1
An optimized molecular model for ammonia
2
Bernhard Eckl, Jadran Vrabec∗, and Hans Hasse
3
Institut für Technische Thermodynamik und Thermische
4
Verfahrenstechnik, Universität Stuttgart, 70550 Stuttgart,
5
Germany
6
Abstract
7
An optimized molecular model for ammonia, which is based on a pre-
8
vious work of Kristóf et al., Mol. Phys. 97 (1999) 1129–1137, is pre-
9
sented. Improvements are achieved by including data on geometry and
10
electrostatics from ab initio quantum mechanical calculations in a prelimi-
11
nary model. Subsequently, the parameters of the Lennard-Jones potential,
12
modelling dispersive and repulsive interactions, are optimized to exper-
13
imental vapour-liquid equilibrium data of pure ammonia. The resulting
14
molecular model shows mean unsigned deviations to experiment of 0.7 %
15
in saturated liquid density, 1.6 % in vapour pressure, and 2.7 % in en-
16
thalpy of vaporization over the whole temperature range from triple point
17
to critical point. This final model is used to predict thermophysical prop-
18
erties in the homogeneous liquid, vapour and supercritical region, which
19
are in excellent agreement with a high precision equation of state that
20
was optimized to a set of 1147 experimental data points. Furthermore, it
21
is shown that the final model is capable to predict the second virial coef-
22
ficient over wide temperature range and the radial distribution functions
23
in the liquid state properly, while no structural information is used in the
24
optimization procedure.
∗ Tel.:
+49-711/685-66107, Fax: +49-711/685-66140, Email: [email protected]
1
25
Keywords: Molecular modelling; ammonia; vapour-liquid equilibrium; critical
26
properties; virial coefficient; radial distribution function
27
1
28
Molecular modelling and simulation is a powerful tool for predicting thermo-
29
physical properties that is becoming more accessible due to the ever increasing
30
computing power and the progress in methods and simulation tools. For appli-
31
cations in process engineering, reliable predictions are needed for a wide variety
32
of properties [1, 2, 3].
Introduction
33
All thermophysical properties are specified by the molecular model. There-
34
fore, a balanced modelling procedure, i.e. selection of model type and parame-
35
terization, is crucial. Unfortunately, thermophysical properties usually depend
36
on the model parameters in a highly non-linear manner. So the development
37
of new molecular models, which are able to describe thermophysical properties
38
with sufficient accuracy for industrial applications, is a time-consuming task.
39
In this paper, a procedure is proposed that uses information from ab initio
40
quantum mechanical calculations to accelerate the modelling process. As an
41
example, ammonia is studied here.
42
Ammonia is a well-known chemical intermediate, mostly used in fertilizer
43
industries; another important application is its use as a refrigerant. Due to its
44
simple symmetric structure and its strong intermolecular interactions it is also
45
of high academic interest both experimentally and theoretically.
46
Different approaches can be found in the literature to construct an inter-
47
molecular potential for ammonia to be used in molecular simulation. Jorgensen
48
and Ibrahim [4] as well as Hinchliffe et al. [5] used experimental bond distances
49
and angles to place their interaction sites. Jorgensen and Ibrahim [4] fitted a
50
12-6-3 potential plus four partial charges to results from ab initio quantum me-
51
chanical calculations, which they derived for 250 orientations of the ammonia
52
dimer using the STO-3G minimal basis set. To obtain reasonable potential en-
2
53
ergies for liquid ammonia compared to experimental results, they had to scale
54
their potential by an arbritrary factor 1.26.
55
Hinchliffe et al. [5] used a combination of exponential repulsion terms, an
56
attractive Morse potential, and four partial charges to construct the intermolec-
57
ular potential. The parameters were determined by fitting to a total of 61 points
58
on the ammonia dimer energy surface at seven different orientations, which were
59
calculated using the 6-31G* basis set. However, Hinchliffe et al. [5] pointed out
60
that the parameterization is ambiguous concerning the selection of dimer con-
61
figurations and the employed interaction potential.
62
In a later work, Impey and Klein [6] reparameterized the molecular model
63
by Hinchliffe et al. [5]. They switched to an ”effective” pair potential using
64
one Lennard-Jones (12-6) potential at the nitrogen nucleus site to describe the
65
dispersive and repulsive interactions. The parameters were optimized to the
66
radial distribution function gN−N of liquid ammonia measured by Narten [7].
67
Kristóf et al. [8] used this model to predict vapour-liquid equilibrium prop-
68
erties and found systematic deviations in both vapour pressure and saturated
69
densities. So they decided to develop a completely new molecular model. They
70
used experimental bond distances and angles to place the interaction sites as
71
well. All other parameters of their model, i.e. the partial charges on all atoms
72
and the parameters of the single Lennard-Jones (12-6) potential, were adjusted
73
to vapour-liquid coexistence properties. With this model, Kristóf et al. [8]
74
reached a description of the vapour-liquid equilibrium (VLE) of ammonia that
75
has a reasonable accuracy.
76
For their simulations, Kristóf et al. [8] used the Gibbs ensemble Monte Carlo
77
(GEMC) technique [9, 10] with an extension to the N pH ensemble [11, 12].
78
GEMC based methods often have difficulties simulating strongly interacting
79
fluids, yielding relatively large statistical uncertainties. In a preliminary study
80
to the present work, the model of Kristóf et al. [8] was used together with the
81
Grand Equilibrium method [34] and gradual insertion [38] for calculating the
82
VLE. The results of this study show much smaller statistical errors than those
3
83
of Kristóf et al. [8] and are slightly outside the error of the original work. Also
84
systematic deviations to the experimental VLE data are observed. This is the
85
case particularly for vapour pressure and critical temperature, cf. Figures 1 to
86
3.
87
In the present work, therefore, two new molecular models for ammonia are
88
proposed. Both are based on the work of Kristóf et al. [8] and include data on
89
geometry and electrostatics from ab initio quantum mechanical calculations.
90
The paper is structured as follows: First, a procedure is proposed for the
91
development of an initial molecular model. This model, referred to by prelim-
92
inary model in the following, is then adjusted to experimental VLE data until
93
a desired quality is reached. The resulting model, denoted as final model in
94
the following, is used subsequently to predict thermal and caloric properties at
95
other states than the VLE as well as structural properties.
96
2
97
Selection of Model Type and Parameterization
98
The modelling philosophy followed here is to keep the molecular model as simple
99
as possible. Therefore, the molecule is assumed to be rigid and non-polarizable,
100
i.e. a single state-independent set of parameters is used. Hydrogen atoms are
101
not modeled explicitly, a united-atom approach is used.
102
For both present models, the preliminary and the final model, a single
103
Lennard-Jones site was assumed to describe the dispersive and repulsive interac-
104
tions. The electrostatic interactions as well as hydrogen bonding were modeled
105
by a total of four partial charges. This modelling approach was found to be
106
appropriate for other hydrogen bonding fluids like methanol [13], ethanol [14],
107
and formic acid [15] and was also followed by Impey and Klein [6] and Kristóf
108
et al. [8] for ammonia.
109
110
Thus, the potential energy uij between two ammonia molecules i and j is
given by
4
"
uij (rij ) = 4ε
σ
rij
12
−
σ
rij
6 #
+
4 X
4
X
a=1 b=1
qia qjb
,
4π0 rijab
(1)
111
where a is the site index of charges on molecule i and b the site index of charges
112
on molecule j, respectively. The site-site distances between molecules i and j
113
are denoted by rij for the single Lennard-Jones potential and rijab for the four
114
partial charges, respectively. σ and ε are the Lennard-Jones size and energy
115
parameters, while qia and qjb are the partial charges located at the sites a and
116
b on the molecules i and j, respectively. Finally, 0 denotes the permittivity of
117
the vacuum.
118
One aim of the proposed modelling procedure is the independence on the
119
availability of specific experimental information on the molecular structure so
120
that it can be used with arbitrary molecules. Therefore, no experimental bond
121
lengths or angles were used here in contrast to [4, 5, 6, 8]. Instead, the nucleus
122
positions were calculated using the software package GAMESS (US) [16]. A
123
geometry optimization was performed on the Hartree-Fock, i.e. self-consistent
124
field (SCF), level using the basis set 6-31G, which is a split-valence orbital
125
basis set without polarizable terms. The resulting geometry (rNH = 1.0136 Å,
126
^HNH = 105.99◦ ) is in good agreement with experimental data (rNH = 1.0124 Å,
127
^HNH = 106.67◦ ) [17].
128
The nucleus positions from the ab initio calculation were directly used to
129
specify the positions of the five interaction sites of the present molecular model.
130
At the nitrogen nucleus site and at each of the hydrogen nucleus sites a partial
131
charge was placed. The Lennard-Jones site coincides with the nitrogen nucleus
132
position, cf. Table 1.
133
To obtain the magnitude of the partial charges, another subsequent quantum
134
mechanical calculation was performed. It was done on Møller-Plesset 2 level
135
using the polarizable basis set 6-311G(d,p) and the geometry from the previous
136
step. By default, quantum mechanical calculations are performed on a single
137
molecule of interest in vacuum. It is widely known, that the gas phase dipolar
138
moments significantly differ from the dipole moment in the liquid state. As it
5
139
was found in prior work [18, 19], molecular models yield better results for VLE
140
properties when a ”liquid-like” dipolar moment is applied. Therefore, the single
141
molecule was calculated here within a dielectric cavity utilizing the COSMO
142
(COnducter like Screening MOdel) method [20] to mimic the liquid state. The
143
partial charges were chosen to yield the resulting dipole moment of 1.94 Debye,
144
the parameters are given in Table 1.
145
The preliminary model combines these electrostatics with the Lennard-Jones
146
parameters of Kristóf et al. [8], so no additional experimental data was used.
147
To obtain the final model, the two Lennard-Jones parameters σ and ε were ad-
148
justed to experimental saturated liquid density, vapour pressure, and enthalpy
149
of vaporization using a Newton scheme. These properties were chosen for the
150
adjustment as they all represent major characteristics of the fluid region. Fur-
151
thermore, they are relatively easy to be measured and are available for many
152
components of technical interest.
153
3
154
3.1
155
VLE results for the final model are compared to data obtained from a reference
156
equation of state (EOS) [21] in Figures 1 to 3. These figures also include the
157
results, that we calculated using the preliminary model and the model of Kristóf
158
et al. [8]. The present numerical simulation results together with experimental
159
data [21] are given in Table 2, technical simulation details are given in the
160
Appendix.
Results and Discussion
Vapour-Liquid Equilibria
161
The reference EOS [21] used for adjustment and comparison here, was fitted
162
to a set of 1147 experimental data points. It is based on two older EOS from
163
the late nineteen seventies [22, 23] and is recommended by the NIST, being
164
included in their reference EOS database REFPROP [24]. The uncertainties of
165
the equation of state are 0.2 % for the density, 2 % for the heat capacity, and
166
2 % for the speed of sound, except in the critical region. The uncertainty for
6
167
the vapour pressure is 0.2 % only. Due to the excellent quality of this reference
168
EOS [21], it is regarded to be as good as experimental data here.
169
The model of Kristóf et al. [8] shows noticeable deviations from the reference
170
EOS. The mean unsigned errors over the full VLE temperature range are 1.9 %
171
in saturated liquid density, 13 % in vapour pressure and 5.1 % in enthalpy
172
of vaporization. Even without any further adjustment to experimental data
173
a better description was found using the preliminary model. The deviations
174
between the simulation results and reference EOS are 1.5 % in the saturated
175
liquid density, 10.4 % in the vapour pressure and 5.1 % in the enthalpy of
176
vaporization.
177
With the final model, a further significant improvement was achieved. The
178
mean unsigned deviations in the saturated liquid density, the vapour pressure
179
and the enthalpy of vaporization are 0.7, 1.6, and 2.7 %, respectively. In Fig-
180
ure 4, the relative deviations of the model from Kristóf et al. [8], the preliminary
181
model, and the final model are shown for the whole VLE temperature range from
182
triple point to critical point.
183
Mathews [25] reports experimental critical values of temperature, density
184
and pressure for ammonia: Tc =405.65 K, ρc =13.8 mol/l, and pc =11.28 MPa.
185
Following the procedure suggested by Lotfi et al. [26] the critical properties
186
Tc =395.82 K, ρc =14.0 mol/l, and pc =11.26 MPa for the model of Kristóf et al.
187
[8] were calculated, where the critical temperature is underestimated by 2.4 %.
188
For the preliminary model Tc =403.99 K, ρc =14.1 mol/l, and pc =11.67 MPa
189
were obtained and for the final model Tc =402.21 K, ρc =13.4 mol/l, and
190
pc =10.52 MPa. The latter two give reasonable results for the critical tem-
191
perature, while the final model underpredicts the critical pressure slightly (by
192
6.7 %).
193
3.2
194
In many technical applications thermodynamic properties in the homogeneous
195
fluid region are needed. Thus, the new molecular model was tested regarding
Homogeneous Region
7
196
its predictive capabilities for such states.
197
Thermal and caloric properties were predicted with the final model in the
198
homogenous liquid, vapour and supercritical fluid region. In total, 70 state
199
points were studied, covering a large range of states with temperatures up to
200
700 K and pressures up to 700 MPa. In Figure 5, relative deviations between
201
simulation and reference EOS [21] in terms of density are shown. The deviations
202
are typically below 3 % with the exception of the extended critical region, where
203
a maximum deviation of 6.8 % is found.
204
Figure 6 presents relative deviations for the enthalpy between simulation
205
and reference EOS [21]. In this case, deviations are very low for low pressures
206
and high temperatures (below 1–2 %). Typical deviations in the other cases are
207
below 5 %.
208
These results confirm the proposed modelling procedure. By adjustment to
209
experimental VLE data only, quantitatively correct predictions in most of the
210
technically important fluid region are obtained.
211
3.3
212
The virial expansion gives an equation of state for low density gases. For ammo-
213
nia it is a good approximation for gaseous states below 0.1 MPa with a maximum
214
error of 2.5 %. Virial coefficients can easily be derived from the intermolecular
215
potential [27, 28, 29]. The second virial coefficient is related to the molecular
216
model by [30]
Second Virial Coefficient
Z
B = −2π
0
∞
uij (rij , ωi , ωj )
exp −
kB T
2
rij
drij ,
−1
(2)
ωi ,ωj
217
where uij (rij , ωi , ωj ) is the interaction energy between two molecules i and j,
218
cf. Equation (1). kB denotes Boltzmann’s constant and the hi brackets indicate
219
an average over the orientations ωi and ωj of the two molecules separated by
220
the centre of mass distance rij .
221
The second virial coefficient was calculated here by evaluating Mayer’s f -
222
function at 363 radii from 2.4 to 8 Å, averaging over 5002 random orientations
8
223
at each radius. The random orientations were generated using a modified Monte
224
Carlo scheme [31, 2]. A cut-off correction was applied for distances larger than
225
8 Å for the LJ potential [32]. The electrostatic interactions need no long-range
226
correction as they vanish by angle averaging.
227
Figure 7 shows the second virial coefficient predicted by the final model in
228
comparison to the reference EOS [21]. A good agreement was found over the
229
full temperature range with a maximum deviation of −4.3 % at 300 K.
230
3.4
231
Due to its scientific and technical importance, experimental data on the micro-
232
scopic structure of liquid ammonia are available. Narten [7] and Ricci et al.
233
[33] applied X-ray and neutron diffraction, respectively. The results of Ricci
234
et al. [33] show less scatter and are available for all three types of atom-atom
235
pair correlations, namely nitrogen-nitrogen (N-N), nitrogen-hydrogen (N-H),
236
and hydrogen-hydrogen (H-H). Thus, they were used for comparison here. In
237
Figure 8, these experimental radial distribution functions for liquid ammonia
238
at 273.15 K and 0.483 MPa are compared to present predictive simulation data
239
based on the final model.
Structural Quantities
240
It was found that these structural properties are in very good agreement,
241
although no adjustment was done regarding structural properties. The atom-
242
atom distance of the first three layers was predicted correctly, while only a minor
243
overshooting in the first peak was found. Please note that the first peak of the
244
experiment in gN−H and gH−H show intramolecular, and not intermolecular,
245
pair correlations, which are not covered by the regarded models.
246
In the experimental radial distribution function gN−H , the hydrogen bonding
247
of ammonia can be seen between 2 and 2.5 Å. Due to the simplified approxima-
248
tion by off-centric partial charges, the molecular model is not capable to describe
249
this effect completely. But even with this simple model a small shoulder at 2.5 Å
250
was obtained.
9
251
4
Conclusion
252
A new molecular model is proposed for ammonia. This (final) model was de-
253
veloped using a modelling procedure, which speeds up the modelling process
254
and can be applied without the need of specific experimental information on
255
the molecular structure. The interaction sites are located at the atom positions
256
obtained by ab initio quantum mechanical calculations on the Hartree-Fock
257
level. The electrostatic interactions, here in the form of partial charges, were
258
parameterized according to high-level, i.e. Møller-Plesset 2, ab initio quantum
259
mechanical results. The latter are obtained by calculations within a dielectric
260
continuum to mimic the (stronger) electrostatic interactions in the liquid phase.
261
The partial charges of the present ammonia models were specified to yield the
262
same dipole moment as quantum mechanics. The Lennard-Jones parameters of
263
the final model were adjusted to VLE data, namely vapour pressure, saturated
264
liquid density, and enthalpy of vaporization.
265
A description of the VLE of ammonia was reached within relative devia-
266
tions of a few percent. Additionally, covering a large region of states, a good
267
prediction of both thermal and caloric properties was found at other states by
268
comparison to a reference EOS [21].
269
Predicted structural quantities, i.e. radial distribution functions in the liq-
270
uid state, are in very good agreement with experimental neutron diffraction
271
data. This shows that molecular models adjusted to macroscopic thermody-
272
namic properties also give reasonable results on microscopic properties. Note
273
that this is not true vice versa in most cases. With the present final model, a
274
similar quality in describing the atomic radial distribution functions is obtained
275
as with the model of Impey and Klein [6] which was adjusted to these data.
276
On the other side, the predictions of macroscopic properties like vapour pres-
277
sure on the basis of the model by Impey and Klein are poor [8]. Thus, a good
278
description of macroscopic properties, like the vapour pressure curve, is a more
279
demanding criterion and, therefore, should be used for adjustment of molecular
10
280
models.
281
5
282
The authors gratefully acknowledge financial support by Deutsche Forschungs-
283
gemeinschaft, Schwerpunktprogramm 1155 ”Molecular Modelling and Simu-
284
lation in Process Engineering”. The simulations were performed on the na-
285
tional super computer NEC SX-8 at the High Performance Computing Centre
286
Stuttgart (HLRS) under the grant MMHBF.
Acknowledgement
We also want to thank Xijun Fu for setting up and running the simulations
287
288
shown in Figures 5 and 6.
289
6
290
The Grand Equilibrium method [34] was used to calculate VLE data at eight
291
temperatures from 240 to 395 K during the optimization process. At each
292
temperature for the liquid, molecular dynamics simulations were performed in
293
the N pT ensemble using isokinetic velocity scaling [32] and Anderson’s barostat
294
[35]. There, the number of molecules is 864 and the time step was 0.58 fs except
295
for the lowest temperature, where 1372 molecules and a time step of 0.44 fs
296
were used. The initial configuration was a face centred cubic lattice, the fluid
297
was equilibrated over 120 000 time steps with the first 20 000 time steps in the
298
canonical (N V T ) ensemble. The production run went over 300 000 time steps
299
(400 000 for 240 K) with a membrane mass of 109 kg/m4 . Widom’s insertion
300
method [36] was used to calculate the chemical potential by inserting up to 4 000
301
test molecules every production time step.
Appendix
302
At the lowest two temperatures, additional Monte Carlo simulations were
303
performed in the N pT ensemble for the liquid. There, the chemical poten-
304
tial of liquid ammonia was calculated by the gradual insertion method [38].
305
The number of molecules was 500. Starting from a face centred cubic lattice,
11
306
15 000 Monte Carlo cycles were performed for equilibration and 50 000 for
307
production, each cycle containing 500 displacement moves, 500 rotation moves,
308
and 1 volume move. Every 50 cycles 5 000 fluctuating state change moves,
309
5 000 fluctuating particle translation/rotation moves, and 25 000 biased particle
310
translation/rotation moves were performed, to determine the chemical poten-
311
tial. These computationally demanding simulations yield the chemical potential
312
in the dense and strong interacting liquid with high accuracy, leading to small
313
uncertainties in the VLE.
314
For the corresponding vapour, Monte Carlo simulations in the pseudo-µV T
315
ensemble were performed. The simulation volume was adjusted to lead to an
316
average number of 500 molecules in the vapour phase. After 1 000 initial N V T
317
Monte Carlo cycles, starting from a face centred cubic lattice, 10 000 equili-
318
bration cycles in the pseudo-µV T ensemble were performed. The length of the
319
production run was 50 000 cycles. One cycle is defined here to be a number
320
of attempts to displace and rotate molecules equal to the actual number of
321
molecules plus three insertion and three deletion attempts.
322
The cut-off radius was set to 17.5 Å throughout and a centre of mass cut-off
323
scheme was employed. Lennard-Jones long-range interactions beyond the cut-off
324
radius were corrected as proposed in [32]. Electrostatic interactions were ap-
325
proximated by a resulting molecular dipole and corrected using the reaction field
326
method [32]. Statistical uncertainties in the simulated values were estimated by
327
a block averaging method [39].
328
For the simulations in the homogeneous region, molecular dynamics sim-
329
ulations were performed with the same technical parameters as used for the
330
saturated liquid runs.
331
For the radial distribution functions a molecular dynamics simulation was
332
performed with 500 molecules. Intermolecular site-site distances were divided
333
in 200 slabs from 0 to 13.5 Å and averaged over 50 000 time steps.
12
334
335
336
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384
385
Press, Oxford, 1987).
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102, 7650 (1995).
386
[34] J. Vrabec and H. Hasse, Mol. Phys. 100, 3375 (2002).
387
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388
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389
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390
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391
[39] H. Flyvbjerg and H.G. Petersen, J. Chem. Phys. 91, 461 (1989).
15
Table 1: Parameters of the final ammonia model. The electronic charge is
e = 1.6021 · 10−19 C.
Interaction
Site
N
H(1)
H(2)
H(3)
x
Å
0.0
0.9347
-0.4673
-0.4673
y
Å
0.0
0.0
0.8095
-0.8095
z
Å
0.0757
-0.3164
-0.3164
-0.3164
16
σ
Å
3.376
—
—
—
ε/kB
K
182.9
—
—
—
q
e
-0.9993
0.3331
0.3331
0.3331
Table 2: Vapour-liquid equilibria of ammonia: simulation results using the final
model (Sim) compared to data from a reference quality equation of state [21]
(EOS) for vapour pressure, saturated densities and enthalpy of vaporization.
The number in parentheses indicates the statistical uncertainty in the last digit.
T
K
240
280
315
345
363
375
385
395
pSim
MPa
0.12(1)
0.60(2)
1.65(4)
3.37(4)
5.22(5)
6.37(6)
7.88(5)
9.54(7)
pEOS
MPa
0.102
0.551
1.637
3.457
5.101
6.485
7.845
9.422
ρ0Sim
mol/l
40.26(1)
36.98(2)
33.76(3)
30.45(4)
28.17(6)
26.18(7)
24.05(9)
20.9 (1)
ρ0EOS
mol/l
40.032
36.939
33.848
30.688
28.368
26.502
24.608
22.090
17
ρ00Sim
mol/l
0.066(5)
0.280(8)
0.74 (1)
1.55 (1)
2.56 (2)
3.17 (3)
4.27 (5)
5.66 (9)
ρ00EOS
mol/l
0.0527
0.257
0.744
1.624
2.544
3.459
4.554
6.272
∆hvSim
kJ/mol
24.11(1)
21.56(1)
18.96(2)
16.19(3)
13.93(5)
12.48(6)
10.49(9)
8.1 (1)
∆hvEOS
kJ/mol
23.31
21.07
18.57
15.79
13.65
11.89
10.08
7.66
392
393
List of Figures
1
Saturated densities of ammonia. Simulation results: N model of
preliminary model, • final model.
394
Kristóf et al. [8], this work:
395
— reference EOS [21]. Critical data from simulation: model of
396
Kristóf et al. [8],
397
2
◦ final model.
+ experimental critical point [25]. 20
Vapour pressure of ammonia. Simulation results: N model of
preliminary model, • final model.
398
Kristóf et al. [8], this work:
399
— reference EOS [21]. . . . . . . . . . . . . . . . . . . . . . . . .
400
3
Enthalpy of vaporization of ammonia. Simulation results: N model
of Kristóf et al. [8], this work:
402
model. — reference EOS [21]. . . . . . . . . . . . . . . . . . . . .
4
preliminary model,
•
401
403
final
tween simulation and reference EOS [21] (δz = (zSim −zEOS )/zEOS ):
405
N model of Kristóf et al. [8], this work:
406
•
407
density, bottom: enthalpy of vaporization. . . . . . . . . . . . . .
5
preliminary model,
final model. Top: vapour pressure, centre: saturated liquid
erence EOS [21] (δρ = (ρSim − ρEOS )/ρEOS ) in the homogeneous
410
region:
411
curve. The size of the bubbles denotes the relative deviation as
412
indicated in the plot. . . . . . . . . . . . . . . . . . . . . . . . . .
6
◦ simulation data of the final model, — vapour pressure
erence EOS [21] (δh = (hSim − hEOS )/hEOS ) in the homogeneous
415
region:
416
curve. The size of the bubbles denotes the relative deviation as
417
indicated in the plot. . . . . . . . . . . . . . . . . . . . . . . . . .
419
7
24
Relative deviations for the enthalpy between simulation and ref-
414
418
23
Relative deviations for the density between simulation and ref-
409
413
22
Relative deviations of vapour-liquid equilibrium properties be-
404
408
21
◦
simulation data of final model, — vapour pressure
Second virial coefficient:
25
• simulation data of the final model,
— reference EOS [21]. . . . . . . . . . . . . . . . . . . . . . . . .
18
26
420
421
8
Partition functions of ammonia: — simulation data of the final
model, ◦ experimental data [33]. . . . . . . . . . . . . . . . . . .
19
27
Figure 1: Eckl et al.
20
Figure 2: Eckl et al.
21
Figure 3: Eckl et al.
22
Figure 4: Eckl et al.
23
Figure 5: Eckl et al.
24
Figure 6: Eckl et al.
25
Figure 7: Eckl et al.
26
Figure 8: Eckl et al.
27