The Sensitivity of Specific Heat Capacity and Heat Transfer Coeffiecient on Nanoparticle Growth in...

7/28/2019 The Sensitivity of Specific Heat Capacity and Heat Transfer Coeffiecient on Nanoparticle Growth in Plsma Chemica

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THE SENSITIVITY OF SPECIFIC HEAT CAPACITY AND HEAT TRANSFER

COEFFIECIENT ON NANOPARTICLE GROWTH IN PLASMA CHEMICAL

REACTOR

Oluranti Sadiku1 and Emmauel Rotimi Sadiku

Department of Mechanical Engineering, Faculty of Engineering and the Built Environment,

Tshwane University of Technology, Pretoria, South Africa.

CSIR Campus Building 14D

Postnet Suit # 186, Private Bag x025 Lynnwoodridge 0040, Republic of South Africa.

e-mail1:[email protected]; [email protected]

Cell Phone No: +27783572455

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

1
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]


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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

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

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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

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

3
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/inteng.html#c2http://hyperphysics.phy-astr.gsu.edu/Hbase/thermo/firlaw.html#c1http://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/inteng.html#c2http://hyperphysics.phy-astr.gsu.edu/Hbase/thermo/firlaw.html#c1


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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

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

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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

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

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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

The population balance equation consists of the following nonlinear partial integro-

differential equation.

t

n

+

v

nxzvG

)),,((+

z

ncz

- ( ) 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 isz

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

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assumed to follow the classical Becker-Doring theory, given by the expression below

(Pratsinis). 16

),2/exp()9/2()2/(*2/13/122/1

11

2

InSkSmTksnI Bs =

(2)

Where 1s is the monomer surface area and*

k which is the number of monomer in the

critical nucleus and is given by:

34

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

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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

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

1

v Second 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

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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

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=PC J mol1 K 1 Heat capacity of process fluid

MW3100.14 =

g kg mol1 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

#10023.6 23=avNmol 1

Avogadros constant

R=8.314 J mol 1 K 1 Universal gas constant

JkB231038.1 = K 1 Boltzmanns constant

= 0.5 Sigma

merg

065.1= Geometric radius

mer 065.01 = Monomer radius

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6/3dpvg = Geometric volume

mEs 191587.10 = Monomer surface area

wT = 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

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 thermal

conductivity.

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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

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

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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.

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

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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.

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

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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 that

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.

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