstly, we investigate direct mutual correlation
between the three parameters of Arrhenius (TA, ln As
and Ea) and, the melting temperature Tm and boiling
temperature Tb. For that, in order to measure and test
any eventual association between the target parameters, we have applied the correlations tests of Spearman’s rank [39], where the null hypothesis supposes
the independence of the variables. Table 3 presents the
result of the test for all possible pairs. Also, we present
in Fig. 2 the pairwise scatter plots, which is a good
graphical method to show any possible correlation.
Based on the graphs of Fig. 2, we deduce that there
is no clear direct relationship between any of the pairwise analysis. In addition, the results of the Spearman’s rank correlations tests show that there is no
relationship between any of the target parameters.
Thus, we deduce that there is no direct correlation
between the Arrhenius parameters (TA, ln As and Ea)
and the temperatures Tm and Tb.
Nevertheless, instead of that, we think that a complicated association may exist. For that, we have tried
several possible complicated relationships between
two or more transformed parameters from which we
found the most interesting result indicating a strong
nonlinear correlation (Fig. 3) may exist between Ea
and (1/TA – 1/Tb). Regarding results given in Figs. 1
and 2b, Tables 2 and 3, and the finite values of the two
parameters of Arrhenius straight line (Eq. 1) whether
T tends to zero or to infinity, we can ascertain that the
curvature of Fig. 3 mathematically admits two asymptotes, one vertical very near of zero and a horizontal
one at infinity.
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A
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HYPERBOLIC CORRELATION BETWEEN
33
Table 1. Arrhenius parameters of some pure solvents studied at previous works and from the literature [14–19, 23–37],
Arrhenius activation energy Ea (kJ mol–1), the logarithm of the entropic factor ln(As/Pa s), the melting point (Tm/K), the
boiling point (Tb/K), the Arrhenius temperature (TA/K) and Arrhenius activation temperature (T*/K)
No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
Tm
Tb
lnAs
Ea
TA
T*
K
K
–
kJ mol–1
K
K
257.15
273.15
288.35
191.15
165.15
183.15
158.15
150.05
260.65
279.62
178.15
182.60
278.65
228.15
178.15
248.65
182.55
143.45
180.15
225.35
250.23
189.55
156.85
289.75
178.45
183.35
266.85
257.15
214.15
293.15
159.15
293.15
293.15
159.15
159.15
214.15
212.15
275.65
253.15
468.15
373.15
478.45
289.75
339.15
390.85
372.15
351.65
475.75
353.89
341.88
371.15
353.15
405.15
409.15
417.15
371.15
309.25
383.75
412.25
349.87
350.15
307.75
391.15
329.20
390.85
457.28
468.15
461.35
503.15
465.15
503.15
503.15
465.15
465.15
461.35
425.00
483.15
438.55
–14.945
24.955
200.83
3001.4
–13.414
–16.486
–15.048
–10.474
–13.925
–15.860
–10.699
–12.404
–12.831
–11.798
–12.367
–11.812
–10.695
–11.027
–11.145
–11.302
–10.886
–11.135
–10.975
–11.152
–11.728
–11.446
–11.308
–11.097
–13.689
–13.564
–14.945
–22.128
–16.210
–20.681
–16.438
–16.485
–21.510
–21.857
–18.266
–10.780
–12.442
–10.914
15.835
26.777
20.025
6.9072
19.742
24.852
7.2626
15.142
14.461
9.1501
11.292
10.940
8.7094
9.1110
9.8360
8.6196
6.0998
9.0229
8.7485
10.329
9.9183
7.5203
11.213
7.4406
19.114
19.997
24.955
47.765
33.359
43.910
33.904
34.033
45.933
46.763
37.551
9.0530
16.410
9.7590
141.98
195.34
160.05
79.310
170.52
188.46
81.639
146.82
135.56
93.281
98.023
111.39
97.939
99.375
106.14
91.723
67.393
97.461
95.872
111.39
101.72
79.021
119.26
80.643
167.94
177.32
200.83
259.62
247.51
255.36
248.07
248.31
256.84
257.32
247.25
101.00
158.63
107.54
1904.5
3220.5
2408.5
830.74
2374.4
2989.0
873.49
1821.2
1739.3
1100.5
1358.1
1315.8
1047.5
1095.8
1183.0
1036.7
733.64
1085.2
1052.2
1242.3
1192.9
904.48
1348.6
894.9
2298.9
2405.1
3001.4
5744.8
4012.2
5281.1
4077.7
4093.3
5524.5
5624.3
4516.3
1088.8
1973.7
1173.7
Pure Solvent
n-octanol
Water
Benzyl alcohol
Ethylamine
Tetrahydrofuran
1-butanol
2-butanol
1-chlorobutane
Methyl benzoate
Cyclohexane
n-hexane
Heptane
Benzene
Chlorobenzene
ethylbenzene
o-xylene
n-heptane
n-pentane
Toluene
m-xylene
Carbone tetrachloride
Ethylacetate
Diethylether
Aceticacid
Acetone
Butyl Alcohol
Aniline
n-octanol
Propylene Glycol
Butane-1,4-diol
Butane-1,2-diol
1,4-Butanediol
1,4-Butanediol
1,2-Butanediol
1,2-Butanediol
Propylene Glycol
N,N-dimethylformamide
Formamide
N,N-dimethylacetamide
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A
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MLIKI et al.
Table 1. (Contd.)
No.
Tm
Tb
lnAs
Ea
TA
T*
K
K
–
kJ mol–1
K
K
Pure Solvent
40
41
42
2-Methoxyethanol
Water
N,N-dimethylacetamide
188.15
273.15
253.15
397.65
373.15
438.55
–12.602
–13.284
–10.896
15.185
15.510
9.4260
144.93
140.42
104.05
1826.3
1865.4
1133.7
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
2-Ethoxyethanol
N,N-dimethylacetamide
1,4-dioxane
Water
Isobutyric acid
Water
Ethanol
Water
Methanol
Water
p-xylene
Dimethylsulfoxide
o-xylene
Dimethylsulfoxide
Ethylene glycol
1,4-dioxane
Water
Triethyl amine
Water
Glycerol
TEGMME*
n-Heptane
Propargylalcohol
Allylalcohol
t-butanol
2-propanol
1-propanol
Bromobenzene
Chlorobenzene
Ethylbenzene
Benzene
Dimethylsulfoxide
3-amino-1-propanol
Isoamylalcohol
2-propanol
Ethanol
1,4-dioxane
N-methylacetamide
183.15
253.15
284.15
273.15
226.15
273.15
159.15
273.15
175.55
273.15
285.65
290.65
248.65
290.65
260.15
284.15
273.15
158.15
273.15
293.15
229.15
182.15
220.15
144.15
298.84
183.65
149.15
242.15
228.15
178.15
278.65
290.65
284.15
156.15
183.65
159.15
284.15
300.15
408.15
438.55
374.15
373.15
426.65
373.15
351.15
373.15
337.75
373.15
411.15
462.15
417.15
462.15
470.15
374.15
373.15
361.95
373.15
455.15
395.15
371.15
387.65
370.15
355.55
355.15
370.15
429.15
405.15
409.15
353.15
462.15
458.65
403.15
355.15
351.15
374.15
478.15
–12.682
–10.934
–11.853
–13.443
–11.200
–13.383
–12.166
–13.232
–11.528
–13.334
–10.761
–11.872
–11.044
–12.002
–16.146
–11.430
–13.742
–11.248
–13.389
–24.143
–13.638
–12.613
–12.607
–12.866
–18.476
–15.032
–13.415
–13.450
–10.406
–10.807
–13.254
–10.975
–18.036
–14.322
–16.403
–12.997
–11.607
–13.155
15.803
9.7973
12.660
15.920
11.126
15.749
13.204
15.433
9.9340
15.640
8.3920
14.058
9.5730
14.333
29.941
11.669
16.684
8.2120
15.786
59.608
21.245
14.344
15.105
15.305
32.029
21.950
17.792
16.771
7.9331
8.5016
14.955
11.721
36.059
21.640
25.430
15.680
12.074
19.128
149.87
107.77
128.47
142.43
119.48
141.54
130.50
140.28
103.64
141.07
93.800
142.42
104.26
143.63
223.03
122.78
146.02
87.810
141.80
296.95
187.36
136.78
144.10
143.07
208.50
175.63
159.51
149.98
91.687
94.617
135.71
128.44
240.46
181.72
186.46
145.10
125.11
174.88
1900.7
1178.3
1522.6
1914.7
1338.1
1894.2
1588.1
1856.2
1194.8
1881.1
1009.3
1690.8
1151.4
1723.9
3601.1
1403.5
2006.6
987.68
1898.6
7169.2
2555.2
1725.2
1816.7
1840.8
3852.2
2640.0
2139.9
2017.1
954.13
1022.5
1798.7
1409.7
4336.9
2602.7
3058.5
1885.9
1452.2
2300.6
RUSSIAN JOURNAL OF PHYSICAL CHEMISTRY A
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HYPERBOLIC CORRELATION BETWEEN
35
Table 1. (Contd.)
No.
Tm
Tb
lnAs
Ea
TA
T*
K
K
–
kJ mol–1
K
K
Pure Solvent
81
82
83
84
85
2-Methoxyethanol
Water
Propylene carbonate
1,2-diethoxyethane
Acetonitrile
188.15
273.15
218.15
199.15
222.15
397.65
373.15
513.15
394.15
354.65
–12.662
–13.307
–11.729
–10.819
–10.793
14.504
15.568
14.192
7.5690
6.9895
137.77
140.70
145.53
84.142
77.885
1744.4
1872.4
1706.9
910.34
840.64
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
Tetrahydrofuran
Methanol
Octane
Pyridine
Benzene
Water
Propylene Glycol
N, N-dimethylacetamide
Water
2-propanol
2-mehoxyethanol
Isoamyl alcohol
Glycerol
1,4-Dimethylbezene
n-decane
n-eicosane
Benzene
n-tetracosane
165.15
175.55
216.15
232.00
278.65
273.15
214.15
253.15
273.15
183.65
397.65
156.00
563.00
286.30
243.30
310.00
278.65
324.00
339.15
337.75
398.75
388.55
353.15
373.15
461.35
438.55
373.15
355.15
397.65
403.15
455.15
411.15
447.25
616.25
353.15
664.55
–10.393
–11.629
–12.886
–12.635
–9.7233
–11.900
–18.266
–10.914
–13.443
–15.030
–12.822
–14.320
–24.140
–10.760
–11.355
–12.675
–13.250
–12.253
6.7382
10.198
13.275
14.925
5.5473
12.043
37.551
9.7590
15.920
21.950
15.704
21.640
59.610
8.3920
10.633
18.630
14.960
18.962
77.977
105.47
123.90
142.08
68.618
121.71
247.25
107.54
142.43
175.63
147.31
181.72
296.95
93.800
112.63
176.78
1135.7
186.13
810.42
1226.5
1596.6
1795.1
667.20
1448.4
4516.3
1173.7
1914.7
2640.0
1888.8
2602.7
7169.4
1009.3
1278.9
2240.7
1799.3
2280.6
* TEGMME is an abbreviation of triethylene glycol monomethyl ether.
Thus, using statistical modeling techniques to find
the expression which best fits this relationship, we suggest the following general equations relying Ea and
(1/TA – 1/Tb) for both directions which may hold well.
a1
Ea =
+ a3,
⎛1
⎞
1
⎜T − T ⎟ − a2
⎝ A
b⎠
(12)
Table 2. Descriptive statistics of temperatures` parameters:
Arithmetic mean (T1), Confidence Interval (CI) and coefficient of variation (CV)
Parameters
T1/K
CI (mean)
CV (%)
Tm
229.16
217.72–240.60
22
Tb
403.23
391.87–414.58
12
TA
146.19
133.89–158.49
37
T*
2093.55
1784.59–2402.52
65
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⎛1
b1
1⎞
⎜T − T ⎟ = E − b + b3,
⎝ A
b⎠
a
2
(13)
where ai (I = 1,2,3) and bj (j = 1,2,3) are the models’
parameters.
Indeed, according to the Table 4, which presents
the result of statistical estimations of suggested models, only a3 is statistically non-significant and tend
thus to zero, all the other estimated parameters are statistically significant, the R-squared tends to one and
the value of Fisher statistics are very high for both
models. This result allows as expecting a very good
predictive power and quality of approximation during
their practical use.
Note that their relationship is not necessarily bijective i.e., the equations resuming their relationship in
the two directions are not necessarily inverse functions
(Ea = f(1/TA – 1/Tb) or 1/TA – 1/Tb = g(Ea)). For that
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MLIKI et al.
we have presented a best model for each direction
independently.
To confirming the good quality of approximation
of the suggested equations, Eq. (12) and Eq. (13), we
have estimated Ea by replacing the experimental values
of (1/TA – 1/Tb) in the Eq. (12) and we have estimated
(1/TA – 1/Tb) by replacing the experimental values of
Ea in Eq. (13). Table 5 provides descriptive statistics of
estimated versus experimental values for the two variables, in addition to the Average Percentage Deviation
(APD) which is an excellent indicator of the approximation quality.
The descriptive statistics indicates distinctly that
the experimental data values are close to the corresponding ones estimated from Eqs. (12) and (13).
Also, the values of the Average Percentage Deviations
(APD) are very low, 3.7% and 3.2% for Ea and (1/TA –
1/Tb) respectively, showing the small discrepancy
between the estimated and the experimental values
(Table 5).
Nevertheless, in order to affirm the obtained
results, we have to apply some statistical test of populations’ comparison. So, we have compared the estimated values with the experimental ones of the two
variables using the Wilcoxon signed-rank test [40],
which is an equality tests on matched data. Note that
the null hypothesis supposes that the distributions of
estimated and experimental values are the same.
Result of the test is presented in the Table 6.
The test of signed-rank of Wilcoxon leads to agree
the null hypothesis for all variables showing that the
estimated and the experimental distributions of both
variables are significantly and statistically the same,
which justify the excellent predictive power of the
Eqs. (12) and (13).
Table 3. Spearman rank correlation tests
Parameters
Ea
lnAs
TA
Prob > |t |
Tm
0.25
0.03
Tb
0.41
0.00
Tm
–0.16
0.18
Tb
–0.25
0.03
Tm
0.24
0.06
Tb
0.49
0.00
These results can also be revealed graphically. In
fact, we display in Figs. 4 and 5 the experimental
observations of one variable on the x-axis versus
simultaneously the estimated and experimental observations of the other variable on the y-axis. Thus, we
can see frankly that the gap between estimated and
experimental values is showing a small discrepancy.
For providing some physical meaning to the suggested empirical equations (Eq. (12) and Eq. (13)),
fruitful for Chemical Engineering, we can propose
some variable transformations to provide new semiempirical equations interesting for theoreticians as follows:
Ea =
Rα1
+ E01,
⎛1
1⎞− 1
−
⎜T T ⎟ T
⎝ A
b⎠
01
(14)
Rβ1
⎛1
1⎞
1
(15)
⎜T − T ⎟ = E − E + T ,
⎝ A
b⎠
a
02
02
where R is the ideal gas constant, α1 and β1 are dimensionless constants, E01 and E02 are the minimal values
70
600
Tm, K
Tb, K
(a)
Spearman rho
60
Ea, kJ mol−1
(b)
- R.lnAs/3
Ea and - R.lnAs/3
(Tm and Tb)/K
500
400
300
200
100
50
40
30
20
10
0
10
20
30
40
50
60
0
50
100
150
200
250
300
TA/K
Fig. 2. Scatter plots of the direct mutual correlation between the Arrhenius parameters (TA, ln As and Ea) and the melting temperature Tm and boiling temperature Tb of some pure liquids. (Ea/kJ mol–1; –Rln(As/Pa s)/3 in J K–1 mol–1).
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HYPERBOLIC CORRELATION BETWEEN
37
0.015
12
1/TA − 1/Tb
(1/TA − 1/Tb)exp
0.010
8
(1/TA − 1/Tb)est
A
(1/TA − 1/Tb)/kK−1
10
0.005
6
4
0
0
20
2
10
20
30
40
50
60
Ea, kJ mol−1
Fig. 3. Scatter plots for correlations between Ea and
(1/TA – 1/Tb).
that the activation energy can theoretically take for the
set of studied viscous liquids, and T01 and T02 are the
minimal values that the difference between the reciprocal of Arrhenius and boiling temperatures (1/TA –
1/Tb) in Eqs. (14) and (15) can theoretically be taken
3 for the set of studied liquids group and which it is schematized by the dashed line in Fig. 3.
For practical use of the Eq. (14) and the Eq. (15),
we report in the Table 7 the estimated values of the
new parameters. Thus, the practical form of the proposed equations, Eq. (14) and Eq. (15), can be defined
as follows:
Ea =
0.0661
,
⎛1
⎞
−4
1
⎜T − T ⎟ − 1.36 × 10
⎝ A
b⎠
Ea, kJ mol−1
60
Fig. 4. Graphical comparison between the experimental
and the estimated values of (1/TA – 1/Tb) as function of
the experimental values (Ea)exp.
0
0
40
(16)
We add that mathematically, the relationship
between the two equations (14) and (15) requires that
there is equality by couple of similar parameters, i.e.
(α1 = β1), (E01 = E02) and (T01 = T02). Nevertheless,
the small and negligible difference between estimated
values of each of these couples of parameters is due to
several reasons such as the measurement and estimation errors, the heterogeneity of the studied temperature range and the implicit dependence of these
parameters to specific properties of the studied solvents. Also, though the quality of the estimates is very
good, the values are very small for which the error of
the estimates does not make it possible to obtain them
with such precision. In addition, there is a distinct difference between the theoretical and experimental values of {(1/TA – 1/Tb)th and (1/TA – 1/Tb)exp}, where
these last ones are due principally to statistical distribution of observations, errors of calculation and measurements, and to the nature of the studied group of
fluids for which its specific features can play an substantial role in this fact.
Causal Correlation Origin
⎛1
−4
1⎞
0.0596
⎜T − T ⎟ = E − 0.8521 + 2.6 × 10 ,
⎝ A
b⎠
a
(17)
where Ea is in (kJ mol–1) and Ti in K.
As a preliminary theoretical explanation of the
causal correlation between the boiling point (Tb/K)
and the Arrhenius temperature (TA/K) we can give the
Table 4. Result of econometric estimations of the models (12) and (13)
Equation
(3)
(4)
Parameter
a1/kJ K–1 mol–1
a2/K–1
a3/kJ mol–1
0.0661 (53.21)
0.000136 (3.99)
tends to 0
b1/J
K–1
mol–1
0.0596 (30.220)
b2/kJ
mol–1
0.8521 (4.84)
b3/K–1
0.00026 (3.2)
Values between parentheses are t-statistics
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R-squared
F-statistics
0.994
7467
0.995
7471
38
MLIKI et al.
Table 5. Descriptive statistics on experimental versus estimated values
Mean
σ
Experimental
17.41
11.24
6.10
59.61
Estimated
17.43
11.10
5.76
63.89
APD (%)
3.7
3.3
0.2
16.5
Experimental
0.0052
0.0024
0.0012
0.0116
Estimated
0.0052
0.0025
0.0013
0.0116
APD (%)
3.2
2.7
0.1
Variables
Ea/kJ mol–1
(1/TA – 1/Tb)
following points: (i). The vaporization phenomenon is
a transition between two thermodynamic states where
molecules escape from the free surface layer of liquid
towards the free space [12, 13]. (ii). The activation
energy (Ea) value is necessary to the transfer between
two transition states where molecules move from one
layer to another adjacent layer in the frame of shear
flow of Newtonian fluid [14–17]. (iii). Similar causal
correlation is observed in previous works suggesting an
estimation of the boiling temperature of some liquid
components forming some binary fluid mixtures
through the viscosity-temperature and viscosity-composition dependences [23–37] and using some mathematical derivations of partial molar activation energies. (iv). We precise that the novel notion of partial
molar activation energies is justified, as a first approximation, by the observation of similar compositiondependence between the variation of the viscosity activation energy (Ea) and the enthalpy (ΔH*) of activation of viscous flow deduced from the temperatureGibbs free energy dependence for each mole fraction
of fifteen studied binary mixtures in previous works
[23–37].
Table 6. Results of Wilcoxon signed-rank test: experimental versus estimated values
Variable
z
Ea
(1/TA – 1/Tb)
Prob > |z|
–0.22
0.84
0.90
0.37
Table 7. The suggested parameters’ optimal values of the
Eq. (14) and Eq. (15)
Equation
Eq. 14
Eq. 15
Parameter
α1
T01/K
0.00795
7353
β1
T02/K
E02/kJ mol–1
0.00717
3846
0.8521
E01/kJ mol–1
0
Min
Max
13.4
CONCLUSION
Statistical and econometric techniques are used in
this paper in order to investigate the existence of any
eventual causal association between parameters of the
Arrhenius-type equation, the melting and boiling
points of some classical Newtonian liquids. For
empirical analysis, 103 data set of viscosity of pure
Newtonian fluids studied at atmospheric pressure and
at different temperatures from the literature is used.
Several analyses of mutual and complex possible correlations are made allowing as finding original and
interesting result. A significant strong nonlinear correlation between the Arrhenius activation energy (Ea),
the Arrhenius temperature (TA) and the boiling point
(Tb) is seen. Consequently, several statistical and
econometric estimations have been made to find the
model which best fit this association. Finally we have
suggested original equations modeling this relationship, Eqs. (12) and (13). In order to validate the proposed equation, we have made several statistical analyses for comparing the experimental values of parameters with the corresponding estimated values
obtained using the Eqs. (12) and (13). Indeed, a
descriptive statistics indicates clearly that the experimental data are close to the corresponding values
estimated from Eqs. (12) and (13). The values of the
Average Percentage Deviations (APD) are very low
showing the small discrepancy between the estimated
and the experimental values. The application of Wilcoxon signed-rank test confirmed that the experimental and the estimated distributions of the two
variables are significantly and statistically the same,
which justify the good predictive power of the suggested equations.
For giving some physical significance to the proposed empirical equations to be useful for chemical
engineering, we have made some variables transformations to obtain new semi-empirical equations,
Eqs. (14) and (15), which may be very interesting for
the theoreticians. Also, for practical use, we have proposed the Eqs. (14) and (15) which allow the predic-
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HYPERBOLIC CORRELATION BETWEEN
Ea, kJ mol
80
60
(Ea)exp
(Ea)est
40
20
0
0
0.005
0.010
0.015
(1/TA 1/Tb)exp
Fig. 5. Graphical comparison between the experimental
and the estimated values of (Ea) as function of the experimental values (1/TA – 1/Tb)exp.
tion of the boiling temperature through information
on viscosity Arrhenius parameters of liquid phase and
may be thus very useful in the handling of engineering
data especially for the study of hydraulic components
and systems efficiency.
Finally, we hope that this present work open a window for theoreticians to give some theoretical justifications for the understanding of the liquid state and
some meaningful insight into molecular interactions.
ACKNOWLEDGMENTS
Authors thank Dr. R.H. Kacem (FSEGN, University of
Carthage, Tunisia) for helpful discussions and for some
econometric and statistical calculations.
DISCLOSURE STATEMENT
Authors declare that there is no conflict of interest.
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