Molecular Simulation of NH3/IL mixture for
Absorption Refrigeration Cycle
Faculty Mechanical, Maritime
and Materials Engineering
Report Number: 2714
ABHISHEK KABRA
4414357
Challenge the future
Molecular Simulation Of NH3/Ionic Liquid Mixture
For Absorption Refrigeration Cycle
by
Abhishek Kabra
in partial fulfillment of the requirements for the degree of
Master of Science
in Mechanical Engineering
at the Delft University of Technology,
Process & Energy Faculty 3mE,
Sustainable Process and Energy Technology
August, 2016
Supervisor:
Dr.ir. C.A. Infante Ferreira
Thesis committee: Prof.Dr.ir. T.J.H. Vlugt
Dr.ir. R. Pecnik
Dr.ir. W. Haije
MSc. T.M. Becker
MSc. M. Wang
An electronic version of this thesis is available at http://repository.tudelft.nl/
Abstract
A molecular based Monte Carlo (MC) simulation method is used to predict the performance
of the absorption refrigeration cycle involving NH3/IL (refrigerant/absorbent) pair as a
working fluid. To investigate the thermodynamic performance of the absorption
refrigeration cycle, various properties such as density, excess enthalpy or absorption heat,
heat capacity and solubility of refrigerant in the absorbent are required. For this reason, MC
simulations as an alternative to experiments, are used to compute the required properties.
MC simulations in the osmotic ensemble are used to compute the solubility of NH3 in ILs. The
ideal part of the heat capacity of pure IL is computed using quantum mechanical calculations
and the residual part along with the density are computed using MC simulations in the NPT
ensemble. The excess enthalpy for the mixture is calculated using MC simulations in the
osmotic ensemble in combination with the NPT ensemble. The simulations are performed at
temperatures ranging from 303 K to 393 K and pressures ranging from 4 to 18 bar. The
performance parameters such as Coefficient of Performance (COP) and circulation ratio (f) of
NH3 paired with two ILs, [emim][Tf2N] and [emim][SCN], are investigated using MC
simulations and compared with the results obtained from correlated experimental data. The
results from the MC simulations are in reasonable agreement with those obtained from the
correlated experimental data. Hence, MC simulations can be used as an inexpensive
alternative for preliminary design considerations involving potential working fluids for
absorption refrigeration cycles in the absence of available experimental data.
i
Acknowledgements
Firstly, I would like to express my sincere gratitude to my supervisor Dr.ir. C.A. Infante
Ferreira for the continuous support and encouragement of my thesis project. His guidance
has not only helped me learn better but has also inspired me continuously.
I would also like to thank my daily supervisor, MSc. Tim M. Becker for his willingness to help
me during any hour of the day. The timely advice and suggestions given by him helped me
to finish the thesis on time. His support and guidance gave my thesis the shape and depth
that it has now. I could not have imagined having a better mentor than him for my project.
I am also extremely thankful to my other daily supervisor, MSc. Meng Wang for helping me
throughout the project. He was always available whenever I ran into a trouble or had a
question about my work. I am also grateful to Dr. Mahinder Ramdin for his valuable
comments during the process of writing this thesis. I would also like to express my gratitude
to the committee members for devoting their time and effort in evaluating my thesis report.
Last, but not the least, I express my deepest gratitude to my family for showering their love
and support on me always. Their faith in me has been a constant source of inspiration for
me.
Delft, University of Technology
August 30, 2016
Abhishek Kabra
iii
Table of Contents
Nomenclature
1. Introduction ........................................................................................................................... 1
2. Ionic Liquids in Absorption Refrigeration Cycles ................................................................. 5
2.1 Ionic Liquids ...................................................................................................................... 5
2.2 Ionic Liquids in Absorption Refrigeration Cycles .............................................................. 6
2.3 Absorption Refrigeration Cycles ....................................................................................... 7
2.4 Thermodynamic Model .................................................................................................... 9
2.4.1 Enthalpy calculation ................................................................................................ 10
2.4.2 Cycle performance................................................................................................... 11
3. Monte Carlo Molecular Simulations ................................................................................... 13
3.1 Molecular Simulation approach ..................................................................................... 13
3.1.1 The Metropolis method........................................................................................... 14
3.2 Technical Details - System set-up................................................................................... 15
3.3 Force fields ..................................................................................................................... 18
3.4 MC moves or trial moves ............................................................................................... 18
3.5 Ensembles....................................................................................................................... 19
3.5.1 The NPT ensemble ................................................................................................... 19
3.5.2 The Osmotic ensemble ............................................................................................ 24
4. Simulation Details ............................................................................................................... 27
4.1 Simulation details for calculating the Heat Capacities................................................... 27
4.2 Simulation details for calculating the Solubility ............................................................. 28
5. Results and Discussions ...................................................................................................... 31
5.1 Thermodynamic properties from MC simulations ......................................................... 31
5.1.1 Density - validation of the force fields .................................................................... 31
5.1.2 Solubility of NH3....................................................................................................... 33
5.1.3 Heat Capacity........................................................................................................... 38
5.1.3 Excess Enthalpy ....................................................................................................... 44
5.2 Properties for Absorption refrigeration cycle ................................................................ 46
5.2.1 Enthalpy calculations ............................................................................................... 46
5.2.2 Circulation ratio, f .................................................................................................... 46
5.2.3 Coefficient of Performance ..................................................................................... 49
v
vi
Table of Contents
5.3 Sensitivity Analysis ......................................................................................................... 51
5.3.1 Heat capacity of an IL .............................................................................................. 51
5.3.2 Excess enthalpy ....................................................................................................... 52
5.3.3 Solubility of NH3 in an IL .......................................................................................... 52
6. Conclusions .......................................................................................................................... 55
Appendix A - Fluctuation formula ......................................................................................... 57
Appendix B - Ideal Heat Capacity from Quantum Mechanics .............................................. 59
Appendix C - Force Field Parameters ..................................................................................... 63
Appendix D - Comparison of the experimental data and the NRTL model .......................... 69
Appendix E - Simulation Details ............................................................................................. 71
Appendix F - Conformers ....................................................................................................... 73
Appendix G - Excess Enthalpy ................................................................................................ 75
Bibliography ............................................................................................................................ 77
Nomenclature
List of Symbols
CP
CV
E
f
f
h
h,H
hE
∆mix h
∆hlatent
I
k
K
K
kB
m
n
N
NA
o
P
P
p
q
Q
Q
r
r
r
r
rc
R
s
T
Tr
U, Utotal
U
Macroscopic Property
Ensemble average of a macroscopic property
Heat Capacity at constant Pressure
Heat Capacity at constant Volume
Total Energy
Circulation ratio
Fugacity
Planck constant
Specific Enthalpy
Excess Enthalpy or Absorption Heat
Mixing Enthalpy
Latent Heat of pure Ammonia
Configurational Enthalpy
Hamiltonian function
Moment of Inertia
force constant
Kinetic Energy
Vibrational mode
Boltzmann Constant
Mass of a particle or a molecule
Mass flow rate
New configuration
Number of molecules or particles
Avogadro constant
Old configuration
Probability of system to be in a certain configuration
Pressure
Momenta of molecules
Point charge
Heat load
Partition function of an ensemble
Position of a molecule
Distance between point charges
Distance between particles
Bond Length
Cut-off distance
Universal Gas constant
Scaled co-ordinates
Temperature
Reduced Temperature ratio
Potential energy of a system
Coulombic Interactions
vii
[-]
[-]
[J kg-1 K-1]
[J kg-1 K-1]
[J]
[kg s-1 kg-1 s]
[Pa]
[Js]
[J kg-1]
[J kg-1]
[J]
[J kg-1]
[J kg-1]
[J]
[kg m2]
[K Å-2]
[J]
[-]
[kg m2 s-2 K-1]
[g mol-1]
[kg s-1]
[-]
[-]
[mol-1]
[-]
[-]
[bar]
[kg m s-1]
[C]
[W]
[-]
[Å]
[Å]
[Å]
[Å]
[Å]
[J K-1 mol-1]
[-]
[K]
[-]
[J]
[J]
viii
UVDW
∆U
V
∆V
X
W
Nomenclature
van der Waals potential
Change in the potential energy of the system
Volume
Change in the volume
Mol fraction
Average property
Mass fraction
Pumping Power
[K]
[J]
[Å3]
[Å3]
[mol mol-1]
[-]
[kg kg-1]
[W]
Greek Symbols
ω
θ
ρ
λ
μ
ω
χ
Ɛ
ϵ
ϵo
ψ
β
Ʌ
σ
σ
Θ
ν
Accentric factor
Angle between bonds
Average density
Coupling parameter
Chemical Potential
degeneracy
Dihedral angle
Depth of the potential well
Fractional molecule
Electrical permittivity of free space
Improper
Reciprocal temperature
Thermal de Broglie wavelength
Size parameter
Symmetry number
Vibrational Temperature
Vibrational frequency
Subscripts
A
C
el
E
G
i, j
i
i
i
IL
K
mix
n
NH3
0
o
Absorber
Condenser
Electronic
Evaporator
Generator
ith, jth component of a mixture
ith state point of the absorption cycle
ith energy level
ith microstate
Ionic Liquid
Vibrational mode
mixture
New configuration
Ammonia
Reference state
Old configuration
[-]
[degree]
[kg m-3]
[-]
[J kg-1]
[-]
[degree]
[K]
[-]
[A2s4 kg-1 m-3]
[-]
[s2 m-2 kg-1]
[m]
[Å]
[-]
[K s cm-1]
[cm-1]
Nomenclature
o
P
rot
r
s
solvent
solute
trans
vib
ν
ix
Nominal value of force field parameters
Pump
Rotational
Refrigerant
Weak solution of ionic liquid coming out of the absorber
Solvent molecules
Solute molecules
Translational
Vibrational
vibrational frequency
Superscripts
accep
ext
exp
int
ig
move
N
res
sim
T
VDW
V
Change in the configuration of the system is accepted
Intermolecular contributions
Experimental data
Intramolecular contributions
Ideal gas part
Change in the configuration of the system
Number of molecules
Residual part
Simulation result
Temperature
van-der Waals
Volume
Abbreviations
A
[bmim]
[BF4]
C
CFCMC
CBMC
COP
DFT
[emim]
E
EOS
EXP
FF
G
H2O
Absorber
1-butyl-3-methylimidazolium
Tetrafluoroborate
Condenser
Continuous Fractional Component Monte Carlo
Configurational-Bias Monte Carlo
Coefficient of Performance
Density Functional Theory
1-ethyl-3-methylimidazolium
Evaporator
Equation of State
Expansion valve
Force field
Generator
Water
HC
Hydrocarbons
HFC
HX
IEA
Hydro fluorocarbons
Heat Exchanger
International Energy Agency
x
ILs
LJ
LiBr
MC
MD
MS
NH3
NIST
NPT
NVT
[PF6]
RK
[SCN]
[Tf2N]
VLE
WL
Nomenclature
Ionic Liquids
Lennard-Jonnes
Lithium Bromide
Monte Carlo
Molecular Dynamics
Molecular Simulation
Ammonia
National Institute of Standards and Technology
isobaric-isothermal ensemble
canonical ensemble
hexafluorophosphate
Redlich-Kwong
Thiocyanate
bis(trifluoromethylsulfonyl)imide
Vapour liquid equilibrium
Wang Landau
1
Introduction
According to the International Energy Agency (IEA), the total energy demand for applications
such as space heating and cooling, food storage, etc., in the United States and the European
Union is in the magnitude of exajoule (1EJ =
J) and is expected to grow further at an
increased rate [1,2]. At the same time, waste heat is generated by many systems as a byproduct of their operation governed by the laws of thermodynamics. To deal with such a rapid
growth of energy demand and utilization, much attention has been focused on absorption
refrigeration cycles. This is because of their effective utilization of low grade waste heat and the
resulting decrease of primary energy consumption [3-6]. Absorption cycle utilizes this waste
heat and transforms it to a higher temperature level in contrast to commonly used vapour
compression cycle which utilizes mechanical compression driven by electricity [7]. Additionally,
heat from renewable sources, such as solar, can also be applied as the driving heat for such
cycles [7,8]. In general, absorption cycles benefit from applying low grade heat instead of
electricity to drive the cycle [7].
The thermodynamic performance of an absorption cycle depends on its working pair which is
usually a binary solution of refrigerant and absorbent with a large boiling point difference
[3,7,9-10]. The refrigerant has a relatively low boiling point, whereas the absorbent has a high
boiling point [3]. The commonly used working pairs are aqueous solutions of lithium bromide
(H2O/LiBr) for refrigeration temperatures above 0oC and aqueous solutions of ammonia
(H2O/NH3) for refrigeration temperatures below 0oC [3,6-7,9]. In case of H2O/LiBr, LiBr acts as
absorbent whereas in case of H2O/NH3, H2O acts as absorbent. However, these commonly used
working pairs are characterized by some detriments. The H2O/LiBr system has some drawbacks
such as crystallization, corrosion and low (sub-atmospheric) operating pressure whereas the
H2O/NH3 system has drawbacks such as difficulty in separation (rectification process), low
system performance and high driving pressure [3,6-7,9]. Therefore, the search for new
propitious working pairs capable to overcome these disadvantages has become the research
focus of many studies in the past few decades [3,7,9,11].
A special class of absorbents, namely ionic liquids (ILs), have been proposed as an alternative to
overcome the detriments of the traditional working pairs [6,9,12]. ILs are molten salts which
are often liquid at room temperature [6,13]. They consist of an organic cation and an organic or
inorganic anion [3,6,14]. The possible use of ILs was discovered a long time ago but was very
limited until the late 1990s. The progressive trend changed quickly as a result of an article
published by Freemantle, emphasizing on the potential applications of ILs as "green" solvents to
replace conventional organic solvents [15]. ILs are promising absorbents because of their
favorable properties such as negligible vapour pressure, high thermal and chemical stability,
high thermal decomposition temperature, non-flammability, typically toxic, low corrosive
character and solvation properties [3,5-7,9,11]. Furthermore, billions of ILs can be designed and
1
2
1. Introduction
synthesized by selecting different anion and cation combinations, with a diverse spectrum of
physicochemical properties [3,5-7].
To investigate the thermodynamic performance of an absorption refrigeration cycle, various
properties such as enthalpy, heat capacity and solubility of refrigerant in the absorbent are
required at different state points in the cycle [3,6,9,11]. However, very little experimental data
for such properties are available for certain ILs. For most of the other ILs, a lack of reliable data
still exists [16-18]. Also the experimental data are not available for certain required conditions
such as high temperatures and pressures. Given the enormous number of possible ILs, it is
impractical to experimentally test even a small fraction of all potential ILs [17,19]. For this
reason, Monte Carlo (MC) simulations are used to compute the required properties [17-20].
MC simulations are a powerful predictive tool which can be used to compute equilibrium
thermodynamic and transport properties of ILs [19-22]. It can also be used to predict the
solubility of gas mixtures for which experimental measurements are difficult to conduct [19]. In
this work, MC simulation techniques are used for the determination of thermodynamic
properties of working pairs.
In recent years, many research groups have investigated NH3/IL systems as replacement for the
traditional systems in absorption refrigeration cycles. Since 2006, Yokozeki and Shiflett [3,16, 23
-25] from DuPont have published a variety of papers on the use of ILs as absorbents. The
solubilities of NH3 in ILs were measured and new working pairs were proposed such as NH3/
[emim][Tf2N], NH3/[bmim][Bf4], NH3/[bmim][Pf6], etc. [16,26-27]. Lagache et al. [20] published a
journal article on methods to predict the heat capacity by MC simulations. Since 2007, Zheng et
al. [3,9,28-30] have conducted numerous experimental and theoretical studies on IL working
pairs. The group published several papers on thermodynamics of the solution and methods for
screening of NH3/IL system [3,9,28-30]. Numerous studies have recently been published on ILs
as absorbent highlighting their importance as novel solvents [31-33].
Considerable research has been devoted to the prediction of thermodynamic properties by
theoretical and experimental methods. At the same time less attention has been paid to
simulation based predictions of the thermodynamic properties as an alternative to
experiments. Furthermore, the accuracy of these simulation results in predicting the
thermodynamic performance of the cycle is still unknown. The objective of this study is to use
MC simulation techniques to determine the thermodynamic properties of working pairs to
investigate their thermodynamic performance in the single-effect absorption refrigeration
cycle. To achieve the objective, this study is divided into several parts which are organized as
follows:
Thesis outline
Chapter 2 presents a brief introduction of ILs and the absorption refrigeration cycle. The cycle
model along with the assumptions and the equations used for investigating the thermodynamic
performance of single-effect absorption cycle are described in this chapter.
Thesis outline
3
Chapter 3 provides the background and the methodology of the MC simulations. It introduces
ensembles and the procedures required for calculating various thermodynamic properties
mentioned in chapter 2.
Chapter 4 provides the simulation details such as force field parameters for the working pairs,
mixing rule, thermodynamic conditions, number of cycles, cut-off distance, tail-correction etc.,
used in the simulations to compute the thermodynamic properties.
Chapter 5 presents the results and discussion for the thermodynamic properties computed
from simulations namely density, heat capacity, excess enthalpy along with the solubility. It
then provides the comparison of the thermodynamic performance of various NH3/IL working
pairs investigated in this work. A sensitivity analysis of the properties affecting the
thermodynamic performance of the cycle is also carried out.
Finally, in Chapter 6 concluding remarks are given.
2
Ionic Liquids in
Absorption Refrigeration Cycles
Over the last two centuries, the two most commonly used working pairs in commercial
absorption refrigeration cycles are NH3/H2O and H2O/LiBr. However, due to the several
drawbacks of these pairs discussed later in this chapter, continuous efforts have been made to
replace these pairs. In the last ten years, IL based working pairs have gained a lot of scientific
attention as emerging environmentally benign solvents or green solvents [12,15,34].
2.1 Ionic Liquids
ILs are molten salts which are liquid at room temperature and have a negligible vapour pressure
[6,13,15]. They consist of an organic cation with an organic or inorganic anion [3,6,14,15]. The
number of cations and anions that can be selected for the design and synthesis of an IL is
enormous. Cations and anions can be selected for a specific need, which enables the design of
ILs with specific physicochemical properties. This is called tunability of ILs.
ILs have gained popularity due to their interesting properties. They have a high thermal
stability. For example, some ILs like 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)
imide ([emim][Tf2N]) and 1-butyl-3-methylimidazolium tetrafluroborate ([emim][BF4]) have
been heated up to 400 oC with no degradation [36]. ILs are highly miscible with polar
substances due to their ionic character. They exhibit ionic interactions such as van der Waals,
dipole-dipole interactions and hydrogen bonding. Consequently, ILs are good candidates for
solvents due to their high affinity for polar and non-polar materials [15]. Due to their negligible
vapour pressure, ILs are considered as possible replacements of traditional industrial solvents
which are often volatile organic compounds. This would prevent environmental pollution due
to emission of volatile organic compounds into the atmosphere. Hence, ILs are considered as
green solvents [12,15]. Moreover they are almost nonflammable and highly electrochemically
stable [15,35].
Many researchers have developed an interest in ILs because of their unique properties,
resulting in an expansion of its application field. So far, applications of ILs have been reported in
electrochemistry [38], analytical chemistry [39], catalysis [15,37], separation technology [37].
ILs are nowadays applied at industrial scale for following applications: Degussa: paint additives
[40], BASF: BASIL, aluminum plating and cellulose dissolution [40], batteries and solar cells.
However, the main challenges to the large scale industrial applications of ILs are their high
toxicity because of the presence of halogens, the high cost which is between 2 and 100 times
the cost of organic solvents and the relatively high viscosity [40]. Before considering their
5
6
2. Ionic Liquids in Absorption Refrigeration Cycle
implementation in industry, toxicity and biodegradability of ILs are important factors to
determine their impact on the environment.
2.2 Ionic Liquids in Absorption Refrigeration Cycle
ILs are promising absorbents in absorption refrigeration cycles due to the aforementioned
properties such as tunability, negligible vapour pressure and high thermal stability. They are
also less corrosive than conventional high melting salts such as LiBr. The applicability of
refrigeration and absorption chiller technologies is limited due to the severe corrosiveness and
crystallization of conventional LiBr/H2O working pair [7]. LiBr crystallizes at 25 oC from aqueous
solution containing more than 65% LiBr [7]. Due to corrosion problems, efficient multi-effect
absorption cycles using driving temperatures above 150 oC are not used in the petrochemical
industry [7]. For low temperature applications expensive, corrosion proof material is required.
The NH3/H2O working pair also faces the need for the rectification of the ammonia-water
mixture [4,7]. In this context, IL based working fluids have gained immense popularity.
In recent years, many research groups have investigated systems based on IL as one of the
working fluids. Since 2006, Yokozeki and Shiflett [3,16,23-25] from DuPont have published a
variety of papers on the use of ILs as absorbents. The solubilities of NH3 in ILs were measured
and new working pairs were proposed such as NH3/[emim][Tf2N], NH3/[bmim][Bf4], NH3/
[bmim][Pf6], etc. [16,26-27]. The group also calculated their thermodynamic performance in a
single-effect absorption cycle. Since 2007, Zheng et al. [3,9,28-30] have conducted numerous
experimental and theoretical studies on IL based working pairs. The group published several
papers on the evaluation and selection method, the thermophysical property measurement
and the modeling of NH3/IL, H2O/IL, HFC/IL, HC/IL and alcohol-IL systems [3,9,28-30]. They also
compared the thermodynamic performance of these IL based systems [3]. Kotenko et al. [41]
calculated the thermodynamic performance of these ILs and compared it with the performance
of the NH3/ H2O system.
In this work, NH3 (refrigerant) is paired with two ILs (absorbents) namely, 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide ([emim][Tf2N]) and 1-ethyl-3-methylimidazolium
thiocyanate ([emim][SCN]). These two ILs are chosen because the imidazolium class of IL is
most widely investigated and reported in literature [3-6,16,19,23-26]. It is useful to compare
and validate our results with the results reported in the literature [3,6,16,23,24,26]. In addition,
according to many researchers, these ILs are expected to have a thermodynamic performance
close to the traditional NH3/H2O working pair [3,6,16,23,24,26]. The structure, molecular weight
and decomposition, melting and glass crystallization temperatures of both ILs are provided in
Table 2.1.
7
Table 2.1 : Properties of [emim][Tf2N] and [emim][SCN].
Properties
[emim][Tf2N]
[emim][SCN]
C8H11F6N3O4S2
391.31
174899-82-2
1-ethyl-3-methylimidazolium
bis(trifluoromethylsulfonyl)imide
[emim]
[Tf2N]
C7H11N3S
169.25
331717-63-6
1-ethyl-3-methylimidazolium
thiocyanate
[emim]
[SCN]
1.52
1.12
806.1
1244.9
32.6
0.1818
717.3
1013.6
22.3
0.3931
254.15-257.15
170-195
258-271
> 623
< 253.15
174.5-179
267
523
Structure
General
Chemical Formula
Molecular Weight (g/mol)
CAS-number
IUPAC name
Cation
Anion
Density (20oC and 1 atm )
(g/cm3)
Critical Parameters42,43
Boiling Temperature (K)
Critical Temperature (K)
Critical Pressure (bar)
Accentric Factor
Temperatures44-48
Melting (K)
Glass Transition (K)
Crystallization (K)
Decomposition (K)
2.3 Absorption Refrigeration Cycle
Figure 2.1 depicts a schematic representation of the single-effect absorption refrigeration cycle
using a refrigerant/IL mixture as working pair. The system consists of an absorber (A), a
condenser (C), an evaporator (E), a generator (G), a liquid pump (P), a solution heat exchanger
(HX) and two expansion valves (EXP). The cycle principle is illustrated in Figure 2.2 by plotting
the vapour pressure (Ln P) against temperature (1/T).
8
2. Ionic Liquids in Absorption Refrigeration Cycle
Figure 2.1: Schematic of a single-effect absorption system
Figure 2.2: The absorption cycle on a ln(P) - 1/T diagram
In Figures 2.1 and 2.2, state points 1-8-9-10-1 represent the refrigerant circulation loop, and
state points 2-4-7-5-2 the solution circulation loop. In the absorber, refrigerant vapour from the
evaporator (state point 1) is exothermally absorbed by releasing the heat QA. The strong IL
solution at state point 5 absorbs refrigerant vapour and results in a weak IL solution at state
point 2. The solution is then pressurized by the liquid pump. The solution heat exchanger
preheats the weak IL solution of state point 3 to state point 4 using the high temperature
strong IL solution flow from the generator. A superheated refrigerant vapour at high
temperature and pressure (state point 8) is separated from the weak IL solution by heating the
2.4 Thermodynamic Model
9
Table 2.2: Main components and the processes of an absorption refrigeration cycle [6].
Components
Evaporator
States
10 ------> 1
Absorber
1, 5 -------> 2
Pump
Solution HX
2 -------> 3
3 -----> 4, 7 -----> 6
Generator
4 -------> 7, 8
Condenser
Expansion device
8 -------> 9
6 -----> 5, 9 -----> 10
Thermodynamic Process
Heat absorption from a device (QE)
Refrigerant absorption into IL/Heat
rejection to ambient (QA)
Isentropic Pressurization
Regenerative pre-heating
Refrigerant desorption from IL/Heat input
from source (QG)
Heat rejection to ambient (QC)
Isenthalpic expansion
generator with the heat load QG. The refrigerant vapour then enters the condenser, liquefies
into liquid refrigerant (state point 9) and releases the heat QC. Thereby, the mixture solution
becomes the strong IL solution (state point 7) and returns to the absorber through the solution
heat exchanger (state point 6) and the expansion device. The high pressure liquid refrigerant
(state point 9) then passes through the expansion valve (state point 10) and goes into the
evaporator. In the evaporator, the refrigerant is vaporized (state point 1) and the heat load QE
is removed. Table 2.2 summarizes the main components and their processes in an absorption
refrigeration cycle.
2.4 Thermodynamic Model
To simulate the thermodynamic performance of the single-effect absorption refrigeration cycle
with NH3/IL as working pair, the following assumptions are made to simplify the calculations
[3,6,11]:
1. The flow through the components is under steady state.
2. The operating pressures in the condenser and the generator are equal (PC=PG), and
similarly, the operating pressures in the evaporator and the absorber are equal (PE=PA).
3. The refrigerant leaves the evaporator (state point 1) as saturated vapour at the dew point
with T=TE=T1 and the condenser (state point 9) as saturated liquid at the bubble point with
T=TC=T9.
4. The solution leaves the generator in equilibrium as saturated liquid at its end generation
temperature, i.e., T7=TG.
5. The refrigerant enters the condenser (state point 8) as superheated vapour at the
temperature T=T8. The refrigerant leaves the generator at the temperature equal to the end
generation temperature, i.e., T8=TG.
6. The condition at state 2 (solution outlet to the absorber) is a subcooled weak IL solution
with a degree of subcooling set to 5 K.
7. The pressure drop due to friction in the connecting pipelines is neglected.
8. The minimal heat transfer temperature difference of the solution HX is set to 5 K.
9. The heat losses through connecting pipes and solution heat exchanger are negligible.
10. No chemical reaction takes place between NH3 and the IL upon mixing.
10
2. Ionic Liquids in Absorption Refrigeration Cycle
The first step in the cycle calculations is to obtain the saturated vapour pressure of a pure
refrigerant (NH3), PE and PC, at given temperatures of TE and TC. Thus, states 1 and 9 can be
determined. Consequently, the pressure levels for the absorber and the generator can be set
using the second assumption. Using the fifth and the sixth assumption, the given temperatures
TA and TG, and the pressure levels obtained above, the fractions of each component for both
weak and strong IL solutions can be determined. Thus, states 2 and 7 can be determined. Using
pump work and the eight assumption, states 3 and 6 can be determined. Then using the heat
balance of solution HX, state 4 can be determined. With the fourth assumption, state 8, which
is pure refrigerant vapour from the generator at superheated state, can be determined. From
isenthalpic expansion, states 5 and 10 can be determined.
2.4.1 Enthalpy calculation
In this section, a method to calculate the enthalpy at each state point of the cycle is discussed.
For a multi-component system, the total enthalpy of the mixture, hmix is expressed as [3,6],
(2.1)
where,
and
are the enthalpy of pure NH3 and IL,
and
are the mole fractions of
NH3 and IL, respectively and
is the excess enthalpy or the absorption heat.
The enthalpy data of the pure NH3 is directly obtained from NIST's Refprop [49]. The enthalpy
of pure IL can be written as a combination of its ideal part,
,and its residual part,
, and is
expressed as,
(2.2)
The ideal and residual enthalpies for pure IL can be obtained from their respective heat
capacities as,
(2.3)
where,
is the reference enthalpy.
and
are ideal and residual heat capacities of the
pure IL, respectively.
= 238.8 kJ/kg at T0 = 250.15 K and P0 = 1 MPa is adopted as an arbitrary
reference state for equation (2.3).
The excess enthalpy, , can be derived from the residual enthalpies using an equation of state
(EOS) and the mixing rule based on the experimental vapour-liquid equilibrium (VLE) data with
the help of following expression [16,24,26],
(2.4)
A cubic EOS of the generic Redlich-Kwong (RK) type is used to calculate the residual enthalpies.
The detailed procedure is described in the literature [16,24,26]. The critical parameters and the
2.4 Thermodynamic Model
11
binary interaction parameters required for calculating the residual enthalpies in equation (2.4)
are taken from literature [16,24,26].
2.4.2 Cycle performance
The overall energy balance for the single-effect absorption refrigeration cycle is given as,
(2.5)
where,
is the pumping power. Two important parameters for evaluating the thermodynamic performance of single-effect absorption refrigeration cycle are the coefficient of
performance (COP) and the circulation ratio, f [3,6,11,16,23]. From the mass balance in the
absorber or generator, we have
(2.6)
because the pure absorbent (IL) in the inflow and the outflow remains the same. The quantity
of IL remains same because of its negligible vapour pressure. This provides a circulation ratio or
mass flow rate ratio, f, defined as,
(2.7)
where,
and
are mass fraction of the absorbent in strong and weak IL solution flowing
out of the generator (G) and absorber (A), respectively.
is the mass flow rate of the weak IL
solution flowing out of the absorber and
is the mass flow rate of the refrigerant (NH3). In
equation (2.7),
and
are obtained from VLE (solubility) data.
According to the heat balance, the heat loads of each of the components, i.e., evaporator,
condenser, generator and absorber, are given by,
(2.8a)
(2.8b)
(2.8c)
(2.8d)
The pumping power,
, is calculated as,
(2.9)
12
2. Ionic Liquids in Absorption Refrigeration Cycle
The coefficient of performance (COP) of a single-effect refrigeration cycle can be defined as,
(2.10)
Equation (2.10) can also be expressed in terms of the enthalpy, h, and the circulation ratio, f ,as,
(2.11)
where,
stands for enthalpy at state point i of the cycle in Figure 2.2.
To calculate the COP of the cycle, properties like enthalpy and circulation ratio are required.
According to equation (2.3) and (2.7), the heat capacity and excess enthalpy are required to
calculate the enthalpy and the VLE data (solubility) are required to calculate the circulation
ratio. Unlike conventional organic liquids or working pairs for which a lot of experimental data
are available, much is unknown about the behavior and properties of ILs. There are various
tools available to investigate the required thermodynamic properties such as conducting
experiments for solubilities and heat capacities [16,17,23,26,29,33], group contribution
methods for heat capacities [50,51] and EOS model for excess enthalpies [16,23,26,29]. These
tools have been used for decades. However, one such tool which can also be used to calculate
the thermodynamic properties are MC molecular simulations. In last ten years, a lot of work has
been done using molecular simulations [17-20,32]. Molecular simulations are capable of making
quantitative predictions of thermodynamic properties of ILs. It can act as an important tool for
predicting properties under conditions where experiments are difficult to conduct, such as high
temperatures and pressures. In this work, MC molecular simulations are used to predict the
thermodynamic properties of the working pair.
3
Monte Carlo Molecular Simulations
Molecular simulations are computer experiments to calculate the macroscopic properties of a
molecular system in equilibrium using the principles of statistical mechanics. MC simulations
play an important role in quantitative prediction of the thermodynamic and transport
properties especially under conditions where physical experiments are difficult to conduct.
Simulations are also useful for predicting thermodynamic properties of mixtures such as
solubilities which are much harder to determine experimentally. Simulations can help to
develop an understanding of the physical chemistry of newly evolving materials such as ILs.
3.1 Molecular Simulation approach
A system is defined by the number of molecules N at some defined thermodynamic state. The
statistical nature of a molecular system determines that an observed macrostate is an average
of a series of microstates in which the system can exist. In the classical approach, a microstate
is defined by the instantaneous set of positions and momenta (rN, pN) of all molecules in the
system. The collection of all such possible microstates is called an ensemble and the
corresponding average over all possible microstates of the system is the ensemble average [52].
To describe the basic principles of the MC simulation, we focus on simulation of a system with a
fixed number of molecules N, in a given volume V at a temperature T, i.e., the NVT ensemble.
The physical properties of a system in equilibrium can be determined from the ensemble
average as [52],
(3.1)
where
stands for the ensemble average of a macroscopic property
in some
microscopic state i. The Hamiltonian function
of the system corresponds to the
total energy of an isolated system as a function of the momenta and coordinates of the
constituent molecules:
, where
is the kinetic energy and
is the potential
energy. T is the temperature and
is the Boltzmann's constant. 1/kBT is also referred to as β
[52]. In equation (3.1),
is expressed as a function of coordinates and momenta. The
integration over momenta can be carried out analytically and independently from coordinates
as it is not multidimensional. The integration over coordinates is however difficult to be
computed analytically because of its multidimensional nature [52]. Therefore, equilibrium
properties of a system that comprises of N molecules at constant volume V and temperature T
can be determined by the configurational average as [52],
13
14
3. Monte Carlo Molecular Simulations
=
(3.2)
where the Boltzmann factor, exp [-U(rN)/kBT ] is proportional to the probability of the system to
be in a specific configuration as a function of the configuration's potential energy
and
temperature T. The partition function of the system at constant number of molecules, volume,
and temperature, i.e., the NVT ensemble is [52],
(3.3)
where
is the thermal de Broglie wavelength. The partition function is related to the
macroscopic properties of a system in thermodynamic equilibrium. All thermodynamic
properties of a system (for e.g. energy, entropy etc.) can be determined from the partition
function. However, the integration over all possible microstates is in practice impossible.
Numerical techniques are required to compute the multidimensional integral analytically [52].
One such technique is the MC importance sampling algorithm developed by Metropolis et al.
[53].
The basic idea behind importance sampling is to sample many points in the region where the
Boltzmann factor is large. Random sampling would in principle approximate macroscopic
properties calculated from configurational averages as in equation (3.2). However, random
sampling with equal probability for all configurations has the problem that most of the
configurations have a negligible Boltzmann factor [52]. This would require a huge numbers of
samples to obtain any statistically meaningful results. The MC importance sampling algorithm is
used in this work for determining the macroscopic properties.
3.1.1 The Metropolis method
As explained, it is in general not possible to evaluate the integral in equation (3.2) by random
sampling, Metropolis et al. [53] provided a solution to this problem, proposing an algorithm in
which configurations are generated with a probability proportional to the Boltzmann factor.
Different configurations are generated by performing molecular trial moves, e.g. molecular
translation or rotation. Every trial move is an attempt to change the configuration of the system
from a configuration o (old) to n (new). The general approach is to first prepare the system in a
configuration which we denote by o. Next we generate a new trial configuration which we
denote by n, by adding a small random displacement to o. Whether the trial configuration
should be accepted or rejected is decided by the "acceptance rules" [52].
In MC simulations, obeying detailed balance is a sufficient condition for sampling the correct
distribution [52]. The condition of detailed balance is given by,
(3.4)
3.2 Technical Details - System set-up
15
where,
:
probability of the system to be in configuration o.
:
probability for attempting a change in the configuration of the system
from o to n.
:
probability that a change in the configuration of the system from o to n is
accepted.
Usually the probability of attempting a forward and backward move is the same i.e.
[52]. This simplifies equation (3.4) to,
(3.5)
Hence, the transition from a configuration o to n is given by,
(3.6)
In the Metropolis scheme, the probability of accepting a change in the configuration of the
system from o to n is chosen as follows [53],
(3.7)
In other words,
if Pn < Po
if Po ≤ Pn
Hence, the acceptance rule for a trial change from configuration o to n in the NVT ensemble is
given by [52],
(3.8)
where U(o) and U(n) are the potential energies of configuration o and n, and
= U(n) -U(o) is
the energy difference between both configurations. A configuration change associated with a
negative difference in the potential energy ( < 0) will always be accepted. Changes carrying
out a positive difference in energy (
> 0) will be accepted with a probability proportional to
3.2 Technical Details - System set-up
The simulation cell and boundary conditions: In order to simulate bulk phases, periodic
boundary conditions are employed so that the results of N particle model system can be
extrapolated to macroscopic bulk values [54]. The simulation box with all its particles, is
replicated throughout the space as shown in Figure 3.1.
16
3. Monte Carlo Molecular Simulations
Figure 3.1: Schematic representation of periodic boundary conditions
The boundary of the periodic box does not have any significance and will not affect any
property of the system. However, the shape and orientation plays an important role in
determining the property under study [22]. If we use periodic boundary conditions, the smallest
perpendicular width of the periodic box has to be larger than twice the spherical cut-off [54] or
in other words, cut-off should be less than half the diameter of the periodic box. Cut-off
distance is the distance beyond which all the intermolecular interactions are truncated or not
considered.
Equilibration and production runs: To start the simulation, the system is initialized by inserting
N molecules into the simulation box at random positions. The ability to insert molecules is
important for an initialization process. All initial conditions are in principle acceptable as the
equilibrium properties of the system do not depend on the initial conditions. During the
initialization period, simulation is performed to rapidly achieve an equilibrium molecular
arrangement. The final step is to start the actual production run where all the statistics are
collected and properties under study are measured. The production run is started immediately
after the equilibration of the system. For MC simulations, the duration can be measured in form
of ‘MC cycles’. It is usually preferable to use the final (well equilibrated) configuration of an
earlier simulation at a nearby state point as the starting configuration for a new run [52].
Interactions (short/long ranged): Interaction basically comprises of two kinds of interactive
forces namely, van der Waals ( e.g. Lennard- Jones potential) and Coulombic interactions.
van der Waals: The most common functional form for the van der Waals interaction is the
Lennard-Jones potential which is a mathematically simple model that approximates the
interaction between a pair of neutral atoms or molecules. It is most commonly expressed as
[54],
(3.9)
where, is depth of the potential well or the 'strength' parameter, is the 'size' parameter or
the finite distance at which the inter-particle potential is zero and r is the distance between the
particles as shown in Figure 3.2.
3.2 Technical Details - System set-up
17
Figure 3.2: The Lennard-Jones potential (blue line). Energy calculations are considered within the cut-off
distance and the neglected energy called the 'tail correction' is approximated (red area).
The Lennard-Jones parameters are used directly in generic force fields (discussed in next
section) and a 'mixing rule' is applied to compute the interaction between different type of
atoms. Two most commonly used mixing rules are [54]:

Arithmetic (also called Lorentz-Berthelot)
(3.10)

Geometric (also called Jorgensen)
(3.11)
The van der Waals potentials can be truncated at a certain distance called as cut-off distance rc
beyond which the interactions are considered sufficiently small as shown in Figure 3.2 [54]. It
results in a shorter computational time. The error that results when the interactions are
ignored can be made arbitrarily small by choosing cut-off distance sufficiently large [52]. If the
interactions are not around zero for r cut-off, truncation of interactions at cut-off will result in
large error in
which can be corrected by adding a tail correction to
as shown in
Figure 3.2 and equation (3.12) [52].
(3.12)
where N is the number of the particles and
is average number density (assuming
homogeneous density). The last term in equation (3.12) is the tail correction. The cut-off
distance and whether to use tail correction or not should be considered as part of the force
field. Common approaches include to simply truncate and shift the van der Waals potential to
18
3. Monte Carlo Molecular Simulations
zero at the cut-off and ignore all interactions beyond the cut-off or use switching function
where the energy is forced to smoothly go to zero [54].
Coulombic interactions: The total electrostatic potential energy of interaction between point
charges qi at the positions ri is given by [54],
(3.13)
where is the electrical permittivity of free space. The Ewald summation technique is used to
solve the coulombic interactions [52,54].
3.3 Force fields
The accurate prediction of macroscopic properties by a molecular simulation depends on the
quality of the force field (FF) used to model the interactions in the system [22]. The standard
molecular mechanics FF, with the functional form as Taylor expansion in bonds, bends (angles),
dihedrals (torsions), etc. is described as [17-19,21,22,32],
(3.14)
where the first four terms denote the intramolecular contributions and the remaining term
denotes the intermolecular contributions to the total energy of the system,
, respectively.
The intramolecular terms for bond stretching and proper and improper angle bending are
described by harmonic potentials with force constants
and
and nominal values of
and
. The dihedral motion is described by a Fourier cosine series where is the
dihedral angle and the remaining parameters are fit to reproduce the torsional energy. The
intermolecular potential is comprised of Lennard-Jones and Coulumbic energy terms [22]. The
FF tries to reproduce a range of equilibrium properties, including vapor–liquid coexistence
densities, vapor pressures, enthalpies of vaporization, and critical properties. Equation (3.14) is
referred to as a force field. FFs have developed over the years and many parameters exist for a
wide range of structures. These parameters determine the quality of the FF.
3.4 MC moves or trial moves
Two general trial moves used to create all possible configurations are translational and
orientational (rotational) moves. Trial moves that involve only moves of the molecular centre of
mass are known as translational moves. A possible method for creating a trial move is to add
small random displacement to the coordinates of the molecular centre of mass. A large
displacement will result in a configuration with high energy and the trial move will probably be
rejected. Orientational (rotational) moves change the molecular orientation when we are
simulating molecules rather than atoms. However, if the molecules under consideration are not
3.5 Ensembles
19
rigid i.e. flexible molecules then it involves trial moves that change the internal degrees of
freedom of a molecule, i.e., bond length and possibly some bond angles. However,
determination of
by equation (3.1) or (3.2) becomes difficult in case of larger flexible
molecules. During MC simulations, both translational and orientational moves can be
performed simultaneously [52].
3.5 Ensembles
Various ensembles can be simulated using MC simulations. The basic ensembles are NPT,
NVT, VT etc. Many others ensembles can be derived from these basic ensembles such as the
Gibbs ensemble, the osmotic ensemble etc. An ensemble is basically a collection of groups of
microstates with the same macroscopic conditions (e.g. N, V and T). The corresponding average
of an observable over all possible microstates is "ensemble average" as stated in section 3.1.
3.5.1 The NPT ensemble
The NPT ensemble is also called the isothermal-isobaric ensemble. It samples the phase space
of a system with a constant number of particles N at constant pressure P and temperature T.
This ensemble plays an important role as real experiments are usually carried out under
constant pressure and temperature. Using this ensemble, heat capacity at constant pressure, Cp
is calculated in this work. The isothermal-isobaric ensemble partition function is given by [54],
(3.15)
where, the factor C is included to make the partition function dimensionless. In addition to the
particle moves performed at constant T (same as NVT ensemble), changes in the volume
(volume trial moves: Vn = Vo +
) are also attempted. However, often an alternative move is
used, where volume of the simulation box is changed not in V but in ln (V). This is because for
specified T and P that place systems near liquid-gas phase coexistence, the volume fluctuations
can be large as the system traverses between two phases. As a result, the simple volume
increments in V can become inefficient.
Compared to the canonical ensemble (NVT), in the NPT ensemble, the volume is an additional
degree of freedom. So the probability distribution function must include volume as variable as
it can take any value at thermal equilibrium. Hence, the probability that a system is in a
determined configuration o (old) with volume ln V and coordinates sN is proportional to [54],
(3.16)
Given this probability distribution function, moves that are accepted to satisfy detailed balance
are performed by particle moves (changes in the position/orientation of molecules or molecular
configuration in case of flexible molecules) and volume moves (changes in volume of the
system, i.e., size of the simulation box). Particle moves that do not involve a change in the
volume of the system are equivalent to a change in the NVT ensemble and the acceptance
20
3. Monte Carlo Molecular Simulations
criterion for a trial change from a configuration o (old) to n (new) is equivalent to one
presented in section 3.1.1 [54],
(3.17)
The acceptance criterion for a trial change in ln V is given by [54],
(3.18)
In the thermodynamic limit (V → ), the relative fluctuation of volume is negligible and the
difference between NPT ensemble and NVT ensemble vanishes.
Residual Heat Capacity using NPT ensemble
In this work, the residual heat capacity using MC simulation in the NPT ensemble is calculated
following the procedure described by Lagache et al. [20]. The total heat capacity at constant
pressure is defined as
(3.19)
where
is the enthalpy which is defined as
.
is the
intermolecular potential energy,
is the intramolecular potential energy and is the kinetic
energy [20]. The total heat capacity can be split into two parts namely, ideal gas part and
residual part. Equation (3.19) can then be written as
(3.20)
Thereby, introducing,
(3.21)
(3.22)
where,
is the ideal gas enthalpy which is a function of temperature
only and
is the residual enthalpy [20]. The ideal gas part is
determined by the intramolecular interactions while the residual part describes the interaction
between molecules [20].
The residual heat capacity can be determined from MC simulation in the NPT ensemble using
the fluctuation formula expressed as equation (3.23) [18,20]. This equation expresses the
derivative of some average property X, i.e.,
with respect to temperature . For the NPT
ensemble it is,
(3.23)
3.5 Ensembles
21
where,
is the configurational enthalpy [20]. The derivation of equation
(3.23) for the NPT ensemble is provided in Appendix A.
Substituting
in equation (3.22), it can also be written as
(3.24)
Using equation (3.23), the derivative of
and
vs.
is obtained as
(3.25)
Equation (3.25) is then used in the NPT ensemble simulations to calculate the residual heat
capacity [18,20]. In principle equation (3.25) is another form of equation (3.22) obtained by
substituting
.
So far, the ideal gas part of the total heat capacity was not considered. In principle, the ideal
gas part can be computed from MC simulations [20,22,32]. But the atomistic force fields used in
MC simulation result in large errors when calculating intramolecular contributions to enthalpy,
because of some approximations which are made while developing the force fields [22]. These
approximations are the harmonic approximation which results in an overestimation of the
vibrational energy of molecules and the neglect of off-diagonal coupling terms [22]. Hence,
using MC simulations for the ideal gas part which are sensitive to intramolecular contributions,
would result in erroneous predictions of the ideal gas part of the heat capacity [20,22]. The
ideal gas part of the heat capacity is then obtained from quantum mechanical calculations. This
is done via a frequency analysis of the optimized cation and anion structures [18,22,32].
Ideal Heat Capacity using Quantum Mechanical calculations
The partition function of a system with a fixed number of molecules N, a volume V and a
temperature T, i.e., the NVT ensemble is [61],
(3.26)
where, Ei corresponds to total energy of the system in microstate i. The intermolecular and the
intramolecular energies constitute the total energy of the system. For an ideal system, it is
assumed that there are no interactions between the molecules as they are reasonably far apart.
As a result, the intermolecular energy is ignored. Hence, for an ideal system the total energy
corresponds to total intramolecular energy. At the molecular level, the total intramolecular
energy of the system, E in a microstate is the sum of electronic, vibrational, rotational and
translational energies [61],
22
3. Monte Carlo Molecular Simulations
(3.27)
The electronic energy corresponds to energy levels of electrons in atoms, ions or molecules.
The vibrational and rotational energy refer to vibrational and rotational degrees of freedom of
a system. The translational energy results from the movement of the center of mass. Equation
(3.26) can be rewritten as,
(3.28)
The energies are assumed to be independent of each other [61]. As a result, the summation in
equation (3.28) can be done separately. Equation (3.28) then can be rewritten as,
(3.29)
Using equation (3.26),
(3.30)
The total intramolecular energy of the system in equilibrium can be calculated by using the
definition of the ensemble average explained in section 3.1 as,
(3.31)
where, is the probability of the system to be in a specific configuration as a function of the
configuration's energy and temperature T. Using equation (3.26), equation (3.31) can be
rewritten as,
(3.32)
The total intramolecular energy per mole of the system at constant V and fixed N can be
obtained from the partition function as [61],
(3.33)
where, R = NAkB is the universal gas constant. The heat capacity at constant volume, Cv can be
calculated as,
(3.34)
The heat capacity at constant pressure for ideal gas part can be calculated as,
(3.35)
3.5 Ensembles
23
To compute
with equation (3.35), the electronic, rotational, vibrational and translational
partition functions are required. The contribution to the heat capacity at constant volume due
to electronic motion is zero as there is no temperature dependent term in the electronic
partition function. For a single atom or molecule, the translational partition function in the NVT
ensemble is [61],
(3.36)
where, m is the mass of an atom or a molecule and h is the Planck's constant. Using equations
(3.33) and (3.34), the contribution to the heat capacity at constant volume due to translation is,
(3.37)
The vibrational partition function of a system is defined as [61],
(3.38)
where,
is the vibrational frequency and K is the vibrational mode. The characteristic
vibrational temperature is defined as
. Using equations (3.33) and (3.34), the
contribution to the heat capacity at constant volume due to vibration is,
(3.39)
For a single atom,
and its contribution to the heat capacity at constant volume is zero
as it does not depend on temperature. For the general case of a non-linear polyatomic
molecule, the rotational partition function is [61],
(3.40)
where,
is the symmetry number and I is the moment of Inertia of the molecule. Using
equations (3.33) and (3.34), the contribution to the heat capacity at constant volume due to
vibration is,
(3.41)
The derivations of equations (3.37), (3.39) and (3.41) are provided in Appendix B. The ideal gas
part of the heat capacity at constant pressure,
is then calculated by using equations (3.37),
(3.39), (3.41) in combination with equation (3.35).
24
3. Monte Carlo Molecular Simulations
3.5.2 The Osmotic Ensemble
An ensemble effective in simulating solubilities is the osmotic ensemble. It simulates a
thermodynamic equilibrium between an infinite reservoir of fluid or fluid mixture (guest) (NH3
in this work) and host molecules (ILs in this work) confined in the simulation box. The control
parameters for this ensemble are the temperature, T, the pressure, P, the fixed number of
solvent or ILs molecules in simulation box, Nsolvent, and the fugacity of the solute or NH3 in the
reservoir, fsolute. In this ensemble, the volume of the system, V, and the number of the particles
of solute, Nsolute, are not fixed but allowed to fluctuate in response to applied pressure. In this
work, the recently developed Continuous Fraction Component Monte Carlo (CFCMC) method is
used in osmotic ensemble to calculate the solubility of NH3 in both the ILs [55].
Solubility calculations using Continuous Fraction Component Monte Carlo
(CFCMC) method
The accuracy and precision of phase equilibria calculations using MC simulations depend on
insertions and deletions of molecules from a system to either evaluate the chemical potential
or change of composition [55]. Standard MC methods suffer from a major drawback that
insertion and deletion probabilities become very low at high densities [54]. Therefore, new
open system MC procedures based on advanced biasing/sampling methods were designed to
overcome this drawback partially [54,55]. An example of such a method is the Continuous
Fraction Component Monte Carlo (CFCMC) method developed by Shi and Maginn [55]. In this
method, the gradual insertions and deletions of molecules are done by using a continuous
coupling parameter (λ) and an adaptive biasing potential [55]. The ensemble gets expanded by
a "fractional" molecule, which is coupled to the system through the parameter λ. The λparameter is confined within the interval [0, 1]. The corresponding strength of the intermolecular interactions (Lennard-Jones and Coulombic interactions) between the fractional
molecule and the surrounding molecule are scaled using the same parameter λ. The intermolecular interactions are scaled in such a way that they show the correct behavior at the
limits of λ = 0 and λ = 1, i.e., conventional intermolecular potential at λ = 1 and no interactions
in the limit of λ → 0 [54]. The CFCMC is a method in which continuous composition changes are
carried out by λ moves in addition to the conventional MC moves for relaxation of the system
(translation, rotation and volume changes) [54,55]. A schematic representation of the osmotic
ensemble using CFCMC method is shown in Figure 3.3.
A λ change will result is three possible outcomes [54]:

λ remains between 0 and 1: There is no change in the number of the particles, nor in
the positions, nor in the intra-molecular energies. Only inter-molecular energy
changes.

λ becomes larger than 1: When λ exceed unity, λ = 1 + ϵ with 0 < ϵ < 1 which results in
a transformation of the current "fractional" molecule to a "whole" molecule and a new
fractional molecule with λ = ϵ is randomly inserted to the system.
3.5 Ensembles
25
Figure 3.3: A schematic representation of the osmotic ensemble using the CFCMC method [56].

λ becomes smaller than 0: When λ falls below 0, λ = -ϵ, which results in current
"fractional" molecule being deleted from system and a new fractional molecule is
randomly inserted with a new λ = 1 - ϵ.
There is a free energy associated with a change in λ value. Whenever the free energy barrier
for λ transitions is large, the system can become "stuck" at values of λ between 0 and 1 [55].
Many systems show such behaviour where λ changes are hard. Hence, an additional bias factor
or potential on λ is used, where each λ has an associated bias factor. It improves the probability
of transitions among adjacent λ values. These bias factors are determined using an iterative
algorithm proposed by Wang and Landau, also known as Wang-Landau algorithm. A careful
calibration of bias potential can make the λ histograms (graph of probability of λ vs λ) flat which
is a must for the result of the simulation to be correct and hence can avoid the system getting
stuck in certain λ-ranges [54].
4
Simulation Details
MC simulations are performed to investigate the heat capacity of ILs, the excess enthalpy, and
the solubility of NH3 in ILs namely, 1-ethyl-3-methylimidazolium bis(trifluoro-methylsulfonyl)
imide ([emim][Tf2N]) and 1-ethyl-3-methylimidazolium thiocyanate ([emim] [SCN]). The
structure, molecular weight and decomposition, melting and glass crystallization temperatures
of both ILs are provided in Table 2.1. The accurate prediction of the properties by MC
simulation depends on the quality of the applied force field. Here, a classical force field
developed by Maginn et al. [17-19,22,32,57] is used for the ILs. The functional form of the
potential energy to describe the system for the above mentioned IL molecules is explained by
equation (3.14) in section 3.3. It includes intramolecular interactions such as bond-stretching,
angle-bending, and torsions. Intermolecular interactions are described via the Lennard-Jones
potential (LJ) and electrostatic interactions are considered via the Ewald summation technique
with a relative precision of 10-5 [52,54]. Force field parameters for ILs are taken from Maginn
et. al [17,32,57] and are provided in Appendix C. A schematic representation of [emim], [Tf 2N]
and [SCN] with atom labels for force field parameters is also provided in Appendix C. The
TraPPE force field is used for NH3 [58]. The parameters are also provided in Appendix C. The
NH3 molecules (solute) are considered rigid in the simulations. The anion and the alkyl part of
the cation of the IL molecules (solvent) are considered flexible whereas the ring of the cation is
considered rigid. The total charge on the anion and cation of the IL is ±0.8 (scaled). The LJ
parameters for the interactions between unlike atoms are obtained by using the LorentzBerthelot combining rules [17-19,22,52,54]. The LJ interactions are truncated and shifted at 12
Å and no tail correction is used.
The MC simulations are performed using the molecular simulation software package RASPA
[59]. The evaporation temperature TE, absorber temperature TA, condensation temperature TC,
and end generation temperature TG are set to 10 oC, 30 oC, 35 oC and 100 oC, respectively. The
simulations are performed at temperatures ranging from 283.15 K to 393.15 K and pressures
ranging from 4 to 19 bar. The initial configurations are generated by randomly inserting IL
molecules in the simulation box. Periodic boundary conditions are used to approximate a
continuous system [52,54]. The number of ILs molecules are chosen to satisfy the estimated
densities and to assure that the simulation box is always larger than twice the cutoff distance
[52,54].
4.1 Simulation details for calculating the Heat Capacities
Ideal Heat Capacity: The ideal gas part of the heat capacity is computed by quantum
mechanical calculations using the Gaussian software [62]. The basic procedure to calculate the
ideal gas part of the heat capacity is explained in section 3.5. The simulations are performed for
[emim], [Tf2N] and [SCN] in the ideal gas state. The possible conformers of the ions and their
relative energies are analyzed using molecular mechanics. A conformer is an isomer of a
27
28
4. Simulation Details
molecule that differs from another isomer by the rotation about a bond in the molecule.
Rotation about bonds are restricted by an energy barrier. To interconvert one conformer to
another, the energy barrier has to be crossed. The calculations are performed for the most
stable conformer(s). To calculate the ideal gas part of the heat capacity, the calculations are
performed using optimized geometries of the most stable conformer(s) at desired
temperatures. The Density Functional Theory (DFT) methods and in particular the B3LYP
functional with 6-31+G(2df,p) basis set is used for the computation. The DFT is a quantum
mechanical method that describes the molecule based on its electron density. The DFT
methods and in particular B3LYP functional is most commonly used [68]. In addition to the good
accuracy of these methods, they require significantly less time than higher level of theory [68].
The basis set is a mathematical representation of atomic orbitals (electron distribution) around
a molecule. Polarization (indicated by 2df,p) and diffusion (indicated by +) functions are also
added to the basis set for an accurate calculation of the anion's frequencies. Paulechka et al.
[68] concluded that 6-31+G(2d,f) basis set is satisfactory for calculation of the frequencies of
the ions. Hence, the same basis set is used in this work. The scaling factor of 0.965 is used for
computation, consistent with B3LYP/6-31+G(2df,p) level of theory [63]. The scaling factor is
used to scale the ion frequencies and produce a good agreement with the experimental
frequencies.
Residual Heat Capacity: The residual part of the heat capacity of [emim][Tf 2N] and [emim][SCN]
is computed by performing MC simulations in the NPT ensemble. A brief description of the NPT
ensemble can be found in section 3.5.1. The simulations are performed with 55 and 70 IL
molecules for [emim][Tf2N] and [emim][SCN], respectively. The system is equilibrated by
conducting an equilibration run of 105 MC cycles. Once the system has reached equilibrium, the
simulations are continued with a production run of 22 to 24 million cycles. The probabilities of
randomly selected moves are: 1 % volume change move, 33 % rotational move, 33 %
displacement move and 33 % CBMC move (partial regrowth).
4.2 Simulation details for calculating the Solubility
The solubility of NH3 in [emim][Tf2N] and [emim][SCN] is computed by conducting MC
simulations in the osmotic ensemble. NH3 molecules are inserted or removed via the
Continuous Fractional Component Monte Carlo (CFCMC) technique developed by Shi and
Maginn [55]. A brief description of the osmotic ensemble and the CFCMC method can be found
in section 3.5.2. The simulations are performed with 52 and 70 IL molecules for [emim][Tf2N]
and [emim][SCN], respectively. An equilibration run of 105 MC cycles is performed to obtain a
flat λ histogram (graph of coupling parameter λ vs. probability of λ) followed by a production
run of 5 million cycles. The Wang-Landau (WL) sampling scheme is used to obtain the required
biasing function [60]. The probabilities of randomly selected moves are: 1 % volume change
move, 33 % λ-change (partial insertion or deletion), 33 % rotational move, 33 % displacement
move and 33 % CBMC move (partial regrowth).
In RASPA, the number of MC steps in a cycle is defined as the total current number of
molecules in the system or simulation box. To compute the residual heat capacity and solubility
4.2 Simulation details for calculating the Solubility
29
from the simulation, block averages are used. The confidence intervals are obtained by
assuming statistical uncertainties of twice the standard deviation. To simulate millions of
production cycle, multiple parallel simulations with less production cycles are performed at the
same conditions. Subsequently, computed properties can be averaged over these conducted
simulations.
5
Results and Discussions
This chapter is divided into 2 sections. In first section, the thermodynamic properties such as
density, VLE data (solubility), excess enthalpy or absorption heat, and heat capacity computed
from MC simulations are reported and discussed. In second section, properties such as enthalpy
and circulation ratio required for determining the performance of the absorption refrigeration
cycle are reported. The heat capacity and excess enthalpy results from MC simulations are used
to calculate the enthalpies. The VLE (solubility) results from MC simulations are used to
calculate the circulation ratio. Thereafter, using enthalpies and circulation ratio, the COP of the
absorption refrigeration cycle is calculated. The effect of the end generation temperature, TG,
on the circulation ratio and the COP is also investigated. A sensitivity analysis is performed in
the end to analyze the effect of the heat capacity of an IL, the solubility of NH 3 in an IL and the
excess enthalpy on the performance of the absorption refrigeration cycle.
5.1 Thermodynamic properties from MC simulations
5.1.1 Density - validation of the force fields
The force fields used in this work are validated by comparing the computed densities of both
ILs with the available experimental data. The experimental and the computed densities as a
function of temperature for [emim][Tf2N] and [emim][SCN] are shown in Figure 5.1 (a) and (b),
respectively. The numerical results are presented in Table 5.1. The computed liquid densities
are in good agreement with the experimentally measured values for both ILs. Deviations
between experimental and computed densities range from +1.6% at 290 K to +2.1% at 393 K for
[emim][Tf2N] and from -1.1 % at 293.15 K to -2.3% at 363.15 K for [emim][SCN]. However, for
both ILs, computed densities agree very well with densities computed by Maginn et al. [65] and
Nguyen et al. [64] using the same force fields. The estimated uncertainty in this work are within
0.5% of the computed densities for both ILs. The force field of [emim][Tf 2N] overestimates the
densities whereas the force field of [emim][SCN] underestimates the densities. The linear trend
of the density with temperature is reproduced well. The computed densities for both ILs
decrease with an increase in temperature over the whole temperature range. This is because as
temperature increases, molecules of the ILs tend to move apart because of an increase in the
kinetic energy. This results in an increase in the volume of the system which decreases the
density. Over the whole temperature range, the density of [emim][Tf2N] is higher than the
density of [emim][SCN]. This is due to the larger size of [Tf2N] in comparison to [SCN] [3].
For NH3, the Trappe force field is preferred over the force field used by Maginn et al. [19]. The
Trappe force field describes the VLE of ammonia accurately in comparison to the force field
used by Maginn et al. This is shown in Figure 5.2.
31
32
5. Results and Discussions
Figure 5.1: Comparison of computed (open black squares) and experimental (filled black squares) [16,67] densities
obtained for (a) [emim][Tf2N], and (b) [emim][SCN]. The simulation results of Maginn et al. [65] and Nguyen et al.
[64] are represented by open colored symbols. The straight lines are linear fits to the simulation results and the
experimental data. Error bars are smaller than symbol size for [emim][SCN] experimental data.
5.1 Thermodynamic properties from MC simulations
33
Table 5.1: Comparison of computed and experimental densities for [emim][Tf2N] and [emim][SCN].
[emim][Tf2N]
T/K
290
300
310
323
333
343
353
363
373
383
393
ρ
exp
-3
ρ
exp
-3
/ kg m
1526.3
1516.1
1505.9
1492.6
1482.3
1472.1
1461.9
1451.6
1441.4
1431.1
1420.9
ρ
sim
/ kg m-3
1551.3
1544.5
1534.1
1523.4
1513.0
1503.1
1493.1
1481.7
1470.9
1460.1
1450.1
ρ
sim
ρa / kg m-3
5.8
7.3
6.8
5.9
6.7
7.2
6.5
4.3
6.9
5.5
4.0
deviation / %
1.6
1.9
1.9
2.1
2.1
2.1
2.1
2.1
2.0
2.0
2.1
[emim][SCN]
T/K
293
298
303
313
323
333
343
353
363
/ kg m
1119.1
1115.9
1112.9
1106.8
1100.8
1094.9
1089.1
1084.1
1078.7
/ kg m-3
1107.1
1109.9
1103.8
1099.1
1089.2
1081.6
1075.8
1063.5
1052.8
ρa / kg m-3
5.6
3.1
5.6
4.1
4.2
2.7
3.9
4.1
4.3
deviation / %
-1.1
-0.5
-0.8
-0.7
-1.1
-1.2
-1.2
-1.9
-2.3
a:simulation uncertainity. Experimental densities for [emim][Tf2N] and [emim][SCN] are taken from references
[16,67], respectively.
5.1.2 Solubility of NH3
The solubility of NH3 in [emim][Tf2N] and [emim][SCN] is computed using MC simulations in the
osmotic ensemble at 308.15, 347.15, 373.15, and 393.15 K and for pressures between 4 bar and
19 bar. The computed results are given in Table 5.2 and Table 5.3. The experimental data of
Yokozeki et al. [16,26] are correlated with a cubic EOS of the generic Redlich-Kwong (RK) type.
The same experimental data are correlated with NRTL model by Wang et al. [78]. The
comparison between the experimental data and the NRTL correlated data by Wang et al. [78]
for both ILs is provided in Appendix D. Using the NRTL model, the experimental data are
computed at desired conditions. The comparison between the NRTL correlated data and the
results computed from MC simulations is shown in Figure 5.3 and Figure 5.4 for [emim][Tf2N]
and [emim][SCN], respectively. Tables 5.2 and 5.3 also give the statistical uncertainty of the
computed results and a comparison with the NRTL correlated experimental data.
34
5. Results and Discussions
Figure 5.2: Comparison of experimental (black circles) and computed vapour and liquid coexistence densities for
NH3. The densities are computed using the Trappe force field (magenta triangles) and the force field used by
Maginn et al. (blue squares) [19]. Simulation error bars are smaller than the symbol size. The experimental data
are taken from the NIST Chemistry Webbook.
The results from the MC simulations are in reasonable quantitative agreement with the NRTL
correlated experimental data. The results indicate high solubilities of NH3 in these ILs. It can be
observed that simulations produce a solubility trend that is qualitatively similar to that
observed in the correlated data. It can be seen from Figure 5.3 and Figure 5.4 that solubility
increases with increase in pressure at constant temperature and decreases with increase in
temperature at constant pressure. The force field of [emim][Tf2N] and [emim][SCN]
overestimates the solubility of NH3 at different temperatures and pressures. The average
relative deviation between correlated and computed results is 17.4% at 308.15 K, 17.2% at
347.15 K, and 28% at 373.15 K for [emim][Tf2N], and 28.9% at 308.15 K, 19.5% at 347.15 K and
25% at 373.15 K for [emim][SCN]. Quantitative deviations in the solubilities can be attributed to
the used force fields. The relative deviations decrease at higher pressures indicating that these
force fields perform better at higher loading of NH3. At higher loading of NH3, ammoniaammonia interactions become more important. The Trappe force field is designed for phase
equilibrium data and describe these interactions well. This results in a smaller relative deviation
at higher loading. The magnitude of these average deviations are similar to the deviations
observed by Maginn et al. [19] and Urukova et al. [66] simulating highly soluble gases in ILs such
as SO2 and CO2 in various ILs. Maginn et al. [19] also computed NH3 absorption in [emim][Tf2N].
The force field of [emim][Tf2N] used in this work is similar to the one used by Maginn et al. The
computed results of Maginn et al. at 347.5 K are shown in Figure 5.3. Maginn et al. under
predicts the solubility of NH3. This is because Maginn et al. used a different force field for NH3.
5.1 Thermodynamic properties from MC simulations
35
Figure 5.3: Computed (coloured symbols) and correlated (black symbols) isotherms for NH3 in [emim][Tf2N].
Squares are at 308.15 K, diamonds are at 347.15 K, triangles are at 373.15 K, and circles are at 393.15 K. Curves are
guides to the eye. Experimental data (open symbols) at 348 K are taken from reference [16]. Experimental data at
308.15 K, 347.15 K, 373.15 K, and 393.15 K are calculated using NRTL model (filled black symbols) [78]. Error bars
are not available for the NRTL correlated experimental data. Dotted line (blue) corresponds to Maginn's simulation
result at 347.5 K. Error bars for Maginn's simulation results are smaller than symbol size.
Figure 5.4: Computed (coloured symbols) and correlated (black symbols) isotherms for NH3 in [emim][SCN].
Squares are at 308.15 K, diamonds are at 347.15 K, triangles are at 373.15 K, and circles are at 393.15 K. Curves are
guides to the eye. Experimental data (open symbols) at 348 K and 372.8 K are taken from reference [26].
Experimental data at 308.15 K, 347.15 K, 373.15 K, and 393.15 K are calculated using NRTL model (filled black
symbols) [78]. Error bars are not available for the NRTL correlated experimental data.
36
5. Results and Discussions
Figure 5.5: Computed (coloured symbols) and correlated (black symbols) isotherms at 308.15 K (circles), 347.15 K
(diamonds) and 373.15 K (squares) for NH3 in two ILs containing the [emim] cation but two different anions [Tf 2N]
(magenta) and [SCN] (green) showing similar solubilities. The NRTL correlated experimental data are taken from
references [16,26,78].
The relative deviations observed in this work are similar to the deviations observed by Maginn
et al. at lower pressures but at higher pressures, the deviations observed by Maginn et al.
increase whereas they decrease in this work. The uncertainty in the simulations is between 1%
and 4% (in mole % of NH3) for both ILs. The experimental uncertainty for simulated pressure
range is less than 2% for [emim][Tf2N], and 4.5% for [emim][SCN] [16,26]. However, the
experimental uncertainty is almost 100% of the experimental value in case of pressures lower
than 2 bar for [emim][Tf2N] indicating the high inaccuracy of experiments at certain conditions
[16]. The same can be seen in Figure 5.3. There is no experimental data available for
[emim][SCN] at pressures lower than 2 bar [26]. More details of the simulations such as λhistogram, convergence of number of molecules and potential energy for few conditions are
provided in Appendix E.
It can be seen that at a given temperature and pressure, there is a little difference in the
solubility of NH3 for the two ILs. The solubility of NH3 in [emim][Tf2N] and [emim][SCN] at
308.15 K, 347.15 K and 373.15 K are shown in Figure 5.5. This indicates that the cation plays a
larger role than the anion in determining the solubility of NH 3. This is due to the fact which is
previously explained by Maginn et al. [19] that NH3 interacts more strongly with the acidic
protons on [emim] leading to a stronger hydrogen bonding interaction between the solute and
the cation [19]. As a result, similar solubilities should be expected for NH 3 in ILs having the same
cation but different anion. Similar conclusion is reported by Maginn et al. [19] after performing
an energy analysis between NH3-[emim] and NH3-[Tf2N].
5.1 Thermodynamic properties from MC simulations
37
Table 5.2 Comparison of computed and correlated solubilities of NH3 (1) in [emim][Tf2N] (2), where x1 is
the mole fraction on NH3.
T/K
308.15
347.15
373.15
393.15
P / bar
4.2938
5.0
6.1505
10.19
6.83
10.19
13.5
18.4
10.19
11.67
13.5
15.15
6.83
10.19
13.5
18.4
x1sim / %
59.5
65.7
73.7
91.1
43.6
56.5
65.6
77.3
40.3
44.7
49.2
52.2
24.3
32.7
39.6
49.3
x1a / %
3.1
3.1
2.9
3.5
3.7
3.3
3.1
1.8
3.1
2.6
3.1
3.5
1.7
2.6
1.9
2.3
x1exp / %
48.5
54.7
62.9
83.4
33.5
47.6
58.4
70.1
28.1
33.7
40.6
45.6
6.6
12.5
22.3
38.8
deviation / %
22.8
20.1
17.1
9.2
30.1
18.6
12.3
10.4
43.9
32.4
21.1
14.5
268.3
161.8
77.1
26.9
a: simulation uncertainty. The experimental data are taken from reference [16]. Experimental data at 308.15 K,
347.15 K, 373.15 K and 393.15 K are calculated using parameters fitted to NRTL model by Wang et al. [78].
Furthermore, MC simulations can also be used to predict solubilities at higher temperatures
and pressures which otherwise are difficult to determine experimentally. These conditions such
as high end generation temperatures are crucial for absorption cycle design and its
performance. Since there is no experimental data available for higher temperatures, the
accuracy of extrapolating the solubility from NRTL model is unknown. This can be seen from
Figure 5.3 that extrapolating the solubility of NH3 in [emim][Tf2N] using NRTL model does not
work well at 373.15 K and 393.15 K. The experimental data range is till 347.6 K. By extrapolating
the data, the average relative deviation observed at 393.15 K is 133.5%. However, for NH3 in
[emim][SCN] the experimental data range is till 372.8 K. Computed results might be more
reliable at higher temperatures as force fields can describe the densities well even at higher
temperatures as seen in Figure 5.1. It is demonstrated herein that, with current force fields, it is
possible to predict qualitatively the solubility of NH3 in ILs. The computed results present
reasonable estimates for the solubility of NH3 in [emim][Tf2N] and [emim][SCN]. These results
encourage the performance of molecular simulation for the prediction of the solubility of NH3 in
other ILs. None of the force field parameters used in this study were fitted to the NRTL
correlated experimental solubility data. Therefore computed results are true predictions and
for higher accuracy, improvements in the force field parameters are desired. One possible
solution could be to modify the mixing rule between IL and NH3.
38
5. Results and Discussions
Table 5.3 Comparison of computed and correlated solubilities of NH3 (1) in [emim][SCN] (2), where x1 is
the mole fraction on NH3.
T/K
308.15
347.15
373.15
393.15
P / bar
4.2938
5.0
6.1505
10.19
6.83
10.19
13.5
18.4
10.19
11.67
13.5
15.15
6.83
10.19
13.5
18.4
x1sim / %
61.8
66.1
74.1
93.1
43.8
55.1
63.5
72.8
39.3
42.7
47.1
50.2
22.1
30.2
36.4
45.1
x1a / %
3.1
2.8
2.2
2.6
2.4
2.4
2.7
2.8
2.3
2.3
2.3
2.3
1.2
1.8
2.1
1.7
x1exp / %
45.5
50.1
56.8
79.4
34.9
45.5
53.7
64.5
30.0
33.8
38.1
41.9
14.5
20.9
26.9
34.8
deviation / %
35.8
32.2
30.4
17.2
25.5
21.2
18.2
12.8
31.0
26.3
23.3
19.8
52.8
43.9
35.2
29.5
a: simulation uncertainty. The experimental data are taken from reference [26]. Experimental data at 308.15 K,
347.15 K, 373.15 K and 393.15 K are calculated using parameters fitted to NRTL model by Wang et al. [78].
5.1.3 Heat Capacity
As explained in section 3.5.1, the ideal gas part of the heat capacity is calculated from
intramolecular energy contributions to enthalpy. In the calculation of the ideal gas part of the
heat capacity, it is assumed that the ions of an IL are in isolated state and there are no
interactions between them. Hence, the ideal gas part of the heat capacity can be calculated
separately and independently for each of the ions of an IL. The ideal gas part of the heat
capacity for an IL is then obtained by adding the ideal gas part of the heat capacity of the
corresponding ions. The calculations are performed for the most stable conformer of each of
the ions. If the energy is similar for two or more stable conformers, the calculations are
performed for all the stable conformers. For the [emim] cation, two main conformers are
found. A planar conformer in which the methyl and the ethyl group lies in the plane with the
cation ring. A non-planar conformer in which CH3 part of the ethyl group lies out of the plane of
the cation ring. For the [emim] cation, the non-planar conformer is found to be the most stable
conformer [68]. For [Tf2N] anion, three main conformers are found. A cis conformer in which
both the CF3 groups lie on the same side of the SNS plane. A trans conformer in which the CF3
groups lie on the opposite sides of the SNS plane. A third conformer in which one of the CF 3
groups lies in the SNS plane. Among the three conformers, the trans conformer of the [Tf 2N]
anion is found to be the most stable conformer [68]. For the [SCN] anion, there is only one
possible configuration. The figures of the conformers are provided in Appendix F.
5.1 Thermodynamic properties from MC simulations
39
Figure 5.6: Ideal gas part of the heat capacity from quantum mechanical calculations for [emim][Tf2N] (black) and
[emim][SCN](blue). Results of Paulechka et al. [68] at 298.15 K and 400 K are shown by red squares. The straight
lines are linear fits to the results.
Table 5.4: Ideal gas part of the heat capacity of the ions and the ILs.
T/K
298
303.15
308.15
313.15
323.15
333.15
343.15
353.15
360.15
363.15
373.15
383.15
393.15
400
[emim]
Cp / J mol-1 K-1
135.1
137.3
139.3
141.4
145.5
149.7
153.8
157.9
160.8
162.0
166.1
170.1
174.1
177.1
ig
[Tf2N]
Cp / J mol-1 K-1
226.1
228.2
230.2
232.2
236.0
239.7
243.3
246.8
249.1
250.1
253.3
256.4
259.4
262.7
ig
[SCN]
Cp / J mol-1 K-1
43.7
44.0
44.2
44.4
44.9
45.3
45.7
46.1
46.3
46.4
46.8
47.1
47.4
47.8
ig
[emim][Tf2N]a
Cpig / J mol-1 K-1
361.2
365.4
369.5
373.6
381.5
389.4
397.1
404.7
409.9
412.1
419.4
426.5
433.5
439.9
[emim][SCN]b
Cpig / J mol-1 K-1
178.9
181.2
183.5
185.8
190.4
195.0
199.5
204.0
207.1
208.5
212.9
217.2
221.5
224.9
a: Column 2 + Column 3; b: Column 2 + Column 4.
The ideal gas part of the heat capacity of the ions and the ILs computed from quantum
mechanical calculations are presented in Table 5.4. Figure 5.6 shows the ideal gas part of the
heat capacity for both ILs. The results presented in Table 5.4 are in fair agreement with the
results from the literature [68].
40
5. Results and Discussions
Paulechka et al. [68] computed the thermodynamic properties of [emim][Tf2N] in the ideal gas
state. The group calculated the ideal gas part of the heat capacity of [emim][Tf 2N] as 353.1 J
mol-1 K-1 at 298 K and as 426.2 J mol-1 K-1 at 400 K. The values are in good agreement with the
values calculated in this work. The difference in the value is due to different scaling factor used
by Paulechka et al. [68]. Figure 5.6 shows the ideal gas part of the heat capacity increases with
an increase in temperature. This is due to an increase in the vibrational energy of the ions. The
ideal gas part of the heat capacity is higher for [emim][Tf 2N] as compared to [emim][SCN] over
the whole temperature range. The reason is the larger size of [Tf 2N] as compared to [SCN]
which results in more degrees of freedom for [Tf2N].
The residual heat capacity of [emim][Tf2N] and [emim][SCN] is computed from MC simulations
in the NPT ensemble at temperatures ranging from 303.15 K to 393.15 K in steps of 10 K. For
303.15 K to 333.15 K, the residual heat capacity is computed at 6.1505 bar (evaporation
pressure) and for 343.15 K to 393.15 K, it is computed at 13.508 bar (condensation pressure).
The computed results are presented in Table 5.5 for [emim][Tf2N] and [emim][SCN]. Figures 5.7
and 5.8 give a comparison of computed residual heat capacities with residual heat capacities
theoretically calculated from equation (5.1) for both ILs. Equation (5.1) is derived by applying
the principle of corresponding states and the thermodynamic equations [69]. The
thermodynamic equations used in derivation of equation (5.1) are the residual function of CV,
the departure function of CP, and a thermodynamic relation of CP in terms of CV and partial
differential of P with respect to T and V. The definitions of the residual and departure functions
along with detailed procedure are explained in reference [69],
(5.1)
where, R is universal gas constant, is accentric factor and Tr is reduced temperature ratio. The
critical parameters for both ILs used in equation (5.1) are taken from Valderrama et al. [42,43].
The group developed a group contribution model for determination of the critical properties of
ILs. However, they were unable to validate this directly due to lack of experimental data on
critical properties of ILs [42,43,69]. ILs start to decompose at temperatures significantly lower
than their critical point. As a result, it is difficult to measure their critical properties
experimentally. Hence, the uncertainty associated with equation (5.1) is high.
It can be seen from Figure 5.7 that the absolute values of computed residual heat capacities for
[emim][Tf2N] are about twice as large as those obtained from equation (5.1). Similarly for
[emim][SCN], the computed residual heat capacities are higher than the values obtained from
equation (5.1). It is difficult to give an explanation for this difference between the computed
and the calculated value. A similar issue is reported by Verevkin et al. [70] while comparing
experimental, theoretical and computed residual heat capacities of the [Cnmim][Tf2N] family (n
varies from 1 to 18). They calculated residual heat capacities with four different methods. In the
first method, they calculated it from experimental measurements of the vaporization
enthalpies. The vaporization enthalpies were calculated at two different average temperatures
5.1 Thermodynamic properties from MC simulations
41
Figure 5.7: Residual heat capacity for [emim][Tf2N] from MC simulations (blue) and from equation (5.1) (black).
The curve is guide to the eye.
Figure 5.8: Residual heat capacity for [emim][SCN] from MC simulations (blue) and from equation (5.1) (black).
The curve is guide to the eye.
from two different experimental methods. The ratio of the difference of the vaporization
enthalpies and the temperatures give the residual heat capacity. In the second method, they
calculated the ideal gas part of the heat capacity from quantum mechanical calculations. The
difference between the ideal gas part and the total heat capacity gives the residual heat
capacity.
42
5. Results and Discussions
Table 5.5: Residual heat capacity of [emim][Tf2N] and [emim][SCN] computed from MC simulations in
the NPT ensemble.
T/K
303.15
313.15
323.15
333.15
343.15
353.15
363.15
373.15
383.15
393.15
Cp
res
[emim][Tf2N]
/ J mol-1 K-1 Cpres a / J mol-1 K-1
105.4
26.3
109.1
23.8
117.6
28.5
119.3
32.9
125.7
38.2
126.3
32.5
128.3
35.4
134.4
35.2
135.8
36.3
138.9
38.1
Cp
res
[emim][SCN]
/ J mol-1 K-1 Cpres b / J mol-1 K-1
105.1
13.1
106.6
14.9
109.5
13.9
109.5
13.1
110.2
13.0
109.5
9.2
110.9
13.0
109.7
12.4
108.8
9.8
108.5
10.3
a, b: simulation uncertainties.
In the third method, they used the basics of statistical thermodynamics and volumetric
properties to calculate the residual part of the heat capacity. In the fourth method, they directly
calculate it from MD simulation. They reported that the residual heat capacities computed from
MD simulation are in good agreement with those obtained from quantum mechanical
calculations, but disagree with the experimentally or theoretically calculated residual heat
capacities [70]. The computed and quantum mechanical calculated residual heat capacities are
almost twice as large as those obtained from experimental or theoretical calculations [70].
According to Verevkin et al. the disagreement could be attributed to the simple rigid rotorharmonic oscillator approximation used in quantum mechanical calculations because of the
large size of ILs. However there is no rigid rotor approximation made in simulations [70]. It
could also be because of the assumption that the ideal gas part of the heat capacity of the ions
can be calculated independently. There could be some coupling effects between the ions.
When the ions are close to each other, there will be some changes in the degrees of freedom
which will affect the ideal gas part of the heat capacity. A further plan to investigate these
effects is to calculate the ideal gas part of the heat capacity of an ion pair or clusters of ion
pairs.
It should be noted that statistical uncertainties are high while computing the residual heat
capacities from MC simulations. The average statistical uncertainty is about 26% and 12% of the
computed result for [emim][Tf2N] and [emim][SCN], respectively. This can be seen in Figures 5.7
and 5.8. The computed residual heat capacity appears to have an increasing trend with
temperature for [emim][Tf2N] and nearly flat profile with temperature for [emim][SCN].
However due to the uncertainties, it is not entirely clear if this trend is real.
The total heat capacity at constant pressure for [emim][Tf2N] and [emim][SCN] is then obtained
by adding the ideal gas part and the residual part. A comparison between the computed heat
capacity and the experimental data as a function of temperature is shown in Figures 5.9 and
5.10.
5.1 Thermodynamic properties from MC simulations
43
Figure 5.9: Comparison of computed molar heat capacities (blue) with experimental heat capacities (black) for
[emim][Tf2N]. The curves are guides to the eye fit through the simulation results. Experimental data are taken from
references [48,69,71,72].
Figure 5.10: Comparison of computed molar heat capacities (blue) with experimental heat capacities (black) for
[emim][SCN]. Maginn et al. [57] simulation results are shown by magenta squares. The curves are guides to the eye
fit through the simulation results. Experimental data are taken from references [33,67].
It can be seen from Figures 5.9 and 5.10 that the computed results for both ILs are in good
agreement with the experimental data. Average deviation between the experimental and the
computed heat capacities are around 4% and 2.5% for [emim][Tf2N] and [emim][SCN]. The
44
5. Results and Discussions
different experimental data sets for [emim][Tf2N] have a high degree of scatter as seen from
Figure 5.9. The uncertainties associated with the experiments are also high. The computed
results agree with those of Ferreira et al. and Paulechka et al. [48,71,72] within the simulation
uncertainties. Similarly for [emim][SCN], within the simulation uncertainties, results agree well
with those of Navarro et al. and Ficke et al. [33,67]. The MC simulations are able to predict the
increase in the heat capacity as temperature increases. This is shown in Figures 5.9 and 5.10. It
can be concluded that the MC simulations appear to give heat capacity estimates that are
consistent with the experiments.
5.1.4 Excess Enthalpy
The excess enthalpies for experimental COP calculations are obtained from a cubic EOS of the
generic Redlich-Kwong (RK) type as explained in section 2.4.1. In Table 5.6, the excess
enthalpies calculated from the RK-EOS for the desired cycle conditions are presented. The
excess enthalpies can also be computed from MC simulations. The molar enthalpy of mixing,
, is computed from MC simulations as [19],
(5.2)
where,
is the molar enthalpy of the mixture.
and
are the molar enthalpies of pure
NH3 and IL, respectively.
is the mole fraction of NH3 in the mixture. The molar enthalpies
of pure NH3 and the IL are computed from the MC simulations in the NPT ensemble whereas
the molar enthalpy of the mixture is computed from the MC simulations in the osmotic
ensemble. The enthalpies in both ensemble are computed using h = U + PV, where U is the total
energy and V is the molar volume. The enthalpies of pure NH3 and the IL are computed at T and
P corresponding to the condition of the mixture. However, the pure NH 3 at mixture condition is
in gaseous state. Hence, the latent heat of pure NH3,
, at the same T and P is added to
the mixing heat to get the excess enthalpies as,
(5.3)
where,
is weight fraction of NH3 in the mixture. The derivation of the equation (5.3) is
provided in Appendix G. The excess enthalpies computed from MC simulations are presented in
Table 5.6. As seen from Table 5.6, the excess enthalpies computed from the RK-EOS
significantly differ from the excess enthalpies computed from MC simulations. The excess
enthalpies computed from the EOS look inconsistent, whereas the excess enthalpies from MC
simulations are consistent and negative in all cases. The trend of the results from MC
simulations is consistent with the trend reported by Maginn et al. [19]. They concluded that the
magnitude of the enthalpy of mixing increases as the temperature decreases and as
concentration of NH3 increases [19,31]. The same can be seen from the results reported in
Table 5.6. However, the values from the EOS could be both positive and negative as seen from
Table 5.6. It is well known that equations of state have difficulties in predicting the excess
properties [73]. The excess properties are important for a fundamental understanding of mixing
processes.
5.1 Thermodynamic properties from MC simulations
45
Table 5.6: Excess enthalpies for NH3/[emim][Tf2N] and NH3/[emim][SCN] computed from MC simulations
and RK-EOS.
ILs
T/K
P / bar
[emim]
[Tf2N]
308.15
313.15
347.15
373.15
393.15
[emim]
[SCN]
308.15
313.15
347.15
373.15
393.15
/ kg kg-1
0.1086
0.0404
0.0766
0.0404
0.0277
/ kJ kg-1
6.15
13.5
13.5
13.5
13.5
/ kJ mol-1
-16.7 1.3
-15.0 0.8
-13.8 1.1
-10.7 0.9
-7.0 0.2
6.15
13.5
13.5
13.5
13.5
-16.5
-15.5
-13.4
-9.7
-6.2
0.2235
0.0822
0.1480
0.0822
0.0544
0.8
1.4
0.9
0.6
0.3
/ kJ kg-1
/ kJ kg-1
-144.2
-72.4
-94.6
-51.5
-28.7
/ kJ kg-1
1122.3
1099.3
913.8
715.6
480.3
-22.3
-27.9
-24.6
-22.6
-15.4
7.3
3.9
-7.3
-2.3
-10.3
-293.8
-158.9
-184.9
-99.4
-54.7
1122.3
1099.3
913.8
715.6
480.3
-43.0
-68.5
-49.6
-40.6
-28.6
10.4
4.3
-7.5
-14.3
-4.5
a: excess enthalpies computed from MC simulations using equations (5.2) and (5.3). b: excess enthalpies computed
from RK-EOS as explained in section 2.4.1. The weighted average molecular mass of the mixture of NH 3 and IL is
used to convert the mixing heat from kJ/mol to kJ/kg. Latent heat of pure ammonia is taken from NIST Refprop
[49].
Theoretically, it is possible to predict the excess properties of mixtures from the properties of
their constituent components using an EOS along with mixing rules and from activity coefficient
models such as NRTL, UNIFAC. However in many cases such calculations are difficult or
inaccurate. Yokozeki et al. [76] used the NRTL model to predict the excess enthalpies and
mentioned that accurate predictions are very difficult because they are derived from the
temperature derivative of the model which is empirical. The EOS are also known to have
difficulties in handling the phase behaviour of certain systems such as polar compounds like IL
[73,74]. Numerous efforts have been made to develop mixing rules and EOS models for such
systems. But still they are not able to satisfactorily represent the excess properties [73].
Furthermore, the excess property calculations are sensitive to the values of the parameters
obtained from different approaches such as EOS or activity coefficient models [74]. This led to
the conclusion as reported by Fermeglia et al. [74] that prediction of excess enthalpies using
parameters from different approaches is likely to be very difficult. In many cases excess
property calculation can be difficult or inaccurate due to the complex structure of the mixtures
such as NH3/IL [75]. The negative values of the excess enthalpies from MC simulations are
indicating that there exist some attraction within the mixture [80]. There is a wide range of
possible interactions such as molecular association, hydrogen bonding, dipole-dipole
interactions between components of the mixture. As a consequence, deviations from an ideal
behaviour of mixture properties occur. These deviations are defined by excess properties which
are very sensitive to intermolecular interactions [75]. The simulation force fields which are
developed specifically to capture such interactions could yield reasonable estimates of the
excess properties.
46
5. Results and Discussions
Table 5.7: Fitted parameters of computed
ILs
[emim][Tf2N]
[emim][SCN]
a/ J mol-1 K-1
-168.07
-93.459
for [emim][Tf2N] and [emim][SCN].
b/ J mol-1 K-2
2.8714
1.8528
c/ J mol-1 K-3
-2.5159 x 10-3
-1.9754 x 10-3
The trend of the simulation results for excess enthalpies reported in Table 5.6 shows that
results can be considered reliable and hence, MC simulations can successfully be used to
predict the excess properties.
5.2 Properties for Absorption refrigeration cycle
5.2.1 Enthalpy calculations
The computed total heat capacities (mol based) for both ILs are fitted to a quadratic polynomial
and the fitted parameters are listed in Table 5.7. The enthalpies of the pure IL at different state
points in the cycle are then calculated by using equation (2.3). For COP calculations from the
correlated experimental data, the experimental heat capacities are also fitted to a polynomial.
The fitted parameters for experimental heat capacities are taken from Wang et al. [78]. The
enthalpies of the pure NH3 at different state points are obtained from NIST's Refprop [49].
5.2.2 Circulation ratio, f
The circulation ratio, f, is defined as the ratio of the mass flow rate of the weak IL solution from
absorber to the mass flow rate of refrigerant (refer equation 2.7). It is an important design and
performance optimizing parameter as it is directly related to the size and cost of the
equipments. It is closely related to the required pumping work. The solubility results from the
simulations in the osmotic ensemble are used to calculate f for both ILs. These values are then
compared with the values calculated from the NRTL correlated experimental solubility data.
The comparison between f calculated from the NRTL correlated experimental data and from
MC simulations for both ILs is presented in Table 5.8. Figures 5.11 and 5.12 show the variation
of f with respect to the end generation temperature, TG, for NH3/[emim][Tf2N] and NH3/[emim]
[SCN], respectively.
It can be seen from Table 5.8, Figures 5.11 and 5.12 that there are high deviations between f
calculated from the NRTL correlated experimental data and MC simulations. The average
deviation for NH3/[emim][Tf2N] is around 46% whereas it is around 66% for NH3/[emim][SCN].
This is due to the deviations in the solubilities of NH3 obtained from the simulations as
compared to the experimental solubilities calculated from the NRTL model. The deviations in
the value of f are higher for NH3/[emim][SCN] as compared to NH3/[emim][Tf2N] because the
deviations in the solubility of NH3 in [emim][SCN] are higher as compared to NH3 in
[emim][Tf2N]. As seen from Table 5.8, Figures 5.11 and 5.12, the deviations are lower at high
end generation temperatures and increase with decrease in the end generation temperature.
At low end generation temperatures, higher solubilities of ammonia are obtained as seen from
Tables 5.2
5.2 Properties for Absorption refrigeration cycle
47
Table 5.8: Comparison of the NRTL correlated experimental data results and the simulation results for
circulation ratio, f at different end generation temperatures, TG.
simulation
wG
wA
0.923
0.891
0.959
0.891
0.972
0.891
fexp
fsim
74
100
120
experimental
wG
wA
0.942
0.931
0.971
0.931
0.987
0.931
83.8
24.3
17.5
74
100
120
0.895
0.942
0.964
0.851
0.918
0.945
78.5
16.2
12.0
ILs
TG / o C
[emim]
[Tf2N]
[emim]
[SCN]
0.884
0.884
0.884
0.776
0.776
0.776
28.8
14.1
12.0
deviation
/%
65.3
41.9
31.4
16.3-110.5
10.8-19.2
9.8-14.8
11.4
6.5
5.6
85.4
59.8
53.3
7.8-20.9
5.4-7.9
4.8-6.4
fa
and
are mass fraction of the IL in strong and weak IL solution flowing out of the generator (G) and absorber
a
(A), respectively. f is range of the circulation ratio due to uncertainty in the simulation results. f is calculated using
equation (2.7). The experimental data are calculated using parameters fitted to the NRTL model by Wang et al.
[78]. Mole fractions of NH3 in both ILs are computed from the NRTL model and MC simulations in the osmotic
ensemble. These mole fractions are then converted to corresponding mass fractions of NH3. Mass fractions of IL is
o
o
o
then calculated as
. TC = 35 C, TA = 30 C, TE = 10 C, PE = 6.15 bar and PC = 13.5 bar.
and 5.3. As a result, lower mass fractions of IL are obtained. This results in a lower molecular
weight of the mixture as the end generator temperature decrease. Since the difference in
correlated and computed solubilities are low at lower end generation temperatures, the
difference in the molecular weight of the mixture is also low. This results in higher mass fraction
difference between correlated and simulation results at lower end generation temperature and
hence in higher deviations in f as the end generation temperature decrease.
For both ILs, the simulations overestimate the solubility of NH3 as observed in section 5.1.2
which results in low mass fractions of ILs. As a result, a lower value of f as compared to the
value calculated from correlated data, is obtained according to equation (2.7). Hence, a higher
COP value is expected from the simulation results in comparison to the value obtained from
correlated data. However, considering the simulation uncertainties in calculating the solubility,
there is a possible range of values of f obtained from the simulation results as shown in Figures
5.11 and 5.12. The error bars are quite large at low end generation temperatures. Figures 5.11
and 5.12 show that simulation results are able to reproduce the trend of the variation of f with
respect to the end generation temperature. f decreases with the increase in the end generation
temperature for both ILs and then reduces to constant value at high end generation
temperatures. High end generation temperatures reduce the mass fraction of NH3 or increase
the mass fraction of IL in the strong IL solution, wG, as seen from Table 5.8. This results in an
increase in the difference of mass fractions of IL in strong and weak IL solutions (denominator
of equation (2.7)) at a higher rate as compared to increase in the mass fraction of IL in strong IL
solution (numerator of equation (2.7)). At low end generation temperatures, the difference in
the mass fractions of IL in weak and strong IL solution is small. As a result, a high value of f is
obtained at low end generation temperatures.
48
5. Results and Discussions
Figure 5.11: Comparison of the circulation ratio, f, calculated from the NRTL correlated experimental data (black
squares) and from the simulation results (blue squares) for NH3/[emim][Tf2N] working pair. f for NH3/H2O working
pair calculated from NRTL correlated experimental data are shown with black diamonds. The NRTL correlated
experimental data are taken from references [16,78].
Figure 5.12: Comparison of the circulation ratio, f, calculated from the NRTL correlated experimental data (black
squares) and from the simulation results (blue squares) for NH3/[emim][SCN] working pair. f for NH3/H2O working
pair calculated from NRTL correlated experimental data are shown with black diamonds. The NRTL correlated
experimental data are taken from references [26,78].
5.2 Properties for Absorption refrigeration cycle
49
High circulation ratio increases the solution pumping power and the generator heat input
requirements according to equations (2.8) and (2.9), respectively. As a result, a lower COP value
will be achieved. Therefore, it is highly impractical to run the absorption cycle at lower TG. It can
be seen from Table 5.8 that f of NH3/[emim][Tf2N] is higher in comparison to f of NH3/
[emim][SCN]. This means higher pumping work and generator heat input is required from
NH3/[emim][Tf2N] in comparison to NH3/[emim][SCN]. It can be seen from Figures 5.11 and
5.12 that f of NH3/IL working pair is significantly higher than f of conventional NH3/H2O working
pair. Hence lower COP value for NH3/IL working pair should be expected in comparison to
NH3/H2O working pair.
5.2.3 Coefficient of Performance
The cycle calculations for the COP are performed using the enthalpies in addition to the
solubility results obtained from MC simulations. Results of the present systems for the cycle
performance calculated from the NRTL correlated experimental data and MC simulation results
are compared in Table 5.9. Figure 5.13 shows the variation of the COP computed from two
different methods with respect to the end generation temperature, TG, for both ILs. The COP of
the well known working pair, NH3/H2O is also calculated from the NRTL correlated experimental
data and is shown in Figure 5.13. It can be seen from Table 5.9 and Figure 5.13 that the COP
results from MC simulations are in reasonable agreement with the results obtained from
correlated experimental data. Average deviation of around 5.5% is observed between
correlated and simulation obtained COP values for both ILs. The smaller value of f is obtained
from MC simulations in comparison to the value obtained from correlated experimental data as
reported in Table 5.8. This is compensated by higher values of the enthalpy difference between
solutions entering (state point 4) and leaving (state point 7) the generator obtained from MC
simulation results in comparison to the correlated experimental data. This difference is mainly
due to the deviations in the excess enthalpies obtained from the simulations and the correlated
experimental data as the deviations in the heat capacities are not significant. Since the
deviations in f are dominant as compared to the deviations in the enthalpies, the value of f(h7 h4) as computed from simulations remain smaller in comparison to the value obtained from the
correlated experimental data. This results in a decrease of the generator heat according to
equation (2.8) where h8 is enthalpy of pure NH3 which is same in both the cases. As a result, MC
simulations overestimate the COP values in comparison to the values obtained from correlated
experimental data.
Figure 5.13 shows that with increasing end generation temperature, TG, the COP of the working
pair becomes smooth after an initial rise. The COP of the system increases until TG = 90 oC and
becomes almost flat for end generation temperatures higher than 90 oC. For the given range of
TG, the lowest COP is observed around 74 oC for both the ILs. This is because of high generator
heat input as a result of high circulation ratio, f at 74 oC. As TG increases, f decreases resulting in
a decrease of generator heat input as seen from Table 5.9. As a result the COP increases initially
according to equation (2.10) since the evaporation heat is constant (for a fixed TE). As seen from
Table 5.9, the pump work is almost negligible in comparison to the generator heat input.
50
5. Results and Discussions
Table 5.9: Comparison of the NRTL correlated experimental and the simulation results for Coefficient of
Performance, COP at different end generation temperatures, TG.
ILs
Method
Experimental
[emim]
[Tf2N]
MC
simulations
Experimental
[emim]
[SCN]
MC
simulations
o
TG / oC
f
74
100
120
74
100
120
83.8
24.3
17.5
28.8
14.1
12.0
74
100
120
74
100
120
78.5
16.2
12.0
11.4
6.5
5.6
o
o
QE /
-1
/ kJ kg
1106.1
1106.1
QG /
WP /
/ kJ kg
1691.0
1372.5
1372.0
1572.1
1320.3
1349.5
/ kJ kg-1
45.3
13.2
9.5
16.4
8.0
6.7
COP
COPa
0.63
0.79
0.80
0.69
0.83
0.82
---------0.55-0.72
0.81-0.86
0.79-0.83
1387.6
1331.9
1364.5
1331.4
1273.2
1312.4
57.4
11.9
8.8
9.0
5.0
4.5
0.76
0.82
0.81
0.82
0.86
0.84
---------0.80-0.87
0.83-0.90
0.82-0.86
-1
a
Cycle conditions: TC = 35 C, TA = 30 C, TE = 10 C, PE = 6.15 bar and PC = 13.5 bar. COP is range of the Coefficient of
Performance due to uncertainty in the simulation results.
Figure 5.13: Comparison of the COP calculated from the NRTL correlated experimental data (black symbols) and
from the simulation results (coloured symbols) and its variation with end generation temperature, TG. Blue squares
correspond to NH3/[emim][Tf2N] and magenta circles correspond to NH3/[emim][SCN] working pair. The COP for
NH3/H2O working pair calculated from NRTL correlated experimental data are shown with black diamonds. The
o
o
NRTL correlated experimental data are taken from references [16,26,78]. Cycle conditions: TC = 35 C, TA = 30 C, TE
o
= 10 C, PE = 6.15 bar and PC = 13.5 bar.
5.3 Sensitivity Analysis
51
It is observed that when TG is higher than a certain temperature, a further increase in the TG
does not enhance the COP anymore as seen from Figure 5.13. This can be attributed to the fact
that more input heat for the generator is required as seen from Table 5.9. The maximum COP
for both the systems is obtained around TG = 100 oC. As a result it is advantageous to operate
the system at TG = 100 oC. It can be seen from Figure 5.13 that simulation results are able to
reproduce this variation of COP with TG. Considering the uncertainties in the circulation ratio
and the enthalpies obtained from the solubility and the heat capacity results, it is possible to
obtain a range of the COP values for a particular TG as provided in Table 5.9.
From Figure 5.13 it can be seen that for a given TG, the COP of the NH3/[emim][SCN] working
pair is a little higher than the COP of the NH3/[emim][Tf2N] working pair. This is due to the
lower generator heat input requirement for NH3/[emim][SCN] on account of a lower circulation
ratio, f as can be seen from Table 5.9. The COP of the present NH3/IL systems is somewhat
lower than the NH3/H2O system. This is mainly because of the significant lower values of the
circulation ratio for the NH3/H2O system in comparison to the NH3/IL systems as seen from
Figures 5.11 and 5.12. However, for the NH3/H2O system an additional large rectifier unit is
required to separate the water vapours from the vapours of NH3. This would result in an
additional power requirement which will lower the performance. There is no such requirement
for the NH3/IL systems because of a negligible vapour pressure of the IL. Hence, in actual
applications, the present NH3/IL systems may compete with the cycle performance of the
traditional NH3/H2O system although detailed system analysis and cost estimations remain to
be investigated.
5.3 Sensitivity Analysis
A sensitivity analysis of the properties affecting the circulation ratio, f, and the COP of the
absorption refrigeration cycle is carried out. The properties include the heat capacity of an IL,
the solubility of NH3 in an IL, and the excess enthalpy. The main objective is to examine the
effects of these properties on the standard COP and f obtained from the correlated
experimental data.
5.3.1 Heat capacity of an IL
Figure 5.14 shows the effect on the COP of the absorption refrigeration cycle by increasing the
fitted experimental heat capacities of both the ILs by 10%. As seen from Figure 5.14, increasing
the heat capacity of ILs by 10% result in an average decrease in the COP of the cycle by around
1% and 1.5% for [emim][Tf2N] and [emim][SCN], respectively. Increasing the heat capacity of
the IL results in an increase in the enthalpy difference between the solution entering (state
point 4) and leaving (state point 7) the generator. This results in an increase in the generator
heat input according to equation (2.8). As a result, the COP decreases. There is no effect on the
circulation ratio as it does not depend on the heat capacity of an IL.
52
5. Results and Discussions
Figure 5.14: Sensitivity of the COP with respect to the heat capacity of the IL. Coloured symbols correspond to the
o
o
COP values by increasing the fitted experimental heat capacities by 10%. Cycle conditions: TC = 35 C, TA = 30 C, TE
o
= 10 C, PE = 6.15 bar and PC = 13.5 bar.
5.3.2 Excess enthalpy
Figure 5.15 shows the effect on the COP of the absorption refrigeration cycle by increasing the
excess enthalpies calculated from an EOS by 15%. As seen from Figure 5.15, increasing the
excess enthalpies by 15% result in an average increase in the COP of the cycle by around 1.25%
for both the ILs. Increasing the excess enthalpies result in a decrease in the enthalpy difference
between the solution entering (state point 4) and leaving (state point 7) the generator. This
results in a decrease in the generator heat input according to equation (2.8). As a result, the
COP increases. There is no effect on the circulation ratio as it does not depend on the excess
enthalpy.
5.3.3 Solubility of NH3 in an IL
Figures 5.16 and 5.17 show the effect on the circulation ratio and the COP of the absorption
refrigeration cycle by increasing the NRTL correlated solubilities of NH3 in both ILs by 2%. As
seen from Figure 5.16, increasing the solubilities by 2% result in an average decrease in f by
around 7.5% and 5.5% for NH3/[emim][Tf2N] and NH3/[emim][SCN], respectively. Increasing the
solubility of NH3, reduces the mass fraction of an IL which results in a decrease in f according to
equation (2.7). The decrease in f also decreases the generator heat input according to equation
(2.8). As a result, the COP of the cycle increases. The average increase of around 2.5% and 2% is
observed in the COP of the cycle for NH3/[emim][Tf2N] and NH3/[emim][SCN], respectively.
5.3 Sensitivity Analysis
53
Figure 5.15: Sensitivity of the COP with respect to the excess enthalpy. Coloured symbols correspond to the COP
o
o
values by increasing the excess enthalpies obtained from an EOS by 15%. Cycle conditions: TC = 35 C, TA = 30 C, TE
o
= 10 C, PE = 6.15 bar and PC = 13.5 bar.
Figure 5.16: Sensitivity of the circulation ratio, f, with respect to the solubility of NH3 in an IL. Coloured symbols
o
o
correspond to the f values by increasing the solubility of NH 3 by 2%. Cycle conditions: TC = 35 C, TA = 30 C, TE = 10
o
C, PE = 6.15 bar and PC = 13.5 bar.
54
5. Results and Discussions
Figure 5.17: Sensitivity of the COP with respect to the solubility of NH 3 in an IL. Coloured symbols correspond to
o
o
o
the COP values by increasing the solubility of NH3 by 2%. Cycle conditions: TC = 35 C, TA = 30 C, TE = 10 C, PE = 6.15
bar and PC = 13.5 bar.
Out of the three properties, the solubility of NH3 in an IL has the most significant effect on the
performance of the absorption refrigeration cycle. It is observed from Table 5.7 that increasing
the solubility of NH3 also increases the excess enthalpy. The sensitivity analysis shows that
increasing the solubility of NH3 and the excess enthalpy increase the COP of the cycle. The
solubility of NH3 and the excess enthalpy calculated from the MC simulation are higher in
comparison to the correlated experimental data. As a result, higher COP values are obtained
from the results of MC simulations in comparison to the results from the correlated
experimental data.
6
Conclusions
An approach based on the computational prediction of the material properties from MC
simulations is used to predict the performance of the single effect absorption refrigeration
cycle involving two NH3/IL (refrigerant/absorbent) working pairs. MC simulations are used to
predict the properties of the working pair at relevant state points in the cycle, and the resulting
coefficient of performance (COP). The approach requires no experimental or correlated data of
the thermodynamic properties for its implementation, and instead utilizes available molecular
force fields for the components of the working pair. In order to estimate the accuracy of MC
simulations in predicting the thermodynamic performance, the COP and the circulation ratio of
NH3/[emim][Tf2N] and NH3/[emim][SCN] working pairs are computed using MC simulations. The
results are compared with those obtained using the NRTL correlated experimental data. The
results show that average deviation of around 5.5% is observed between the correlated and the
computed COP for both ILs. The average deviations for the circulation ratio are around 46% for
NH3/[emim][Tf2N] and 66% for NH3/[emim][SCN]. The MC simulations over-predict the COP and
under-predict the circulation ratio in comparison to the value calculated from the correlated
experimental data. These deviations are mainly due to the higher excess enthalpies and the
overestimation of the solubilities of NH3 in both ILs by MC simulations.
This study shows that MC simulations can be used to get an first estimate of the COP of the
single effect absorption refrigeration cycle. This means that molecular simulations can be used
in two areas [77]. The first is in property prediction. MC simulations have been shown to be
capable of making qualitative as well as quantitative predictions of the thermodynamic
properties of pure compounds and mixtures such as density, heat capacity, solubility etc.
Property predictions will be essential under conditions where experiments are difficult to
perform such as high temperatures and pressures. Simulations can also be useful for predicting
mixture properties such as excess enthalpies. These measurements are much harder to carry
out experimentally and theoretically but are not difficult to conduct in a simulation. Overall,
there is reasonable quantitative agreement between the results from the correlated
experimental data and the results computed from MC simulations. The agreement is good for
the density and the heat capacity. The simulations tend to overestimate the solubilities, with
average relative deviations ranging from 17-29%. The magnitude of these deviations are similar
to the deviations observed in many other studies. The excess enthalpy computed from MC
simulations also seems to be consistent in comparison to those computed from the RK-EOS.
However, the qualitative trends and the temperature dependence of these properties are
captured well by MC simulations. It is shown that extrapolating the experimental solubility data
does not work well at temperatures outside the range of the experimental data. These
temperatures could be important for the performance of the absorption cycles. The sensitivity
analysis also shows that the solubility has the most significant effect on the performance of the
absorption refrigeration cycle. This work highlights the application of molecular simulation in
55
56
6. Conclusions
this area. The second area where MC simulations are useful is in providing the qualitative
insight and to help develop an understanding into the nature of ILs and their interactions such
as hydrogen bonding of the cation with NH3.
Despite the encouragements to use MC simulations for such applications, some problems still
remain. Accurate and new force fields must be developed and validated for a larger range of
compounds as there are a number of ILs for which force field parameters do not exist. For
conventional compounds, it is possible to compute solubilities and heat capacities and compare
these with the available experimental data. This is not an option for the ILs where only
experimental liquid densities are available. The liquid density is a poor choice to be used for
validation of the force fields as force fields having varying parameters can give the same liquid
density [77]. Parameters exclusively based on liquid density data do not always yield accurate
results when compared to the experimental data as observed in this work. It has been argued
that the vaporization enthalpies and the melting point which are becoming readily available
now, are a better experimental quantity to compare against [77]. However, liquid densities are
the only experimentally determined material properties that are widely available for the ILs.
Indeed the next step could be to develop force field parameters for an accurate prediction of
the properties. The interplay between NH3 and the IL force fields seems to be a problem as
well. One possible solution could be to modify the interaction parameters of the IL with the NH3
force fields to reproduce the available experimental solubilities.
Finally, there is a great opportunity for the development and use of MC simulations in the area
of ILs and absorption refrigeration cycles. This approach can be used as an inexpensive
alternative for preliminary design considerations involving potential working fluids for
absorption refrigeration cycles in the absence of available experimental data. Perhaps this work
will serve as encouragement to apply simulation based study to investigate the performance of
potential working pairs.
Appendix A
Fluctuation formula
The isobaric-isothermal partition function of a system, i.e., NPT ensemble composed of N
molecules is given by
(A.1)
where,
is the volume of the system, is the thermal de Broglie wavelength, is the
configurational enthalpy,
.
is the intermolecular potential energy,
and
is the intramolecular potential energy. are the dimensionless degree of freedom and
n the number of degrees of freedom.
is also referred to as 1/TkB where
is the
Boltzmann's constant.
The physical properties of a system at thermal equilibrium can be determined from the
ensemble average as,
(A.2)
Now, taking the derivative of equation (A.2) with respect to
results in,
(or temperature) at constant P
(A.3)
Substituting,
in equation (A.3) we get,
(A.4)
57
Appendix B
Ideal Heat Capacity from Quantum Mechanics
Translational Partition function
For a single atom or molecule, the translational partition function in NVT ensemble is
approximated as [61],
(B.1)
where, m is the mass of an atom or a molecule and h is Planck's constant.
Taking log of both sides,
(B.2)
(B.3)
(B.4)
Differentiating equation (B.4) with respect to temperature T at constant V and fixed N gives,
(B.5)
Using equation (3.38) and (B.5), the contribution to the total intramolecular energy due to
translation is,
(B.6)
(B.7)
Using equation (3.39) and (B.7), the contribution to the heat capacity at constant volume due to
translation is,
(B.8)
Vibrational Partition function
The vibrational partition function of a system with uncoupled vibrational modes (with system's
other degrees of freedom) is defined as [61],
59
60
Appendix B
(B.9)
where, is the vibrational frequency and K is the vibrational mode. A characteristic vibrational
temperature is defined as
.
Taking log of both sides,
(B.10)
(B.11)
(B.12)
(B.13)
Differentiating equation (B.13) with respect to temperature T at constant V and fixed N gives,
(B.14)
Taking
in the numerator of second term to denominator gives,
(B.15)
Using equation (3.38) and (B.15), the contribution to the total intramolecular energy due to
vibration is,
(B.16)
(B.17)
Substituting,
in equation (B.17) gives,
(B.18)
Appendix B
61
Using equation (3.39) and (B.18), the contribution to the heat capacity at constant volume due
to vibration is,
(B.19)
Rotation Partition function
For a single atom,
and its contribution to the heat capacity at constant volume is zero
as it does not depend on temperature. For the general case of non-linear polyatomic molecule,
the rotational partition function is [61],
(B.20)
where,
is symmetry number and I is the moment of Inertia of the molecule.
Taking log of both sides,
(B.21)
(B.22)
Differentiating equation (B.22) with respect to temperature T at constant V and fixed N gives,
(B.23)
Using equation (3.38) and (B.23), the contribution to the total intramolecular energy due to
rotation is,
(B.24)
(B.25)
Using equation (3.39) and (B.25), the contribution to the heat capacity at constant volume due
to rotation is,
(B.26)
Appendix C
Force Field Parameters
Figure C.1: Schematic of the cation [emim] and anion [Tf2N] and [SCN] with atom labels used in
the simulation [57].
63
64
Appendix C
Table C.1: Atom types and the partial charges for cation and anions of two ILs [17,57].
atom ID
atom type
Partial charge
(e)
na
cc
na
cc
cd
h5
h4
h4
c3
h1
h1
h1
c3
h1
h1
c3
h
hc
hc
0.206
-0.101
0.072
-0.131
-0.151
0.178
0.190
0.183
-0.296
0.142
0.142
0.142
-0.035
0.089
0.089
-0.120
0.067
0.067
0.067
n2
s6
s6
o
o
o
o
c3
c3
f
-0.6374
0.9719
0.9719
-0.4847
-0.4847
-0.4847
-0.4847
0.2483
0.2483
-0.1107
[emim]+
N1
C1
N2
C2
C3
H1
H2
H3
C4
H4
H5
H6
C5
H7
H8
C6
H9
H10
H11
[Tf2N]N1
S1
S2
O1
O2
O3
O4
C1
C2
F1
Appendix C
65
Table C.1: (Continued)
atom ID
atom type
Partial charge
(e)
F2
F3
F4
F5
F6
f
f
f
f
f
-0.1107
-0.1107
-0.1107
-0.1107
-0.1107
s
c1
n1
-0.554
0.317
-0.563
[SCN]S1
C1
N1
Table C.2: Lennard-Jones parameters (width σ and depth ϵ) and atomic mass for cation and
anions of two ILs [17,57].
atom type
mass
(amu)
element
ϵ
(K)
σ
(Å)
14.010
12.010
12.010
1.008
1.008
12.010
1.008
1.008
N
C
C
H
H
C
H
H
85.547
43.277
43.277
7.548
7.548
55.052
7.901
7.901
3.250
3.400
3.400
2.421
2.511
3.400
2.471
2.650
14.010
32.060
16.000
12.010
N
S
O
C
85.547
125.805
105.676
55.052
3.250
3.564
2.960
3.400
[emim]+
na
cc
cd
h5
h4
c3
h1
hc
[Tf2N]n2
s6
o
c3
66
Appendix C
Table C.2: (Continued)
atom type
mass
(amu)
element
ϵ
(K)
σ
(Å)
f
19.000
F
30.696
3.118
32.060
12.010
14.010
S
C
N
125.805
105.676
85.547
3.564
3.400
3.250
[SCN]s
c1
n1
Table C.3: Equilibrium bond distances (ro) and harmonic bond force constants (kb) for cation
and anions of two ILs [17,57].
atom type atom type
kb
(K/Å2)
ro
(Å)
168427.0
169031.0
152526.0
169736.0
1.456
1.093
1.535
1.092
197438.4
338504.7
120411.2
232448.0
1.575
1.430
1.820
1.315
187097.0
510516.0
1.630
1.138
[emim]+
c3-na
c3-h1
c3-c3
c3-hc
[Tf2N]n2-s6
o-s6
c3-s6
c3-f
[SCN]c1-s
c1-n1
Appendix C
67
Table C.4: Equilibrium angles (θo) and harmonic bond force constants (kθ) for cation and anions
of two ILs [17,57].
atom type atom type atom type
kθ
θo
(K/rad2)
(degree)
37062.0
25040.0
36690.0
25272.0
25111.0
32146.0
31481.0
34692.0
36690.0
25272.0
33077.0
23747.0
31481.0
23747.0
19716.0
23334.0
23329.0
19482.0
109.33
122.10
109.42
119.66
109.45
128.01
125.09
109.90
109.42
119.66
112.81
129.11
125.09
129.11
109.55
110.05
110.07
108.35
38854.2
55390.8
46434.3
61531.7
54480.5
40059.9
46757.4
127.77
112.6
102.93
118.47
104.03
110.75
108.16
41158.0
141.00
[emim]+
na-cc-na
h5-cc-na
cc-cd-na
h4-cd-na
h1-c3-na
cc-na-cd
c3-na-cc
cc-na-cc
cd-cc-na
h4-cc-na
c3-c3-na
cc-cd-h4
c3-na-cd
cd-cc-h4
h1-c3-h1
c3-c3-hc
c3-c3-h1
hc-c3-hc
[Tf2N]s6-n2-s6
n2-s6-o
c3-s6-n2
o-s6-o
c3-s6-o
f-c3-s6
f-c3-f
[SCN]n1-c1-s
68
Appendix C
Table C.5: Dihedral parameters (charmm) for cation and anions of two ILs [17,57].
atom type atom type atom type atom type
kχ
n
(K)
[emim]+
hc-c3-c3-na
h1-c3-c3-hc
78.500
78.500
3
3
0
-489.8
95.58
-292.06
193.45
96.45
2
3
3
2
1
3
[Tf2N]s6-n2-s6-o
s6-n2-s6-c3
f-c3-s6-n2
s6-n2-s6-c3
s6-n2-s6-c3
f-c3-s6-o
[SCN]N/A
Table C.6: Bonded and nonbonded parameters (Trappe) for Ammonia used in the simulations
[58].
atom type
mass
(amu)
n
h3
m
14.0067
1.00790
0.00
atom type atom type
n-h3
m-n
Nonbonded
partial
element
charge
(e)
N
H
M
0.00
0.41
-1.23
Bonded
atom type r
atom type (Å)
atom type
1.012
h3-n-h3
0.080
h3-n-m
ϵ
σ
(K)
(Å)
185.0
1.0
1.0
3.420
1.0
1.0
θ
(degree)
106.7
67.9
Appendix D
Comparison of the experimental data with the
NRTL model
Figure D.1: Comparison of the experimental (black symbols) and the NRTL correlated (blue symbols) isotherms for
NH3 in [emim][Tf2N]. Circles are at 298.4 K, diamonds are at 323.4 K, and squares are at 347.6 K. Curves are guides
to the eye. Experimental data are taken from reference [16]. The NRTL fitted parameters are taken from Wang et
al. [78].
Figure D.2: Comparison of the experimental (black symbols) and the NRTL correlated (blue symbols) isotherms for
NH3 in [emim][SCN]. Circles are at 298.1 K, squares are at 322.6 K, diamonds are at 348 K and triangles are at 347.6
K. Curves are guides to the eye. Experimental data are taken from reference [26]. The NRTL fitted parameters are
taken from Wang et al. [78].
69
Appendix E
Simulation details
Figure E.1: Convergence for the total potential energy, U, and the number of NH3 molecules, N+λ, as a function of
number of cycles and the probability of λ, P(λ) as a function of λ for NH 3 in [emim][Tf2N]. Note that the total
number of NH3 molecules plus the fractional molecule contribution λ is shown. The simulation is at 393.15 K and
6.83 bar. Multiple simulations are performed for same T and P as shown with different colours. Result for a
particular T and P is computed as averages of these multiple simulations.
Figure E.2: Convergence for the total potential energy, U, and the number of NH3 molecules, N+λ, as a function of
number of cycles and the probability of λ, P(λ) as a function of λ for NH 3 in [emim][Tf2N]. Note that the total
number of NH3 molecules plus the fractional molecule contribution λ is shown. The simulation is at 373.15 K and
10.19 bar. Multiple simulations are performed for same T and P as shown with different colours. Result for a
particular T and P is computed as averages of these multiple simulations.
71
72
Appendix E
Figure E.3: Convergence for the total potential energy, U, and the number of NH3 molecules, N+λ, as a function of
number of cycles and the probability of λ, P(λ) as a function of λ for NH 3 in [emim][SCN]. Note that the total
number of NH3 molecules plus the fractional molecule contribution λ is shown. The simulation is at 393.15 K and
13.50 bar. Multiple simulations are performed for same T and P as shown with different colours. Result for a
particular T and P is computed as averages of these multiple simulations.
Figure E.4: Convergence for the total potential energy, U, and the number of NH3 molecules, N+λ, as a function of
number of cycles and the probability of λ, P(λ) as a function of λ for NH 3 in [emim][SCN]. Note that the total
number of NH3 molecules plus the fractional molecule contribution λ is shown. The simulation is at 347.15 K and
6.83 bar. Multiple simulations are performed for same T and P as shown with different colours. Result for a
particular T and P is computed as averages of these multiple simulations.
Appendix F
Conformers
a.
b.
c.
Figure F.1: Conformers of the [Tf2N] anion, (a) cis; (b) one CF3 group in SNS plane; (c) trans.
a.
b.
Figure F.2: Conformers of the [SCN] anion, (a) planar; (b) non-planar
73
Appendix G
Excess Enthalpy
For a multi-component system, the total enthalpy of the mixture, h is expressed as [3,6],
(F.1)
where,
and
are the enthalpy of pure NH3 and IL,
and
are the mole fractions of
NH3 and IL, respectively and
is the mixing heat. At mixture conditions of T and P, the pure
NH3 in equation (F.1) is in gaseous state.
The total enthalpy of the mixture is also expressed as,
(F.2)
where,
is the excess enthalpy or the absorption heat. In equation (F.2), the saturated liquid
enthalpy of pure NH3 at same T ,
, is used to calculate the total enthalpy of the mixture.
Equation (F.1) can also be written as,
(F.3)
where,
we get,
is latent heat of pure NH3 at same T and P. Comparing equation (F.2) and (F.3),
(F.4)
On mass basis, equation (F.4) can also be written as,
(F.5)
where,
is weight fraction of NH3.
75
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