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7/29/2019 Modeling to Investigate the Sensitivity of Specific Heat Capacity and Heat Transfer Coeffiecient on Nanoparticle Gr
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MODELING TO INVESTIGATE THE SENSITIVITY OF SPECIFIC HEAT
CAPACITY AND HEAT TRANSFER COEFFIECIENT ON NANOPARTICLE
GROWTH IN HIGH TEMPERATURE REACTOR
Abstract
A mathematical model is developed to simulate the comprehensive systemsof nanoparticlegrowth. The model developed is the detailed population balance equation (PBE). The
thermophysical properties of nanoparticles, especially specific heat capacity, overall heat
transfer coefficient were investigated. This article presents a study on the sensitivity of
overall heat transfer coefficient and the specific heat capacity during nanoparticle growth
with the simultaneous chemical reaction, nucleation, condensation and coagulation under
high temperature. The influence of the overall heat transfer coefficient and specific heat
capacity on modeling of nanoparticle growth for the understanding of thermal stability on
particle growth was investigated.
Keywords: Population balance equation, Particle size distribution, Average particle diameter,
Thermal conductivity, overall heat transfer coefficient and specific heat capacity.
1.Introduction
Nanotechnology is mainly defined by size and comprises the visualisation,
characterisation, production and manipulation of structures which are smaller than 100
nanometers (nm). They are particles with one or more dimensions on the nanoscale. When the
particle size is decreased to the nanoscale range, fundamental physical and chemical
properties appear to change, often resulting in completely new and different
physical/chemical property than before. For example, titanium dioxide particles lose their
white colour and become colourless at decreasing size ranges below 50 nm. Other particle
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types, known for their electrical insulating properties, may become conductive at the nanosize
scale, or low soluble substances can increase their solubility when their sizes are below 100
nm1, 2. The behaviour of nanoparticles is similar to the behaviour of a gas or a vapour and it is
related to the size of the particles, which depends on their formation mechanism and the
diffusion forces. Air diffusion is the principal mode of transport of particles smaller than 100
nm. The speed at which particles diffuse is determined by their 'coefficient of diffusion',
which is inversely proportional to their diameter.
Coagulation of very small particles quickly leads to the formation of larger particles
in lower concentration. Particles ranging from 1 to 100 nm tend to agglomerate quickly to
form larger diameter particles. When they reach a size of ~100 nm, they grow at a slower
pace, up to 2,000 nm. This zone of slow growth, between 100 and 2, 000 nm, is called the
'accumulation mode'. The primary aerosol particles that come into contact with each other
adhere together, due to short distance forces (a few atoms diameter) to form loosely larger
particles or agglomerates. The aerosol coagulation process is caused by the relative motion
among the particles. When the movement is due to the Brownian effect, the process is called
Brownian or thermal coagulation; this is a spontaneous and it is an ever-present phenomenon
for aerosols. If the relative motion is caused by external forces (such as gravity, electrical or
aerodynamic forces), the process is called kinematic coagulation.3
Nanostructured or nanophase materials are nanometre-sized solid substances
engineered on the atomic or molecular scale to produce either new or enhanced physical
properties not exhibited by conventional bulk solids. Therefore, particles less than 100nm in
diameter exhibit properties differently from those of conventional solids. Properties that
mostly determine the thermal performance of materials for heat transfer applications are the
thermal conductivity, viscosity, specific heat capacity, overall heat transfer coefficient and
density. The transfer of heat is normally from a high temperature object to a lower
2
http://hyperphysics.phy-astr.gsu.edu/Hbase/thermo/heat.html#c1http://hyperphysics.phy-astr.gsu.edu/Hbase/thermo/temper.html#c1http://hyperphysics.phy-astr.gsu.edu/Hbase/thermo/heat.html#c1http://hyperphysics.phy-astr.gsu.edu/Hbase/thermo/temper.html#c17/29/2019 Modeling to Investigate the Sensitivity of Specific Heat Capacity and Heat Transfer Coeffiecient on Nanoparticle Gr
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temperature object. Heat transfer changes the internal energy of both systems involved,
according to the First Law of Thermodynamics. This is the application of conservation of
energy principle to heat and the thermodynamic process. Thermal conductivity is the quantity
of heat transmitted during a given time, through a material in a direction normal to a surface
area of the particle. This is due to the temperature difference under steady state conditions,
when the heat transfer is dependent only on the temperature gradient. Fluids such as air,
water, ethylene glycol, and mineral oils are typically used as heat transfer media in
applications such as power generation, chemical production, automobiles, computing
processes, air conditioning and refrigeration. However, their heat transfer capabilities are
limited by their very low thermal conductivity. These fluids have almost two orders of
magnitude lower in thermal conductivity when compared to metals, resulting in low heat
removal efficiencies.4
Many studies involving suspension in mill and microsized particle in various fluids
have been done. Ahuja5, 6 did show that by suspending 50-100m sized polystyrene particles
in glycerine, the thermal conductivity is lowered below that of glycerine. The lower thermal
conductivity followed the existing heterogeneous mixed media models, like that of Hamilton
and Crosser 7. However, convective heat transfer rate of the mixture in the laminar flow
increased by a factor of 2, without any increase in friction losses. The same work also
investigated, from the theoretical standard point, the effect of varying the particle size,
density and other factors that might influence this enhancement. Ahuja suggested that the
physical mechanisms of heat transfer enhancement for this mixture are due to centrifugal
fantype churning, due to rotation of particles in the shear gradient and a good dispersion of
particle in the flow thereby creating more of this churning. Application of such coolants to
real systems has proved difficult due to the inherent inability to keep these particles dispersed
3
http://hyperphysics.phy-astr.gsu.edu/Hbase/thermo/inteng.html#c2http://hyperphysics.phy-astr.gsu.edu/Hbase/thermo/firlaw.html#c1http://hyperphysics.phy-astr.gsu.edu/Hbase/thermo/inteng.html#c2http://hyperphysics.phy-astr.gsu.edu/Hbase/thermo/firlaw.html#c17/29/2019 Modeling to Investigate the Sensitivity of Specific Heat Capacity and Heat Transfer Coeffiecient on Nanoparticle Gr
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in fluids and the resultant settling and clogging potential. Therefore, these fluids are hardly
considered for industrial applications. 5, 6
Over the years, a significant amount of data has been gathered on the thermal conductivity of
nanofluids. Typical materials used for nanoparticles include metals such as copper, silver,
and gold and metal oxides such as alumina, titania, and iron. Carbon nanotubes have also
been used to enhance the thermal conductivity of liquids. Experimental data on the thermal
conductivity of nanofluids varies widely and the mechanisms responsible for the thermal
conductivity enhancement are under debate. Masahide et al 8 experimentally investigated the
thermal performance of a new type of self-rewetting fluid heat pipe containing aqueous 1-
butanolic solution with microwave-assisted synthesis of very dilute polyvinylpyrrolidone
(PVP)-capped silver nanoparticles (nano-self-rewetting fluid). This was done by investigating
the temperature dependencies of surface tension and viscosity of 1-butanol containing
chemically synthesized PVP-capped silver nanofluids. Zhang et al 9 employed homogeneous
atomic chain to demonstrate atomistic green function approach with one degree of freedom
per atom in order to determine the thermal conductance. Yulong et al 10 worked on the heat
conduction, convective heat transfer under both natural and forced flow conditions, and
boiling heat transfer in a nucleate regime. Syam Sundar and Sharma 11 developed flow heat
transfer coefficient and friction of Al2O3 nanoparticles dispersed in water and ethylene glycol
in circular tube. The heat transfer coefficient and friction factor characteristics of Al 2O3
nanofluid in circular tube were numerically studied. The thermo-physical properties of the
nanofluid are considered for heat transfer coefficient and friction factor by assuming that
nanofluid is a single-phase fluid. Sami et al 12 studied situations in which nanoparticles in a
fluid are strongly heated, thereby generating high heat fluxes. The situation they studied is
relevant to experiments in which a fluid is locally heated by using selective absorption of
radiation by solid particles. They explored the phenomenon of heat transfer in the vicinity of
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strongly heated nanoparticles, using molecular dynamics simulations of atomically realistic
models or of more coarse-grained LJ monoatomic fluids. Garg et al 4 developed a technique
for chemical synthesis of copper in ethylene glycol nanofluid and measured its thermal
conductivity and viscosity. Yang et al. 13 studied the convection heat transfer performance of
the graphite nanofluids in laminar flow through a circular tube. The experimental results
show that the nanoparticles increase the heat transfer coefficient of the fluid system in
laminar flow, but the increase is significantly less than that predicted by current correlation,
based on static thermal conductivity measurements. Afshar et al 14 analytically solved the
NavierStokes and energy equations for fluid flow in a microchannel in slip flow regime
during which time, temperature and velocity profiles were evaluated. They concluded that the
key to improve the heat transfer is to add nanoparticles so that part of heat is being
transported from the microchannel without any additional pressure drop. Wen et al 15
employed the molecular dynamics simulations with the quantum corrected SuttonChen type
of the many-body force field to investigate the energetic and structural evolution of platinum
nanoparticles with different shapes during the heating process. The shapes of nanoparticles
include cube, octahedron, truncated octahedron, and sphere. Their study addresses the shape
effects on thermal characteristics of platinum nanoparticles;however the modeling of process
sensitivity of overall heat transfer coefficient and specific heat capacity for the understanding
of thermal stability on particle growth has not been investigated. In this article, the objective
is to investigate the sensitivity of the overall heat transfer coefficient and specific heat
capacity on modeling of nanoparticle growth for the understanding of thermal stability on
particle growth. Thermal conductivity is possibly the most important element in this study
because it points out the nanoparticle with high heat transfer potential.
2.0 Process description
2.1Population balance model
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The population balance equation consists of the following nonlinear partial integro-
differential equation.
tn
+
vnxzvG
)),,(( +
zncz
- ( ) vvvI )(
= ( ) ( ) ( ) ( ) ( ) vdtzvnvvtzvnvdtzvvntzvnxvvvo
v
o
~,,~~,),,(~,,~,,~,~,~
2
1
(1)
The first term on the left hand side of equation (1) describes the change in the number
concentration of particle volume interval v, v+dv and in the spatial interval,z.z+dz, n (v, z, t)
denotes the particle size distribution function, v is particle volume, tis time,z [0, L] is the
spatial coordinate,L is the length of the process. The second term on the left hand side gives
the loss or gain of particles by condensational growth, the third term on the left hand side
which is
z
ncz
corresponds to the convective transport of aerosol particles at fluid velocity
zc and the fourth term on the left hand side accounts for the formation of new particles of
critical volume v by nucleation rate I . ( ) vvvI )( , also accounts for gain and
loss of particles by condensation. ),,( xzvG , )( *vI and )~,~,~( xvvv are the
nonlinear scalar functions and is the standard Dirac function. The nucleation rate )( *vI is
assumed to follow the classical Becker-Doring theory, given by the expression below
(Pratsinis). 16
),2/exp()9/2()2/( *2/13/122/1112
InSkSmTksnIBs
=
(2)
Where 1s is the monomer surface area and*
k which is the number of monomer in the
critical nucleus and is given by:
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3
4
6
=
InS
k
, Where Tkv B/3/2
1= and is the surface tension. (3)
The mathematical equation that describes the dimensionless temperature is given by Kalani
and Christofides 17 thus:
)(21 TTTETCCBzd
Tdc
d
Tdwzl ++=
(4)
The term on the left hand side describes the dimensionless temperature with
dimensionless time. The first term on the right hand side describes the dimensionless
temperature with dimensionless distance due to convection by chemical reaction. The second
term on the right hand side gives the temperature change in the concentration of reactants due
to energy release in the chemical reaction. The last term on the right hand side describes the
temperature change in the heat transfer. Heat transport and heat release due to chemical
reactions leads to spatial and temporal temperature distributions in chemical reactors. This
prediction is based on the first law of thermodynamics which says that the total energy of a
given system obeys a conservation law. Table 1 and 2 gives the dimensionless variable and
process parameters used for the simulation respectively.
2.2 THE LAW OF HEAT CONDUCTION
The law of heat conduction states that the time rate of heat transfer through a material
is proportional to the negative temperature gradient and the area in the right angles, to that
gradient, through which heat is flowing. This law is also known as Fouriers law. The
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differential form of this law will be considered which look at the heat flow rate intensity
(flow of energy per unit area per unit time). The differential form of Fourier's Law of thermal
conduction shows that the heat flow rate intensity is equal to the product of thermal
conductivity k, and negative local temp gradient. The heat flow rate is the amount of energy
that flows through a particular surface per unit area per unit time. Fourier law for one
dimensional form in its x direction is given as:
dx
dTkq
x= (5)
Table 1: Dimensionless Variables by Kalani and Christofides 17
N= 0M / sn , V= 1M / sn 1v Aerosol concentration and volume
2V = 2M / sn
2
1vSecond aerosol moment
=(2 1m /k TB ) 12/1
sns
Characteristic time for particle growth
K=(2 k TB /3 sn) ,
)//('
snII =
Coagulation coefficient and Nucleation rate
1/1 rKn = Knudsen number
1
' / vvv gg = Dimensionless geometric volume
1
'/ rrr gg = Dimensionless geometric radius
Lzz /= Dimensionless distance
Lcc zzl /= , /t= Dimensionless velocity and time
= t/ Dimensionless time
Table 2: Process model parameters for the simulation study
L=1.5m Reactor length
D=0.1m Reactor Diameter
P 0 =101000 pa Process pressure
T 0 =2000 K Inlet temperature
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y10 = 0.4 Inlet mole fractions of O2
y 20 = 0.6 Inlet mole fractions of TiCl4
U=160 J m 112 Ks Overall coefficient of heat transfer
JR 88000= mol1 Heat of reaction
25.1615=P
C J mol 1 K 1 Heat capacity of process fluid
MW3100.14 =g kg mol 1 Mol wt. of process fluid
K=11.4m 3 mol 1 s 1 Reaction rate constant
kg5107.6 = m 1 s 1 Viscosity of process fluid
004.900162.0log906.0/4644)(log ++= TTTmmHgPs PVT relation
08.0= N m 1 Surface tension
329
1 1033.5 mv
= Monomer volume
#10023.623
=av
N mol 1 Avogadros constant
R=8.314 J mol 1 K 1 Universal gas constant
JkB23
1038.1 = K 1 Boltzmanns constant
= 0.5 Sigma
merg 065.1= Geometric radius
mer 065.01 = Monomer radius
6/3dpvg = Geometric volume
mEs 191587.10 = Monomer surface area
w
T
= 1600K Wall temperature
Re = 2000 Reynolds number
3.0 RESULTS AND DISCUSSION
The mechanisms that govern the effective thermal conductivity of nanoparticles suspensions
may be associated with many factors that include heat transport within the individual
particles and the Brownian motion of individual particles. The thermal conductivity is often
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treated as a constant; however, this is not always true. For this investigation, the thermal
conductivity increases slightly with dramatic increase in average particle diameter. Figure 1
shows the steady state profile of the average particle diameter as a function of the thermal
conductivity. The thermal conductivity of a material generally varies with temperature; the
variation can be small over a significant range of temperatures for some common materials.
Figure 2 shows the variation of the thermal conductivity with dimensionless temperature.
Fig. 0. The steady state profile of the average particle diameter against the thermalconductivity.
Fig. 2. The steady state profile of the thermal conductivity against the dimensionless
temperature.
The maximum diameter of particles is determined by the competition between the
heating of the particles due to the energy released in the reactor and the heat dissipation (loss
of energy over time) from the particles surface to the surroundings. The average particle
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diameter increases with increase in the residence time. It can be seen from Figure 3 below,
that when the particles are nucleated, a primary particle with diameter of ~2nm is produced at
constant time. After a certain number of particles have been produced, the frequency of bi-
particle collision increases due to the heating of particles, resulting in a sharp increase in
particle diameter. The particles gain heat from their surrounding as they approach the heated
surface, and when they leave the thermal region, they give up their heat to the bulk.
Fig. 3. The steady state profile of the average particle diameter against time
Particles established a large local temperature gradient due their lateral motion,
resulting from their contact and rebound to the heat transfer surface. Figures 4 and 5 show the
average particle diameter and particle volume as functions of the overall heat transfer. The
average particle diameter and the particle volume increase with increasing overall heat
transfer.
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Fig. 4. The steady state profile of the average particle diameter against overall heat transfer
Fig. 5. The steady state profile of the particle volume against overall heat transfer
The steady state profile of the average particle diameter against the heat capacity of
the process fluid is given in figure 6 below. The result shows that the average particle
diameter increases steadily with heat capacity in the region between 1700-1730 J/mol.Kand
thereafter remains constant. The heat capacity does not have much influence on the average
particle diameter as the average particle diameter approaches a limiting value of 11.82nm.
The heat capacity has significant effect on the wall temperature. The wall temperature
increased with increasing heat capacity as shown in figure 7.
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Fig. 6. The steady state profile of the average particle diameter against heat capacity
Fig. 7. The wall temperature of the reactor against heat capacity
4. Summary
In summary, modeling of the sensitivity analysis of the overall heat transfer
coefficient and specific heat capacity for the understanding of thermal stability on particle
growth has been investigated. The influence of the overall heat transfer coefficient and
specific heat capacity on modeling of nanoparticle growth for the understanding of thermal
stability on particle growth was also investigated. The heat transfer behaviour shows that13
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nanoparticle properties have influence on the thermal conductivity in the nucleate region as
particle diameter increases with increasing overall heat transfer. The sensitivity of the
variation of the overall heat transfer on the average particle diameter and particle volume
shows that the thermal conductivity is an important element in the growth of nanoparticle.
This is because it points out the potential of high heat transfer on nanoparticle. The sensitivity
of the variation of the heat capacity on the wall temperature also shows that the thermal
conductivity is an important element in the growth of nanoparticle because nanoparticle
formation with high wall temperature increases the growth of particle. Further work should
be done experimentally on investigating the sensitivity analysis of the overall heat transfer
coefficient and specific heat capacity for the understanding of thermal stability on particle
growth.
References
1. D.B. Warheit, Toxicol. Sci. 101, 183 (2008)
2. V. Colvin,Nature Biotechnology. 21, 1166 (2003)
3. J.A.Schwarz, I. Cristian, and K.P. Contescu, Dekker Encyclopedia of Encyclopaedia of
Nanoscience and Nanotechnology, 1st ed., pp. 42, New York, (Marcel Dekker) Taylor and
Francis, (2004).
4.J.Garg,B. Poudel, M. Chiesa, J. B. Gordon, J. J Ma, J. B. Wang, Z. F. Ren, Y. T. Kang, H.
Ohtani,J.Nanda, G. H. McKinley and G. Chen,J. Appl. Phys. 103, 074301, (2008)
5. A. S. Ahuja,J. Appl. Phys. 46, 3408, (1975)
6. A. S. Ahuja,J. Appl. Phys. 46, 747, (1975)
7. R. L. Hamilton and O. K. Crosser,Ind. & Eng. Chem. Fund. 1, 187 (1962)
8. S. Masahide, A. Yoshiyuki, U. Yuki, D. P. Roberto, S. Raffaele and C. Anselmo, Thermal
performance of self-rewetting fluid heat pipe containing dilute solutions of polymer-capped
14
7/29/2019 Modeling to Investigate the Sensitivity of Specific Heat Capacity and Heat Transfer Coeffiecient on Nanoparticle Gr
15/15
silver nanoparticles synthesized by microwave-polyol process. Proceedings of International
Transport Phenomena VI, Volterra, Italy, paper no: ITP-09-19, pp 1-7. October 4-9, (2009)
9. W. Zhang, T. S. Fisher and N. Mingo,Numerical heat transfer, Part B. 51, 333 (2007)
10. D. Yulong, C. Haisheng, W. Liang, Y. Chane-Yuan, H. Yurong, Y. Wei, P.L. Wei, Z.
Lingling, and H. Ran,Kona, 25, 23, (2007)
11. L. Syam Sundar and K. V. Sharma,Int. J. Dynamic. of fluids, 4, 121 (2008)
12. M. Samy, S. Sergei, J. Laurent, K. Pawel and B. Jean-Louis, Proceedings of the National
Academy of Science, 106, 15113 (2009)
13. Y. Yang, Z. George Zhang, E. A. Grulke, W. B. Anderson and G. Wu, International
Journal of Heat and Mass Transfer, 48, 1107 (2005)
14. H. Afshar, M. Sham, S. M. M. Nainian, and G. Ahmadi,International Communications
in Heat and Mass Transfer, 36, 1060 (2009)
15. Y. Wen, H. Fang, Z. Zhu and S. Sun, Phys. Lett, A 373, 272 (2009)
16. S. E. Pratsinis,Journal of Collide Interface Science, 124, 416 (1988)
17. A. Kalani and D. P. Christofides, Chem. Eng. Sci. 54, 2669 (1999)
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